We study three different discrete-time queueing systems, which accommodate two types of customers, named type 1 and type 2. New customers arrive independently from slot to slot, but the numbers of arrivals of both types in any slot are possibly mutually dependent; their joint probability generating function ( pgf ) is A(z1,z2). The service times of all customers are deterministically equal to one time slot. We first consider a scenario ( Option A) with one single server which is to be shared by the two customer types. Here, we assume that type-1 customers have absolute service priority over type-2 customers. Moreover, the server is subject to random server interruptions, which occur independently from slot to slot. We derive a functional equation for the steady-state joint pgf U(z1, z2) of the numbers of type-1 and type-2 customers in the system. Relying on the application of Rouch & eacute;'s theorem, we are able to explicitly solve the functional equation for arbitrary arrival pgfs A(z1,z2), but more elegant results are obtained for some specific choices of A(z1,z2). Next, we focus on two different scenarios ( Option B and Option C) where both customer types have their own dedicated server. Here, there are no service priorities involved. In Option B, the two servers experience simultaneous interruptions, whereas in Option C, only one of the servers is subject to interruptions. Again, we derive functional equations for the pgf U(z1, z 2 ). Although solving these equations for arbitrary arrival pgfs A(z1, z2) seems infeasible, we succeed in finding exact closed-form solutions for specific choices of A(z1,z2). Remarkably, the results obtained for the single-server priority system in Option A can be used to solve a specific instance of Option B, where the arrivals of both types of customers during any time slot are partly identical. It turns out that (fully or partly) identical arrivals also allow explicit solutions for Option C. In addition, we also provide other examples where the functional equations for Options B and C can be solved explicitly.

In this paper, we revisit the discrete-time two-queue randomly alternating service system, where one common server is shared by two queues by allocating the server, independently from slot to slot, with fixed probabilities to either queue. Arrivals of new customers into the two-queue system occur independently from slot to slot, but may be mutually dependent within a slot. They are characterized by the joint probability generating function (pgf) A(z1,z2) of the numbers of arrivals in both queues during one time slot. The service times of all customers are equal to exactly one time slot. We extend various existing results with respect to the queueing behavior of this system. Specifically, we show that the exact solutions that were previously found for the steady-state joint pgf U(z1, z2) of the system contents in both queues for the scenarios of independent Bernoulli arrivals, identical Bernoulli arrivals, global geometric arrivals, global geometric group arrivals, and the superposition of identical Bernoulli arrivals and global geometric (group) arrivals, are all special cases of a more general result, which is valid for a whole class of arrival pgfs A(z1,z2) that (among others) encompasses the aforementioned specific arrival scenarios. However, the defined class is much broader than this, and our new result allows the solution for entirely new arrival pgfs as well. We illustrate this abundantly with a large number of detailed examples. The proof of the general result is a mainly algebraic one and, unlike earlier studies, does not require intricate techniques from complex-function analysis.