We consider a discrete population of users with homogeneous service demand who need to decide when to arrive to a system in which the service rate deteriorates linearly with the number of users in the system. The users have heterogeneous desired departure times from the system, and their goal is to minimise a weighted sum of the travel time and square deviation from the desired departure times. Users join the system sequentially, according to the order of their desired departure times. We model this scenario as a non-cooperative game in which each user selects his actual arrival time. We present explicit equilibria solutions for a two-user example, namely the Subgame Perfect and Cournot Nash equilibria and show that multiple equilibria may exist. We further explain why a general solution for any number of users is computationally challenging. The difficulty lies in the fact that the objective functions are piecewise-convex, i.e., non-smooth and non-convex. As a result, the minimisation of the costs relies on checking all arrival and departure order permutations, which is exponentially large with respect to the population size. Instead we propose an iterated best-response algorithm which can be efficiently studied numerically. Finally, we compare the equilibrium arrival profiles to a socially optimal solution and discuss the implications.
Bibliographical noteFunding Information:
We thank Yoni Nazarathy for his comments and advice and an anonymous reviewer for his/her most helpful comments. We are grateful to the Australia-Israel Scientific Exchange Foundation (AISEF) for supporting the first author’s visit to The Swinburne University of Technology. This work was supported by the Australian Research Council (ARC) Future Fellowships Grant FT120100723 , and the Israel Science Foundation Grant no. 1319/11 .
© 2016 Elsevier B.V.
- Arrival time game
- Linear slowdown
- Piecewise-convex programming
- Processor sharing
- Sequential timing game
ASJC Scopus subject areas
- Computer Science (all)
- Modeling and Simulation
- Management Science and Operations Research
- Information Systems and Management