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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 (R2 > 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. [ABSTRACT FROM AUTHOR]

Queues are often associated with uncertainty or unreliability, which can arise from chance or climatic events, phase changes in system behaviour, or inherent randomness. Knowing the probability distribution of the number of customers in a queue is important for estimating the risk of stress or disruption to routine services and upstream blocking, potentially leading to exceeding critical limits, gridlock or incidents. The present paper focuses on time-varying queues produced by transient oversaturation during demand peaks where there is randomness in arrivals and service. The objective is to present practical methods for estimating a probability distribution from knowledge of the mean, variance and utilisation (degree of saturation) of a queue available from computationally efficient, if approximate, time-dependent calculation. This is made possible by a novel expression for time-dependent queue variance. The queue processes considered are those commonly used to represent isolated priority (M/M/1) and signal-like (M/D/1) systems, plus some statistical variations within the common Pollaczek-Khinchin framework. Results are verified by comparison with Markov simulation based on recurrence relations. [ABSTRACT FROM AUTHOR]

Many analyses of traffic signal queues use Webster and Cobbe's formula, which combines the net effect of the red/green cycle with a term representing stochastic effects, idealised as an M/D/1 queue process having random arrivals and uniform service. Several authors have noted that this component should depend not only on demand intensity but also on throughput capacity in each green period, although an extra empirical term may partially allow for this. Extending the service interval in M/D/1 (M = Markovian, i.e. random, D = deterministic, i.e. uniform, 1 = one server) enables the effect to be reproduced, but no exact expressions for its moments are found. Approximate formulae for the extended mean exist but are accurate only near saturation. The paper derives novel approximations for the equilibrium mean and also variance and utilisation, using functions linking traffic intensity with green period capacity. With three moments, equilibrium probability distributions can be estimated for which a method based on a doubly nested geometric distribution is described. [ABSTRACT FROM PUBLISHER]