Machine interference problem involving unsuccessful switchover for a group of repairable servers with vacations

Tzu-Hsin Liu; Jau-Chuan Ke; Ching-Chang Kuo

doi:10.3934/jimo.2020027

Article Contents

2021, Volume 17, Issue 3: 1411-1422. Doi: 10.3934/jimo.2020027

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Machine interference problem involving unsuccessful switchover for a group of repairable servers with vacations

Department of Applied Statistics, National Taichung University of Science and Technology, Taichung, Taiwan

^* Corresponding author: Jau-Chuan Ke
^* Corresponding author: Jau-Chuan Ke

Received: January 2019

Revised: June 2019

Early access: February 2020

Published: May 2021

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Abstract

The purpose of this paper is to explore the multiple-server machine interference problem with standby unsuccessful switchover and Bernoulli vacation schedule. Failure times of operating and standby machines are assumed to have exponential distributions and repair times of the failed machines and vacation times of servers are also assumed to have exponential distributions. After the completion of service, the server either goes for a vacation or may continue serving for the next machine. The vacation policy we considered is a single vacation policy. In practice, the switchover may experience a significant failure. The matrix analytical method and recursive method are employed to obtain the steady-state probability vectors, and closed-form expressions of some important system characteristics are obtained. The problem of cost optimization dealt with a number of numerical examples is provided by the Quasi-Newton method, the pattern search method, and the Nelder-Mead simplex direct search method. Expressions of various system characteristics are derived. Sensitivity analysis is performed numerically for system parameters. This paper presents the first time that machine interference problem with unsuccessful switchover for a group of repairable servers with vacations has been obtained, which is quite useful for the decision makers.

Keywords:

Bernoulli vacation; Cost,
Standbys unsuccessful switchover,
Recursive method,
Quasi-Newton method,
Pattern search method,
Nelder-Mead simplex direct search method.

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

Citation:

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Figure 1. Plot of the average cost function versus the mean service rate and mean vacation rate

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Table 1. The average cost function for given $ (W, S) $ with $ M = 6 $, $ \lambda = 0.5 $, $ \alpha = 0.2\lambda $, $ \theta = 0.02 $, $ \beta = 0.02 $

$ (S, W) $	(1, 1)	(1, 2)	(1, 3)	(1, 4)	(1, 5)	(1, 6)	(1, 7)	(1, 8)
$ TAC $	99.72	98.43	99.20	100.74	102.52	104.32	106.04	107.65
$ (S, W) $	(1, 9)	(1, 10)	(1, 11)	(1, 12)	(2, 1)	(2, 2)	(2, 3)	(2, 4)
$ TAC $	109.13	110.49	111.74	112.88	92.32	$ \mathbf{90.90} $	92.45	94.40
$ (S, W) $	(2, 5)	(2, 6)	(2, 7)	(2, 8)	(2, 9)	(2, 10)	(2, 11)	(2, 12)
$ TAC $	96.70	98.92	100.97	102.83	104.51	106.02	107.39	108.62
$ (S, W) $	(3, 1)	(3, 2)	(3, 3)	(3, 4)	(3, 5)	(3, 6)	(3, 7)	(3, 8)
$ TAC $	97.43	94.99	96.04	97.96	100.03	102.00	103.83	105.47
$ (S, W) $	(3, 9)	(3, 10)	(3, 11)	(3, 12)	(4, 1)	(4, 2)	(4, 3)	(4, 4)
$ TAC $	106.95	108.28	109.46	110.53	98.44	96.39	97.75	99.94
$ (S, W) $	(4, 5)	(4, 6)	(4, 7)	(4, 8)	(4, 9)	(4, 10)	(4, 11)	(4, 12)
$ TAC $	102.17	104.22	106.04	107.65	109.07	110.33	111.45	112.44
$ (S, W) $	(5, 9)	(5, 10)	(5, 11)	(5, 12)	(6, 1)	(6, 2)	(6, 3)	(6, 4)
$ TAC $	109.33	110.60	111.72	112.72	95.48	94.17	96.15	98.83
$ (S, W) $	(6, 5)	(6, 6)	(6, 7)	(6, 8)	(6, 9)	(6, 10)	(6, 11)	(6, 12)
$ TAC $	101.39	103.66	105.63	107.35	108.84	110.16	111.33	112.37
$ (S, W) $	(7, 2)	(7, 3)	(7, 4)	(7, 5)	(7, 6)	(7, 7)	(7, 8)	(7, 9)
$ TAC $	91.92	94.27	97.24	100.01	102.43	104.53	106.35	107.93
$ (S, W) $	(7, 10)	(7, 11)	(7, 12)	(8, 3)	(8, 4)	(8, 5)	(8, 6)	(8, 7)
$ TAC $	109.32	110.54	111.64	91.97	95.24	98.25	100.85	103.10
$ (S, W) $	(8, 8)	(8, 9)	(8, 10)	(8, 11)	(8, 12)	(9, 4)	(9, 5)	(9, 6)
$ TAC $	105.03	106.71	108.19	109.49	110.65	92.94	96.20	99.01
$ (S, W) $	(9, 7)	(9, 8)	(9, 9)	(9, 10)	(9, 11)	(9, 12)	(10, 5)	(10, 6)
$ TAC $	101.41	103.48	105.28	106.85	108.24	109.48	93.93	96.95
$ (S, W) $	(10, 7)	(10, 8)	(10, 9)	(10, 10)	(10, 11)	(10, 12)	(11, 6)	(11, 7)
$ TAC $	99.53	101.75	103.67	105.36	106.84	108.16	94.73	97.49
$ (S, W) $	(11, 8)	(11, 9)	(11, 10)	(11, 11)	(11, 12)	(12, 7)	(12, 8)	(12, 9)
$ TAC $	99.87	101.92	103.72	105.31	106.71	95.32	97.86	100.06
$ (S, W) $	(12, 10)	(12, 11)	(12, 12)
$ TAC $	101.98	103.67	105.17

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Table 2. The minimum average cost function for varying values of $ \lambda $ with $ M = 6 $, $ \alpha = 0.2\lambda $, $ \theta = 0.02 $, $ \beta = 0.02 $

QN method
$ \lambda $	0.2	0.4	0.6
$ TAC $	68.34	85.47	95.95
$ (W^, S^) $	(1, 1)	(2, 2)	(2, 2)
$ (\mu^, \delta^) $	(2.37, 5.43)	(2.38, 2.14)	(3.09, 4.09)
Iterations	1204	1427	1583
CPU Time	6.26	5.62	5.64
MN method
$ TAC $	68.34	85.47	95.95
$ (W^, S^) $	(1, 1)	(2, 2)	(2, 2)
$ (\mu^, \delta^) $	(2.37, 5.43)	(2.38, 2.14)	(3.09, 4.09)
Iterations	6741	6074	6176
CPU Time	6.39	5.77	5.62
PS method
$ TAC $	68.34	85.47	95.95
$ (W^, S^) $	(1, 1)	(2, 2)	(2, 2)
$ (\mu^, \delta^) $	(2.37, 5.43)	(2.38, 2.14)	(3.09, 4.09)
Iterations	11679	13179	13294
CPU Time	22.44	23.57	25.80

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