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 \(\gamma ^m \delta ^n\) . Using a transform-based approach, we prove that this is certainly the case if the steady-state joint probability generating function \(U(z_1,z_2)\) 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 \(z_1\) and \(z_2\) , respectively, whose zeroes \(\hat{z_1}\) and \(\hat{z_2}\) all have multiplicity one. We show that the decay rates \(\gamma \) and \(\delta \) appearing in u(m, n) are the inverse values of (some of) the zeroes \(\hat{z_1}\) and \(\hat{z_2}\) , but, in general, there may be zero-pairs \((\hat{z_1}, \hat{z_2})\) that do not contribute a bivariate geometric term in u(m, n). For two specific classes of discrete-time two-queue systems, we prove that, when \(U(z_1,z_2)\) has the prescribed form, only the zero-pairs that are zero-tuples of the kernel of the system contribute a term in u(m, n). In an extended series of examples, we then demonstrate that, within the two classes, specific instances (corresponding with specific arrival processes) that comply with the condition on \(U(z_1,z_2)\) indeed exist. In some examples, we can use existing solutions, but in other cases, we also construct entirely new solutions, thereby identifying several new solvable two-queue models. We observe that, in most cases, the zero-pairs that do contribute terms in u(m, n) are mutually connected, in the sense that each of them has its first or second component in common with at least one other pair that contributes, but we also construct a remarkable example where this is not the case.
