A stochastic model for a general load-sharing system under overload condition

APPLIED STOCHASTIC MODELS IN BUSINESS AND INDUSTRYAppl. Stochastic Models Bus. Ind. 2010; 26:624–638Published online 21 December 2009 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/asmb.824

A stochastic model for a general load-sharing system underoverload condition

Won Young Yun1 and Ji Hwan Cha2,∗,†

1Department of Industrial Engineering, Pusan National University, Busan 609-735, Korea2Department of Statistics, Ewha Womans University, Seoul 120-750, Korea

SUMMARY

A load-sharing parallel system functions if at least one unit in the system is functioning and the survivingunits share the load. In most of research on load-sharing system, the performance of the system has beenstudied only for the case when the lifetimes of components in the system follow exponential distributions.In this paper a load-sharing parallel system is considered when the lifetimes of the units in the systemare any continuous random variables. The reliability function of the system is derived and the problemof load allocation is also considered. Copyright q 2009 John Wiley & Sons, Ltd.

Received 5 May 2009; Revised 10 August 2009; Accepted 8 November 2009

KEY WORDS: load-sharing system; usual stochastic order; proportional hazards type model; virtual age;optimal load sharing

1. INTRODUCTION

In a load-sharing system, the induced load is shared by the working units in the system duringits operation. In the real world, many systems follow the structure of load-sharing, for example,electric generators sharing an electrical load in a plant, bolts used to hold a machine member, andpumps or valves in a hydraulic system, and so on.

There has been much research on various load-sharing systems. Kapur and Lamberson [1]obtained the system reliability of a load-sharing parallel system with two dissimilar units. Kecce-cioglu [2] also obtained the system reliability of a load-sharing parallel system with two and threecomponents. Scheuer [3] studied k-out-of-n:G system, in which the failure rate of each componentdepends on the number of the failed components. Shechner [4] considered the system reliabilityunder the assumption that the failure rate of components is a linear function of load. Lin et al. [5]

∗Correspondence to: Ji Hwan Cha, Department of Statistics Ewha Womans University, Seoul, 120-750 Korea.†E-mail: [email protected]

Contract/grant sponsor: Korean Government (MOEHRD, Basic Research Promotion Fund); contract/grant number:KRF-2007-521-C00063

Copyright q 2009 John Wiley & Sons, Ltd.

STOCHASTIC MODEL FOR A GENERAL LOAD-SHARING SYSTEM 625

Figure 1. Load-sharing system with two units.

obtained the system reliability of a parallel system by a Markov model. Yinghui and Jing [6]developed a new model based on capacity flow model for a load-sharing k-out-of-n:G systemconsisting of different components that have exponential life distributions. See also, for example,Wang et al. [7] and Shao and Lamberson [8]. In most of research on load-sharing system, theperformance of the system has been studied only for the case when the components constitutingthe system have exponential distributions, i.e. the case of constant failure rates. In this paper, weconsider a load-sharing parallel system when the lifetimes of the units in the system follow anycontinuous random variables. Consider the following load-sharing parallel system with two units(Figure 1).

When the system starts its operation, the total load L is shared by units 1 and 2 with assignmentproportions � and 1−�, respectively. When one unit fails, the other unit takes the full load L andcontinues its operation. Throughout this paper, we assume that total load L is fixed and thus, withoutloss of generality, we may set L≡1. Under the above-described model, the reliability function ofthe load-sharing parallel system is derived. Obviously, in this case, the system performance will besignificantly affected by the assignment of given load, i.e. the value of �. Therefore, the problemof optimal assignment of load is also considered.

This paper is organized as follows. In Section 2, the probabilistic frame for modelling thelifetime distributions of the units under partial and full load will be constructed. In the load-sharingsystem under consideration, a surviving unit takes full load after the failure of the other unit. Thus,in this case, there is one change of stress level, i.e. from lower stress to higher stress, for thesurviving system, which will change the operating characteristic of the unit. In order to expresscorresponding lifetime of the unit under varying environment, a parametric failure time regressionmodel in accelerated life testing will be adopted. Under the proposed model, the lifetimes ofthe surviving unit before and after the change of stress level are ordered in the sense of usualstochastic order. As, in many cases, the load has not any impact on the lifetime of an equipmentwhen it is ranged up to nominal level, the proposed model would be more appropriate for theequipments under their overloads. Related discussions and remarks will be made in the final section.In Section 3, the explicit formula for the reliability function of the system will be derived. Someparticular cases are considered and optimal assignment problem will be investigated in Section 4.In Section 5, another method for describing corresponding lifetime of the surviving unit undervarying environment, which is based on the proportional hazards type model, will be adopted andstudies on the reliability function of the system and the corresponding load assignment problemwill be performed. The method considered in this section will be compared with that based on theusual stochastic order in the previous sections. Finally, in Section 6, some concluding remarks arediscussed.

