Reliability Characteristics of k-out-of-n Warm Standby Systems

We study reliability characteristics of the k-out-of-n warm standby system with identical components subject to exponential lifetime distributions. We derive state probabilities of the warm standby system in a form that is similar to the state probabilities of the active redundancy system. Subsequently, the system reliability is expressed in several forms that can provide new in-sights into the system reliability characteristics. We also show that all properties and computational procedures that are applicable for active redundancy are also applicable for the warm standby redundancy. As a result, it is shown that the system reliability can be evaluated using robust algorithms within O(n-k+1) computational time. In addition, we provide closed-form expressions for the hazard rate, probability density function, and mean residual life function. We show that the time-to-failure distribution of the k-out-of-n warm standby system is equal to the beta exponential distribution. Subsequently, we derive closed-form expressions for the higher order moments of the system failure time. Further, we show that the reliability of the warm standby system can be calculated using well-established numerical procedures that are available for the beta distribution. We prove that the improvement in system reliability with an additional redundant component follows a negative binomial (Polya) distribution, and it is log-concave in n. Similarly, we prove that the system reliability function is log-concave in n. Because the k-out-of-n system with active redundancy can be considered as a special case of the k-out-of-n warm standby system, we indirectly provide some new results for the active redundancy case as well.