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A Characterization of Optimal Queueing Regimes

Marco Scarsini, Eran Shmaya

TL;DR

The paper characterizes universally optimal queueing regimes for an M/M/1 with strategic entry decisions, introducing an abstract regime framework and a necessary-and-sufficient condition for universality: a new arrival may join at the back of the queue only if the past could not have supported a larger queue, with preemption emerging as essential in some cases. It shows Hassin's sufficient condition is not necessary by including the priority-slots regime and constructs a new universally optimal regime using a scoring priority mechanism. The results connect social-optimum thresholds with equilibrium behavior under any parameter set, providing design principles for parameter-free queue protocols that achieve welfare optimality across settings. This advances the theory of strategic queues by formalizing universal optimality and delivering concrete regime examples and proofs that unify prior insights on FCFS, preemption, and priority-based schemes. The findings have implications for capacity design and regulatory policies in service systems where requester behavior adapts to incentives and externalities.

Abstract

We consider an M/M/1 queueing model where customers can strategically decide to enter or leave the queue. We characterize the class of queueing regimes such that, for any parameters of the model, the socially efficient behavior is an equilibrium outcome.

A Characterization of Optimal Queueing Regimes

TL;DR

The paper characterizes universally optimal queueing regimes for an M/M/1 with strategic entry decisions, introducing an abstract regime framework and a necessary-and-sufficient condition for universality: a new arrival may join at the back of the queue only if the past could not have supported a larger queue, with preemption emerging as essential in some cases. It shows Hassin's sufficient condition is not necessary by including the priority-slots regime and constructs a new universally optimal regime using a scoring priority mechanism. The results connect social-optimum thresholds with equilibrium behavior under any parameter set, providing design principles for parameter-free queue protocols that achieve welfare optimality across settings. This advances the theory of strategic queues by formalizing universal optimality and delivering concrete regime examples and proofs that unify prior insights on FCFS, preemption, and priority-based schemes. The findings have implications for capacity design and regulatory policies in service systems where requester behavior adapts to incentives and externalities.

Abstract

We consider an M/M/1 queueing model where customers can strategically decide to enter or leave the queue. We characterize the class of queueing regimes such that, for any parameters of the model, the socially efficient behavior is an equilibrium outcome.
Paper Structure (12 sections, 4 theorems, 5 equations, 1 figure)

This paper contains 12 sections, 4 theorems, 5 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Related Work
  3. Notation
  4. Environment
  5. Social Optimum
  6. Queuing Regimes
  7. Strategies and Equilibrium
  8. Universally Optimal Regimes
  9. Proof of the Main Result
  10. Preliminaries
  11. The Proof
  12. List of symbols

Key Result

Theorem 8

The following two conditions are equivalent for a queuing regime:

Figures (1)

  • Figure 1: Suppose that before time $t_{0}$ the state is $x$, at time at time $t_{0}$ a customer arrives and the customer at position $i$, say, immediately reneges, and at time $t_{1}$ service is completed and nobody reneges. Round $k$ starts at time $t_{0}$ and ends at time $t_{1}$. The state during round $k$ is $x_{k}=\rho_{i}(\alpha(x))$. Round $k+1$ starts at time $t_{1}$ and ends with the next arrival or service at time $t_{2}$. The state during round $k+1$ is $x_{k+1}=\xi(x_{k})$.

Theorems & Definitions (18)

  • Remark 1
  • Example 2: FCFS
  • Example 3: LCFS
  • Example 4: PS
  • Definition 5
  • Example 6: FCFS
  • Example 7: Priority slots
  • Theorem 8
  • Corollary 9
  • proof
  • ...and 8 more