The effect of green time on stochastic queues at traffic signals

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.