On Stress-Strength Reliability with a Time-Dependent Strength

e 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 �xed 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.


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 de�ned 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.is 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.en the lifetime of the system is represented as          >   . ( e most important characteristic in reliability analysis is the reliability function of a system which is de�ned as the probability of surviving at time , that is,      >  . ( is function is also known as the survival function in the reliability literature and its exact formulation is of special importance in engineering applications.e reliability function for the lifetime random variable given in (1) is e 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,  2  ≤  1   1 for all  1   2 and (ii)   ; that is, stress remains �xed over time (static).e �rst 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

Reliability and Ordering Properties
Under the assumptions (i), (ii)  and  are independent and the reliability function can be formulated as where  = {  },    = {  } and    = 1 −   .
e following example illustrates the computation of reliability function for the stochastic strength process given with its analytical form.at is, the strength aging deterioration process is expressed as a function of time, and a random variable.
Example 1.Let  be de�ned by where  follows Pareto distribution with c.d.f.   = 1 −  2 ,   ,   .en the c.d.f. of  is Let  have a c.d.f. =   ,     1,   .en using (4) we have Differentiating 1 −  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.
e process de�ned 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 de�ned in Example 1 it can be easily seen that an increase in  leads to a decrease in MTTF of the system.Since  =   1, the larger  the harsher the stress and hence the smaller the reliability.is 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 ℎ  and ℎ  , respectively.Note that the hazard and reversed hazard rates are de�ned, respectively, as For more details on stochastic orderings refer to Shaked and Shanthikumar [5].e following concepts will be useful for the next section.If the conditions given in �e�nition 2 hold with  1  1 and  2  1 then  is said to be totally positive of order 2 (TP 2 ); and  is said to be reverse regular of order 2 (RR 2 ) if they hold with  1  1 and  2  −1.
Proposition 3. Let   denote the lifetime of the system whose stress-strength pair is    ,   1 2. en Proof.e proof of (a) immediately follows because {  ≤ }      ],   1 2, and    is increasing in .e proofs of (b) and (c) can be obtained as an application of eorems 1.B.14, 1.B.52, and 1.C.17 in Shaked and Shanthikumar [5].ese results are obtained using basic composition formula of Karlin [6]

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   ,   1 2   .e components are subjected to a common random stress .If   denotes the lifetime of the th component then the joint survival function of  1   2      is given by If the components are independent then we have e following result can be proved using the basic composition formula of Karlin [6]  where   denotes the smallest th in   ,  2 , … ,   , showing that    # of orderings for which the th failure causes system failure ×   − . (25) 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     , e function given by ( 33) is a mixture of independent variate d.f.'s with equal marginals; that is, the random vector   ,  2 , … ,    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 e proof is now completed by conditioning on .

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 stressstrength reliability and its estimation for aged power system components subjected to voltage surges using Weibull process.

.
Example 4. Consider the process de�ned in Example 1 with     1 −  − ,    and let   have a c.d.f.       ,  <  < 1,   1 2. In this case   is also RR 2 in  .If  1 ≤  2 then  1 1   2         From (19) it follows that the lifetimes of the components are dependent even if the strengths of them are independent.is positive dependence among the lifetimes arises from common environmental stress characterized by .ere are many types of positive dependence.e likelihood ratio (or TP 2 ) dependence as the strongest notion of positive dependence is de�ned as follows.Let  1 ,  2 have the joint probability density  1   2 .en recall from �e�nition 2 that  1   2  is TP 2 if .e random variables  1 and  2 are said to be likelihood ratio (or TP 2 ) dependent if their joint density is TP 2 .
together with   1   2    ℎ 1   1  ℎ 2   2     (21) where ℎ        and     {   ≤ },   1 2. If ℎ 1   1  is TP 2 (RR 2 ) in  1   and ℎ 2   2  is TP 2 (RR 2 ) in   2 , then  1 and  2 are likelihood ratio dependent.Example 6.Let      −   ,   ,   1 2 and   be an exponential random variable with c.d.f.     1 −  −   ,   .Also assume that the common random stress  has c.d.f.    , 0 <  < .In this case the joint survival function of   and  2 is found to be for   , 2,   and  2 are likelihood ratio dependent.Consider a system  with  components which has two possible states;    if the system is functioning and   0 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    0 if it has failed.e 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     ,  2 , … ,    represents the lifetime of the system.Let   ,  2 , … ,   denote the i.i.d.lifetime random variables with continuous distribution.Samaniego [12] introduced the signature of a coherent system     ,  2 , … ,    as the vector     ,  2 , … ,   , where        ,  2 , … ,        ,   , … , , (24) 2 , … ,    can be represented as     ,  2 , … ,    and   ,  2 , … ,    are i.i.d. with c.d.f.    {    },   , 2, … ,  then Proof.Under the assumption that   ,  2 , … ,    are i.i.d. the joint survival function of   ,  2 , … ,   is    ,  2 , … , eorem 7. Let   denote the lifetime of the th component whose strength is   ,   , 2, … , , that is,    inf {   ≥ 0,     }.If  denotes the structure function of the coherent system with lifetime , that is,          .