قابلیت اطمینان و تجزیه و تحلیل حساسیت سیستم با چندین ایستگاه سرویس غیر قابل اعتماد و شکست سوئیچینگ آماده به کار

This paper presents the reliability and sensitivity analysis of a system with M primary units, W warm standby units, and R unreliable service stations where warm standby units switching to the primary state might fail. Failure times of primary and warm standby units are assumed to have exponential distributions, and service times of the failed units are exponentially distributed. In addition, breakdown times and repair times of the service stations also follow exponential distributions. Expressions for system reliability,RY(t)RY(t), and mean time to system failure, MTTF are derived. Sensitivity analysis, relative sensitivity analysis of the system reliability and the mean time to failure, with respect to system parameters are also investigated.

In machine repair problems, most studies about the reliability of a system assume that the switchover from warm standby units to primary units is always perfect and the service stations are reliable. Although these assumptions simplify the analysis of the problem, they might not reflect certain real situations. For instance, a warm standby unit with a lower failure rate might not be able to switch over to a primary unit successfully, and it might also need a longer warm-up time. In this paper, we investigate the reliability of a system in which the switching failure might occur and service stations might also break down. Although the concept of imperfect switching was discussed by Lewis [1], to the best of our knowledge, it has never been considered in the reliability problem in a system with multiple unreliable service stations. Under the assumption of perfect switching, Wang [2] proposed the M/M/1 machine repair problem with two different types of a single service station subject to breakdowns. Later, Wang and Kuo [3] extended the result to the M/Ek/1M/Ek/1 machine repair problem with a single unreliable server. Wang [4] also developed steady-state analytic solutions for the M/M/R machine repair problem with spares and R unreliable service stations. Cao [5] derived reliability measures of a machine service model with a single unreliable service station. Wang and Sivazlian [6] presented the reliability characteristics of a repairable system with warm standbys. Ke and Wang [7] and [8] analyzed the reliability characteristics of a repairable system with warm standby units in which failed units balk with a constant probability and renege according to an exponential distribution. Lately, Wang et al. [9] analyzed the reliability and sensitivity of a repairable system with warm standbys and multiple unreliable service stations. On the other hand, the reliability of the system with a switching device which might fail was also discussed. Chung [10] gave the reliability and availability of k operating machines and s cold standbys with multiple repair facilities and multiple critical and non-critical errors when the switching mechanism is subject to failure. Gurov and Utkin [11] presented the transient behavior of repairable and unrepairable cold standby systems with conversion switches. They established mathematical models of systems by a set of integral equations. Pan [12] discussed a non-repairable system with one or two standby components, one switch, and one sensor. Coit [13] derived a solution methodology to determine optimal design configurations for non-repairable systems with cold standbys, non-constant hazard functions, and imperfect switching. In this paper, we examine the reliability characteristics of a system with M identical primary units operating simultaneously in parallel, W warm standby units, and R unreliable service stations. This model is an extension of Wang et al. [9] model with the consideration of imperfect switching. In particular, we study the impact of the switching failure on the reliability function and the mean time to failure. In addition, the sensitivity and the relative sensitivity of the system reliability as well as of the mean time to failure with respect to system parameters are investigated. The rest of the paper is organized as follows. In Section 2, we formulate the problem and provide notation subsequently used throughout the paper. In Section 3, explicit expressions for reliability function, RY(t)RY(t), and the mean time to system failure, the MTTF are derived using Laplace transform techniques. Sensitivity and relative sensitivity analysis of RY(t)RY(t) and the MTTF are also developed in terms of the system parameters. In Section 4, numerical results are provided to illustrate the sensitivity and the relative sensitivity of RY(t)RY(t) and of the MTTF with respect to system parameters. Conclusions are presented in the last section.

In this paper, we discuss a repairable system with M primary units, W warm standby units, and R unreliable service stations when warm standby units switching to the primary state might fail. Explicit expressions for the MTTF and RY(t)RY(t) are provided. The numerical results indicate that the switching failure has a significant effect on the MTTF and RY(t)RY(t), especially for a system when primary units have lower failure rates. In addition, the sensitivity analysis and relative sensitivity analysis show that the impact of switching failure probability q to the reliability function is more significant than of μμ and ββ. Overall, the order of the sensitivity to the system reliability is λ>α>q>β>μλ>α>q>β>μ, while the order of the relative sensitivity to the system reliability is λ>μ>β>α>qλ>μ>β>α>q.