Predicting queue variability to enable analysis of overload risk

Abstract

Predicting the risk of traffic demands and delays exceeding critical limits at road junctions, airports, hospitals, etc, requires knowing how both mean and variance of queue size vary over time. Microscopic simulation can explore variability but is computationally demanding and gives only sample results. A computationally efficient approximation to the mean is used in many modelling tools, but only empirical extensions for variance in particular situations have been available. The paper derives theoretical formulae for time-dependent and equilibrium variance, believed to be novel and to apply generally to queues covered by the Pollaczek-Khinchin mean formula, and offering possible structural insights. These are applied in an extended approximation giving mutually consistent mean and variance estimates with improved accuracy. Tests on oversaturated peak demand cases are compared with Markov probabilistic simulation, demonstrating accuracy (R 2 >0.99) for typical random, priority-like (M/M/1) and traffic-signal-like (M/D/1) queues. Implications for risk analysis, planning and policy are considered.