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 insights 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 (Pólya) 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.