IEEE TRANSACTIONS ON RELIABILITY, VOL. 51, NO. 1, MARCH 2002 17

A Reliability Test-Plan for Series Systems WithComponents Having Stochastic Failure Rates

J. Hariharan Nair and Sanjeev V. Sabnis

Abstract—This paper proposes a reliability test plan for a seriessystem, by considering the parameter of the exponential distri-bution to be a random variable having uniform distribution over[0 ], = 1 2 . . . . Explicit expressions are obtained forthe optimal values of the , when the number of components inthe system is 2. The general solution, albeit implicit, has also beenobtained when the number of components in a given system is3.Mathematical programming is used to find the optimal solutionand to illustrate it with numerical results.

Index Terms—component testing, consumer risk, cost minimiza-tion, Kuhn–Tucker conditions, producer risk, system reliability.

ACRONYMS1

Cdf cumulative distribution functionr.v. random variable

NOTATION

highest value of [producer, consumer] risknumber of components in the series systemsystem reliability for a unit time perioda specified proper fraction; if the system isless than , the system is considered unsatisfac-tory to warrant rejectiona specified proper fraction; if , thesystem is satisfactory enough to warrant accep-tancefailure rate of component

parameter of the uniform distribution

total number of allowable component failuresoptimal value oftotal time on test for componentoptimal value of

cost of testing componentper unit timetotal number of failures for componentuntil

: total number of component failures, a realization of r.v.

Cdf of given

Manuscript received December 28, 1998; revised October 10, 2000.The authors are with the Department of Mathematics, Indian Insti-

tute of Technology, Bombay, Mumbai - 400 076 India (e-mail: {HNair;SVS}@math.iitb.ernet.in).

Publisher Item Identifier S 0018-9529(02)03783-1.

1The singular and plural of an acronym are always spelled the same.

given the value ofuniform distribution over the rangemean of a Poisson r.v. such thatwhere

I. INTRODUCTION

I IN TODAY’S competitive world, it has become imperativefor manufacturers to produce highly reliable systems. To en-

sure the desired reliability, the manufacturers are required totest their product. Testing invariably involves a lot of expendi-ture; thus manufacturers have always shown interest in test planswhich are cost effective. With this motivation, many researchers,e.g., [2]–[12], have proposed test plans over the years.

Reference [2] considers ancomponent series system andminimizes the total test-cost while guaranteeing that the proba-bility of rejecting an acceptable system is minimum. This paperassumes that componentis tested for time and that no fail-ures occur during this period. Later, [3], [4] extended [2] by re-laxing both of these assumptions. That is, [3], [4] devised testplans to obtain an optimal testing cost by: i) minimizing both theproducer and consumer risks, and ii) allowing the replacementof failed component by a -identical component. This resultwas further generalized in [5], [7], [8], [10]–[12]; these papersdeal with constructing newer test plans for series and parallelsystems under types I and II censoring. For example, [7] con-siders a series system withcomponents; componenthavingexponential distribution with unknown parameter, and with

being a known upper bound on. The optimal testing timesdepend on both and . Also, [8] construct system-basedcomponent test plans for minimizing the total test cost for a par-allel system with highly reliable components; it considers a typeI censoring scheme, maximum likelihood estimators, and sumrule statistics for analyzes.

Consider a series system havingcomponents. This paperaims to construct a reliability test plan for a series system suchthat its is a r.v. having uniform distribution over . Thisresult in this paper differs from the earlier results in that noneof the previous results treat the parameter of the life distributionof components as a r.v. The need for considering this situationarises when a failed component with failure rate, say,, is re-placed by a component with similar failure rate. Thusbut ; this could be due to the environmental effects oncomponents or the standby (idle) status of the component. Thechoice of uniform distribution over a finite interval is motivatedby the fact that the distribution is not unreasonable and that itmakes the analysis simpler.

0018-9529/02$17.00 © 2002 IEEE

18 IEEE TRANSACTIONS ON RELIABILITY, VOL. 51, NO. 1, MARCH 2002

Constructing a reliability test plan entails ascertaining the op-timal values of s which minimize the total testing cost whileensuring that

rejecting the plan

and

accepting the plan

The use of the exact expression of system reliabilityrendersthe problem intractable and hence a second order approximationis used. This approximation holds very well ifs are very small( ).

Section II lists the assumptions made for all of the mathe-matics. Section III states the problem. Section IV constructs atest plan for -component systems. Section V obtains explicitexpressions for optimal for 2-component systems and givesthe corresponding numerical results. Section VI provides thegeneral solution.

