Periodic solution to the time-inhomogeneous multi-server Poisson queue

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Abstract

We derive the periodic family of asymptotic distributions and the periodic moments for number in the queue for the multi-server queue with Poisson arrivals and exponential service for time-varying periodic arrival and departure rates, and time-varying periodic number of servers. The method is a straight-forward application of generating functions.

Introduction

In many queueing applications, arrival rates and number of servers vary periodically. Examples include call centers, emergency dispatching, and retail establishments. For such queues with the average arrival rate less than the average effective service rate, periodic asymptotic probabilities pj(t), the probability of j in the queue at time t, t[0,T) where T is the length of the period, exist. We present a method for computing these probabilities using one period of data. We also derive formulae for the rising factorial moments of the number in the queue as a function of time within the period. This is an extension of results obtained for the transient solution for the time-dependent Poisson queue given in [10]. As noted in [15] and references cited there, computational methods and approximation techniques involved in time-inhomogeneous queueing problems have long been regarded as challenging. In this paper, we present a relatively simple method for computing these quantities for queues with periodically varying rates via the numerical solution of integral equations covering only one time period.

Literature surveys in the area of queues with time-varying rates are available in [12], [9]. Treatment of the single-server Poisson queue with time-varying arrival rate and constant service dates back at least to A.N. Kolmogorov's paper on the waiting time distribution in 1931 [7], [8]. Additional references and an overview of many mathematical tools related to queues with time-varying rates is available in [11].

Early work in the area of time-dependent Poisson queues with periodic rates includes work by Hasofer for the single-server queue with general service [3], [4]. In Do Le Minh's paper [12] treating the discrete-time queue with time-varying compound Poisson input and general service, he describes Hasofer's work as elaborate, but having limited usefulness in computational applications. More recent work in this area includes Breuer's work on the BMAP/PH/c periodic queue.

We assume a pre-emptive discipline so that when the number of servers decreases, customers who were in service return to the queue. This contrasts with the exhaustive discipline in which the server completes service of customers being served before leaving. See [5] for recent work on an approach for modeling the Mt/M/ct queue with the exhaustive discipline.

Section snippets

The family of generating functions

Consider a queueing system with time-varying periodic arrival and service rates, λ(t), and μ(t), respectively, and time-varying periodic number of servers, c(t). For ease of exposition, we consider the case where the period is of length 1. c^=maxtc(t) is the maximum number of servers in the period.

Define Pi,j(s,t) as the probability that the system is in state j at time t given that it was in state i at time s and pj(t)=limnPi,j(s,n+t),0t<1,ninteger.For each t in [0,1), pj(t), j=0,1,2, is

Moments

In this section, we provide formulae for the queue length process {Q(t),t0} with initial distribution given by P{Q(0)=n}=pn(0), where pn(0) are the periodic asymptotic probabilities for the beginning of the period. We define the functions: qn(m)(t)=j=nqn(m-1)(t) for m=0,1,2, and nZ with qn(0)(t)=pn(t). Then q(1)(t)=1, q0(2)(t)=E[Q(t)]+1 and more generally, for m1,q0(m+1)(t)=E[(Q(t)+1)(Q(t)+2)(Q(t)+m)]/m!=1m!k=0m(-1)m-ks(m+1,k)E[Qk(t)],where the s(k,j) are the Stirling numbers of the

Examples

We consider examples with time-varying periodic arrival rates for a single-server queue, a two-server queue and a queue which alternates between one and three servers. For the one and two-server queues, we show the simpler form taken by the integral equations needed to solve for pc^-1(t).

Conclusion

Although the formulas given can be quite complicated, the method requires only one period of data so that periodic asymptotic results may be found without having to truncate an infinite system of differential equations, and without having to compute transient solutions over a long time horizon. Properties of modified Bessel functions may be exploited to provide estimates for the kernel functions in the integral equations derived herein. The formulae for the rising factorial moments of the

Acknowledgements

This paper is part of a cooperative effort between Cleveland State University and the City of Cleveland Division of Police. In particular, I want to thank Deputy Chief Prioleau Green of the Police Division, and Felipe Martins, John Holcomb and Teresa LaGrange at Cleveland State University.

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