Laplace-Stieltjes transform of the system mean lifetime via geometric process model

Gökhan Gökdere; Mehmet Gürcan

doi:10.1515/math-2016-0034

Enjoy 40% off

academic books on De Gruyter Brill *

Article Open Access

Laplace-Stieltjes transform of the system mean lifetime via geometric process model

Gökhan Gökdere and Mehmet Gürcan

Published/Copyright: June 10, 2016

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information Explore this Subject

$Open Mathematics$

From the journal Open Mathematics Volume 14 Issue 1

Abstract

Operation principle of the engineering systems occupies an important role in the reliability theory. In most of the studies, the reliability function of the system is obtained analytically according to the structure of the system. Also in such studies the mean operating time of the system is calculated. However, the reliability function of some systems, such as repairable system, cannot be easily obtained analytically. In this case, forming Laplace-Stieltjes transform of the system can provide a solution to the problem. In this paper, we have designed a system which consists of two components that can be repairable with the aging property. Firstly, the Laplace-Stieltjes transform of the system is formed. Later, the mean operating time of the system is calculated by means of Laplace-Stieltjes transform. The system’s repair policy is evaluated depending on the geometric process. This property provides the aging of the system. We also provide special systems with different marginal lifetime distributions to illustrate the theoretical results in this study.

Keywords: Cold standby system; Laplace-Stieltjes transform; System mean lifetime; Geometric process

MSC 2010: 62K05; 65R10; 90B25

1 Introduction

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.

Definition 1.1

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(X>α)≥P(Y>α).

Furthermore, a stochastic process {X_n, n = 1, 2, } is called stochastically decreasing if X_n ≥ _stX_n + 1 and stochastically increasing if X_n ≤ _stX_n + 1.

Definition 1.2

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(aⁿ^-1t), 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:

If α > 1, then the geometric process {X_n, n = 1, 2, …} is stochastically decreasing.
If 0 < α < 1, then the geometric process {X_n, n = 1, 2, …} is stochastically increasing.
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.

2 Model assumptions

In this section, some assumptions concerning failures and repairs are given. Initially, a two component cold standby repairable system is set up, in which the component 1 operates while the component 2 is in cold standby state.

Assumption 2.1

Failure time of the component k is the random variableXk(n)with the distribution function

FXk(n)(t)=Fk(mkt),

where k = 1, 2; t ≥ 0, mk={an,k=1an−1,k=2, and a > 1 is the ratio of the GP for component k. The finite meanofXk(n)isEXk(n)=Tkmk.

Assumption 2.2

Repair time of component k is the random variableYk(n)with the distribution function

GYk(n)(t)=Gk(bn−1t),

where k = 1, 2; t ≥ 0. The finite mean ofYk(n)isEYk(n)=1bn−1μk, where 0 < b < 1 is the ratio of the GP for component k.

Assumption 2.3

LetXk(1),Xk(2),... be successive working times andYk(1),Yk(2),...be successive repair times of the component k, (k = 1, 2). It is supposed that all these random variables are independent.

3 The mean lifetime of the system

In this section, we have studied a system consisting of two components which can be repaired. There are some difficulties to obtain the reliability function of a repairable system due to the repair time of the faulty components. When any component of the system breaks down and needs repair, it increases the operating time of the system. In such conditions the repaired component’s performance is lower than the new component. Apart from that, the repairing time increases after each break down and repair. Based on the above discussion, our proposed model for repairable system is defined as below.

Let τ₁₂(a,b) denote the system mean lifetime under the assumption that at t = 0, the component 1 begins to work and the component 2 is inactive (on standby). Let τ(a,b) denote the system mean lifetime measured from the moment when the component 1 fails. It is evident that τ₁₂(a,b) = X₁ + τ(a,b), where X₁ and τ(a,b) are independent random variables. Also, X₁ is the random variable with the distribution function F_X_₁ with finite mean EX₁ = T₁.

