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aaef24a 8eccedb aaef24a 8eccedb aaef24a f006b5f aaef24a 8fe65d4 aaef24a | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 | # /// script
# requires-python = ">=3.13"
# dependencies = [
# "altair",
# "asimpy",
# "marimo",
# "polars==1.24.0",
# ]
# ///
import marimo
__generated_with = "0.20.4"
app = marimo.App(width="medium")
@app.cell(hide_code=True)
def _():
import marimo as mo
import random
import altair as alt
import polars as pl
from asimpy import Environment, Process, Queue
return Environment, Process, Queue, alt, mo, pl, random
@app.cell(hide_code=True)
def _(mo):
mo.md(r"""
# Tandem Queue Blocking
## *Variability Travels Downstream*
Two processing stages are arranged in series: Stage 1 feeds work into a bounded buffer, which feeds Stage 2. Both stages have the same mean service rate $\mu$, and the arrival rate $\lambda < \mu$ so neither stage is overloaded on average. However, Stage 1 has high variance (hyperexponential service); Stage 2 has zero variance (deterministic service).
Even though both stages have identical mean throughput and the system is underloaded, Stage 2 sits idle for a substantial fraction of time when the buffer between them is small. The idle fraction only vanishes as the buffer size $K \to \infty$.
High service-time variance at Stage 1 produces bursts of output—many jobs finish close together—followed by droughts. With a small buffer, the burst overflows (blocking Stage 1) and the drought starves Stage 2. Both effects reduce system throughput below what we would intuitively expect.
## Analysis
For a two-stage tandem queue with a finite buffer of capacity $K$, the blocking probability at Stage 1 and the starvation probability at Stage 2 depend on the full service-time distributions, not just their means. The Kingman approximation gives the mean wait in a single G/G/1 queue as:
$$W_q \approx \frac{\rho}{1-\rho} \cdot \frac{c_a^2 + c_s^2}{2} \cdot \frac{1}{\mu}$$
where $c_a^2$ and $c_s^2$ are the squared coefficients of variation of inter-arrival and service times respectively. For a hyperexponential service distribution with $c_s^2 \gg 1$, waiting times are far higher than the M/M/1 formula predicts.
In a tandem network, this extra variability propagates: the departure process of Stage 1 (which is the arrival process for Stage 2) has higher variance than Poisson when Stage 1 has high service variance. This is *Departure Process Variability Propagation* and is a key driver of manufacturing and supply-chain [bullwhip effects](https://en.wikipedia.org/wiki/Bullwhip_effect).
## Buffer as a Variability Absorber
The buffer acts as a shock absorber. Each unit of additional buffer capacity $K$ reduces the starvation probability at Stage 2 by absorbing burst output from Stage 1. The marginal benefit decreases as $K$ grows, leading to a classic diminishing-returns relationship. Practitioners use this to size work-in-progress inventory (WIP) buffers in manufacturing cells.
""")
return
@app.cell(hide_code=True)
def _(mo):
mo.md(r"""
## Implementation
- A `Source` process generates jobs with exponential inter-arrival times into an unlimited input queue.
- `Stage1` pulls from the input queue, applies a hyperexponential service delay, and pushes to a bounded `Queue(max_capacity=K)`. If the buffer is full, Stage 1 blocks (back-pressure).
- `Stage2` pulls from the bounded buffer, applies deterministic service, and records completion times.
- Stage 2 idle time is measured as the wait inside `queue.get()` (time spent waiting for work to appear).
The simulation sweeps $K$ from 1 to 21 and reports Stage 2 idle fraction.
""")
return
@app.cell(hide_code=True)
def _(mo):
sim_time_slider = mo.ui.slider(
start=0,
stop=100_000,
step=1_000,
value=20_000,
label="Simulation time",
)
arrival_rate_slider = mo.ui.slider(
start=0.3,
stop=0.95,
step=0.05,
value=0.8,
label="Arrival rate",
)
seed_input = mo.ui.number(
value=192,
step=1,
label="Random seed",
)
run_button = mo.ui.run_button(label="Run simulation")
mo.vstack([
sim_time_slider,
arrival_rate_slider,
seed_input,
run_button,
])
return arrival_rate_slider, seed_input, sim_time_slider
@app.cell
def _(arrival_rate_slider, seed_input, sim_time_slider):
SIM_TIME = int(sim_time_slider.value)
ARRIVAL_RATE = float(arrival_rate_slider.value)
SEED = int(seed_input.value)
MEAN_SERVICE = 1.0
return ARRIVAL_RATE, MEAN_SERVICE, SEED, SIM_TIME
@app.cell
def _(MEAN_SERVICE, random):
def high_variance_service():
if random.random() < 0.80:
return random.expovariate(5.0)
return random.expovariate(1.0 / 4.5)
def low_variance_service():
return MEAN_SERVICE
return high_variance_service, low_variance_service
@app.cell
def _(ARRIVAL_RATE, Process, random):
class Source(Process):
def init(self, buffer):
self.buffer = buffer
async def run(self):
jid = 0
while True:
await self.timeout(random.expovariate(ARRIVAL_RATE))
await self.buffer.put(jid)
jid += 1
return (Source,)
@app.cell
def _(Process, high_variance_service):
class Stage1(Process):
def init(self, inp, out, idle_tally):
self.inp = inp
self.out = out
self.idle_tally = idle_tally
async def run(self):
while True:
idle_start = self.now
job = await self.inp.get()
