QUEUES WITH TIME-DEPENDENT ARRIVAL RATES

Suppose that the arrival rate A(t) of customers to a service facility increases with time at a nearly constant rate, dA(t)/dt = a, so as to pass through the saturation condition, ,A(t)= =t = service capacity, at some time which we label as t = 0. The stochastic properties of the queue are investigated here through use of the diffusion approximation (Fokker-Planck equation). It is shown that there is a characteristic time T proportional to a-2/3 such that if t 0, It 1> T, the queue is approximately normally distributed with a mean of the order L larger than that predicted by deterministic queueing models. Numerical estimates are given for the mean and variance of the distribution for all t. The queue distributions are also evaluated in non-dimensional units.