Non-cooperative queueing games on a network of single server queues

Corine M Laan,Judith Timmer,Richard J Boucherie

doi:10.1007/s11134-020-09681-9

Abstract

This paper introduces non-cooperative games on a network of single server queues with fixed routes. A player has a set of routes available and has to decide which route(s) to use for its customers. Each player’s goal is to minimize the expected sojourn time of its customers. We consider two cases: a continuous strategy space, where each player is allowed to divide its customers over multiple routes, and a discrete strategy space, where each player selects a single route for all its customers. For the continuous strategy space, we show that a unique pure-strategy Nash equilibrium exists that can be found using a best-response algorithm. For the discrete strategy space, we show that the game has a Nash equilibrium in mixed strategies, but need not have a pure-strategy Nash equilibrium. We show the existence of pure-strategy Nash equilibria for four subclasses: (i) N-player games with equal arrival rates for the players, (ii) 2-player games with identical service rates for all nodes, (iii) 2-player games on a 2times 2-grid, and (iv) 2-player games on an Atimes B-grid with small differences in the service rates.

Highlights

In this paper, we introduce and analyze a new type of queueing games: non-cooperative games on a network of single server queues with fixed routes
For the continuous strategy space, we show that a unique pure-strategy Nash equilibrium exists that can be found using a best-response algorithm
We show the existence of pure-strategy Nash equilibria for four subclasses: (i) N -player games with equal arrival rates for the players, (ii) 2-player games with identical service rates for all nodes, (iii) 2-player games on a 2 × 2-grid, and (iv) 2-player games on an A × B-grid with small differences in the service rates

Summary

Introduction

We introduce and analyze a new type of queueing games: non-cooperative games on a network of single server queues with fixed routes. The strategy profile uniquely determines the routes that are available for the customers of all players as well as the rate at which these customers arrive to the nodes along their route. To this end, let a customer of type j, r be a customer of player j that follows route r. Strategy profile p = ( p(1), ..., p(N)) is feasible if λi < μi for all i ∈ C This is the stability condition for the network of single server queues under which the equilibrium distribution for the number of customers at the nodes exists [11, p.

Continuous strategy space

Discrete strategy space

Conclusions