Penetration of Radio Signals Into Buildings in the Cellular Radio 2719
Environment
E. H. Walker

Transmission Errors and Forward Error Correction in Embedded 2735
Differential Pulse Code Modulation
D. J. Goodman and C.-E. Sundberg

CCITT Compatible Coding of Multilevel Pictures 2765
H. Gharavi and A. N. Netravali

Performance of the Queueing Network Analyzer 2817
W. Whitt
PAPERS BY BELL LABORATORIES AUTHORS 2845

Calculation of Modes in an Optical Fiber Using
the Finite Element Method and EISPACK

This paper presents a method for computing the propagation modes of a
circular optical fiber. Finite element analysis reduces Maxwell’s equations to
standard eigenvalue equations involving symmetric tridiagonal matrices. Rou-
tines from the Eigensystem Package (EISPACK) compute their eigenvalues
and eigenvectors, and from these the waveforms, propagation constants, and
delays (per unit length) of the modes are obtained. An extension allows loss
of leaky modes to be calculated. Examples indicate that the method is reliable,
economical, and comprehensive, applying to both single and multimode fibers.

determine pulse dispersion along the fiber). The technique combines
the Finite Element Method (FEM) and routines from the Eigensystem
Package (EISPACK)’ to achieve an efficient calculation of these modal
quantities for both single and multimode fibers.

The most popular approach to modal calculations for multimode
fibers has been the Wentzel, Kramers, Brillouin (WKB) method,
where Maxwell’s equations are approximated by an easily integrated
first-order differential equation.2 WKB analysis provides a simple
model for understanding optical transmission through a fiber® and has
guided fiber design.*

Accuracy of the WKB method declines substantially with the num-
ber of propagating modes. For single-mode fibers the WKB method is
not suitable, and instead, various analytic and numerical techniques
have been used.

Analytic calculations have centered about the step-index profile and
the infinitely extending parabolic profiles. Modal quantities have been
expressed in terms of Bessel functions for the single step’ and, also,
the double step.® Parabolic profiles, analyzed by analogy with the
harmonic oscillator,® have well-known expressions for their modal
quantities. Coupled with perturbation analysis,’° these results cover a
broad range of profiles. But numerical approaches permit an even
more comprehensive treatment.

Several numerical procedures have extended the analysis of the
step-index profile. The solution in the core comes from a numerical
integration; the solution in the cladding (where the index is assumed
constant) is well known. Boundary conditions at the core-cladding
interface link the two solutions and lead to a set of linear equations
involving the propagation constant as a parameter. A search of the
corresponding determinant for zeroes gives the propagation constants
and, subsequently, the waveforms and delays.

In Ref. 11 this procedure is applied to Maxwell’s first-order vector
equations, and important results have been obtained for multimode’””?
and single-mode’*”’ fibers. A similar scheme’® deals with two coupled
second-order differential equations equivalent to Maxwell’s equations.
The second-order scalar wave equation approximates Maxwell’s equa-
tions by neglecting the relatively small gradient index terms. In Ref.
17 the basic procedure is applied to the scalar wave equation, and a
variety of results have been obtained for single-mode fibers. !®9

for the vector equations. Approximate solutions in the core and clad-
ding are matched at the interface by performing a determinant search,
as described before. The FEM has also been applied”? to diverse single-
mode waveguides by approximating the index profile by piecewise
constant functions and then enforcing boundary conditions across the
numerous interfaces. The result is a matrix eigenvalue equation in
generalized form. Both of these FEM approaches seem limited to a
small number of modes for economical operation.

The approach in this paper is characterized by a sequence of three
steps. First, Maxwell’s equations are transformed to ordinary differ-
ential equations in eigenvalue form. In one case, gradient index terms
are neglected, and in the second, they are considered to first order so
that their effect can be monitored. Next, finite element analysis, using
the Galerkin technique,” reduces these differential equations to matrix
equations in standard eigenvalue form. The matrices are symmetric
and tridiagonal, and their positive eigenvalues correspond to the
propagation modes. The routine BISECT in the EISPACK? library
delivers the eigenvalues and TINVIT, also in EISPACK, delivers the
corresponding eigenvectors. The eigenvalues give the propagation
constants of the modes, the eigenvectors give the waveforms, and a
combination of the two give the delays.

Like many other numerical techniques, this method can treat any
uniform, circular fiber and meet usual standards of accuracy. Also, the
effect of gradient index terms can be monitored. But by casting the
equations in standard eigenvalue form, modern techniques of compu-
tational linear algebra (as used in EISPACK) can achieve substantial
cost advantage over other numerical approaches. Typically, this
method will process for a multimode fiber 25 modes per second on the
Cray-1* computer.

The calculation procedure is derived in the next section. Effects of
material dispersion are incorporated into the analysis, and calculation
of loss for leaky modes is also considered. Results are given in Section
III for a variety of single and multimode examples. These results, as
discussed in Section IV, illustrate the reliability, economy, and scope
of the method.

This section derives from Maxwell’s equations an algorithm that
computes the propagation modes of an optical fiber. The algorithm is
straightforward and can be carried out on any computer that has
access to the EISPACK' or similar routines.

The fiber is assumed perfectly straight and circular, and uniform
along its length. The cylindrical coordinate system (r, 0, z) is defined
so that the z-axis coincides with the fiber axis. The index of refraction
can then be expressed by the function n(r), the index profile. The
profile can be any bounded function in the core (r < R,), but it is
constant (n,.) in the cladding. The geometry is shown in Fig. 1.

where «, denotes the free-space permittivity. The permeability yu is
assumed throughout to be yu,, the free-space value.

Maxwell’s equations relate the electric field HE and the magnetic
field H by

where w denotes the frequency of excitation (in rad/s). Taking the
curl of the second equation eliminates E to give

with n the index of refraction, c the velocity of light in vacuum, and )
the free-space wavelength of the excitation.

where H, is a vector field independent of z, and 8 is the propagation
constant of the mode. Substituting this into the field equation gives,
in cylindrical coordinates,

where g, means dg/dr. The transverse (i.e., r and 8) components of H
are uncoupled from the longitudinal (i.e., z) component and satisfy an
eigenvalue equation with eigenvalue 6”. The corresponding operators
are indicated as a 2 X 2 submatrix in the full 3 X 3 matrix.

where the functions hy and h, depend only on r. The two forms
correspond to different polarizations. Substituting these into eq. (7)
gives

where the + sign depends on the polarization.
The case where m’ = 0 reduces to two uncoupled equations:

for the Transverse Magnetic (TM) modes for which H, is identically
zero, and

i a ee re
(2,4 b+) b= Bh, (11)
for the Transverse Electric (TE) modes for which E, is identically
zero. Often, gradient index terms (those involving g,) are neglected on
the basis that profile variations are relatively small.* The difference
or splitting in 6? for the TM and TE modes measures the accuracy of

where (A — B)fp = 6f2. The second term of eq. (12) is neglected
because its first order perturbation contribution is zero, as in general

respectively. If two eigenvalues are equal or nearly equal, a degenerate-
perturbation calculation may be required.

To first order, the modes when m’ # 0 are either the EH, with
transverse H fields of the form

kdrd_  (m’ +1) (m' + 1) tH ;
(' dr kdr r ae ie
or the HE,,, with
_ [f cos m’6 f sin m’0
Hor = (/ sin m’6 and —f cos m’6)’ (19)
where f satisfies
Redo d? (m= 1)? m1) ale — 92
(' dr k dr ez op 1 eas a
With m = m’ +1, eqs. (18) and (20) can be combined as
kdrd m_m 7, ee
(pore Ry 4 elm oi (21)

The modes are indexed by the angular parameter m. For m = 0
there are two polarizations for each HE,, mode. This group includes
the fundamental mode HE),;, which propagates in single-mode fibers.
For m = 1 there are two polarizations for the HE:, modes, also the
TM modes and the TE modes. For m > 1 there are two polarizations
for the HE n+1,.. modes and two for the EH »-1,, modes.

When gradient index terms are neglected, eq. (9) reduces to the
scalar wave equation

(:2, dr er ea (22)
For m = 0 each solution represents two polarizations, i.e., two modes;
for m # 0 each solution represents two polarizations and two angular
harmonics, m’ = m + 1, 1.e., four modes.

The scalar wave equation and its counterparts [eqs. (10) and (21)]
assume the form of a time-independent Schroedinger equation in two
dimensions. In particular, the scalar wave equation can be expressed
as

The quantity (k? — k3) plays the role of a potential that is 0 in the
outer cladding and beyond; (6? — k4) is an eigenvalue.
Results on Schroedinger’s operators” imply that the propagation

modes correspond on a one-to-one basis to the positive eigenvalues
and the number of such modes is finite (see Ref. 23, p. 366). This fact
implies that increasingly accurate estimates of the propagation modes
can be obtained by ever finer discretization of the equations. By
contrast, finer discretizations introduce ever more negative eigenval-
ues corresponding to the continuum of radiation or unbound modes.

The Finite Element Method (FEM) can solve to desired accuracy
the differential equations just derived. The present discussion special-
izes to the scalar wave equation [eq. (23)]; modifications needed for
the equations containing gradient index terms are indicated in the
appendix.

The solution function f(r) is approximated by a piecewise linear
function. This can be expressed in terms of interpolation functions
(shown in Fig. 2) as

The first and last terms of the series are affected by end conditions
on f(r). At the center (r = 0) of the fiber f(r) and its radial derivative
must be bounded and well defined; so f(0) = 0 when m > 0 and df/dr
(0) = 0 or equivalently f(0) = f(6) to first order in 6 when m = 0.

At the other end, R = Lé represents a truncation radius in the
cladding, and R + 6 represents the truncated part in a way that will
now be explained. Assuming that the cladding extends indefinitely,
the solution there is

where K,, denotes the mth order modified Bessel function of the
second kind,” a is an unknown coefficient,

f'/f = nKn(nR)/Kn(nR). (28)
A Taylor’s expansion of f to first order about r = R yields
f(R + 6) = f(R) + 6f’(R)
= f(R)Q + dnKn(nR)/Kn(nR)). (29)

Most modes of a multimode fiber are unaffected by the end condition
in eq. (29) because their waveforms are negligibly small beyond the
core. But the waveform of a single-mode fiber can extend beyond the
core and then the end condition needs to be enforced.

Sample values of f(r), the coefficients in eq. (25), are estimated by
the Galerkin weighted residual method.” Although a piecewise linear
approximation cannot satisfy the scalar wave equation, it can satisfy
weighted averages of the equation. In the Galerkin technique the
weightings are chosen as the basis functions N,(r) j = 1, ---, L. In
terms of inner products, defined for any functions p(r) and q(r) as

for j= 1, ---, L. The first term comes from an integration by parts.
Substituting the piecewise linear approximation of f [from eq. (25)]
into the Galerkin equations gives L equations for L + 2 sample values,
but the end conditions eliminate two of these values. The result is L
equations in L unknowns, expressed as the matrix equation,

(A — m?B + C)f = 67(8? — k2)Df. (32)
The column vector f denotes the sample values,
f =[f(ri), «++, ADI. (33)
The L X L matrix A has jl element
_ _(@N: aj).
aj = (a > dr ) (34)

for all j and / except, to accommodate the end conditions, ayo must be
added to a); when m = 0 and az,74:f14:/f, must be added to az, for all

The latter three matrices are evaluated by “lumping the masses,”
meaning that the integrals are evaluated numerically and the integra-
tion points are chosen as the sample points.”” The trapezoidal rule
then yields 0 for the off-diagonal elements and for the main diagonal
gives

bn = 6°/l Ci = I[k?(r1) = k2]64 dn = 16? (36)
for!=1,---,L. The matrix A can be evaluated exactly. It is symmetric
and tridiagonal (i.e., a, = 0 if |! — 7| > 1) and has

forl!=1,---, L. When m = 0, ay is —1.56”.
Eq. (32) converts to a standard symmetric eigenvalue problem by
multiplying both sides by the diagonal matrix D~'” and putting

standard form relative to the vector g.
The key matrix, T, is symmetric and tridiagonal. For m # 0

Equation (39) represents the FEM reduction of the scalar wave
equation. The other differential equations, as indicated in the appen-
dix, reduce to the same form, with T symmetric and tridiagonal. The
sample distance 6 is discussed in Section III.

The propagation modes are calculated by solving for the positive
eigenvalues (4) of 7. The associated propagation constants are

When the end condition is enforced, the T matrix depends on @ in its
(L, L) entry, but a simple iteration procedure yields the proper 8. The
associated waveforms are given by f or g.

The modal delays per unit length (7,) are the reciprocal of the group
velocities,

where g is assumed to be normalized (i.e., g-g = 1). This follows
standard procedure for taking the derivative of an eigenvalue with
respect to a parameter.”° Equations (40), (41), and (42) for T imply
that dT/dw” is a diagonal matrix. Using the equivalence between w*d/
dw? and —\2d/d)?, eq. (47) becomes

To form the T matrix, the index profile n(r) must be available for
any excitation wavelength (A). Sellmeier expansions”’ of the form

i=1
have been fitted to measured values of refractive index over the range
of wavelengths 0.8 um to 1.5 um for bulk samples of pure SiOs, 13.5-
percent Ge doped SiOz, and 1 percent F doped SiO, (denoted here as

a, b, and c, respectively). Also, results in Ref. 28 indicate that the
index is approximately linear with concentration. Therefore, the index
profile of an optical fiber having a graded Ge dopant is taken as

where C,(r) denotes the concentration profile for Ge, relative to 13.5
percent. For dual dopants, Ge and F, the index profile is

The concentration profiles in these equations may be specified or
may be deduced from the index profile at a reference wavelength. Once
the concentration profiles are available, the index profile and its ?
derivative can be determined from eq. (50) or eq. (51) for any wave-
length.

The T matrix can be expressed in dimensionless form by replacing
the sample spacing by

where RF, denotes the core radius and L, the number of samples in the
core. This gives

where now C,(r) represents the Ge concentration normalized with
respect to n,, the index at the center. The latter term in eq. (54) can
usually be neglected because n; ~ nq and often, C,(r) ~ 1 for most r.
Then the 7 matrix can be expressed in terms of the usual V number,

where now an iteration on V, is required. A similar expression, involv-
ing three V numbers, holds for dual dopants.
The effective index (n,) is obtained from V, in eq. (56). The delay

d
cc — na) — d° ne (nj — nap|t (58)
Matrix T is always symmetric and tridiagonal. The routine BISECT
in the EISPACK’ library computes the positive eigenvalues for such
matrices and TINVIT, also in EISPACK, computes the associated
eigenvectors g. These routines operate efficiently and reliably.

When the index profile drops below n,; over part of its range, then
leaky modes may exist. These modes have n, <n, and complex prop-
agation constants. The corresponding attenuation accounts for radia-
tion loss as the waveform spreads out radially. As is well known, leaky
modes also can arise when m # 0 from the negative term, — (rm?/r’),
in the propagation equation.

Leakage loss is calculated by truncating the waveform at the begin-
ning of the outer cladding and enforcing the proper end condition. In
terms of the complex propagation constant y, eq. (27) for n changes
to

and the end condition [in eq. (43)] becomes complex. Now, the T
matrix has a real (7) and an imaginary (T;) part, but matrix T;
consists of all zeroes except the (L, L) diagonal element (call it x).
First-order perturbation theory estimates an eigenvalue of T as

where py, is an eigenvalue of T, and g; is the Lth component of the
associated normalized eigenvector g. Again, an iteration is required
because T depends on y.

In this section modal calculations based on the results of Section II
are illustrated in 10 examples. The next section summarizes and
evaluates the results.

3.1 Infinite parabolic profile
The parabolic profile that extends without radial limit has index

for all r, where R denotes the nominal radius of the fiber core. Under
the scalar wave approximation, its modal waveforms are Laguerre
Gaussian functions, its propagation constants are

Bum = [Rani — 2Qkan,(2A)?/R]”, (62)
and its modal delays (neglecting material dispersion) are
Tum = (Bum + Ran}/Bym)/2w, (63)
where the mode number is
Q=2n+mt+1 m,n=0,1,--:, (64)

with m the angular harmonic and yu the radial harmonic. Each Q
represents a group of modes that have the same propagation constant
and delay (see Ref. 9).

Modal delays were calculated assuming parameter values A = 0.013,
R = 25 um, and A = 1.3 wm. Convergence with respect to truncation
radius (TR) was complete (within 0.1 ps/km) for each mode for TR =
1.5R. Convergence with respect to the number (L;) of sample points
in the core was within i ps/km in the rms delay for L, = 400, as
indicated in Table I. An accuracy of 1 ps/km determines the bandwidth
within 5 percent for a 10-GHz X km fiber and 0.5 percent for a 1-GHz
xX km fiber. The percent errors of the computed delays (with L, = 500)
are shown in Fig. 3.

The most accurately computed mode within a mode group had radial
number yu = 0; their waveforms have no zero-crossings and are most
easily approximated by piecewise linear functions. The least accurately
computed mode (which was off by 5.5 ps/km) had the largest pv. As
another measure of accuracy, the spread in the computed delays for
each Q is also given in Table I.

Table I—Convergence results for
infinite parabolic profile in units of

Fig. 3—Computation errors in modal delays for infinite parabolic profile.

0.005 and R = 5 um, but again \ = 1.3 um. The calculated effective
index of the HE,, mode was accurate to seven decimal places and the
delay to 0.1 ps/km when TR was 2.5R and L, was 300. This precision
translates to nm accuracy in the zero-dispersion wavelength. The
beam radius (BR) defined by the condition

was accurate to four decimal places, as indicated in Table II. The
number of samples in the core can be less in the single-mode case
because the HE,, waveform does not oscillate; the truncation radius
needs to be greater because the waveform extends farther into the
cladding.

Assuming parameter values of A = 0.013, R = 25 um, A = 1.3 wm, and
a = 2 (the parabolic case), effective index (n.) and delay (7z) were
computed for each propagation mode under the scalar wave approxi-
mation. The rms delay (trms) converged to 1.793 ns/km within 0.3
percent and the rms delay with the two highest-order-mode groups
deleted (7‘pms) converged to 113 ps/km within 0.9 percent when L, =
300 and TR = 0.7R.

The delays spread almost 10 ns/km for Q = 14 and 3 ns/km for Q =
13, but less than 1 ns/km for the others. The spread in n, is negligible,
reaching a maximum of 0.01 percent.

Figure 5 shows n, and 7, plotted jointly on an expanded delay scale
with outliers excised. To compare with WKB results, n, and 7, for the
corresponding infinite parabola are indicated. The reference delays
exceed all in their mode group: by as little as 0.2 ps/km for each of
the first nine mode groups, but by more than 34 ps/km and 86 ps/km
for Q = 13 and 14, respectively.

Gradient index terms cause modal delays to spread still further.
Figure 6 shows the delay differences caused by these terms for each
mode of the parabolic fiber. Of the 112 modal delays only 14 differed
by more than 20 ps/km from the corresponding scalar wave values.
The TM mode nearest cutoff had the largest difference (33 ps/km).

EFFECTIVE INDEX
Fig. 5—Plots of 7, versus n, for cladded and infinite parabolic profiles.

Fig. 6—Absolute differences in modal delays due to gradient index terms for parabolic
fiber.

The change in trms was 5.9 ps/km and in 7’gms it was less than 1.5
ps/km. The error is less than 1.5 percent in either case. In the examples
that follow gradient index terms are neglected.

Material dispersion causes the bandwidth of power-law fibers to
depend sharply on the excitation wavelength (A). Here and in the
following examples, the profile is assumed known at the reference
wavelength of \ = 0.8 um; the concentration profile and the index
profile at other wavelengths are determined, as discussed in the
previous section.

Bandwidth is defined as the half-power frequency of the transfer
function over some distance. When intramodal dispersion is neglected,
the transfer function is

where 7, denotes the delay and a?, the power of the nth mode. Defining
the rms delay (7Rms) by

TRMS = b in ie = Tag) / x oa) (68)
and the average delay by
Tay =) AnTn / y an, (69)
then to second order in the w(t, — Tav), the bandwidth is
BW = 1/(277Rs), (70)

Bandwidth was calculated on this basis over a range of wavelengths
for the parabolic fiber of example 2, assuming the tapered modal-
power distribution shown in Fig. 7. Lower-order modes are weighted
more heavily than the higher and the highest are omitted as the higher
modes are increasingly vulnerable to microbending and cladding ab-
sorption.

Figure 8 shows the spectral plot of bandwidth. The peak value
occurs at \ = 0.963 wm, compared to 0.983 um reported in Ref. 13.

The spectral plot is also shown for the case where material dispersion
is neglected. The figure shows that the bandwidth is lower and essen-

Fig. 8—Plots of bandwidth versus wavelength for parabolic fiber with and without
material dispersion, using tapered modal-power distribution.

tially constant except for small discontinuities associated with new
modes cutting in. Other calculation indicates that bandwidth increases
to a peak at 9.5 GHz X km for a = 1.98 compared to an optimum a = |
1.97 estimated by WKB analysis.‘

Material dispersion causes the optimum a of power-law fibers to
depend on excitation wavelength. Figure 9 shows bandwidth versus a
when A = 0.013 and R = 25 um for \ = 0.82 wm and A = 1.32 um. The
optimum a’s are 2.07 for the former and 1.88 for the latter, compared
to values of 2.081 and 1.884, respectively, reported in Ref. 13. The
peak bandwidths are 5.48 GHz x km and 6.02 GHz Xx km, respectively.

The fractional power that propagates in the cladding can be used to
derive an alternate modal-power distribution. Modes with more than
0.1 percent of their power in the cladding will lose most of their power
over 1 km for typical cladding losses of 1 dB/m. A realistic modal

Fig. 9—Plots of bandwidth (in power-law fibers) versus a for \ = .82 wm and \ =
1.32 um, using tapered modal-power distribution.

distribution would neglect these modes and for simplicity could weight
the others equally.

Figure 10 shows bandwidth versus a for the two cases of example 5,
assuming the new distribution. The optimum a is lower by 0.01 and
the peak bandwidth is increased by 40 to 50 percent in either case.
Table III indicates the modes neglected in the bandwidth calculation
when A = 1.32 wm and a = 1.88.

Layer structure in actual fibers perturbs the ideal profile. In this
example power-law profiles are approximated by steps that span equal
areas and match the ideal profile at the mid-area points of the layers
(see Fig. 11). Bandwidth at \ = 1.32 um (using the modal distribution
in example 6) is shown versus a and the number of steps (NS) in Fig.
12. The graphs indicate that the optimum a is essentially independent
of NS, but the peak bandwidth and its sharpness decrease for smaller
NS.

Fig. 10—Plots of bandwidth versus a for \ = .82 um and A = 1.32 um assuming equal
excitation of modes having less than 0.1 percent power in cladding.

Delay (per km) of the HE,, mode was computed versus ) for an
actual single-mode fiber based on its measured index profile. The
profile measurement was performed on the associated preform and is
shown in Fig. 13. The depression of the inner cladding is due to F
doping.

As the calculation presumes radially symmetric profiles, the right
and left sides were considered separately. About 750 sample points
were used in either case.

The radial scale of the fiber profile was assumed to depend linearly
on that of the preform. The fiber core radius (R) was estimated from
the preform core radius (RP), fiber diameter (DF), and preform
diameter (DP) as

where the 1-percent addition estimates the SiO, loss in the outer
cladding during the draw process.

Figure 14 shows computed differential delay versus ) for the right
and left profiles together with measurements. It was assumed that
measurement matched calculation exactly at \ = 1.32 um, because
only relative delays were measured. The zero-dispersion wavelength
(at the delay minimum) was computed as \, = 1.3127 um for the left
profile and \, = 1.3113 um for the right, compared to measurement of
ho = 1.3114 pm.

Leakage loss (in dB/m) of the TE mode of the single-mode fiber in
the preceding example was calculated for both sides of the profile as a
function of \ and is shown in Fig. 15. The difference between the left
and right profiles becomes evident in this figure. A loss of 4 dB/m has
been identified”? with effective cutoff, and corresponds to cutoff wave-
lengths, A, = 1.17 wm for the left profile and \, = 1.23 um for the right,
compared to a measurement of 1.28 um. This discrepancy is discussed
in the next section.

The equivalent step approximation (where the index of the core and
the inner cladding are area-weighted averages of the profile) gives
Ao = 1.808 um and i, = 1.21 um. The X, value straddles the previous
calculated values, but the A, value is somewhat less.

The dimensionless formulation allows design curves to be generated
efficiently. Figure 16 shows V, versus V for the HE,, mode of three
power-law profiles. The specific parameters (R, A, \) determine V and,
hence, V. and dV,/dV. Delay versus \ is computed according to eq.
(58) and X, is estimated at the minimum of the delay curve. Figure 17
shows A, versus R for two values of A for triangular profiles (a = 1).

A method for computing modes of a circular optical fiber has been
presented. The finite element method reduces Maxwell’s equations to
the standard eigenvalue problem, involving tridiagonal matrices. Rou-
tines from EISPACK exploit the tridiagonal form to compute the
eigenvalues and eigenvectors efficiently. From these the modal quan-
tities are obtained.

Using a piecewise linear approximation of the waveform is necessary
to get tridiagonal form. Although piecewise quadratic and cubic ap-
proximations of the waveform can lead to smaller matrices, tridiagonal

form is lost and the eigencalculations would be less efficient. Extension
to elliptical and other nonradially symmetric fibers leads to similar
difficulty in the eigen calculations.

The method applies to any circular fiber and can account for
gradient index terms to first order. As illustrated in the 10 examples
of Section III, the calculations can provide information on the radial
distribution of propagating power, on pulse dispersion (arising from
material, intermodal, and intramodal dispersion), and on leakage loss.

The accuracy of the method was tested for the infinite parabolic
profile of the first example. After convergence was established, the
modal quantities were compared with the actual values known through
analysis. Agreement was excellent for both a multimode and a single-
mode profile.

The WKB and scalar wave approximations were tested for the
parabolic fiber in the next two examples. The cladding of the fiber
altered the delays of the higher-order modes substantially, a facet
missed by WKB analysis. The gradient index terms contributed less
than 1.5 percent to the rms delay; and so, for most purposes the scalar
wave approximation suffices for this fiber. The validity of these

approximations may change in other fibers; the value of the method
is that it permits the test.

Material dispersion was incorporated into the calculation in the
remaining examples. The variation of refractive index with wavelength
and dopant concentration was modeled by Sellmeier expansions based
on measurements of bulk samples with certain dopant concentration.”"
A linear dependence of index on concentration was assumed, but is
not essential to the method. Irreversible thermal and stress effects*”
might also be incorporated.

Bandwidth was estimated in examples 4 and 5 for power-law fibers
assuming a tapered modal-power distribution. With the higher-order
mode groups deemphasized, results essentially agreed with WKB
analysis. The usual sharp peak in bandwidth occurred in the spec-
tral plot for the parabolic (a = 2) fiber in example 4 and in the
a-dependence in example 5. Calculation of optimum a agreed reason-
ably well in these cases with other numerical work and WKB calcu-
lations.

The bandwidth calculation was repeated in example 6 using a
different modal-power distribution. Modes with more than 0.1 percent
of their power in the cladding were neglected, as being too lossy to
maintain power. The optimum a’s were essentially the same, but peak

Fig. 14—Measurement and calculations for left and right profiles of differential delay
versus wavelength.

bandwidths were somewhat higher. This example could be expanded
to simulate differential mode attenuation, mode mixing, concatenation
of dissimilar fibers, and other longitudinal variations.

Perturbations of a profiles usually lower bandwidth. Example 7
concerned the effect of layer structure on bandwidth and showed the
expected decline with a smaller number of layers. Other perturbations
such as ripple can also be treated. As a measure of its efficiency, the
method used about two seconds per profile on the Cray-1 for this
example.

The last three examples concerned single-mode fibers. The first two
of these involved an actual fiber whose profile was measured in the
preform stage. Calculation of delay (per km) versus \ matched meas-
urement extremely well, but calculated leakage loss exceeded meas-
urement, giving a 7-percent average underestimate of cutoff wave-
length (A,). Other calculations of leakage involving only slightly de-
pressed inner claddings, where \, matches measurement to within 1
percent, suggest that the discrepancy may involve material changes in
the F-doped glass caused by the draw process (consistent with obser-

WAVELENGTH IN MICROMETERS
Fig. 15—Plots of computed leakage loss versus wavelength for left and right profiles.

Fig. 16—Plots of V. versus V for the HE,, mode of three power-law profiles.
2690 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1983

Fig. 17—Plots of computed zero-dispersion wavelength (\,) versus core radius (R)
for two values of A, for triangular profiles.

vations in Ref. 31). Such changes need to be understood when pre-
dicting transmission performance from a preform profile. Similar
values for \, and \, were obtained for the equivalent step profile (with
depressed inner cladding), but the lower i, indicates the effect of
profile structure.

The dimensionless formulation permits the greatest efficiency in
getting design curves. Zero-dispersion wavelength (\,) for triangular
profiles is calculated in example 10 showing the expected shift®” to
higher wavelengths. Spot size or cutoff wavelength can be obtained
with similar efficiency.

In summary, the calculation method is reliable, and relatively in-
expensive. In the context of circular fibers it is comprehensive, capable
of simulating diverse effects.

The author is grateful for useful discussions with I. A. White and
A. J. Ritger on multimode fibers, W. T. Anderson and P. F. Glodis on

The author is also grateful to J. S. Nobles and D. L. Philen for fiber
measurements and to H. W. Friedrichsen for programming support.

