Exact reliability formula for a linear consecutive kout-of-n : F system and relayed consecutive systems with a change point for any k ≤ n , with stress-strength application

This paper presents exact formulas for the reliability of linear consecutive k-out-of-n: F, and relayed consecutive k-out-of-n: F systems, having a change point at position c, 1 ≤ c ≤ n, for any k ≤ n. A change point at position c, means that the components after this point have reliabilities that are different from those before or at position c. The components are assumed to be independent. Practically, the change in the components reliabilities may be due to change in the stress applied. Assuming a change in stress, exact formulas of the stress-strength reliability of the systems are derived, considering two cases. The first case assumed strength and stress having the same form of distributions, while the second case assumed strength and stress having different forms of distributions. Estimation of the stress-strength reliability for both cases is discussed. Applications to both cases are considered with numerical illustration.


Notations
Number of components of the system.

𝑘
Minimum number of consecutive failed components required for system failure.

𝑁(𝑗, 𝑠, 𝑟 )
The number of ways in which  identical balls can be placed in  distinct urns subject to the requirement that at most  balls are placed in any one urn.

𝑐
Position of the change point of the system, 1 ≤  ≤ .P Is a vector [ 1 ,  2 ]. /// Linear consecutive k-out-of-n: F system with a change point at position .///; P /// with reliability  1 for components from 1 to , and reliability  2 from  + 1 to .R w (, , ; P) Reliability of a ///; P, given that the component at position  is working.R  (, , ; P) Reliability of a ///; P, given that the component at position  is failed.R(, , ; P) Reliability of a ///; P.

Introduction
A linear consecutive k-out-of-n: F system consists of  components arranged linearly, the system fails if and only if  or more consecutive components fail.There exist many practical applications of this type of systems, for example, the telecommunications system with  relay stations, and the oil pipeline system with  pump stations, given by Chiang and Niu (1981).Many researches in the literature concerned such systems.Derman et al. (1982) presented a formula for the reliability of a linear consecutive k-outof-n: F system with identical and independent components with reliabilities , given by R(, ; ) = ∑ (,  −  + 1,  − 1)   −  =0 . ( Lambiris and Papastavridis (1985) derived an expression for the numbers (,  −  + 1;  − 1) in (1), with  −  + 1 ≥ 0, as follows Other different formulas for the reliability of a linear consecutive k-out-of-n: F system are obtained by different methods, see Chiang and Niu (1981), Hwang (1982), Kossow and Preuss (1989), Ge and Wang (1990), Jung and Kim (1993), Chao et al. (1995), Mokhlis (2001), and Gökdere et al. (2016).
Also, relayed linear consecutive k-out-of-n: F systems are of practical importance.There are two types of relayed linear consecutive k-out-of-n: F systems: The first type is a relayed-unipolar linear consecutive k-out-of-n: F system which is a system consisting of  components arranged in a line such that the system fails if the first component fails or at least  consecutive components fail; while the second type is a relayed-bipolar linear consecutive k-out-of-n: F system, where the  components are arranged in a line such that the system fails if the first or last component fails, or at least  consecutive components fail.The notation of relayed-unipolar and bipolar consecutive k out-of-n: F systems is given by Hwang (1988).
In the reliability theory, stress-strength reliability of a system with random strength  under the random stress , is the probability that the strength exceeds the stress.Stressstrength analysis was first considered by Birnbaum (1956) and developed by Birnbaum and McCarty (1958).Johnson (1988) presented stress-strength models for the system reliability.To the authors' knowledge few researches have considered the stress-strength reliability problem of consecutive k-out-of-n systems.Eryılmaz (2008) presented the multi component stress-strength reliability of consecutive k-out-of-n: G system, for both cases in which there is a change and no change in strength.Zhao et al. (2018) studied a two-stage shock model with self-healing mechanism.A change point is introduced to describe the two-stage failure processes of the system.They proposed three preventive maintenance policies for the system under different monitoring conditions.
Akici (2010) discussed a linear consecutive k-out-of-n: F system with a change point, using the longest run statistic under the condition of special values of , 2 ≥ .Akici (2010) mentioned a practical example of this model, which is the gas pipeline from Russia to Turkey.This pipeline system is under two different stresses  1 (water) and  2 (soil).The pipeline starts from Russia passes across the Black Sea, enter Turkey from Samsun, and ends in Ankara, in Samsun it leaves the sea and enters soil, which is the change point of this system, and there are many other practical examples.Akici (2010) discussed this situation for special values of  satisfying 2 ≥ .However, in practice  could take any value between 1 and , not necessarily, 2 ≥ .So, in this paper we study the reliability of a linear consecutive k-out-of-n: F system with a change point at position , when  could take any possible value i.e., 1 ≤  ≤ .Also, the reliabilities of the relayed-unipolar and relayed-bipolar linear consecutive k-out-of-n: F systems with a change point at position  are derived for any , 1 ≤  ≤ .Exact formulas for the linear and relayed linear (unipolar and bipolar) consecutive k-out-of-n: F systems are derived conditioning on the state (working or failed) of the component at position , for any values of , 1 ≤  ≤ .Formulas for R (:) (, , ), R (:) (, , ), and R (:) (, , ) are derived, for any distributions of strength of components () and stresses   (),  = 1,2.As application, we discussed two cases: Case I and Case II.Case I, when () and   (),  = 1,2 have the same form (general exponential form), in this case exact formulas of R (:) (, , ), R (:) (, , ), and R (:) (, , ) are obtained.For illustrating these results numerically, the negative exponential distribution is taken as an example of the general exponential form.Case II, when () and   (),  = 1,2 have different forms (generalized Lindley distribution for strength and negative exponential distribution for stresses).The Generalized Lindley distribution includes as special cases the exponential, gamma and Lindley distributions, and it is used in stress-strength reliability modeling and analyzing lifetime data, see Elbatal et al. (2013).
The paper is organized as follows: In Section 2, we obtain explicit forms for R(, , ; P), R  (, , ; P), and R  (, , ; P).In Section 3, we present the stressstrength reliability of the systems generally for any continuous distributions, assuming the change of components reliabilities is due to change in the stress applied.In Section 4, we obtain exact formulas for R (:) (, , ), R (:) (, , ), and R (:) (, , ) for case I, and case II.In Section 5, the maximum likelihood estimators of the stress-strength reliabilities are obtained for both cases I & II.In Section 6, a numerical illustration of the theoretical results is presented, to show the effect of , , and the parameters of the distributions on the reliabilities.Also, the performance of the estimators is detected.In Section 7, a conclusion is presented.

