Explicit Solutions for Coupled Parallel Queues

We consider a system of two coupled parallel queues 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 into both queues within a slot may be mutually dependent. Their joint probability generating function (pgf) is indicated as A(z1,z2) and characterizes the whole model. 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 numbers of customers present in both queues is a formidable task. Only for very specific choices of the arrival pgf A(z1,z2) are explicit results known. In this paper, we identify a multi-parameter, 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 that 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 observe that our generic model encompasses most of the previously known results as special cases.