Reliability of k-out-of-n standby systems with gamma distributions

Cold standby redundancy is used as an effective mechanism for improving system reliability. However, methods for analyzing the reliability of k-out-of-n cold standby systems, particularly with components having age-dependent hazard (failure) rates, are limited. In this paper, using the concepts of counting processes, we propose an efficient method to evaluate the reliability of k-out-of-n cold standby systems. This proposed method considers gamma distributions for component lives and the effects of switch failures on system reliability. The main advantage of this counting process-based method is that it reduces a complex problem involving multiple integrals into an equivalent simple problem involving one-dimensional convolution integrals. We consider the gamma distribution for three reasons: (1) it can be used to model increasing, decreasing, and constant hazard rates, similar to the Weibull distribution, (2) it can be used to approximate several component failure time distributions, and (3) it has well established algorithms for calculating the convolutions that are used in the counting process-based method. In addition, we can find closed-form solutions to the convolution of the gamma distribution when its shape parameter is equal to an integer. Hence, we show that all steps involved in finding the reliability of k-out-of-n cold standby system are simple. We demonstrate the proposed method and its computational efficiency using a numerical example.