Strategic behaviour in a tandem queue with alternating server
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Strategic behaviour in a tandem queue with alternating server Nimrod Dvir1
· Refael Hassin1 · Uri Yechiali1
Received: 30 October 2019 / Revised: 27 April 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract This paper considers an unobservable two-site tandem queueing system attended by an alternating server. We study the strategic customer behaviour under two thresholdbased operating policies, applied by a profit-maximizing server, while customers’ strategic behaviour and server’s switching costs are taken into account. Under the Exact-N policy, in each cycle the server first completes service of N customers in the first stage (Q 1 ), then switches to the second stage (Q 2 ) and then serves those N customers before switching back to Q 1 to start a new cycle. This policy leads to a mixture of Follow-the-Crowd and Avoid-the-Crowd customer behaviour. In contrast, under the N-Limited policy, the server switches from Q 1 to Q 2 also when the first queue is emptied, making this regime work-conserving and leading only to Avoidthe-Crowd behaviour. Performance measures are obtained using matrix geometric methods for both policies and any threshold N , while for sequential service (N = 1) explicit expressions are derived. It is shown that the system’s stability condition is independent of N , and of the switching policy. Optimal performances in equilibrium, under each of these switching policies, are analysed and compared through a numerical study. Keywords Strategic behaviour · Tandem queues · Alternating server · Threshold policy · Switching cost · Equilibrium · Follow-the-Crowd · Avoid-the-Crowd Mathematics Subject Classification 60K25 · 60K30 · 90B22 · 91A10
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Nimrod Dvir [email protected] Refael Hassin [email protected] Uri Yechiali [email protected]
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Department of Statistics and Operations Research, School of Mathematical Sciences, Tel-Aviv University, Tel-Aviv, Israel
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Queueing Systems
1 Introduction This study analyses a queueing system where customers are served in two phases by the same server. There is a separate queue for each phase, and the server alternates between the two queues. The server incurs a switching cost for every change in the queue being served.1 Arriving customers are strategic and act to maximize their utility. Their decision to join or not is based on the system’s known parameters, while the system’s state is unobservable. Customers are homogeneous; they incur a waiting cost which is linear in their sojourn time in the system and gain a fixed value upon service completion in the second phase. Both queues are first-come first-served (FCFS), whereas the server determines the operating policy (that is, when to serve in each queue). We consider two common threshold-based operating policies (regimes): (i) ExactN and (ii) N-Limited. The first is a strict policy in which the server switches the queue operated only after the number of customers served reaches a fixed threshold. The second is a more adaptable policy, where the server switches when
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