Abstract

The study of stress-strength reliability in a time-dependent context needs to model at least one of the stress or strength quantities as dynamic. We study the stress-strength reliability for the case in which the strength of the system is decreasing in time and the stress remains fixed over time; that is, the strength of the system is modeled as a stochastic process and the stress is considered to be a usual random variable. We present stochastic ordering results among the lifetimes of the systems which have the same strength but are subjected to different stresses. Multicomponent form of the aforementioned stress-strength interference is also considered. We illustrate the results for the special case when the strength is modeled by a Weibull process.

1. Introduction

Stress-strength models are of special importance in reliability literature and engineering applications. A technical system or unit may be subjected to randomly occurring environmental stresses such as pressure, temperature, and humidity and the survival of the system heavily depends on its resistance. In the simplest form of the stress-strength model, a failure occurs when the strength (or resistance) of the unit drops below the stress. In this case the reliability is defined as the probability that the unit’s strength is greater than the stress, that is, , where is the random strength of the unit and is the random stress placed on it. This reliability has been widely studied under various distributional assumptions on and . (See, e.g., Johnson [1] and Kotz et al. [2] for an extensive and lucid review of the topic.)

In the aforementioned simplest form, stress and strength quantities are considered to be both static. Dynamic modeling of stress-strength interference might offer more realistic applications to real-life reliability studies than static modeling and it enables us to investigate the time-dependent (dynamic) reliability properties of the system. Let and denote the stress that the system is experiencing and strength of the system at time , respectively. Then the lifetime of the system is represented as

The most important characteristic in reliability analysis is the reliability function of a system which is defined as the probability of surviving at time , that is, This function is also known as the survival function in the reliability literature and its exact formulation is of special importance in engineering applications.

The reliability function for the lifetime random variable given in (1) is

The function given by (3) has been investigated in several papers. Whitmore [3] computed the function under the assumptions that and are independent Brownian motions. Ebrahimi [4] studied the properties of assuming the strength of the system is decreasing in time.

In this paper, we study assuming (i) is decreasing in time, that is, for all and (ii) ; that is, stress remains fixed over time (static). The first assumption is common in reality because the system’s strength could degrade due to aging. In Section 2, we provide some stochastic ordering results among the lifetimes of the systems which have the same strength but are subjected to different stresses. In Section 3, stress-strength interference is considered for multicomponent systems. Finally, in Section 4 some results are presented for the special case when the strength is modeled by a Weibull process.

2. Reliability and Ordering Properties

Under the assumptions (i), (ii) and are independent and the reliability function can be formulated as where , and .

The following example illustrates the computation of reliability function for the stochastic strength process given with its analytical form. That is, the strength aging deterioration process is expressed as a function of time, and a random variable.

Example 1. Let be defined by where follows Pareto distribution with c.d.f. , , . Then the c.d.f. of is
Let have a c.d.f. , , . Then using (4) we have
Differentiating w.r.t. using the rule of differentiation under the integral sign, the p.d.f. of is found as
Using (8) in (7) we obtain
Integrating both sides of the last equation over we get which is the MTTF of the system.

The process defined by (5) can be considered in a more general form given by where and is a nondecreasing function. In this case the reliability function and MTTF of the system are found to be

For the system defined in Example 1 it can be easily seen that an increase in leads to a decrease in MTTF of the system. Since , the larger the harsher the stress and hence the smaller the reliability. This can be theoretically established using the concept of stochastic ordering as in the following lines.

We investigate the behaviour of the lifetime of the system under different stresses in terms of stochastic ordering. In this context, we consider the following stochastic orderings between the lifetimes. Let and be two random variables having cumulative distributions and , densities and , hazard rates and , and reversed hazard rates and , respectively. Note that the hazard and reversed hazard rates are defined, respectively, as

Usual Stochastic Ordering. is said to be smaller than in usual stochastic ordering if for all , or equivalently for all increasing functions for which the expectations exist. This relation is denoted by .

Hazard Rate Ordering. is said to be smaller than in hazard rate ordering if for all or is a nondecreasing function of and we write .

Reversed Hazard Rate Ordering. is said to be smaller than in the reversed hazard rate order if for all or is a nondecreasing function of and we write .

Likelihood Ratio Ordering. is said to be smaller than in likelihood ratio ordering (written as ) if is nondecreasing function of .

We have the following implications among these orderings: For more details on stochastic orderings refer to Shaked and Shanthikumar [5].

The following concepts will be useful for the next section.

Definition 2. A function is said to be sign regular of order 2 (SR₂) if and whenever , , and for .

If the conditions given in Definition 2 hold with and then is said to be totally positive of order 2 (TP₂); and is said to be reverse regular of order 2 (RR₂) if they hold with and .

Proposition 3. Let denote the lifetime of the system whose stress-strength pair is , . Then(a)if then ;(b)if and is RR₂ in then ;(c)if and is RR₂ in then .

Proof. The proof of (a) immediately follows because , , and is increasing in . The proofs of (b) and (c) can be obtained as an application of Theorems 1.B.14, 1.B.52, and 1.C.17 in Shaked and Shanthikumar [5]. These results are obtained using basic composition formula of Karlin [6].

Example 4. Consider the process defined in Example 1 with , and let have a c.d.f. , , . In this case
For any and which implies the RR₂ property of . If then . Thus by the Proposition 3 we have . Similarly, is also RR₂ in . If then and hence by the Proposition 3 we have .

