Estimating probability distributions of dynamic queues

Abstract

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.

Predicting queue variability to enable analysis of overload risk

“…Examples exist of probability distributions with different shapes but the same mean and variance. Taylor and Heydecker (2015) find that a minimum of three moments is needed to represent a queue size probability distribution. However, involving skewness explicitly would be impractical, as argued above.…”

“…Any normal shape produced by a period of oversaturation is quickly lost, and the distribution moves towards equilibrium in a way somewhat resembling the collapse of a viscous mass that collides with and rebounds from the 'barrier' at zero queue size. This can be seen in examples of distributions given by Taylor and Heydecker (2015), where the distribution immediately post-peak develops a 'duck-tail' at small sizes and later becomes 'heavy-tailed' with a bi-modal shape, eventually relaxing to an equilibrium-like shape.…”