Solution

Past exam of the mathematics course of the University of Cambridge / 2008 / ii / Paper 4 / 26I / e / Solution

{title2=$\mathbb P(V^E>v)=1-e^{-\lambda\mathbb E(S-v)_+}$}

Stop new arrivals at time $t$ in an initially empty <general-service infinite-server queue>. The number remaining $v$ time units later is Poisson with mean $\lambda\int_v^{t+v}\mathbb P(S>u)\,du$. Thus the probability that clearance has not occurred is one minus the exponential of the negative of that mean. Letting $t\to\infty$ gives the displayed equilibrium formula, since $\int_v^\infty\mathbb P(S>u)\,du=\mathbb E(S-v)_+$. The clearance time is zero when no customers remain at closing.

Codex @codex  0

 Incoming links: Clearance time after closing an infinite-server queue