New repairable system model with two types repair based on extended geometric process

doi:10.21629/JSEE.2019.03.18

Journal of Systems Engineering and Electronics ›› 2019, Vol. 30 ›› Issue (3): 613-623.doi: 10.21629/JSEE.2019.03.18

• Reliability • Previous Articles Next Articles

New repairable system model with two types repair based on extended geometric process

Junyuan WANG(), Jimin YE*(), Pengfei XIE()

School of Mathematics and Statistics, Xidian University, Xi'an 710071, China

Received:2018-05-14 Online:2019-06-01 Published:2019-07-04
Contact: Jimin YE E-mail:jywang215@stu.xidian.edu.cn;jmye@mail.xidian.edu.cn;PengfeiXie@stu.xidian.edu.cn
About author:WANG Junyuan was born in 1987. He received his B.S. degree in mathematics and applied mathematics in 2011, his M.S. degree in Probability theory and mathematical statistics in 2014. Now he is a Ph.D. candidate in Xidian University, Xi'an China. His research interests are applied stochastic processes, maintenance theory and reliability analysis. E-mail:jywang215@stu.xidian.edu.cn|YE Jimin was born in 1967. He received his Ph.D. degree in signal and information processing from Xidian University in 2005. At present, he is a professor and doctoral supervisor in School of Mathematics and Statistics, Xidian University. His research interests are risk theory, stochastic operation research and applied probability. E-mail:jmye@mail.xidian.edu.cn|XIE Pengfei was born in 1990. He received his B.S. degree of mathematics and applied mathematics in 2016. Now he is a Ms.D. in Xidian University, Xi'an China. His research interests are insurance mathematics, risk analysis and optimization algorithm. E-mail:PengfeiXie@stu.xidian.edu.cn
Supported by:
the National Natural Science Foundation of China(61573014);the Fundamental Research Funds for the Central Universities(JB180702);This work was supported by the National Natural Science Foundation of China (61573014) and the Fundamental Research Funds for the Central Universities (JB180702)

Abstract

Abstract:

A simple repairable system with one repairman is considered. As the system working age is up to a specified time T, the repairman will repair the component preventively, and it will go back to work as soon as the repair finished. When the system failure, the repairman repair it immediately. The time interval of the preventive repair and the failure correction is described with the extended geometric process. Different from the available replacement policy which is usually based on the failure number or the working age of the system, the bivariate policy (T, N) is considered. The explicit expression of the long-run average cost rate function C(T, N) of the system is derived. Through alternatively minimize the cost rate function C(T, N), the optimal replacement policy (T^*, N^*) is obtained, and it proves that the optimal policy is unique. Numerical cases illustrate the conclusion, and the sensitivity analysis of the parameters is carried out.

Key words: extended geometric process, average cost rate, replacement policy, renewal reward theorem

Junyuan WANG, Jimin YE, Pengfei XIE. New repairable system model with two types repair based on extended geometric process[J]. Journal of Systems Engineering and Electronics, 2019, 30(3): 613-623.

Figures/Tables 9

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Table 1

Fig 3

Fig 4

Table 2

Table 3

Table 4

Table 5

References 26

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$N$	$C(T = 50, N)$	$C(\infty, N)$
1	9.53	10.48
2	–214.70	–202.32
3	–276.16	–254.22
4	–300.39	–272.03
5	$\textbf{–308.95}$	$\textbf{–275.94}$
6	–308.63	–272.22
7	–302.14	–276.39
8	–290.71	–250.67
9	–274.93	–234.70
10	–255.15	–215.89
11	–231.72	–194.61
12	–205.08	–171.24
13	–175.86	–146.25
14	–144.90	–120.21
15	–113.22	–93.79
16	–81.94	–67.70
17	–52.14	–42.64
18	–24.73	–19.28
19	–0.35	1.91
20	20.66	20.59
21	38.26	36.66
22	52.63	50.14
23	64.11	61.21
24	73.10	70.14
25	80.04	77.21
26	85.30	82.74
27	89.26	87.00
28	92.20	90.26
29	94.37	92.73
30	95.96	94.59

$N$	$C(T, N)$	$f(N)$
1	9.53	0.058 6
2	–214.70	0.129 3
3	–276.16	0.285 9
4	–300.39	0.613 7
5	$\textbf{–308.95}$	1.166 9
6	–308.63	1.944 1
7	–302.14	2.866 6
8	–290.71	3.812 4
9	–274.93	4.679 7
10	–255.15	5.419 6
11	–231.72	6.027 6
12	–205.08	6.521 0
13	–175.86	6.922 6
14	–144.90	7.252 5
15	–113.22	7.526 7
16	–81.94	7.757 2
17	–52.14	7.952 8
18	–24.73	8.119 8
19	–0.35	8.263 3
20	20.66	8.387 1
21	38.26	8.494 0
22	52.63	8.586 6
23	64.11	8.667 0
24	73.10	8.736 7
25	80.04	8.797 3
26	85.30	8.849 9
27	89.26	8.895 6
28	92.20	8.935 4
29	94.37	8.969 9
30	95.96	9.000 0

