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Reliability analysis for a \(k/n(F)\) system with repairable repair-equipment. (English) Zbl 1205.90104
Summary: The reliability and replacement policy of a \(k/n(F)\) (i.e. \(k\)-out-of-\(n: F\)) system with repairable repair-equipment is analyzed. We assume that both the working and repair times of all components in the system and the repair-equipment follow exponential distributions, and the repairs on the components are perfect whereas that on the repair-equipment is imperfect. Under these assumptions, by using the geometric process, the vector Markov process and the queueing theory, we derive reliability indices for such a system and discuss its properties. We also optimize a replacement policy \(N\) under which the repair-equipment is replaced whenever its failure number reaches \(N\). The explicit expression for the expected cost rate (i.e. the expected long-run cost per unit time) of the repair-equipment is derived, and the corresponding optimal replacement policy \(N^{*}\) can be obtained analytically or numerically. Finally, a numerical example for policy \(N\) is given.

90B25 Reliability, availability, maintenance, inspection in operations research
60K25 Queueing theory (aspects of probability theory)
Full Text: DOI
[1] Barlow, R.E.; Proschan, F., Mathematical theory of reliability, (1965), Wiley New York · Zbl 0132.39302
[2] Linton, D.G.; Saw, J.G., Reliability analysis of the k-out-of-n:F system, IEEE trans. reliab., R-23, 2, 97-103, (1974)
[3] Phillips, M.J., k-out-of-n:G systems are preferable, IEEE trans. reliab., R-29, 2, 166-169, (1980) · Zbl 0437.60065
[4] Gupta, H.; Sharma, J., State transition matrix and transition diagram of k-out-of-n:G system with spares, IEEE trans. reliab., R-30, 4, 395-397, (1981) · Zbl 0484.90046
[5] Kenyon, R.L.; Newell, R.J., Steady-state availability of k-out-of-n:G system with single repair, IEEE trans. reliab., R-32, 2, 188-190, (1983) · Zbl 0522.90030
[6] Nakagawa, T., Optimization problems in k-out-of-n system, IEEE trans. reliab., R-34, 3, 248-250, (1985) · Zbl 0563.90049
[7] McGrady, P.W., The availability of a k-out-of-n:G network, IEEE trans. reliab., R-34, 5, 451-452, (1985) · Zbl 0579.90043
[8] Moustafa, M.S., Transient analysis of reliability with and without repair for k-out-of-n system with two failure modes, Reliab. eng. sys. safe., 53, 31-35, (1996)
[9] Lam, Y., A note on the optimal replacement problem, Adv. appl. prob., 20, 479-482, (1988) · Zbl 0642.60071
[10] Lam, Y., Geometric processes and replacement problem, Acta math. appl. sin., 4, 366-377, (1988) · Zbl 0662.60095
[11] Lam, Y., A repair replacement model, Adv. appl. prob., 22, 494-497, (1990) · Zbl 0702.60075
[12] Zhang, Y.L., A bivariate optimal replacement policy for a repairable system, J. appl. prob., 31, 1123-1127, (1994) · Zbl 0811.60072
[13] Zhang, Y.L., An optimal geometric process model for a cold standby repairable system, Reliab. eng. sys. saf., 63, 107-110, (1999)
[14] Zhang, Y.L., A geometric process repair model with good-as-new preventive repair, IEEE trans. reliab., 51, 2, 223-228, (2002)
[15] Zhang, Y.L., An optimal replacement policy for a three-state repairable system with a monotone process model, IEEE trans. reliab., 53, 4, 452-457, (2004)
[16] Lam, Y.; Zhang, Y.L.; Zheng, Y.H., A geometric process equivalent model for a multistate degenerative system, Eur. J. oper. res., 142, 21-29, (2002) · Zbl 1081.90536
[17] Zhang, Y.L.; Yam, R.C.M.; Zuo, M.J., Optimal replacement policy for a deteriorating production system with preventive maintenance, Inter. J. sys. sci., 32, 10, 1193-1198, (2001) · Zbl 1006.90038
[18] Zhang, Y.L.; Yam, R.C.M.; Zuo, M.J., Optimal replacement policy for a multistate repairable system, J. oper. res. soc., 53, 336-341, (2002) · Zbl 1103.90328
[19] Zhang, Y.L.; Yam, R.C.M.; Zuo, M.J., A bivariate optimal replacement policy for a multistate repairable system, Reliab. eng. sys. safe., 92, 535-542, (2007)
[20] Lam, Y.; Zhang, Y.L., Analysis of a two-component series system with a geometric process model, Nav. res. log., 43, 491-502, (1996) · Zbl 0848.90060
[21] Lam, Y.; Zhang, Y.L., Analysis of a parallel system with different units, Acta math. appl. sin., 12, 408-417, (1996) · Zbl 0886.60086
[22] Lam, Y.; Zhang, Y.L., A shock model for the maintenance problem of a repairable system, Comput. oper. res., 31, 1807-1820, (2004) · Zbl 1073.90012
[23] Wu, S.; Clements-Croome, D., Optimal maintenance policies under different operational schedules, IEEE trans. reliab., 54, 2, 338-346, (2005)
[24] Wu, S.; Clements-Croome, D., A novel repair model for imperfect maintenance, IMA J. manage. math., 17, 235-246, (2006) · Zbl 1126.90016
[25] Wang, G.J.; Zhang, Y.L., A shock model with two-type failures and optimal replacement policy, Inter. J. sys. sci., 36, 4, 209-214, (2005) · Zbl 1085.90016
[26] Wang, G.J.; Zhang, Y.L., Optimal periodic preventive repair and replacement policy assuming geometric process repair, IEEE trans. reliab., 55, 1, 118-122, (2006)
[27] Zhang, Y.L.; Wang, G.J., A bivariate optimal repair-replacement model using geometric process for cold standby repairable system, Eng. optim., 38, 5, 609-619, (2006)
[28] Zhang, Y.L.; Wang, G.J., A geometric process repair model for a series repairable system with k dissimilar components, Appl. math. model., 31, 1997-2007, (2007) · Zbl 1167.90474
[29] Zhang, Y.L.; Wang, G.J., A deteriorating cold standby repairable system with priority in use, Eur. J. oper. res., 183, 278-295, (2007) · Zbl 1127.90334
[30] Lam, Y.; Zhang, Y.L.; Liu, Q., A geometric process model for an \(M / M / 1\) queueing system with a repairable service station, Eur. J. oper. res., 168, 100-121, (2006) · Zbl 1077.90014
[31] Ross, S.M., Stochastic processes, (1996), Wiley New York · Zbl 0888.60002
[32] Takacs, L., Introduction to the theory of queues, (1962), Oxford University Press New York · Zbl 0118.13503
[33] Kleinrock, L., Queueing systems, volume 1 theory, (1975), John Wiley & Sons New York
[34] Lam, Y., Calculating the rate of occurrence of failure for continuous-time Markov chains with application to a two-component parallel system, J. oper. res. soc., 45, 528-536, (1995) · Zbl 0838.90057
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