 Residual service time

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]