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

  • * Corresponding author: Jinting Wang

    * Corresponding author: Jinting Wang

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 Full Text(HTML) Figure(8) / Table(1) Related Papers Cited by
  • 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.

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


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  • Figure 1.  State transition diagram for the observable case

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

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

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

    Figure 5.  State transition diagram for unobservable case

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

    Figure 7.  Equilibrium arrival rates in Case 2b

    Figure 8.  Throughput in Case 2b

    Table 1.  Comparison of relevant literature

    Literature Negative customer Vacation Retrial
    IR Catastrophe Multiple vacations Working vacation $ N $-policy
    [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 $
     | Show Table
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  • [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.
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    [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. GoliA. 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. LiJ. Wang and F. Zhang, Equilibrium customer strategies in Markovian queues with partial breakdowns, Computers & Industrial Engineering, 66 (2013), 751-757. 
    [16] X. LiJ. 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. SunS. 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. WangJ. 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. WangX. 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. ZhangJ. 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.
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