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.

翻译：受英格兰和威尔士监狱人口需求预测的驱动，本文描述了一种利用无限服务队列研究服务系统的方法，其中系统初始状态非空，且初始个体的已服务时间未知。通过将人群分为初始内容与新到达者，我们可以分别或联合地对这些子群体应用多种技术，从而实现短期队列长度预测以及更长期的考量，例如管理拥塞和分析潜在干预措施的影响。本文重点关注非齐次泊松到达过程下 $M_t/G/\infty$ 队列的瞬态行为，我们的分析考虑了各种可能的简化，包括近似方法。我们利用公开数据在贝叶斯框架下展示了该方法在该领域中的应用，以进行模型推理。