A generalized result for the discrete-time two-queue randomly alternating service system

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) $A(z_1,z_2)$ 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 $U(z_1,z_2)$ 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 $A(z_1,z_2)$ 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.