Copyright q 2009 John Wiley & Sons, Ltd. Appl. Stochastic Models Bus. Ind. 2010; 26:624–638DOI: 10.1002/asmb

626 W. Y. YUN AND J. H. CHA

2. MODELLING LIFETIME DISTRIBUTIONS FOR UNITS IN LOAD-SHARING SYSTEM

The ‘Accelerated Failure Time’ regression model is the most widely used parametric failure timeregression model in accelerated life testing (cf. Section 3 of Meeker and Escobar [9]). Acceleratedlife tests (ALT) are frequently used in practice to obtain information on the life distribution orperformance over time of highly reliable products in an affordable amount of testing time. DuringALT, test units are used more frequently than usual or are subjected to higher level of stresses,such as temperature and voltage, than usual. The information obtained from the ALT is thenused to predict product performance in the usual level of environment. Nelson [10] provides anextensive and comprehensive source for background material, practical methodology, basic theory,and examples for accelerated testing. Meeker and Escobar [9] proposed a general model to takecare of the age conversion. Denote X as the lifetime of a component used in the usual level ofenvironment and F(t) as its Cdf. Let f (t) and r(t)= f (t)/F̄(t) be the pdf and FR of X . Alsolet r.v. XA be the lifetime of a component operated during ALT and FA(t) be the correspondingCdf. According to their model it is assumed that

FA(t)=F(�(t)) ∀t�0, (1)

where the non-decreasing function �(·) depends on the accelerated environment, �(t)�t for allt�0, and �(0)=0. Clearly, the model given in (1) implies that

XA�stX. (2)

Here, the notation ‘�st’ denotes the usual stochastic order, that is, we say that Z1 is smaller thanZ2 in the usual stochastic order, denoted as Z1�stZ2 (or F1�stF2), if F2(t)�F1(t), for all t�0,where F1(t) and F2(t) are the Cdfs of Z1 and Z2, respectively. As a distribution function of anabsolutely continuous random variable is a continuous function that increases from 0 to 1, therelationship defined in (1) is equivalent to the relationship (2).

Motivated by the ideas used in ALT we propose a model described below. Let Xi , Fi (t), andri (t) be the lifetime, the corresponding Cdf and failure rate function of unit i under full loadcondition, i=1,2, respectively. Furthermore, denote XPi as the lifetime of unit i under partialload condition, and FPi (t) as its Cdf, i=1,2. It will be assumed throughout this paper that

FP1(t)=F1(g1(�)t) ∀t�0 (3)

and

FP2(t)=F2(g2(1−�)t) ∀t�0, (4)

where the function gi (�) satisfies 0�gi (�)�1, gi (0)=0, gi (1)=1, and gi (�) is strictly increasingin �, i=1,2. Then models given in (3) and (4) certainly imply that Xi�stXPi , i=1,2. Note thatEquations (3) and (4) present a specific case (with a scale transformation function that is linearin t) of the general ALT model.

On the other hand, suppose that the unit i has been operating during (0,u] without failure underpartial load and just has started its operation at time t=u under full load condition, i=1,2. LetUiu denote this unit, i=1,2. From the time instant u, the unit Uiu will be operated under full loadcondition. Then it is assumed that the unit i immediately after the transformation of environmentalcondition at time t=u (the unit Uiu) has its age (under full load condition) wi (u), where wi (u)



is non-decreasing, wi (u)�u, for all u�0 and wi (0)=0, i=1,2. Thus, the survival function of theunit Uiu working under full load condition is given by

exp

{−

∫ t

0ri (wi (u)+s)ds

}= F̄i (wi (u)+ t)

F̄i (wi (u))

= P(Xi>wi (u)+ t |Xi>wi (u)), t�0, (5)

i=1,2. The meaning of Equation (5) is that the performance of a unit i which has been operatedunder partial load condition during (0,u] is the same as it has been operated in the full loadcondition during (0,wi (u)], i=1,2. Hence wi (u) in Equation (5) represents the ‘virtual age’ ofUiu, i=1,2. The concept of the virtual age in the problem of general repair was studied by Kijima[11]. In Finkelstein [12], some other kinds of virtual age were also defined and discussed from apractical point of view.

