86 IEEE TRANSACTIONS ON RELIABILITY, VOL. R-l9, NO. 3, AUGUST 1970

Random FailureDUNCAN BADENIUS, MEMBER, IEEE

Abstract-Random failure is perhaps the most abused term in reliability c subset;and related engineering interests. Past definitions, being logically more n intersection set, or conjunction;descriptive than constructive, have not been truly definitive. An attempt iois made here to offer one that is: the class of failures statistically independent i implication;of past history. This definition is based upon the mutual implication between mutual implication;the stochastic processes producing such events and the unique characteriza- Pr{.} probability (of);tion thereof by their independence. R the set of reals or real line;

Reader Aids: 2i intensity function and/or constant;Purpose: Tutorial H Lebesgue measure.Special math needed for explanations: Probability theorySpecial math needed for results: NoneResults useful to: Reliability engineers. BASIC RANDOMNESS AND MATHEMATICAL PROBABILITY

Preliminary to constructing the definition of randomINTRODUCTION failure, the foundations of basic randomness itself will be

IF FAILURE rate is the most often used term in reliability, considered. In what follows, note that failures are con-random failure easily is its most abused one. In a recent sidered as a special subset of events.

paper [1] the author discussed certain questions sur- Conceptually, randomness is not essential to the purelyrounding the former term, with a view to their resolution. mathematical theory of probability, nor is it used in itsIt is proposed here to do the same for random failure, rigorous axiomatic development [3]. There, probabilitybut chiefly to construct a suitable definition thereof. The is defined in terms of a few fundamental postulates fromfollowing example illustrates that the term is not readily which the whole theory proceeds, but that make no re-made precise. ference to randomness as an essential part thereof. Yet,A standards committee [2] not long ago defined random paradoxically, the concept is rooted in two definitions of

failure as follows: "Any failure whose cause and/or probability, separate from the axiomatic definition butmechanism make its time of occurrence unpredictable." still valid and often used, as given by the so-called statisticalHere, reference to causation (and/or mechanism) is irrele- and classical schools of probability. These are, respectively,vant and misleading. Every failure necessarily has a physical the frequency theoretic and range theoretic definitions ofcause, but there is no Jekyll and Hyde cause, say, half probability. Even here the purely mathematical role ofphysical and half stochastic, mysteriously making things randomness is slight to nonexistent. Indeed, its real rolefail at random. Also, randomness epitomizes unpredicta- is ontological: it is probability theory's raison d'etre.bility, and vice versa, i.e., the terms are synonymous. Without it, the latter would be merely an elaborate exerciseHence, the definition is circular. Finally, there is an apparent in deductive reasoning.disparity between the fact that unpredictable literally The frequency theoretic definition defines probabilitymeans not predictable, and the fact that random failures as the limiting value of a relative frequency of a randomlydo figure in reliability predictions. At best, the language (sic) distributed series of occurrences [3]. Here the require-is ambiguous. ment of randomness is sequentially oriented.' The rangeRandom failures herein are defined as the class of theoretic definition, simply stated, defines probability as

failures statistically independent of past history. This the ratio of the number of "favorable" unit-alternativesdefinition is precise, noncircular, and unambiguous. What to the total number thereof, favorable and unfavorable [3].follows treats its etymology and justification. Unit-alternatives are the simplest possible outcomes such

that for every pair: 1) neither is certain nor impossible,NOTATION 2) their joint occurrence is mutually exclusive, and 3) they

are indistinguishable except by some identifying means notThe following notation iS used in some of the sections: 4 .-~ 'T~1II1influencing their occurrence as outcomes. (Note that whileA, B, etc. sets, statements, etc; this definition may be hard to apply, it is not circular.)s ~~set membership; Given 1), equality of possibilities (sic) additionally onlyManuscript received July 21, 1970; revised October 1, 1970. requires that 2) be satisfied. (This differs from Laplace [5]The author is with Republic Aviation Division, Fairchild Hiller, who, in effect, assigns equipossibility to the case of unit-

Farmingdale, N.Y. 11735.alentvscasnsoeseuainndofsonvrEditor's Note: Though considered quite controversial by the referees, alentvscuigsoepclton ndofsonvr

this paper is being published to stimulate discussion and help clarify theconcept for engineers. Correspondence is invited onl the topic. Church 14] hasinvestigated the problem ofdefiningarandom sequence.

BADENIUS: RANDOM FAILURE 87

his real meaning.) Finally, satisfaction of 1) only is simply processes. Purely random events accordingly may be definedbasic randomness. This leads directly to the following as follows.(most general) definition of random events: the class of all Definition: Purans are the class of all events whoseevents whose occurrence is neither certain nor impossible. occurrence is -statistically independent of past history,Accordingly, what randomness represents in the frequency i.e., whose point process realizations are either Bernoulliantheoretic definition of probability, is satisfied in the range or Poissonian.theoretic definition by unit-alternatives with the property Remark: By "events" it is understood to mean randomof basic randomness, as just defined. The three cases cited events.illustrate that there are different kinds of randomness. Hence, purans are a subclass of random events, andThere are also different degrees of randomness, as many randomfailures in turn are a subclass of purans. Further,as there are degrees of probability, but inversely. For it is suggested that (in the context of failure) "purans"instance: rare random events are highly random. is a better name for random failures.

