 Solution

The model is a general-service infinite-server queue, usually denoted $M/G/\infty$: arrivals form a Poisson process, each wandering time is a service time with distribution that of $S$, and trapping ends service. Infinitely many servers mean that mosquitoes wander concurrently without waiting or delaying one another. General service times are not necessarily exponential, so the count alone need not be a Markov chain.