1. B. T. Smith et al., Matrix Eigensystem Routines—EISPACK Guide, New York:
Springer Verlag, 1976.

2. P. M. Morse and H. Feshback, Methods of Theoretical Physics, New York: McGraw-
Hill, 1953, p. 1092.

4. D. Gloge and E. A. J. Marcatili, “Multimode Theory of Graded-Core Fibers,”
B.S.T.J., 52, No. 9 (November 1973), pp. 1563-78.

5. C. Pask, “Exact Expressions for Scalar Modal Eigenvalues and Group Delays in
Power-Law Optical Fibers,” J. Opt. Soc. Am., 69, No. 11 (November 1979), pp.
1599-1603.

6. G. A. E. Crone and J. M. Arnold, “Anomalous Group Delay in Optical Fibers,” Opt.
Quant. Elect., 12, No. 6 (November 1980), pp. 511-7.

8. S. Kawakami and S. Nishida, “Characteristics of a Doubly Clad Optical Fiber with
a Low-Index Inner Cladding,” IEEE Trans., QE-11 (December 1974), pp. 879-87.

10. R. A. Sammut and A. K. Ghatak, “Perturbation Theory of Optical Fibers with
Power-Law Core Profile,” Opt. Quant. Elect., 10, No. 6, (November 1978), pp.
475-82.

11. G.E. Peterson, A. Carnevale, U. C. Paek, and D. W. Berreman, “An Exact Numerical
Solution to Maxwell’s Equations for Lightguides,” B.S.T.J., 59, No. 7 (September
1980), pp. 1175-96.

12. G. E. Peterson, A. Carnevale, and U. C. Paek, “Comparison of Vector and Scalar
Modes in a Lightguide with a Hyperbolic Secant Index Distribution,” B.S.T.J.,
59, No. 9 (November 1980), pp. 1681-91.

13. G. E. Peterson, A. Carnevale, U. C. Paek, and J. W. Fleming, “Numerical Calculation
of Optimum a for a Germania-Doped Silica Lightguide,” B.S.T.J., 60, No. 4 (April
1981), pp. 455-70.

14. U. C. Paek, G. E. Peterson, and A. Carnevale, “Dispersionless Single-Mode Light-
guides with a Index Profiles,” B.S.T.J., 60, No. 5 (June 1981), pp. 583-98.

15. U. C. Paek, G. E. Peterson, and A. Carnevale, “Electromagnetic Fields, Field
Confinement, and Energy Flow in Dispersionless Single-Mode Lightguides with
Graded-Index Profiles,” B.S.T.J., 60, No. 8 (October 1981), pp. 1727-48.

16. Y. Kokubun and K. Iga, “Mode Analysis of Graded-Index Optical Fibers Using a
Scalar Wave Equation Including Gradient-Index Terms and a Direct Numerical
Integration,” J. Opt. Soc. Am., 70, No. 4 (April 1980), pp. 388-94.

17. W. L. Mammel and L. G. Cohen, “Numerical Prediction of Fiber Transmission
Characteristics from Arbitrary Refractive-Index Profiles,” Appl. Opt., 21, No. 4
(February 1982), pp. 699-703.

18. S.J. Jang, L. G. Cohen, W. L. Mammel, and M. A. Saifi, “Experimental Verification
of Ultra-Wide Bandwidth Spectra in Doubly-Clad Single-Mode Fiber,” B.S.T.WJ.,
61, No. 3 (March 1982), pp. 385-90.

19. L. G. Cohen, W. L. Mammel, and S. Lumish, “Tailoring the Shapes of Dispersion
Spectra to Control Bandwidths in Single-Mode Fibers,” Optics Letters, 7, No. 4
(April 1982), pp. 183-5.

20. K. Ikamoto and T. Okoshi, “Vectorial Wave Analysis of Inhomogeneous Optical
Fibers Using Finite Element Method,” IEEE Trans., MTT-26 (February 1978),
pp. 109-14.

21. C. Yeh, K. Ha, S. B. Dong, and W. P. Brown, “Single-Mode Optical Waveguides,”
Appl. Opt., 18, No. 10 (May 1979), pp. 1490-504.

22. D. S. Burnett, Finite Element Analysis, From Concept to Applications, New York:
Addison-Wesley, to be published.

23. M. Reed and B. Simon, Methods of Modern Mathematical Physics IV: Analysis of
Operators, New York: Academic Press, 1978.

24. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Washington,
D. C.: National Bureau of Standards, 1970, Chapter 9.

26. T. Kato, Perturbation Theory a Linear Operators, New York: Springer-Verlag,
1966, p. 125 and p. 391.

27. J. W. Fleming, “Material Dispersion in Lightguide Glasses,” Elect. Lett., 14, No. 11
(May 1978), pp. 326-8.

28. S. E. Miller and A. G. Chynoweth, ed., Optical Fiber Telecommunications, New
York: Academic Press, 1979, Chapter 7, p. 188.

30. P. L. Chu and T. Whitbread, “Measurement of Stresses in Optical Fiber and
Preform,” Appl. Opt., 21, No. 23 (December 1, 1982), pp. 4241-5.

31. M. J. Saunders, “A Comparison of Single-Mode Refractive Index Profiles in Pre-
forms and Fibers,” 8th ECOC, paper C-17 (September 1982).

32. K. I. White, “Design Parameters for Dispersion-Shifted Triangular-Profile Single-
Mode Fibers,” Elect. Lett., 18, No. 17 (August 19, 1982), pp. 725-7.

Gradient index terms, involving the quantity g, = d/drink?, occur
in eqs. (10) and (21). Although the index or its derivative may be
discontinuous, they can be approximated by smooth functions, so k?
and g, can be taken as well defined, continuous functions of r. This
appendix concerns the changes needed in the T matrix to accommo-
date these terms.

Equation (10) converts to symmetric form when h, = kf and both
sides are divided by k. The result is

in place of matrix A specified in eq. (34). The first quantity is aj as
before, the second adds to the C matrix of eq. (35), and the last two
when evaluated by the trapezoidal rule contribute only to the first side
diagonals. The term g,/r is handled in the same way as k?. Combining
these contributions gives a new 7 matrix that is symmetric and
tridiagonal.

The rest proceeds in the same way to give a symmetric, tridiagonal T
matrix.

Material dispersion is incorporated by expressing the index in terms
of concentration functions as before. Radial derivatives needed for the
gradient index terms themselves involve derivatives of the concentra-
tion function. The w” derivatives of these terms, needed for the modal
delays, are found by differentiating the coefficients of the concentra-
tion functions.

Terrence A. Lenahan, S.B. and S.M. (Electrical Engineering) 1964, Mas-
sachusetts Institute of Technology; Ph.D. (Applied Mathematics) 1970, Uni-
versity of Pennsylvania; Bell Laboratories, 1970—. Mr. Lenahan has done
studies of various areas of mathematical physics including electromagnetic
propagation, elasticity, and fluid mechanics. He has recently been interested
in the propagation of light in optical fibers. Member, American Mathematical
Society, SIAM, Sigma Xi.

Measurements of 800-MHz Radio Transmission
Into Buildings With Metallic Walls

In this paper we describe an experiment conducted to measure 800-MHz
attenuation into buildings. This information is needed for refining the config-
uration and design of portable radiotelephone systems that will accommodate
low-power portable sets. Signal levels have been measured in and around three
small buildings and a house, using an instrumentation van with an erectable
27-foot-high antenna. These buildings and the house all have metallic mate-
rials in their walls and thus are expected to exhibit high attenuation. The van
was parked at one location with respect to the three buildings, and at nine
different locations, at distances ranging from 400 feet to 1600 feet, from the
house. We found that small-scale signal envelope variations are approximately
Rayleigh distributed. For the house, large-scale distributions of the small-scale
signal medians are approximately log-normal. Median signal levels outside the
house decrease as d~*°, where d is the distance from the van. Inside the house,
levels decrease as d~*° for first-floor locations and as d~*° for second-floor
locations. Average signal levels at 1000 feet, relative to free space, are —12.5
dB outside, —18.5 dB for the first floor, —16.5 dB for the second floor, and
—28.9 dB for the basement. Other statistics of the signal levels and of
attenuation into the house are also given in the paper. Cross-polarization
couplings of —10 dB to 0 dB were measured in and around the three small
buildings. The small-scale signal medians inside the three buildings range
from 2 dB above to 24 dB below the averages of the signal medians outside
buildings.

I. INTRODUCTION
The attenuation of radio signals propagating into buildings has a
significant effect on the performance of portable radiotelephone sys-

tems. Frequencies in the 800-MHz to 900-MHz range are good can-
didates for such systems. Most earlier propagation measurements at
these frequencies in the shadowing and multipath environment around
buildings were made for mobile radio systems.” Ranges for the mobile
radio data are generally greater than one mile. Battery power limita-
tions on portable systems will likely restrict such systems to ranges
on the order of 1000 feet. Earlier measurements made in buildings are
limited in scope,** are oriented towards lower frequencies,” or are
directed toward high-power portable sets with ranges greater than one
mile.

One important portable radiotelephone environment comprises sub-
urban residential areas characterized by discrete houses and other
small buildings with densities ranging from less than one house per
acre to a few houses per acre. Houses and small buildings with metallic
materials in their walls are expected to exhibit high extreme values of
attenuation.

In the residential environment, fixed radio terminals that commu-
nicate with portable sets could be placed at convenient locations
outside buildings. These fixed terminals will be referred to as Portable
Radiotelephone Terminals or PORTs.

We have implemented an experiment to provide 800-MHz atten-
uation information needed for refining the configuration and design
of portable radiotelephone systems that will accommodate low-power
portable sets. Measurements were made in and around three small
buildings and a house. The buildings and the house all have metallic
materials in their walls. An instrumentation van was parked at differ-
ent locations to simulate different PORT with distances ranging from
200 to 1600 feet. A 27-foot-high erectable antenna on the van simulated
an unobtrusive PORT antenna. A portable signal source was moved
in and around a building and signal levels were received and recorded
in the van.

*The actual frequency of the measurements is 815 MHz; however, the statistical
results are not sensitive to small changes in frequency. Therefore, when frequency is
referred to relative to the measurements, it will be rounded to 800 MHz.

The transmitting antenna is a half-wavelength coaxial sleeve dipole
attached to the top of the hand-held unit. The de power is provided
by a self-contained nickel cadmium battery through a voltage regula-
tor. The regulator minimizes output power drift due to normal battery
discharge. The transmitter output is 0.8 watt. The output varies less
than 0.3 dB and 700 Hz over continuous one-hour periods that include
ambient temperature changes of 0°C to 25°C.

The instrumentation van (movable PORT) is a modified motor
home, containing a 5-kw ac generator. An uninterruptable power
supply isolates the instrumentation from generator voltage fluctua-
tions that otherwise could affect measurement accuracy. The van is
shown in Fig. 1.

Two 27-foot antenna masts are installed in pivoting mounts so they
can be stored horizontally for transport and can be erected at the test
site. They are 14.2 feet apart and are adjusted to vertical using built-
in bubble levels. An 815-MHz collinear receiving antenna is mounted
on one mast. A bracket on the other mast holds the 815-MHz signal
source to provide a reference signal for calibrating the receiver. The
centers of the two antennas are at the same height when erected.

The measuring receiver is an 815-MHz frequency-modulated (FM)
communications receiver, modified to detect the received signal enve-
lope. The receiving antenna is a collinear array (four dipole elements,
5.8-dB gain over dipole, 18-degree vertical beam width) mounted
vertically at the top of the tilt-over mast on the instrumentation van.
In the receiver modification, an 11.7-MHz intermediate frequency (IF)
output is extracted before the limiter, down-converted to 13 kHz,
bandpass-filtered (BW3as = 8 KHz), and linearly detected. The mod-
ified receiver has a —123 dBm sensitivity for 0 dB output s/n from the
linear envelope detector and a 45-dB measuring range between levels
6 dB above the noise level and 3 dB below saturation. The measuring
range is linear within +1 dB over a 35-dB range at high signal levels.
The entire receiver is enclosed in a sealed brass box to provide radio
frequency (RF) isolation. A variable RF attenuator reduces the input
signal in 1-dB steps to prevent overloading of the receiver.

The analog receiver output drives a 12-bit-resolution digital storage
oscilloscope and an integral 5-1/4-inch flexible disc drive for data
storage. The oscilloscope is set to record 2048 samples in a 20-second
measurement period. Up to 16 tracks of 2048 samples each can be

Fig. 1—The instrumentation van with the receiving antenna mast and the reference
signal mast erected. The center of the four-element collinear receiving antenna is 27
feet above ground at the top of the mast mounted on the right side of the van near the
front. The signal source is at the top of its mast mounted on the left side near the rear.
The center of the signal-source dipole is also 27 feet above ground.

recorded on each disc. Data for each parked position of the van, i.e.,
one PORT location, are stored on a separate disc.

The recorded data are transferred to a desktop computer. At the
time of transfer the following steps are performed:

1. Hand-recorded log information is appended to the signal strength
data. The log information includes such items as the address of the
house, the position of the van, the path azimuth, the path length, and
the receiver input attenuator setting.

2. The data are scaled to convert the recorded signal voltages to dB
relative to 0 dB at the reference location 14.2 feet from the receiving
antenna. The scaling takes into account recorded dc offsets, recorded
reference levels, and the receiver input attenuator settings.

3. Medians and cumulative distributions of signal level are calcu-
lated from the scaled data.

4, 'The scaled data are stored on flexible discs within the computer
for further analysis.

Van locations were selected using tax maps covering the immediate
vicinity of the house. Points were chosen equally spaced in azimuth
around the house for each of three radii at about 400, 800, and 1600
feet. Road layout and terrain irregularities influenced the final choices
of vehicle placement. A single van location was used for the measure-
ments in the other three small buildings.

Measurements were coordinated between the van and the measure-
ment location over a 450-MHz voice link. The link comprises a 25-
watt FM transceiver in the van and a 2-watt handie-talkie carried by
the person making the measurements.

A typical procedure for measuring a building starts with the van
parked at an appropriate position. The van location must be fairly
level. If necessary, wooden ramps are put under wheels to aid in
leveling. The portable transmitter is installed on its mast as a local
reference source. The mast is erected and is plumbed to vertical.
Similarly, the receiving antenna on the opposite side of the vehicle is
erected and plumbed. Keeping both masts plumbed on the level van
assures a fixed distance between the reference and receiving antennas
for calibration purposes. The linear detector IF input is grounded, and
the dc level is adjusted and recorded on a disc track. Next, the detector
input is ungrounded and the receiver RF attenuator is adjusted so the
RF reference level is within the receiver operating range. This level,
which serves as a calibration reference at the fixed 14.2 foot distance,
is then recorded.

The signal source is removed from the mast and taken to the building
for signal level measurements. The unit is hand-held at arm’s length,

for a scan height of 4.5 feet.® At a selected location within the building,
a 20-second raster scan is made by moving the transmitter in a
horizontal plane at 2.5 ft/s.° The 4-foot square scanned area consists
of 12 parallel linear scans separated by 4-inch increments. During the
scan period 2048 data points are taken at 100 samples/s.

During the scanning period, the oscilloscope is monitored to see
that signal amplitudes are within the receiver operating range. If not,
the RF input attenuation is adjusted and the scan is repeated. The
remaining locations within the building and immediately outside are
similarly scanned and recorded. After the measurements are made,
the transmitter is reinstalled on the reference mast, erected and
plumbed, and the dc level and RF signal reference level are again
recorded. The closure error between beginning and ending reference-
level recordings is usually <0.5 dB. If the closure is >1 dB, the
measurements are repeated. Such high closure error occasionally oc-
curs when the transmitter battery has discharged below the regulation
limit of the voltage regulator.

Motion of the signal source through the 4-foot-square areas inside
and outside of buildings results in small-scale signal variations. The
variations are caused by multipath propagation.”® Inside houses and
in areas shadowed from the van, where propagation is dominated by
reflection and scattering, the variations in the received signal envelope
are approximately Rayleigh distributed (see Ref. 6 and Section IV of
this paper). Received signal minima are separated on the order of one
half wavelength.® The medians (or means) of these small-scale varia-
tions are approximately stationary over the small areas but the medi-
ans for areas in different rooms in a building can be significantly
different. Thus, the signal statistics can be modeled as a combination
of a small-scale, quasi-stationary process (multipath) superimposed
on a large-scale process (shadowing). This model is like the models
used for mobile radio propagation.”"”

Only a single parameter is needed to describe the small-scale Ray-
leigh-distributed signal variation. The median can be determined if
somewhat over half of the samples are above the measurement thresh-
old. The mean, however, will be biased by the receiver noise level
unless significantly more than half the samples are above the thresh-
old. Therefore, medians of received signal variations for the 4-ft-
square areas are used to characterize the small-scale signal variations
at different measurement locations. The lowest median signal meas-
ured was 44 dB above the receiver noise level for the data presented
in Sections III and IV. The lowest median cross-polarized signal
measured was 38 dB above the noise level.

Building attenuation for a location inside a building can be defined
as the difference between the signal level at the location and a
“representative” signal level outside the building. Both levels are taken
to be small-scale medians, measured in decibels, relative to a common
reference level.

There are at least two possibilities that could be used for the
representative level outside. The first choice is to use the average level
of the signal in which the building is immersed. This level can be
approximated by taking the average of several median levels, in
decibels, measured at different locations surrounding the building.
This decibel averaging is appropriate for large-scale variations of the
median that are log-normally distributed. Large-scale variations are
shown in Section IV to be approximately log-normally distributed.
This definition of representative level seems appropriate for system
considerations because service would have to be provided all around
the exterior of a building.

The second possible choice for a representative level is to use the
level of the signal incident on the building from the PORT (or received
at the PORT from the incident region, since reciprocity holds). The
incident level can be determined readily when the PORT is in line-of-
sight of the building and there is little multipath from surrounding
buildings. This was the case for the measurements in Ref. 6. However,
when there are many intervening buildings between the PORT and
the subject building and there is considerable multipath from sur-
rounding buildings, the signal level incident on the building is not
easily defined. This second choice of representative level appears least
appropriate at the longest distances where the shadowing and multi-
path are most significant. These are also the distances of most concern
in system considerations.

In Section III, comparisons are made of attenuations determined
using both choices for representative levels. However, the more appro-
priate average outside level is used for the statistics in Section IV.

itl. ATTENUATIONS AND CROSS-POLARIZATION COUPLINGS FOR
THREE BUILDINGS

The first building, shown in Fig. 2, is 20 feet by 38 feet, of corrugated
steel construction and mounted on a concrete platform. There are
metal screened doors inside the main metal doors, and all of the nine
windows are metal screened, with the exception of three in the rear.
The building inside is a single large area and is about one-third filled
with equipment.

The second building is shown in Fig. 3. It is 21.5 feet by 28 feet, is
covered with aluminum siding and has one solid door and nine win-

Fig. 2—Corrugated steel building on Crawford Hill as seen from the instrumentation
van position.

Fig. 3 SAntehin Control building on Crawford Hill. The photograph was taken
looking toward the building along the path from the van.

dows. All except the three largest windows are metal screened. The
building interior is divided roughly in half on the long dimension and
is about half filled with equipment.

Fig. 4—Wooden building on Crawford Hill. The propagation path from the van was
into the corner to the left in the photograph.

and contains vapor barrier foil insulation in the walls. This building,
shown in Fig. 4, was empty. In front it has two doors with unscreened
windows and three other unscreened windows. The room with double
doors to the right in Fig. 4 is attached to an exterior wall. Measure-
ments were not made in the attached room.

Signal levels were measured at four locations outside and two or
three locations inside of the three buildings from a single van (PORT)
location. These levels are summarized in Table I. The distances
indicated in the table (i.e., 225 feet, 330 feet, and 845 feet) are the
path length from the van antenna to the fronts of the buildings. The
centers of the four outside locations for each building were about 5 to
10 feet away from the midpoints of the four sides of the building. The
outside locations are numbered clockwise looking down on the building
from above, starting with the side closest to the van. The small-scale
median signal levels in decibels and the decibel average for the four
outside locations are tabulated in the column labeled “Level/dB”.
These levels are relative to 0 dB at the signal level reference 14.2 feet
from the van receiving antenna. The column labeled 1/r?/dB contains
the signal levels that would occur in free space (1/r?) at the distance
of the measurement location. These levels are also relative to 0 dB at
the 14.2 foot reference location. The 1/r? level in the row labeled Avg.
is the free space value at the building midpoint.

For the wooden building positioned at 330 feet from the van and for
the building with metal siding positioned at 845 feet, the signal
medians measured in front of the buildings are 1 to 2 dB greater than

the free-space values. The signal median in front of the all-metal
building positioned at 225 feet is about 1 dB below the free-space
value. These values are all consistent with the effects of a single
reflection from the relatively flat dry ground between the van and the
buildings.’® The 27-foot van antenna height and a reflection coefficient
phase angle of nearly 180 degrees, appropriate for small angles between
the incident wave and smooth dry ground, yield signal maxima at
distances above the ground of about 3 feet for 330 feet from the van
and of about 8 feet for 845 feet from the van. For the same conditions,
a minimum occurs about 4 feet above the ground at 225 feet from the
van. Recall, the signal source is scanned about 4.5 feet above the
ground. Therefore, at 330 feet, the scan is near a broad maximum that
could be as much as 6 dB above free space if the ground were perfectly
reflecting. The measured level at 330 feet is 2 dB above free space. At
845 feet, the scan should be down on the side of an 8-foot-high
maximum. The measured level at 845 feet is 1.8 dB above free space.
At 225 feet, the scan is near the first minimum above the ground. The
measured level at 225 feet is 1.2 dB below free space.

The interior locations of the buildings are arbitrarily numbered in
the column labeled “Location Inside”. The column labeled “Relative
to Average Attenuation in decibels” contains the differences between
the median signal levels measured at the locations and the average of
the four outside median levels for the building. Positive attenuation
indicates the signal level inside is smaller than the average level
outside.

were made at Location 1 inside the metal-sided building. The spread
of the resulting three medians was only +0.5 dB around the average
value listed in the table. Two separate measurements were made at
Location 2 outside the wooden building. The resulting two medians
were within +0.3 dB of the average value listed. Thus, the measure-
ment repeatability is good and is consistent with the +0.5 dB standard
deviation of the medians expected for the statistical fluctuation re-
sulting from the limited number (~150) of independent samples in a
4-foot-square area. Measurements were made at the same locations in
and around the metal building on three different days that had
different ground moisture conditions, etc. The three medians for each
location were averaged and the difference was taken between the
location average and the individual medians. The standard deviation
of the differences was 1 dB.

The column labeled “Relative to Incident Attenuation in decibels”
contains the differences between the median levels at the inside
locations and the median level for the outside location that is closest
to the van. These outside closest locations have the largest signal
levels measured for their corresponding buildings. The outside level is
corrected for free space (1/r”) for the distance between the outside
location and the inside location being considered. This is the second
definition of building attenuation described in Section 2.3. This second
definition of attenuation is essentially the same definition of building
attenuation that was used in Ref. 6 for the same all-metal building
listed first in Table I (the values in Ref. 6 were not corrected for 1/
r”), The second definition attenuation values are within 1 or 2 dB of
the values in Ref. 6 for Locations 1 and 3 inside the metal building.
At Location 2, however, the attenuation in Table I is 6 dB less than
the earlier value. Items inside the building have been rearranged since |
the earlier measurements, but no reason for such a large change is
evident. The second definition attenuation into the all-metal building
is greater than attenuation from the front (van side) to the back of
that building (10.0 dB front-to-back attenuation including 1/r? cor-
rection). The second definition attenuation into the other buildings is
less than the front-to-back attenuation (21.7 dB for the metal-sided
bulding and 15.0 dB for the wooden one). This suggests that the
dominant mechanism for signal propagation behind the metal building
is scatter and/or reflection from objects behind and to the side of that
building rather than passage through the building itself. The dominant
mechanism for propagation behind the other two buildings is not
evident from these simple measurements since either passage through
multiple walls or scatter and/or reflection is consistent with the result.

For Location 1 in the wooden building, the first definition attenua-
tion is negative. This indicates that the median level inside at that

location is greater than the average of the four outside medians. This
is a reasonable situation because the signal levels at one side and at
the back of that building are much lower than the signal level inside
Location 1. This inside location has unscreened windows between it
and the outside in the direction of the van. As mentioned in Section
II, this first definition of attenuation seems more appropriate for use
in system analysis since a system would have to serve the outside
locations at all sides of a building. Of course, the distribution of outside
levels relative to the outside average is also needed for a complete
assessment of system performance. ,

The building attenuations by either definition are greatest for the
all-metal building with metal screened windows. Since both of the
other buildings have metal in their walls, attenuation into them
probably depends strongly on coupling through the nonscreened win-
dows.

Cross-polarization coupling in multipath propagation is significant
in reducing the effects of the random orientation of portable radiote-
lephones.’* Cross-polarization coupling is defined as 20 log(E,/F;),
where F, is the average or median field magnitude of the polarization
aligned with the transmitted polarization and E, is the average (or
median) field magnitude of the polarization orthogonal (crossed) to
E,. An indication of the cross-polarization coupling can be obtained
by orienting the signal-source dipole antenna horizontally and scan-
ning a measurement location with the dipole pointed towards the van
(end-on orientation). The scan can be repeated with the dipole per-
pendicular to the direction of the van (broadside orientation).

If the multipath propagation were uniformly distributed in azimuth
around the measurement location and were confined to a horizontal
plane, horizontally polarized multipath would produce a median re-
ceived signal level in a scanned horizontal dipole that was 3 dB less
than the median that would be produced by the same multipath in a
scanned loop oriented with its plane horizontal. The 3-dB decrease
results from the nonuniform directivity pattern of the dipole in any
plane containing the dipole. The median level received by the loop in
the horizontally polarized multipath would be the same as the median
level received by a scanned vertical dipole in vertically polarized
multipath of the same average intensity. The multipath signal varia-
tions in all cases would be Rayleigh distributed.

For the measurement situation, equal median signal levels for end-
on and broadside horizontal scans would be consistent with multipath
having a uniform azimuthal distribution. Then, under the assumption
that the propagation directions are confined to a horizontal plane, the

cross-polarization coupling would be 3 dB greater than the signal
difference A = L, — Ly, where L, is the median level of the signal from
a scan with the source oriented vertically, and L;, is the median level
of the signal from a scan with the source oriented horizontally. The
medians have a statistical fluctuation with a standard deviation on
the order of +0.5 dB because of the limited number of independent °
samples. Therefore, for differences between the medians of the two
horizontal scans of one or two decibels, they can be taken as equal
and their average value can be used to determine A.

Table II summarizes the cross-polarization measurements made in
and around the three buildings. The medians of the end-on and
broadside scans are tabulated in columns labeled End-on and Broad-
side. The values are in decibels relative to the median of the signal
scan at the same location with the source antenna oriented vertically.
That is, columns End-on and Broadside indicate A for end-on and
broadside scans. The column labeled cross-polarization is 3 dB greater
than the average A for end-on and broadside scans.

The locations inside the metal building would yield a small positive
value for cross-polarization coupling. This probably indicates a break-
down of the assumption that the multipath propagation is confined to
a horizontal plane, so the coupling is taken as 0 dB. If the multipath
propagation were uniformly distributed in all directions in three di-
mensions, the orientation of the antenna would be irrelevant. Since
the data show only a small bias towards stronger median signal for
the vertical antenna in the metal building, an alternative assumption
for that building would seem to be uniformly distributed multipath
propagation in all directions in three dimensions.

The cross-polarization values in Table II are all greater than —10
dB and most are greater than —6 dB. Another point worth noting is
that the locations with the lowest signal levels (greatest attenuation)
also have the largest cross-polarization coupling with values greater
than —6 dB and usually greater than —3 dB. Since cross-polarization

coupling increases the effectiveness of diversity in mitigating the
effects of random portable set orientation and multipath propagation,'*
this trend could be significant in determining system performance.

The house is a two-story colonial located on a level, one acre lot. It
is in a newly developed area with a density of one house per acre and
with relatively few trees. The house has an area of 2400 square feet,
consisting of living room, dining room, kitchen, and den on the first
floor, and four bedrooms on the second floor. It also has a basement
and a two-car attached garage. It is constructed with wood, with
aluminum siding on three sides and nonmetallic siding on the front.
All exterior walls contain insulation faced with a metal foil vapor
barrier.

The cumulative distributions of the envelope variations of multipath
propagation for a scan of a 4-foot-square location are expected to be
Rayleigh.”* Figure 5 shows the distributions measured at four locations
in and around the house. A Rayleigh distribution is a straight line
with the particular slope indicated on the figure. The distributions are
good approximations to the Rayleigh distribution for many locations
inside the house and behind the house from the van. A few locations
on the same side of the house as the van and only 400 feet from it
experience essentially line-of-sight propagation. At these few locations
the signal variation is small and the envelope distribution significantly
departs from Rayleigh, as indicated by the example plotted as squares
on the figure. The medians of the four distributions on the figure have
been normalized to 0 dB. The two distributions that show the greatest
departure from Rayleigh were selected as the extremes that have the
greatest and least spread in attenuation of any of the distributions in
the data set.

The open data points in Fig. 6 are the medians of the measured
signal envelopes for the small-scale locations outside the house. The
medians are plotted versus the distance between the van antenna and
the location. The median levels are in decibels relative to the signal
level at the van reference. The outside locations are in front of the
midpoints of the four outside walls of the house. The decibel averages
of the four locations for each van position are indicated by the solid
data points.

Fig. 5—Measured signal envelope distributions from four small (4-ft square) areas.
On these coordinates, Rayleigh distributions are straight lines with the slope indicated.
Distances between the van and the measurement location are indicated in the key.

