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 $A(z_1,z_2)$ and characterizes the whole model. In general, determining the steady-state joint probability mass function (pmf) $u(m,n),m,n\ge 0$ or the corresponding joint pgf $U(z_1,z_2)$ of the numbers of customers present in both queues is a formidable task. Only for very specific choices of the arrival pgf $A(z_1,z_2)$ are explicit results known. In this paper, we identify a multi-parameter, generic class of arrival pgfs $A(z_1,z_2)$, for which we can explicitly determine the system-content pgf $U(z_1,z_2)$. We find that, for arrival pgfs of this class, $U(z_1,z_2)$ has a denominator that is a product, say $r_1\left(z_1\right)r_2\left(z_2\right)$, of two univariate functions. This property allows a straightforward inversion of $U(z_1,z_2)$, 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.