IEEE TRANSACTIONS ON RELIABILITY, VOL. 49, NO. 1, MARCH 2000 85

Generic Rules to Evaluate System-Failure FrequencySuprasad V. Amari, Member, IEEE

Abstract—Frequency of failure of a system with -independentcomponents can be obtained from the system availability (unavail-ability) expression and failure and repair rates of the components.Although, Grouped Variable Inversion is an efficient technique tofind the system availability, there is no convenient method to con-vert the “availability expression obtained by this technique” intoan “expression for system-failure frequency.” This paper presentgeneric rules to find system-failure frequency, particularly, whenthe availability or unavailability expression of a system is obtainedusing this technique. The rules are straightforward, and produceappreciably shorter expressions for system-failure frequency. Ex-amples illustrate the simplicity and efficiency of the proposed rules.

Index Terms—Disjoint product, sum of disjoint products, systemavailability, system-failure frequency.

I. INTRODUCTION

Acronyms

GVI grouped variable inversion (method)SVI single variable inversion (method)

Notation

label of component, [failure, repair] rate of component

system-failure frequency “mean number ofsystem-failures” per time-unit

, Boolean variable indicating that componentis[working, failed]; is the complement of

, steady state [availability, unavailability] of compo-nent ; Pr component is [working, failed]

, system [availability, unavailability]For convenience, both and are used in the paper;however, note that .

is a key parameter in reliability and risk analysis; it canbe evaluated from a or [5], and failure and repair rates ofits components; usually is used, but sometimes is used inthis context. Other performance measures of a system, such asmean up-time, mean down-time, and mean cycle-time can bedetermined easily from , , of the system [7].

The methods available [5], [6] to convert into a functionof are limited to the case where is expressed as the sum

Manuscript received November 1, 1999.The author is with the Reliability Engineering Centre; Department of Indus-

trial Engineering and Management; Indian Institute of Technology; Kharagpur721302 India (e-mail: suprasada@pune.tcs.co.in).

Responsible editor: S. Rai.Publisher Item Identifier S 0018-9529(00)06206-0.

of products of component availabilities or products of compo-nent unavailabilities, or their mixed products. These methodsapply only when is obtained using the I-E method [7] or SVI(a disjoint product method) [1]. Recent literature on reliabilityor availability evaluation of systems with-independent compo-nents [3], [4], [8] shows that GVI is efficient in terms of compu-tation time and accuracy as well as in getting appreciably fewerdisjoint products. Although, GVI is efficient for finding, thereis no convenient method in this case to convertinto . Thisis because, for GVI the disjoint products involve the comple-ments of products of component availabilities or complementsof products of component unavailabilities or both. Therefore,this paper proposes generic rules to convertobtained througha GVI technique into an expression of.

II. STEADY-STATE FREQUENCY OFFAILURE

A. Assumptions

1) The system is composed of-independent components. Itsstructure function is -coherent.

2) The system and its components have 2 states: up (working)and down (failed).

3) The failure and repair rates of the components are constant.4) All components are repairable. Repaired components are

good-as-new.5) The system is in steady-state.

Example 1 demonstrates the evaluation ofusing SVI andGVI; the number of terms using SVI is more than in using GVI[3].

B. Example 1 [3]

Let:

be the minimal path sets of a system.1) SVI:

(1)

2) GVI:

(2)

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86 IEEE TRANSACTIONS ON RELIABILITY, VOL. 49, NO. 1, MARCH 2000

For (2), there is no direct method to getfrom . Followingsimple rules, (2) can be expressed as in either (1) or (3).

(3)

Ref. [5], [6] have the rules for calculating when or isexpressed in the mixed products of theand . The rules arelabeled I and II.

Case 1: is expressed in terms of mixed productsand

Rule I: To get from , multiply every termby ; and treat constants

as if they were not present. (I)Case 2: is expressed in terms of mixed products

andRule II: To get from , multiply every term

by ; and treat constantsas if they were not present. (II)

Hence, from (3), using rule I, for example 1:

(4)

Both (3) and (4) have 7 terms. Similarly, these rules were appliedto (1) to get ; there were 9 terms.

Since, rules I and II can not deal with the products of inversionof several variables, contains more terms. Section III providessimple, easy-to-remember rules to findfrom , using GVI.