The solid line is the linear-least-squares regression fit to the 36
medians from the four locations for each of the nine van positions.
The least-squares regression fit to the averages of the four locations
for each van position yields the same solid line. Signal level varies
with distance as d~*° for the solid line. The dotted line represents
free-space propagation (d~”) relative to 0 dB at the van reference. At
1000 feet from the van, the signal level represented by the solid line
is 12.5 dB lower than the level would be in free space, i.e., the average
excess attenuation over free space is 12.5 dB at 1000 feet. The distance
dependence exponent is probably reliable only to several tenths be-
cause of the large spread in the data and the relatively small number
of data points. The signal level outside was correlated to distance with
a coefficient of 0.8.

The lower dashed line is the linear regression line for the data near
400 feet and 800 feet only. The upper dashed line is for the data near
800 feet and 1600 feet. The distance dependence for the 400-foot to

Fig. 6—Medians of the small-scale envelope variations for different outside measure-
ment locations plotted versus distances between the locations and the van antenna.
Open data points are medians for individual outside locations. Solid points are averages
of the medians for the four measurement locations for each of the van locations. Points
represented by the same symbols at nearly the same distance designate the four
measurements associated with a particular van position. Signal levels are with respect
to 0 dB at the signal reference located 14.2 feet from the van receiving antenna. The
dotted line represents free-space propagation (1/r*) relative to 0 dB at the signal
reference. The solid line and the dashed lines are linear-least-squares regression lines,
as discussed in the text.

800-foot distances is d~*'; the dependence for the 800-foot to 1600-
foot distances is d~*. Thus, the signal decreases faster with distance
in the first several hundred feet than it does farther from the house.
This is reasonable because, for the first few hundred feet, there are
few obstructions along the path and propagation approaches line-of-
sight. After several hundred feet, the number of intervening houses
increases. Reflection from the ground also causes the signal to decrease
more rapidly with distance than it does in free space.

The data points at 400 feet that are greater than free space values
deserve comment. These points are from locations that are not shad-
owed from the van, i.e., they are within line-of-sight. A single ground
reflection from a 27-foot-high antenna 400 feet away would produce a
signal maximum about 4 feet above the ground. The signal maximum
would be greater than the single-path free-space value. (It would be 6
dB greater for a reflection coefficient of unity.) The signal levels a few
dB above free space are not unexpected 4.5 feet above the ground in

locations that have large reflecting surfaces (walls of houses) nearby
in addition to the ground.

Figure 7 is the cumulative distribution of the data from Fig. 6 after
the distance dependence, d~**, is removed. The calculated standard
deviation for the data points is 9.0 dB. A straight line on the coordi-
nates in Fig. 7 represents a log-normal distribution of signal level. The
measured distribution is a reasonable approximation to a log-normal
distribution.

The distance dependences for the linear-least-square fits to the
medians for measurement locations inside the house were somewhat
different for the different floors of the house. The variation with
distance was d~*° for the first floor, d~*° for the second floor, and d~*”
for the basement. Since only one location was measured in the base-
ment, this value for the basement may contain considerable statistical

aoe 5 10 20 30 40 50 60 70 80 90 95 98 99
PERCENTAGE OF LOCATIONS WITH NORMALIZED LEVEL = ORDINATE
Fig. 7—Cumulative distribution of the medians of the small-scale envelope variations
for outside locations after the values for the solid regression line in Fig. 6 are subtracted

out. The solid line and the dashed line in this figure represent log-normal distributi
with standard deviations of 9 dB and 10 dB, respectively. Sere edna ve

error. The variation with distance for the composite set of data from
the entire house (eight measurement locations inside the house and
nine van positions) was d~*°. The less rapid decrease in signal level
with distance for the second floor compared with the first floor is
reasonable because the height of the second floor results in less
attenuation from intervening houses. Since the decrease in signal level
with distance is less rapid for the locations inside the house than
outside, the apparent building attenuation will decrease somewhat
with distance.

At 1000 feet, the excess attenuation over the free space values are
18.5 dB for the first floor, 16.5 dB for the second floor, 29.9 dB for the
basement, and 19 dB for the composite data set for the entire house.
The average building attenuation at 1000 feet, which is the difference
between the linear regression values outside and inside, is then 6.0 dB
for the first floor, 4.0 dB for the second floor, 16.4 dB for the basement,
and 6.5 dB for the entire house.

The building attenuation considered in this section uses the first
definition in Section 2.3, i.e., attenuation is with respect to the average
of the four outside median levels. Building attenuations for the house
are plotted versus distance in Fig. 8. The open data points represent
data from the first floor; the solid data points and the points marked
B represent data from the second floor and the basement, respectively.
As noted in the previous section, the attenuation is weakly dependent
on distance. Also, the attenuation into the basement is significantly
greater than the attenuation into the other two floors. The linear
regression lines are labeled on the figure for several combinations of
the data. The variation of attenuation with distance is d~°® for the
first floor, d~}° for the second floor, d~'* for the basement, and d7}°
for the first and second floor together and for the entire house. The
attenuation at 1000 feet taken from the regression lines is 6.0 dB for
the first floor, 4.0 dB for the second floor, and 16.4 dB for the basement.
These 1000-ft attenuation values are the same as those obtained in
the previous section from the separate linear regression lines to the
signal levels inside and outside.

Means (m) and standard deviations (c) for several different group-
ings of the attenuation data are tabulated in Table III. It appears that
most of the difference in the distance dependence for the first and
second floors results from the different attenuation averages at 1600
feet.

The different distance dependences and different average attenua-
tion for the different floors present a dilemma if they are real effects
not confined to this data set. For radio system analysis, a single

Fig. 8—Building attenuation by the first definition, i.e., relative to the average of the
median levels for the four outside locations, plotted versus distance. Open data points
and X points are from the first floor, solid data points are from the second floor, and
points marked B are from the basement. All points marked with a particular symbol
are from the same measurement location in the house but are for different van positions.
The lines are linear-least-squares regression lines through different groupings of points.

Fig. 9—Cumulative distribution of the building attenuations from Fig. 8 for the first
and second floors after the values for the first- and second-floor regression line (solid
line in Fig. 8) are subtracted out. The straight line represents a log-normal distribution
with a standard deviation of 4.4 dB.

distance dependence and average attenuation are desirable; however,
if only the composite regression line for the whole house were removed
from the data, the variation remaining in the residual attenuation data
would have a significantly larger standard deviation (o = 6.2 dB) than
if the individual regression lines for each floor were removed separately
(o = 4.1 dB). Also, the composite line for the entire house depends on
the number of measurements made on each floor. For systems analysis,
these numbers of measurements should be related to the probability
of calls being made on those floors (or at those locations); but these
probabilities are unknown.

The separation in decibels between the regression lines on Fig. 8 for
the first and second floors is not extreme over the 4 to 1 distance

covered. Also, the attenuation averages for these floors for the three
distances are not grossly different (all within +2 dB). Therefore, it
seems reasonable to simplify the description of the attenuation by
combining the first and second floor data. The composite regression
line for this combination is also shown on the figure. When this
composite regression line is removed from the first and second floor
data, the resulting cumulative distribution of attenuation is as shown
in Fig. 9. The standard deviation of the attenuation data plotted in
Fig. 9 is 4.4 dB, not significantly different from the 4.1-dB standard
deviation for all the data with the regression line for each floor removed
separately. The distribution in Fig. 9 is a reasonable fit to a log-normal
distribution, especially considering the limited size of the data set.
Combining the basement data with the data from the other floors is
not reasonable because of the large difference in average attenuations.
Since this difference can be expected to exist for all houses, the
attenuation into basements will probably have to be described sepa-
rately from the attenuation for the rooms on other floors.

Signal levels were measured in and around three small buildings
from a single position of an instrumentation van. Measurements were
also made in and around a house from nine different van positions
ranging from 400 feet to 1600 feet away. The received signal envelope
is approximately Rayleigh distributed over most small-scale areas
about 4 feet square. For the three buildings, building attenuation of
the small-scale signal medians ranged from —2 to 24 dB relative to
the average signal level outside each building.

For the house, some of the parameters of the medians of the small-
scale envelope variations and of the building attenuation defined in
this paper are summarized in Table IV. After the distance dependence
is removed from the median signal levels outside of the house, the
distribution of the residual large-scale signal-level variations is ap-

the house
First
&
Out- First Second Base- Second
Parameter side Floor Floor ment Floors Units
Signal-level distance dependence —4.5 —3.9 —3.0 —3.2 — —
exponent
Signal-level relative to free —-125 -185 -—-165  -—28.9 — dB
space at 1000 ft.
Building attenuation exponent _ —0.6 —1.5 —-14 -1.0 —
Average building attenuation at — +6.0 +40 +164 +5.1 £=dB
1000 ft.

proximately log-normal with a standard deviation of 9 dB. The stand-
ard deviation of the building attenuation for the first and second floors
is 4.4 dB after distance dependence is removed. Because of the large
average attenuation into the basement of the house, it does not appear
reasonable to combine the basement attenuation statistics with the
statistics for the rooms above ground level.

Cross-polarization coupling is strong inside and outside of the three
buildings. For all locations measured, the coupling is greater than —10
dB and for most locations it is —6 dB or greater.

Initial design of the van masts and mounts was done by W. I.
Tohlman and H. H. Hoffman. We wish to thank H. W. Arnold for
permitting the measurements reported in this paper to be made in his
house. The continued support of L. J. Greenstein and D. O. Reudink
is greatly appreciated.

1. D. C. Cox, “Co-Channel Interference Considerations in Frequency-Reuse Small
Coverage-Area Radio Systems,” IEEE Trans. Commun., COM-30 (January 1982),
pp. 135-42.

. J. M. Durante, “Building Penetration Loss at 900 MHz,” Conference Proceedings,
IEEE VTG Conference, 1973, p. 1-7.

. P. I. Wells and P. V. Tryon, “The Attenuation of UHF Radio Signals by Houses,”
U. S. Dept. of Commerce Report, OT Report 76-98, August 1976; and IEEE
Trans. Veh. Technol., VT-26 (November 1977), pp. 358-62.

. J. Shefer, “Propagation Statistics of 900 MHz and 450 MHz Signals Inside Build-
ings,” Microwave Mobile Radio Symposium, March 7-9, 1973, Boulder, Colorado.

. H. H. Hoffman and D. C. Cox, “Attenuation of 900 MHz Radio Waves Propagating
Into a Metal Building,” IEEE Trans. Ant. Propag., AP-30 (July 1982), pp. 808-11.

. D. Mitchell and K. G. Van Wynen, “A 150 MC Personal Radio Signaling System,”
B.S.T.J., 40, No. 5 (September 1961), pp. 1239-57.

. G. V. Waldo, “Report on the Analysis of Measurements and Observations of New
York City UHF-TV Project,” IEEE Trans. Broadc., 9 (1963), pp. 7-36.

10. K. Tsujimura and M. Kuwabara, “Cordless Telephone System and its Propagation
Characteristics,” JEEE Trans. Veh. Technol., VT-26 (November 1977), pp.
367-71.

11. M. Komura, T. Hogihira, and M. Ogasawara, “New Radio Paging System and Its
Propagation Characteristics,” IEEE Trans. Veh. Technol., VT-26 (November
1977), pp. 362-6.

12. D. C. Cox, “Multipath Delay Spread and Path Loss Correlation for 910 MHz Urban
Mobile Radio Propagation,” IEEE Trans. Veh. Technol., VT-26 (November
1977), pp. 340-4.

13. P. A. Matthews, Radio Wave Propagation, VHF and Above, London: Chapman and
Hall, Ltd., 1965, Chapter 2.

14. D. C. Cox, Antenna Diversity Performance in Mitigating the Effects of Portable
Radiotelephone Orientation and Multipath Propagation, IEEE Trans. Commun.,
COM-31 (May 1983), pp. 620-8.

Donald C. Cox, B.S., M.S., University of Nebraska, Lincoln, 1959 and 1960,
respectively; Ph.D., Stanford University, 1968, all in Electrical Engineering;

Honorary Dr. of Science, University of Nebraska, Lincoln, 1983; Stanford
University, 1963-1968; Bell Laboratories, 1968—. From 1960 to 1963, Mr. Cox
worked on microwave communications system design at Wright-Patterson
AFB, Ohio. From 1963 to 1968 he was at Stanford University doing tunnel
diode amplifier design and research on microwave propagation in the tropo-
sphere. From 1968 to 1973 he was a Member of Technical Staff at Bell
Laboratories, Holmdel, New Jersey, doing research in mobile radio propaga-
tion and on high-capacity mobile radio systems. He is now Supervisor of a
group doing propagation and systems research for portable-radio telephony
and for millimeter-wave satellite communications. Fellow, IEEE; member,
Commissions B, C, and F of USNC/URSI, Sigma Xi, Sigma Tau, Eta Kappa
Nu and Pi Mu Epsilon; Registered Professional Engineer, Ohio and Nebraska.

Roy R. Murray, B.S. (Electronic Engineering), 1975, Monmouth College;
Bell Laboratories, 1965—. Mr. Murray has worked on high-speed multi-level
digital modulators and, more recently, on UHF radio propagation into build-
ings. Currently, he is a member of the Telecommunication Systems Research
Department. Member, Eta Kappa Nu.

Penetration of Radio Signals Into Buildings in the
Cellular Radio Environment

Penetration of radio signals from Advanced Mobile Phone Service cell site
transmitters has been measured in fourteen office and industrial buildings in
the Chicago area. Signal levels on the first floor of buildings averaged 14 dB
less than reference levels in the adjacent streets. This penetration loss was
found to decrease with increasing height. Standard deviations of penetration
loss ranging from 5 to 11 dB attest to the diversity of architecture and floor
arrangements. Other relationships useful in planning for portable phone
terminal applications are derived from the data and are presented in this
paper. The data include over 4000 measurements, each being the average of
1024 samples of the local field taken over an eight-second period as the
instrumentation traveled about 20 feet over the measurement path. The
measurement path in each building was traversed for each of the several cell
sites that transmitted signals of adequate strength into the buildings.

The use of portable and hand-held terminals in 850-MHz cellular
radio systems involves a radio field environment inside buildings that
differs from the more familiar highway and street propagation envi-
ronment of the mobile terminal. The laws that define propagation
over open and urban terrain are complicated by a penetration loss in
the transmission of the signal into the building’s interior. The char-
acterization of this loss, as encountered in large office and commercial

buildings, was the purpose of an extensive series of measurements
made in fourteen buildings served by the Advanced Mobile Phone
Service (AMPS) Developmental Cellular System in Chicago. These
measurements were then compared with the penetration loss in a
single-family residence.

The measurements were made in January 1980 to serve as a basis
for evolving system requirements related to the use of portable phone
terminals in AMPS systems.

Indoor portable service in cellular systems offers a possible improve-
ment over conventional systems. The multiplicity of serving cell sites
should in many cases illuminate all sides of those buildings that enjoy
a relatively exposed or unshadowed environment. It was expected that
this effect would be somewhat less pronounced in buildings that are
more sheltered in the urban core; the data support this conjecture.
The data also support expectations that areas with windows would
have lower penetration losses than areas without windows; the differ-
ence is about 6 dB. First-floor penetration losses averaged 14.2 dB.
The loss decreased with height at 1.9 dB per floor, very close to the
2.0- and 2.5-dB rates reported in Refs. 1 and 2. The penetration loss
measured in the aluminum-sided ranch house, as a matter of interest,
is 7.3 dB, a value close to the findings of Cox et al. in Ref. 3.

Most of the measurements made in the Cellular Test Bed (Ref. 4)
and reported in Ref. 5 were of the signal as received by a mobile
transceiver. The measurements reported here include the entire dis-
tribution of setup channel signal levels throughout the measurement
areas for each cell site transmitter that qualified as a server. As such,
the data are perfectly applicable to conventional mobile portable and
paging applications. The effect is that of taking measurements with
the base station in several different locations.

The Chicago AMPS Developmental Cellular System, as described
in Ref. 4, uses ten separate setup channels for paging and mobile call
origination in the Chicago service area. The channel signals are
radiated continuously from omnidirectional antennas. The instrumen-
tation used for the measurements reported in this paper can be tuned
manually to any of these frequencies to measure one channel at a
time. Figure 1 shows nominal coverage contours for the setup channels
of the seven cell sites that served the selected buildings. The related
voice-channel coverage is shown in Fig. 2 of the paper by D. L. Huff.*
The center of each contour represents a cell site location. In general,
the cell sites are spaced about 15 miles apart.

A group of buildings in the Chicago area was selected to represent
a range of physical characteristics including location, architecture,

© COMMERCIAL, INDUSTRIAL, NOMINAL SETUP
OR OFFICE BUILDING CHANNEL COVERAGE

Fig. 1—AMPS Developmental Cellular System locations of 15 buildings and seven
serving cell sites.

and local environment. Signal strength measurements were made
along planned routes on selected floors of these buildings. A Propa-
gation Measuring Set (PMS) was used to measure the strength of
signals transmitted from cell sites.

Building penetration loss for a given floor area is defined to be the
difference between the average of these measurements and an average
of measurements made on the outside at street level.

Outside signal strength was measured at street level around the
perimeter of the building, along the closest available path to the
building’s outside walls. These paths included driveways, streets, or
parking lots, as required to achieve proximity to the building under
test.

Fig. 2—Differences in level between the strongest and weakest serving channels on
the same floor.

The purpose of the measurement program, as stated previously, was
to determine the extent to which signals available from AMPS cell
sites penetrate buildings. A first-order determination was desired;
effects of 5 dB or more on the loss measurements were considered to
be significant. A detailed investigation of the structure of interior
fields in small buildings is reported by Cox in Ref. 3.

The measurements were subject to the effects of the following
different conditions:

1. Different types of outside wall construction, such as steel-framed
glass, brick, block masonry, etc.

2. Urban vs. suburban areas to identify difference between buildings
in exposed locations and buildings sheltered in the dense urban core.

6. Different types of window treatment currently used to reflect
sunlight and heat. =

Time was an important factor in selecting the coverage objective for
each building. Test runs were chosen to require about three hours of
operations in any given building. The total time of recorded data is a
small percentage of that..

Six significant categories of interior areas were identified. These
were defined as open, enclosed, and hallways—each with and without
windows. Typical of the areas that were considered to be “open” were
cafeterias, lobbies, and large office spaces or conference rooms. Typical
of “enclosed” areas were small, walled areas with space for only a few
occupants, such as small office areas or workrooms.

thought to represent a worst case. Higher floors, thought to represent
better penetration, were included with less regularity. Test routes were
designed to distribute measurements over the floor areas in a uniform
manner.

The buildings chosen in the Chicago AMPS service area were those
where multiple cell sites could provide outside building signal levels of
at least —90 dBm.

In the downtown section of Chicago, tests were made in three
buildings. These represent the urban group. The downtown buildings
in which tests were made can be characterized as “single tenant
buildings”.

In the suburban area around Chicago, tests were made in eleven
commercial buildings and one residential dwelling. Only the data from
commercial buildings were included in subsequent analyses.

From the standpoint of architectural variety the program provided
measurements in a warehouse, a manufacturing plant, open-space
offices, cafeterias, lobbies, small offices, and hallways. Outside wall
construction varied from unwindowed concrete slab to steel-framed
windows occupying about 90 percent of the outside surface.

The locations of the buildings where tests were conducted are shown
on the map of Fig. 1. The circles show the idealized boundaries of
regions of cell site contro] channel coverage. Table I shows the distri-
bution of floor levels in buildings where tests were conducted and the
cell sites whose transmissions served the various floor levels. A more
detailed description of each of the buildings is given in the appendix.

Several factors were considered when selecting floors and floor areas
for measurement. From a test sampling standpoint the first and second
floors were chosen when available since building penetration was
expected to be poorest at the lower levels. Higher floor levels were
included in the tests to provide data on the sensitivity of building
penetration to height. On any particular floor, areas were chosen that
represent the different types of interiors; test paths were chosen to be
uniformly distributed over the floor. In two instances measurements
were attempted in basement areas but signal strength levels were
below the threshold of the PMS.

A portable Propagation Measurement Set (PMS) designed for field
strength measurements was adapted for this test program by mounting
it in a small “tea” cart equipped with a battery, battery charger, and a
mobile antenna mounted on a 2-foot by 2-foot ground plane. The

io aAmNye CW NF Pe WN RO PDH FH CONK NR OOWH OW WHF ANH
PAPA PP PPPS PAM POPS OS

* CVL-Cloverdale; EOL-Eola; LMT-Lemont; LNS-Lyons; CNL-Canal; MGV-Mor-
ton Grove; BEV-Beverly.
ground plane was positioned 4-1/2 feet above floor level, at approxi-
mately shoulder height.

The PMS has an internal calibration system that calibrates the
receiver over an 80-dB range in 1-dB increments. In the measurement
mode, the basic unit of measurement is a one-second power average
derived from a log-amplifier output sampled at a 128-Hz rate. The
data output of the set can be adjusted to provide one average value,
also derived as a power average, for a group of 1 to 128 of these one-
second averages, selectable in binary increments. The measurement
range of the set is —40 dBm to approximately —122 dBm.

The PMS uses a paper tape printer for hard copy output. The output
data include frequency, a sequential “major marker,” the power aver-
age, and the one-second sample variance for the group of one-second
samples contributing to each longer-term mean. In this application, a
mean value was generated each eight seconds. The “major marker” is
used to key measurements to the time of day and to logged operator
observations.

Prior to a series of measurements, the PMS is calibrated and the
calibration results are printed on paper tape. A received signal strength
meter in the PMS provides a visual indication of the receiver output
without operating the paper tape recorder. The measurement team
uses it to verify test set operation and to choose the group of control
channels for measurement. The set also contains start, stop, and pause
controls to limit data output to areas of interest.

To increase the confidence level of outside measurements, two
methods were used where possible to determine outside signal strength.
One method used the Mobile Telephone Laboratory (MTL) vehicle,
which has the capability of power averaging instantaneous signal
strength samples. This facility is described by Huff in Ref. 4. The
MTL receivers were sampled at a rate of 32 samples per second. The
short-term (1/2-second) MTL power averages were grouped in seg-
ments representing building faces and then further power averaged to
provide a single mean value for each face segment. The segment means
were then decibel averaged over all faces to develop the outside
reference.

The second method used the PMS at the time of the in-building
measurements to spot check MTL data as a guard against system
variations.

MTL measurements were not available for the buildings in the
Chicago downtown area, the AMA building, the IBT Headquarters
building at Randolph, and the Bell Training Center at West Adams.
The PMS was used to collect both inside and outside measurements.

The PMS was transported to the building locations in a step van.
While the PMS was in the van, the van rooftop antenna was connected
to it, and street-level data were recorded along the building perimeter.

The measurements collected along the entire perimeter were decibel
averaged to provide the outside reference.

The largest possible variety of areas was included in the routing
chosen for the PMS measurements in each particular building. In
measuring areas within a building the data were marked so that the
different interior types could be distinguished. The PMS has a provi-
sion for including a marker number in the data output. As the PMS
measurements were made along each route, an accompanying com-
mentary was made on a cassette recorder. This commentary provided
information about the type of building interior area being measured
and the associated major marker number. The major marker infor-
mation and the interior type were used subsequently in coding the
data for computer program input.

Also, this commentary was useful in flagging data that should be
discarded and in coordinating the data with the floor area to which it
applied.

As we noted earlier, the PMS has the capability to generate average
values that can be selected to represent from 1 to 128 seconds of real
time. The 8-second average was chosen for output data in these
measurements. That value represents a compromise between the vol-
ume of data points and the number of feet of travel per data point.
With the operational procedures used in these tests, the 8-second
averages result in one data point for about each 20 feet of travel. A
total data volume of approximately 4000 data points was accumulated
for the entire series of in-building measurements.

During the initial planning phase, the expected setup channel signal
strength in the vicinity of each building was predicted from propaga-
tion contours of the cell sites in the Chicago AMPS System (see Fig.
1). Before measuring each building floor, the operator of the PMS
noted the received signal strength indicator (RSSI) value at several
locations on the floor. If it was determined that a high percentage of
the area was served at signal strengths above the PMS receiver
threshold, the operator proceeded to collect data at each setup channel
frequency over the entire prescribed route for the selected building
floor.

In the measurements made in the first building in the measurements
program (Bell Laboratories, Naperville), a high degree of repeatability
was noted when the measurements from PMS output tapes were
compared for repeated runs. Based on these visual observations it was
decided that a single pass at each frequency of interest would provide
sufficient accuracy for measurements in the remaining areas.

A comparison of repeated measurements over a common route is
shown in Table II. The results indicate run averages are repeatable
within approximately 1 dB.

As the first step in data processing, all measurements are coded to
associate them with major marker numbers. The coding identifies the:

. Serving frequency
. Time of day
. Number of 1-second samples included in each average
. Type of interior area (six possible types)
. Flags indicating:
(a) repeated measurements
(b) data to be rejected because of operational trouble
(c) omission of N data points (beginning or end) for reasons of
ambiguity.
The time-of-day information permits verification of the major marker
sequence and data group duration.

Data become ambiguous when the transition from one major marker
to the next is accomplished without halting the measurements system.
The ambiguous data point (8-second sample) contains portions of data
from two major markers; it is deleted from the database.

1. Initial processing of the data (Pass I) develops statistics for all
data for each separate category (building, floor, area type, and channel
frequency). Invalid data points are eliminated. A summary listing is
provided that includes the number of entries, the number of entries
below PMS threshold, the average power level, and the standard
deviation for each group of data from categories of areas. In addition
to this summary, histograms for each group are developed.

2. The second pass introduces the outside street-level reference
information and combines it with the Pass I data. This produces a
display of the data for each frequency with its differential level relative
to the outside reference (penetration loss).

3. The third pass combines the penetration loss for all the meas-
urement frequencies on common floors, as generated in Pass II. This
allows penetration loss to be used as a base for further combining to
classify losses by area type, floor level, etc.

4. The fourth pass combines the penetration loss data from Pass
III across different buildings. In the combining process similar data
from different buildings are weighted to compensate for differences in
the data volume collected in each specific building. The weighted
average loss for a class of data from several buildings represents equal
contribution from each building. Pass IV generates information about
the effects of classes of building locations (suburban vs. urban) and
penetration loss by floor and area type, for all buildings.

The conclusions made as a result of our extensive measurements
are as follows:

1. Urban vs. suburban—First-floor penetration loss measurements
were processed for three different groupings of buildings. The urban
(downtown Chicago) building penetration for three buildings shows
an average value of 18.0 dB. The penetration loss distribution for
suburban buildings has an average value of 13.1 dB. These statistics
are listed in Table III. The comparison indicates a loss for the urban
group that is approximately 5 dB greater.

2. Penetration loss sensitivity to transmitter location—Different
penetration losses are measured on common floors of common build-
ings as a function of the specific serving-site identity. These differences
arise when a building is served by multiple cell sites. The distribution
of the “max-min” decibel values for such multisite penetration loss
differences is plotted in Fig. 2. The average max-min penetration loss
difference for all the data is 5.4 dB with a sigma of 4.6 dB. This

suggests that building penetration is a relatively strong function of
the direction of illumination.

3. Penetration loss sensitivity to interior type—Table IV compares
all interiors with and without windows for all buildings and all floors.
The presence of windows is shown to reduce the average penetration
loss by 6 dB, with a sigma of 5.2 dB. Not included in the data is the
office area of the Chrysler building in Elk Grove where the copper-
sputtered glass windows blocked out all measurable signal levels. For
the windowed data category, Table IV lists penetration loss for three
combinations of interior types. Open interior areas are shown to have
3 GB less penetration loss than hallways.

4, Floor height effect—A list of building penetration loss values
ranked by floor height is shown in Table II. The high values of loss
for the twelfth and fifteenth floors of a single building are the result
of the shadowing effect of adjacent buildings. While these buildings
effectively sandwiched the subject building, the adjacent street areas
were relatively open. This may be a relatively common occurrence in
the urban core and would appear to substantiate the observations of
Durante.’ Figure 3 is a scatter diagram with a straight line fit for
penetration loss vs. floor level. The dots indicate data points that
represent decibel averages of penetration losses. For each floor, there
is a data point for each channel that served that floor. Only floor
levels having three or more data points are included in the diagram
and the building described above was omitted. The mean values for
all data on each floor are indicated by “X’s.” The “least-squares”
straight line fit to these means is also shown. The slope of the line is
—1.9 dB per floor. The first floor intercept is 10.4 dB. While this loss
rate agrees closely with the findings in Refs. 1 and 2, the first-floor
penetration loss is about 10 dB lower. Subsequent measurements by
Bell Laboratories in the Newark, N.J., Cellular Test Bed have con-
firmed the ranges of loss values and loss rates presented in this paper.
It is assumed that differences in penetration loss with respect to values
reported previously are probably due to. differences in the methods of
establishing the outside street-level reference.

Table 1V—Penetration loss comparisons
Windowed/nonwindowed areas = 6.0 dB (Standard Deviation = 5.2 dB)
For windowed areas:

Enclosed/Hallways = 0 dB (Standard Deviation = 6.0 dB)
where open areas are in conjunction with enclosed areas
Open/Enclosed = 1.0 dB (Standard Deviation = 4.8 dB)
where open areas are in conjunction with hallways
Open/Hallways = 3.1 dB (Standard Deviation = 5.2 dB)

The test team members for operations and data measurement were
M. Patarca, L. F. Smith, and J. G. Rebele. Data keypunching and
coding from PMS output tapes and corresponding annotation was
performed by M. Patarca. The software and the processing of data to
produce the results of the measurements program was completed by
A. Some. K. K. Kelly collaborated in the selection and location of
candidate buildings. Messrs. J. T. Kennedy, G. A. Lothian and J. R.
Nevarez arranged basing the operations in Chicago and obtained the
permissions necessary to access the buildings for measurements.

