Coupled queues whose interior stationary joint content distribution is a finite sum of bivariate geometric terms
Year: 2025
Type: article
Abstract: Triggered by earlier work on random walks in the quarter-plane, we study the issue of two-queue systems whereby, at least for states ( m , n ) in some interior part of the state space, the stationary joint system-content distribution u ( m , n ) can be expressed as a finite linear combination of bivariate geometric terms of type $$\gamma ^m \delta ^n$$ γ m δ n . Using a transform-based approach, we prove that this is certainly the case if the steady-state joint probability generating function $$U(z_1,z_2)$$ U ( z 1 , z 2 ) of the two system contents can be expressed as a bivariate rational function of its two arguments, with mutually prime numerator and denominator, whereby the denominator is the product of two univariate polynomials in $$z_1$$ z 1 and $$z_2$$ z 2 , respectively, whose zeroes $$\hat{z_1}$$ z 1 ^ and $$\hat{z_2}$$ z 2 ^ all have multiplicity one . We show that the decay rates $$\gamma $$ γ and $$\delta $$ δ appearing in u ( m , n ) are the inverse values of (some of) the zeroes $$\hat{z_1}$$ z 1 ^ and $$\hat{z_2}$$ z 2 ^ , but, in general, there may be zero-pairs $$(\hat{z_1}, \hat{z_2})$$ ( z 1 ^ , z 2 ^ ) that do not contribute a bivariate geometric term in u ( m , n ). For two specific classes
Source: Top
Authors: Herwig Bruneel, Arnaud Devos
Institution: Ghent University
Language: English
Subfield: Management Information Systems
Domain: Social Sciences
Open Access status: hybrid
Estimated APC paid: $2,990
OpenAlex ID: W4417412201