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

For equilibrium, let $t\to \infty$ in part (e). Monotone convergence of the tail integral and the finite mean giveand consequentlyThis limiting argument uses the equilibrium age and residual-service distribution produced by the queue, not just a fresh independent copy of $S$ for each mosquito present. Replacing their residual service by fresh service would generally be wrong unless the service law is memoryless.