Explicit Solutions for Coupled Parallel Queues

Author:

Bruneel Herwig1ORCID,Devos Arnaud1ORCID

Affiliation:

1. SMACS Research Group, Department of Telecommunications and Information Processing, Ghent University, 9000 Ghent, Belgium

Abstract

We consider a system of two coupled parallel queues with infinite waiting rooms. The time setting is discrete. In either queue, the service of a customer requires exactly one discrete time slot. Arrivals of new customers occur independently from slot to slot, but the numbers of arrivals into both queues within a slot may be mutually dependent. Their joint probability generating function (pgf) is indicated as A(z1,z2) and characterizes the whole model. In general, determining the steady-state joint probability mass function (pmf) u(m,n),m,n≥0 or the corresponding joint pgf U(z1,z2) of the numbers of customers present in both queues is a formidable task. Only for very specific choices of the arrival pgf A(z1,z2) are explicit results known. In this paper, we identify a multi-parameter, generic class of arrival pgfs A(z1,z2), for which we can explicitly determine the system-content pgf U(z1,z2). We find that, for arrival pgfs of this class, U(z1,z2) has a denominator that is a product, say r1(z1)r2(z2), of two univariate functions. This property allows a straightforward inversion of U(z1,z2), resulting in a pmf u(m,n) which can be expressed as a finite linear combination of bivariate geometric terms. We observe that our generic model encompasses most of the previously known results as special cases.

Publisher

MDPI AG

Reference62 articles.

1. The equilibrium distribution for a clocked buffered switch;Jaffe;Probab. Eng. Informational Sci.,1992

2. The compensation approach applied to a 2 × 2 switch;Boxma;Probab. Eng. Informational Sci.,1993

3. On the determination of the stationary distribution of a symmetric clocked buffered switch;Ramaswami;Proceedings of the ITC-15, Teletraffic Contributions for the Information Age,1997

4. On the asymmetric clocked buffered switch;Cohen;Queueing Syst.,1998

5. Queueing models with multiple waiting lines;Adan;Queueing Syst.,2001

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