The approach introduced in this section and that given in Finkelstein [12–14] is somewhatdifferent, but, basically there are many similarities in the adopted stochastic methodologies, espe-cially in the concept of the virtual age. See also Finkelstein [15] for related topics.

3. EVALUATION OF SYSTEM RELIABILITY FUNCTION

Throughout this paper, we assume that the lifetimes of the units under partial load conditions, XP1and XP2, are independent. Let X∗

i be the excess lifetime of unit i after the failure of the otherunit, i=1,2. Define the random variable I as

I ={0 if unit 2 fails before the failure of unit 1 (XP2�XP1),

1 if unit 2 survives the failure time of unit 1 (XP2>XP1).

Then given the event {XP2=u, I =1}, X∗1 =0 with probability 1. Given {XP2=u, I =0}, the

conditional distribution of X∗1 is given by

P{X∗1>t |XP2=u, I =0}= F̄1(w1(u)+ t)

F̄1(w1(u)), t�0.

On the other hand, given the event {XP1=u, I =0}, X∗2 =0 with probability 1 and, given

{XP1=u, I =1}, the conditional distribution of X∗2 is given by

P{X∗2>t |XP1=u, I =1}= F̄2(w2(u)+ t)

F̄2(w2(u)), t�0.

Furthermore, we see that

P(I =0|XP2=u)= F̄1(g1(�)u), P(I =1|XP2=u)=F1(g1(�)u),

and

P(I =0|XP1=u)=F2(g2(1−�)u), P(I =1|XP1=u)= F̄2(g2(1−�)u).



The following result gives the reliability function of the load-sharing system.

Theorem 1The reliability function of the general load-sharing system is given by

R(t) = exp

{−

∫ g1(�)t

0r1(u)du

}·exp

{−

∫ g2(1−�)t

0r2(u)du

}

+∫ t

0exp

{−

∫ t−u

0r1(w1(u)+s)ds

}·exp

{−

∫ g1(�)u

0r1(s)ds

}g2(1−�) f2(g2(1−�)u)du

+∫ t

0exp

{−

∫ t−u

0r2(w2(u)+s)ds

}·exp

{−

∫ g2(1−�)u

0r2(s)ds

}g1(�) f1(g1(�)u)du. (6)

ProofThe reliability function can be expressed as

P(T>t)

= P(XP1>t, XP2>t)+P(0�XP2�t, XP2+X∗1>t, I =0)+P(0�XP1�t, XP1+X∗

2>t, I =1)

= F̄P1(t) · F̄P2(t)+∫ t

0P(X∗

1>t−u|XP2=u, I=0)P(I=0|XP2=u) ·g2(1−�) f2(g2(1−�)u)du

+∫ t

0P(X∗

2>t−u|XP1=u, I =1)P(I =1|XP1=u) ·g1(�) f1(g1(�)u)du,

which yields the desired result. �

4. SOME PARTICULAR CASES AND OPTIMAL LOAD ALLOCATION

In this section some particular cases of the model are investigated. Under the settings defined inSection 2, for example, the distribution of XP1 is given by

FP1(t)=F1(g1(�)t) ∀t�0.

Then

E[XP1]=∫ ∞

0F̄1(g1(�)t)dt=

∫ ∞

0

1

g1(�)F̄1(u)du= 1

g1(�)E[X ].

The practical meaning of the settings in Section 2 can be interpreted as follows. In particular, ifg1(�)=�, and the allocation proportion is given by �= 1

2 , then E[XP1]=2E[X ]. In this case, ifthe load is reduced to half, then the mean lifetime becomes double. If g1(�)=�2, and the allocationproportion is given by �= 1

2 , then E[XP1]=4E[X ]. That is, if the load is reduced to half, then themean lifetime becomes four times of the original one. Similar interpretations can also be appliedto g2(�).



