Strategic joining in a single-server retrial queue with batch service

Ke Sun; Jinting Wang; Zhe George Zhang

doi:10.3934/jimo.2020120

Article Contents

2021, Volume 17, Issue 6: 3309-3332. Doi: 10.3934/jimo.2020120

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Strategic joining in a single-server retrial queue with batch service

1.
Department of Mathematics, Beijing Jiaotong University, Beijing, 100044, China
2.
Department of Management Science, School of Management Science and Engineering, Central, University of Finance and Economics, Beijing, 100081, China
3.
Department of Decision Sciences, Western Washington University, Bellingham, WA 98225, USA
4.
Beedie School of Business, Simon Fraser University Burnaby, BC V5A 1S6, Canada

^* Corresponding author: Jinting Wang
^* Corresponding author: Jinting Wang

Received: October 31, 2019

Revised: March 31, 2020

Early access: June 2020

Published: November 2021

This work was supported in part by the National Natural Science Foundation of China (Grant nos. 71871008 and 71571014) and the Fundamental Research Funds for the Central Universities under grant 2019YJS196

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Abstract

We consider a single-server retrial queue with batch service where potential customers arrive according to a Poisson process. The service process has two stages: busy period and admission period, which are corresponding to whether the server is in service or not, respectively. In such an alternate renewal process, if arrivals find busy period, they make join-or-balk decisions. Those joining ones will stay in the orbit and attempt to get into server at a constant rate. While, if arrivals find the server is in an admission period, they get into the server directly. At the end of each admission period, all customers in the server will be served together regardless of the size of the batch. The reward of each arrival after completion of service depends on the service size. Furthermore, customers in the orbit fail to get into the server before the end of each service cycle will be forced to leave the system. We study an observable scenario that customers are informed about the service period upon arrivals and an unobservable case without this information. The Nash equilibrium joining strategies are identified and the social- and profit- maximization problems are obtained, respectively. Finally, the optimal joining strategies in the observable queue and the comparison of social welfare between the two queues are illustrated by numerical examples.

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Mathematics Subject Classification: Primary: 60K25; Secondary: 90B22.

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Figure 1. The alternate renewal process in the retrial queue with batch service

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Figure 2. Transition rate diagram of the observable retrial queue with batch service

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Figure 3. Transition rate diagram of the unobservable retrial queue with batch service

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Figure 4. The three joining probabilities in observable case with respect to $ \lambda $, $ \mu = 1, \gamma = 2, k = 1.5, \theta = 5 $

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Figure 5. The three joining probabilities in observable case with respect to $ \mu $, $ \lambda = 0.7, \gamma = 2, k = 1.5, \theta = 5 $

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Figure 6. The three joining probabilities in observable case with respect to $ \gamma $, $ \lambda = 0.8, \gamma = 2, k = 2, \theta = 5 $

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Figure 7. The comparison of social benefits with respect to $ \lambda $, $ \mu = 1, \gamma = 2, k = 2, \theta = 5 $

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Figure 8. The comparison of social benefits with respect to $ \mu $, $ \lambda = 0.7, \gamma = 2, k = 1.5, \theta = 5 $

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Figure 9. The comparison of social benefits with respect to $ \gamma $, $ \lambda = 0.8, \mu = 1, k = 1.5, \theta = 5 $

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