Stochastic modeling and optimization of discrete-time cold standby repairable systems with unreliable repair facility and retrial mechanism. (English) Zbl 07900339
Summary: This study develops novel reliability models for discrete-time cold standby repairable systems with unreliable repair facility and retrial mechanism. In view of the situation that the state of some engineering systems cannot be continuously monitored, discrete distribution is used to describe the distribution of random variables involved in this paper. For the case that multiple events can occur simultaneously in the discrete-time system model, the priority order of multiple events occurring simultaneously is defined. Two different models are proposed based on two types of priority rules. The retrial mechanism of the components is considered in the case that the failure information of the system components cannot be successfully transmitted to the repair facility. The preventive maintenance (PM) and repair after the breakdown of the repair facility are considered. The key reliability indices are derived by using the iterative algorithm of the difference equation and the generative function method. The algorithms for calculating system state probability, key reliability indices and cost-benefit ratio (CBR) are designed. To demonstrate the impact of each parameter on the system reliability indices, numerical results are provided. The optimal repair rate configuration scheme is chosen using the Particle Swarm Optimization (PSO) algorithm to achieve the lowest possible CBR. The impact of the system model with or without PM on stationary availability and CBR is shown. In addition, the impact of the number of system components on a series of system performance indices is analyzed, and the optimal number of components of the system corresponding to the minimum value of CBR is obtained.
MSC:
| 90B25 | Reliability, availability, maintenance, inspection in operations research |
| 62N05 | Reliability and life testing |
| 90B22 | Queues and service in operations research |
References:
| [1] | Chen, W. L., System reliability analysis of retrial machine repair systems with warm standbys and a single server of working breakdown and recovery policy, Syst. Eng., 21, 1, 59-69, 2018 |
| [2] | Yen, T.-C.; Wang, K.-H.; Wu, C.-H., Reliability-based measure of a retrial machine repair problem with working breakdowns under the F-policy, Comput. Ind. Eng., 150, Article 106885 pp., 2020 |
| [3] | Gao, S.; Wang, J. T., Reliability and availability analysis of a retrial system with mixed standbys and an unreliable repair facility, Eng. Syst. Saf., 205, Article 107240 pp., 2021 |
| [4] | Li, M. J.; Hu, L. M.; Peng, R.; Bai, Z. X., Reliability modeling for repairable circular consecutive \(k\)-out-of-\(n\): F systems with retrial feature, Reliab. Eng. Syst. Saf., 216, Article 107957 pp., 2021 |
| [5] | Wang, Y.; Hu, L. M.; Yang, L., Reliability modeling and analysis for linear consecutive \(k\)-out-of-\(n\): F retrial systems with two maintenance activities, Reliab. Eng. Syst. Saf., 226, Article 108665 pp., 2022 |
| [6] | Hu, L. M.; Liu, S. J.; Peng, R., Reliability and sensitivity analysis of a repairable \(k\)-out-of-\(n\): G system with two failure modes and retrial feature, Commun. Stat.-Theory Methods, 51, 9, 3043-3064, 2022 · Zbl 07535578 |
| [7] | Wang, K. H.; Wu, C. H.; Yen, T. C., Comparative cost-benefit analysis of four retrial systems with preventive maintenance and unreliable service station, Reliab. Eng. Syst. Saf., 221, Article 108342 pp., 2022 |
| [8] | Wang, Y.; Hu, L. M.; Zhao, B., Stochastic modeling and cost-benefit evaluation of consecutive \(k\)-out-of-\(n\): F repairable retrial systems with two-phase repair and vacation, Comput. Ind. Eng., 175, Article 108851 pp., 2023 |
| [9] | Kang, J.; Hu, L. M.; Peng, R.; Li, Y.; Tian, R. L., Availability and cost-benefit evaluation for a repairable retrial system with warm standbys and priority, Stat. Theory Related Fields, 7, 2, 164-175, 2023 · Zbl 07710178 |
| [10] | Li, M. J.; Hu, L. M.; Wu, S. M.; Zhao, B.; Wang, Y., Reliability assessment for consecutive-\(k\)-out-of-\(n\) : F retrial systems under Poisson shocks, Appl. Math. Comput., 448, Article 127913 pp., 2023 · Zbl 1511.90130 |
| [11] | Nakagawa, T.; Osaki, S., The discrete Weibull distribution, IEEE Trans. Reliab., 24, 5, 300-301, 1975 |
| [12] | Salvia, A. A.; Bollinger, R. C., On discrete hazard functions, IEEE Trans. Reliab., 31, 5, 458-459, 1982 · Zbl 0507.62084 |
| [13] | Xekalaki, E., Hazard functions and life distributions in discrete time, Commun. Stat.-Theory Methods, 12, 21, 2503-2509, 1983 · Zbl 0552.62008 |
| [14] | Stein, W. E.; Dattero, R., A new discrete Weibull distribution, IEEE Trans. Reliab., 33, 2, 196-197, 1984 · Zbl 0563.62079 |
