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

• * 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)

• 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.

 Citation:

• 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 $\surd$ $\surd$
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