Document Type : Research Paper


1 Ph.D. Candidate, Faculty of Industrial and Mechanical Engineering, Qazvin Branch, Islamic Azad University, Qazvin, Iran

2 Associate Professor, Faculty of Industrial and Mechanical Engineering, Qazvin Branch, Islamic Azad University, Qazvin, Iran,

3 Professor, Department of Industrial Management, Faculty of Management and Accounting, Allameh Tabataba’i University, Tehran, Iran


Due to the high sensitivity in applying of electronic and mechanical equipment, creating any conditions to increase the reliability of a system is always one of the important issues for system designers. Hence, making academic models much closer to the real word applications is very attractive. In the most studies in the reliability area, it is assumed that the failure rates of the system components are constant and have exponential distributions. This distribution and its attractive memory less property provide simple mathematical relationships in order to obtain the system reliability. But in real word problems, considering time-dependent failure rates is more realistic to model processes. It means that, the system components do not fail with a constant rate during the time horizon; but this failure rate changes over the time. One of the most useful statistical distributions in order to model the time-dependent failure rates is the Weibull distribution. This distribution is not a memory less one, so it was impossible to apply simple and explicit mathematical relationships as the same as exponential distributions for the reliability of a system. Therefore, researchers in this field have used simulation technique in these circumstances which is not an exact method to get near-optimum solutions. In this paper, for the first time, it is tried to obtain a mathematical equation to calculate the reliability function of a system with time-dependent components based on Weibull distribution. Also, in order to validate the proposed method, the results compared with exact solution that exists in literature.


Banjevic D, Jardine AK. Calculation of reliability function and remaining useful life for a Markov failure time process. IMA journal of management mathematics. 2006 Apr 1;17(2):115-30.
Carrasco JM, Ortega EM, Cordeiro GM. A generalized modified Weibull distribution for lifetime modeling. Computational Statistics and Data Analysis 2008; 53(2):450-462.
Coit DW, Smith AE. Genetic algorithm to maximize a lower-bound for system time-to-failure with uncertain component Weibull parameters. Computers and Industrial Engineering 2002; 28;41(4):423-440.
Elmahdy EE, Aboutahoun AW. A new approach for parameter estimation of finite Weibull mixture distributions for reliability modeling. Applied Mathematical Modelling 2013; 37(4):1800-1810.
Eryilmaz S. Reliability of a k-out-of-n system equipped with a single warm standby component. IEEE Transactions on Reliability 2013; 62(2):499-503.
Fazlollahtabar H, Saidi-Mehrabad M, Balakrishnan J. Integrated Markov-neural reliability computation method: A case for multiple automated guided vehicle system. Reliability Engineering & System Safety. 2015; 135:34-44.
Flammini F, Marrone S, Mazzocca N, Vittorini V. A new modeling approach to the safety evaluation of N-modular redundant computer systems in presence of imperfect maintenance. Reliability Engineering and System Safety 2009; 94(9):1422-1432.
Gray JN. Continuous-time Markov methods in the solution of practical reliability problems. Reliability engineering 1985; 11(4):233-52.
Guilani PP, Azimi P, Niaki ST, Niaki SA. Redundancy allocation problem of a system with increasing failure rates of components based on Weibull distribution: A simulation-based optimization approach. Reliability Engineering and System Safety 2016; 152:187-96.
Guilani PP, Sharifi M, Niaki ST, Zaretalab A. Reliability evaluation of non-reparable three-state systems using Markov model and its comparison with the UGF and the recursive methods. Reliability Engineering and System Safety 2014; 129:29-35.
Hisada K, Arizino F. Reliability tests for Weibull distribution with varying shape-parameter, based on complete data. IEEE transactions on Reliability 2002 ;51(3):331-336.
Huang CC, Yuan J. A two-stage preventive maintenance policy for a multi-state deterioration system. Reliability Engineering and System Safety 2010; 95(11):1255-60.
Huang L, Xu Q. Lifetime reliability for load-sharing redundant systems with arbitrary failure distributions. IEEE Transactions on Reliability 2010; 59(2):319-330.
Jiang P, Lim JH, Zuo MJ, Guo B. Reliability estimation in a Weibull lifetime distribution with zerofailure field data. Quality and Reliability Engineering International 2010; 26(7):691-701.
Landon J, Özekici S, Soyer R. A Markov modulated Poisson model for software reliability. European Journal of Operational Research. 2013; 229(2):404-10.
Mendes, A.A., D.W. Coit, and J.L.D. Ribeiro, Establishment of the optimal time interval between periodic inspections for redundant systems. Reliability Engineering & System Safety, 2014. 131: p. 148-165.
Montoro-Cazorla D, Pérez-Ocón R. Constructing a Markov process for modeling a reliability system under multiple failures and replacements. Reliability Engineering & System Safety. 2018 (in press).
Nadarajah S, Kotz S. On some recent modifications of Weibull distribution. IEEE Transactions on Reliability 2005; 54(4):561-562.
Nourelfath M, Châtelet E, Nahas N. Joint redundancy and imperfect preventive maintenance optimization for series–parallel multi-state degraded systems. Reliability Engineering and System Safety 2012; 103:51-60.
Peng X, Yan Z. Estimation and application for a new extended Weibull distribution. Reliability Engineering and System Safety 2014; 121:34-42.
Pham H, Lai CD. On recent generalizations of the Weibull distribution. IEEE transactions on reliability 2007; 56(3):454-8.
Ross, SM. A first course in probability. Pearson 2014.
Soro IW, Nourelfath M, Aït-Kadi D. Performance evaluation of multi-state degraded systems with minimal repairs and imperfect preventive maintenance. Reliability Engineering and System Safety 2010; 95(2):65-69.
Wang Z, Mourelatos ZP, Li J, Baseski I, Singh A. Time-dependent reliability of dynamic systems using subset simulation with splitting over a series of correlated time intervals. Journal of Mechanical Design. 2014;136(6):061008.
Wu S, Zhang L, Lundteigen MA, Liu Y, Zheng W. Reliability assessment for final elements of SISs with time dependent failures. Journal of Loss Prevention in the Process Industries. 2018;51:186-99.