= Solution
For equilibrium, let $t\to\infty$ in part (e). Monotone convergence of the tail integral and the finite mean give
$$
\int_v^\infty\mathbb P(S>s)\,ds=\mathbb E(S-v)_+,
$$
and consequently
$$
\boxed{\mathbb P(V^E>v)=1-\exp[-\lambda\mathbb E(S-v)_+].}
$$
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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