قابلیت اطمینان و تجزیه و تحلیل حساسیت یک سیستم قابل جبران و پوشش ناقص تحت شرایط فشار سرویس

This paper investigates reliability and sensitivity analysis of a repairable system with imperfect coverage under service pressure condition. Failure times and repair times of failed units are assumed to be exponentially distributed. As a unit fails, it may be immediately detected, located and replaced with a coverage probability c by a standby if one is available. When the repairmen are under the pressure of a long queue, the repairmen may increase the repair rate to reduce the queue length. We derive the explicit expressions for reliability function and mean time to system failure (MTTF). Various cases are analyzed to study the effects of different parameters on the system reliability and MTTF. We also accomplish sensitivity analysis and relative sensitivity analysis of the reliability characteristics with respect to system parameters.

Uncertainty is one of the important issues in management decisions. One of the most useful uncertainty measures is system reliability. The reliability of a system with standbys plays an important role in power plants, manufacturing systems, industrial systems and technical systems. Keeping a stable operating quality and a high level of reliability or availability is often a fundamental necessity. It maybe switches incompletely an existing spare module to a failed unit. When a failed unit is not detected, located and recovered, it needs time to be found and cleared. Therefore, we study the reliability of a system with multiple active units when covering a failed unit imperfectly. When the repairmen are under the pressure of a long queue, the repairmen may increase the repair rate to reduce the queue length. It may be impossible to switch in an existing spare module and then recover from a failure. Faults such as these are called to be not covered, and the probabilities of successful recovery on the failure of an active unit (or standby unit) is denoted by c. Quantity c which includes the probabilities of successful detection, location, and recovery from a failure is known as the coverage factor or coverage probability (see Trivedi [11]). A standby unit is called a ‘warm standby’ if its failure rate is nonzero and is less than the failure rate of an active unit. Active and warm standby units can be considered to be repairable. We continue with the assumption that the coverage factor is the same for active and standby unit failures. This paper differs from previous works in that: (i) the reliability problem with standby units has distinct characteristics which are different from the machine repair problem with standby units; (ii) it considers multiple imperfect coverage and reboot delay; and (iii) it considers the service pressure condition to prevent a long queue. The purpose of this article is to accomplish three objectives. The first objective is to develop the explicit expressions for reliability function, RY(t), and mean time to system failure, MTTF using Laplace transform techniques. The second objective is to perform sensitive analysis and relative sensitivity analysis of RY(t) and MTTF along with specified values of the system parameters. The third objective is to provide the numerical results to illustrate the sensitivity and the relative sensitivity of RY(t) and MTTF with respect to system parameters. Buzacott and Shanthikumar [2] reviewed queueing models that can be utilized to design manufacturing systems. The problem of machine failures and repairs in the context of queueing models has been investigated by several researchers. Govil and Fu [9] provided an excellent overview of the contributions of queueing models applied to manufacturing systems. In most papers, the queueing problems of the system discussed are more than the reliability problems of the system. Past work may be divided into two parts according to the system studied from the viewpoint of the queueing theory or from the viewpoint of the reliability. The major literature we review is the viewpoint of the reliability system. Cao and Cheng [3] first introduced reliability concept into a queueing system with a repairable service station where the lifetime of the service station is exponentially distributed and its repair time has a general distribution. Wang and Sivazlian [16] presented the reliability characteristics of a system consisting of M operating machines, S warm standbys and R repairmen. They have established relations between system reliability and the number of spares, the number of repairmen, the failure rate and the repair rate. Meng [10] compared the mean time to system failure for four redundant series. He obtained a general ordering relationship between the MTTF of these four systems. Wang and Kuo [14] investigated the reliability and availability characteristics of four different series system configurations with mixed standby components. They have provided a systematic methodology to develop the MTTF and the steady-state availability of four configurations with mixed standby units. Galikowsky et al. [4] and Wang and Pearn [15] investigated the cost benefit analysis of series system with cold standby components and warm standby components, respectively. They developed the explicit expressions for the MTTF and the steady-state availability. Ke and Wang [6] extended Wang and Sivazlian's [16] model by considering the balking and reneging in a repairable system. They provided the explicit expressions for the reliability characteristics of a repairable system with warm standby units plus balking and reneging. The concept of coverage factor and its effect on the reliability and availability model of a repairable system has been introduced by several authors such as Arnold [1], Trivedi [11], and Wang and Chiu [13]. The idea of imperfect coverage we discussed in this paper has been proposed by Trivedi [11]. Moreover, Wang and Chiu [13] analyzed the cost benefit analysis of availability systems with warm standby units and imperfect coverage. The concept of reboot delay and its effect on the reliability and availability model of a repairable system has been introduced by Trivedi [11]. Recently, Wang and Chen [12] investigated the reliability and availability analysis of a repairable system with standby switching failures. When the repairmen are under the pressure of a long queue, they may increase the repair rate to reduce the queue length. The concept of the service pressure coefficient was first introduced by Hiller and Lieberman [5]. Recently, Ke et al. [8] studied the reliability and sensitivity analysis of a system with multiple unreliable servers and standby switching failures. Ke et al. [7] first studied the reliability analysis of a system with standbys subjected to switching failure and presented a contour of the MTTF which is useful for the decision makers.

In this paper, we investigate a repairable system with warm standby units and multiple imperfect coverage under the service pressure condition. We provide the explicit expressions for RY(t) and MTTF. The numerical results show that the impact of service pressure coefficient a should not be ignored for a repairable system when the active units have lower failure rate. Moreover, the sensitivity analysis and the relative sensitivity analysis indicate that the impact of service pressure coefficient a to RY(t) and MTTF is more significant than of c and β. Overall, the order of the sensitivity to RY(t) is λ > μ > a > α > β ≈ c (K = 1). Moreover, the order of the sensitivity to MTTF is λ > μ > a > α > c > β (K = 1), and the order of the relative sensitivity to MTTF is c > λ > μ > a > α > β (K = 1).