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

Abstract

In this paper, we analyze a discrete-time queueing model with two types (classes) of customers and two servers, one for each customer class. Although each server can only process one type of customers, all customers are accommodated in one single queue and served in their order of arrival, irrespective of their types. The numbers of customers arriving in the system from time slot to time slot are independent, but the types of consecutive customers are not necessarily independent. Specifically, we assume that a first-order Markovian correlation ("interclass correlation") exists between the types of subsequent customers in the arrival stream. The fact that multiple customers of the same type may arrive back-to-back and customers have to be served in their order of arrival, causes occasional under-utilization of the service capacity of the system, because some customers may not be able to reach their server owing to the presence of customers of the opposite type in front of them.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.