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July 2017, 13(3): 1467-1481. doi: 10.3934/jimo.2017002
Performance analysis and optimization of a pseudo-fault Geo/Geo/1 repairable queueing system with N-policy, setup time and multiple working vacations
Zhanyou Ma 1, , Pengcheng Wang 2, and Wuyi Yue 3,
Department of Applied Mathematics, College of Science, Yanshan University, Qinhuangdao 066004, China
School of Transportation Science and Engineering, Beihang University, Beijing 100191, China
Department of Intelligence and Informatics, Konan University, Kobe 658-8501, Japan
The reviewing process of the paper was handled by Yutaka Takahashi as Guest Editor.
Received September 2015 Published December 2016
In this paper, we consider a discrete time Geo/Geo/1 repairable queueing system with a pseudo-fault, setup time, $N$-policy and multiple working vacations. We assume that the service interruption is caused by pseudo-fault or breakdown, and occurs only when the server is busy. If the pseudo-fault occurs, the server will enter into a vacation period instead of a busy period. At a breakdown instant, the repair period starts immediately and after repaired the server is assumed to be as good as new. Using a quasi birth-and-death chain, we establish a two-dimensional Markov chain. We obtain the distribution of the steady-state queue length by using a matrix-geometric solution method. Moreover, we analyze the considered queueing system and provide several performance indices of the system in steady-state. According to the queueing system, we first investigate the individual and social optimal behaviors of the customer. Then we propose a pricing policy to optimize the system socially, and study the Nash equilibrium and social optimization of the proposed strategy to determine the optimal expected parameters of the system. Finally, we present some numerical results to illustrate the effect of several parameters on the systems.
Keywords: Pseudo-fault, breakdown, setup time, multiple working vacation, matrix-geometric solution, optimization.
Mathematics Subject Classification: Primary: 60K25; Secondary: 90B22.
Citation: Zhanyou Ma, Pengcheng Wang, Wuyi Yue. Performance analysis and optimization of a pseudo-fault Geo/Geo/1 repairable queueing system with N-policy, setup time and multiple working vacations. Journal of Industrial & Management Optimization, 2017, 13 (3) : 1467-1481. doi: 10.3934/jimo.2017002
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Figure 1. The schematic diagram for the model description
Figure 2. The relation of $P_B$ to $\mu_v$ and $N$
Figure 3. The relation of $E[L]$ to $p$ and $N$
Figure 4. The relation of $E[L_q]$ to $\mu_b$ and $\beta$
Figure 5. The relation of $P_{q2}$ to $p$ and $\alpha$
Figure 6. Individual benefit $U_I$ versus arrival rate $p$
Figure 7. Social benefit $U_S$ versus arrival rate $p$
Table 1. The relation of $E[W]$ to $q$ and $\theta$
$\theta$ The expected waiting time $E[W]$
$q=0$ $q=0.05$ $q=0.1$ $q=0.15$ $q=0.2$ $q=0.25$ $q=0.3$
0.3 20.207 22.379 24.622 26.885 29.148 31.430 33.798
Table 2. The relation of $E[L]$ to $\mu_b$ and $\gamma$
$\gamma$ The expected queue length $E[L]$
$\mu_b=0.6$ $\mu_b=0.65$ $\mu_b=0.7$ $\mu_b=0.75$ $\mu_b=0.8$ $\mu_b=0.85$ $\mu_b=0.9$
0.6 15.069 12.820 11.560 10.757 10.207 9.809 9.508
0.8 13.100 11.697 10.830 10.250 9.838 9.529 9.288
Table 3. Comparison of individually and socially optimal arrival rate
Vacation parameter $\theta$ Individually optimal arrival rate $p^e$ Socially optimal arrival rate $p^*$
0.2 0.348 0.204
Table 4. Numerical results for admission fee
Vacation parameter $\theta$ Socially maximum benefit $U_S^*$ Admission fee $f$
0.2 28.726 140.813
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