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Many-Server Queueing Systems with Heterogeneous Strategic Servers in Heavy Traffic

Burak Büke, Goncalo dos Reis, Vadim Platonov

TL;DR

The study develops a mean-field framework for many-server queues with heterogeneous, strategic servers in heavy traffic, introducing a measure-valued fairness process to capture idleness distribution across diverse server abilities. It demonstrates how staffing, routing, and strategic behavior interact to produce asymmetric Nash equilibria and analyzes regime selection between purely quality-driven and quality-and-efficiency-driven systems. A generalized random routing policy is characterized in the purely QD regime, showing how to realize target idleness densities via an $h$-random policy and a fixed-point equation for the induced $L_F$. The results provide practical insights into fairness, staffing, and regime choice in heterogeneous call centers and identify open challenges in designing skill-based routing and extending the model to multi-queue settings.

Abstract

In most service systems, the servers are humans who desire to experience a certain level of idleness. In call centers, this manifests itself as the call avoidance behavior, where servers strategically adjust their service rate to strike a balance between the idleness they receive and effort to work harder. Moreover, being humans, each server values this trade-off differently and has different capabilities. Drawing ideas on mean-field games we develop a novel framework relying on measure-valued processes to simultaneously address strategic server behavior and inherent server heterogeneity in service systems. This framework enables us to extend the recent literature on strategic servers in four new directions by: (i) incorporating individual choices of servers, (ii) incorporating individual abilities of servers, (iii) modeling the discomfort experienced by servers due to low levels of idleness, and (iv) considering more general routing policies. Using our framework, we are able to asymptotically characterize asymmetric Nash equilibria for many-server systems with strategic servers. In simpler cases, it has been shown that the purely quality-driven regime is asymptotically optimal. However, we show that if the discomfort increases fast enough as the idleness approaches zero, the quality-and-efficiency-driven regime and other quality driven regimes can be optimal. This is the first time this conclusion appears in the literature.

Many-Server Queueing Systems with Heterogeneous Strategic Servers in Heavy Traffic

TL;DR

The study develops a mean-field framework for many-server queues with heterogeneous, strategic servers in heavy traffic, introducing a measure-valued fairness process to capture idleness distribution across diverse server abilities. It demonstrates how staffing, routing, and strategic behavior interact to produce asymmetric Nash equilibria and analyzes regime selection between purely quality-driven and quality-and-efficiency-driven systems. A generalized random routing policy is characterized in the purely QD regime, showing how to realize target idleness densities via an -random policy and a fixed-point equation for the induced . The results provide practical insights into fairness, staffing, and regime choice in heterogeneous call centers and identify open challenges in designing skill-based routing and extending the model to multi-queue settings.

Abstract

In most service systems, the servers are humans who desire to experience a certain level of idleness. In call centers, this manifests itself as the call avoidance behavior, where servers strategically adjust their service rate to strike a balance between the idleness they receive and effort to work harder. Moreover, being humans, each server values this trade-off differently and has different capabilities. Drawing ideas on mean-field games we develop a novel framework relying on measure-valued processes to simultaneously address strategic server behavior and inherent server heterogeneity in service systems. This framework enables us to extend the recent literature on strategic servers in four new directions by: (i) incorporating individual choices of servers, (ii) incorporating individual abilities of servers, (iii) modeling the discomfort experienced by servers due to low levels of idleness, and (iv) considering more general routing policies. Using our framework, we are able to asymptotically characterize asymmetric Nash equilibria for many-server systems with strategic servers. In simpler cases, it has been shown that the purely quality-driven regime is asymptotically optimal. However, we show that if the discomfort increases fast enough as the idleness approaches zero, the quality-and-efficiency-driven regime and other quality driven regimes can be optimal. This is the first time this conclusion appears in the literature.
Paper Structure (21 sections, 21 theorems, 135 equations, 18 figures, 1 table)

This paper contains 21 sections, 21 theorems, 135 equations, 18 figures, 1 table.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. Notation
  4. Literature on Strategic Behavior in Queues
  5. A Many-Server Queueing Model with Strategic and Heterogeneous Servers
  6. Dynamics of the System Processes
  7. The Fairness Process
  8. Convergence of the Scaled System Length Processes
  9. A Generalized Random Routing Policy in the Purely Quality-Driven Regime
  10. Strategic Servers with Individual Preferences
  11. The Expected Long Run Proportion of Idleness Experienced by Individual Servers
  12. Characterization of Nash Equilibria for $h$-Random Policies
  13. Numerical Experiments
  14. Concluding Remarks
  15. Proofs for Theorems
  16. ...and 6 more sections

Key Result

Proposition 3.4

Suppose $1/2\leq \alpha <1$ and let $\eta_\alpha^{SSF}, \eta_\alpha^{FSF}, \eta_\alpha^{LISF}$ and $\eta_\alpha^{RR}$ be the limiting fairness processes corresponding to slowest-server-first, fastest-server-first, longest-idle-server-first and uniformly random routing, respectively. Then, for all $t

Figures (18)

  • Figure 6.1: The convergence of best response rates to the equilibrium
  • Figure 6.2: The sensitivity of the equilibrium mean service rate and the number of servers to various parameters
  • Figure 6.3: The sensitivity of the equilibrium mean service rate and the number of servers to staffing level $\beta$
  • Figure 6.4: The equilibrium distributions for different parametric setups
  • Figure B.1: Integral on the left-hand side of \ref{['eq:g_equilibrium_equation']} as function of $L_F$ for various values of $r$ and $\beta$.
  • ...and 13 more figures

Theorems & Definitions (43)

  • Proposition 3.4
  • Theorem 4.1
  • Theorem 4.2
  • Remark 4.3
  • Proposition 5.1
  • Definition 5.2
  • Lemma 5.3
  • Theorem 5.4
  • Lemma 5.5
  • Corollary 5.6
  • ...and 33 more