Now the virtual age function wi (u) discussed in Section 2 will be defined. Similar to thecumulative exposure model given in Nelson [10], if we assume

F1(w1(u))=FP1(u)=F1(g1(�)u) ∀u�0 (7)

and

F2(w2(u))=FP2(u)=F2(g2(1−�)u) ∀u�0, (8)

that is, if we assume that the virtual age wi (u) would produce the same population cumulativefraction of units failing as the age u does under partial load condition, then we obtain the followingrelationships:

w1(u)=F−11 FP1(u)=F−1

1 F1(g1(�)u)=g1(�)u ∀u�0, (9)

and

w2(u)=F−12 FP2(u)=F−1

2 F2(g2(1−�)u)=g2(1−�)u ∀u�0. (10)

Therefore, if the conditions (7) and (8) are satisfied, then we may let w1(u)=g1(�)u, ∀u�0 andw2(u)=g2(1−�)u, ∀u�0. In the remaining part of this paper, unless otherwise specified, weassume the relationships in (9) and (10).

4.1. Exponential distribution

In this subsection we consider the case when the lifetime distributions of units are exponential. Weassume that r1(t)=�1, t�0, and r2(t)=�2, t�0. Note that, in this case, the results do not dependon the functions w1(u) and w2(u).

In this case, from (6), the reliability function is given by

R(t) = exp{−g1(�)�1t}·exp{−g2(1−�)�2t}

+∫ t

0exp{−�1(t−u)}·exp{−g1(�)�1u}·g2(1−�)�2 exp{−g2(1−�)�2u}du

+∫ t

0exp{−�2(t−u)}·exp{−g2(1−�)�2u}·g1(�)�1 exp{−g1(�)�1u}du

= exp{−g1(�)�1t−g2(1−�)�2t}+ g2(1−�)�2(1−g1(�))�1−g2(1−�)�2

×exp{−�1t}[exp{(1−g1(�))�1t−g2(1−�)�2t}−1]

+ g1(�)�1(1−g2(1−�))�2−g1(�)�1

exp{−�2t}[exp{(1−g2(1−�))�2t−g1(�)�1t}−1]. (11)

In particular, when gi (�)=�, i=1,2, the reliability function in (11) turns to be

R(t)= 1

�1−�2[�1 exp{−�2t}−�2 exp{−�1t}].



Figure 2. The mean lifetime function m(�).

The mean lifetime in this case is then given by

E[T ]=∫ ∞

0R(t)dt= �1+�2

�1�2.

In this case, it is of interest that the reliability function and the mean lifetime is independent of �,that is, the performance of the load-sharing system is the same regardless of the load allocationpolicy.

Now we consider the special case when �1=1, �2=2, and gi (�)=�2, i=1,2. From (11), thereliability function R(t) can be readily obtained. As the mean lifetime of the system is a functionof �, define m(�)≡E[T ]=∫ ∞

0 R(t)dt and consider �∗ which maximizes m(�). Then, in this case,m(�) is given by

m(�)= 5�2−8�+6

2(3�2−4�+2),

and its derivative is given by

m′(�)= 4(�2−4�+2)

2(3�2−4�+2)2.

Now, m′(�)=0 when �=2−√2≈0.585 and m′(�)>0, for �<0.585, m′(�)<0, for �>0.585. Thus,

m(�) has its unique maximum at �∗ =0.585 and, in this case, the mean lifetime is given bym(0.585)=2.207112. The graph for m(�) is given in Figure 2.

Observe that the mean lifetimes of the system at �=0 and �=1 are identical. In these cases, theload-sharing system corresponds to the corresponding ‘cold-standby’ systems, respectively, andthe mean lifetimes of the system in these cases are given by just sum of mean lifetimes of twocomponents, 1.0+0.5=1.5. Also we can see that m(�)>1.5, for all 0<�<1. Thus, it can be saidthat, at least in view of mean lifetime, any load-sharing system is better than ordinary cold standbysystem.



4.2. Weibull distribution

In this subsection we consider the case when the lifetime distributions of units follow Weibulldistributions. In this case the results depend on wi (u), i=1,2.

Suppose that r1(t)=3t2, t�0, and r2(t)=6t2, t�0. Then, from (6), the reliability function isgiven by

R(t) = exp

{−

∫ g1(�)t

03u2 du

}exp

{−

∫ g2(1−�)t

06u2 du

}

+∫ t

0exp

{−

∫ t−u

03(g1(�)u+s)2 ds

}·exp

{−

∫ g1(�)u

03s2 ds

}

×g2(1−�)6(g2(1−�)u)2 exp

{−

∫ g2(1−�)u

06s2 ds

}du

+∫ t

0exp

{−

∫ t−u

06(g2(1−�)u+s)2 ds

}·exp

{−

∫ g2(1−�)u

06s2 ds

}

×g1(�)3(g1(�)u)2 exp

{−

∫ g1(�)u

03s2 ds

}du

= exp{−(g1(�)t)3−2(g2(1−�)t)3}

+∫ t

06(g2(1−�))3u2 ·exp{−(t−(1−g1(�))u)3−2(g2(1−�)u)3}du

+∫ t

03(g1(�))3u2 ·exp{−2(t−(1−g2(1−�))u)3−(g1(�)u)3}du.

