Strategic behavior and optimization in a hybrid M/M/1 queue with retrials
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Strategic behavior and optimization in a hybrid M/M/1 queue with retrials Yoav Kerner1
· Ophir Shmuel-Bittner1
Received: 2 November 2019 / Revised: 23 September 2020 / Accepted: 11 October 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In standard queues, when there are waiting customers, service completions are followed by service commencements. In retrial queues, this is not the case. In such systems, customers try to receive service at a time of their choosing, or the server seeks the next customer for a non-negligible time. In this note, we consider a hybrid model with both a finite standard queue and an orbit. While in the orbit, customers try to join the standard queue in their own time. We assume that the retrial rate is a decision variable, and study both the Nash equilibrium and the socially optimal retrial rates, under a cost model that considers both waiting costs and retrial costs. Keywords Retrial queues · Strategic behavior · Nash equilibrium · Social optimization
1 Introduction Service systems surround us in the routines of our daily lives. Sometimes, customers do not find a place in a queue, so they have to leave and return later. A natural question is when to return. The decision of when to return is influenced by the future availability of the system, which also depends on the decisions of other customers. In this paper, we study an M/M/1 system with finite capacity. Customers who arrive and find the queue fully occupied are directed to an external orbit. While waiting in the orbit, each customer decides independently when to try rejoining the queue. Trying is associated with a cost, regardless of the outcome. Of course, waiting is also associated with a cost. If one tries too early, it is quite likely that there is still no room available. On the
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Yoav Kerner [email protected] Ophir Shmuel-Bittner [email protected]
1
Department of Industrial Engineering and Management, Ben Gurion University of the Negev, Beer Sheva, Israel
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Queueing Systems
other hand, waiting for too long imposes waiting costs, and seats that become available might be occupied by others. The latter impact of others on one’s own costs leads us to observe this decision-making problem from a game-theoretic perspective. Game theory, a theory that models multi-agent systems in which each agent acts selfishly, was introduced 70 years ago as a field in microeconomics and mathematics. The first to bring queueing theory and game theory together, by importing decision-making tools into the study of queueing systems, was Naor [18]. Naor studied the performance of a basic queueing system, where customers decide selfishly whether or not to join it. He identified that a threshold strategy for balking from an observable first-come, first-served (FCFS) M/M/1 queue is an equilibrium strategy. Naor also designed a procedure that computes the socially optimal joining strategy. Edelson and Hildebrand [5] explored the properties of the unobservable M/M/1 system in which the customers do not o
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