The system described above fails when for some n ≥1 either event

An={X2(j)>Y1(j),X1(j)>Y2(j),j=1,2,....,n−1;X^2(n)≤Y1(n)}

or event

Bn={X2(j)>Y1(j),X1(j)>Y2(j),j=1,2,....,n−1;X2(n)>Y1(n),X^1(n)≤Y1(n)}

occurs. Here X^1(n) and X^2(n) are random variables which have the same distribution as X1(n) and X2(n) given that X1(n)<Y2(n) and X2(n)<Y1(n), respectively. A possible process for events A_n and B_n are shown in Figure 1 and 2, respectively as follows

$Fig. 1 A diagram for event An in cycle n.$

Fig. 1

A diagram for event A_n in cycle n.

$Fig. 2 A diagram for event Bn in cycle n.$

Fig. 2

A diagram for event B_n in cycle n.

Let

αik(n)=P{Xk(n)>Yi(n)}=∫0∞GY1(n)(t)dFXk(n)(t)(1)

where i = 1, 2; k = 1, 2; i ≠ k. Note that, α₁₂(n) is the probability that after the n-th repair the working component 2 does not fail while the component 1 is under (n + 1)-th repair, and α₂₁(n) is the probability that after the (n + 1)-th repair, the working component 1 does not fail while the component 2 is under n-th repair. Then, using w(1), the probability that the system breakdown in the cycle n is obtained as

α=P{(X2(n)≤Y1(n))∪[(X2(n)>Y1(n))]∩(X1(n)≤Y2(n))}=(1−α12(n))+[α12(n)(1−α21(n))]=1−α12(n)α21(n)(2)

Thus using (1) and (2), the probabilities of events A_n and B_n are found to be

P{An}=∏j=1n−1(1−α(j))α¯21(n),

and

P{Bn}=∏j=1n−1(1−α(j))α12(n)α¯21(n),

where α¯12(n)=1−α12(n) and α¯21(n)=1−α21(n). For the system mean lifetime measured from X₁ we have

P{τ(a,b)=ζ1+ζ2+...+ζn−1+X^2(n)}=∏j=1n−1(1−α(j))α¯12(n),(3)

P{τ(a,b)=ζ1+ζ2+...+ζn−1+X2(n)+X^1(n)}=∏j=1n−1(1−α(j))α12(n)α¯12(n),(4)

where ζj=X2(j)+X1(j), j = 1, 2, …, n – 1 and for n = 1 the sum and product vanishes.

4 Construction of LS transform of the system mean lifetime

In this section, we obtain LS transform of the system mean lifetime. The system consists of two components. Each component can be repairable and has aging property. We introduce the following notations;

qik(p)(s)=∫0∞e−stGYi(p)(t)dFXk(p)(t),(5)

q¯ik(p)(s)=∫0∞e−stG¯Yi(p)(t)dFXk(p)(t)=fXk(p)(s)−qik(p)−qik(p)(s),(6)

fXk(p)(s)=∫0∞e−stdFXk(p)(t),(7)

where i = 1, 2; k = 1, 2; i ≠ k; p ≥ 1 and G¯Yi(p)(t)=1−GYi(p)(t). Using the formula of total expectation and (3)-(4) we have

Ee−sτ(a,b)=∑n=1∞(Ee−s(ζ1+ζ2+...+ζn−1+X^2(n))∏j=1n−1(1−α(j))α¯12(n)+Ee−s(ζ1+ζ2+...+ζn−1+X2(n)+X^2(n))∏j=1n−1(1−α(j))α12(n)α¯12(n)).(8)

Lemma 4.1

LetXk(j)be independent random variables. ForP{Xk(j)<t|Xk(j)>Yi(j)}, we have

Ee−sXk(j)=qik(j)(s)αik(j).(9)

Proof. Using the definition P{Xk(j)<t|Xk(j)>Yi(j)}, we have

Ee−sXk(j)=∫0∞e−stdP{Xk(j)<t|Xk(j)>Yi(j)}=1P{Xk(j)>Yi(j)}∫0∞e−stdP{Xk(j)<tXk(j)>Yi(j)}.