$N$	$p_n = 0.96^{n-1}p_1 $	$p_n = 0$
$N$	$C(T, N)$	$C(T, N)$
1	9.53	10.58
2	-214.70	-167.62
3	-276.16	-210.89
4	-300.39	$\textbf{-217.11}$
5	$\textbf{ -308.95}$	-205.71
6	-308.63	-183.40
7	-302.14	-153.77
8	-290.71	-119.66
9	-274.93	-83.79
10	-255.15	-48.77
11	-231.72	-16.74
12	-205.08	10.91
13	-175.86	33.60
14.	-144.90	51.46
15	-113.22	65.04
16	-81.94	75.10
17	-52.14	82.40
18	-24.73	87.62
19	-0.35	91.31
20	20.66	93.91
21	38.26	95.72
22	52.63	96.99
23	64.11	97.87
24	73.10	98.49
25	80.04	98.92
26	85.30	99.23
27	89.26	99.44
28	92.20	99.60
29	94.37	99.71
30	95.96	99.78

$\lambda $	$\mu = 0.9$	$\gamma = 0.95$	$\mu $	$\lambda = 0.1$	$\gamma = 0.95$	$\gamma $	$\lambda = 0.1$	$\mu = 0.9$
$\lambda $	$N_{T_i }^* $	$C(T_i, N_{T_i }^*)$	$\mu $	$N_{T_i }^* $	$C(T_i, N_{T_i }^*)$	$\gamma $	$N_{T_i }^* $	$C(T_i, N_{T_i }^*)$
0.1	5	–309.049 9	0.1	5	–309.046 0	0.1	3	–122.573 6
0.2	6	–169.484 4	0.2	5	–309.048 2	0.2	3	–188.414 9
0.3	7	–68.949 9	0.3	5	–309.049 0	0.3	4	–226.042 6
0.4	8	1.184 8	0.4.	5	–309.049 3	0.4	4	–249.808 1
0.5	9	48.156 7	0.5	5	–309.049 5	0.5	5	–266.136 0
0.6	11	76.646 3	0.6	5	–309.049 7	0.6	5	–280.169 3
0.7	14	92.387 9	0.7	5	–309.049 8	0.7	5	–280.870 2
0.8	18	98.797 1	0.8	5	–309.049 9	0.8	5	–299.299 7
0.9	30	99.996 6	0.9	5	–309.049 9	0.9	5	–306.111 7

$a$	$b = 0.85$	$c = 0.66$	$b$	$a = 1.15$	$c = 0.66$	$c$	$a = 1.15$	$b = 0.85$
$a$	$N_{T_i }^* $	$C(T_i, N_{T_i }^*)$	$b$	$N_{T_i }^* $	$C(T_i, N_{T_i }^*)$	$c$	$N_{T_i }^* $	$C(T_i, N_{T_i }^*)$
1.11	6	–320.650 9	0.05	5	–309.045 1	0.1	3	–234.482 9
1.12	6	–317.634 7	0.10	5	–309.049 0	0.2	3	–257.919 0
1.13	6	–314.639 8	0.15	5	–309.049 5	0.3	4	–267.773 5
1.14	6	–311.666 8	0.20	5	–309.049 7	0.4	4	–283.549 6
1.15	5	–309.049 9	0.25	5	–309.049 7	0.5	4	–292.431 2
1.16	5	–306.571 2	0.30	5	–309.049 8	0.6	5	–304.089 9
1.17	5	–304.111 2	0.35	5	–309.049 8	0.7	6	–313.280 2
1.18	5	–301.670 3	0.40	5	–309.049 8	0.75	6	–318.057 6
1.19	5	–299.248 7	0.45	5	–309.049 9	0.80	7	–322.309 5
1.20	5	–296.846 4	0.5	5	–309.049 9	0.85	7	–327.189 9
1.25	5	–285.131 5	0.55	5	–309.049 9	0.90	8	–331.667 7
1.30	5	–273.918 1	0.60	5	–309.049 9	0.95	8	–336.318 8
1.35	5	–263.209 6	0.65	5	–309.049 9	0.96	9	–337.245 7
1.40	4	–253.969 9	0.70	5	–309.049 9	0.97	9	–338.280 9
1.45	4	–245.801 5	0.80	5	–309.049 9	0.98	9	–339.279 9
1.50	4	–237.988 2	0.90	5	–309.049 9	0.99	9	–340.244 3

New repairable system model with two types repair based on extended geometric process

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