Strategic shield against external shocks in a Markovian queue with vulnerable server

Lingjiao Zhang; Jinting Wang

doi:10.3934/jimo.2022095

Article Contents

2023, Volume 19, Issue 5: 3483-3508. Doi: 10.3934/jimo.2022095

This issue Previous Article A product service supply chain network equilibrium considering risk management in the context of COVID-19 pandemic Next Article Research on optimal pricing decisions of the service supply chain oriented to strategic consumers

Strategic shield against external shocks in a Markovian queue with vulnerable server

Lingjiao Zhang and
Jinting Wang^,

Department of Management Sciences, School of Management Science and Engineering, Central University of Finance and Economics, Beijing 100081, China

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

Received: January 2022

Revised: April 2022

Early access: June 2022

Published: May 2023

This work was supported in part by the National Natural Science Foundation of China under grant no. 71871008, and the Emerging Interdisciplinary Project of CUFE (Grant No. 21XXJC010)

Abstract / Introduction Full Text(HTML) Figure(8) / Table(1) Related Papers Cited by

Abstract

A threshold-type queue-length control strategy is proposed in the paper to investigate the influence of external shocks to a queueing system where customers are strategic and the server is vulnerable. An empty system is required to initiate service only when the number of waiting customers reaches a given threshold. External shocks occur according to a Poisson process, and once occur, the server breaks down and the customer being served is forced to leave the system. Arriving customers have to decide whether to join the system or not based on a reward-cost structure under different levels of information. The focus is on examining the equilibrium performance of the system under the interaction between the server's states and customers' joining decisions for different information levels. The equilibrium threshold in the observable queue and mixed joining probability in the unobservable queue are obtained. The optimal value of threshold $ N $ is discussed by taking strategic customer behavior and vulnerability of the server into consideration. These findings have important managerial implications on the evaluation of the shield threshold for the system with external attacks and unreliability factor, and also on optimal operation management of the system.

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

Citation:

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Figure 1. State transition diagram for the observable case

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Figure 2. Throughput with $ N $ in fully observable case

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Figure 3. $ \Delta $ Throughput with $ \alpha $ in fully observable case

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Figure 4. $ \Delta $Throughput with $ \rho $ in almost observable case

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Figure 5. State transition diagram for unobservable case

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Figure 6. Throughput with $ N $ and $ \alpha $ in almost unobservable case

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Figure 7. Equilibrium arrival rates in Case 2b

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Figure 8. Throughput in Case 2b

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Table 1. Comparison of relevant literature

Literature	Negative customer		Vacation			Retrial
Literature	IR	Catastrophe	Multiple vacations	Working vacation	$ N $-policy	Retrial
[2],[3]		$ \surd $
[8],[9],[10]					$ \surd $
[13],[19]	$ \surd $
[14]		$ \surd $				$ \surd $
[18]	$ \surd $			$ \surd $
[20]			$ \surd $		$ \surd $
[21]	$ \surd $					$ \surd $
[23]					$ \surd $	$ \surd $
[24]				$ \surd $
This paper	$ \surd $				$ \surd $