Literally, then, to speak of random failure is to speak Remark: Detailed discussion of the Bernoulli andof the commonplace, or formally trivial. That the term Poisson processes is outside the scope of this paper;has a special meaning, beyond its literal meaning, is however, a few particulars are noteworthy. The Poissonnot immediately evident nor universally accepted. This process may be considered as a generalization of thespecial meaning is that of pure as opposed to mere Bernoulli process, from discrete independent trials withrandomness. constant success probability, to continuous time with con-

stant process intensity, i.e., in reliability applications,PURANS (PURELY RANDOM EVENTS) constant hazard rate.7 The mathematical bridge, over which

Purans are purely random events. This is what random the Bernoulli passes to reach the Poisson, is the generalizedfailure, in its nontrivial sense, signifies. Says Lindley exponential function ax. In books on pure mathematics8 it is

"I think a point process2 having the following property is defined, for all real x, as ax exp (x ln a). In the Bernoullithe only one that can be truly called random. Basic property: case, a = p the success probability, x n integral numberFor any fixed t let A be any event referring to the process of trials. In the Poisson case, In a = In p is the processprior to t and B any event referring to it after t: then A intensity (= hazard rate), x = for continuous time (theand B must be independent"3 [7]. This shall be taken here positive eals). But 0 p . , . p . , pas a definition of independence of past history. Lindley i > 0 say, so that pn = exp (- t). The left-hand side of thecontinues: "The probability definition will describe how last equation is the expression for reliability over n Bernoullithese points are attained. If we have the basic criterion4 trials the right-hand side that for exponential reliabilityfor the case of a continuous line we obtain the Poisson over time. From a purely mathematical standpoint, then,process: if a lattice is used then we have the Bernoulli the generalization from n to t has always existed.process'"5 [7]. Elsewhere he remarks: "The purely randomprocess is one in which... the future is independent of the PURANS AND PROCESS CHARACTERIZATIONpast ... the basic requirement is that the probability of an Consider three statements.9 A: The process is such thatincident in a region is independent of any event concerning the past and future, given the present, are independent.incidents outside the region" [10]. B: The intervals between successive incidents have inde-Doob seems to take a similar view: "If events occur pendent exponential densities.'0 C: The distribution of a

in accordance with the Poisson law ... they are sometimes number of incidents in a fixed interval is Poisson. Proofdescribed as 'purely random,' or in the physical literature that A -* B and that A -* C is routine." A weak'2 formsometimes simply as 'random"' [11]. of B -* A is proved in Raiffa-Schlaifer [9]. Subject to aNo doubt it is from the latter usage that the special limitation, to be mentioned shortly, Renyi [13] has

meaning of the term random failure arose. Though ad- proved that C -+ A. Both of these last two inferencesmittedly ambiguous and misleading to some, this usage (i.e., B -* A and C -+ A) are known as process charac-of the term plainly does have an historical precedent. terizations, the latter, and more usual, being by means

It may be shown6 that independence of past history of a distribution. Says Lindley: "It is worth remembering...is a unique property of the Bernoulli and Poisson that the Poisson distribution characterizes the Poisson

process. That is, not only does the process yield the dis-tribution, but the distribution with mean proportional

2A stochastic process whose realizations consist of a series of pointevents [6]. In the following t represents time.

3That is, statistically independent. This implies that Pr{A n~B} = 7 See [1] for a definition and derivation.Pr{A}Prf{B} 8 For instance, Hardy [12].

4 Cantor's axiom that there exists a one-to-one correspondence between 9 These details were supplied by Lindley [7].the set of real numbers and the points of a straight line. 10° Referred to by Raiffa-Schlaifer [9] as gamma-Il densities, the ex-

5Few references discuss the Bernoulli process in any detail, though the ponential being regarded as a special case of the gamma distribution.Poisson process usually always is. However, Schlaifer [8] and Raiffa- l' See Lindley[10].Schlaifer [9] discuss both; [8] is elementary, and [9] is advanced. 12 "Weak"' because tbe independence of every type of statement is6 For instance, Raiffa-Schlaifer [9]. not established [7].