self.idle_tally.append(self.now - idle_start)
await self.timeout(high_variance_service())
await self.out.put(job)
return (Stage1,)
@app.cell
def _(Process, low_variance_service):
class Stage2(Process):
def init(self, inp, idle_tally, completions):
self.inp = inp
self.idle_tally = idle_tally
self.completions = completions
async def run(self):
while True:
idle_start = self.now
await self.inp.get()
idle = self.now - idle_start
self.idle_tally.append(idle)
await self.timeout(low_variance_service())
self.completions.append(self.now)
return (Stage2,)
@app.cell
def _(Environment, Queue, SIM_TIME, Source, Stage1, Stage2):
def simulate(buffer_capacity):
env = Environment()
input_q = Queue(env)
middle_q = Queue(env, capacity=buffer_capacity)
s2_idle = []
completions = []
Source(env, input_q)
Stage1(env, input_q, middle_q, [])
Stage2(env, middle_q, s2_idle, completions)
env.run(until=SIM_TIME)
n = len(completions)
idle_total = sum(s2_idle)
return {
"buffer_capacity": buffer_capacity,
"throughput": n / SIM_TIME,
"stage2_idle_frac": idle_total / SIM_TIME,
"n_completed": n,
}
return (simulate,)
@app.cell
def _(SEED, pl, random, simulate):
random.seed(SEED)
df = pl.DataFrame([simulate(buffer_capacity=k) for k in [1, 2, 3, 5, 8, 13, 21]])
return (df,)
@app.cell(hide_code=True)
def _(ARRIVAL_RATE, MEAN_SERVICE, mo):
mo.md(f"""
## Results
Arrival rate: {ARRIVAL_RATE}, mean service per stage: {MEAN_SERVICE}
- Stage 1: hyperexponential (high variance: 80% short mean=0.2, 20% long mean=4.5)
- Stage 2: deterministic (zero variance)
> As buffer capacity grows, Stage 2 idle fraction falls toward 0%.
""")
return
@app.cell
def _(df):
df
return
@app.cell
def _(alt, df):
throughput_line = (
alt.Chart(df)
.mark_line(point=True)
.encode(
x=alt.X("buffer_capacity:Q", title="Buffer capacity (K)"),
y=alt.Y("throughput:Q", title="Throughput (jobs/time)"),
tooltip=["buffer_capacity:Q", "throughput:Q"],
)
.properties(title="Tandem Queue: Throughput vs. Buffer Capacity")
)
idle_line = (
alt.Chart(df)
.mark_line(point=True, strokeDash=[4, 4], color="orange")
.encode(
x=alt.X("buffer_capacity:Q"),
y=alt.Y("stage2_idle_frac:Q", title="Stage 2 idle fraction"),
)
)
(throughput_line + idle_line).resolve_scale(y="independent")
return
@app.cell(hide_code=True)
def _(mo):
mo.md(r"""
## Understanding the Math
### Coefficient of variation
The coefficient of variation (CV) of a random variable $X$ with mean $\mu$ and standard deviation $\sigma$ is defined as $c = \sigma / \mu$. It measures spread relative to the mean. A CV of 0 means the variable is deterministic: every value equals $\mu$. A CV of 1 means the spread equals the mean (the exponential distribution has CV exactly 1). A CV greater than 1 means the distribution is bursty: occasional very large values dominate, even if most values are small. The squared CV $c^2 = \sigma^2/\mu^2$ appears frequently in queueing formulas.
### Why high CV at Stage 1 creates bursts and droughts
With a hyperexponential service distribution ($c_s^2 \gg 1$), Stage 1 sometimes completes several jobs in rapid succession (a burst) and sometimes spends a very long time on a single job (a drought). During a burst, jobs pile up in the buffer between stages. During a drought, Stage 2 exhausts the buffer and has to wait for Stage 1 to finish. This wastes capacity even though the system is underloaded on average.
### The buffer as a shock absorber
A buffer of capacity $K$ can hold at most $K$ jobs between the two stages. It absorbs burst output from Stage 1 and releases it steadily to Stage 2. With a small buffer, a burst overflows (blocking Stage 1) or a drought empties the buffer (starving Stage 2). As $K$ grows, both effects weaken and Stage 2 idle time falls. However, the marginal benefit of each extra unit of buffer decreases. Real factories choose $K$ to balance the cost of holding inventory against the cost of machine starvation.
### Kingman's approximation
For a single-stage queue with general service and arrival distributions, the approximate mean waiting time is:
$$W_q \approx \frac{\rho}{1-\rho} \cdot \frac{c_a^2 + c_s^2}{2} \cdot \frac{1}{\mu}$$
Here $c_a^2$ is the squared CV of inter-arrival times and $c_s^2$ is the squared CV of service times. Notice that the formula separates the utilization effect (the $\rho/(1-\rho)$ term) from the variability effect (the $(c_a^2 + c_s^2)/2$ term). When Stage 1 has $c_s^2 \gg 1$, wait time is far higher than the basic M/M/1 formula predicts, even at the same mean throughput. The mean alone does not tell you enough.
### Supply-chain connection
In manufacturing, Stage 1 corresponds to a supplier and Stage 2 to a production line. High variability at the supplier forces the factory to hold large work-in-progress (WIP) buffers, tying up capital and floor space. The [Toyota Production System](https://en.wikipedia.org/wiki/Toyota_Production_System) explicitly targets CV reduction as the primary tool for shrinking necessary WIP by making every process more deterministic through standardized work and small batch sizes. The math here explains exactly why: lower $c_s^2$ directly reduces $W_q$ and the required buffer size $K$.
""")
return
if __name__ == "__main__":
app.run()
|