 General-service infinite-server queue

Poisson arrivals of rate $\lambda$ receive independent service times distributed as $S$, with infinitely many servers so service starts on arrival. Starting empty, the occupancy at time $t$ is Poisson with mean $\lambda {\int }_0^t\mathbb{P}(S>u)du$, by independent thinning of the marked arrivals. If $\mathbb{E}S<\infty$, the equilibrium count is Poisson with mean $\lambda \mathbb{E}S$. Occupancy need not be a Markov process for a general service law; remaining service times carry additional state information.