88 IEEE TRANSACTIONS ON RELIABILITY, AUGUST 1970

to the length of the interval, can only arise from the process. name for them. This term here has been assigned a preciseA common way of testing whether a process is Poisson is meaning, and so, consequently, has random failure.to test whether the number of incidents has a Poissondistribution ... and then use the characterization" [10]. ACKNOWLEDGMENTThe aforementioned limitation (i.e., on C -* A) is as

follows.'3 Let N(X) denote the number of events of a Special thanks are due to Prof. D. V. Lindley, Universitystochastic point process on the line R lying within X c R, College, London, England, and to Dr. R. E. Miles, Aus-and lXI denote the Lebesgue measure of X (if it exists). tralian National University, Canberra, Australia, for theirNext, the Poisson process of intensity i on R, denoted by generous help with the mere technical mathematical details7t(,), is that point process for which 1) N(X) has a Poisson of this paper. Responsibility for the paper per se, however, isdistribution with mean )JXJ for all Borel subsets"4 X c R, the author's.and 2) N(X), .. , N(Xm) are mutually independent ifXl, , X. are disjoint subsets of R (m = 2, 3, .). Thefollowing then may be asserted. 1) Let Y denote the class REFERENCESof subsets ofR that are unions of a finite number of intervals. [l] D. Badenius, "Failure rate/MTBF," IEEE Trans. Reliability,Renyi [13] has proved that if, for a point process in R vol. R-19, pp. 66-67, May 1970.

N'XasaPoisondstriutiowit ma .X 'for al [2] J. J. Naresky, "Reliability definitions," IEEE Standards Committee,N(X) has a Poisson distribution with mean ~iXI for all IEEE Newsletter, G-R, vol. 13, April 1968.X E 1Y then it is 2t(/). 2) Let I denote the class of intervals [3] G. H. von Wright, "Probability," in Encyclopedia Britannica, vol. 18,of R. It is not true that, if N(X) is Poisson distributed 1962, pp. 529-532.

with mean .X forallXEI,then hepointprocess [4] A. Church, "On the concept of a random sequence," Bull. Amer.with mean AXIX for all X E I, then the point process iS Math Soc., vol. 46, pp. 130-135, 1940.necessarily n(AX). Counterexamples are found in Goldman [5] P. S. Laplace, A Philosophical Essay on Probabilities (English transl.)[17] and Moran [18]. New York: Dover 1951, pp. 11-12.

The definition of purely random events given herein [6] D. R. Cox and H. D. Miller, The Theory of Stochastic Processes.The definition of purely random events given herein New York: Wiley, 1965.depends only upon the mutual implication A +-+ B. It is [7] D. V. Lindley, private communication, 1968-1969.independent of statement C. Statement C clearly has im- [8] R. Schlaifer, Probability and Statistics for Business Decisions. New

York: McGraw-Hill, 1959.portant practical implications, but these are not within [9] H. Raiffa and R. Schlaifer, Applied Statistical Decision Theory.the scope of this paper to explore. Boston: Harvard University, 1961.

[10] D. V. Lindley, Introduction to Probability and Statistics from aBayesian Viewpoint, pt. 1. London: Cambridge University Press,

CONCLUSION 1965, ch. 2.[11] J. L. Doob, Stochastic Processes. New York: Wiley, 1953, p. 400.

Random failures hold no special mystery. If they seem [12] G. H. Hardy, A Course of Pure Mathematics, 10th ed. London:peculiarly unpredictable, it is because their occurrence is Cambridge University Press, 1960.independent of other similar occurrences, thus patternless, [13] A. Renyi, "Remarks on the Poisson process," Studia Scientiarum

and, morever,usualyhgr. It id Mathematicarum Hungarica, vol. 2, 1967, pp. 119-123.and, moreover, usually highly improbable. It is desired [14] R. E. Miles, private communication, Dep. Statistics, Australianand planned for that they should be rare events.'5 Indeed, National Univ. Canberra, 1968.their comparative rarity is what seemingly removes their [15] B. V. Gnedenko, The Theory of Probability (English transl.), 4th ed.

convese, sccesful ocurrnce, rom he arna o randm- E

New York: Chelsea, 1967.converse, successful occurrence, from the arena of random- [16] A. Papoulis, Probability, Random Variables, and Stochastic Processes.ness. Purans, i.e., as a subclass thereof, is a less ambiguous New York: McGraw-Hill, 1965.

[17] J. R. Goldman, "Stochastic point processes: limit theorems," Ann.Math. Stat., vol. 38, 1967, pp. 771-779.

13 These details were supplied by Miles [14]. [18] P. A. P. Moran, "A non-Markovian quasi-Poisson process," Studia'4 Subsets that are elements of a Borel field of events. See, for instance, Scientiarum Mathematicarum Hungarica, vol. 2, 1967, pp. 425-429.

Gnedenko [15] or Papoulis [16]. [19] F. A. Haight, Handbook of the Poisson Distribution, ORSA 11.15 The Poisson distribution itself is sometimes called the law of rare New York: Wiley, 1967, p. 24.

events [19], [20], which is based upon the circumstances in which it is a [20] Z. W. Birnbaum, Introduction to Probability and Mathematicallimiting case of the binomial distribution. Statistics. New York: Harper, 1962, p. 151.