Periodic solution to the time-inhomogeneous multi-server Poisson queue
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 , the probability of in the queue at time , where 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 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 queue with the exhaustive discipline.
Section snippets
The family of generating functions
Consider a queueing system with time-varying periodic arrival and service rates, , and , respectively, and time-varying periodic number of servers, . For ease of exposition, we consider the case where the period is of length 1. is the maximum number of servers in the period.
Define as the probability that the system is in state at time given that it was in state at time and For each in , , is
Moments
In this section, we provide formulae for the queue length process with initial distribution given by , where are the periodic asymptotic probabilities for the beginning of the period. We define the functions: for and with . Then , and more generally, for ,where the 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 .
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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2018, Applied Mathematical ModellingCitation Excerpt :They use the approximations and/or perturbation methods to identify several different time ranges where the behavior of the queue is different. Markovian single and multiple server queues with time-varying parameters have been analyzed by many authors (See e.g., [2,10–12,15,20–22,30,32,33] and the references therein). Extra references and outline of numerous mathematical techniques concerning queueing systems, where all the parameters depend on time, are accessible in Massey [24], Schwarz et al. [28] and Stolletz [29].
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