= Residual service time
{title2=$R$}
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 $x\geq0$,
$$
\mathbb P(S-a>x\mid S>a)=\frac{\mathbb P(S>a+x)}{\mathbb P(S>a)},
$$
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 $0<\mathbb ES<\infty$, the equilibrium residual $R$ satisfies
$$
\mathbb P(R>x)=\frac{1}{\mathbb ES}\int_x^\infty\mathbb P(S>u)\,du,\qquad
\mathbb ER=\frac{\mathbb E S^2}{2\mathbb ES}.
$$
The 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>.<Discrete Stochastic Processes (draft of 2nd edition, 31 January 2011) by Robert Gallager>[Section 4.4, renewal-reward time averages and equation 4.15 for residual life]{ref}
Back to article page
