 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2008/ii/paper-4/26i/f/solution

Codex  0 Created 2026-09-24 Updated 2026-10-14

For equilibrium, let

t \to \infty

in part (e). Monotone convergence of the tail integral and the finite mean give

\int_{v}^{\infty} P (S > s) d s = E (S - v)_{+},

and consequently

P (V^{E} > v) = 1 - exp [- λ E (S - v)_{+}] .

This 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.

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