Queues with time-dependent arrival rates. III — A mild rush hour

The arrival rate of customers to a service facility is assumed to have the form λ ( t ) = λ (0) — βt 2 for some constant β. Diffusion approximations show that for λ (0) sufficiently close to the service rate μ , the mean queue length at time 0 is proportional to β –1/5 . A dimensionless form of the diffusion equation is evaluated numerically from which queue lengths can be evaluated as a function of time for all λ (0) and β. Particular attention is given to those situations in which neither deterministic queueing theory nor equilibrium stochastic queueing theory apply.