From (1), it follows that

Ee−sXk(j)=1αik(j)∫0∞e−std[∫0∞P{Xk(j)<t,Xk(j)>Yi(j),Xk(j)=uj}dFXk(j)(uj)]=1αik(j)∫0∞e−std[P{uj<t,uj>Yi(j)}dFXk(j)(uj)+∫t∞P{uj<t,uj>Yi(j)}dFXk(j)(uj)].

The right hand-side of the above equality tends to zero as u_j < t. Hence it takes the form

Ee−sXk(j)=1αik(j)∫0∞e−std[∫0tP{uj>Yi(j)}dFXk(j)(uj)]=1αik(j)∫0∞e−std[∫0tGYi(j)(uj)dFXk(j)(uj)]=1αik(j)∫0∞e−stGYi(j)(t)dFXk(j)(t)

From this and (5) we get the proof. □

Lemma 4.2

Let X^k(n) be independent random variable. ForP{Xk(n)<t|Xk(n)<Yi(j)}, we have

Ee−sX^k(n)=∫0∞e−stdP{Xk(n)<t|Xk(n)<Yi(j)}=q¯ik(n)(s)α¯ik(n).(10)

Proof. It can be obtained similarly as Lemma 4.1. □

Taking into consideration the Assumption 2.3 and substituting (9) and (10) in (8) we find that

Ee−sτ(a,b)=∑n=1∞[q¯12(n)(s)+q12(n)(s)q¯21(n)(s)]∏j=1n−1[q12(j)(s)q21(j)(s)].

In view of τ₁₂(a, b) = X₁ +τ(a, b) and fX1(s)=Ee−sX1=∫0∞e−stdFX1(t), the LS transform of the system ₀mean lifetime τ₁₂(a,b)is obtained as

Ee−sτ12(a,b)=fX1(s)(∑n=1∞[q¯12(n)(s)+q12(n)(s)q¯21(n)(s)]∏j=1n−1[q12(j)(s)q21(j)(s)]).(11)

With the LS transform, the mean lifetime of the system τ₁₂(a, b) is expressed in terms of the integrals q_ik(p)(s) and q¯ik(p)(s), i = 1, 2; k = 1, 2; i ≠ k. The LS transform of the system mean lifetime given by the equation (11) is required to calculate the expected value of the system’s mean lifetime.

We will use the following special function and integral in the next section to obtain explicit expression of the Laplace-Stieltjes transform of the system mean lifetime for a Weibull distributional example. The function is the error function defined by (Equation 8.250.1 in [3])

Φ(x)=2π∫0xe−t2dt.(12)

The integral is

∫0∞e−x24β−γxdx=πβeβγ2[1−Φ(γβ)](13)

where β > 0 (Equation 3.322.2 in [3]).

5 Numerical examples

In probabilistic design it is common to use parametric statistical models to illustrate the theoretical results for the proposed model. In this section, we apply our model to a Weibull and Gamma distributional examples.

5.1 A Weibull distributional example

Assume that the working time of the component k yields exponential distribution while the repair time of the component k yields Weibull distribution, i.e.

FX1(t)=1−eλ1t,FX1(n)(t)=F1(ant)=1−eλ1ant,FX2(n)(t)=F2(an−1t)=1−eλ1an−1t,GYk(n)(t)=Gk(bn−1t)=1−e(μkbn−1t)βk,

where t ≥ 0, a > 1, 0 < b < 1, μ_k > 0, β_k > 0; k = 1, 2. In order to obtain the explicit expression of the LS transform of the system mean lifetime expediently, let β₁ = β₂ = 2, we can get

fX1(s)=∫0∞e−std(1−e−λ1t)=λ1s+λ1q12(n)(s)=∫0∞e−st(1−e−(μ1bn−1t)2)d(1−e−λ2an−1t)=λ2an−1s+λ2an−1−πλ2an−12μ1bn−1e(An)2[1−Φ(An)]q21(n)(s)=∫0∞e−st(1−e−(μ2bn−1t)2)d(1−e−λ1ant)=λ1ans+λ1an−πλ1an2μ2bn−1e(Bn)2[1−Φ(Bn)]q¯12(n)(s)=∫0∞e−st(e−(μ1bn−1t)2)d(1−e−λ2an−1t)=πλ2an−12μ1bn−1e(An)2[1−Φ(An)]q¯21(n)(s)=∫0∞e−st(e−(μ2bn−1t)2)d(1−e−λ1ant)=πλ1an2μ2bn−1e(Bn)2[1−Φ(Bn)]

where An=s+λ2an−12μ1bn−1 and Bn=s+λ1an2μ2bn−1. In above, we also used equation (12) and (13). Substituting the above equations into (11), we can get the explicit expression of the LS transform of the system mean lifetime as follows

