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

During a rush hour, the arrival rate ?(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 ?(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) = d?(t)/dtevaluated at time t= 0 when ?(t) = µ.The maximum of E{X(t)}, therefore, occurs when ?(t) again is equal to µat time t1as predicted by deterministic queueing theory, but is larger than given by the deterministic theory by this same constant (0.95)L(provided t1is sufficiently large). It is shown here that the maximum queue, suptX(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 ?(t) is quadratic in t.