Performance analysis of a discrete-time two-class global-FCFS queue with two servers and geometric service times

In this paper, we assume that the service times of both types of customers are independent, geometrically distributed random variables. The paper extends earlier work where all the service times were assumed to be of fixed length, either equal to 1 slot each, or equal to multiple slots. The fact that, in the present paper, service times are of variable length, entails that customers being served simultaneously can overtake each other, thus disturbing the original arrival order. This phenomenon did not occur in previous studies with fixed-length service times, and represents the main new element of the paper. It also complicates the analysis of the system considerably. Nevertheless, we are able to derive explicit expressions for the probability generating functions and the mean values of the main performance measures of the system, in terms of the original system parameters and one root of a non-linear equation. Our results reveal the impact of the interclass correlation and the variable nature of the service times on the achievable throughput, the (mean) number of customers in the system, the (mean) customer sojourn times, the (mean) unfinished work in the system, and related quantities.