Explicit Solutions for Coupled Parallel Queues

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.