Coupled queues with server interruptions: Some solutions

Abstract

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 $A(z_1,z_2)$. 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 $U(z_1,z_2)$ of the numbers of type-1 and type-2 customers in the system. Relying on the application of Rouché’s theorem, we are able to explicitly solve the functional equation for arbitrary arrival pgfs $A(z_1,z_2)$, but more elegant results are obtained for some specific choices of $A(z_1,z_2)$.

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 $U(z_1,z_2)$. Although solving these equations for arbitrary arrival pgfs $A(z_1,z_2)$ seems infeasible, we succeed in finding exact closed-form solutions for specific choices of $A(z_1,z_2)$. 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.