3. Multicomponent Setup

In the previous sections we analyzed stress-strength reliability for a single component system. Most of the engineering systems consist of several components and the components might have different statistical properties. Multicomponent stress-strength reliability in a static form has been studied in various papers including Bhattacharyya and Johnson [7], Chandra and Owen [8], Johnson [1], Pandey et al. [9], Eryilmaz [10], and Eryilmaz [11].

Assume that a system consists of components and the deteriorating strength of the th component at time is denoted by the process , . The components are subjected to a common random stress . If denotes the lifetime of the th component then the joint survival function of is given by If the components are independent then we have

From (19) it follows that the lifetimes of the components are dependent even if the strengths of them are independent. This positive dependence among the lifetimes arises from common environmental stress characterized by . There are many types of positive dependence. The likelihood ratio (or TP₂) dependence as the strongest notion of positive dependence is defined as follows. Let , have the joint probability density . Then recall from Definition 2 that is TP₂ if for and . The random variables and are said to be likelihood ratio (or TP₂) dependent if their joint density is TP₂.

The following result can be proved using the basic composition formula of Karlin [6] together with where and , .

Proposition 5. If is TP₂ (RR₂) in and is TP₂ (RR₂) in , then and are likelihood ratio dependent.

Example 6. Let , , and be an exponential random variable with c.d.f. , . Also assume that the common random stress has c.d.f. , . In this case the joint survival function of and is found to be
Since is RR₂ for , and are likelihood ratio dependent.

Consider a system with components which has two possible states; if the system is functioning and if the system has failed. Since the state of the system is determined by the states of its components we can write , where if the th component is functioning and if it has failed. The function is called structure function. A system with structure function is coherent if is nondecreasing in each argument, and each component is relevant to the performance of the system. If the components’ lifetimes are denoted by , then represents the lifetime of the system.

Let denote the i.i.d. lifetime random variables with continuous distribution. Samaniego [12] introduced the signature of a coherent system as the vector , where where denotes the smallest th in , showing that A general formula for the reliability function of any coherent structure consisting of components can be given by using the concept of “signature” if the components are independent and identical. Samaniego [12] (see also [13]) showed that the reliability function of a coherent system can be represented as Navarro et al. [14] (see also Navarro and Rychlik [15]) proved that the representation (26) also holds whenever has an absolutely continuous exchangeable joint distribution. The following theorem provides the reliability function of any coherent structure consisting of components.

Theorem 7. Let denote the lifetime of the th component whose strength is , , that is, . If denotes the structure function of the coherent system with lifetime , that is, and are i.i.d. with c.d.f. , then where

Proof. Under the assumption that are i.i.d. the joint survival function of is
The function given by (33) is a mixture of independent -variate d.f.’s with equal marginals; that is, the random vector is positive dependent by mixture (PDM). PDM d.f.’s are exchangeable. (See, e.g., Shaked [16] for the concept of PDM and associated exchangeability). Since the representation (26) also holds for exchangeable lifetimes we get (27) with
The usage of inclusion-exclusion principle for the probability inside the sum gives
The proof is now completed by conditioning on .

4. Weibull Stress-Strength Model

In this section we study the stress-strength reliability for the Weibull process which can be used to model the decreasing strength of a unit. Chiodo and Mazzanti [17] studied stress-strength reliability and its estimation for aged power system components subjected to voltage surges using Weibull process.

Let be a Weibull process whose one-dimensional distribution is where the shape parameter is assumed to be time independent and the intensity function is decreasing in time with . Similarly, assume that the stress random variable has a Weibull distribution with c.d.f.

Under these assumptions the reliability function is found to be

The following results can be obtained from Proposition 3.

Corollary 8. Let denote the lifetime of the system whose stress-strength pair is , , where has a Weibull distribution with scale parameter and shape parameter and is a Weibull process whose distribution is given by (32). Then, since is RR₂ in , if then .

Corollary 9. Under the same assumptions of Corollary 8, the function is RR₂ in . Indeed, because is decreasing, , and hence . For and , which implies that is RR₂ in . Therefore, from Proposition 3, if then .

Remark 10. If then which is the survival function of the log-logistic distribution. That is, the lifetime of the system has log-logistic distribution with scale parameter and shape parameter . If then the MTTF of the system is found to be

Theorem 11. Let be a Weibull process with intensity associated with th component, . Assume that the common random stress has a Weibull distribution with c.d.f. given by (33). Then the joint survival function of is and the survival copula associated with (39) belongs to the Clayton family and is given by

Proof. Using (19) one can write
By the definition of survival copula (see, e.g., Nelsen [18, page 32]) which is known to be a Clayton copula (see, e.g., Nelsen [18, page 152]). Thus the proof is completed.

Remark 12. For , , (39) becomes the survival function of the multivariate log-logistic distribution generated by the Clayton family of copulas.

Example 13. Consider the system consisting of components whose deteriorating strengths are modeled by a Weibull process with the common intensity function . Suppose that these components are subjected to a common random stress which has c.d.f. given by (33). Then, the lifetimes of the components are exchangeable and we have
If, for example, a system has a 2-out-of-3 structure, that is, the system functions if and only if at least two components function, then since using Theorem 7, the reliability of the system is found to be

Acknowledgment

The author thanks the anonymous referee for his/her helpful comments and suggestions.