First, we consider the case when gi (�)=�1/3, i=1,2. The mean lifetime of the systemm(�)≡E[T ]=∫ ∞

0 R(t)dt for this case is given in Figure 3.In this case, the mean lifetime of the system is equivalently maximized at �=0 and �=1, and it

holds that m(�)<m(0)=m(1), for all 0<�<1. Thus, in this case, the ordinary cold standby systemis better than any load-sharing system. The maximum mean lifetime of the system (�=0 or �=1)is given by just sum of mean lifetimes of two components, 1·�( 43 )+2−1/3 ·�( 43 )≈1.602. For thisspecial case, by the interpretation of gi (�) given in the first part of this section, if gi (�)=�1/3,i=1,2, and the allocation proportion is given by �= 1

2 , then E[XP1]≈1.26E[X ]. Thus, in thiscase, it can be said that the merit obtained by adopting a load-sharing structure is too minor andit would be better to apply an ordinary cold standby structure.

On the other hand, suppose that gi (�)’s are given by gi (�)=�2, i=1,2, in the above example.In this case, the mean lifetime of the system m(�) is given in Figure 4.

As is seen from Figure 4, there exists �∗ which maximizes the mean lifetime of the system, whichis given by �∗ =0.529. In this case, the maximum mean lifetime is given by m(0.529)=2.864612.





5. MODELLING BASED ON THE PROPORTIONAL HAZARDS TYPE MODEL

In the previous sections, in order to express corresponding lifetime of the unit under varyingenvironment, the usual stochastic order between the lifetimes of the surviving unit before andafter the change of stress level is assumed (cf. (1) and (2)). In this section we consider a strongercondition for the relationship between the lifetimes of the unit under different environments.

5.1. Modelling and evaluation of system reliability

Let rPi (t) be the failure rate function of XPi , i=1,2. In this section we assume that

rP1(t)=k1(�)r1(t) ∀t�0 (12)



and

rP2(t)=k2(1−�)r2(t) ∀t�0, (13)

where the function ki (�) satisfies 0�ki (�)�1, ki (0)=0, ki (1)=1, and ki (�) is strictly increasing in�, i=1,2. Observe that the above assumed model ((12) and (13)) is the proportional hazards typemodel, which is very popular in survival and reliability analysis. Note that the above relationshipimplies the usual stochastic order defined in Section 2 and thus can be considered as a strongercondition. Then, under the setting in (12) and (13), it follows that

FP1(t)=1−exp

{−

∫ t

0k1(�)r1(u)du

}, t�0

and

FP2(t)=1−exp

{−

∫ t

0k2(1−�)r2(u)du

}, t�0.

Then the reliability function of the load-sharing system is given in the following theorem.

Theorem 2The reliability function of the general load-sharing system is given by

R(t) = exp

{−

∫ t

0k1(�)r1(u)du

}·exp

{−

∫ t

0k2(1−�)r2(u)du

}

+∫ t

0exp

{−

∫ t−u

0r1(w1(u)+s)ds

}·exp

{−

∫ u

0k1(�)r1(s)ds

}

×k2(1−�)r2(u)exp

{−

∫ u

0k2(1−�)r2(s)ds

}du

+∫ t

0exp

{−

∫ t−u

0r2(w2(u)+s)ds

}·exp

{−

∫ u

0k2(1−�)r2(s)ds

}

×k1(�)r1(u)exp

{−

∫ u

0k1(�)r1(s)ds

}du. (14)

ProofThe result can be obviously obtained from the proof of Theorem 1. �

Now we consider how the functions wi (u), i=1,2, can be obtained under the setting discussedin this section. As in Section 4, following the idea in the cumulative exposure model, we have thefollowing relationships:

1−exp

{−

∫ w1(u)

0r1(s)ds

}=F1(w1(u))=FP1(u)=1−exp

{−k1(�)

∫ u

0r1(s)ds

}



and

1−exp

{−

∫ w2(u)

0r2(s)ds

}=F2(w2(u))=FP2(u)=1−exp

{−k2(1−�)

∫ u

0r2(s)ds

}.

