Queues with non-stationary input stream: Ross's conjecture

Abstract

Characteristics of queues with non-stationary input streams are difficult to evaluate, therefore their bounds are of importance. First we define what we understand by the stationary delay and find out the stability conditions of single-server queues with non-stationary inputs. For this purpose we introduce the notion of an ergodically stable sequence of random variables. The theory worked out is applied to single-server queues with stationary doubly stochastic Poisson arrivals. Then the interarrival times do not form a stationary sequence (‘time stationary’ does not imply ‘customer stationary’). We show that the average customer delay in the queue is greater than in a standard M/G/1 queue with the same average input rate and service times. This result is used in examples which show that the assumption of stationarity of the input point process is non-essential.