Coupled queues whose interior stationary joint content distribution is a finite sum of bivariate geometric terms

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 $$\gamma ^m \delta ^n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>γ</mml:mi> <mml:mi>m</mml:mi> </mml:msup> <mml:msup> <mml:mi>δ</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> </mml:math> . Using a transform-based approach, we prove that this is certainly the case if the steady-state joint probability generating function $$U(z_1,z_2)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>U</mml:mi> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>z</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>z</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> 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 $$z_1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>z</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> and $$z_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>z</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> , respectively, whose zeroes $$\hat{z_1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:msub> <mml:mi>z</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>^</mml:mo> </mml:mover> </mml:math> and $$\hat{z_2}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:msub> <mml:mi>z</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>^</mml:mo> </mml:mover> </mml:math> all have multiplicity one . We show that the decay rates $$\gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>γ</mml:mi> </mml:math> and $$\delta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>δ</mml:mi> </mml:math> appearing in u ( m , n ) are the inverse values of (some of) the zeroes $$\hat{z_1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:msub> <mml:mi>z</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>^</mml:mo> </mml:mover> </mml:math> and $$\hat{z_2}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:msub> <mml:mi>z</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>^</mml:mo> </mml:mover> </mml:math> , but, in general, there may be zero-pairs $$(\hat{z_1}, \hat{z_2})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mover> <mml:msub> <mml:mi>z</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>^</mml:mo> </mml:mover> <mml:mo>,</mml:mo> <mml:mover> <mml:msub> <mml:mi>z</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>^</mml:mo> </mml:mover> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> that do not contribute a bivariate geometric term in u ( m , n ). For two specific classes</jats