A Queueing Model for the Ambulance Ramping Problem with an Offload Zone

TL;DR

This paper develops an analytically tractable, multi-server, multi-class non-preemptive priority queueing model with an offload zone (APOT) to study ambulance ramping. It derives exact and approximate distributions for ambulance queue length and waiting time, and characterizes the offload delay rate $\overline{\Omega}=30\lambda_{amb}\overline{W}_{amb}$, providing a closed-form algebraic approximation that closely matches the exact result. The approach combines generating-function techniques, a pole/cut decomposition, and an exponential-tail Ansatz, with discrete-event simulation (DES) used to validate both exact and approximate results across performance measures such as means, 90th percentiles, and occupancy of the APOT. The findings show a monotone improvement in the offload rate with APOT size $M$, with diminishing marginal gains, and offer practical insights for APOT sizing under realistic ED-ramping conditions. The framework is extendable to more complex priority structures and dynamic arrival patterns, enabling broader policy analysis for ambulance offload strategies.

Abstract

This work develops a methodology for studying the effect of an offload zone on the ambulance ramping problem using a multi-server, multi-class non-preemptive priority queueing model that can be treated analytically. A prototype model for the ambulance/emergency-department interface is constructed, which is then implemented as a formal discrete event simulation, and is run as a regenerative steady-state simulation for empirical estimation of the ambulance queue-length and waiting-time distributions. The model is also solved by analytical means for explicit and exact representations of these distributions, which are subsequently tested against simulation results. A number of measures of performance is extracted, including the mean and 90th percentiles of the ambulance queue length and waiting time, as well as the average number of ambulance days lost per month due to offload delay (offload delay rate). Various easily computable approximations are proposed and tested. In particular, a closed-form, purely algebraic expression that approximates the dependence of the offload delay rate on the capacity of the offload zone is proposed. It can be evaluated directly from model input parameters and is found to be, for all practical purposes, indistinguishable from the exact result.

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