Chaper Basic Reliabiliy Mahemaics The basics of mahemaical heory ha are relevan o he sudy of reliabiliy and safey engineering are discussed in his chaper. The basic conceps of se heory and probabiliy heory are explained firs. Then he elemens of componen reliabiliy are presened. Differen disribuions used in reliabiliy and safey sudies wih suiable examples are explained. The reamen of failure daa is given in he las secion of he chaper.. Classical Se Theory and Boolean Algebra A se is a collecion of elemens having cerain specific characerisics. A se ha conains all elemens of ineres is known as a universal se, denoed by U. A subse refers o a collecion of elemens ha belong o a universal se. For example, if universal se U represens employees in a company, hen female employees is a subse A of U. For graphical represenaion of ses wihin he frame of reference of universal se, Venn diagrams are widely used. They can be very convenienly used o describe various se operaions. The Venn diagram in Figure. shows he universal se wih a recangle and subse A wih a circle. The complemen of a se A (denoed by Ā) is a se which consiss of he elemens of U ha do no belong o A. A U Figure. Venn diagram for subse A
6 Basic Reliabiliy Mahemaics.. Operaions on Ses Le A and B be any subses of he universal se U, he union of wo ses A and B is a se of all he elemens ha belong o a leas one of he wo ses A and B. The union is denoed by and read as OR. Thus A B is a se ha conains all he elemens ha are in A, B or boh A and B. The Venn diagram of A B is shown in Figure.. A B Figure. Venn diagram for A B The inersecion of A and B is he se of elemens which belong o boh ses. The inersecion is denoed by and read as AND. The Venn diagram of A B is shown in Figure.3. A B Figure.3 Venn diagram for A B Two ses A and B are ermed muually exclusive or disjoin ses when A and B have no elemens in common, i.e., A B =, where denoes an empy se. This can be represened by a Venn diagram as shown in Figure.4. A B Figure.4 Venn diagram for muually exclusive evens
. Classical Se Theory and Boolean Algebra 7.. Laws of Se Theory Some imporan laws of se heory are enumeraed in Table.. Table. Laws of se heory Name Descripion Ideniy law A = A; A U = U A = ; A U = A Idempoency law A A = A A A = A Commuaive law A B = B A A B = B A Associaive law A (B C) = (A B) C A (B C) = (A B) C Disribuive law A (B C) = (A B) (A C) A (B C) = (A B) (A C) Complemenaion law De Morgan s laws A A = U A A = Φ A = A ( A B) = A B ( A B) = A B..3 Boolean Algebra Boolean algebra finds is exensive use in evaluaion of reliabiliy and safey procedures due o consideraion ha componens and sysem can presen in eiher success or failure sae. Consider a variable X ha denoes he sae of a componen and assume represens success and represens failure. Then, he probabiliy ha X is equal o P(X = ) is called he reliabiliy of ha paricular componen. Depending upon he configuraion of he sysem, i will also have a success or failure sae. Based on his binary sae assumpion, Boolean algebra can be convenienly used. In Boolean algebra all he variables mus have one of wo values, eiher or. There are hree Boolean operaions, namely, OR, AND, and NOT. These operaions are denoed by +, (do), and (superbar over he variable) respecively. A se of posulaes and useful heorems are lised in Table., where x, x, x 3 denoe variables of a se X.
8 Basic Reliabiliy Mahemaics Table. Boolean algebra heorems Posulae/heorem Remarks x + = x Ideniy x = x x + x = x Idempoence x x = x = and = x = x Involuion x + xx = x x ( x + x) = x Absorpion x + ( x + x3) = ( x + x) + x3 x ( x x ) = ( x x ) x Associaive 3 3 ( x = x x + x) ( x = x + x x) De Morgan s heorem Consider a funcion f(x, x, x 3,, x n ) of n variables, which are combined by Boolean operaions. Depending upon he values of consiuen variables x, x,, x n, f will be or. As hese are n variables and each can have wo possible values or, n combinaions of variables will have o be considered for deerminaion of he value of f. Truh ables are used represen he value of f for all hese combinaions. A ruh able is given for a Boolean expression f(x, x, x 3 ) = x x + x x 3 + x x 3 in Table.3. In reliabiliy calculaions, i is necessary o minimize he Boolean expression in order o eliminae repeiion of he same elemens. The premise of all minimizaion echniques is he se of Boolean algebra heorems menioned in he Table.. The amoun of labor involved in minimizaion increases as he number of variables increases. Geomeric mehods and he famous Karnaugh s map is applicable only up o six variables. Nowadays, sophisicaed compuerized algorihms are available for calculaion wih large number of variables. Table.3 Truh able x x x 3 f
. Conceps of Probabiliy Theory 9. Conceps of Probabiliy Theory The word experimen is used in probabiliy and saisics o describe any process of observaion ha generaes raw daa. An experimen becomes a random experimen if i saisfies he following condiions: i can be repeaable, he oucome is random (hough i is possible o describe all he possible oucomes) and he paern of occurrence is definie if he experimen is repeaed a large number of imes. Examples of a random experimen are ossing of a coin, rolling a die, and failure imes of engineering equipmen from is life esing. The se of all possible oucomes of a random experimen is known as a sample space and is denoed by S. The sample space for random experimen of rolling a die is {,, 3, 4, 5, and 6}. In he case of life esing of engineering equipmen, he sample space is from o. Any subse of sample space is known as an even E. If he oucome of he random experimen