Reliability Formulas
Assume we have a linear consecutive k-out-of-n: F system with independent components, having a change point at position .This means that the first  components are identical with reliability  1 ( 1 = 1 −  1 ), while the remaining ( − ) components are also identical but with a different reliability  2 ( 2 = 1 −  2 ).The reliability of this system, R(, , ; P), is given by the following theorem.

Proof
The state of the component at position  is either working or failed.Conditioning on the state of the component at the change point , we have R(, , ; P) =  1 R  (, , ; P) +  1 R  (, , ; P).
Now, we try to find R  (, , ; P) and R  (, , ; P).Suppose that we have  failed components in the system,  failures from these  appearing before the change point c, and the remaining  −  failures appearing after .
(ii) For computing R  (, , ; P), we argue as follows: The component at position  fails with probability of failure  1 , in this case the position of  affects the system failure.Then we have the following three cases for the system to operate: (a)  failures from the  failures before  are directly located before the position , and  failures from the  −  failures after , are directly located after , this situation is depicted by Figure 2, this case for any position of , and 1 ≤  ≤  − 2.
The following corollary presents the formulas of R  (, , ; P) and R  (, , ; P).

Stress-Strength Reliability
In this section we obtain R (:) (, , ), R (:) (, , ), and R (:) (, , ).Suppose that a change in the stress occurs after the component at position .Thus, assume we have two common independent stresses   ,  = 1,2, with cumulative distribution functions   (),  = 1,2, where the common stress  1 is imposed on components 1 to , and the common stress  2 is imposed on components  + 1 to .The strengths   ,  = 1, … ,  of the components are independent and identically distributed, having cumulative distribution function ().The stress-strength reliability of /// is given by the following theorem. where

Proof
We can easily show that (12)  .

