Markov Modelling Approach for Queues with Correlated Service Times -- the $M/M_D/2$ Model

TL;DR

This paper introduces a new Markov modelling approach for queues with correlated server services, specifically applied to the $M/M_D/2$ model, enabling easier analysis of system performance impacts.

Contribution

A novel Markov modelling method for queueing systems with correlated services is proposed and applied to the $M/M_D/2$ model, providing analytic solutions for stationary distributions.

Findings

Proved the queueing process is a Markov chain.

Derived an analytic stationary distribution for the $M/M_D/2$ system.

Highlighted the impact of service correlation on system performance.

Abstract

Demand for studying queueing systems with multiple servers providing correlated services was created about 60 years ago, motivated by various applications. In recent years, the importance of such studies has been significantly increased, supported by new applications of greater significance to much larger scaled industry, and the whole society. Such studies have been considered very challenging. In this paper, a new Markov modelling approach for queueing systems with servers providing correlated services is proposed. We apply this new proposed approach to a queueing system with arrivals according to a Poisson process and two positive correlated exponential servers, referred to as the $M/M_D/2$ queue. We first prove that the queueing process (the number of customers in the system) is a Markov chain, and then provide an analytic solution for the stationary distribution of the process, based on which it becomes much easier to see the impact of the dependence on system performance compared to the performance with independent services.