1. Simple deterministic queue model

• $D(t)$: Cumulative departures
• $A(t)$: Cumulative arrivals
• $R$: fixed rate

<aside> 📖 Queue occupancy $Q(t)$

$Q(t)=A(t)-D(t)$

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<aside> 📖 Queue delay $d(t)$

The time spent in the queue by a byte that arrived at time $t$, assuming the queue is served first-come-first-served (FCFS/FIFO).

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• Example

2. Small packets reduce end to end delay

Breaking message into packets allows parallel transmission across all links, reducing end to end latency.

<aside> 📖 Send in one packet

$\text{End-to-end delay } \\t=\sum_i\left( \frac M {r_i} + \frac {l_i} c \right)$

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<aside> 📖 Send in one packet

$\text{End-to-end delay } \\ t=\sum_i\left( \frac M {r_i} + \frac {l_i} c \right) + \left( \frac Mp-1 \right)\frac p{r_{min}}$

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3. Statistical multiplexing

Statistical multiplexing lets us carry many flows efficiently on a single link.

<aside> 📖 Basic idea

The average rate of many flows tends to be convergent to a fixed rate.

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<aside> 📖 Features

• Statistical multiplexing means the egress link need not run at rate NR.
• The buffer absorbs brief periods when the aggregate rate exceeds R.
• Because the buffer has finite size, losses can occur. </aside>

$\text{Statistical Multiplexing Gain} = \frac {2C} R$