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This article discusses the stochastic behavior and reliability properties for the inactivity times of failed components in coherent systems under double monitoring. A mixture representation of reliability function is obtained for the inactivity times of failed components, and some stochastic comparison results are also established. Furthermore, some sufficient conditions are developed in terms of the aging properties of the inactivity times of failed components. Finally, some numerical examples are presented to illustrate the theoretical results.

In many significant safety systems, the fault tolerance of a fail-safe system has been a fundamental design attributed to obtaining more higher reliability. However, in most practical scenarios, the components in a fail-safe system suffer from additional random shocks, and one component’s failure may increase the failure probability of other surviving components. This article analyzes the effects of survived probabilities and components lifetimes on the reliability of fail-safe systems with statistically dependent components subject to random shocks in terms of the usual stochastic and hazard rate orderings. Sufficient conditions are established by means of transformation functions, underlying component lifetime distributions, and majorization orders. Some numerical examples are also presented for illustrating our theoretical findings.

In this paper, we propose a copula-based approach to study the allocation problem of hot standbys in series systems composed of two heterogeneous and dependent components. By assuming that the lifetimes of components and spares are dependent and linked via a general survival copula, optimal allocation strategies are presented for the case of one and two redundancies at the component level. Further, redundancies allocation mechanisms are also compared between the allocations at the component level and the system level. For the case of one hot standby, we find that the performance of the redundant system at the component level is always worse than that at the system level. For the case of two hot standbys, the reversed allocation principle (i.e., Barlow-Proschan principle) is valid. Numerical examples and applications are also provided as illustrations. A real application on improving tensile strength of cables in high voltage electricity transmission network systems is presented for showing the applicability of our results.

In this manuscript, we gain a mixture representation for reliability function of the residual lifetime of unfailed components in a coherent system under periodical inspections, given that the number of failed components before time t1 is r(≥0), but the system is still operating at time t1, and the system eventually failed at time t2(>t1). Some aging properties and stochastic orders of the residual lifetime on survival components are also established. Finally, some numerical examples and graphs are given in order to confirm the theoretical results.