### Abstract

In this paper we introduce non-cooperative games on a Jackson network. A player has a set of routes available and has to decide which routes to use for its customers. Each player's goal is to minimize the expected sojourn time of its customers. We consider two cases. First, each player is allowed to divide his customers over multiple routes. This results in a game for which it can be shown that a unique pure-strategy Nash equilibrium exists. This Nash equilibrium can be found by using a best-response algorithm.

Second, each player may only select a single route for its customers. This results in a game with finite strategy spaces. In general, such games need not have a pure-strategy Nash equilibrium, as shown by an example. We show the existence of pure-strategy Nash equilibria for four subclasses of games on a Jackson network: (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 x 2-grid, and (iv) 2-player games on an A x B-grid with small differences in the service rates.

Second, each player may only select a single route for its customers. This results in a game with finite strategy spaces. In general, such games need not have a pure-strategy Nash equilibrium, as shown by an example. We show the existence of pure-strategy Nash equilibria for four subclasses of games on a Jackson network: (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 x 2-grid, and (iv) 2-player games on an A x B-grid with small differences in the service rates.

Original language | English |
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Place of Publication | Enschede |

Publisher | University of Twente |

Number of pages | 16 |

Volume | 2066 |

Publication status | Published - Jan 2019 |

### Publication series

Name | Memorandum of the Department of Applied Mathematics |
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ISSN (Print) | 1874-4850 |

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### Keywords

- Non-cooperative games
- Routing
- Pure-strategy Nash equilibria
- Jackson networks

### Cite this

Laan, C. M., Timmer, J., & Boucherie, R. J. (2019).

*Non-cooperative queueing games on a Jackson network*. (Memorandum of the Department of Applied Mathematics). Enschede: University of Twente.