From these relationships, we obtain

w1(u)=�−11 (k1(�)�1(u)) (15)

and

w2(u)=�−12 (k2(1−�)�2(u)), (16)

where �i (t)=∫ t0 ri (u)du, i=1,2.

5.2. Relationship between two methods

In this subsection, we discuss the relationship between the method for modelling lifetime distri-bution of the unit under partial load condition (which is based on the proportional hazards typemodel) considered in this section and that based on the usual stochastic ordering in the previoussections.

From (3) and (4), we have

rP1(t) = g1(�)f1(g1(�)t)

1−F1(g1(�)t)

= g1(�)r1(g1(�)t)

=(g1(�) ·r1(g1(�)t)

r1(t)

)·r1(t), (17)

and

rP2(t) = g2(1−�)f2(g2(1−�)t)

1−F2(g2(1−�)t)

= g2(1−�)r2(g2(1−�)t)

=(g2(1−�) ·r2(g2(1−�)t)

r2(t)

)·r2(t). (18)

Thus, in (17) and (18), if (g1(�) ·r1(g1(�)t)

r1(t)

)=k1(�) (19)

and (g2(1−�) ·r2(g2(1−�)t)

r2(t)

)=k2(1−�), (20)

where k1(�) and k2(1−�) are independent of t , respectively, then the model in Section 2 becomesthe model considered in this section.



Note that Equations (19) and (20) (with right-hand sides not depending on t) hold only forWeibull distributions [16]. For example, consider the failure rate functions in the previous examples,r1(t)=3t2, t�0, and r2(t)=6t2, t�0. In this case, these failure rate functions satisfy the conditions(19) and (20) with (g1(�))3=k1(�) and (g2(1−�))3=k2(1−�).

5.3. Some particular cases

In the following, as in Section 4, some particular cases of the model are investigated. If the lifetimedistribution of the system is exponential or Weibull distributions, then the failure rate functions inthese cases satisfy the conditions (19) and (20). Thus, these cases can be covered by the modelsconsidered in Section 4. Thus, in the following, we consider a particular case where the modellingmethod based on the proportional hazards type model can uniquely be applied.

Suppose that r1(t)=2t+2, t�0, and r2(t)=4t+4, t�0. Under this specific setting, we have

�1(t)= t2+2t, �−11 (t)=√

t+1−1

and

�2(t)=2(t2+2t), �−11 (t)=√

t/2+1−1.

Then, from (15) and (16),

w1(u)=√k1(�)(u2+2u)+1−1, w2(u)=

√k2(1−�)(u2+2u)+1−1.

For the specific failure rate functions r1(t) and r2(t) given above, from (14), the reliability functionis given by

R(t) = exp{−k1(�) ·(t2+2t)−k2(1−�) ·2(t2+2t)}

+∫ t

0k2(1−�) ·(4u+4) ·exp{−(t−u+w1(u)+1)2+(w1(u)+1)2

−k1(�) ·(u2+2u)−k2(1−�) ·2(u2+2u)}du

+∫ t

0k1(�) ·(2u+2) ·exp{−2(t−u+w2(u)+1)2+2(w2(u)+1)2

−k2(1−�) ·2(u2+2u)−k1(�) ·(u2+2u)}du.

First, we consider the case when ki (�)=�, i=1,2. Under this setting, for example, the failurerate function of XP1 is given by

rP1(t)=�r1(t) ∀t�0.

In this case, if we set �≡ 12 then the probability of instantaneous failure becomes half of the full

load case, if �≡ 13 then that becomes one-third of the full load case, and so on. The mean lifetime

of the system m(�) is given in Figure 5.In this case, the mean lifetime of the system is equivalently maximized at �=0 and �=1. Thus,

it can be concluded that the ordinary cold standby system is better than any load-sharing system.





For this special case, by the interpretation of ki (�) given above, it can be said that the meritobtained by adopting a load-sharing structure is not enough and so it would be better to apply anordinary cold standby structure.

On the other hand, suppose that ki (�)’s are given by ki (�)=�2, i=1,2, in the above example.In this case, the mean lifetime of the system m(�) is given in Figure 6.