Corollary 2
Let  1 and  2 be two common stresses with cumulative distribution functions  1 () and  2 (), imposed on components 1 to  and components  + 1 to , respectively.Assume that  1 and  2 are independent.Let   ,  = 1, … ,  denote the strengths of components 1, … , .Assume that   ′  are independent and identically distributed.For any  ≤ , and 1 ≤  ≤ , the stress-strength reliability of a unipolarrelayed and bipolar-relayed /// system, is given respectively by R .

The Exact Stress-Strength Reliability Formulas for Some Special Distributions
The exact reliability formulas of R (:) (, , ), R (:) (, , ), and R (:) (, , ) are obtained in two cases: case I, when the strength and the stresses have the same form of distributions.As an application of this case, we consider the general exponential form.
Case II, when the strength and the stresses have different forms of distributions, and as application, we consider the generalized Lindley distribution for strength, while the stresses have a negative exponential distribution.
The distribution with form ( 16) is called general exponential form distribution, see Mokhlis et al. (2017).Many distributions have cumulative distribution functions satisfying the form in (16).Table 1 presents some examples of these distributions.

Case II
Suppose that the distributions of stresses  1 and  2 are negative exponential, with cumulative distribution functions are given by Let the distribution of the strength  is a generalized Lindley distribution, with probability distribution and cumulative distribution function given respectively by ]  − , and where (, ) is a lower incomplete gamma.We can easily see that () is a mixture of two gamma distributions, with parameters (, ) and (, ) with probabilities The above integral can be obtained numerically, and hence the expression for R (:) (, , ), R (:) (, , ), and R (:) (, , ).

Estimation of The Stress-Strength Reliability
Case I As an application of the general exponential form, we take for simplicity (, ) = .
It is clear from Tables (2) to ( 7) that all reliabilities increase as  increases for all cases of , and for the different values of the parameters.The position of change point  also influence the reliability according to the rate of stress before or after this point and number of components under this stress.We also see the effect of the strength and stresses parameters, on the reliability of the system in both cases I, and II.For case I, for the strength parameters, we can see that the increase of  causes decrease in the reliability, while for stresses parameters, as  1 or

Conclusions
In this paper explicit expressions for R(, , ; P), R  (, , ; P), and R  (, , ; P) are obtained conditioning on the state of the change point at position  (working or failed).Then consequently, R (:) (, , ), R (:) (, , ), and R (:) (, , ) are presented when the change in the components reliabilities is due to change in the stress.This means that the components from 1 to  are subjected to a common stress,  1 , while the components from  + 1 to  are subjected to a different common stress,  2 .The strengths of all components 1 to  are independent and identical.The stress-strength reliabilities of linear consecutive k-out-of-n: F system, unipolar-relayed and bipolar-relayed linear consecutive k-out-of-n: F systems are obtained for any distribution of stresses   ,  = 1,2, and strength .As application, two cases are discussed: Case I and Case II.In Case I, the stresses and strength are assumed to have the same form of distributions (as an example, general exponential form, in ( 16)).Exact formulas are obtained in this case, showing that the forms of the stress-strength reliabilities for all systems do not involve the parameter .In Case II, the stresses and strength are assumed to have different forms of distributions (as an example, negative exponential distribution for stresses and generalized Lindley distribution for strength).Numerical illustrations are applied through simulation studies for both cases, to detect the effect of position of the change point , the value of  with respect to , and different distributions parameters, on the stress strength reliabilities.For both Cases I and II, the simulation studies showed that R (:) (, , ), R (:) (, , ), and R (:) (, , ) increase as  increases for any value of , and for the different values of the parameters.It is also shown that the position of change point  influences the reliability, depending on the rate of stress before or after this point and the number of components under that stress.Also, the stress-strength reliability of all systems is sensitive to the distributions parameters involved in its form.The maximum likelihood estimators R ̂(:) (, , ), R ̂(:) (, , ) and R ̂(:) (, , ) of R (:) (, , ), R (:) (, , ) and R (:) (, , ), are their mean square errors are also calculated to indicate the accuracy of the estimation.