Ee−sτ12(a,b)=λ1s+λ1[limN→∞∑n=1N{(πλ2an−12μ1bn−1e(An)2[1−Φ(An)]) +(λ2an−1s+λ2an−1−πλ2an−12μ1bn−1e(An)2[1−Φ(An)])(πλ1an2μ2bn−1e(Bn)2[1−Φ(Bn)])} ×∏j=1n−1(λ2aj−1s+λ2aj−1−πλ2aj−12μ1bj−1e(Aj)2[1−Φ(Aj)])(πλ1aj2μ2bj−1e(Bj)2[1−Φ(Bj)])](14)

When the above series is convergent, the remaining terms after certain N are neglected in the numerical calculations. From (14) and Eτ12(a,b)=−ddsEe−sτ12(a,b)(0) we can get the mean time to failure of the system for Weibull distributional example. Table 1 contains some numerical results for selected values of the parameters λ₁, λ₂, a, b, μ₁, μ₂ and N. We have used the programming language Wolfram Mathematica 9 for differentiating (14) and calculations.

Table 1

Mean time to failure of the system for Weibull distributional example.

N	a	b	λ₁	λ₂	μ₁	μ₂	Eτ₁₂(a, b)
6	1:2	0:8	0:3	0:5	0:6	0:7	12:5534
9	1:2	0:8	0:3	0:5	1:6	2:3	19:4406
48	1:01	0:99	0:1	0:1	1:0	1:0	116:7784

5.2 A Gamma distributional example

Assume that the working time of the component k yields exponential distribution while the repair time of the component k yields Gamma distribution, i.e.

FX1(t)=1−eλ1t,FX1(n)(t)=F1(ant)=1−eλ1ant,FX2(n)(t)=F2(an−1t)=1−eλ2an−1t,GYk(n)(t)=Gk(bn−1t)=1−∑i=01(μkbn−1t)i!e−μkbn−1t.

where t ≥ 0, a > 1, 0 < b < 1, λ_k > 0, μ_k > 0; k = 1, 2. In the following, we will compute several integrations so that the expressions of fX1(s),q12(n)(s),q21(n)(s),q¯12(n)(s),q¯21(n)(s) can be derived, respectively.

∫0∞e−std(1−e−λ1t)=λ1s+λ1∫0∞e−st(1−∑i=01(μ1bn−1t)ii!e−μ1bn−1t)d(1−e−λ2an−1t)=λ2(μ1)2an−1b2n−2(s+λ2an−1)(s+μ1bn−1+λ2an−1)2,∫0∞e−st(1−∑i=01(μ2bn−1t)ii!e−μ2bn−1t)d(1−e−λ1ant)=λ1(μ2)2anb2n−2(s+λ1an)(s+μ2bn−1+λ1an)2,∫0∞e−st(∑i=01(μ1bn−1t)ii!e−μ1bn−1t)d(1−e−λ2an−1t)=λ2an−1(1+s+μ1bn−1+λ2an−1)(s+μ1bn−1+λ2an−1)2,∫0∞e−st(∑i=01(μ2bn−1t)ii!e−μ2bn−1t)d(1−e−λ1ant)=λ1an(1+s+μ2bn−1+λ1an)(s+μ2bn−1+λ1an)2.

