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New Perspectives on the Erlang-A Queue

Andrew Daw, Jamol Pender

TL;DR

This work addresses nonstationary Erlang-A queues with abandonments by deriving non-asymptotic approximations for all moments, the MGF, and the CGF, together with explicit error bounds. The approach hinges on convexity-based arguments using Jensen's and the FKG inequality to produce bounds and to obtain a stochastic representation as a shifted Poisson (or $M/M/\infty$) queue. The results include exact steady-state and nonstationary characterizations of the MGF/CGF and extend to nonstationary Erlang-B and Erlang-C under stability. Numerical experiments validate the accuracy of the bounds and demonstrate the practical relevance for staffing and performance analysis.

Abstract

The non-stationary Erlang-A queue is a fundamental queueing model that is used to describe the dynamic behavior of large scale multi-server service systems that may experience customer abandonments, such as call centers, hospitals, and urban mobility systems. In this paper, we develop novel approximations to all of its transient and steady state moments, the moment generating function, and the cumulant generating function. We also provide precise bounds for the difference of our approximations and the true model. More importantly, we show that our approximations have explicit stochastic representations as shifted Poisson random variables. Moreover, we are also able to show that our approximations and bounds also hold for non-stationary Erlang-B and Erlang-C queueing models under certain stability conditions. Finally, we perform numerous simulations to support the conclusions of our results.

New Perspectives on the Erlang-A Queue

TL;DR

This work addresses nonstationary Erlang-A queues with abandonments by deriving non-asymptotic approximations for all moments, the MGF, and the CGF, together with explicit error bounds. The approach hinges on convexity-based arguments using Jensen's and the FKG inequality to produce bounds and to obtain a stochastic representation as a shifted Poisson (or ) queue. The results include exact steady-state and nonstationary characterizations of the MGF/CGF and extend to nonstationary Erlang-B and Erlang-C under stability. Numerical experiments validate the accuracy of the bounds and demonstrate the practical relevance for staffing and performance analysis.

Abstract

The non-stationary Erlang-A queue is a fundamental queueing model that is used to describe the dynamic behavior of large scale multi-server service systems that may experience customer abandonments, such as call centers, hospitals, and urban mobility systems. In this paper, we develop novel approximations to all of its transient and steady state moments, the moment generating function, and the cumulant generating function. We also provide precise bounds for the difference of our approximations and the true model. More importantly, we show that our approximations have explicit stochastic representations as shifted Poisson random variables. Moreover, we are also able to show that our approximations and bounds also hold for non-stationary Erlang-B and Erlang-C queueing models under certain stability conditions. Finally, we perform numerous simulations to support the conclusions of our results.

Paper Structure

This paper contains 16 sections, 24 theorems, 96 equations, 14 figures.

Table of Contents

  1. Introduction
  2. Main Contributions of the Paper
  3. Organization of the Paper
  4. The Erlang-A Queueing Model
  5. Mean Field Approximation is Identical to the Fluid Limit
  6. Inequalities for the Moments of the Erlang-A Queue
  7. Inequalities for the Mean
  8. Inequalities for the $m^{th}$ Moment
  9. Numerical Results
  10. Inequalities and Characterizations for Generating Functions of the Erlang-A Queue
  11. An Inequality for the Moment Generating Function of the Erlang-A Queue
  12. An Inequality for the Cumulant Moment Generating Function of the Erlang-A Queue
  13. Characterization of the Moment Generating Function in Steady-State
  14. Characterization of the Nonstationary Moment Generating Function
  15. Numerical Results
  16. ...and 1 more sections

Key Result

Lemma 3.1

Let $f: \mathbb{R}^2 \to \mathbb{R}$ be a continuous function in both variables. If we assume that initial value problem has a unique solution for the time interval [0,T] and then $x(t) \geq y(t)$ for all $t \in [0,T]$.

Figures (14)

  • Figure 1: $\lambda(t) = 10 + 2 \cdot \sin (t)$, $\mu = 1$, $Q(0) = 0$, $c=10$. $\theta = 0.5$ (Left) and $\theta = 2$ (Right).
  • Figure 2: $\lambda(t) = 10 + 2 \cdot \sin( t)$, $\mu = 1$, $\theta = 0.5$, $Q(0) = 0$, $c=10$.
  • Figure 3: $\lambda(t) = 10 + 2 \cdot \sin( t)$, $\mu = 1$, $\theta = 2$, $Q(0) = 0$, $c=10$.
  • Figure 4: $\lambda(t) = 100 + 20 \cdot \sin( t)$, $\mu = 1$, $\theta = 0.5$, $Q(0) = 0$, $c=100$.
  • Figure 5: $\lambda(t) = 100 + 20 \cdot \sin( t)$, $\mu = 1$, $\theta = 2$, $Q(0) = 0$, $c=100$.
  • ...and 9 more figures

Theorems & Definitions (49)

  • Remark 2.1
  • Lemma 3.1: A Comparison Lemma
  • Lemma 3.1: A Comparison Lemma
  • proof
  • Theorem 3.2
  • proof
  • Corollary 3.3
  • proof
  • Corollary 3.4
  • proof
  • ...and 39 more