II. A SSUMPTIONS

1) There are -independent components connected in se-ries.

2) The life-length of component is exponential with rate-parameter for . These life-lengths are-independent.

3) The of component is a r.v. having distribu-tion for ; these s are -independent.The s are -independent of the failure times.

4) During testing, any failed component is immediately re-placed by an i.i.d. component.

5) The total cost is a linear function of the componenttest-times. [This assumption is very practical because: a)components are tested separately; and b) fuel, electricity,human resources, etc., can have a constant cost-rate.]

6) The component interfaces within the system are perfect;thus it is not necessary to test them.

III. PROBLEM STATEMENT

Consider a series system havingdistinct components. As-sumptions 1–3 imply

(1)

The optimization problem in is

Minimize (2)

subject to

Rejecting the system (3)

Accepting the system (4)

The objective is to derive an optimum component-test-plan fora system with these characteristics. This involves determining

the optimal values of which minimize the objective functionin (2) subject to (3) and (4).

To derive the optimum component-test-plan, the data are col-lected by testing componentfor , and if it fails before ,replacing it by another component whose failure rate has thesame distribution as that of component. The distribution of ,given , Poisson with parameter , .Most earlier efforts to construct reliability test plans hinge onone of the crucial properties of the Poisson distribution: the Cdfof a Poisson random variable with mean,, is strictly decreasingin .

IV. A NALYSIS BASED ON AN UNBIASED ESTIMATOR OF

The optimization problem in Section III becomes intractableif the in (1) is used. Therefore, consider a second order ap-proximation of :

(5)

The approximation holds for sufficiently small. This implies that the system is highly reliable. It

follows that

This implies that is an unbiased estimator of

Use a decision rule based on , which has been used in[7], [8], etc. The rule is:

Accept the system iff ; reject it otherwise.Therefore, the problem is to determine theand such that

is minimum subject to the constraints:

(6)

(7)

Constraints (6) and (7) can be rewritten as

(8)

(9)

NAIR AND SABNIS: A RELIABILITY TEST-PLAN FOR SERIES SYSTEMS 19

Because the integrals in (8) and (9) can not be evaluated explic-itly, stronger conditions are used. For given, , , as inSection III, there exists an such that

for all (10)

for all (11)

and

has a Poisson distribution with mean.is strictly decreasing in so that (10) and (11) can be

written as

for all

for all

These inequalities are equivalent to

(12)

(13)

Rewrite (12) and (13) as

Use of Kuhn–Tucker conditions on the first inequality yields

The problem is restated:

Minimize (14)

subject to

(15)

(16)

A. Special Case

Without loss of generality let:

Thus s are i.i.d. r.v. From inequalities (12) and (13) it followsthat

Thus the optimal are

for

This is logically appealing because if allcomponents are i.i.d.then the component with the minimum cost would be tested forthe maximum possible amount of time and other componentswould not be tested at all.

V. CASE:

Notation:for

solution of (21)solution of (22)

The inequality constraints in (15) and (16) become

The optimization problem becomes

Minimize (17)

subject to

(18)

(19)

The constraint represented by (18) is the interior of an ellipsewhich lies on the axis with center at (0, 0), and majorand minor axes equal to, respectively,

The constraint (19) corresponds to the exterior part of an ellipseon the axis , with center at (0, 0) and major and minoraxes equal to, respectively,

20 IEEE TRANSACTIONS ON RELIABILITY, VOL. 51, NO. 1, MARCH 2002

There is no feasible solution to the optimization problem in (17)if the elliptical boundary of an ellipse corresponding to (18) iscompletely contained in that of the ellipse corresponding to (19).Thus, a feasible region does not exist if

This suggests that the feasible region exists whenever, for some,

Because

is a strictly increasing function of , and [1]

(20)

The coordinates of points in which the two ellipses

(21)

(22)

intersect are

The shaded regions AEB and CFD in Fig. 1 are the feasibleregions when , , , ,

. Each of the feasible regions is symmetrical aboutthe line . The objective functionfor some is an ellipse centered at (0, 0). Without loss ofgenerality, let . The problem reduces to choosing: 1) a

such that

is a minimum; and 2) which belongs to feasible regionAEB CFD

(23)

For any belonging to the feasible region,

(24)

Fig. 1. Feasible region for a 2-component series.