| [15] | Alfa, A. S.; Castro, I. T., Discrete time analysis of a repairable machine, J. Appl. Probab., 39, 3, 503-516, 2002 · Zbl 1031.60077 |
| [16] | Bracquemond, C.; Gaudoin, O., A survey on discrete lifetime distributions, Int. J. Reliab. Qual. Sa., 10, 01, 69-98, 2003 |
| [17] | Eryilmaz, S., Discrete time cold standby repairable system: Combinatorial analysis, Commun. Stat.-Theory Methods, 45, 24, 7399-7405, 2016 · Zbl 1349.62501 |
| [18] | Dembińska, A.; Nikolov, N. I.; Stoimenova, E., Reliability properties of \(k\)-out-of-\(n\) systems with one cold standby unit, J. Comput. Appl. Math., 388, Article 113289 pp., 2021 · Zbl 1459.62190 |
| [19] | Jasiński, K., Some conditional reliability properties of \(k\)-out-of-\(n\) system composed of different types of components with discrete independent lifetimes, Metrika, 84, 8, 1241-1251, 2021 · Zbl 1496.62173 |
| [20] | Ruiz-Castro, J. E.; Pérez-Ocón, R.; Fernández-Villodre, G., Modelling a reliability system governed by discrete phase-type distributions, Reliab. Eng. Syst. Saf., 93, 11, 1650-1657, 2008 |
| [21] | Ruiz-Castro, J. E.; Fernández-Villodre, G.; Pérez-Ocón, R., A multi-component general discrete system subject to different types of failures with loss of units, Discrete Event Dyn. S, 19, 1, 31-65, 2009 · Zbl 1169.93368 |
| [22] | Ruiz-Castro, J. E.; Fernández-Villodre, G.; Pérez-Ocón, R., Discrete repairable systems with external and internal failures under phase-type distributions, IEEE Trans. Reliab., 58, 1, 41-52, 2009 |
| [23] | Wang, J. T.; Cao, J. H.; Li, Q. L., Reliability analysis of the retrial queue with server breakdowns and repairs, Queueing Syst., 38, 363-380, 2001 · Zbl 1028.90014 |
| [24] | Moreno, P., A discrete-time retrial queue with unreliable server and general server lifetime, J. Math. Sci., 132, 5, 643-655, 2006 · Zbl 1411.60138 |
| [25] | Tang, Y. H.; Yun, X.; Huang, S. J., Discrete-time \(G e o^X/G/1\) queue with unreliable server and multiple adaptive delayed vacations, J. Comput. Appl. Math., 220, 439-455, 2008 · Zbl 1147.60056 |
| [26] | Choudhury, G.; Ke, J.-C.; Tadj, L., The N-policy for an unreliable server with delaying repair and two phases of service, J. Comput. Appl. Math., 231, 1, 349-364, 2008 · Zbl 1166.60055 |
| [27] | Tang, Y. H.; Yu, M. M.; Yun, X.; Huang, S. J., Reliability indices of discrete-time \(\text{Geo}^{\text{X}}/\text{G} /1\) queueing system with unreliable service station and multiple adaptive delayed vacations, J. Syst. Sci. Complex., 25, 6, 1122-1135, 2012 · Zbl 1282.90052 |
| [28] | Jain, M.; Sharma, G. C.; Sharma, R., Maximum entropy approach for discrete-time unreliable server \(\text{Geo}^{\text{X}}/\text{Geo} /1\) queue with working vacation, Int. J. Math. Oper. Res., 4, 1, 56-77, 2012 · Zbl 1390.90220 |
| [29] | Sharma, G.; Priya, K., Analysis of G-queue with unreliable server, Opsearch, 50, 3, 334-345, 2013 · Zbl 1353.90043 |
| [30] | Bhagat, A.; Jain, M., \(N\)-Policy for \(M^x /\) G \(/ 1\) unreliable retrial G-queue with preemptive resume and multi-services, J. Oper. Res. Soc. China, 4, 4, 437-459, 2016 · Zbl 1365.90043 |
| [31] | Gao, S.; Wang, J. T.; Van Do, T., Analysis of a discrete-time repairable queue with disasters and working breakdowns, RAIRO-Oper. Res., 53, 4, 1197-1216, 2019 · Zbl 1439.60085 |
| [32] | Barlow, R.; Hunter, L., Optimum preventive maintenance policies, Oper. Res., 8, 1, 90-100, 1960 · Zbl 0095.34304 |
| [33] | Nakagawa, T., Optimum policies when preventive maintenance is imperfect, IEEE Trans. Reliab., 28, 4, 331-332, 1979 · Zbl 0426.90040 |
| [34] | Lie, C. H.; Chun, Y. H., An algorithm for preventive maintenance policy, IEEE Trans. Reliab., 35, 1, 71-75, 1986 · Zbl 0595.90032 |
| [35] | Nakagawa, T., Modified discrete preventive maintenance policies, Nav. Res. Logist. Q., 33, 4, 703-715, 1986 · Zbl 0601.90061 |
| [36] | Sandoh, H.; Hirakoshi, H.; Nakagawa, T., A new modified discrete preventive maintenance policy and its application to hard disk management, J. Qual. Maintenance Eng., 4, 4, 284-290, 1998 |
| [37] | Chalabi, N.; Dahane, M.; Beldjilali, B., Optimisation of preventive maintenance grouping strategy for multi-component series systems: Particle swarm based approach, Comput. Ind. Eng., 102, 440-451, 2016 |
| [38] | Taleb, S.; Aissani, A., Preventive maintenance in an unreliable M/G/1 retrial queue with persistent and impatient customers, Ann. Oper. Res., 247, 1, 291-317, 2016 · Zbl 1358.90037 |
| [39] | Barron, Y.; Yechiali, U., Generalized control-limit preventive repair policies for deteriorating cold and warm standby Markovian systems, IISE Trans., 49, 11, 1031-1049, 2017 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