III. T HE GENERIC RULES

To improve readability, , , , are used in therules for , , , , respectively. The proofis based on mathematical induction, and is available from theauthor.

Let:

i.e.,

i.e.,

(5)

Case 3: is expressed in terms of mixed productsand

Rule III: To get from , multiply every termby . (III)

Case 4: is expressed in terms of mixed productsand

Rule IV: To get from , multiply every termby . (IV)

Case 5: is expressed in terms of mixed productsand

Rule V: To get from , multiply every termby . (V)

Case 6: is expressed in terms of mixed productsand

Rule VI: To get from , multiply every termby . (VI)

Rule III (IV) applies when ( ) is obtained using path-sets,whereas rule V (VI) applies when ( ) is obtained using cut-sets. In general (say, for fault-trees),or can be obtained asthe sum of mixed products of , , , ,and constant coefficients.

If there exist -identical components/modules, then there canexist repeated sub-products within a product, i.e., there can existhigher powers of sub-products within a product (e.g., a series-parallel system with-identical units in each subsystem). There-fore, the more generic rules are:

Case 7: is expressed in terms of mixed products, , , ,

and a constant .Rule VII: To get from , multiply every term

by

. (VII)Case 8: is expressed in terms of mixed products

, , , ,and a constant .

Rule VIII: To get from , multiply every term

by

. (VIII)In all cases, the constants should be treated as if they were

not present. However, the constant coefficients remain in theircorresponding positions as shown in rules VII and VIII. For ex-ample, consider:

the constant to be ignored is 1. Hence—

For example 1, can be found from (2), using rule III:

(6)

After some mathematical manipulation, theexpression in (6)can be expanded in the form of (4) [2], since they are identical.

Example 2: is given in (7). The fault-tree of the system isdescribed in [2].

(7)

Using rule VIII—

(8)

AMARI: GENERIC RULES TO EVALUATE SYSTEM-FAILURE FREQUENCY 87

The number of terms (products) inequals the number of termsin the availability (unavailability) expression. Therefore, it isstraightforward to prove that rules III–VIII will produce fewerterms, because these rules can take expressions of sum of prod-ucts of mixed polynomials as input, and this kind of expres-sion contains fewer terms. Since the computation time is pro-portional to the number of terms, these rules produces fewerterms and require less computation resources.

ACKNOWLEDGMENT

The author would like to thank the Dr. S. Rai, Editor; ananonymous referee; and Dr. W. Schneeweiss for their usefulcomments on earlier versions of this paper.

REFERENCES

[1] J. A. Abraham, “An improved algorithm for network reliability,”IEEETrans. Reliability, vol. R-28, pp. 58–61, Apr. 1979.

[2] S. V. Amari, “Reliability, Risk and Fault-Tolerance of Complex Sys-tems,” Ph.D. dissertation, Indian Institute of Technology, Kharagpur,1998.

[3] K. D. Heidtmann, “Smaller sums of disjoint products by subproduct in-version,”IEEE Trans. Reliability, vol. 38, pp. 305–311, Aug. 1989.

[4] S. Rai, M. Veeraraghavan, and K. S. Trivedi, “A survey of efficient re-liability computation using disjoint products approach,”Networks, vol.25, pp. 147–163, 1995.

[5] W. G. Schneeweiss, “Addendum to: Computing failure frequency viamixed products of availabilities and unavailabilities,”IEEE Trans. Reli-ability, vol. R-32, pp. 461–462, Dec. 1983.

[6] D. Shi, “General formula for calculating the steady-state frequency ofsystem failure,”IEEE Trans. Reliability, vol. R-30, pp. 444–447, 1981.

[7] C. Singh and R. Billinton, “A new method to determine the failure fre-quency of a complex system,”IEEE Trans. Reliability, vol. R-23, pp.231–234, 1974.

[8] S. Soh and S. Rai, “CAREL: Computer Aided Reliability Evaluation fordistributed computing networks,”IEEE Trans. Parallel and DistributedSystems, pp. 199–213, Apr. 1999.

Suprasad V. AmariFor biography, see IEEE TRANS.RELIABILITY , vol. 46, Dec.1997, p. 522.