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; Pengcheng Wang; Wuyi Yue

doi:10.3934/jimo.2017002

Article Contents

2017, Volume 13, Issue 3: 1467-1481. Doi: 10.3934/jimo.2017002

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Performance analysis and optimization of a pseudo-fault Geo/Geo/1 repairable queueing system with N-policy, setup time and multiple working vacations

1.
Department of Applied Mathematics, College of Science, Yanshan University, Qinhuangdao 066004, China
2.
School of Transportation Science and Engineering, Beihang University, Beijing 100191, China
3.
Department of Intelligence and Informatics, Konan University, Kobe 658-8501, Japan

Received: September 2015

Published: September 2017

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Abstract

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.

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

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Figure 1. The schematic diagram for the model description

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Figure 2. The relation of $P_B$ to $\mu_v$ and $N$

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Figure 3. The relation of $E[L]$ to $p$ and $N$

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Figure 4. The relation of $E[L_q]$ to $\mu_b$ and $\beta$

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Figure 5. The relation of $P_{q2}$ to $p$ and $\alpha$

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Figure 6. Individual benefit $U_I$ versus arrival rate $p$

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Figure 7. Social benefit $U_S$ versus arrival rate $p$

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Table 1. The relation of $E[W]$ to $q$ and $\theta$

$\theta$	The expected waiting time $E[W]$
$\theta$	$q=0$	$q=0.05$	$q=0.1$	$q=0.15$	$q=0.2$	$q=0.25$	$q=0.3$
0.3	20.207	22.379	24.622	26.885	29.148	31.430	33.798
0.5	19.942	22.019	24.140	26.250	28.318	30.351	32.396
0.7	19.843	21.884	23.958	26.009	28.006	29.947	31.875

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Table 2. The relation of $E[L]$ to $\mu_b$ and $\gamma$

$\gamma$	The expected queue length $E[L]$
$\gamma$	$\mu_b=0.6$	$\mu_b=0.65$	$\mu_b=0.7$	$\mu_b=0.75$	$\mu_b=0.8$	$\mu_b=0.85$	$\mu_b=0.9$
0.4	32.139	17.645	14.184	12.449	11.387	10.673	10.165
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

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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
0.4	0.378	0.228
0.8	0.392	0.240

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Table 4. Numerical results for admission fee

Vacation parameter $\theta$	Socially maximum benefit $U_S^*$	Admission fee $f$
0.2	28.726	140.813
0.4	30.796	135.071
0.8	33.101	133.754

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