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Motivated by demand prediction for the custodial prison population in Englandand Wales, this paper describes an approach to the study of service systemsusing infinite server queues, where the system has non-empty initial state andthe elapsed time of individuals initially present is not known. By separatingthe population into initial content and new arrivals, we can apply severaltechniques either separately or jointly to those sub-populations, to enableboth short-term queue length predictions and longer-term considerations such asmanaging congestion and analysing the impact of potential interventions. Thefocus in the paper is the transient behaviour of the $M_t/G/\infty$ queue witha non-homogeneous Poisson arrival process and our analysis considers variouspossible simplifications, including approximation. We illustrate the approachin that domain using publicly available data in a Bayesian framework to performmodel inference.

Using infinite server queues with partial information for occupancy prediction