1. J. M. Durante, “Building Penetration Loss at 900 MHz,” Conference Proceedings,
IEEE VTG Conf., 1973, pp. 1-7.

2. J. Dietz et. al., “Examination of the Feasibility of Conventional Land Mobile
Operations at 950 MHz,” FTC Office of the Chief Engineer, Research Division,

3. D. C. Cox, R. R. Murray, and A. W. Norris, “Measurements of 800-MHz Radio
Transmissions into Buildings With Metallic Walls,” B.S.T.J., this issue.
4. Special issue entitled “Advanced Mobile Phone Service,” B.S.T.J., 58, No. 1 (Jan-

1-11, transmitted to the FCC by Illinois Bell, June 8, 1977-November 26, 1979.
6. L. P. Rice, “Radio Transmission into Buildings at 35 and 150 mc,” B.S.T.J., 38, No.
1 (January 1959), pp. 197-210.

Masonry with some corrugated siding. The
interior has some windowed office space but
a large portion is devoted to cereal processing
equipment.

Low light transmission glass in metal frame-
work. Mostly windowed halls with interior
office space. _

Open area without nearby structures. Lo-
cated at the bottom of a hill to the East.
Brick and glass with high-rise dormitory area.
Windowed office space.

5th floor—Cloverdale, Eola, Lemont, Lyons
9th floor—Cloverdale, Eola, Lemont, Lyons

Stand-alone with other multistory buildings
spaced several hundred yards distant.

1st floor—Cloverdale, Eola, Lemont, Lyons
2nd floor—Cloverdale, Eola, Lemont, Lyons.

Stand-alone with other multistory buildings
spaced several hundred yards distant.
Narrow windows with concrete in a ratio of
approximately 50 percent. Interior office
space uses half-height dividers. First floor is
a lobby area.

Concrete slabs, only one floor. The interior
is a large warehouse with metal storage bins
for automobile parts. An attached office
building had no measurable signal because
copper-sputtered windows constituted 100
percent of outside surface.

Corner buildings in area with other multi-
story buildings of about same height.

Corner building without nearby structures.
Brick and smaller glass windows typical of
factory construction. First floor is open area
with heavy machinery; second floor is open
area with assembly lines; third floor is office
space with windowed space.

Corner building in area with other multistory
buildings of similar height.

Reflective glass windows and concrete. The
interior office area is open with three-quarter
partitions.

Bounded on four sides by city streets. Neigh-
boring buildings are multistory.

Concrete and glass with lobby on first floor
and windowed work and office spaces on up-
per floors. Office partitions are three-quarter-
high dividers.

Neighboring multistory buildings in a busi-
ness district. The building front is open to
the street and both sides abut neighboring
buildings. The rear of the building opens into
a service courtyard.

Cell site signals: 1st floor—Canal
2nd floor—Canal
12th floor—Canal
15th floor—Canal.

The outside reference measurements for this building were incom-
plete because a complete perimeter path was not available. The
measurements for this building are not combined with the results
from other buildings.

The penetration loss for this building based on the available
street-level reference measurements is 7.3 dB with a sigma of 6.7

Edward H. Walker, B.S.E.E., 1956, Newark College of Engineering;
Western Electric Company, 1940-1957; Bell Laboratories, 1958-1981.
Mr. Walker has been involved in the design of radio circuits and the
development of military radar systems. Since 1973 he has been respon-
sible for the experimental evaluation of audio quality of the AMPS
System and of mobile and portable terminals. His work on building
penetration was completed prior to his retirement from Bell Laboratories
in 1981. Member, IEEE.

Transmission Errors and Forward Error
Correction in Embedded Differential
Pulse Code Modulation

We have derived formulas for the combined effects of quantization and
transmission errors on the performance of embedded Differential Pulse Code
Modulation (DPCM), a source code that can be used for variable-bit-rate
speech transmission. Our analysis is more general and more precise than
previous work on transmission errors in digital communication of analog
signals. Special cases include conventional DPCM and Pulse Code Modulation
(PCM). Our main result is a signal-to-noise ratio formula in which the effects
of source characteristics (input signal, codec design parameters) and the effects
of transmission characteristics (modulation, channel, forward error correction)
are clearly distinguishable. We also present, in computationally convenient
forms, specialized formulas that apply to uncoded transmission through a
random-error channel, transmission through a slowly fading channel, and
transmission with part or all of the DPCM signal protected by an error-
correcting code. Numerical results show how channel coding can have different
effects on conventional and embedded DPCM. They also show how the binary-
number representation of quantizer outputs influences performance.

Embedded coding can play a valuable role in variable-bit-rate speech
transmission. With an embedded code the analog-to-digital (a/d) and

digital-to-analog (d/a) converters operate at a constant, high bit rate,
and the transmission system controls the instantaneous rate. Proposed
applications for variable-bit-rate operation include a digital private
branch exchange,’ digital speech interpolation,” packet-switched voice
transmission,® and mobile radio.*

Sophisticated versions of Differential Pulse Code Modulation
(DPCM) are promising speech codes for these and other environ-
ments.°’ However, conventional DPCM is not suited to variable-bit-
rate transmission because the decoder amplifies the effects of bit-rate
adjustments. On the other hand, a slightly modified form of DPCM
avoids this problem and produces an embedded code.®

Figure 1 shows the codec (coder, decoder) structure of embedded
DPCM. Although up to E bits/sample can be transmitted, the signals
presented to the two integrators have a resolution of only M bits/
sample, the minimum bit rate of the channel. While Fig. 1 is a useful
guide to practical implementations, Fig. 2, which is equivalent, is
easier to analyze. It shows the quantizer at the encoder as a successive-
approximation combination of two quantizers: a “minimal” quantizer
with M bits/sample and a “supplemental” quantizer with E-M bits/
sample, operating on the error signal of the minimal quantizer.*

In embedded DPCM, all of the bits from the minimal quantizer
arrive at the decoder; the transmission system can delete some or all
of the supplemental bits. With S bits/sample of the supplemental
quantizer transmitted to the decoder, the rate is D = M + S bits/
sample, and the quantizing distortion is very close to that of a conven-
tional codec with D bits/sample.

Errors in the two bit streams have different effects on the decoder
output. Errors in the M, minimal bits, are enhanced by the decoder
integrator, which has no effect on errors in the S, supplemental bits.
This situation compares favorably with conventional DPCM, where
all errors are integrated at the decoder. It also has implications for
forward error correction in embedded DPCM. Figures 1 and 2 will be
further explained in Section II.

Our principal contribution in this paper is an analysis of the com-
bined effects of granular quantizing distortion and transmission errors
on the mean-square error of embedded DPCM. The analysis
is quite general: special cases include Pulse Code Modulation (PCM)
(M = 0) and conventional DPCM (S = 0). The formulas for the noisy-

Fig. 1—Embedded DPCM encoder and decoder. Because both predictors operate on
the same signal (with resolution M bits/sample) performance is unaffected by errors
due to bit-rate adjustment. The channel can transmit between M bits/sample and E
bits/sample.

is DELETE D/A SUPPLEMENTAL
E-M BITS = E-M BITS r
BITS PCM CODEC
TRANSMISSION
D=S+M
BITS
PREDICTOR

A/D — ANALOG-DIGITAL
D/A — DIGITAL-ANALOG
DPCM CODEC — DIFFERENTIAL PULSE CODE MODULATION CODER-DECODER
PCM CODEC — PULSE-CODE MODULATION CODER-DECODER

Fig. 2—Embedded DPCM encoder and decoder. E-bit analog-to-digital conversion is
shown as a two-stage, successive-approximation process. All M minimal bits are trans-
mitted. S, the number of supplemental bits transmitted, can range from 0 to E-M.

channel performance of conventional DPCM are new, and the spe-
cialization to PCM is more precise than previous work on the sub-
ject,!°-!? which includes several approximations that are accurate for
multibit (= 6-bit) quantizers, but are rather imprecise at lower bit
rates. The method of analysis has the advantage of separating source
effects from transmission effects. The source effects include the char-
acteristics of the analog input signal and the codec design parameters.
The transmission effects include modulation and demodulation, the
channel, forward error correction, and diversity reception.

The main result is eq. (61), in which the transmission effects are
contained in the discrete probability function P(l), where / is an index
of binary error patterns. The other symbols in (61) are source param-
eters and functions of source parameters. After deriving (61) we apply
it to specific transmission environments and present, in Table VI,
specialized formulas that are convenient for numerical computation.

In all, there are 78 formulas in Sections III through VI, most of
them intermediate steps in derivations of a few key results. Anticipat-
ing that few readers will require all these details we provide here a
summary of the analysis and we display a few numerical results.
Sections I, II, and VII contain the main ideas of our work and sufficient
information to allow readers to perform, on hand calculators, compu-
tations similar to the ones we present.

Section III introduces the notations for the signals and errors in the
M-bit minimal DPCM codec and the S-bit supplementary PCM codec
of Fig. 2. The analysis of Section III leads to (2), which expresses the
sampled-data error sequence as a function of quantization errors,
transmission errors, and integrator characteristics. Section IV begins
the analysis of the mean-square value of (2) by deriving (35), the ratio
of the mean-square codec input to the mean-square value of the
encoder difference signal. Section V defines A factors, which are
conditional mean squares of the errors due to specific binary error
patterns, and derives (61), the general signal-to-noise ratio (s/n)
formula. Section VI adapts (61) to specific transmission models and
provides guides to numerical computation.

Figure 3 shows the performance of embedded DPCM in four trans-
mission environments, all of them employing Coherent Phase Shift
Keying (CPSK) modulation at 32 kb/s in a white-Gaussian-noise
channel. The encoder operates at 32 kb/s (8-kHz sampling, 4 bits/
sample), and in format 1 all of this information is transmitted. Figure
3 indicates that when the channel s/n falls below 10 dB, the audio
s/n deteriorates rapidly. In format:2, the least significant bit of each
DPCM code word is deleted, and the remaining 3 bits/sample are

protected by a rate 3/4 convolutional code. Although there is more
quantizing noise than in format 1 (the s/n is 6 dB lower in the absence
of transmission errors), the convolutional code permits accurate re-
ception of the transmitted bit stream at channel s/n’s down to 3 dB.
Going one step further with this approach to channel coding, we have
format 4, in which 16 kb/s of speech data are transmitted under the
protection of a rate 1/2 code. The threshold of essentially error-free
performance is now extended down to a channel s/n of about 0 dB.

In code format 3 the speech transmission rate is 24 kb/s, as in
format 2, but now only 2 of the 3 bits/sample are protected by the
convolutional code, which has rate 2/3. The threshold of curve 3 in
Fig. 3 is about 1 dB lower than that of curve 2. On the other hand,
format 3 is slightly worse than format 2 in intermediate channel
conditions (s/n’s between 3 and 5 dB). Over this range, format 2 is
essentially error free, while format 3 is affected by errors in the
unprotected third bit of each code word. The effect is small, however,
because these errors are not amplified at the decoder.

With conventional, rather than embedded, DPCM, the correspond-
ing picture, Fig. 4, is rather different, especially with respect to format
3. Here channel errors in the unprotected third bit are amplified by
the integrator at the decoder. The result is a noticeably lower output
s/n relative to format 2 (all three bits protected) when the channel
s/n is between 3 and 6 dB. On the other hand, in clear channels the
greater accuracy of prediction in the conventional encoder causes the
output s/n of conventional DPCM at 24 kb/s (formats 2 and 3) and

Figure 5, which applies to 24 kb/s speech transmission with a rate
2/3 code, summarizes the performance differences between conven-
tional and embedded DPCM. Conventional DPCM has somewhat
lower quantizing noise, which is reflected in the higher s/n in good
channels. In intermediate conditions, when errors in the unprotected

Fig. 4—Performance of conventional DPCM in four transmission environments.

Fig.5—A 24-kb/s speech transmission with a rate of 2/3 code (Format 3) used to
summarize performance differences between embedded and conventional DPCM.

bit influence performance, embedded DPCM is better because the
effects of these errors are not amplified at the decoder. In very difficult
channels, errors in the two coded bits dominate performance and the
s/n’s of conventional and embedded DPCM are virtually equal.

Figure 1 shows the signal processing operations that take place in
embedded DPCM encoding, transmission, and decoding. While the
analog-to-digital converter at the encoder generates E bits/sample,
the resolution of the signal presented to the integrator is limited to M
bits/sample, where M is the minimum bit rate of the transmission
system. The transmitted bit rate, D, can vary between M and E. At
the receiver E—D filler bits are appended to the incoming signal. As in
the encoder, E-M bits are deleted at the integrator input so that in
the absence of transmission errors the encoder integrator and the
decoder integrator produce the same approximation signal. When this
signal is added to the full-resolution (D bits) quantizing error, the sum
has nearly the quality of a conventional DPCM signal with D bits/
sample.

While Fig. 1 demonstrates practical implementations, Fig. 2, which
is equivalent, is easier to analyze. It represents the analog-to-digital
conversion as a two-step, successive-approximation process. First the
input to the converter is represented by M bits/sample. Then the error
of this representation is processed by another analog-to-digital con-
verter with E-M bits/sample. Taken together, the two digital signals
comprise an E bits/sample representation of the DPCM difference
signal. All of the M bits of the minimal analog-to-digital converter are
transmitted. The other E-M bits are subject to deletion by the trans-
mission system. At the receiver the minimal, M-bit signal is processed
by a conventional DPCM decoder. The result is added to the supple-
mental, S = D — M bit representation of the DPCM error signal to
produce the system output.

To analyze Fig. 2, we introduce Fig. 6, which shows the signals that
appear in the analysis and defines their notations. We are interested
in the overall error signal

the difference between decoder output and encoder input. In particular
we will derive the formula

estk) = q'slk) - as{k) A/D — ANALOG-DIGITAL
ep lk) = q'p(k) - ap (k) D/A — DIGITAL-ANALOG Vk = aq (k-1) + acgag(k-2) +» + + + aay (k-K)

Fig. 6—The signal-processing operations of Fig. 2 and the notation used to analyze
them. Three new digital-to-analog converters define signals that appear in the analysis.

where p(k) is the quantization noise of the two-stage, D-bit analog-
to-digital conversion, ep(k) is the effect of a transmission error on the
entire D-bit transmitted code word, and ey(k) is the effect of a
transmission error on the minimal, M-bit DPCM code word. The
coefficients b; are related to the predictor coefficients ai, a2, --- ax
according to (12).
Formally,

eu(k) = qu(k) — gm (k), (3)
the difference between the quantized inputs to the decoder and encoder
integrators. To define np(k) and ep(k), we view the combined code
word with M + S = D bits as a digital representation of &(k) = x(k) —
y(k). A D-bit digital-to-analog converter would produce the quantized
signal gp(k), and so we have the definition of quantization error:

At the receiver, where the D bits/sample are possibly corrupted by
transmission errors, a digital-to-analog converter would produce qp(k).
The transmision error is

ep(k) = qn(k) — gn(k). (5)
In the remainder of Section III we derive eq. (2); in Section IV and V
we analyze its mean square.
3.2 Derivation of the error sequence

Here the signal analysis is facilitated by the transform notation of
Table I. The reader may verify that the output of the minimal encoder,
Qwu(z), is related to input and quantizing noise® by

Qmu(z) = [X(z) + Na(z)][1 — F(2)]. (6)
Table |—Transform notation for codec signals
Encoder Signal Description Decoder Signal
X(z) Encoder input, decoder output X’(z)
Minimal decoder output ‘u(z)
Y(z) Approximation signal Y’(z)
Qu(z) M-bit representation of X(z) — Y(z) Q‘u(z)
Na(z) Qu(z) — [X(z) — Y(z)]
Quantizing error in Qy(z)
Q’ulz) — Qu(z) En(z)
Transmission error in Q‘y4(z)
Qs(z) S-bit representation of —Ny(z) Q's(z)
No(z) Qs(z) + Nu(z)
Quantizing error in Qs(z) Es(2)
Q(z) — Qs(z) ny
Transmission error in Q(z)
K
F(z) Predictor }az™ F(z)
i=1

» 7... Q(z)
5 A eae (7)
which leads to
XG OR aN ee, (8)
1 — F(z)
In the supplemental encoder,
Qs(z) = —Na(z) + No(z), (9)
and at the decoder
Qs(z) = —Nu(z) + Np(z) + Es(z). (10)
Combining (8) and (10) we have the output of the entire decoder,
E
X'(z) = Xu(z) + Qs(z) = X(z) + No(z) + Es(z) + 7 are (11)
To transform (11) to time-domain notation, we defined 6,, i = 0, 1,
2, ---, to be the inverse z transform of 1/(1 — F), such that
1 ae
fre iy)
Then we have
x’(k) = x(k) + np(k) + es(k) + Y bieu(k — i), (13)
i=0

To analyze the mean value of (16), we assume that the sequence {x(k)}
is drawn from a stationary ergodic random process. In our derivations
we ignore all correlations in (16) between nonsimultaneous samples.
That is, we assume
E{[no(k) + en(Blem(k- i} x0; i= (17)
and
Elem(k — it)em(k-—j)}= 0; 1 Fj. (18)
Equation (17) indicates that the overall error (quantizing plus channel
distortion) in the kth sample is uncorrelated with errors in other
samples of the minimal M-bit quantized samples. Equation (18) states
that errors in different minimal samples are uncorrelated. These
approximations are accurate because the sequence of samples at the
input to a DPCM quantizer is decorrelated by the differential coding
process and because transmission errors affecting different code words
are independent or only weakly correlated.
The approximations, (17) and (18), remove the last two sums from
the expected value of (16), leaving

The expectations in (19) are related to the quantization and trans-
mission of &(k), the DPCM difference signal. In Section V, we present
a complete theory of the errors due to these operations. While this
theory relates these errors to a = E{£*(k)}, we are ultimately interested

in the s/n of the codec input, x(k):
s/n = E{x*(k)}/E{e*(k)} = o3/oe. (21)
To find this quantity, we will now derive o2/o; and then combine it
with the results of Section V. To begin the derivation, we refer to Fig.

The first term in (24) depends on the spectral properties of x(k) and
on the predictor. The ratio of o2 to this quantity is called the prediction
gain,

K K
G = ¥; > ajajr;-;, (26)
j=0 i=0
in which ag = 1, aj = —a;,, i = 1, 2, --- , K, and r, is a normalized
covariance coefficient of the stationary input,
rn = E{x(k)x(k + n)}/o%. (27)

In evaluating the second and third terms of (24), we ignore corre-
lation between different quantizing-noise samples and correlations
between quantizing-noise samples and samples of the codec input.
Thus we use (18) and the approximation

The noise component of (29), which depends on the quantizer
overload point and on the statistical properties of &(k), is analyzed in
Section V, where we restrict our attention to granular quantizing noise
and derive o2(B), the noise power of a B-bit quantizer with unity
overload point. If the actual overload point is Emax, the noise power of
the M-bit minimal quantizer is

and the approximation would be exact if ny,(k) were uniformly distrib-
uted over the range —Ay,/2 to Ay/2. Table II presents o2(B) numeri-
cally and indicates the fractional error due to the above approximation.
A parameter in Table II is the dimensionless load factor

L = Emax/o¢. (33)
Combining (29), (31), and (33), we arrive at
ot = G'o2 + apL?o2(M)o:, (34)
or the quantity we set out to derive:
o;/o% = G[1 — apL?o4(M))]. (35)

V. QUANTIZATION NOISE AND TRANSMISSION NOISE IN PCM
5.1 Granular and overload conditions

To analyze (19), we study, statistically, the quantization and trans-
mission of the DPCM difference signal £(k). In this type of study it is
customary to separate the quantizing error into two components:
overload distortion and granular noise. In speech communication this
distinction is valuable for predicting subjective quality.’*"* Moreover,
in analyzing DPCM the distinction is essential because, except for a
codec with an ideal integrator,” (F(z) = z~', which is pathologically
vulnerable to transmission errors), there is no theory for computing
the mean-square slope-overload distortion. Thus our analysis sepa-
rates the transmission of clipped samples of &(k) from samples subject
to granular distortion. Our theory pertains only to the transmission
of unclipped samples. For those samples we add two different distor-
tions, quantizing noise and noise due to transmission errors. Unlike
slope overload, both of these impairments are essentially uncorrelated
with the signal. This gives us confidence that the mean-square sum is
a reasonable quality measure.

where é(k) is the quantizer input and &,,,. is the overload point of the
uniform DPCM quantizer. The probability of overload is poy and pz, =
1 — Poy is the probability of granular quantization. By definition, the

quantizer is overloaded at time k if the quantization error exceeds half
of a quantization step, i.e., if

Our remaining analysis will be confined to the second expectation
on the right side of (36) and in particular to the ratio,

To be concise in the remainder of this paper, we will omit the granular
condition, | &(k)| < &max, from our notation of expected values.

To facilitate numerical evaluation of s/n’s, we will present three
tables of normalized error terms. The normalization relates these
errors to a quantizer with a unity overload point and an input with
probability density function p,(-). If the quantizer of interest has an
overload point of &max and the input has the probability density p;(-),
the relevant errors are table entries scaled by £ax. The two probability
densities are related by

To confine our attention to the granular quantization condition, we
perform our averages with respect to the conditional probability den-
sity

The model is illustrated in Fig. 7. The signal u = £/émax is processed
by a B-bit analog-to-digital converter with overload point 1 and step
size

The digital output of the a/d is i, and the corresponding quantized
signal is u;, which is related to u by the graph in Fig. 7 and by

Fig. 7—Model for analyzing quantizing noise and the effects of transmission errors.
The graph and table define the quantizer and two binary number representations.

with the transformation of i to 1’ characterized by a binary-error
pattern with index 1, | being an integer in the range 0, 2? — 1. To relate
the error effect to l, we refer to the natural-binary representation of |
and specify that a 1 in the bth least-significant position of | causes an
inversion of the bth most-significant* bit in the binary representation
of 1. Thus | = 0 refers to error-free transmission (i = 1’); | = 1 refers
to an error only in the most significant bit; | = 5 refers to errors in
the first and third most significant bits; etc.

With u the quantizer input and / the binary error pattern, we denote
the received sample in Fig. 7, uj. It is helpful to separate the complete
error u, — u into quantization-noise and transmission-noise compo-
nents as follows,

5.3 Conditional expectations of transmission-error effects
5.3.1 The general approach

Our goal is to evaluate the mean-square of (44) over the joint
distribution of input statistics and binary-error patterns. The key to
our analysis is the definition of A factors, which are conditional mean-
square errors, each related to a specific binary-error pattern, |. By
analyzing these conditional errors, we separate the effects of source
characteristics from the effects of transmission characteristics. The
source effects are embodied in the A factors; the transmission effects
are embodied in probabilities of error patterns. These probabilities
govern the weighted addition of the A factors to produce the final
result.

This approach to analyzing transmission impairments was intro-
duced by Rydbeck and Sundberg,’ ” who were mainly concerned with
quantizers with 6 to 8 bits/sample. This high resolution admitted
various approximations that are inaccurate in the 2- to 4-bit quantizers
of greatest interest for embedded DPCM transmission. Thus we pro-
ceed to a precise calculation of two types of A factors: conditional
expectations related to the isolated effects of digital transmission
errors and conditional expectations that include correlations between
transmission errors and quantizing noise. In high-resolution quantiz-
ers this correlation is negligible, and the two types of A factors are
essentially equal.

To compute the mean-square value of (44), conditioned on error
pattern J, we will identify three important quantities: o2(B), the

* This reversal of the bit ordering of / relative to the binary representation of i will
facilitate bookkeeping in subsequent computations.

granular-noise power of a B-bit quantizer; A;(B), the mean-square
effect of error pattern / on the quantized signal u; and A,(B), the
overall effect of error pattern / on the mean-square error of the analog
output Uj.

To derive computationally convenient expressions for o2(B), A,(B),
and A,(B), we defined the integrals

in which p; is the lower boundary and »;4; is the upper boundary of
quantizing interval 1:

The first integral (45) is the probability of using interval 1. The second
integral (46) is the average quantization error in interval 1. If B is
large, Ag is small, and q; ~ 0 because u; is in the center of the
quantization interval. The third integral (47) is the mean-square signal
when the quantizer is in the granular condition.

Now we write the definitions followed by computational formulas
for the quantizing noise and the effects of error pattern /:

A\(B), the difference between the total noise and the quantizing noise,
includes the correlation between quantization effects and transmis-
sion-error effects. In multibit quantizers (B > 4) this correlation is
small, and A,(B) ~ A,(B), an assumption inherent in previous work on
PCM. Because low-resolution quantizers are of interest in DPCM, we
take account of this correlation in our present work.

Table III displays formulas for p;, qi, and o2, that apply to inputs
with Gaussian and exponential probability density functions. Because
the input, u, of the normalized quantizer is related to the input, &, of
the quantizer with overload point £ max by u = &/Emax, We have

where L is the dimensionless load factor defined in (33). Because L is
a familiar quantizer design quantity, we have written the formulas in
Table III as functions of L.

With the formulas in Table III, it is a simple matter to compute
o2(B) precisely. However, for B = 4 the approximation

is very accurate (within 3 percent of the exact value for L < 5.6). For
B = 2 and 3, Table II shows the exact values of o2(B) and the
approximation errors

for L = 1.78, 3.16, 5.62 (L? = 10+ 5 dB).
To compute A,(B), A(B), it is necessary to know uj, — u;, which
depends on the binary number representation of u;.

We consider two representations: natural-binary and sign-magni-
tude, both defined in Fig. 7. Although in general the A,(B) and A,(B)
depend on p; and q;, there are some special cases that are important
and illuminating. For example, in the natural-binary code, the single-
error A factors are independent of the signal statistics and of the
quantizer. An error in the most significant bit causes uj, — u; = +1
provided it is the only error in the B-bit code word. Thus Ai(B) = 1.
Likewise, any isolated error in the second most significant bit causes
an output error of +1/2, and in general a single binary error in position
b causes the mean-square error

AB) = (1/4)°7; =o", (56)
In the sign-magnitude code, isolated binary errors in positions b =
2, 3, ---, B have the same effects as corresponding errors in the

natural-binary code. However, an isolated error in the most significant
position transforms u; to u,, = —u;. The mean-square effect is

Table I!I—Formulas for computing A factors and quantizing noise B-bit transmission

Referring to (52) and Fig. 7, we have the mean-square difference
between £’ and &, over all possible error patterns

where P(l) is the probability of error pattern |. The effect of the
transmission errors on the quantized version of & is Exax vei
P(1)A,(B). Returning to (19), we have two expectations: the first is the
combined (quantizing and transmission) noise of a D-bit signal; the
second expectation is the noise due to transmission errors in the M-
bit minimal signal. Thus, we can write (19) as

where £2... = L’o? is related to o2 by
2 2 Vo
™™ G[l — apL?o2(M)]
The s/n, which is the principal subject of this paper, is, therefore,

With the exception of the two summations in the denominator, all of
the quantities in (61) are properties of the input signal and the codec
design parameters. These summations,
2P_-1 ‘ QM_1
oo = Py P(I)A(D) + bp Py P(I)A(M) (62)
comprise the effects of transmission errors on the performance of
embedded DPCM. We analyze them in Section VI.

The 2” probabilities, P(l), of binary-error patterns are properties of
the digital transmission system, which includes a modulator, a channel,
a demodulator, possibly a codec for forward error correction, and
possibly a means for combining different versions (diversity branches)
of the received signal. Depending on these components, the P(J)
exhibit properties that facilitate evaluation of the sums in (61). In the
following subsections we consider three paradigms: (1) random errors
with statistically independent transmission of all bits; (2) slow fading

with the bit-error probability constant over each code word, but
independent from word to word; and (3) channel coding that makes
all error patterns equally likely.

Errors in all bits are statistically independent of each other and
occur with probability, P. The probability of error pattern | depends
only on w, the Hamming weight of J, i.e., the number of ones in the
B-bit binary representation of /. Thus,

This expansion leads us to express the summations in (61) as poly-
nomials in P. The coefficients of the polynomial involve the sums of
all A factors with a fixed weight, w. Let us denote these sums S,,(B)
where, for example,

where |, is the set of all error patterns with Hamming weight, w.
Combining (63) and (65), we can write

discovered and proved that for any input probability distribution,
T,,(B) = T.,(B) = 0 for w = 3. Thus the tranmission term in (61) is

This formula is valid for channels with random binary errors, and
Table IV presents values of T,(B), T:(B), T,(B), and T,(B) for three
quantizer load factors, B = 2-6 bits and Gaussian and exponential
inputs.

Now the binary-error probability is a random variable that is con-
stant over each code word but varies from word to word. In this case
the effects of digital errors can be calculated as in (69) but with the
average values P” replacing P’. These averages are computed over the
distributions of channel s/n’s that govern the random fluctuation of
P from one code word to the next.

To analyze the performance of embedded DPCM protected by an
error-correcting channel code, we make three simplifying approxima-
tions. The first one, which pertains to the error-correcting code, states
that when there is a decoding error, all error patterns are equally
likely. Thus we assume that if the C most significant DPCM bits are
protected by the code,

1
where Py is the word-error probability and P, is the binary-error
probability of the channel code. They are related by

The other two approximations apply when C < D, so that the C
most significant DPCM bits are protected and the other D-C bits are
uncoded. To simplify computations for this case, we (1) ignore simul-
taneous errors in the protected and unprotected parts of the D-bit
word and (2) ignore multiple errors in the unprotected part. We
consider separately three different relationships among C, the number
of coded bits; D, the length of the entire DPCM code word; and M,
the number of bits in the minimal quantizer.

