1. Explicit Solutions for Coupled Parallel Queues.
- Author
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Bruneel, Herwig and Devos, Arnaud
- Subjects
WAITING rooms ,GENERATING functions ,CONSUMERS ,PROBABILITY theory - 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. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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