From Figure 6, it can be seen that there exists �∗ which maximizes the mean lifetime ofthe system, which is given by �∗ =0.568. In this case, the maximum mean lifetime is given bym(0.568)=0.779044.



6. CONCLUDING REMARKS

There has been much research on reliability and availability of load-sharing systems. However, inmost of these research, the distributions of the units composing the systems are assumed to beexponential. Assuming arbitrary lifetime distributions, we have proposed a method of modellinga general load-sharing system and studied its performance measures. The stochastically orderedlifetimes of a unit under different environments are modelled via usual stochastic order. Based on theproposed model, the reliability function of the system is derived. Furthermore, the optimal allocationproblem is also considered. Another method for modelling stochastically ordered lifetimes of aunit under different environments, which is based on the proportional hazards type model, hasbeen applied and the reliability function and optimal allocation policy are also derived. Then theyare compared with those in the first case.

Useful interpretations on the proposed stochastic models have been made based on the investi-gation of physical meanings and effects of the functions gi (�), ki (�), and wi (u), i=1,2. Thoseinterpretations could help modelling the functions in real applications. In many cases, if an equip-ment is working under a load which is specified in its specification, the load changing should nothave any impact on lifetime of the equipment. For example, when generator load is ranged up tonominal level it has no impact on the lifetime. It is the same also for electronic equipments, circuitboards, etc. Therefore, from a practical point of view, the proposed model could be applied to theequipments under their overloads which is greater than maximal load specified by the specification.

As the conditions on the functions gi (�), ki (�), and wi (u), i=1,2, described in Sections 2 and 5are not too restrictive, the proposed model can be considered as quite general, and various specificparametric models could be developed based on much experiences and extensive real data. In realapplications, the proposed model would give a guide of modelling a load-sharing system withunits of general distributions and the results obtained in this paper would provide the performancemeasures of the system.

ACKNOWLEDGEMENTS

The authors thank the referees for helpful comments and careful reading of this paper, which haveimproved the presentation of this paper considerably. This work was supported by the Korea ResearchFoundation Grant funded by the Korean Government (MOEHRD, Basic Research Promotion Fund) (KRF-2007-521-C00063).

REFERENCES

1. Kapur KC, Lamberson LR. Reliability in Engineering Design. Wiley: New York, 1977.2. Keccecioglu D. Reliability Engineering Handbook. Prentice-Hall: New Jersey, 1991.3. Scheuer EM. Reliability of a m-out-of-n system when component failure induces higher failure rates in survivors.

IEEE Transactions on Reliability 1988; 37:73–74.4. Shechner Z. A load sharing model: the linear break down rule. Naval Research Logistics 1984; 31:137–144.5. Lin HH, Chen KH, Wang RT. A multivariate exponential shared-load model. IEEE Transactions on Reliability

1993; 42:165–171.6. Yinghui T, Jing Z. New model for load-sharing k-out-of-n: G system with different components. Journal of

Systems Engineering and Electronics 2008; 19:748–751.7. Wang KS, Huang JJ, Tsai YT, Hsu FS. Study of loading policies for unequal strength shared-load system.

Reliability Engineering and System Safety 2000; 67:119–128.8. Shao J, Lamberson LR. Modeling a shared-load k-out-of-n:G system. IEEE Transactions on Reliability 1991;

40:205–209.



9. Meeker WQ, Escobar LA. A review of recent research and current issues of accelerated testing. InternationalStatistical Review 1993; 61:147–168.

10. Nelson W. Accelerated Testing: Statistical Models, Test Plans, and Data Analysis. Wiley: New York, 1990.11. Kijima M. Some results for repairable systems with general repair. Journal of Applied Probability 1989; 26:

89–102.12. Finkelstein MS. On statistical sand information-based virtual age of degrading system. Reliability Engineering

and System Safety 2007; 92:676–681.13. Finkelstein MS. Wearing-out of components in a variable environment. Reliability Engineering and System Safety

1999; 66:235–242.14. Finkelstein MS. Mortality in varying environment. Working Paper of the Max Planck Institute for Demographic

Research, WP 2004-029, Rostock, Germany, 2004.15. Finkelstein MS. A note on some aging properties of the accelerated life model. Reliability Engineering and

System Safety 2001; 71:109–112.16. Cox DR, Oakes D. Analysis of Survival Data. Chapman and Hall: New York, 1984.


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A stochastic model for a general load-sharing system under overload condition