In this case P(/) may be calculated according to (70) for all D-bit
error patterns, / = 1, 2, ---, 2?"'. This value of P(l) is constant
throughout the first sum in (62). In the second sum we have the
probability of M-bit error patterns. For each M-bit error pattern there
are 2?™ J)-bit patterns. Hence P(/) in the second sum of (62) is higher
by the factor 2?” than P(l) in the first sum. Because each sum in
(61) is a constant probability times a sum of A factors, we write

Table V contains numerical values of Agum and Asum for the sets of
conditions of interest to us here.

6.3.2 Entire minimal code word, parts of the supplemental code word
protected (M < C < D)

In this event we assume that all of the unprotected bits have the
binary-error probability P and that as before the protected bits have
binary-error probability P.. Furthermore, we set to 0 the probability
of simultaneous errors in the protected and unprotected parts of the
code word. (These errors occur with probability related to P.P.) We
also set to 0 the probability of multiple errors in the unprotected part
of the code word (which occur with probability less than P?). Thus we
break the first sum in (61) into two parts. The first part accounts for

errors in the first C (protected) bits when the other D-C bits are error
free, 1 = 1, 2, ---, 2° — 1. The second part accounts for single errors
in the remaining D-C bits when the first C bits are error free, | =
2°, 20+! ..., 22-1. The result is

for the probability of a single error in the unprotected part of the code
word. A further approximation A»i(D) = Asi(D) simplifies computation
of the second term of (74) because, for natural-binary and sign-
magnitude representations, (56) applies for b > 1. This allows us to
derive

Just as we decomposed the first sum in (61) into two parts in the
previous case, we similarly decompose the second sum when some of
the M minimal bits are unprotected. The result is

The useful computational formulas (61), (62), (69), (72), (77), and
(78) are summarized in Table VI. In this section we apply these
formulas to illustrate some of the properties of embedded DPCM and
its relationship to conventional DPCM.

All of our numerical results pertain to a Gauss-Markov input signal
with adjacent-sample correlation r, = 0.85. The codec uses single
integration with coefficient a, = 0.85 and the load factor, L = V10.
For this configuration the coding gain is G = 3.6. If the embedded
codec has a minimal quantizer with M = 2 bits, Cgource in Table VI is
0.31. For conventional DPCM Cyource = 0.35 with 3 bits/sample and

Csaite G1 — apL?a2(M)])
s/n — o2(D) rae 2° Cia tas L?
es Notation
General Formula for Single
Symbol Description Formula Integration
G Coding gain (26) (1 — 2ayr; + az)!
ap Predictor gain (30) a?
L Load factor (33)
M Minimal codec bits
D Transmitted bits
bp 1 + bp is integrator gain (20), (21) a?/(1 — a?)
rn (B) Boos autocorrelation 5}
oO uantizing noise 49 -2B
ot Transmission-error effects (62) arifa oe Table: Tl
(b) Error Formulas*
Transmission Format Error Effect o%

0.36 with 4 bits/sample. Thus the quantizing-noise penalty of the
embedded codec is 0.54 dB when 3 bits are transmitted, and 0.68 dB
when 4 bits are transmitted. As indicated in Ref. 8 these penalties
increase for higher values of L and a,. They decrease rapidly as M
increases.

In our numerical examples the modulation is coherent phase shift
keying (CPSK) so that in a white-Gaussian-noise channel the binary
error probability is

codes. For all of them, we use the following truncated union bound to
calculate binary-error an
d+4

where m = 1, 2, 3 for the rate 1/2, 2/3, and 3/4 codes, respectively,
and the coefficients w; and the free distance, d, characterize the
convolutional coder and decoder. The codes considered here are punc-
tured codes!* with constraint length 5 (16 states in the decoder mem-
ory). Table VII contains their coefficients and free distances. The
combination of (79) and (81) with the formulas i in Table VI produces
the curves in Figs. 3, 4, 5, and 8.

For transmission environments other than CPSK in a white-Gaus-
sian-noise channel, there are formulas for P and P, to be used in place
of (79) and (81). There are many families of modulation schemes,
channel conditions, error-correcting codes, and reception techniques
that are of practical interest. This paper provides the tools for studying
their effects on the performance of embedded and conventional
DPCM. This is a subject worthy of further investigation.

Without forward-error correction, the noise due to transmission
errors is dominated by the effects of single errors in the most signifi-
cant part of the transmitted code word. With the natural-binary
representation, an error in the most significant bit always causes a
noise impulse of half the peak-to-peak range of the quantizer (56).
With the sign-magnitude representation, an error in the sign bit
inverts the polarity of the quantized signal, thereby producing a noise
impulse of approximately twice the magnitude of the quantizer input

Table ViI—Source and channel code formats,
convolutional code properties

Format 1 Format 2 Format 3 Format 4
Source code
bits/sample 4 3 3 2
bits/second 32K 24K 24K 16K
bits/sample
protected 0 3 2 2
Channel code
rate No code 3/4 2/3 1/2
Free distance, d 4 5 7
Weight wz 22 25 4
Wa+1 Error 0 112 12
Wa+2 Properties 1687 357 20
Wa43 0 1858 72
Wat4 66964 8406 225

Fig. 8—Performance of embedded DPCM with sign-magnitude and natural-binary
representations of quantizer outputs.

(57). Consequently, quantizers employing the sign-magnitude repre-
sentation are somewhat less affected by transmission errors than
quantizers with the natural-binary representation when the input
probability distribution has its mode at zero. This is illustrated in Fig.
8, which pertains to uncoded 32 kb/s embedded DPCM transmission.
When transmission errors are the dominant distortion, signals repre-
sented in the natural-binary format are about 2 dB noisier than signals
represented by the sign-magnitude format.

With forward error correction, all error patterns are equally likely,
and the two representations have essentially the same s/n.

1. F.S. Boxall, “A Digital Carrier-Concentrator System with Elastic Traffic Capacity,”
IEEE Trans. Commun., COM-22, No. 10 (October 1974), pp. 1636-42.

2. J. A. Sciulli and S. J. Campanella, “A Speech Predictive Encoding Communication
System for Multichannel Telephony,” IEEE Trans. Commun., COM-21 (July
1973), pp. 827-35.

3. T. Bially, B. Gold, and S. Seneff, “A Technique for Adaptive Voice Flow Control in
Integrated Packet Networks,” IEEE Trans. Commun., COM-28, No. 3 (March
1980), pp. 325-33.

4. D. J. Goodman and C.-E. Sundberg, “Combined Source and Channel Coding for
Variable-Bit-Rate Speech Transmission,” B.S.T.J., 62, No. 7 (September 1983).

5. Proceedings of the 1982 IEEE International Conference on Acoustics Speech and

Signal Processing, Paris, May 1982. Session S8, pp. 954-83, contains four papers
on adaptive DPCM at 32 kb/s.

. J.-M. Raulin, et al., “A 60 Channel PCM-ADPCM Converter,” IEEE Trans.
Commun., COM-30, No. 4 (April 1982), pp. 567-73.

. D. W. Petr, “32 Kbps ADPCM-DLQ Coding for Network Applications,” Record of
the IREE Global Telecommunications Conference, Miami, Florida, November
1982, pp. 239-43.

. D. J. Goodman, “Embedded DPCM for Variable Bit Rate Transmission,” IEEE
Trans. Commun., COM-28, No. 7 (July 1980), pp. 1040-6.

. N.S. Jayant, “Variable Rate ADPCM Based on Explicit Noise Coding,” B.S.T.J.,
62, No. 3 (March 1983), pp. 657-77.

11. C.-E. Sundberg and N. Rydbeck, “Pulse Code Modulation with Error-Correcting
Codes for TDMA Satellite Communication Systems,” Ericsson Technics, 32, No.
1 (1976), pp. 1-56.

12. N. Rydbeck and C.-E. Sundberg, “PCM/TDMA Satellite Communication Systems
with Error-Correcting and Error-Detecting Codes,” Ericsson Technics, 32, No. 3
(1976), pp. 195-247.

13. D. J. Goodman, B. J. McDermott, and L. H. Nakatani, “Subjective Evaluation of
PCM Coded Speech,” B.S.T.J., 55, No. 8 (October 1976), pp. 1087-109.

14, B. McDermott, C. Scagliola, and D. J. Goodman, “Perceptual and Objective Eval-
uation of Speech Processed by Adaptive Differential PCM,” B.S.T.J., 57, No. 5
(May-June 1978), pp. 1597-618.

15. L. J. Greenstein, “Slope Overload Noise in Linear Delta Modulators with Gaussian
Inputs,” B.S.T.J., 52, No. 3 (March 1978), pp. 387-421.

16. G. C. Clark and J. B. Cain, Error-Correction Coding for Digital Communications,
New York: Plenum Press, 1981, pp. 237-8, and p. 403.

David J. Goodman, B.E.E., 1960, Rensselaer Polytechnic Institute; M.E.E.,
1962, New York University; Ph.D. (Electrical Engineering), 1967, Imperial
College, London; Bell Laboratories, 1967—-. Mr. Goodman has studied various
aspects of digital communications including analog-to-digital conversion, dig-
ital signal processing, subjective assessment of voiceband codecs, and spread
spectrum modulation for mobile radio. He is Head, Communications Methods
Research Department. In 1974 and 1975 he was a Senior Research Fellow,
and in 1983 a Visiting Professor at Imperial College, London, England.
Member, IEEE.

Carl-Erik W. Sundberg, M.S.E.E., 1966, and Dr. Techn., 1975, Lund
Institute of Technology, University of Lund, Sweden; Bell Laboratories,
1981-1982. Mr. Sundberg is an Associate Professor in the Department of
Telecommunication Theory, University of Lund, and a consultant in his field.
He is Director of the consulting company SUNCOM, Lund. During 1976 he
was with the European Space Research and Technology Centre (ESTEC),
Noordwijk, The Netherlands, as an ESA Research Fellow. He has been a
Consulting Scientist at LM Ericsson and SAAB-SCANIA, Sweden, and at
Bell Laboratories. His research interests include source coding, channel coding
(especially decoding techniques), digital modulation methods, fault-tolerant
systems, digital mobile radio systems, spread spectrum systems, and digital
satellite communication systems. He has published a large number of papers
in these areas during the last few years. Senior Member, IEEE; member, SER,
Sveriges Elektroingenjorers Riksforening.

The Comite Consultatif International Telegraphique et Telephonique
(CCITT) has recently recommended a code for two-level (black and white)
graphics transmission. A large number of pictures in graphics communication
contain areas that cannot be represented adequately by only two shades of
gray. We describe techniques by which a composite picture, containing an
arbitrary mixture of two- and multilevel areas, can be coded by schemes that
are compatible with the CCITT code. First, the composite picture is segmented
automatically into two types of areas: one requiring only two levels (text,
drawings, etc.) and the other requiring multilevels (for example, photos). A
Differential Pulse Code Modulation (DPCM) scheme is then used to code the
multilevel areas. Code assignment for the outputs of the DPCM quantizer are
based on the local conditional statistics, and the bit stream is processed to
change the statistics of the run lengths so that the CCITT run-length code
becomes efficient. Results of computer simulations are presented in terms of
quality of processed pictures and the required bit rate. Simulations show that
our CCITT: compatible scheme is as efficient as an incompatible but optimum -
DPCM coding scheme.

Simultaneous developments (algorithmic as well as systems) have
taken place for many years in coding and transmission of two-level
(black and white) document facsimile, and multilevel (many shades of
gray) pictures.” The former type of pictures require very high spatial
resolution to preserve the sharpness and have been coded by one-

dimensional run-length coding and two-dimensional edge difference
coding [Comite Consultatif International Telegraphique et Telepho-
nique (CCITT) one- and two-dimensional codes]. On the other hand,
multilevel pictures contain gradual luminance transitions, and there-
fore require lower spatial resolution. They have been coded by Differ-
ential Pulse Code Modulation (DPCM) and transform methods. Most
pictures used in business facsimile systems and audiographics confer-
encing contain a mixture of two-level and multilevel segments or
subpictures. Coding such pictures using two-level techniques would
not be adequate from the point of view of the picture quality, and
using multilevel techniques would generate enormous data rates. Thus,
it is of interest to devise schemes that automatically divide a picture
into segments, each segment with a specified amplitude (gray shades)
and spatial resolution and code each segment as best suited for it.
Another practical requirement is that of compatibility. A coding
scheme that handles a mixture of two-level and multilevel segments
should be upwardly compatible with the CCITT standard schemes for
two-level pictures. System cost will be reduced if the scheme for two-
level multilevel pictures uses hardware blocks that are also used by
the two-level picture coder. We present such a scheme below. Principal
characteristics of our scheme are:

2. Automatic segmentation of pictures into two-level and multilevel
segments

3. High coding efficiency by preprocessing the multilevel segments
to fit the CCITT codes

5. Nonlossy (information preserving) coding of two-level segments,
and lossy coding of multilevel segments.

The function of the segmentor is to classify each picture element as
one of the three:

3. Gray — Multilevel
Figure 1 shows a neighborhood of the current picture element (pel)
used for segmentation. We assume that each pel, obtained from the
scanner, is specified by many shades of gray (e.g., 8 bits). The size of
the neighborhood can be arbitrary. If it is too small, then many

Fig. 1—Picture elements used for segmentation of the current pel. The size of the
neighborhood is not necessarily 5 X 5 as suggested in the figure.

discontinuous segments of gray pixels will be generated. On the other
hand, if the neighborhood is too large, then the ability to resolve small
gray areas is lost. We slide this window of neighborhood over the pels
along a scan line and classify each pel. We consider the boundary pels
of the picture separately. Two thresholds, ¢,; and ty (t; < te), are
selected. It is hypothesized that most black pels will have intensity
less than t,, white pels will have intensity greater than t2, and gray pel
intensities may lie anywhere. Within the neighborhood let

We define a state, S, consisting of three components: S,, S:, and S3.
A picture is segmented on the basis of the value of S. Let

S=S,+ 82+ 83, (1)
where
S, = 1, if ng > m + ne
= 0, otherwise
S2 = 1, if previous pel is gray
= 0, otherwise
and
S3 = 1, if t,; < intensity of present pel < te
= 0, otherwise.

S = 2 => current pel gray
S < 2 and intensity of current pel > T = > current pel white
S < 2 and intensity of current pel < T = > current pel black.

T is a threshold used to distinguish black elements from white ones,
once they are known to be of the two-level type. If the range (te — ¢)
is decreased by increasing t; and decreasing t2, then more elements
will be regarded as two-level and the quality of picture may suffer, but
this will also decrease the bit rate.

We evaluated the performance of the segmentor, in particular its
dependence on the block size and (t2 — t,) by computer simulation.
Since there are no standard mixtures of two-level and multilevel
images, we created our own by taking a 512 X 512 gray-level image
(shown in Fig. 2) and superimposing it on the CCITT documents four
and five. Since this 512 X 512 original was scanned at low resolution
(compared to 200 pels/inch used for CCITT documents), it contains
significant sharp transitions that would not be present in a photograph
scanned at 200 pels/inch. Also, because the original gray-level picture
and the CCITT documents are rather “clean”, segmentor works quite

well. However, this may not be a typical situation if a nonideal scanner
was used. We, therefore, added random noise to the entire composite
picture. This noise had a variance of 425 (on an 8-bit scale, 0-255).
Table I shows the performance of the segmentor with respect to block
size for a composite picture made from CCITT document 4. Here t; =
28, tg = 195. As we view such a segmented picture we realize that a
5 X 5 block may be too small. A 9 X 9 block appears quite adequate
even when the added noise variance reaches 758. Higher block sizes
result in a larger number of contiguous gray pels, thereby decreasing
the number of segments. Figure 3 shows a segmented picture. Due to
equipment limitations we show only a 512 xX 512 section of the

Fig. 3—A segmented picture. Pels classified black and white are reproduced with
intensities 30 and 215, respectively. Gray-level pels are reproduced with 8-bit intensities.

composite picture. The segmentor has adequately separated the two-
level areas from the multilevel areas.

In most cases, areas of the picture that are segmented to be gray do
not need as much spatial resolution as the two-level segments. If the
two-level picture is at a very high spatial resolution (e.g., 200 pels/
inch), then without any significant loss of quality, spatial resolution
can be reduced in gray areas. Following is a scheme for subsampling
and interpolation. A subsampling pattern is shown in Fig. 4. Interpo-
lation is performed by averaging four surrounding pels, as in Fig. 4.
Although we show only 2:1 subsampling, higher subsampling ratios
may be used if the quality requirements are not very high. Also, two-
dimensional subsampling may be performed, but this may increase the
complexity.

After the pels (black, white, or gray) are classified and the gray
areas to be transmitted are determined, a DPCM coder is used for
gray areas. The resulting bit stream from the DPCM coder is pre-
processed, multiplexed with bits from the two-level segments, and then
coded by a CCITT one-dimensional or two-dimensional coder. Ad-
dresses for the segments of gray pels are coded separately and multi-
plexed with the coded data to transmit on the channel. A block diagram
for the transmitter portion is shown in Fig. 5. Details of the algorithm
are given below. Only a nonsubsampled case is illustrated; a subsam-
pled case follows trivially.

The purpose of gray segment coding is to convert an 8-bit/pel signal
representing gray areas into a coded 3-bit/pel signal, which can then
be preprocessed and run-length coded. This procedure reduces the bit
rate for gray pels to about 2 bits/pel.

Fig. 4—Subsampling and interpolation pattern used in gray areas. Only one-dimen-
sional subsampling is considered.

CCITT — COMITE CONSULTATIF INTERNATIONAL
TELEGRAPHIQUE ET TELEPHONIQUE

wee 6—Configuration of pels used for prediction. Only a nonsubsampled case is
shown.

It is assumed in this figure that all elements A, B, C are gray elements.
Appropriate modification is made if some of these are two-level ele-
ments.

The prediction error is quantized by a symmetric seven-level quan-
tizer with the transfer characteristics given in Fig. 7. For most pictures
with a resolution of 100 pels/inch, this appears adequate, although in
some cases dynamic range may not be sufficient. Subjective studies
are needed to optimize the characteristics for a given set of pictures.

Efficiency can be improved further by adapting the prediction and
quantization.

To reduce statistical redundancy and create a bit stream that can
be coded compatibly with the CCITT code, seven levels of the quan-
tizer output are mapped into a three-bit code. First, a table of 49 states
is constructed by looking at the seven outcomes of the quantized
prediction values for both elements A and B (in Fig. 6). Given a state,
the code words for the present pel are arranged in order of conditional
frequency of occurrence. Such statistics are precomputed for a set of
pictures. The code word that is most frequent (for a given state) is
given the code [000], the next highest is given code [001], etc. In
addition, to decrease the probability of occurrence of isolated ‘1’, if
the last bit of the code word for A is a ‘1’, then the entire code word
for the present pel is complemented (i.e., ‘0’ — ‘1’, and ‘1’ — ‘0’). The
table of 49 states and the corresponding code words are shown in
Table II.

The code words for various states (e.g., runs) for the CCITT scheme
are already defined based on the statistics. The statistics of the states
for the gray-level segments are quite different. As an example, Fig. 8
shows histograms of the runs for black and white pels on which the
one-dimensional CCITT code is based. The same figure also shows
the histograms of the runs of the bits from gray-level picture (only
512 x 512) with the code assignment of the previous section but
without any bit complementing. It is clear that the histograms are not
similar in shape, and therefore using the CCITT code for runs of bits
from gray segments would not be efficient. Since our experience shows
that the two-dimensional CCITT code is not efficient for the gray
segments, we give below a method of preprocessing that makes efficient
use of the one-dimensional CCITT code. Let n2(i) and n°(i) be the
histograms of the runs of black elements for the CCITT code and the
gray-level segments, respectively. Also let c(i) be the code assigned to
the ith run by the CCITT coder. Let j(i) be the sequence that is
arranged in descending order of the histogram function h2({), i.e.,

Similarly, arrange the gray-level histogram h(i) with the function
j*(t). Then the code word for a length j*(z) of the gray segment is the
same as c(j(i)). Thus, we arrange the two histograms in descending
order and choose the code to be the same for entries of both the

; ; Decreasing Frequency of Occurrence —>
Quantizer Quantizer
No. Levelfor B LevelforA 000 001 011 111 110 100 010

CCITT — COMITE CONSULTATIF INTERNATIONAL
TELEGRAPHIQUE ET TELEPHONIQUE

Fig. 8—Histograms of the (a) white and (b) black runs used in the 1D CCITT code
and the processed gray-level pictures.

rearranged histograms. We found that in reality much of the gain in
coding efficiency can be obtained by exchanging the code words for a
few run lengths. This leads to simple preprocessing. Figure 9 shows
the results of preprocessing on the histograms. It is clear that, if bit
complementing is not used, after the preprocessing the code set is
more attached to the histograms and therefore leads to more efficient
code. However, if bit complementing is used, the histogram without
any preprocessing is not too different from the CCITT histogram.
Therefore, when bit complementing is used, the advantages of pre-
processing are not large. Although this is the case for the picture we
considered, more experiments are needed to evaluate statistics of
typical pictures and usefulness of the preprocessing for such statistics.

To encode positional information of the boundaries of a segmented
picture, each composite line is considered as a sequence of alternating
black and white runs corresponding to the lengths of two-level and
gray-level segments, respectively. This is then coded by the two-

CCITT — COMITE CONSULTATIF INTERNATIONAL
TELEGRAPHIQUE ET TELEPHONIQUE

Fig. 9—Effect of preprocessing on the histograms of the (a) white and (b) black runs.

dimensional CCITT code and is transmitted at the beginning of each
composite line.

The bits resulting from the above procedure for gray areas are
multiplexed pel by pel with those of the two-level pels. Our experiments
show that while it is advantageous to encode two-level areas by two-
dimensional code, most of the two-dimensional correlation in gray
segments is removed by two-dimensional prediction and code assign-
ment. Therefore, gray areas are coded by one-dimensional code. Since
the number of gray-level pels may vary from line to line, in order to
maintain proper registration of two-level pels (for two-dimensional
coding), a sample count is maintained and is used to initialize the two-
dimensional coder once it comes out of the gray segment within a line.

Results of computer simulations are given in Tables III and IV.
Table III shows results for 512 X 512 gray-level picture, and Table IV
shows results for composite pictures with CCITT standard documents
4 and 5. It is clear from Table III that for the gray-level picture,
without any preprocessing or bit complementing, coding efficiency is

Coded bits
Docu- Docu-
No. Coding algorithms ment 4 ment 5
1. One-dimensional CCITT code on noncomposite docu- 870803 547853
ment
2. Two-dimensional CCITT code on noncomposite docu- 577527 286911
ment

3. Two-dimensional code for two-level, one-dimensional 1169270 893288
code for gray level (no complementing)

rather low. This is a result of the mismatch of the run-length statistics.
Considerable improvement is obtained by preprocessing the run
lengths before applying the CCITT coder. Even higher improvement
is obtained by the bit-complementing technique. Much of the mis-

match between the statistics is removed by the complementing tech-
nique, and therefore additional improvement obtained by preprocess-
ing the complemented output is marginal. The use of complementing
makes it possible to achieve bit rates that are close to the entropy of
the coded output. Coding of composite pictures shows similar results.
Another interesting conclusion from Table IV is that the two-dimen-
sional CCITT code is not very efficient for gray-level segments of the
composite picture. This is a result of lack of line-to-line correlation
among bits that are outputs of the quantizer. Much of the line-to-line
correlation is already removed by the two-dimensional prediction and
the bit assignment based on conditional statistics.

We have presented an algorithm that can automatically segment
areas of a picture that require only two shades of gray from those that
require many shades of gray. Gray areas are coded in a way that
creates a bit stream that subsequently can be efficiently coded by a
CCITT coder. We find that, for the gray areas, it is possible to achieve
coding efficiencies close to the entropy of the DPCM quantizer output.
Therefore, we conclude that it is possible to encode documents that
contain an arbitrary mixture of two-level and multilevel areas using a
CCITT coder that requires only a preprocessor at the transmitter and
a postprocessor at the receiver.

1. A. N. Netravali, ed., special issue on Digital Encoding of Graphics, Proc. IEEE, 68,
No. 7 (July 1980).

3. R. Hunter and A. H. Robinson, “International Digital Facsimile Coding Standards,”
Proc. IEEE, 68, No. 7 (July 1980), pp. 830-46.

Arun N. Netravali, B. Tech. (Honors), 1967, Indian Institute of Technology,
Bombay, India; M.S., 1969, Ph.D. (Electrical Engineering), 1970, Rice Uni-
versity; Optimal Data Corporation, 1970-1972; Bell Laboratories, 1972—. Mr.
Netravali has worked on problems related to filtering, guidance, and control
for the space shuttle. At Bell Laboratories, he has worked on various aspects
of digital processing and computing. He was a Visiting Professor in the
Department of Electrical Engineering at Rutgers University. He is presently
Director of the Computer Technology Research Laboratory. Mr. Netravali
holds over 20 patents and has had more than 60 papers published. He was the
recipient of the Donald Fink Prize Award for the best review paper published
in the Proceedings of the IEEE. Editorial board, Proceedings of the IEEE;
editor, Transactions on Communications; member, Tau Beta Pi, Sigma Xi;
Senior Member, IENE.

Hamid Gharavi, B.S.E.E., 1970, Tehran Polytechnic; M.S.C. (Digital Com-
munication), Ph.D. (Electrical Engineering), 1979, University of Technology,
Loughborough, England; Research fellow at University of Technology, Lough-
borough, England, 1979-1980; Lecturer at Auckland University, New Zealand,
1980-1981; Bell Laboratories, 1982—. Mr. Gharavi has worked on problems
related to digital modulations for satellite communication. He also has worked
on bandwidth compression of color television signals, source coding, graphics,
and pattern recognition. Member, IEEE.

This paper describes the Queueing Network Analyzer (QNA), a software
package developed at Bell Laboratories to calculate approximate congestion
measures for a network of queues. The first version of QNA analyzes open
networks of multiserver nodes with the first-come, first-served discipline and
no capacity constraints. An important feature is that the external arrival
processes need not be Poisson and the service-time distributions need not be
exponential. Treating other kinds of variability is important. For example,
with packet-switched communication networks we need to describe the conges-
tion resulting from bursty traffic and the nearly constant service times of
packets. The general approach in QNA is to approximately characterize the
arrival processes by two or three parameters and then analyze the individual
nodes separately. The first version of QNA uses two parameters to characterize
the arrival processes and service times, one to describe the rate and the other
to describe the variability. The nodes are then analyzed as standard GI/G/m
queues partially characterized by the first two moments of the interarrival-
time and service-time distributions. Congestion measures for the network as
a whole are obtained by assuming as an approximation that the nodes are
stochastically independent given the approximate flow parameters.

Networks of queues have proven to be useful models to analyze the
performance of complex systems such as computers, switching ma-
chines, communications networks, and production job shops.'’ To
facilitate the analysis of these models, several software packages have

been developed in recent years, e.g, BEST/1,2 CADS,? PANA-
CEA,!°” and one based on Heffes.’? These software packages contain
algorithms for Markov models that can be solved exactly. For some
applications, the model assumptions are at least approximately satis-
fied, so that the analysis can be very helpful. For many other appli-
cations, however, the model assumptions are not even approximately
satisfied, so that the analysis can be misleading.

A natural alternative to an exact analysis of an approximate model
is an approximate analysis of a more exact model. This paper describes
a software package called the Queueing Network Analyzer (QNA),
which was recently developed at Bell Laboratories to calculate ap-
proximate congestion measures for networks of queues. QNA goes
beyond existing exact methods by treating non-Markov networks: The
arrival processes need not be Poisson and the service-time distribu-
tions need not be exponential. QNA treats other kinds of variability
by approximately characterizing each arrival process and each service-
time distribution with a variability parameter. It is also possible to
analyze large networks quickly with QNA because the required calcu-
lations are minimal, the most complicated part being the solution of a
system of linear equations. The current version of QNA is written in
FORTRAN.

Here is a rough description of the model: There is a network of
nodes and directed arcs. The nodes represent service facilities and the
arcs represent flows of customers, jobs, or packets. There is also one
external node, which is not a service facility, representing the outside
world. Customers enter the network on directed arcs from the external
node to the internal nodes, move from node to node along the internal
directed arcs, and eventually leave the system on one of the directed
arcs from an internal node to the external node. The flows of customers
on the arcs are assumed to be random so that they can be represented
as stochastic processes.

If all servers are busy at a node when a customer arrives, then the
customer joins a queue and waits until a server is free. When there is
a free server, that customer begins service, which is carried out without
interruption. Successive service times at each node are assumed to be
random variables, which may depend on the type of customer but
which otherwise are independent of the history of the network and are
mutually independent and identically distributed. After the customer
completes service, he goes along some directed arc from that node to
another node. The customer receives service in this way from several
internal nodes and then eventually leaves the network. A picture of a
typical network (without the external node) is given in Fig. 1.

An important feature of the model is that there may be flows from
node j to node i, as well as flows from node i to node j. This is of

course useful when customers can return to a node where they previ-
ously received service, but it is also useful when customers cannot
return to a node where they previously received service. Then flows
from node j to node i represent different customers than the customers
that flow from node i to node j.

