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Robust Queueing for Single-Server Queues with Abandonment

Wei You

TL;DR

Extensive numerical experiments demonstrate that the refined RQ approximation is accurate across underload, critical loading, and overload, and remains robust relative to existing heavy-traffic and hazard-rate-scaling benchmarks.

Abstract

Single-server queues with customer abandonment arise in call centers and many service systems, but steady-state performance measures remain analytically intractable beyond Markovian assumptions. This paper develops Robust Queueing (RQ) approximations for the mean steady-state virtual waiting time (offered waiting time) in the GI/GI/1+GI model. The approach starts from a reverse-time supremum representation of the virtual waiting time as the reflection of an effective net-input process that accounts for abandonments. We approximate effective net-input increments by their mean plus a robustness parameter times their standard deviation. For the drift, we introduce a Poisson-surrogate compensator and show that the associated correction term is asymptotically negligible in the long-patience regime. For variability, we propose two implementable surrogates: (i) a deterministic time-change approximation that yields a first RQ algorithm, and (ii) a refined algorithm based on a heavy-traffic limit that produces a scale-dependent variance function capturing the variance-reduction effect of abandonment. The resulting steady-state approximation reduces to a one-dimensional fixed point solvable by bisection and takes as input the arrival index of dispersion for counts (IDC), the service-time squared coefficient of variation, and the patience-time distribution. We further show how to extend the method to queues in series by feeding an approximation of the upstream departure IDC into the downstream RQ algorithm. Extensive numerical experiments demonstrate that the refined RQ approximation is accurate across underload, critical loading, and overload, and remains robust relative to existing heavy-traffic and hazard-rate-scaling benchmarks.

Robust Queueing for Single-Server Queues with Abandonment

TL;DR

Extensive numerical experiments demonstrate that the refined RQ approximation is accurate across underload, critical loading, and overload, and remains robust relative to existing heavy-traffic and hazard-rate-scaling benchmarks.

Abstract

Single-server queues with customer abandonment arise in call centers and many service systems, but steady-state performance measures remain analytically intractable beyond Markovian assumptions. This paper develops Robust Queueing (RQ) approximations for the mean steady-state virtual waiting time (offered waiting time) in the GI/GI/1+GI model. The approach starts from a reverse-time supremum representation of the virtual waiting time as the reflection of an effective net-input process that accounts for abandonments. We approximate effective net-input increments by their mean plus a robustness parameter times their standard deviation. For the drift, we introduce a Poisson-surrogate compensator and show that the associated correction term is asymptotically negligible in the long-patience regime. For variability, we propose two implementable surrogates: (i) a deterministic time-change approximation that yields a first RQ algorithm, and (ii) a refined algorithm based on a heavy-traffic limit that produces a scale-dependent variance function capturing the variance-reduction effect of abandonment. The resulting steady-state approximation reduces to a one-dimensional fixed point solvable by bisection and takes as input the arrival index of dispersion for counts (IDC), the service-time squared coefficient of variation, and the patience-time distribution. We further show how to extend the method to queues in series by feeding an approximation of the upstream departure IDC into the downstream RQ algorithm. Extensive numerical experiments demonstrate that the refined RQ approximation is accurate across underload, critical loading, and overload, and remains robust relative to existing heavy-traffic and hazard-rate-scaling benchmarks.
Paper Structure (81 sections, 19 theorems, 342 equations, 5 figures)

This paper contains 81 sections, 19 theorems, 342 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Exact analysis and structural properties.
  4. Heavy-traffic diffusion approximations.
  5. Robust queueing and indices of dispersion.
  6. Contributions and Organization
  7. Preliminary
  8. Review of Robust Queueing for the G/GI/1 Model
  9. The Dynamics of the G/GI/1+GI Model
  10. The Drift of the Effective Net-Input Process
  11. Queues with Poisson arrivals
  12. Queues with a General Renewal Arrival Process
  13. A First Robust Queueing Algorithm
  14. The First RQ Algorithm
  15. Heavy-Traffic Limits
  16. ...and 66 more sections

Key Result

Lemma 1

Under Assumption assumption, suppose that $A(\cdot)$ is a homogeneous Poisson process with rate $\lambda$. Then, for any $0\le s\le t$, Moreover, if $\{Z(u)\}_{u\in\mathbb R}$ is strictly stationary, then for all $0\le s\le t$,

Figures (5)

  • Figure 1: Simulation estimates (solid) and approximations \ref{['eq:V']} (dashed) of the effective IDW in the $H_2(4)/M/1{+}M$ model (left) the $H_2(4)/M/1{+}E_2$ model (right) with $\mu=1$, $c=2$, and $\alpha=2^{-i}$ for $i\in\{0,3,6,9,12\}$.
  • Figure 2: Signed relative error heat maps for the refined RQ approximation (left column), the Ward--Glynn approximation ward2003diffusionward2005diffusion (center column; replaced by the hazard-rate scaling approximation reed2008approximating when $f(0)=0$), and the Huang--Gurvich approximation huang2018beyond (right column), for the $M/M/1{+}M$ model (top row), the $M/M/1{+}E_2$ model (mid row), and the $M/M/1{+}H_2(4)$ model (bottom row).
  • Figure 3: Signed relative error heat maps for the refined RQ approximation (left column), the Ward--Glynn approximation ward2003diffusionward2005diffusion (center column; replaced by the hazard-rate scaling approximation reed2008approximating when $f(0)=0$), and the Huang--Gurvich approximation huang2018beyond (right column), for the $M/LN(1,4)/1{+}H_2(4)$ model (top row) and the $M/LN(1,4)/1{+}E_2$ model (bottom row).
  • Figure 4: Signed relative error heat maps for the refined RQ approximation (left column), the Ward--Glynn approximation ward2003diffusionward2005diffusion (center column, when $f(0)>0$; replaced by the hazard-rate scaling approximation reed2008approximating when $f(0)=0$), and the Huang--Gurvich approximation huang2018beyond (right column), for the $E_2/LN(1,2)/1{+}E_2$ (top row), $H_2(4)/LN(1,2)/1{+}H_2(4)$ (mid row), and $H_2(4)/LN(1,2)/1{+}E_2$ (bottom row) models.
  • Figure 5: Signed relative error heat maps for tandem systems with non-renewal arrivals at Queue 2. Top row: $H_2(4)/E_2/1 \to \cdot/M/1 {+} H_2(4)$. Bottom row: $E_2/H_2(4)/1 \to \cdot/M/1 {+} E_2$. In each row, the left panel corresponds to the (refined) RQ approximation, the middle panel corresponds to the Ward--Glynn approximation when applicable (top row) or its hazard-rate-scaling variant when $f(0)=0$ (bottom row), and the right panel corresponds to the Huang--Gurvich approximation.

Theorems & Definitions (41)

  • Remark 1: Comparison to Brownian queues
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Theorem 1: Heavy-traffic limit for RQ
  • Corollary 1: Calibration of RQ for the underloaded regime
  • Proposition 1: Theorem 1, ward2005diffusion
  • Theorem 2: Heavy-traffic limit for $M/M/1{+}GI$
  • Lemma 4
  • proof
  • ...and 31 more