Using infinite server queues with partial information for occupancy prediction

TL;DR

This paper develops a method for predicting occupancy in service systems modeled as infinite server queues with partial initial information, focusing on transient behavior and applying Bayesian inference to real data.

Contribution

It introduces a novel approach combining queue separation and approximation techniques for occupancy prediction with incomplete initial data.

Findings

Effective short-term queue length predictions achieved.

Demonstrated the approach using real data in a Bayesian framework.

Insights into managing congestion and interventions in service systems.

Abstract

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.