Figure 1 :
Figure 1: System with  failures and the component at position  is working.

Figure 1
Figure 1 shows a system with  failures, such that  failures and  −  − 1 working components are before c, and  −  failures and  −  −  +  working components are after c.The component at position  is operating with reliability  1 , in this case the position of  does not affect the system failure.For the system to operate, we must prevent the appearance of  consecutive failures before and after the change point .The  failures may occupy any position from 1 to  − 1, with no consecutive  failures, and this can be done by (,  − ,  − 1) ways.The remaining  −  failures may occupy any position from  + 1 to  with no consecutive  failures, with ( − ,  −  −  +  + 1,  − 1) ways.For each , each arrangement has probability  1   1 −−1 before the change point , and probability  2 −  2 −−+ after .This means that

Figure 2 :
Figure 2: System with  failures and the component at position  is failed.

Figure 2 1 𝑐 2 𝑛
Figure 2 shows a system with  failures, where  consecutive failures from the  failures are directly located before , and  consecutive failures from the  −  failures are directly located after .The system operates if  +  + 1 ≤  − 1, in this case 0 ≤  ≤  − 2 and 0 ≤  ≤  −  − 2, and at each of positions  −  − 1 and

Figure 3 :
Figure 3: System with  failures and the first  components failed.

Figure 3
Figure 3 shows a system with  failures, the first  components failed, and  consecutive failures occur directly after those  failures.In this case  +  ≤  ≤  − 1, if  of consecutive failures occur directly after the first  failed components, then the system operates if  +  ≤  − 1, then  can take values from 0 to  −  − 1.At position  +  + 1, we must have a working component.The remaining  −  −  failures can occupy any position in the remaining  −  −  − 1 positions, without  consecutive failures.There are ( −  − ,  − ,  − 1) arrangements that satisfy the condition.Since for each , each arrangement has probability  1 −1 before the change point , and  2 −  2 − after this point.Then this possibility occurs with probability

Figure 4 :
Figure 4: System with  failures and the last  −  components failed.

Figure 4
Figure 4 shows a system with  failures, the last  −  components fail and  consecutive failures occur directly before position .In this case we have failures at the last  −  positions, position , and  consecutive failures before .In order the system not to fail, we must have  −  +  + 1 ≤  − 1, this means that 0 ≤  ≤  −  +  − 2 and at position  −  − 1 we must have a working component.The remaining  −  −  +  − 1 failures can occupy any position in the remaining  −  − 2 positions, without  consecutive failures, and this can be done by ( −  +  −  − 1,  − ,  − 1) arrangement.Since for each , each arrangement has probability  1 −+−1  1 − before the change point , and  2 − after , with  −  +  + 1 ≤  ≤  − 1.Then the probability of this case is
illustrate the reliabilities for case II (different forms of distributions), showing the effect of position of the change point , the value of  with respect to , and different parameters, on reliabilities.Using R-programming we compute R (:) (, , ), R (:) (, , ), and R (:) (, , ) with  = 6,  = (2, 3,  4) , and different values of the parameters of the distributions of the stresses and strength, with different values of  = (2,  4).( ≤  ,   > ).
by their corresponding maximum likelihood estimators.For obtaining the maximum likelihood estimator of the parameters, let  1 ,  2 , … ,    ;  = 1,2, and  1 ,  2 , … ,   3 be samples of size   ,  = 1,2, and  3 from stresses and strength distributions.Clearly the maximum likelihood estimators of the parameters   ,  are given by

Table 3 : Exact reliabilities for case I, with
(8)ncrease the reliability increases.From Tables (2 − 4), we see that the highest value of the reliability is attained for the largest values of  1 and  2 , and the smallest value of .Also, for generalized Lindley strength, we see from Tables (5 − 7) as  increases the reliability decreases, but as  or  increase the reliability increases.Also, we have the highest value of the reliability attained when the values of , , and   ( = 1,2) are large and  is small.From Table(8), we see that bias and the mean square errors are small.The mean square error in all cases decreases by increasing the sample size.However, the estimators are satisfactory.