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

Suman Thapa, Yiqiang Q. Zhao

TL;DR

The paper addresses queueing with dependent (correlated) service times by developing a Markovian framework for the two-server system $M/M_D/2$ with MO-BVE service times. It proves the number-in-system process is a CTMC, derives a tractable stationary distribution that is geometric with boundary corrections, and uses it to quantify how positive dependence alters performance relative to the independent-server case. The approach extends conventional birth–death analysis to dependent services, accounting for simultaneous departures due to MO-BED singularities, and provides analytic insights into how dependence impacts metrics like the expected number in the system and in the queue. This work offers a mathematically rigorous, analytically tractable method for assessing dependence in two-server queues with applications to networks and large-scale service systems.

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.

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

TL;DR

The paper addresses queueing with dependent (correlated) service times by developing a Markovian framework for the two-server system with MO-BVE service times. It proves the number-in-system process is a CTMC, derives a tractable stationary distribution that is geometric with boundary corrections, and uses it to quantify how positive dependence alters performance relative to the independent-server case. The approach extends conventional birth–death analysis to dependent services, accounting for simultaneous departures due to MO-BED singularities, and provides analytic insights into how dependence impacts metrics like the expected number in the system and in the queue. This work offers a mathematically rigorous, analytically tractable method for assessing dependence in two-server queues with applications to networks and large-scale service systems.

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 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.
Paper Structure (8 sections, 1 theorem, 9 equations)

This paper contains 8 sections, 1 theorem, 9 equations.

Table of Contents

  1. Introduction
  2. Motivations
  3. Literature review
  4. Main contributions in this paper
  5. Organization of the paper
  6. Bivariate dependent service times
  7. The $M/M_D/2$ model
  8. Markov chain

Key Result

Theorem 1

For the $M/M_D/2$ queueing system with arrival rate $\lambda$ and two heterogeneous dependent servers characterized by the MO-BED with parameters $\mu_1$, $\mu_2$ and $\mu_{12}$ given in (eqn:density of MO-BED-b), the process $\{X(t): t \geq 0\}$ defined in the previous section, representing the num

Theorems & Definitions (3)

  • Definition 2.1
  • Remark 3.1
  • Theorem 1