To be precise about the model, we give a list of basic assumptions.
It is worth noting, however, that work is under way to extend QNA so
that it can analyze systems in which each of the following assumptions
is replaced by obvious alternatives. The general approximation tech-
nique is flexible, so that it is not difficult to modify and extend the
algorithm.

Assumption 1. The network is open rather than closed. Customers
come from outside, receive service at one or more nodes, and eventually
leave the system.

Assumption 2. There are no capacity constraints. There is no limit
on the number of customers that can be in the entire network and
each service facility has unlimited waiting space.

Assumption 3. There can be any number of servers at each node.
They are identical independent servers, each serving one customer at
a time.

Assumption 4. Customers are selected for service at each facility
according to the first-come, first-served discipline.

Assumption 5. There can be any number of customer classes, but
customers cannot change classes. Moreover, much of the analysis in
QNA is done for the aggregate or typical customer (see Sections 2.3
and VI).

e.g., an arrival can cause more than one departure (see Section 2.2).
(Think of messages.)

The general approach is to represent all the arrival processes and
service-time distributions by a few parameters. The congestion at each
facility is then described by approximate formulas that depend only
on these parameters. The parameters for the internal flows are deter-
mined by applying an elementary calculus that transforms the param-
eters for each of the three basic network operations: superposition
(merging), thinning (splitting), and flow through a queue (departure).
These basic operations are depicted in Fig. 2. When the network is
acyclic (e.g., queues in series), the basic transformations can be applied
successively one at a time, but in general it is necessary to solve a
system of equations or use an iterative method. To summarize, there
are four key elements in this general approach:

1. Parameters characterizing the flows and nodes that will be readily
available in applications and that have considerable descriptive power
in approximations of the congestion at each node.

2. Approximations for multiserver queues based on the partial infor-
mation provided by the parameters characterizing the arrival process
and the service-time distribution at each node.

3. A calculus for transforming the parameters to represent the basic
network operations: merging, splitting, and departure.

4. A synthesis algorithm to solve the system of equations resulting
from the basic calculus applied to the network.

Fig. 2—Basic network operations: (a) Superposition or merging. (b) Decomposition
or splitting. (c) Departure or flow through a queue.

eters are the first two moments. However, we actually work with the
mean service time 7 and the squared coefficient of variation c2, which
is the variance of the service time divided by the square of its mean.
The user has the option of working with the service rate p = 7’
instead of 7. For the arrival processes, the parameters are associated
with renewal-process approximations. The first two parameters are
equivalent to the first two moments of the renewal interval (interval
between successive points) in the approximating renewal process. The
equivalent parameters we use are the arrival rate \, which is the
reciprocal of the renewal-interval mean, and the squared coefficient
of variation c2, which is the variance of the renewal interval divided
by the square of its mean.

We obtain the approximation of the flows by applying the general
framework and the basic procedures for approximating point processes
in Whitt,'* incorporating refinements such as the hybrid procedures
developed for merging by Albin.’>’® Of course, the general idea of
simple two-parameter approximations for stochastic point processes
goes back at least to the equivalent random method for approximating
overflow streams (see Wilkinson,'’ Cooper,!® and references there).
Renewal-process approximations for such point processes were intro-
duced by Kuczura’® (also see Rath and Sheng”). Two-parameter
approximations for networks of queues similar to QNA have also been
developed by others, apparently first by Reiser and Kobayashi" (also
see Kuehn,” Sevcik et al.,?? Chandy and Sauer,”* Chapter 4 of Gelenbe
and Mitrani,* and Shanthikumar and Buzacott®). These two-parame-
ter approximations for networks of queues are also similar in spirit to
two-parameter approximations for networks of blocking systems with
alternate routing (see Katz?’).

Some authors have referred to these two-parameter heuristic ap-
proximations for networks of queues as diffusion approximations,*”?
but diffusion processes are not actually used. Diffusion approximations
and associated heavy-traffic limit theorems have motivated some of
the heuristic approximations in the literature and in QNA, and they
are closely related to the asymptotic method for approximating point
processes,‘ but the heuristic approximations in QNA are not the same
as the more complicated diffusion approximations for networks of
queues in Iglehart and Whitt,?° Harrison and Reiman,”’ and Rei-
man.7®?9

a generalization of the product-form solution that is valid for Marko-
vian networks, i.e., in the Markov models the components of the vector
representing the equilibrium number of customers at each node are
stochastically independent, so that the probability mass functions for
the vector is the product of the probability mass functions for the
components. While QNA can be thought of as a decomposition method
or an extended-product-form solution, an effort is made to capture the
dependence among the nodes. The idea is to represent this dependence
approximately through the internal flow parameters.

To see the motivation for QNA, consider the elementary open
network containing a single node with a single server, an infinite
waiting room, and the first-come, first-served discipline. Suppose there
is a single customer class with each customer being served only once
before departing. The standard Markov model of this elementary
network, which is embodied in BEST/1, CADS, and PANACBEA, is
the classical M/M/1 queue,’*”® which has a Poisson arrival process
and an exponential service-time distribution. For the M/M/1 model,
the expected waiting time EW (before the customer begins service) is

where 7 is the mean service time and p is the traffic intensity, which
is assumed to satisfy 0 < p <1.

On the other hand, QNA uses an approximation for the GI/G/1
model to represent this one-node network. The GI/G/1 model has a
renewal arrival process and both the interarrival-time distribution and
the service-time distribution are general. In QNA, the arrival process
is represented by a renewal process partially characterized by two
parameters: the arrival rate \ and the variability parameter cz. The
service-time distribution is also partially characterized by two param-
eters: the mean service time 7 and the variability parameter c2. In
contrast to (1), the formula for the expected waiting time in QNA is

where g = g(p, c2, c2) is either one (when c? = 1) or less than one
(when c2 < 1); see (45). When g(p, c2, c2) = 1, (2) differs from (1) by
the factor (c2 + c2)/2. When the arrival process is Poisson, c2 = 1;
when the service-time distribution is exponential, c2 = 1. Hence, if the
GI/G/1 model is actually an M/M/1 model, (2) reduces to (1). Of
course, the user of QNA can set c2 = c2 = 1 and obtain (1). In fact,
the values c2 = 1 and c2 = 1 are default values that the program uses
if the user does not have variability parameters to provide. Each c?
can assume any nonnegative value: c? = 0 for the degenerate deter-
ministic distribution; c? = k7! for an erlang E,, the sum of k i.i.d.
exponential random variables; and c? > 1 for mixtures of exponential

distributions. Obviously, the difference between (2) and (1) can be
large, so that (2) often significantly reduces the error.

To obtain (2), we studied the GI/G/1 queue partially characterized
by the moments of the interarrival-time and service-time distributions.
Building on previous work by Holtzman,*° Rolski,*! and Eckberg,*” we
investigated the set of possible values of EW given the partial infor-
mation.**3” When c2 = 1, formula (2) is always a possible value, i.e.,
there always is a GI/G/1 system with interarrival-time and service-
time distributions having the specified moments in which (2) is correct.
In general, (2) appears to be a reasonably typical value.

For the single-node example just considered, the arrival process was
a renewal process. More generally, it is natural to think of all the non-
Poisson arrival processes in the model as renewal processes, either
because they are initially renewal processes or because the algorithm
can be-interpreted as approximating general arrival processes by
renewal processes. Hence, with one customer class, it is natural to
think of the model as a generalization of the open Jackson network
M/M/m queues to an open Jackson network of GI/G/m queues. Each
node is approximated by a GI/G/m queue having a renewal arrival
process independent of service times that are independent and iden-
tically distributed with a general distribution. It is significant that
QNA is consistent with the Jackson network theory: If there is a single
class of customers, if all the arrival processes are Poisson, and if all
the service-time distributions are exponential, then QNA is exact.
However, for the general model few analytical results are available, so
approximations are needed.

The software package QNA has a flexible input procedure: the model
will accept more than one kind of input (see Section II). For the
standard input, only limited information is required. Only two param-
eters are needed for each service-time distribution and each external
arrival process. Also, a routing matrix is needed, which gives the
proportion of those customers completing service at facility 1 that go
next to facility j. (The algorithm is based on Markovian routing.)
Hence, for n nodes, the input consists of n? + 4n numbers.

There is also an alternate input by classes and routes. In this scheme
there are different classes of customers and each class enters the
network at a fixed node and passes through a specified sequence of
nodes. For each class, there are two parameters characterizing its
external arrival process and two parameters characterizing the service-
time distribution at each node on its route. With this input by routes,
different classes can have different service-time distributions at a
given node and the same class can have different service-time distri-
butions during different visits to the same node. For a class with n
nodes on its route, the input consists of 3n + 2 numbers. (This includes

QNA also provides a fairly rich output. Several different congestion
measures are calculated for each node: the traffic intensity (utiliza-
tion), the expected number of busy servers (offered load), and the
mean and variance of the equilibrium delay and number of customers
present. In fact, for single-server nodes the delay distribution itself is
described. Congestion measures for the entire network are also calcu-
lated, under the approximation assumption that the nodes are sto-
chastically independent given the approximate flow parameters.
Means and variances of total service times, total delays, and total
sojourn times (response times) are given. When the input is by routes,
these characteristics are given for each customer class. Otherwise,
these characteristics are given for any route requested by the user.

A desirable feature of QNA is the structure of the calculus to
transform the parameters to characterize the internal flows. The
calculus is linear for each network operation, so that the parameters
for the internal flows are determined simply by solving systems of
linear equations. For the rates, the system of linear equations is just
the familiar traffic rate equations occurring in the Jackson network of
M/M/m queues. After having obtained the rates, we obtain the vari-
ability parameters of the internal flows (the squared coefficients of
variations) by solving another system of linear equations. As a by-
product, the existence of a unique nonnegative solution for the flow
parameters is trivially guaranteed. There is no guarantee that an
iterative scheme will converge, and if it does, there is typically no
guarantee that a solution is unique. The linearity also guarantees that
the computation required is not great. Since there is only one linear
equation per node in the network, QNA can be used to analyze large
networks repeatedly at minimal cost.

The linear calculus for transforming the variability parameters
incorporates results of recent studies to improve the accuracy of the
approximations. The general framework for approximating point proc-
esses in Whitt" is used. Significant improvement over previous ap-
proximation methods of this kind has been obtained by paying partic-
ular attention to the difficult superposition operation. For superposi-
tion, we use a modification of the hybrid procedure developed at Bell
Laboratories by Albin.*16°89

We emphasize that QNA is spre enats In applications it is im-
portant to validate the QNA output by comparing it with simulations
and/or measurements. QNA is designed so that it is easy to incorporate
improvements and it is easy to tune QNA for particular applications.

QNA also provides a useful framework for developing new approxi-
mation procedures. Moreover, it is easy to use QNA in conjunction
with other special algorithms available to analyze the nodes or the
flows.

The rest of this paper describes QNA in more detail. The paper is
organized—just as the output is—according to the main steps in the
analysis. The input is described in Section II. Section III describes the
preliminary analysis to eliminate immediate feedback. The procedures
to determine the internal flow parameters are contained in Section
IV, and the procedures to calculate approximate congestion measures
for the nodes are contained in Section V. Section VI contains the
procedures to calculate approximate congestion measures for the net-
work as a whole.

In this section we describe the input options currently available for
QNA. We anticipate more input options in the future. In Section 2.1
we describe the standard input, which is relatively compact. In Section
2.2 we describe a minor modification of the standard input, which
allows for the creation or combination of customers at the nodes. For
example, when a packet completes service at some node, it may cause
several packets to be sent to other nodes. In Section 2.3 we describe
an alternate input for different classes of customers having specified
routes. We also describe the way QNA converts this input by classes
and routes into the standard input of Section 2.1.

With the standard input, there is a single customer class and no
creation of customers at the nodes. Any number of networks can be
processed during a single run, so the user first specifies the number of
networks. Then, for each network, the user specifies the number of
nodes, and for each node the number of servers. For each node in the
network, there are two parameters for the service-time distribution
and two parameters for the external arrival process. Finally, there is
a routing matrix, indicating the proportion of customers that go to
node j from node i. Here is a list of the input data for each network
with the notation we use:

co; = variability parameter of the external arrival process to node j
(squared coefficient of variation of the renewal interval in the
approximating renewal process)

cz, = squared coefficient of variation of the service-time distribution
at node j

qi; = proportion of those customers completing service at node i
that go next to node j.

1 X n vector. The user has the option of inputting 7; or its reciprocal
pj, the service rate at node j. (The same form must be used for all
nodes.)

The user need not specify the variability parameters cj; and c%, in
which case they are set equal to the default value one, corresponding
to the M/M/1 model having a Poisson arrival proccess and an expo-
nential service-time distribution. (Again, this option applies to all
nodes.) Alternatively, the user can specify only the service-time vari-
ability parameters, c2,, in which case either all the arrival-process
variability parameters are automatically set equal to 1, yielding an
M/G/1 approximation for each node, or the QNA algorithm is applied.

QNA has an option to allow creating or combining customers at the
nodes following the completion of service. For example, a message
processed at some node might cause messages to be sent to several
other nodes. Alternatively, messages might be divided into packets
after service at one node and then later recombined into messages
after service at another node. In a job shop, the focus might shift back
and forth between units and lots, e.g., at different nodes we might
consider bottles, six-packs, cases, and even truckloads.

With this option, the user must specify the multiplicative factor y;
of customer creation or combination at node j for each j. There is
customer creation (combination) at node j if y; > 1 (y; < 1). If
customers are neither created nor combined, then y; = 1. If ); is the
overall arrival rate to node j, then the departure rate, after this
modification, is \;; and the rate of departure from the network j is

MVYj ( aoe) an)
k=1
When artificial nodes are used, the creation or combination can also
be placed before service.
To obtain our approximation formulas, we work with the following
models of customer creation and combination. These models require
integer values, but the approximation formulas and the QNA input do

not. For customer creation, we replace each departure from node j
with a batch of size y;. For customer combination, we replace yj"
successive interdeparture intervals by a single one. From these models
it is not difficult to calculate the impact of y;, e.g., the departure rate
at node j is simply multiplied by y;.

QNA provides the option of defining different customer classes.
Each class has its own route or itinerary that specifies the sequence
of nodes visited. Thus, for each class the routing is deterministic. Each
class has an external arrival process that goes to the first node on the
route. As usual, the external arrival process is characterized by rate
and variability parameters. Also, each class may have its own service-
time distribution at each node on its route. The service-time distri-
butions can be different, not only for different classes, but also for
different visits to the same node by the same class. These service-time
distributions are also characterized by rate and variability parameters.
(Alternatively, the user can elect to input the service-time parameters
for each node. Then all classes have the same service-time distribution
‘at each visit to a particular node.)

As with the standard input, the user must specify the number of
nodes and the number of servers at each node. Now we need the
number of routes too. The required data are:

n = number of nodes
m; = number of servers at node j
r = number of routes.
Here is a list of the input data for the kth customer class of a network:
n, = number of nodes on route k
A, = external arrival rate of class k
cz, = variability parameter of the external arrival process for class
k
ny = the jth node visited by customer class:k
Tj = the mean service time of class k at the jth node of its route

QNA converts this input by classes and routes into the standard
input in Section 2.1. It then calculates the parameters of a typical or
aggregate customer. Later, when computing sojourn times or response
times of each customer class, QNA uses the service-time parameters
of that customer class. The first version of QNA assumes as an
approximation that each customer sees independent versions of the
equilibrium distribution at each node. Hence, the waiting time before
beginning service at each node is assumed to be the same for all classes
and all visits.

We now indicate how QNA converts the input by classes and routes
into the standard input of Section 2.1. For this purpose, let 1H be the
indicator function of the set H, i.e., 1H(x) = 1 if x € H and 1H(x) =0
otherwise.

i.e., the external arrival rate at node j, Xo, is the sum of all route
arrival rates , for which the first node on the route is j. (Here the 1H
notation is used for H = {k:nz, = j}.) Similarly, the flow rate from i to
jis

From (4) and (5), we obtain the routing matrix Q. The proportion of
customers that go to j from i is

If node i is an active part of the network, then the denominator will
be strictly positive. Otherwise, QNA gives an error message.

Next, if the service-time parameters are given by routes, we obtain
the service-time parameters for the nodes by averaging:

The denominator in (7) will be strictly positive if node j is ever visited.
Otherwise, as with (6), QNA supplies an error message.

We obtain the node variability parameters c2; using the property
that the second moment of a mixture of distributions is the mixture
of the second moments. Therefore, we have

pj = Aytj/mj. (9)
QNA uses this information to calculate the variability parameters cj;
of the external arrival process. The hybrid approximation for super-
position arrival processes in Section 4.2 is also used here because the
external arrival process to node j is the superposition of the external
arrival processes to node j from the different classes. If \o; = 0, then
cj; does not matter and QNA sets cg; = 1. Otherwise,

(np, Nes Ch Nk, Thi, Coa sey Mens Thnrys Ching). (13)
Here suppose that the r vectors are:
(2,22) Te Weeds 8,33)
(35:3, 25 1,2, 072.1. -1> I-21)

(2,.2, 4; 2,1, 1: 1, 2. 1). (14)
The first route corresponds to a Poisson arrival process at rate 2 to
node 1, with all customers being fed back immediately for a second
service before departing from the network. (Of course, the arrival
process need not actually be Poisson; a Poisson process always has
= 1 but other processes could have c? = 1 too.) The second class
also enters at node 1, then goes to node 2 and back to node 1 before
departing from the network, etc.

By (3), the external arrival rates are \o; = 5 and doz = 2. By (4), the
internal flow rates are \y, = 2, Aig = 3, Agi = 5, and Age = 0. By (5),
the flow rates out of the network are Ayo = 7 sad Azo = 0. By (6), the
routing probabilities are: qi, = 1/6, qi2 = 1/4, doi = 1, and qo2 = 0. By

(7), the mean service times are 7; = 2 and ro = 1. By (8), c2, = 1.67
and c2, = 1.00. Note that both service times at node 2 in (14). have
mean 1 and squared coefficient of variation 1, as with a common
exponential distribution, so we should want rz. = c2, = 1.

To obtain the internal arrival rates, we solve the traffic rate equa-
tions as in Section 4.1, i.e.,

Finally we obtain the variability parameters cé;. First, from (12),
vy, = 25/13 and v2, = 1.0. Then, from (11), w, = 0.629 and wz = 1, so
that c; = 1.38 and c%. = 4. Since there is only one ‘external arrival
process to node 2, we should have v2 = Wz = 1 and c2. = c2 = 4.

Immediate feedback occurs whenever q;; > 0. Since QNA assumes
Markovian routing, each customer completing service at node i is
immediately fed back to node i to be served again with probability q;;.
Each time the customer goes to the end of the line. With the decom-
position method, QNA assumes the customer finds the equilibrium
number of customers at the node each time, with each visit being an
independent experiment.

QNA eliminates immediate feedback by giving each customer, upon
arrival from another node, his or her total service time before going
to a different node. This is equivalent to putting a customer immedi-
ately fed back at the head of the line instead of at the end of the line.
Transitions from node i back to node i are eliminated and the new
probability of a transition to node j becomes the old conditional
probability given that the customer departs from node i. In other
words, each visit to node i from elsewhere plus all subsequent times
immediately fed back are interpreted as a single visit. The service time
is increased to compensate.

The motivation for this procedure is easy to explain. For a multi-
server node with Bernoulli (Markovian) feedback and iid service times
that are independent of a general arrival process (not necessarily
Poisson or renewal), the distribution of the queue length process (but
not the waiting times) is the same after this transformation. Hence,

we calculate the approximate values of the mean and variance of the
equilibrium queue length for the transformed node without feedback
and use them to derive approximate waiting time characteristics. By
Little’s formula,*!*” the expected waiting time is also exact, i.e., the
only error is in the approximation of the arrival process by a renewal
process and the approximations for the characteristics of the GI/G/m
queue; there is no additional error due to the immediate feedback.
The first step of the reconfiguring procedure is quite simple: the
new service time is regarded as a geometric mixture of the n-fold
convolution of the old service-time distribution. The parameters 7;,
c2,, and qj are changed to 7;, ¢3;, and gj when qi > 0:
Tj = Tif (_1 - Gii)
63 = gi + (1 — ques
Git = 0
Gy = 9/1 — qu), J Fl (16)
Afterwards, when calculating congestion measures for node i, QNA
makes further adjustments. When we eliminate immediate feedback
according to (16), we no longer count the times a customer is fed back
immediately as separate visits. Hence, we need to adjust the congestion
measures that are expressed per visit. For example, since the expected
number of visits to node i per visit from outside is (1 — qj)~', to obtain
the expected waiting time per original visit to node i, we multiply the
values of the expected waiting time EW; obtained from (16) by
(1 — qj). Of course, the number of customers at each node is not
affected by the feedback treatment.
Let di, 7:, etc., represent the new adjusted values. In terms of the

where N; represents the equilibrium number of customers at node i
and the T; variables represent the sojourn time per visit at node i.

We obtain the variables 7; and ¢2; in (17) by inverting the operation
in (16), so we receive the original data again. The last five formulas in
(17) involving the second-moment characteristics of W; are based on
the results of N; in the transformed system and heavy-traffic limit
theorems for networks of queues by Reiman.”*? The main quantity
desired is Var(W;); the variable W/ is a preliminary approximation
for W;.

In heavy traffic, the changes in the queue length at the nodes are
negligible during a customer’s sojourn in the network. Hence, if node
i is visited X; times by some customer, then the total sojourn time at
node i, say T/, is distributed approximately (in heavy traffic) as X;Tj,
where X; is independent of T/ and T/ is the sojourn time per individual
visit in (17). (We use 7T/ and T; instead of 7; and T; because we do
not use the description of T; obtained directly from (16) and T; will
differ from T/.) By the independence, ET/? = EX2ET!?. Since X; is
geometrically distributed with mean (1 — qj)~!, c?(X;) = qu, and we
obtain the seventh formula in (17).

The sixth and eighth formulas in (17) just express the formula for
c” in terms of the mean and variance and the fact that the sojourn
time is the sum of a waiting time and a service time. The final formula
for Var(W/) is obtained by approximating W/ by the sum of N;; iid
service times, using standard formulas for the variance of a random
sum (e.g., compute EW?” by first conditioning on N;). Finally, we
obtain the fifth formula for Var(W;) by splitting the variance of T/
into waiting-time and service-time components and dividing by the
expected number of visits to node i. As a consequence, Var(T}) seems
more reliable than Var(W;,). This procedure makes Var(T;), computed
from Var(W,) by adding variance components as in Section VI, agree
with the direct formula for Var(T';) in (17).

Experience indicates that eliminating immediate feedback often
yields a better approximation (see Kuehn” and Sections V and VII of
Whitt*’). It is also often desirable to reconfigure the network to
eliminate almost-immediate feedback, e.g., flows that return relatively
quickly after passing through one or more other nodes (see Section V
of Whitt*°). Further study is needed to understand feedback phenom-
ena and to develop improved approximations.

In this section we indicate how QNA calculates the internal flow
parameters. In Section 4.1 we focus on the flow rates, which are
obtained via the traffic rate equations, just as with the Markov models.
In Section 4.2 we display the corresponding system of linear equations
yielding the variability parameters. The remaining subsections explain
how the variability parameter equations were obtained. The basic
operations of superposition, splitting, and departure are discussed in
Section 4.3, Section 4.4, and Section 4.5, and their synthesis in Section
4.6.

In this step QNA calculates the total arrival rate to each node. Let
\; be the total arrival rate to node j, let y; be the multiplicative factor
of customer creation at node j as specified in Section 2.2, and let 6; be
the departure rate (to other nodes as well as out of the network) at
node j. In general, 6; = djv;. If there is no customer creation, then
+; = 1 and the rate in equals the rate out.

where Ao = (Ao;) is the external arrival-rate vector, Q = (q,) is the
routing matrix, and I' = (y,;) is the diagonal matrix with y;; = y; and
i; = 0 fori # 7. When there is no customer creation, y; = 1 and lr =
I. Of course, (18) is just a system of linear equations. To solve them is
equivalent to inverting the matrix (J — T'Q) in (19). When customers
can be created at the nodes as in Section 2.2, special care should be
taken to be sure that (18) has a solution. We need to have sp(T'Q) <1
where sp(TQ) is the spectral radius of T'Q.

Given the arrival rates, it is possible to solve for the traffic intensities
or utilizations at each node, defined by

If p; 2 1, then the ith node is unstable. If any node is unstable, the
algorithm gives an error message, prints out the traffic intensities, and
stops. The associated offered load at node i, which coincides with the
expected number of busy servers [see p. 400 of Heyman and Sobel*!
or (4.2.3) of Franken et al.*?] is

The parameters a; and p; coincide for a single server, with a; tending
to be more useful as the number of servers, m;, increases (obviously
when m; = ©).

After the arrival rates have been calculated for the nodes, QNA
calculates related quantities for the arcs:

Ay = AViGs —the arrival rate to node j from node i
Diy = Ay/Aj —the proportion of arrivals to j that
came from 1, 1 = 0. (22)

d; = Nii ( - > a) —the departure rate out of the
: ee network from node i
c=) d= the total departure rate out
te of the network. (23)

The heart of the approximation is the system of equations yielding
the variability parameters for the internal flows, i.e., the squared
coefficients of variation for the arrival processes, c3;. (These are
derived in Sections 4.3 through 4.7.) The equations are linear, of the
form

where x;, vj, and w; depend on the basic data determined previously,
e.g., pi, m; and c2;, but not on the variability parameters c3; being
calculated. The parameter 7; is the multiplicative factor of customer
creation or combination, introduced in Section 2.2. The variables x;
and »; are used to specify the departure operation; the variable w; is
used to specify the superposition operation. The variables »; and w;
are weights or probabilities that are used in convex combinations

arising in hybrid approximations for departure and superpositions,
respectively. The variables x;, v;, and w; are included to make modifi-
cation of the algorithm based on (24) easy. The specific values in this
version of QNA are:

It is significant that it is easy to modify this system of equations.
For example, other hybrid procedures for departures or superpositions
can be introduced just by changing »;; and w;, respectively. In this way,
it is easy to calculate and compare the variability parameters for
several different approximation procedures.

The purpose of the following sections is to explain the key approx-
imation equations (24) through (30), which yield the variability param-
eters for the internal flows. The approximations are all based on the
basic methods in Whitt:** the asymptotic method and the stationary-
interval method. We consider the basic operations—superposition,
splitting, and departure—in turn, and then their synthesis.

For superposition, the stationary-interval method is nonlinear so it
presents difficulties.'***?* Moreover, there appears to be no natural
modification that makes it linear. On the other hand, the asymptotic
method is linear. By the asymptotic method, the superposition squared
coefficient of variation c4 as a function of component squared coeffi-
cients of variation c? and the rates ); is just the convex combination

However, neither the asymptotic method nor the stationary-interval
method alone works very well over a wide range of cases, e.g., see
Section ITI of Whitt.*° Albin’””* found that considerable improvement
could be obtained by using a refined composite procedure, which is
based on a convex combination of cX for the asymptotic method and
c& for the stationary-interval method. Her hybrid c# is of the form

Unfortunately, since c% is nonlinear, so is c?;. However, Albin found
that a convex combination of c&{ and the exponential c? of 1 worked
almost as well, having 4-percent average absolute error as opposed to
3 percent. Hence, we use such a hybrid procedure, namely,

where w is a function of p and the rates. Extensive simulation
prompted Albin to suggest the weighting function

Note that if there are k component processes with equal rates then
v = k. The parameter pv can be thought of as the number of component
streams, with it being an equivalent number if the rates are unequal.

However, the weighting function (58) fails to satisfy an important
consistency condition: We should have w = 1 when p = 1; if there is a
single arrival process, the superposition operation should leave the c?
parameter unchanged. Moreover, new theoretical results®® indicate
that the exponent of (1 — p) in (34) should be 2. Hence, we use formula
(33) based on the weight function w in (29).

No approximation is needed for splitting because a renewal process
that is split by independent probabilities (Markovian routing) is again
a renewal process. However, approximation is of course indirectly
associated with this step because the real process being split is typically
not a renewal process and the splitting is often not according to
Markovian routing.

Since a renewal process split according to Markovian routing is a
renewal process, the asymptotic method and the stationary-interval
method coincide. If a stream with a parameter c’? is split into k streams,
with each being selected independently according to probabilities p;,

i= 1, 2, ---, Rk, then the ith process obtained from the splitting has
squared coefficient of variation c? given by
c} = pic? + 1 — pi, (36)

renewal-interval distribution in the split stream is a geometrically
distributed random sum of the original renewal intervals.

For the stationary-interval method with single-server nodes, we
apply Marshall’s formula for the squared coefficient of variation of an
interdeparture time, say ci, in a GI/G/1 queue:***

where EW is the mean waiting time. Since EW appears in (387), the
congestion at the node affects the variability of the departure process.
A stationary-interval method approximation for ci is obtained by
inserting an approximation for EW in a GI/G/1 queue. Our analy-
sis***” suggests that it suffices to use the linear approximation (2)
with g set equal to one. When this is combined with (37), we obtain

A simple extension of (38) for GI/G/m queues that is being used in
the current version of QNA is

cqg@=1+(1— p*)(e3 —1) + Un (cS; — 1). (39)
Note that (39) agrees with (38) when m = 1 and (39) yields cj = 1 as
it should for M/M/m and M/G/ systems for which the stationary
departure process is known to be Poisson. The third term in (39)
approaches 0 as m increases, reflecting the way multiple servers tend
to act as a superposition operation. A basis for further refinements of
(39) is the asymptotic analysis of departure processes in Whitt.” This
asymptotic analysis shows that in some cases the variability of the
departure process depends on the arrival and service processes in a
more complicated way.

