In many engineering systems, cold standby redundancy is an effective way to achieve high system reliability while preserving limited power resources. Cold standby redundancy technique use one or more redundant components that are unpowered, do not consume any energy and do not fail until being activated to replace a faulty online component. Whenever working component fails, then an available cold standby component is immediately powered up to take over the mission task. Some recent works on the research of the cold standby systems are in ([2, 7, 11, 12]).

In perfect repair model, it is assumed that the repair completely restores all properties of failed components. However, this is not always true in real world implementations. In practice, after the repair most repairable components are not “as good as new” because of the stresses. To pay attention to this problem, much work has been done by Brown and Proschan [1], Park [9], Kijima [4], Makis and Jardine [8]. For an imperfect repair model, it is more acceptable to consider that the successive operating times of the component after repair will be even shorter, while the consecutive repair times of the component after its failure will be even longer. For such a stochastic phenomenon, Lam [5, 6] studied a new repair-replacement policy and introduced a geometric process (GP) model. In this model, after the repair the successive operating times of the system are stochastically decreasing, while the consecutive repair times after the failure are stochastically increasing.

The following are the definition of stochastic order and geometric process, respectively, which can be seen from Ross [10] and Lam [6], respectively.

*Given X and Y random variables. For all real α, X is said to be stochastically larger than Y or Y is stochastically smaller than X, if*
$$P\left(X>\alpha \right)\ge P\left(Y>\alpha \right).$$

*Furthermore, a stochastic process* {*X*_{n}, n = 1, 2, } *is called stochastically decreasing if X*_{n} ≥ _{st} *X*_{n} + 1 *and stochastically increasing if X*_{n} ≤ _{st} *X*_{n} + 1.

*Let* {*X*_{n}, *n* = 1, 2, …} *be a sequence of non-negative independent random variables. Under the condition that α is a positive constant, if the distribution function of X*_{n} is F_{n}(*t*) = *F*(*a*^{n}^{-1}*t*), *t* ≥ 0*, then* {*X*_{n}, *n* = 1, 2, …} *is called a geometric process. The positive constant ˛is the ratio of the geometric process*.

It is evident that:

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If *α* > 1, then the geometric process {*X*_{n}, *n* = 1, 2, …} is stochastically decreasing.

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If 0 < *α* < 1, then the geometric process {*X*_{n}, *n* = 1, 2, …} is stochastically increasing.

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If *α* = 1, then the geometric process {*X*_{n}, *n* = 1, 2, …} is a renewal process.

In this paper, we study two non-identical components called the component 1 and component 2 and one repairman. Initially, the component 1 begins to operate and the other component is in cold standby state. As soon as the operating component 1 fails, the standby component is switched on immediately and the component 1 is taken for repair. When the repair of the component 1 is completed and the component 2 is still working, the repaired component returns to the standby pool and is once again available to be used as working component. The system breakdown occurs when the working component fails while the other component is under repair.

The organization of this paper is as follows. In Section 2, we give some assumptions concerning failures and repairs that will be useful throughout the paper. In Section 3, we describe the system mean lifetime measured when the component 1 fails. In Section 4, we present Laplace-Stieltjes (LS) transform of the system mean lifetime. In Section 5, we give a Gamma distributed example and a Weibull distributed example to illustrate the theoretical results for the proposed model. In the last section, we summarize what we have done in the article.

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