This paper studies a system of two parallel discrete-time queues with infinite waiting rooms and unit service times. Both queues have their own dedicated batch server, each having a randomly varying service capacity, where the service capacity of a server during a slot refers to the (maximum) number of customers the server can process during that slot. The service capacities are independent from slot to slot, strictly positive and geometrically distributed. Arrivals of new customers into the two queues occur independently from slot to slot, but are not necessarily mutually independent within a slot. Their joint probability-generating function (pgf) is indicated as A (z₁, z₂). We derive a kernel-type functional equation for the steady-state joint pgf U (z₁, z₂) of the system contents (numbers of customers present, including customers in service, if any) in both queues. Our main objective is to examine whether specific instances of the model exist for which the functional equation can be solved explicitly. We show that this is indeed the case. We first identify three particular cases where the solution can be found easily: independent arrivals, Bernoulli arrivals in one of the queues, infinite service capacity in one of the queues. Next, we establish a necessary and sufficient condition on the algebraic form of the kernel of the system in order to ensure that the queue contents of the two queues (excluding any customers in service) are mutually independent. We then demonstrate that the analysis simplifies considerably when one of the queues has unit service capacity. For this simplified case, we identify two types of arrival pgfs that lead to closed-form solutions, one based on the independent queue contents criterion and the other for identical arrivals. For the most general case, where both servers have geometric service capacities with arbitrary parameters, we first develop a transformation technique that leads to the solution of the functional equation for some cases. We illustrate this approach by means of a practical application in the context of a 2 2 output-buffered packet switch. Finding (other) arrival pgfs that allow (simple) explicit solutions for the most general functional equation turns out to be extremely difficult. Nevertheless, based on the criterion to have independent queue contents, we find several “solvable” (classes of) arrival pgfs, with the drawback, however, that they depend explicitly on the service capacities.
Herwig Bruneel (Mon,) studied this question.