As with superposition, the asymptotic method yields a more elemen-
tary approximation than the stationary-interval method. In fact, the
asymptotic-method approximation for the departure process is just
the arrival process itself, i.e., the asymptotic-method approximation
for c4 is just c2.4* The number of departures in a long interval of time
is just the number of arrivals minus the number in queue, and the
number in queue fluctuates around its steady-state distribution,
whereas the number of arrivals goes to infinity.

It remains to combine the basic methods to form a refined hybrid
procedure. However, limited experience indicates that this refinement
is not as critical as for superposition. The stationary-interval method

alone seems to perform better for departure processes than for super-
position processes.**

The most appropriate view for the departure process—the station-
ary-interval method or the asymptotic method—depends on the traffic
intensities at the next nodes where the departures are arrivals. As the
traffic intensity of the next node increases, the asymptotic-method
approximation for the departure process becomes more relevant. For
example, consider the case of two queues in series with parameters
Nols Cdr; M1, C21, Me, and ¢2). If 42 —> A while uw; remains unchanged, then
p2 — 1 and the second queue is in heavy traffic. Under such heavy
traffic conditions, it has been shown” that the congestion measures
at the second node are asymptotically the same as if the first facility
were removed, i.e., as if the arrival process to the second node were
just the arrival process to the first node. More generally, for any arrival
process it has been proved that the asymptotic method is an asymp-
totically correct approximation for a queue in heavy traffic.”°

Hence, it is natural to tune the departure approximation by using
the traffic intensities in the following nodes. Since the departure
process typically will be split and sent to different nodes with different
traffic intensities, it is appropriate to do the tuning after splitting. Let
c3; be the departure c” at node i. Then

is the c” for the portion of the departures going to node j. We let c?; be
a weighted combination of the approximations obtained by the asymp-
totic method and the stationary-interval method [using (39)]:

where vj; is chosen to satisfy 0 < yj; < 1 and be increasing in p; with
vi; > 1 as p; — 1. However, we have not yet found that positive v;
helps,“ so the current version of the QNA uses (28).

From (38) it is clear that the departure process variability, as
depicted by QNA, is an appropriate weighted average of the arrival-
process variability and the service-time variability. Hence, when the
service time is deterministic, so that cz, = 0, the departure process is
less variable than the arrival process. However, the actual reduction
of variability in a network caused by deterministic service times often
is not as great as predicted by (38) or (39). Hence, we have replaced
(39) by

We treat customer creation or combination as a modification of the
departure process. When there is customer creation at node i, we
replace each departure by a batch of size y;. When there is combination
at node i, we replace each interdeparture interval by the sum of yj;
such intervals. These make more sense for integer values, but we do
not require it. Hence, as described in Section 2.2, the departure rate
from node i is y;A; when the arrival rate is \;. We use the asymptotic
method to obtain the variability parameter. Since the number of
departures from node i in a large time interval is y; times the number
of arrivals, the asymptotic-method approximation of the variability
parameter for customer creation or combination is just to multiply
c3; by y;. (By the asymptotic method, c? = lim,_,.Var N(t)/EN(t); see
Section 2 of Whitt.’) This is done before splitting.

We obtain the basic system of equations (24) through (30) by
combining Sections 4.3 through 4.6 as follows:

The first line is based on superposition, Section 4.3, and the second
line is based on departure, splitting and customer creation, Sections
4.4 through 4.6.

Having calculated the rate and variability parameters associated
with each internal arrival process, we are ready to calculate the
approximate congestion measures for each node. At this point we have
decomposed the network into separate service facilities that are ana-
lyzed in isolation. Each facility is a standard GI/G/m queue partially
characterized by five parameters: the number of servers plus the first

two moments of the interarrival time and the first two moments of
the service time. Instead of the moments we use the arrival rate \, the
mean service time 7, and the squared coefficients of variation c? and
c2, Since we are focusing on a single node, we omit the subscript
indexing the node throughout this section.

There are many procedures that could be applied at this point.
We could fit complete distributions to the parameters, and then
apply any existing algorithm for solving a GI/G/m queue or a special
case. Among the attractive options are procedures for analyzing the
GI/G/1 queue,“* the M/PH/m queue with phase-type service-time
distributions,*”>” the GI/H;,/m queue with hyperexponential service-
time distributions™* and the GI/E,/m queue with Erlang service-
time distributions.” Also available are approximations based on
heavy-traffic and light-traffic limiting behavior.***’ The actual proce-
dures used in this version of QNA, however, are quite elementary. Our
study of the GI/G/1 queue***’ indicates that these elementary proce-
dures are consistent with the limited information available. Since the
arrival process is usually not a renewal process, and since only two
moments are known for each distribution, there is little to be gained
from more elaborate procedures. In fact, a user of QNA should be
cautioned not to rely too heavily on detailed descriptions such as the
tail of the waiting-time distribution. Such detailed descriptions may
prove to be reasonably accurate, but they should certainly be checked
by simulation.

We begin with the steady-state waiting time (before beginning
service), here denoted by W. The main congestion measure is the
mean EW, but we also generate an entire probability distribution for
W. First, the approximation formula for the mean is as in (2):

When c?2 < 1, (44) is the Kraemer and Langenbach-Belz approxima-
tion,*® which is known to perform well.*?°"°? When c?2 > 1, the original
Kraemer and Langenbach-Belz refinement does not seem to help, so

Let the number of customers in the facility, including the one in
service, be denoted by N. The probability that the server is busy at an
arbitrary time, P(N > 0), and the mean EN can be obtained from
Little’s formula (see Section 11.3 of Heyman and Sobel"*’):

Formula (46) is exact even for stationary nonrenewal arrival processes
and (47) is exact given EW.

For the probability of delay, P(W > 0), denoted here by oc, we use
the Kraemer and Langenbach-Belz approximation:*®

Formula (48) also yields the correct value for M/G/1 systems, namely,
p. Additional supporting evidence for (48) is contained in Whitt.

We next focus on the conditional delay given that the server is busy,
denoted by D. Obviously, ED = EW/c. We first give an approximation
formula for the squared coefficient of variation of D, ch. This formula
is the exact formula for the M/G/1 queue, with the service-time
distribution being H? when c? = 1 and E, when c2 = k™’, where H? is
the hyperexponential distribution with balanced means and E, is the
Erlang distribution (see p. 256 of Cohen® and Section 3 of Whitt").
The idea underlying this approximation is that the conditional delay
D in a GI/G/1 queue (rather than the total delay W) depends more
on the service-time distribution than on the interarrival-time distri-
bution. Hence, the M/G/1 formula for cj is used as an approximation
for all GI/G/1 systems. The M/G/1 formula for cj is:

where d? = E(v")/(Ev)? with v being a service-time random variable.
Even E(v*) is available, it can be used in (50), but since E(v?) is not
available with two parameters, we use approximations for d?. The
approximations are based on the H8 and E* distributions.

We obtain formula (52) by considering an Erlang EF, variable, which
can be represented as the sum of k iid exponential random variables
X; with mean 7/k, where 7 is the mean of the E; variable. In this case

which reduces to (52) because c2 = k™ for an E, variable. Note that
(51) and (52) agree at the boundary when c? = 1.

Var(D) = (ED)*c}, = (EW)?cd/o?
E(D’) = Var(D) + (ED). (53)
From D we then obtain second-moment characteristics for W:
E(W*) _ , oH) _ , cb tle
(EW)? (cED)? oO i
Var(W) = (EW)*c and E(W?’) = Var(W) + (EW)?. (54)

We now indicate how QNA calculates an approximate probability
distribution for W. The distribution has an atom at zero as given in
(48) and a density above zero. The density is chosen so that W and D

have the first two moments already determined for them. (This is the
general rule, but it is not quite followed in Cases 2 and 4 below.)

Case 3: 0.501 < c?% < 0.99. Let the distribution of D be the convo-
lution of two exponential distributions with parameters y; and y2
(v1 > ¥2), i.e., let D have density

Case 4: ch < 0.501. Let D have an E, (Erlang) distribution with
mean ED, which has c” = 0.5. Its density is

For deterministic service times, d3 = 1, so that the smallest possible
ch via (50) is (1 + 2p)/3. Hence, Case 4 above will not occur often.

Finally, we come to the second moment and variance of N, the
number in system. For the M/G/1 queue, it is not difficult to compute
E(N?). Since the steady-state number in the facility is equal to the
number of arrivals during a customer’s time in the facility, it is easy
to compute the moments of N from the moments of W; for example,

and o is the probability of delay in (48). The maximum is used in (65)
to avoid dividing by zero. For the M/G/1 queue, Y2 = 1; for GI/M/1
queues, Y2 in (65) provides just the right correction, so that (64) is
exact given the true value of o, EW and Var(W). The correction Y2
in (65) makes (64) too small for D/D/1 queues by a factor of (1 + p)7?,
but (64) is asymptotically correct in heavy traffic: ci) ~ 1 as p > 1 if
either c2 > 0 or c2>0.
From (47) and (64) we immediately obtain

When there is immediate feedback at the node and it is eliminated,
adjustments are necessary in the formulas of this section, as indicated
in Section III.

The first congestion measures for multiserver nodes provided by
QNA are exact. Even for nonrenewal stationary-arrival processes, the
expected number of busy servers is just the offered load [see p. 400 of
Heyman and Sobel*! and (4.2.3) of Franken et al.*”}:

E min{N, m} = a = At (67)
and the traffic intensity or utilization is
p = a/m. (68)
By Little’s formula, as in (47),
EN =a+dEW. (69)

QNA currently provides only a few simple approximate congestion
measures for multiserver nodes. These are obtained by modifying the
exact formulas for the M/M/m model.’® Let characteristics such as
EW<(c?2, c2, m) represent the characteristic as a function of the param-
eters c2, c2, and m, and let characteristics such as EW(M/M/m) be
the exact value for M/M/m system. A simple approximation for EW

EW(c?, c2, m) = ( EW(M/M/m). (70)
Formula (70) has frequently been used for M/G/m queues and is
known to perform quite well in that case. ©’ By virtue of heavy-traffic
limit theorems, we know that (70) is also asymptotically correct for
GI/G/m systems as p — 1 for fixed m. Limited additional study
indicates that (70) is also reasonable for moderate values of p when
c2 > 0.9 and c2 = 0.9, or when c2 < 1.1 and c? < 1.1. The actual
value may be significantly smaller (larger) when c2 < 0.9 and c2> 1.1
(c2 > 1.1 and c2 < 0.9).

The simple approximation (70) is also supplemented by simple
approximations for the second moments of W and N. They are
obtained from:

More detailed and sophisticated approximations for multi-
server nodes are being studied. As we indicated before, a variety of
methods and algorithms can be applied given the parameters of the
arrival process.*”7*”

In this section we describe the approximate congestion measures
calculated by QNA for the network as a whole. In Section 6.1 we
discuss congestion measures representing the system view, e.g.,
throughput and number of customers in the network; in Sections 6.2
and 6.3 we discuss congestion measures representing the customer
view, e.g., number of nodes visited and response times. In fact, there
are actually two different customer views. In Section 6.2 we discuss
the view of an arbitrary, typical, or aggregate customer; in Section 6.3
we discuss the view of a particular customer with a specified route
through the network.

A basic total network performance measure is the throughput, which
we define as the total external arrival rate Xo,

When no customers are created at the nodes, the total external arrival
rate equals the total departure rate from the network, so that there is
little ambiguity about what we mean by throughput. However, when
customers are created or combined at the nodes, as in Section 2.2,
there is more than one possible interpretation. We might be interested
in the rate at which arrivals are processed, i.e., (72). For example, the
customers created at the nodes might be regarded only as extra work
that must be done to serve the arrivals. On the other hand, we might
be interested in the rate at which customers leave the network or in
the rate of service completions. The departure rate from the network
is

A description of the overall congestion is provided by the mean and
variance of the number N of customers in the entire network. In
general,

and, as an approximation based on assuming that the nodes are
independent, we have

Formula (76) is valid for the Markovian models as a consequence of
the product-form solution, but is an approximation in general.

When we turn to the congestion experienced by individual cus-
tomers, there are two very different approaches. The first approach
keeps strict adherence to the model assumptions with the standard
input in Section 2.1, and is based on interpreting the routing matrix
as independent probabilities (Markovian routing). This means that
each time any customer completes service at node i, that customer
proceeds to node j with probability q,, independent of the current state

and history of the network. If the network is cyclic, this means that
every customer has positive probability of visting some nodes more
than once. This is the perspective of an aggregate customer. It might
be that no individual customer actually ever visits the same node more
than once.

If the aggregate view is desired, then the customer experience can
be described by employing the basic theory of absorbing Markov chains
as in Chapter III of Kemeny and Snell. We can regard the external
node as a single absorbing state to which all customers go when they
leave the network or we can have more absorbing states, to distinguish
between network departures from different nodes or different subsets
of nodes. For this interpretation, the routing matrix Q is the transient
subchain associated with the absorbing Markov chain and the inverse
(I — Q)7 in (19) with I = J is the fundamental matrix of the absorbing
chain (see p. 45 of Kemeny and Snell®). Solving the traffic-rate
equations is tantamount to solving for this fundamental matrix.

From the fundamental matrix it is easy to calculate the moments of
the number n,; of visits to any state j starting from any state i (on an
external arrival process). For example, En, is just the (i, j)-th entry
of (I — Q)~". It is also easy to calculate the probability of absorption
into each of the absorbing states starting from any initial distribution.
These various congestion measures are easily obtained working with
n X n matrices.

Suppose that we focus on an arbitrary, typical, or aggregate customer
arriving on an external arrival process. Then that customer enters
node i with probability Xoi/Ao, where Ao is defined in (72) and the
expected number of visits to node i for each customer is

(We have used the fundamental matrix to get ),.) The mean of the
time, 7;, that an arbitrary customer spends in node i during his or her
time in the network is thus

ET; a (EV;) (7; + EW;) (78)
and the expected total sojourn time (time spent in the network from
first arrival to final departure) for an arbitrary customer is thus

ET = >; ET; = > EV;(7; + EW;). (79)
i=1 i=1
The variance of 7; is thus
Var(7’;) = EV«Var(W;) + ric2;) + Var(V;)(EW; + 7;)*. (80)
The term Var(V;) in (80) as well as EV; is easily obtained from the
fundamental matrix. In particular, Var(V;) = EV? — (EV;)? and

where F is the fundamental matrix (I — Q)’, Fag is the n X n matrix
with all off-diagonal entries 0 and diagonal entries the same as F.

To obtain an approximation for the variance of the total sojourn
time in the network, we assume that the sojourn times at the different
nodes are conditionally independent, given any particular routing.
(This is not valid even for all acyclic networks of M/M/1 nodes,®? but
is often approximately true.””°) In particular, for a customer entering
some specified node and making V; visits to node j, 1 <j <n, before
eventually leaving the network,

nV;
T= (3 > Ty), (82)
j=l k=1
where T}, is the sojourn time for the kth visit to node j. Our approxi-
mation assumption is that the variables T;; are mutually independent
given the vector (Vi, Vo, ---, Vn).
Hence,
2

Another approach is to decouple the macroscopic and microscopic
interpretations. This view is common in statistical mechanics. The
total network may exhibit statistical regularity not evidenced in any
single particle (customer). In this view, we think of the total system
evolving as if customers were routed according to independent proba-
bilities, even though individual customers may have very different

routing probabilities, perhaps nonrandom routing or acyclic routing.
For example, we may consider the cyclic network entirely appropriate
for the macroscopic view even though no individual customer ever
visits any node more than once. In order for this view to be realistic,
each individual customer should have a relatively negligible effect on
the total network.

The procedure here is to solve for the equilibrium or macroscopic
behavior of the network first and then afterwards consider particular
customers. The particular customers will have their own routes
through the network and perhaps their own service times at the nodes
along the way. There are two cases, depending on whether the input
is by classes and routes, as in Section 2.3 or the standard input as in
Section 2.1.

First, suppose that we are using the input by classes and routes in
Section 2.3. Then the particular customers correspond to the customer
classes specified in the input. Hence, each customer has a deterministic
route through the network and possibly special service times at the
nodes on the route. In this case, as described in Section 2.3, QNA first
converts the input by classes and routes into the standard input in
Section 2.1. Then QNA solves for the equilibrium behavior. Finally,
congestion measures are calculated for the different classes under the
assumption that they follow their originally specified special routes
and that upon arrival at the nodes on the route they see independent
versions of the equilibrium state of the network. Hence, in the notation
of Section 2.3, for a customer in class k, the expected total service
time is

and the expected total sojourn time or response time is the sum of
(85) and (86). Similarly, for a customer in class k the variance of the
total service time is

With the standard input in Section 2.1, the user must specify the
particular customers to be analyzed. In this case, the user specifies
classes with routes and possibly service times (rate and variability
parameters), but these data are not used in calculating the equilibrium
behavior. The decoupling principle is used with greater force here;
there need not be any consistency between the microscopic and mac-
roscopic views: This additional input does not affect the equilibrium
behavior of the total network.

In the current version of QNA the individual customer routes are
deterministic, so that the additional input required is just as in Section
2.3 and the congestion measures are just as in (85) through (88) in
Section 6.3.1. However, it is possible to modify QNA to allow random
routes. Then the additional input would be just as in Section 2.1; for
each class it would consist of a routing matrix plus parameters for the
arrivals process and service times.

The software package QNA was written by Anne Seery. She used a
subroutine for analyzing the M/M/m queue written by Shlomo Halfin.
It has been a pleasure collaborating with Anne Seery on this venture.
I also appreciate the help from many other colleagues and the contin-
ued support of my management: W. A. Cornell, C. S. Dawson, J. C.
Lawson, C. J. McCallum, Jr., M. Segal, R. E. Thomas, and E. Wolman.

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This paper describes the performance of the Queueing Network Analyzer
(QNA), a software package developed at Bell Laboratories to calculate ap-
proximate congestion measures for networks of queues. QNA is compared with
simulations and other approximations of several open networks of single-
server queues. This paper illustrates how to apply QNA and indicates the
quality that can be expected from the approximations. The examples here
demonstrate the importance of the variability parameters used in QNA to
describe non-Poisson arrival processes and nonexponential service-time dis-
tributions. For these examples, QNA performs much better than the standard
Markovian algorithm, which does not use variability parameters. The accuracy
of the QNA results (e.g., the expected delays) in these examples is satisfactory
for engineering purposes.

This paper is a sequel to Whitt,’ which described the software
package called the Queueing Network Analyzer (QNA). QNA calcu-
lates approximate congestion measures for networks of queues. The
first version of QNA treats open networks of multiserver queues with
the first-come, first-served discipline and no capacity constraints.
QNA is designed to treat non-Markovian models: The arrival processes
need not be Poisson and the service-time distributions need not be
exponential. QNA approximately characterizes other kinds of varia-
bility through variability parameters assigned to each arrival process
and each service-time distribution. The first step in the algorithm is

to solve for the flow rates and the variability parameters of the internal
arrival processes. The second step is to compute approximate conges-
tion measures for each queue separately by regarding it as a standard
GI/G/m queue in which the renewal arrival process and the service-
time distribution are each partially characterized by their first two
moments or, equivalently, the rate and variability parameters. The
third and final step is to calculate congestion measures for the network
as a whole.

This paper describes the performance of QNA by comparing it with
simulations and other approximations of networks of queues. Even
though QNA can analyze multiserver queues, only single-server queues
are considered here. Among the other approximations in each case are
the M/M/1 and M/G/1 approximations, which can be obtained from
QNA by using default options. The M/M/1 approximation, which is
embodied in the Markovian algorithms, is obtained by setting all
variability parameters equal to 1. With the M/M/1 approximation,
the nodes are treated as independent M/M/1 queues with the correct
rates. The M/M/1 approximation yields the exact equilibrium distri-
bution of queue lengths for the Markov model with Poisson external
arrival processes, exponential service-time distributions and one cus-
tomer class. The M/G/1 approximation is obtained by setting the
variability parameter of each arrival process equal to 1 and using the
specified service-time variability parameter c2; then the expected wait-
ing time at each node is (1 + c3)/2 times the M/M/1 value.

The congestion measures we consider in the examples here are the
expected waiting time (before beginning service) and the expected
sojourn time (waiting time plus service time) at a node or in the entire
network. Of course, QNA produces other congestion measures, but we
are comparing with previously published simulation results, which are
mostly limited to expected waiting times and sojourn times.

We begin in Section II with a single GI/G/1 queue and discuss the
implications of previous work on approximations for the GI/G/1
queue.”” In Section III we consider a single queue with a superposition
arrival process and compare QNA with simulations by Albin.®”° In
Section IV we consider a network of eight queues in series analyzed
by Fraker,'’ and in Sections V and VI we consider two networks
analyzed by Kuehn:’” Section V treats a tightly coupled two-node
network and Section VI treats a nine-node network. In Section VII
we treat a five-node network used to model a Bell Laboratories
‘computer system. Finally, in Section VIII we consider a model from
Gelenbe and Mitrani?® for a packet-switched communication network.
The examples in Sections VII and VIII have input by classes and
routes as in Section 2.3 of Whitt.’

expected in applications of QNA. They also demonstrate the impor-
tance of the variability parameters when the external arrival processes
are not nearly Poisson or the service-time distributions are not nearly
exponential. These examples also illustrate how to apply QNA, e.g.,
to model superposition arrival processes (Section III), to eliminate
almost immediate feedback (Section V), and to conduct sensitivity
analyses for the variability (Section VII).

We begin by considering the special network containing a single
service facility, in particular, the GI/G/1 queue with service times and
interarrival times each partially characterized by their first two mo-
ments or, equivalently, by the four parameters 7 (the mean service
time), c2 (the squared coefficient of variation of the service time),
(the arrival rate), and c?2 (the squared coefficient of variation of the
interarrival time).! The subscript indexing the node is suppressed
since there is only one node.

It is useful to consider this model because it has been extensively
studied and is relatively well understood. In many cases we can
analytically determine the quality of the approximations for the
GI/G/1 queue. Hence, we can get an idea about the quality of the
approximations for more general networks. Of course, approximations
for a node in a general network might be worse because the internal
arrival processes usually are not actually renewal processes. On the
other hand, the network operations of superposition and splitting tend
to make stochastic point processes more like Poisson processes, so
that larger networks may actually be better behaved.

The model specification above determines the approximate conges-
tion measures produced by QNA, but the model specification is not
complete since there are many service-time and interarrival-time
distributions with the given parameters. The formulas produced by
the QNA are approximations for all these systems, so it is natural to
ask how the approximate congestion measures compare to the set of
all possible values that are consistent with the partial specification.
Fortunately, it is often possible to identify the set of all possible
values.?° Moreover, it is often possible to locate the more likely values
by identifying the set of all possible values under various natural
constraints on the distribution. When this cannot be done exactly, it
can often be done approximately using bounds.®

We now give a brief summary of results evaluating approximations
for the expected waiting time or, equivalently (by Little’s formula”),
the expected queue length in the GI/G/1 queue based on the four
parameters X, c+, 7, and c2. First, recall that for the M/G/1 queue,
with Poisson arrival process (c2 = 1), the expected waiting time

actually depends on the service-time distribution only through the two
parameters 7 and c2. Nonexponential interarrival-time distributions
tend to be more difficult, however. For the GI/M/1 queue with an
exponential service-time distribution (c2 = 1), the expected waiting
time depends on the interarrival-time distribution beyond the param-
eters \ and c2. Given ) and c?2 in the GI/M/1 queue, the maximum
relative error (upper bound minus lower bound divided by lower bound)
in the mean queue length (number in system including anyone in
service) is exactly c2 (see Ref. 2). A similar, but somewhat less concise,
result holds for the expected waiting time by virtue of Little’s formula.
The maximum relative error for the expected sojourn time (waiting
time plus service time) is also c2. This result suggests more generally
that the reliability of the approximations might decrease when c?
increases, which is consistent with numerical experience.

If we assume that the interarrival-time distribution is not too
irregular, then the maximum relative error becomes much less. In the
H,/M/1 queue with a hyperexponential interarrival-time distribution
(mixture of exponential distributions having c2 > 1), the maximum
relative error in the mean queue length is (c2 — 1)/2 (see Ref. 4).
It turns out that the extremal interarrival-time distributions for
H,/M/1 queues also are extremal for all interarrival-time distributions
that have Increasing Mean Residual Life (IMRL) (also c2? > 1) and all
service-time distributions,’ so that the maximum relative error in the
mean queue length for IMRL/G/1 queues is (c2 — 1)/[2 + p(c2 — 1)].

Other kinds of shape constraints for the GI/M/1 queue have been
investigated by means of nonlinear programming.’ In general, we
conclude that if the distributions are not irregular, then the maximum
relative error in the GI/G/1 queue might be about 0.05 c2, e.g., about
10 percent when c2 = 2.0.

From heavy-traffic limit theorems that describe the queue as p >
1,° where p = dz is the traffic intensity, we know that asymptotically
the queue length and waiting-time distributions depend on the inter-
arrival-time and service-time distributions only through the four pa-
rameters X, c2, 7, and c2. This suggests that more generally the quality
of the approximations might improve as p increases. This is certainly
consistent with experience for the GI/G/1 queue, but not necessarily
for more complex networks, e.g., the tightly coupled two-node network
here in Section V.

The heavy-traffic limit theorems are closely related to diffusion
approximations because diffusion processes emerge as limits in the
heavy-traffic limit theorems. We have recently compared various
diffusion approximations for the expected waiting time in a GI/G/1
queue to known bounds.® We now show how QNA and other related
approximations fit into this framework. Table I here compares four

Table |—Bounds and approximations for the expected waiting time,
EW, in a GI/G/1 queue: Three cases

Notes: 1. In each case the mean service time is 7 = 1.
2. “H” indicates high (greater than or equal to) and “L” indicates low (less than
or equal to) in comparison with the bounds.

approximations for the expected waiting time, EW, with various upper
and lower bounds in the three cases in Table 1 of Ref. 6. The four
approximations are the M/M/1, M/G/1, Kraemer and Langenbach-
Belz,’ and QNA.

The M/M/1 approximation is obtained by replacing both variability
parameters c% and c2 by 1. The M/M/1 approximation is produced by
a direct application of the Markovian software packages. The M/G/1
approximation is obtained by replacing c2 by 1 and using the specified
value of c2. The M/G/1 approximation EW is the exact value for the
approximating M/G/1 system since EW depends on the service-time
distribution only through its first two moments. Both the M/M/1 and
M/G/1 approximations are produced by QNA using default options.

The QNA approximation is the Kraemer and Langenbach-Belz
approximation when c3 < 1 and is slightly greater when c2 > 1.' The
one case in which c2 > 1 in Table I shows that the difference between
the two approximations is small compared to the distance between the
upper and lower Monotone Failure Rate (MFR) bounds.®> The MFR
bounds are for interarrival-time distributions with decreasing failure
rate when c2 = 1 and increasing failure rate when c2 < 1. The MFR
bounds are tight when c2 > 1 but not when c2 < 1 (see Ref. 5). Since
the interarrival-time distribution need not have monotone failure rate,
the range of all possible values is greater, but the MFR bounds indicate
the more likely values.

For the three cases in Table J, the M/M/1 approximation performs
very poorly, falling way outside the bounds. The M/G/1 approximation
always coincides with one of the MFR bounds—the upper bound when
c2 < 1 and the lower bound when c2 = 1—but it would be better to
have an approximation somewhere in the middle between the bounds.
When c2 = 1, the QNA approximation is a convex combination of the

two MFR bounds.° Since the MFR bounds are tight when c2 = 1, the
QNA approximation always yields the exact value of EW for some
GI/G/1 system with the given parameters.’ When c2 < 1, the QNA
approximation is slightly less than the convex combination of the
MFR bounds. The convex combination of the MFR bounds is known
to be an upper bound for E,/G/1 systems,” so that it is appropriate to
use a smaller value. We conjecture that the QNA approximation always
yields an exact value of EW for some GI/G/1 system with the given
parameters when c2 < 1 too.

Table I provides a sample of the comparisons possible using the
previous studies.” © Since QNA coincides with the Kraemer and Lan-
genbach-Belz approximation when c2 < 1 and the Sakasegawa-Yu
approximation when c2 = 1, previous comparisons such as Tables 13
and 14 in Klincewicz and Whitt? also apply to QNA.

In this section we consider one single-server queue with a superpo-
sition arrival process. Such a system with two component arrival
processes is depicted in Fig. 1a. Since only one external arrival process
at each node is allowed in QNA, this model cannot be analyzed directly,
but it is easy to modify the model so that QNA does apply. We added
dummy nodes with very low traffic intensity on each component arrival
process, as shown in Fig. 1b. Since the new dummy nodes have low

Fig. 1—(a) The original queue with a superposition arrival process. (b) The equivalent
network with one external arrival process at each node.

traffic intensity, the rate and variability parameters of the departure
processes from the dummy nodes will be almost identical to the
corresponding parameters of the external arrival processes.

This model has recently been studied quite extensively by Albin®”
and is now relatively well understood. As in Section I, here we consider
one illustrative example, which suggests the accuracy to expect more
generally and shows the importance of the variability parameters.

