A queue subject to extraneous phase changes

Abstract

Many service systems exhibit variations of a random nature in the intensity of the arrival process or of the speed of service or of both. Changes in work shifts, rush hours, interruptions in the arrival process, server breakdowns, etc. all fall into this category. The present study deals with a generalization of the classical M/G/1 queue by considering an extraneous process of phases which can be in one of the states {1, …, m}. During any interval spent in phase i, the arrivals are according to a homogeneous Poisson process of rate λ i and any service initiated during such an interval has a duration distributed according to H i (·). The process of phases is assumed to be an irreducible Markov chain in continuous time and is fully characterized by its initial conditions, by an irreducible stochastic matrix P and by the mean sojourn times σ₁ ^-1, …, σ m ^-1 in each phase. Independently of the queueing aspects, this arrival process is a generalization of the classical Poisson process which can be of interest in modelling simple point processes with randomly fluctuating “arrival” rate. Two approaches to the time dependent study of this queue are presented; one generalizes the imbedded semi-Markov process obtained by considering the queue immediately following departure points; the other approach exploits the relationship between this queue and branching processes. The latter is more elegant from a purely theoretical viewpoint and involves iterates of a general type of matrix function introduced by the author. By making extensive use of the Perron-Frobenius theory of positive matrices the equilibrium condition of the queue is obtained. While retaining a similar intuitive interpretation the equilibrium condition is substantially more complicated than for the M/G/1 model. The recurrence relations which yield the joint distribution of the phase state at time t, the queue length, the total number served and the virtual waiting time at t are exhibited in detail. Via transform techniques a number of limiting and marginal distributions are discussed. The discussion relies heavily on the theory of Markov renewal processes. Throughout the paper and in a final section the author advocates the use of the structural properties of the queue and the resulting recurrence relations to organize the numerical analysis of complex queueing models such as the present one. More explicit results for the case of two phases are given and are compared to results obtained by Yechiali and Naor for a closely related two-phase generalization of the M/M/1 queue.