samples/texts/3179895/page_7.md

a rate assignment $x$, the link loads $x^l, l \in \mathcal{L}$, are given by

$$x^l=\sum _{s\in S}(\sum _{o\in O^s:l\in V_o^s}{x}_o^s+\sum _{o\in O^s}\left(\sum _{d\in D^s:l\in V_d^o}{x}_{o,d}^s\right))(2)x^l\; =\; \backslash sum\_\{s\; \backslash in\; S\}\; \backslash left(\; \backslash sum\_\{o\; \backslash in\; O^s:\; l\; \backslash in\; V\_o^s\}\; x\_o^s\; +\; \backslash sum\_\{o\; \backslash in\; O^s\}\; \backslash left(\; \backslash sum\_\{d\; \backslash in\; D^s:\; l\; \backslash in\; V\_d^o\}\; x\_\{o,d\}^s\; \backslash right)\; \backslash right)\; \backslash quad\; (2)$$

This model is referred to as NM-I in Section 6.

3.2.2 Network Model-II

Under Network Model-II the routers are IP multicast capable. We assume that each overlay node $o \in O^s$ creates a multicast tree for forwarding packets. However, due to the lack of additional required intelligence the data rate to all receivers is the same and is given by $x_o^s = \max_{d \in D^s} x_{o,d}^s$.

Under this model the load of link $l$ can be written as

$$x^l=\sum _{s\in S}(\sum _{o\in O^s:l\in V_o^s}{x}_o^s+\sum _{o\in O^s:l\in T_o^s}{x}_o^s)(3)x^l\; =\; \backslash sum\_\{s\; \backslash in\; S\}\; \backslash left(\; \backslash sum\_\{o\; \backslash in\; O^s:\; l\; \backslash in\; V\_o^s\}\; x\_o^s\; +\; \backslash sum\_\{o\; \backslash in\; O^s:\; l\; \backslash in\; T\_o^s\}\; x\_o^s\; \backslash right)\; \backslash quad\; (3)$$

where $T_o^s$ is set of links in the multicast tree rooted at overlay node $o$ and serving destination nodes in $D^s$.

This model is referred to as NM-IIa in Section 6.

3.2.3 Network Model-III

In this model, in addition to the IP multicast capability we also assume that each router is capable of forwarding packets onto each branch at a different rate. We refer to these routers as “smart” routers to distinguish them from the routers used in the previous model. This is shown in Fig. 3. Under this model a source $s$ can select the individual rates $x_{o,d}^s$ independently for each destination, and each destination $d \in D^s$ will receive the intended rate $x_{o,d}^{s,3}$ instead of $\max_{d' \in D^s} x_{o,d'}^s$ as under Network Model-II. This allows the network operator more flexibility in rate assignment and to better exploit the existence of multiple paths through overlay nodes, while making use of multicast nature of the traffic at the same time.

The link rates can be written as

$$x^l=\sum _{s\in S}(\sum _{o\in O^s:l\in V_o^s}{x}_o^s+\sum _{o\in O^s}\underset{d\in D^s:l\in {\widehat{V}}_d^o}{\max }{x}_{o,d}^s)(4)x^l\; =\; \backslash sum\_\{s\; \backslash in\; S\}\; \backslash left(\; \backslash sum\_\{o\; \backslash in\; O^s:\; l\; \backslash in\; V\_o^s\}\; x\_o^s\; +\; \backslash sum\_\{o\; \backslash in\; O^s\}\; \backslash max\_\{d\; \backslash in\; D^s:\; l\; \backslash in\; \backslash hat\{V\}\_d^o\}\; x\_\{o,d\}^s\; \backslash right)\; \backslash quad\; (4)$$

Here $\hat{V}_d^o$ denotes the set of links along the path from overlay node $o$ to destination $d$ in the multicast tree, which may be different from the path provided by the underlying routing protocol. Under this model it is necessary to adopt a special coding scheme, such as Digital Fountain codes, in order to ensure that all destinations can recover the transmitted data as explained in Section 1. We assume that a suitable coding scheme is adopted. We will refer to this model as NM-III while presenting the experiments.

Due to the large gap between the level of intelligence currently available at the IP routers and the required intelligence assumed in this model, it is unlikely that this type

³This assumes that there are no packet losses along the path.