Queues with time-dependent arrival rates. II — The maximum queue and the return to equilibrium

Abstract

During a rush hour, the arrival rate λ(t) of customers to a service facility is assumed to increase to a maximum value exceeding the service rate μ, and then decrease again. In Part I it was shown that, after λ(t) has passed μ, the expected queue E{X(t)} exceeds that given by the deterministic theory by a fixed amount, (0.95)L, which is proportional to the (–1/3) power of a(t) = dλ(t)/dt evaluated at time t = 0 when λ(t) = μ. The maximum of E{X(t)}, therefore, occurs when λ(t) again is equal to μ at time t₁ as predicted by deterministic queueing theory, but is larger than given by the deterministic theory by this same constant (0.95)L (provided t₁ is sufficiently large). It is shown here that the maximum queue, sup_tX(t) is, approximately normally distributed with a mean (0.95) (L + L₁) larger than predicted by deterministic theory where L, is proportional to the (–1/3) power of a(t₁). We also investigate the distribution of X(t) at the end of the rush hour when the queue distribution returns to equilibrium. During the transition, the queue distribution is approximately a mixture of a truncated normal and the equilibrium distributions. These results are applied to a case where λ(t) is quadratic in t.