From (23) and (24),

Thus the distance to from (0, 0) is “less than or equalto” the distance between the origin and any other feasible point.Therefore, which minimizes the objective function, oc-curs at

The optimum values of and (because ) are

The optimum value of the objective function, and the corre-sponding ellipse, are

(25)

A. Numerical Results

Some numerical results for are given. Two sets ofvalues for and ( , and ,

) and for and : , , and ,are given. From Table I, for , ;

, , the smallest value of for which

is 21. Thus and the corresponding-values are theoptimal values. The optimal values of and are sensitiveto small changes in and . The optimal values of do notdepend on the magnitudes ofand ; instead they depend on

NAIR AND SABNIS: A RELIABILITY TEST-PLAN FOR SERIES SYSTEMS 21

TABLE I

the sign of . Therefore, this test can be appropriate for asystem whose testing costs are widely different.

VI. GENERAL CASE

This section solves the problem for . From Section IV,the optimization problem in (14), inequalities (15), and (16) canbe slightly modified

Minimize

Let

Then the problem can be restated:

Minimize

(26)

(27)

The , are the arithmetic and geometric means of, respectively. The equalities in (26) and (27) corre-

spond to 2 straight lines in and . Fig. 2 is the plot of thesetwo lines for , , , , are, respectively, 5, 0.80, 0.95,0.05, and 0.05.

The triangular-shaped shaded region, CDE, represents thefeasible region. The optimum value of occurs only at

Fig. 2. Feasible region of(z ; z ).

the intersection of these two lines. In view of this, the originalminimization problem reduces to

Minimize

(28)

(29)

This problem is solved using Lagrange multipliers. The La-grange constants , are obtained by solving:

The solution of these equations does not have a closed form.However, for given values of , one can easilysolve these 2 equations for and . The optimal are

for all

REFERENCES

[1] I. K. Altinel, “The design of optimum component test plans in thedemonstration of a series system reliability,”Computational Statisticsand Data Analysis, vol. 14, pp. 281–292, 1992.

[2] S. Gal, “Optimal test design for reliability demonstration,”Operat. Res.,vol. 22, pp. 1236–1242, 1974.

[3] M. Mazumdar, “An optimum procedure for component testing in thedemonstration of series system reliability,”IEEE Trans. Reliability, vol.R-26, pp. 342–345, 1977.

[4] , “An optimum procedure for component testing procedure for aseries system with redundant subsystems,”Technometrics, vol. 22, pp.23–27, 1980.

22 IEEE TRANSACTIONS ON RELIABILITY, VOL. 51, NO. 1, MARCH 2002

[5] J. Rajagopal and M. Mazumdar, “A comparison of several component-testing plans for a parallel system,”IEEE Trans. Reliability, vol. R-36,pp. 419–424, 1987.

[6] , “Design of system based component test plans,” inProc. 2nd In-dustrial Eng. Res. Conf., 1993, pp. 247–251.

[7] , “Designing component test plans for series system reliabilityvia mathematical programming,”Technometrics, vol. 37, pp. 195–212,1995.

[8] , “Minimum cost component test plans for evaluating reliabilityof a highly reliable parallel system,”Naval Res. Logistics, vol. 44, pp.401–418, 1997.

[9] J. Rajagopal, M. Mazumdar, and T. H. Savits, “Some properties of thePoisson distribution with an application to reliability testing,”Prob. inEng. and Informat. Sci., vol. 8, pp. 345–354, 1994.

[10] J. H. Yan and M. Mazumdar, “A comparison of several componenttesting plans for a parallel system,”IEEE Trans. Reliability, vol. R-35,pp. 437–443, 1986.

[11] , “A comparison of several component testing plans for a parallelsystem,”IEEE Trans. Reliability, vol. R-36, pp. 419–424, 1987.

[12] , “A component testing plan for a parallel system with type II cen-soring,” IEEE Trans. Reliability, vol. R-36, pp. 425–428, 1987.

J. Hariharan Nair received the M.Sc. degree in statistics from the University ofKerala (Department of Statistics, Thiruvananthapuram) and the M.Phil. degreefrom the Department of Statistics, University of Poona, Pune, India.

His research interests include reliability theory and statistical modeling. Heis working for Capital International Services, USA.

Sanjeev V. Sabnisreceived the B.Sc. and M.Sc. degrees in statistics from theUniversity of Poona, Pune, India, and the Ph.D. degree in statistics from OldDominion University, Virginia, USA.

He is an Associate Professor with the Department of Mathematics, IndianInstitute of Technology, Bombay. He works in the area of reliability theory andapplied statistics.