Analytic Computation Schemes for the Discrete-Time Bulk Service Queue

A. J. E. M. Janssen, J. S. H. van Leeuwaarden
2005 Queueing systems  
In commonly used root-finding approaches for the discrete-time bulk service queue, the stationary queue length distribution follows from the roots inside or outside the unit circle of a characteristic equation. We present analytic representations of these roots in the form of sample values of periodic functions with analytically given Fourier series coefficients, making these approaches more transparent and explicit. The resulting computational scheme is easy to implement and numerically
more » ... We also discuss a method to determine the roots by applying successive substitutions to a fixed point equation. We outline under which conditions this method works, and compare these conditions with those needed for the Fourier series representation. Finally, we present a solution for the stationary queue length distribution that does not depend on roots. This solution is explicit and well-suited for determining tail probabilities up to a high accuracy, as demonstrated by some numerical examples. keywords: discrete-time bulk service, multi-server, roots, stationary distribution, Szegö curve, Spitzer's identity. Here, time is assumed to be slotted, X n denotes the queue length at the beginning of slot n, A n denotes the number of new arriving customers during slot n, and s denotes the fixed number of customers that can be served during one slot. The sequence of A n is assumed to be independent and identically distributed (i.i.d.) according to a discrete random variable A with probability generating function (pgf) A(z). Without loss of generality we assume throughout that P (A = 0) > 0. The pgf of the stationary queue length in the discrete-time bulk service queue was first derived by Bruneel & Wuyts [4], although the same pgf occurs in earlier work on the D/G/1 queue by Servi [25] and on bulk queues by Powell [24] . The solution requires finding the roots of z s = A(z) within the unit circle. Zhao & Campbell [30] presented a full solution for the stationary queue length distribution in terms of the roots of z s = A(z) outside the unit circle, assuming that A(z) is a polynomial. Chaudhry & Kim [7] used the same technique and presented some numerical work. The technique of finding roots to complete a transform has become a classic one in queueing theory. It started from the analysis of the M/D/s queue by Crommelin [10], whose solution required finding the roots within the unit circle of z s = e λ(z−1) for some value λ < s. Through the years, root-finding turned out to be particularly important in the theory of bulk queues, originating from the work of Bailey [2] and Downton [13], who consider a bulk service queue with Poisson arrivals. For an overview on bulk queues we refer to Powell [24] and Chaudhry & Templeton [8]. Initially, the potential difficulties of root-finding were considered to be a slur on the unblemished transforms, since the determination of the roots can be numerically hazardous and the roots themselves have no probabilistic interpretation. However, Chaudhry and others [6] have made every effort to dispel the scepticism towards root-finding in queueing theory. They emphasize that root-finding in queueing is well-structured, in the sense that the roots are distinct for most models and that their location is well-predictable, so that numerical problems are not likely to occur. While in general this is true for the moments of the stationary queue length, for determining the distribution itself, dependence on the roots might cause some problems, in particular for tail probabilities. In Chaudhry & Kim [7] a comparison is made between using the roots of z s = A(z) either inside or outside the unit circle. The performance of both approaches, though, heavily depends on the model parameters. Since it is therefore difficult to give a fair comparison, we choose to stress their common weakness: their performance inherently depends on how precise the roots of z s = A(z) are determined. Any deviation of the numerically determined roots from their true values results in errors in the computed probabilities. The main purpose of this paper is to present an analytical rather than a numerical framework for dealing with the discrete-time bulk service queue. In particular, we will present explicit expressions for the roots of z s = A(z) and the stationary queue length distribution. Under some mild conditions, we show that the roots of z s = A(z), both inside and outside the unit circle, can be represented as sample values of a periodic function with analytically given Fourier coefficients. In this way, the roots are no longer implicitly defined, and one can determine the roots as accurately as one wishes in a totally transparent way. Another way to determine the roots while maintaining transparency, results from applying successive substitution to a fixed-point equation. This idea originates from the work of Harris et al. [16] on root-finding for the continuous-time G/E k /1 queue, and was presented more formally by Adan & Zhao [1] who distinguished a class of continuous distributions for which the method works. In this paper we further investigate the method for finding the roots of z s = A(z) for discrete distributions whose pgf is A(z). We present necessary conditions for the method to
doi:10.1007/s11134-005-0402-z fatcat:kegbjcfdz5g7phyejefwpzsmjm