Here, we used suitable transformations and simplifications. Now, substituting the above equations into (11), we can get the explicit expression of the LS transform of the system mean lifetime as follows

Ee−sτ12(a,b)=λ1s+λ1[limN→∞∑n=1N{(λ2an−1(1+s+μ1bn−1+λ2an−1(s+μ1bn−1+λ2an−1)2) +(λ1λ2(μ1)2a2n−1b2n−2(1+s+μ2bn−1+λ1an)(s+λ2an−1)(s+μ1bn−1+λ2an−1)2(s+μ2bn−1+λ1an)2) ×∏j=1n−1λ1λ2(μ1)2(μ2)2a2n−1b4n−4(s+λ2an−1)(s+λ1an)(s+μ1bn−1+λ2sn−1)2(s+μ2bn−1+λ1an)2].(15)

When the above series is convergent, the remaining terms after certain Nare neglected in the numerical calculations. From (15) and Eτ12(a,b)=−ddsEe−sτ12(a,b)(0) we can get the mean time to failure of the system for Gamma distributional example. Table 2 contains some numerical results for selected values of the parameters λ₁, λ₂, a, b, μ₁, μ₂ and N. We have used the programming language Wolfram Mathematica 9 for differentiating (15) and calculations.

Table 2

Mean time to failure of the system for Gamma distributional example.

N	a	b	λ₁	λ₂	μ₁	μ₂	Eτ₁₂(a, b)
5	1:2	0:8	0:3	0:5	0:6	0:7	8:193
7	1:2	0:8	0:3	0:5	1:6	2:3	7:439
30	1:01	0:99	0:1	0:1	1:0	1:0	65:455

6 Conclusion

In this article, we have studied a repairable aging cold standby system with a single standby unit when the components are independent. We have presented Laplace-Stieltjes transform of the system mean lifetime and compute its expected value for Weibull distributional example and Gamma distributional example.

Acknowledgement

The authors would like to express gratitude to the referees for their valuable suggestions and comments that led to some improvements in this article.

References

[1] Brown M., Proschan F., Imperfect repair, J. Appl. Prob., 1983, 20, 851-85910.21236/ADA086845Search in Google Scholar

[2] Eryılmaz S., A study on reliability of coherent systems equipped with a cold standby component, Metrika, 2014, 77, 349-35910.1007/s00184-013-0441-0Search in Google Scholar

[3] Gradshteyn I. S., Ryzhik I. M., Table of Integrals, Series and Products, 6th ed., Academic Press, California, 2000Search in Google Scholar

[4] Kijima M., Some results for repairable system with general repair, J. Appl. Prob., 1989, 26, 89-10210.2307/3214319Search in Google Scholar

[5] Lam Y., Geometric processes and replacement problem, Acta. Math. Appl. Sin., 1988a, 4, 366-37710.1007/BF02007241Search in Google Scholar

[6] Lam Y., A note on the optimal replacement problem, Adv. Appl. Prob., 1988b, 20, 479-48210.1017/S0001867800017092Search in Google Scholar

[7] Levitin G., Xing L., Dai Y., Cold-standby sequencing optimization considering mission cost, Rel. Eng. Syst. Safety, 2013, 118, 28–3410.1016/j.ress.2013.04.010Search in Google Scholar

[8] Makis V., Jardine A.K.S., A note on optimal replacement policy under general repair, Eur. J. Oper. Res., 1993, 69, 75-82 G. Gökdere, M. Gürcan10.1016/0377-2217(93)90092-2Search in Google Scholar

[9] Park K. S., Optimal number of minimal repairs before replacement, IEEE Trans. Reliab., 1979, 28, 137-14010.1109/TR.1979.5220523Search in Google Scholar

[10] Ross S. M., Stochastic Processes, Wiley, New York, 1996Search in Google Scholar

[11] Wu Q., Wu S., Reliability analysis of two-unit cold standby repairable systems under Poisson shocks, Appl. Math. Comput., 2011, 218, 171-18210.1016/j.amc.2011.05.089Search in Google Scholar

[12] Xing L., Tannous O., Dugan J. B., Reliability analysis of nonrepairable cold-standby systems using sequential binary decision diagrams, IEEE Trans. Syst., 2012, 42, 715-72610.1109/TSMCA.2011.2170415Search in Google Scholar

Received: 2016-1-21

Accepted: 2016-5-4

Published Online: 2016-6-10

Published in Print: 2016-1-1

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.