The specific model we analyze is the 2GI;/M/1 system with an
exponential service-time distribution and n iid stationary renewal
processes as component arrival processes. The total arrival rate is 1,
so the rate of each component process is n~'. We consider six cases
involving two values of n, n = 2 and 16, and three values of the traffic
intensity p, p = 0.3, 0.7, and 0.9. In each case the component renewal-
interval distribution is H2, i.e., the mixture of two exponentials with
balanced means: one with mean m, realized with probability p and the
other with mean mz, realized with probability 1 — p, where pm, =
(1 — p)mg. In each case the squared coefficient of variation of the
component renewal process interval is c? = 6. This is quite high
variability, so that the component processes are not nearly Poisson.
The three parameters of an Hp, distribution are determined by speci-
fying the mean as n, c? = 6 and balanced means.

This model is taken from Chapter 3 and Appendix 6 of Albin.®
Albin’s approximations are based on the rate and variability parame-
ters just as in QNA. In fact, the superposition approximation in QNA
is a modification of Albin’s procedure.' With Albin’s procedure, the
variability parameter of the superposition process is a convex combi-
nation of the variability parameters obtained from the stationary-
interval method and asymptotic method described in Whitt. The
specific implementation of the stationary-interval method in Whitt!”
and Albin® is part of Kuehn’s” algorithm for approximating networks
of queues. Since Kuehn’s implementation of the stationary-interval
method is nonlinear, a different procedure is used in QNA. In QNA
the variability parameter of the superposition process is a convex
combination of the variability parameters obtained from the asymp-
totic method and a Poisson process.’ Extensive experimentation has
shown, however, that the approximation in QNA is very close to
Albin’s hybrid approximation, which performed very well in many
experiments (about 3-percent average absolute relative percent error).

Several different approximations for the expected waiting time are
compared with simulation in Table II. The simulation results were
obtained in Albin.*® The sample standard deviation is given in paren-
theses below the simulation estimate in Table II to indicate the
statistical reliability of the estimate. The simulation program was
written in FORTRAN using the “Super-Duper” subprogram in Mar-

Table {I—A comparison of approximations and simulation of the
expected waiting time, EW, in a 2GI/M/1 queue with a superposition
arrival process

Notes: 1. The total arrival rate is 1 in each case.
The component renewal processes have H’ (hyperexponential) renewal-inter-
val distributions with mean n, c? = 6, and balanced means.

. The sample standard deviations appear below the simulation estimates in
parentheses.

: he relative percent error appears below the approximation values in paren-
theses.

saglia et al. to generate uniform random numbers.’* A different random
number seed was used for each simulation. The simulations began
with an empty system, but the first 1000 customers were not counted
to allow the system to approach steady-state. Each simulation con-
sisted of 20 batches, with the number of customers per batch depending
on the traffic intensity: 3,000 per batch for p = 0.3, 15,000 per batch
for p = 0.7, and 50,000 per batch for p = 0.9. Even though much more
simulation time was spent on the cases with higher traffic intensities,
the statistical reliability was slightly less.

In Table II, in parentheses below the approximation values are the
relative percent errors (RE), which are defined as

At the bottom of Table II are the average absolute relative percent
errors (ARE), which are defined as

6
ARE = YY |RE;\/6. (2)
i=l
Dividing by the simulation value perhaps inflates the errors when p =

The stationary-interval method and the M/M/1 approximation are
not bad for large n and small p because the superposition process
converages to a Poisson process as n — © and the queue reflects this
if p is not too big, in particular, if n(1 — p)? is sufficiently large."
However, for p = 0.9, these two methods perform poorly. On the other
hand, the asymptotic method performs reasonably well for p = 0.9,
but not well in other cases. In particular, the asymptotic method does
not reflect the convergence to a Poisson process as n — %; it gives the
same answers for n = 2 and 16. So, for fixed p, the asymptotic method
gets worse as n increases.

As Albin determined in extensive experiments,°*” her hybrid approx-
imation is much better than either basic method alone. This example
also illustrates how close QNA is to Albin’s hybrid procedure. For
queues with superposition arrival processes, we conclude that QNA
usually gives reasonable results and strongly dominates the two basic
methods.

In closing this section, we add a caveat. The component processes
in the simulation for Table II, in Albin’s hybrid procedure and in
QNA, are all based on the case of balanced means. However, as
discussed in Whitt,*° given the first two moments, the one-parameter
family of renewal processes with hyperexponential renewal-interval
distributions range from a Poisson process to a batch Poisson process
with geometrically distributed batch size. For a single renewal arrival
process, the expected waiting time, EW, in an H2/G/1 queue also
ranges between these same extremes, i.e., the M/G/1 and the M8/G/1
systems. Unfortunately, no related theory yet exists for superposition
arrival processes. However, we can easily describe what happens in
the two extremes. The superposition of independent Poisson processes
is Poisson and the superposition of independent batch Poisson proc-
esses with geometrically distributed batches having a common mean
is batch Poisson with geometrically distributed batches. Thus, we
conjecture that the maximum and minimum values for EW with a
superposition of iid Hj-renewal processes correspond to the M®/G/1
and M/G/1 systems, respectively. If the conjecture is true, then we
would have the same range of possible values for H»-superposition
arrival processes as for H2-renewal processes. However, if the compo-
nent processes are not too batchy, then the superposition process will
become more Poisson as n increases. We should usually expect the
superposition process to be more nearly Poisson as n increases. To
summarize, this heuristic analysis suggests that the range of possible
values for / W in the 2GI,;/G/1 queue given the basic parameters },
c2, T, c2 may be about the same as for the GI/G/1 queue. In fact, the

superposition operation may actually make the arrival process better
behaved.

In this section we apply QNA to a network of eight single-server
queues in series previously analyzed by Fraker.’! The external arrival
process is Poisson and all service-time distributions are Erlang. Fraker
considered eight cases involving four traffic intensities (9 = 0.3, 0.5,
0.7, and 0.9) and four Erlang service-time distributions (M = F,, Eu,
Es, and D = E..). Each of the traffic intensities and each of the service-
time distributions are assigned randomly to two of the eight nodes.
Fraker developed an approximation for these systems and compared
it with simulations.

Tables III and IV describe Fraker’s first two cases and the approx-
imations for the expected waiting time at each node. The service-time
squared coefficient of variation specifies the Erlang distribution since
c? = k7! for E,. Fraker made three simulation runs of 2500 customers,
discarding the first 500 in each case to damp out the transient effects
of starting the simulation. Statistics were collected for six blocks of
1000 customers each. Unfortunately, this is not enough to produce
very good accuracy, especially for the nodes with higher traffic inten-
sities. (Compare with the simulation length in Section III.) The
statistical reliability can be seen from the results of the six runs
displayed in Fraker.'! (These also appear in Appendix 1 of Whitt.'®)
An idea of the variability can also be seen from node 1 because all the
approximations except the M/M/1 approximation are exact for node
1. When p = 0.9 the length of a 95-percent confidence interval
approximately equals the estimated value; when p = 0.7 the length of

Table II—A comparison of approximations and simulation of the
expected waiting time at each node in Fraker’s model of eight
queues in series: Case 1

Approximation Methods
Squared ee ee
Coeffi- (Marko- (Asymp-  (Lag-1 QNA
Traffic cient of vian Net- _ totic Corre- ——————
Node Intensity, Variation, Simulated work) Method) lations)
No. pj eG Value M/M/1 M/G/1 Fraker EW; Cay

8 0.3 1/4 0.00 0.13 0.08 0.01 0.01 0.33
Note: The arrival process is Poisson with rate 1 and the service-time distributions are

Table IV—A comparison of approximations and simulation of the
expected waiting time at each node in Fraker’s model of eight
queues in series: Case 2

Table V compares the approximations with simulation for the nodes
with traffic intensity p = 0.7 in all eight cases. Since the approxima-
tions are exact for the first node, the first node is not included for the
cases in which p,; = 0.7 (Cases 1, 5, and 6). For the approximations,
the difference between the approximation value and the simulation
value is displayed.

Tables III through V show that QNA performs about the same as
Fraker’s approximation, which is based on lag-1 correlations and is
especially designed for queues with Erlang service times. Both these
approximations performed significantly better than the M/G/1 ap-
proximation, which in turn performs significantly better than the
M/M/1 approximation.

Additional analysis of Fraker’s models plus other queues in series is
contained in Whitt.’ The performance of QNA in these other cases
is consistent with the description here.

In this section we consider a two-node network analyzed by Kuehn?”
and Gelenbe and Mitrani.’® This network is depicted in Fig. 2. It has
one external arrival process, which comes to node 1. Customers com-
pleting service at node 1 leave the system with probability 1/2; other-
wise they go to node 2 and then back to node 1 to be served again. At
node 2 customers are immediately fed back to node 2 for another
service with probability q22, but in most cases qz2 = 0.

Table V—The expected waiting time at the nodes with p; = 0.7 in
Fraker’s eight cases of eight single-server queues in series

Fig. 2—Kuehn’s first example: A network of two queues with one external arrival
process.

three values of the external arrival rate for each case: 9, = 0.15, 0.30,
and 0.45. In each case the mean service time at node 1 is 7; = 1 and
the transition probability from node 1 to node 2 is giz = 1/2, so that
the traffic intensity at node 1 is p: = 2A. For node 1 these three
external arrival rates correspond roughly to light traffic (p; = 0.3),
moderate traffic (p; = 0.6), and heavy traffic (p; = 0.9). In Cases 2 and
3 the traffic intensity at node 2 is the same as at node 1, 1.e., po = 2Xo1,
but in all other cases it is pe = Ao. In these other cases node 2 is
always in relatively light traffic. The external arrival process is always
a renewal process. Each interarrival-time distribution and service-
time distribution is one of four distributions: deterministic (D with c?

The results are described in Tables VII and VIII. The simulation
results and Kuehn’s approximation are taken from Kuehn.” Two
different approximation results are given for QNA in Table VII. The
first column is the standard application of QNA with the network
reconfigured to eliminate immediate feedback in the one case it occurs,
at node 2 in Case 3. (See Section 3 of Ref. 1.) The final column of
Table VII is an adjusted version of QNA to eliminate almost immediate
feedback, which we discuss below.

For this network the quality of the standard QNA approximation is
about the same as Kuehn’s approximation. They both work well for
low and moderate traffic intensities, e.g., about 10-percent average
absolute relative percent error when p,; = 0.6 (Table VIII), but not so
well in heavy traffic. This two-node network presents an obvious
difficulty for QNA. The network is tightly coupled so that many
departures from node 1 rapidly return to node 1 for additional service.
However, since these returning customers first pass through node 2,
there is no immediate feedback, so that QNA does not reconfigure the
network to eliminate the feedback. Nevertheless, it is evident that this
almost immediate feedback for node 1 is very similar to immediate
feedback and the potential exists for better results by reconfiguring
the network to eliminate this feedback too.

In all cases except 2 and 3, almost immediate feedback is eliminated
by applying the standard version of QNA with immediate feedback
elimination twice. The first time we apply QNA to the full network
and the second time we apply QNA with node 2 removed. When node

Notes: 1. In each case 7; = 1 and qy2 = 1/2.
2. In each case the arrival rate assumes one of three values: Ao = 0.15, 0.30, and
0.45,

Table VII—A comparison of approximations and simulation of the
expected total sojourn time (waiting time plus service time) in
Kuehn’s two-node network

Simulation
(with 95-Per- Approximation Methods
External cent Confi-
System _ Arrival dence Inter- QNA
No. Rate, dor vals) M/M/1 M/G/1 Kuehn QNA _ Adjusted

Notes: 1. In each case the traffic intensity at node 1 is p; = 2Ao1.
2. In Cases 2 and 8 the traffic intensity at node 2 is pe = 2o1; otherwise it is
p2 = Xo.

2 is removed, the feedback to node 1 becomes immediate and the
network is reconfigured by QNA to eliminate it. We use the second
run with node 2 removed to determine the expected waiting time per
visit at node 1. We use the first run to determine the expected number
of visits to node 1 and the expected total sojourn time at node 2.

We do not treat nodes 1 and 2 symmetrically in Cases 1 and 4
through 8 because p; = 22 so that pe is relatively small compared to
pi. Customers that return to node 1 via node 2 will not be delayed long
at node 2 before coming back, but customers returning to node 2 via
node 1 will be delayed relatively longer before coming back. If we had
f1 < pz, we would remove node 1 in the second run of the QNA and
focus instead on node 2.

In Cases 2 and 3 the traffic intensities at nodes 1 and 2 are equal,
so the motivation for eliminating almost immediate feedback is less.
What we have done for Table IV is first calculate the congestion

Table VIII—A comparison of approximation methods in Kuehn’s
two-node network with Ao; = 0.3 (p; = 0.6): The average absolute
relative percent error in the expected total sojourn time compared

measures for node 1 via the second run of QNA with node 2 removed
as before. Then we use the results for node 1 to approximate the
variability parameter of the arrival process to node 2. Finally, we
analyze node 2 in isolation with the correct rates and this approximate
arrival variability parameter. This works slightly better than the first
procedure, but neither works well.

The results demonstrate that the standard version of QNA performs
relatively well in Cases 2 and 3 when p; = po. The adjustment to
eliminate almost immediate feedback yields a significant improvement
when p; > pe, but the results after adjustment to eliminate almost
immediate feedback are much worse when p; = pz.

As a refined procedure for this two-node network, we suggest elim-
inating almost immediate feedback at the node with higher traffic
intensity when the traffic intensities differ significantly, and using the
standard QNA algorithm otherwise. The refined procedure in Table
VIII is standard QNA in Cases 2 and 3 and the adjusted QNA in all
other cases.

Table VIII displays the relative percentage errors for all the approx-
imations for the eight cases with Ao: = 0.3 (p: = 0.6). The refined
procedure in the last column yields very good results. Table VIII also
demonstrates that the standard version of QNA is significantly better
than the M/M/1 approximation, but not uniformly better. In some
cases, e.g., in Case 8, errors in opposite directions can cancel for the
M/M/1 approximation.

The improvement from eliminating almost immediate feedback “by
hand” suggests that it would be desirable to develop an automatic
procedure for eliminating almost immediate feedback to incorporate

in QNA, and this is being investigated. It also indicates the potential
for “tuning” QNA for particular applications.

We now consider Gelenbe and Mitrani’s’® experiment, which con-
sists of five cases. The network is as depicted in Fig. 2 except the
routing probability qi2 is not exactly 1/2. The parameter values are
given in Table [X and the results in Table X (pp. 137, 138 of Gelenbe
and Mitrani’*). We only include the best of three approximation
schemes discussed by Gelenbe and Mitrani. Unfortunately, Gelenbe
and Mitrani provided no information about the statistical reliability
of the simulation estimates. Since p; and p2 are nearly equal in each
case, we did not try to eliminate almost immediate feedback.

The M/M/1 approximation values are obviously much too large
because the M/M/1 approximation does not reflect the low variability
of the service times. The M/G/1 approximation is much too low at
node 2 because it does not benefit from the feedback elimination
procedure. The Gelenbe-Pujolle procedure is better than the M/G/1
procedure, but not uniformly so. The QNA approximation is clearly

_ Table IX—The parameter values for Gelenbe and Mitrani’s
experiment with the two-node network in Fig. 2

Table X—A comparison of approximations with simulations: The
expected number of customers at each node in the two-node
network in Gelenbe and Mitrani'?

the best, but it may underestimate the congestion for high traffic
intensities.

We now consider a nine-node network analyzed by Kuehn,’ which
is depicted in Fig. 3. The mean service time at node j is 7; = 1 for each
j. There are three external arrival processes with \o; = 0.5 for each j;
these come to nodes 1, 2, and 3. Kuehn let the three external arrival
processes be Poisson processes. As in Section V, all service-time
distributions are D, E,, M, or H’. Kuehn considered two cases:
homogeneous servers, in which all the service-time distributions are
identical, and heterogeneous servers, in which nodes 1 through 3 have
one service-time distribution and nodes 4 through 9 have another.

Kuehn compared his approximation with the M/M/1 and M/G/1
approximations and simulation for service-time variability parameters
ranging from c2, = 0 to 4. He focused on the expected total sojourn
time (waiting time plus service time) in the network and the expected
sojourn time per visit in node 4. For this network Kuehn found, first,
that the service-time variability parameters are significant (the so-
journ-time measures increase significantly with c2,); second, that his
approximation tracks the simulation well; and, third, that the M/G/1
approximation also works well, but not quite as well as his approxi-
mation.

We obtained similar results applying QNA. The QNA approxima-
tion values are indistinguishable from the approximation values dis-
played graphically by Kuehn, which are consistently within the sim-

Fig. 3—Kuehn’s second example: A network of nine queues with three Poisson
external arrival processes.

ulation confidence intervals. In Table XI we display some of our
results. Here we consider only the case of homogeneous servers in
which c?; = 0, 1, or 4. However, we let the variability parameter of all
external arrival processes by cj; = 0.25, 1.0, or 4.0. We thus obtain
nine cases.

Kuehn did simulations only in the case cé; = 1. The different values
obtained by QNA when cj; # 1 suggest that the congestion measures
in this model are more sensitive to the variability of the service times
than the variability of the interarrival times. Of course, we should
expect that the M/G/1 approximation might perform well when the
variability parameters of the external arrival processes are 1 or close
to 1, but for relatively large and interconnected networks the M/G/1
approximation may perform well for other external arrival processes.
It performs reasonably well here when cj; = 0.25 or 4.0.

In this section we apply QNA to a network with input by customer
classes and routes as in Section 2.3 of Ref. 1. We compare QNA to a
simulation model used in the development of a computer system at
Bell Laboratories. In this model there are five nodes and two customer
classes.

The customer classes correspond to typical functions performed by
the system. The route for each class represents a typical sequence of
operations performed by the system to process one of these functions.

Table XI—Approximations of expected sojourn times
in Kuehn’s nine-node network in Fig. 3: The case of
homogeneous servers, cz identical for all j

In the simulation model, both the routes and the service times at the
nodes are deterministic for each class. Hence, the input for QNA is
just as specified in Section 2.3 of Ref. 1 with the service-time variability
parameters set equal to 0, i.e., c2,; = 0 for each class k and j on class
k’s route. The two routes we consider are given in Tables XII and

XIII. Note that node 1 frequently appears several times in succession,
so that there is immediate feedback at node 1. Also note that the
service times differ at different visits to the same node.

The customer classes arrive according to independent Poisson proc-
esses. In the case we consider the arrival rates of Classes 1 and 2 are
0.00015278 and 0.00030555, respectively.

The QNA, M/G/1, and M/M/1 approximations are compared with
simulation in Table XIV. The simulation values are the average of
three separate runs. The values from these separate runs are displayed
to give an idea of the statistical reliability. The congestion measures
compared are the expected waiting times at the nodes and the expected
total waiting time (excluding service time) on three route segments.
The first segment is the first 25 nodes of the second route; the second
segment is the first 21 nodes on the first route; and the third segment
is eight nodes from node 23 to node 30 on the first route. In Table
XIV the waiting times at the nodes are measured in milliseconds while
the waiting times on the route segments are measured in seconds.

From Table XIV, it is apparent that QNA with immediate-feedback
elimination performs reasonably well, significantly better than the
M/M/1 and M/G/1 approximations. Since the approximating varia-
bility parameters of the arrival processes are very close to 1, QNA
without immediate-feedback elimination is very similar to the M/G/1
approximation. Hence, again we see that eliminating immediate feed-

Section VII
Mean Mean Mean
Service Service Service
Number Node Time Number Node Time Number Node Time

Table XIV—A comparison of approximations and simulation of the
expected waiting times for the model of Section VII

Expected Waiting Time at the Total Expected Waiting
Nodes Time on a Route Segment

Notes: 1. The relative percent errors appear below the approximation in parentheses.
2. The value ci, = 0.47 at node 1 is before adjustment for feedback; after

adjustment it is 0.79.
3. The units of measurement are milliseconds for the nodes and seconds for the
route segments.

back helps. The M/M/1 approximations at nodes 3 and 4 evidently
are too large because the service times are nearly constant. However,
at node 1 the M/M/1 approximation does pretty well, apparently
because two different errors cancel. (We have not displayed the relative
percentage errors at node 2 since it seems of little consequence because
of the low traffic intensity.)

It is interesting to know how the system would perform if the
external arrival processes are not Poisson and if the service times at
the nodes on the routes are not deterministic. With QNA we can easily
perform such sensitivity analyses. We can simply change the variabil-
ity parameters of the external arrival processes and the service times
on the routes. The results of such a study are given in Tables XV and
XVI. Table XV gives the approximating variability parameter of the

arrival process at each node as a function of the external arrival
process c? and the service time c’. It is assumed that both external
arrival processes have the same c?” and that all service times on both
routes have the same c*. From Table XV, it is evident that the
variability of the external arrival process hardly has any effect. Thus,
QNA predicts that this model, and perhaps the system itself, will be
robust to changes in the variability of the arriving traffic. The varia-
bility from outside is evidently dissipated on the long routes through
the network.

Table XV—The approximate variability parameter of the arrival
process at each node determined by the QNA, as a function of the
given variability parameters: The model of Section VII

Table XVI—The approximate service-time variability parameter cj
and mean delay EW; at node j determined by the QNA, as a function
of the variability of each service time on the route: The model of
Section VII

In this section we consider a model of a packet-switched communi-
cation network analyzed in Section 4.3.1 of Gelenbe and Mitrani.*
The basic model has 5 switching nodes and 12 one-way data links, as
depicted in Fig. 4. However, in this model each data link is a server
and the packets waiting for transmission on the link form the queue.
Packets are assumed to arrive at the switching nodes according to
independent Poisson processes. Each packet arriving from outside at
node i has final destination j with probability d;. Each packet with
destination j goes next to node rj from node i. Hence, there is a fixed
route for each origin-destination pair.

We analyze this network using the input by classes and routes in
Section 2.3 of Ref. 1. However, unlike Section VII, here the service-
time parameters are associated with the nodes rather than the routes
(which is an input option in QNA). The network of queues has 12
nodes with one server at each node and 20 routes. As specified by
Gelenbe and Mitrani, the service rate at nodes 1, 2, 7, 8, 11, and 12 is
4,8 (in thousands of bits per second) and the service rate at the other
nodes is 48. Since packet lengths are assumed constant, c2; = 0 for all
j.

For this example the matrices D = (d,;) of destination probabilities
and R = (r;) of next-node routes are:

Fig. 4—Gelenbe and Mitrani’s model of a packet-switched network: The 12 links are
the nodes in the network of queues.

The input data for the routes are given in Table XVII. These data
are obtained from the matrices D and R plus the external arrival rates
of 6.00, 8.25, 7.50, 6.75, and 1.50 at the five switching nodes (p. 141 of
Gelenbe and Mitrani”*).

The results are compared with Gelenbe and Mitrani’s approximation
and simulation in Table XVIII. Since only 6000 packets reached their
destination in the simulation, the statistical reliability of the simula-
tion estimates cannot be very good, cf. Section III. Our analysis is
revealing. First, Gelenbe and Mitrani describe their results as average
buffer queue lengths, which might be thought to exclude the customer
(packet) being served (transmitted). However, the results obviously
include the customer in service. Second, the two M/M/1 approxima-
tions should agree, but they do not. Evidently, the arrival rates at
switches 1 and 2 were not actually as cited in the text.’’ Hence, the
numbers for the one heavily loaded link, link 2, cannot be meaningfully
compared.

It is useful to consider the expected number waiting excluding the
customer in service. This is easily obtained because the probability
that the server is busy is exactly the traffic intensity, p (see Section
11.3 of Heyman and Sobel?*). We thus obtain estimates of the expected
number waiting by subtracting p from the numbers displayed in Table

Table XVII—The input data by routes for Gelenbe and
Mitrani’s model of a packet-switched communication

network
External Arrival Number
Origin-Des- Process Param- of Nodes
Route _tination _ eters on the Node
Number Pair Ar Ck Route Sequence

XVIII. When this is done, some of the simulation estimates become
negative, demonstrating that the input parameters are incorrect or the
statistical reliability of the simulation is not very good. In this case,
probably both problems exist.

We also observe that the traffic intensities at all link queues but
the second are very small, so the numbers displayed in Table XVIII
are mostly estimates of the traffic intensities themselves. Moreover,
because the traffic intensities are small, the variability parameter of
the departure process produced by QNA will be very close to the
variability parameter of the arrival process (see Section 4.5 of Ref. 1).
This would not be true for the second link with traffic intensity 0.835,
but note that all departures from the second link leave the system.
Since the external arrival processes are all Poisson, QNA should and
does perform virtually the same as an M/G/1 approximation. In fact,
since the service times are all constant (cz; = 0), the approximation
reduces to the M/D/1 system. Moreover, we predict that a proper
simulation of this model with the specified parameters will yield values
very close to the M/D/1 approximation.

We also display in Table XIX the point-to-point (origin-destination)
average total service times, delays, and sojourn times (service times
plus delays) produced by QNA. No simulation values were available
for comparison, however. The output is useful to indicate unacceptably
high or low values. It is also useful to determine the separate contri-
butions of service times and delays to sojourn times. Of course, in

Table XVIII—A comparison of approximations and simulation: The
expected number waiting and being served on each link in the
Gelenbe-and-Mitrani model of a packet-switched communication
network depicted in Fig. 4

Approximation Methods
M/M/1
Gelenbe
Traffic In- Simulation and via Gelenbe-

Table XIX—Average point-to-point service times, delays, and sojourn
times for the Gelenbe-and-Mitrani model of a packet-switched
communication network

Poisson Arrivals c? = 1.0 Bursty Arrivals c? = 4.0
Mean Total Mean Total
Mean Total Mean Total Sojourn Mean Total Sojourn
Route Service Time Delay on Time on Delay on Time on

Table XIX delays play a significant role only for routes using the
second link.

We conclude by remarking that the assumption of Poisson arrivals
for packets at each switch made by Gelenbe and Mitrani’® often is not
realistic. Often messages containing many packets arrive according to
a Poisson process, but the packets arrive in a much more bursty
manner. Hence, it is appropriate to use QNA with arrival-process
variability parameters much larger than 1. The last two columns of
Table XIX give the mean delays and sojourn times when the variability
parameters of the external arrival processes are changed from c” = 1.0
to c? = 4.0. When this is done here, the large delays on routes 4 and 8
increase significantly. To a large extent, QNA was motivated by the
need to be able to systematically study the effect of such variability.

I am grateful to Anne Seery for using QNA to generate much of the
data here, as well as for writing the QNA program. I am grateful to E.
B. Zucker for the simulation data in Section VII.

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Ward Whitt, A. B. (Mathematics), 1964, Dartmouth College; Ph.D. (Opera-
tions Research), 1968, Cornell University; Stanford University, 1968-1969;
Yale University, 1969-1977; Bell Laboratories, 1977—. At Yale University,
from 1973-1977, Mr. Whitt was Associate Professor in the departments of
Administrative Sciences and Statistics. At Bell Laboratories he is in the
Operations Research Department in the Network Analysis Center.

Stein A. H., An C. M., A Mean-Value Theorem for Zeta-Functions Associated
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Bartoli P. D., CCITT Message Handling Facilities. Aust Tele R 16(3):53-62, 1982.
Borsuk J. A., Light-Intensity Profiles of Surface-Emitting InGaAsP LEDs—
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Capasso F., Tsang W. T., Williams G. F., Staircase Solid-State Photomultipliers
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Chin A. K., Zipfel C. L., Ermanis F., Marchut L., Camlibel I., Di Giuseppe M. A., Chin
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Ma K. T., Kusic G. L., Tie Line Contingency Studies Based Upon Partial
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Time-Compression Multiplexing (TCM) of Three Broadcast-Quality
TV Signals on a Satellite Transponder
K. Y. Eng, B. G. Haskell, and R. L. Schmidt

Synchronization of Noncolocated TV Signals in a Satellite Time-
Compression Multiplexing System
K. Y. Eng and B. G. Haskell

Equivalent Queueing Networks and Their Use in Approximate Equi-
librium Analysis
A. Kumar

TELBECC—A Computational Method and Computer Program for
Analyzing Telephone Building Energy Consumption and Control
P. B. Grimado

On the Average Product of Gauss-Markov Variables
B. F. Logan, J. E. Mazo, A. M. Odlyzko, and L. A. Shepp

Bandwidth-Conserving Independent Amplitude and Phase Modula-
tion
B. F. Logan

Parallel Fault Simulation Using Distributed Processing
Y. H. Levendel, P. R. Menon, and S. H. Patel

Generation of Syntax-Directed Editors With Text-Oriented Features
B. A. Bottos and C. M. R. Kintala

Performance Analysis of a Preemptive Priority Queue With Applica-
tions to Packet Communications Systems
M. G. Hluchyj, C. D. Tsao, and R. R. Boorstyn

Part 3
THE AR6A SINGLE-SIDEBAND MICROWAVE RADIO SYSTEM:
Prologue
R. E. Markle

System Design and Performance
J. Gammie, J. P. Moffatt, R. H. Moseley, and W. A. Robinson

Radio Transmitter-Receiver Units
R. C. Heidt, E. F. Cook, R. P. Hecken, R. W. Judkins, J. M.
Kiker, Jr., F. J. Provenzano, Jr., and H. C. Wang

Equalization for Multipath Fading
N. O. Burgess, R. C. MacLean, G. J. Mandeville, D. I. McLean,
M. E. Sands, and R. P. Snicer

The Traveling-Wave-Tube Amplifier
J. F. Balicki, E. F. Cook, R. C. Heidt, and V. E. Rutter

Predistortion for the Traveling-Wave-Tube Amplifier
R. P. Hecken, R. C. Heidt, and D. E. Sanford

System Networks
R. L. Adams, J. L. Donoghue, A. N. Georgiades, J. R. Sundquist,
R. E. Sheehey, and C. F. Walker

magazine :: Bell System Technical Journal :: BSTJ V62N09 198311 Part 1