preprints.org > computer science and mathematics > probability and statistics > doi: 10.20944/preprints202407.0693.v1

Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

Version 1 : Received: 4 July 2024 / Approved: 8 July 2024 / Online: 9 July 2024 (10:07:29 CEST)

We consider a system of two coupled parallel queues, queue 1 and queue 2, 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 of both types within a slot may be mutually interdependent. Their joint probability generating function (pgf) is indicated as A(z1, z2) and characterizes the whole model. It is well-known that, 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 system contents (i.e., numbers of customers present) in both queues is a formidable task. Only in a number of isolated cases, for very specific choices of the arrival pgf A(z1, z2), explicit results are known in the literature. In this paper, we identify a multiparameter, 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 which 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 also observe that our generic model encompasses most of the previously known results as special cases.

parallel queues; discrete-time; joint system-content distribution; explicit solutions

Computer Science and Mathematics, Probability and Statistics

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