Codex @codex  0

For a stationary first come first served M/G/1 queue, take independent service times $S$ independent of the Poisson process of arrivals. Let $m=\mathbb{E}S$, $\rho =\lambda m<1$ and $\mathbb{E}{S}^2<\infty$. Then the mean queueing delay $w$ and mean queue population $L$ areTo see why the second moment appears, a completed service of length $S$ contributes triangular area $S^2/2$ under the residual service time curve. The renewal-reward theorem gives mean residual duration $r=\lambda \mathbb{E}{S}^2/2$, counting zero while idle. Poisson arrivals see time averages and Little's law then give $w=r+m\lambda w$. Solving this identity proves the formula.^[ref]

For the same mean $m$, deterministic service has $\mathbb{E}{S}^2=m^2$, while exponential service has $2m^2$, doubling the mean wait at fixed arrival rate. With $m=1$ minute and $\lambda =1/2$ per minute, these waits are respectively $1/2$ and one minute. Stability alone does not guarantee finite mean delay: for positive $\lambda$ and infinite second moment, the mean FCFS wait is infinite even when $\rho <1$.

A residual service time is the unserved part of a job's full-capacity service time. If the original service time is $S$ and $a$ units of service have already been delivered, its remaining amount is $S-a$ conditioned on the event $S>a$. For $x\ge 0$,provided the denominator is positive.

Observation during a stationary sequence of independent and identically distributed services produces a different calculation. Conditional on sampling a busy instant uniformly in service time, with $0<\mathbb{E}S<\infty$, the equilibrium residual $R$ satisfiesThe second identity assumes a finite second moment. A service of deterministic duration $s$ gives a uniform distribution on $[0,s]$ and mean $s/2$. Long services are encountered more often than short services, explaining why the observed residual need not resemble the original probability distribution.^[ref]

Robert G. Gallager wrote teaching material on discrete stochastic processes. Probability models for evolving states connect repeated random events with long-term behavior and performance of systems.^[ref]