# American Institute of Mathematical Sciences

• Previous Article
Time-consistent strategy for a multi-period mean-variance asset-liability management problem with stochastic interest rate
• JIMO Home
• This Issue
• Next Article
Coordination contracts for a dual-channel supply chain under capital constraints
doi: 10.3934/jimo.2020120

## 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

Received  November 2019 Revised  April 2020 Published  June 2020

Fund Project: 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

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.

Citation: Ke Sun, Jinting Wang, Zhe George Zhang. Strategic joining in a single-server retrial queue with batch service. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2020120
##### References:

show all references

##### References:
The alternate renewal process in the retrial queue with batch service
Transition rate diagram of the observable retrial queue with batch service
Transition rate diagram of the unobservable retrial queue with batch service
The three joining probabilities in observable case with respect to $\lambda$, $\mu = 1, \gamma = 2, k = 1.5, \theta = 5$
The three joining probabilities in observable case with respect to $\mu$, $\lambda = 0.7, \gamma = 2, k = 1.5, \theta = 5$
The three joining probabilities in observable case with respect to $\gamma$, $\lambda = 0.8, \gamma = 2, k = 2, \theta = 5$
The comparison of social benefits with respect to $\lambda$, $\mu = 1, \gamma = 2, k = 2, \theta = 5$
The comparison of social benefits with respect to $\mu$, $\lambda = 0.7, \gamma = 2, k = 1.5, \theta = 5$
The comparison of social benefits with respect to $\gamma$, $\lambda = 0.8, \mu = 1, k = 1.5, \theta = 5$
 [1] Jian Zhang, Tony T. Lee, Tong Ye, Liang Huang. An approximate mean queue length formula for queueing systems with varying service rate. Journal of Industrial & Management Optimization, 2021, 17 (1) : 185-204. doi: 10.3934/jimo.2019106 [2] Yukihiko Nakata. Existence of a period two solution of a delay differential equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1103-1110. doi: 10.3934/dcdss.2020392 [3] Hongguang Ma, Xiang Li. Multi-period hazardous waste collection planning with consideration of risk stability. Journal of Industrial & Management Optimization, 2021, 17 (1) : 393-408. doi: 10.3934/jimo.2019117 [4] Attila Dénes, Gergely Röst. Single species population dynamics in seasonal environment with short reproduction period. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020288 [5] Lin Jiang, Song Wang. Robust multi-period and multi-objective portfolio selection. Journal of Industrial & Management Optimization, 2021, 17 (2) : 695-709. doi: 10.3934/jimo.2019130 [6] Qiang Fu, Xin Guo, Sun Young Jeon, Eric N. Reither, Emma Zang, Kenneth C. Land. The uses and abuses of an age-period-cohort method: On the linear algebra and statistical properties of intrinsic and related estimators. Mathematical Foundations of Computing, 2020  doi: 10.3934/mfc.2021001 [7] Yanjun He, Wei Zeng, Minghui Yu, Hongtao Zhou, Delie Ming. Incentives for production capacity improvement in construction supplier development. Journal of Industrial & Management Optimization, 2021, 17 (1) : 409-426. doi: 10.3934/jimo.2019118 [8] Hong Fu, Mingwu Liu, Bo Chen. Supplier's investment in manufacturer's quality improvement with equity holding. Journal of Industrial & Management Optimization, 2021, 17 (2) : 649-668. doi: 10.3934/jimo.2019127 [9] Zonghong Cao, Jie Min. Selection and impact of decision mode of encroachment and retail service in a dual-channel supply chain. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020167 [10] Sujit Kumar Samanta, Rakesh Nandi. Analysis of $GI^{[X]}/D$-$MSP/1/\infty$ queue using $RG$-factorization. Journal of Industrial & Management Optimization, 2021, 17 (2) : 549-573. doi: 10.3934/jimo.2019123 [11] Tien-Yu Lin, Bhaba R. Sarker, Chien-Jui Lin. An optimal setup cost reduction and lot size for economic production quantity model with imperfect quality and quantity discounts. Journal of Industrial & Management Optimization, 2021, 17 (1) : 467-484. doi: 10.3934/jimo.2020043 [12] Wenqin Zhang, Zhengchun Zhou, Udaya Parampalli, Vladimir Sidorenko. Capacity-achieving private information retrieval scheme with a smaller sub-packetization. Advances in Mathematics of Communications, 2021, 15 (2) : 347-363. doi: 10.3934/amc.2020070 [13] Mahdi Karimi, Seyed Jafar Sadjadi. Optimization of a Multi-Item Inventory model for deteriorating items with capacity constraint using dynamic programming. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2021013 [14] Reza Lotfi, Zahra Yadegari, Seyed Hossein Hosseini, Amir Hossein Khameneh, Erfan Babaee Tirkolaee, Gerhard-Wilhelm Weber. A robust time-cost-quality-energy-environment trade-off with resource-constrained in project management: A case study for a bridge construction project. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020158

2019 Impact Factor: 1.366