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References

[1]	A. Aghighi, A. Goli, B. Malmir and E. B. Tirkolaee, The stochastic location-routing-inventory problem of perishable products with reneging and balking, Journal of Ambient Intelligence and Humanized Computing, 2021. doi: 10.1007/s12652-021-03524-y.
[2]	O. Boudali and A. Economou, Optimal and equilibrium balking strategies in the single server Markovian queue with catastrophes, European J. Oper. Res., 218 (2012), 708-715. doi: 10.1016/j.ejor.2011.11.043.
[3]	O. Boudali and A. Economou, The effect of catastrophes on the strategic customer behavior in queueing systems, Naval Res. Logist., 60 (2013), 571-587. doi: 10.1002/nav.21553.
[4]	P. Cao, S. He, J. Huang and Y. Liu, To pool or not to pool: Queueing design for large-scale service systems, Oper. Res., 69 (2021), 1866-1885. doi: 10.1287/opre.2019.1976.
[5]	Y. Dimitrakopoulos and A. N. Burnetas, Customer equilibrium and optimal strategies in an $M/M/1$ queue with dynamic service control, European J. Oper. Res., 252 (2016), 477-486. doi: 10.1016/j.ejor.2015.12.029.
[6]	A. Economou and S. Kanta, Equilibrium balking strategies in the observable single-server queue with breakdowns and repairs, Oper. Res. Lett., 36 (2008), 696-699. doi: 10.1016/j.orl.2008.06.006.
[7]	A. Goli, A. Aazami and A. Jabbarzadeh, Accelerated cuckoo optimization algorithm for capacitated vehicle routing problem in competitive conditions, International Journal of Artif Intell, 16 (2018), 88-112.
[8]	P. Guo and R. Hassin, Strategic behavior and social optimization in Markovian vacation queues, Oper. Res., 59 (2011), 986-997. doi: 10.1287/opre.1100.0907.
[9]	P. Guo and R. Hassin, Strategic behavior and social optimization in Markovian vacation queues: The case of heterogeneous customers, European J. Oper. Res., 222 (2012), 278-286. doi: 10.1016/j.ejor.2012.05.026.
[10]	P. Guo and Q. Li, Strategic behavior and social optimization in partially-observable Markovian vacation queues, Oper. Res. Lett., 41 (2013), 277-284. doi: 10.1016/j.orl.2013.02.005.
[11]	R. Hassin, Rational Queueing, Chapman and Hall, London, 2016.
[12]	R. Hassin and M. Haviv, To Queue or Not to Queue: Equilibrium Behavior in Queueing Systems, Kluwer, Boston, MA, 2003. doi: 10.1007/978-1-4615-0359-0.
[13]	D. H. Lee, Optimal pricing strategies and customers' equilibrium behavior in an unobservable M/M/1 queueing system with negative customers and repair, Math. Probl. Eng., 2017 (2017), Article ID 8910819, 11 pp. doi: 10.1155/2017/8910819.
[14]	K. Li and J. Wang, Equilibrium balking strategies in the single-server retrial queue with constant retrial rate and catastrophes, Quality Technology and Quantitative Management, 18 (2021), 156-178.
[15]	L. Li, J. Wang and F. Zhang, Equilibrium customer strategies in Markovian queues with partial breakdowns, Computers & Industrial Engineering, 66 (2013), 751-757.
[16]	X. Li, J. Wang and F. Zhang, New results on equilibrium balking strategies in the single-server queue with breakdowns and repairs, Appl. Math. Comput., 241 (2014), 380-388. doi: 10.1016/j.amc.2014.05.025.
[17]	P. Naor, The regulation of queue size by levying tolls, Econometrica, 37 (1969), 15-24.
[18]	G. Panda and V. Goswami, Equilibrium joining strategies of positive customers in a Markovian queue with negative arrivals and working vacations, Methodology and Computing in Applied Probability, (2021). doi: 10.1007/s11009-021-09864-8.
[19]	K. Sun and J. Wang, Equilibrium joining strategies in the single server queues with negative customers, Int. J. Comput. Math., 96 (2019), 1169-1191. doi: 10.1080/00207160.2018.1490018.
[20]	W. Sun, S. Li and C.-G. E., Equilibrium and optimal balking strategies of customers in Markovian queues with multiple vacations and $N$-policy, Appl. Math. Model., 40 (2016), 284-301. doi: 10.1016/j.apm.2015.04.045.
[21]	F. Wang, J. Wang and F. Zhang, Strategic behavior in the single-server constant retrial queue with individual removal, Quality Technology and Quantitative Management, 12 (2015), 323-340.
[22]	J. Wang and F. Zhang, Equilibrium analysis of the observable queues with balking and delayed repairs, Appl. Math. Comput., 218 (2011), 2716-2729. doi: 10.1016/j.amc.2011.08.012.
[23]	J. Wang, X. Zhang and P. Huang, Strategic behavior and social optimization in a constant retrial queue with the $N$-policy, European J. Oper. Res., 256 (2017), 841-849. doi: 10.1016/j.ejor.2016.06.034.
[24]	F. Zhang, J. Wang and B. Liu, Equilibrium balking strategies in Markovian queues with working vacations, Appl. Math. Model., 37 (2013), 8264-8282. doi: 10.1016/j.apm.2013.03.049.