An $\epsilon$-Nash equilibrium with high probability for strategic customers in heavy traffic

TL;DR

This paper analyzes strategic customer behavior in heavy traffic multiclass queues, showing that an approximate Nash equilibrium exists with high probability by leveraging heavy traffic phenomena and new diffusion limit results.

Contribution

It introduces a game theoretic model for queue decisions in heavy traffic and proves the existence of an $ ext{epsilon}$-Nash equilibrium with high probability, supported by new diffusion limit results.

Findings

Existence of an $ ext{epsilon}$-Nash equilibrium with probability approaching 1.

Development of new diffusion limit results for finite buffer systems.

Application of Reiman's snapshot principle to predict waiting times in strategic decision-making.

Abstract

A multiclass queue with many servers is considered, where customers make a join-or-leave decision upon arrival based on queue length information, without knowing the scheduling policy or the state of other queues. A game theoretic formulation is proposed and analyzed, that takes advantage of a phenomenon unique to heavy traffic regimes, namely Reiman's snaphshot principle, by which waiting times are predicted with high precision by the information available upon arrival. The payoff considered is given as a random variable, which depends on the customer's decision, accounting for waiting time in the queue and penalty for leaving. The notion of an equilibrium is only meaningful in an asymptotic framework, which is taken here to be the Halfin-Whitt heavy traffic regime. The main result is the identification of an $\epsilon$-Nash equilibrium with probability approaching 1. On way to proving this result, new diffusion limit results for systems with finite buffers are obtained.