Cold standby 1-out-of-n redundant systems are well-established models in system reliability engineering. To date, reliability analyses of such systems have predominantly assumed exponential, Erlang, or Weibull failure distributions for their components. The Lindley distribution and its generalizations represent a significant class of statistical distributions in reliability engineering. Certain generalized Lindley distributions, due to the appealing characteristics of their hazard functions, can serve as suitable alternatives to other well-known lifetime distributions like the Weibull. This study investigates the reliability of a 1-out-of-n cold standby redundant system with perfect and imperfect switching, assuming that the active component failure times follow the Generalized Lindley distribution. We derive a closed-form expression for the system reliability. To achieve this, the distribution of the sum of n independent and identically distributed random variables following the Generalized Lindley distribution is first determined using the moment-generating function approach.

翻译：冷储备1-out-of-n冗余系统是系统可靠性工程中成熟的模型。迄今为止，此类系统的可靠性分析主要假设其部件服从指数分布、爱尔朗分布或威布尔分布等故障分布。Lindley分布及其推广形式是可靠性工程中一类重要的统计分布。某些广义Lindley分布因其风险函数的吸引人特性，可作为威布尔等其他著名寿命分布的合适替代。本研究在假设活动部件故障时间服从广义Lindley分布的前提下，探讨了具有完善与不完善切换功能的1-out-of-n冷储备冗余系统的可靠性。我们推导了系统可靠性的闭式表达式。为此，首先利用矩生成函数方法确定了服从广义Lindley分布的n个独立同分布随机变量之和的分布。