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An M/M/c queue with queueing-time dependent service rates

Bernardo D'Auria, Ivo J. B. F. Adan, René Bekker, Vidyadhar Kulkarni

TL;DR

This paper analyzes an M/M/c queue in which service rates depend on the experienced queueing time via a threshold, leading to two classes of customers and a Markovian description of the virtual waiting time $W(t)$. The authors derive the limiting distribution as a mixture of exponentials by solving integro-differential balance equations and provide both scalar and matrix representations, enabling exact computation of the stationary distribution and mean waiting time. They present a complete solution framework, including a detailed treatment of special cases (notably $c=1$ and $c=2$) and a numerical study demonstrating that differences between $\mu_1$ and $\mu_2$ can have substantial impact on queue performance. The results offer rigorous insights into how delay-dependent service alters performance metrics and can inform managerial decisions in health care, call centers, and networks where slowdown or speedup mechanisms occur.

Abstract

Recent studies indicate that in many situations service times are affected by the experienced queueing delay of the particular customer. This effect has been detected in different areas, such as health care, call centers and telecommunication networks. In this paper we present a methodology to analyze a model having this property. The specific model is an M/M/c queue in which any customer may be tagged at her arrival time if her queueing time will be above a certain fixed threshold. All tagged customers are then served at a given rate that may differ from the rate used for the non-tagged customers. We show how it is possible to model the virtual queueing time of this queueing system by a specific Markov chain. Then, solving the corresponding balance equations, we give a recursive solution to compute the stationary distribution, which involves a mixture of exponential terms. Using numerical experiments, we demonstrate that the differences in service rates can have a crucial impact on queueing time performance.

An M/M/c queue with queueing-time dependent service rates

TL;DR

This paper analyzes an M/M/c queue in which service rates depend on the experienced queueing time via a threshold, leading to two classes of customers and a Markovian description of the virtual waiting time . The authors derive the limiting distribution as a mixture of exponentials by solving integro-differential balance equations and provide both scalar and matrix representations, enabling exact computation of the stationary distribution and mean waiting time. They present a complete solution framework, including a detailed treatment of special cases (notably and ) and a numerical study demonstrating that differences between and can have substantial impact on queue performance. The results offer rigorous insights into how delay-dependent service alters performance metrics and can inform managerial decisions in health care, call centers, and networks where slowdown or speedup mechanisms occur.

Abstract

Recent studies indicate that in many situations service times are affected by the experienced queueing delay of the particular customer. This effect has been detected in different areas, such as health care, call centers and telecommunication networks. In this paper we present a methodology to analyze a model having this property. The specific model is an M/M/c queue in which any customer may be tagged at her arrival time if her queueing time will be above a certain fixed threshold. All tagged customers are then served at a given rate that may differ from the rate used for the non-tagged customers. We show how it is possible to model the virtual queueing time of this queueing system by a specific Markov chain. Then, solving the corresponding balance equations, we give a recursive solution to compute the stationary distribution, which involves a mixture of exponential terms. Using numerical experiments, we demonstrate that the differences in service rates can have a crucial impact on queueing time performance.

Paper Structure

This paper contains 19 sections, 10 theorems, 122 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Model and state description
  3. Limiting distribution: balance equations
  4. Solution of the balance equations
  5. Limiting distribution
  6. Computing the solution
  7. Special cases
  8. Single-server queue
  9. Numerical example for c=2
  10. Numerical insights
  11. Proofs
  12. Proof of Theorem \ref{['th:dFx']}:
  13. Proof of Theorem \ref{['th:dFx2']}:
  14. Proof of Corollary \ref{['cor:bc']}:
  15. Proof of Lemma \ref{['lm:right.eigenvec']}:
  16. ...and 4 more sections

Key Result

Theorem 1

The limiting distribution vector $F$ satisfies the following integro-differential equations:

Figures (7)

  • Figure 1: Sample path of the VQT process $W(t)$ for the case $c=1$.
  • Figure 2: Sketch of the transition diagram for the case $c=2$.
  • Figure 3: Stationary VQT cumulative distribution function for $\lambda=2$, $\mu_1 = 0.75$, $\mu_2=1.12$ and $k=0.45$.
  • Figure 4: Cdf of the stationary VQT for $k=5$, $c=3$, $\lambda = 2$, $\mu_1 =0.8$, and $\mu_2\in\{0.7,0.8,0.9\}$.
  • Figure 5: Stationary VQT density for $k=5$, $c=3$, $\lambda = 2$, $\mu_2 =0.8$, and $\mu_1\in\{0.3,0.6,0.67,0.74,0.8\}$.
  • ...and 2 more figures

Theorems & Definitions (23)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Corollary 1
  • proof
  • Remark 1
  • Lemma 1
  • proof
  • Theorem 3
  • ...and 13 more