Explicit Solutions for Coupled Parallel Queues

Herwig Bruneel () and Arnaud Devos
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Herwig Bruneel: SMACS Research Group, Department of Telecommunications and Information Processing, Ghent University, 9000 Ghent, Belgium
Arnaud Devos: SMACS Research Group, Department of Telecommunications and Information Processing, Ghent University, 9000 Ghent, Belgium

Mathematics, 2024, vol. 12, issue 15, 1-31

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 ( z 1 , z 2 ) 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 ( z 1 , z 2 ) of the numbers of customers present in both queues is a formidable task. Only for very specific choices of the arrival pgf A ( z 1 , z 2 ) are explicit results known. In this paper, we identify a multi-parameter, generic class of arrival pgfs A ( z 1 , z 2 ) , for which we can explicitly determine the system-content pgf U ( z 1 , z 2 ) . We find that, for arrival pgfs of this class, U ( z 1 , z 2 ) has a denominator that is a product, say r 1 ( z 1 ) r 2 ( z 2 ) , of two univariate functions. This property allows a straightforward inversion of U ( z 1 , z 2 ) , 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.

Keywords: parallel queues; discrete time; joint system-content distribution; explicit solutions (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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