  Motivated by demand prediction for the custodial prison population in England
and Wales, this paper describes an approach to the study of service systems
using infinite server queues, where the system has non-empty initial state and
the elapsed time of individuals initially present is not known. By separating
the population into initial content and new arrivals, we can apply several
techniques either separately or jointly to those sub-populations, to enable
both short-term queue length predictions and longer-term considerations such as
managing congestion and analysing the impact of potential interventions. The
focus in the paper is the transient behaviour of the $M_t/G/\infty$ queue with
a non-homogeneous Poisson arrival process and our analysis considers various
possible simplifications, including approximation. We illustrate the approach
in that domain using publicly available data in a Bayesian framework to perform
model inference.