is conained in E hen one can say ha E has occurred. Probabiliy is used o quanify he likelihood, or chance, ha an oucome of a random experimen will occur. Probabiliy is associaed wih any even E of a sample space S depending upon is chance of occurrence which is obained from available daa or informaion. The concep of he probabiliy of a paricular even is subjec o various meanings or inerpreaions. There are mainly hree inerpreaions of probabiliy: classical, frequency, and subjecive inerpreaions. The classical inerpreaion of probabiliy is based on he noion of equally likely oucomes and was originally developed in he conex of games of chance in he early days of probabiliy heory. Here he probabiliy of an even E is equal o he number of oucomes composing ha even (n) divided by he oal number of possible oucomes (N). This inerpreaion is simple, inuiively appealing, and easy o implemen, bu is applicabiliy is, of course, limied by is resricion o equally likely oucomes. Mahemaically, i is expressed as follows: n P ( E) =. (.) N The relaive-frequency inerpreaion of probabiliy defines he probabiliy of an even in erms of he proporion of imes he even occurs in a long series of idenical rials. In principle, his inerpreaion seems quie sensible. In pracice, is use requires exensive daa, which in many cases are simply no available and in oher cases may be quesionable in erms of wha can be viewed as idenical rials. Mahemaically, i is expressed as follows: n PE ( ) = lim N. (.) N The subjecive inerpreaion of probabiliy views probabiliy as a degree of belief, and his noion can be defined operaionally by having an individual make
Basic Reliabiliy Mahemaics cerain comparisons among loeries. By is very naure, a subjecive probabiliy is he probabiliy of a paricular person. This implies, of course, ha differen people can have differen probabiliies for he same even. The fac ha subjecive probabiliies can be manipulaed according o he usual mahemaical rules of probabiliy is no ransparen bu can be shown o follow from an underlying axiomaic framework. Regardless of which inerpreaion one gives o probabiliy, here is a general consensus ha he mahemaics of probabiliy is he same in all cases... Axioms of Probabiliy Probabiliy is a number ha is assigned o each member of a collecion of evens from a random experimen ha saisfies he following properies. If S is he sample space and E is any even in a random experimen:. P(S) =.. P(E). 3. For wo evens E and E wih E E =, P(E E ) = P(E ) + P(E ). The propery ha P(E) is equivalen o he requiremen ha a relaive frequency mus be beween and. The propery ha P(S) = is a consequence of he fac ha an oucome from he sample space occurs in every rial of an experimen. Consequenly, he relaive frequency of S is. Propery 3 implies ha if he evens E and E have no oucomes in common, he relaive frequency of oucomes is he sum of he relaive frequencies of he oucomes in E and E... Calculus of Probabiliy Theory... Independen Evens and Muually Exclusive Evens Two evens are said o be independen if he occurrence of one does no affec he probabiliy of occurrence of oher even. Le us say A and B are wo evens; if he occurrence of A does no provide any informaion abou occurrence of B hen A and B are saisically independen. For example in a process plan, he failure of a pump does no affec he failure of a valve. Two evens are said o be muually exclusive if he occurrence of one even prevens he occurrence of he oher even. If he wo evens A and B are muually exclusive hen hey are saisically dependen. Success and failure evens of any componen are muually exclusive. In a given ime, a pump ha is successfully operaing implies ha failure has no aken place.
. Conceps of Probabiliy Theory... Condiional Probabiliy The concep of condiional probabiliy is he mos imporan in all of probabiliy heory. I is ofen of ineres o calculae probabiliies when some parial informaion concerning he resul of he experimen is available, or o recalculae hem in he ligh of addiional informaion. For wo evens A and B, he probabiliy of A given ha B has occurred is referred o as condiional probabiliy and is denoed by P(A B) = P(A B)/P(B). Figure.5 Venn diagram for A B If he even B occurs hen in order for A o occur i is necessary ha he acual occurrence be a poin in boh A and B, i.e., i mus be in A B (Figure.5). Now, since we know ha B has occurred, i follows ha B becomes our new sample space and hence he probabiliy ha he even A B occurs will equal he probabiliy of A B relaive o he probabiliy of B. I is mahemaically expressed as P( A B) P( A B) =. (.3) P( B) Similarly one can wrie P( A B) P( B A) =. (.4) P( A)...3 Probabiliy for Inersecion of Evens From Equaion.4, one can wrie P ( A B) = P( A) P( B A). (.5) If A and B are independen evens hen he condiional probabiliy P(B A) is equal o P(B) only. Now Equaion.5 becomes simply he produc of probabiliy of A and probabiliy of B: P ( A B) = P( A) P( B). (.6)
Basic Reliabiliy Mahemaics Thus when A and B are independen, he probabiliy ha A and B occur ogeher is simply he produc of he probabiliies ha A and B occur individually. In general he probabiliy of occurrence of n dependen evens E, E,, E n is calculaed by he following expression: P( E E... En ) = P( E ) P( E E ) P( E3 E E )... P( En E E... En ). If all he evens are independen hen he probabiliy of join occurrence is simply he produc of individual probabiliies of evens: P E E... E ) = P( E ) P( E ) P( E )... P( E ). (.7) ( n 3 n...4 Probabiliy for Union of Evens Le A and B be wo evens. From he Venn diagram (Figure.6), as regions,, and 3 are muually exclusive, i follows ha PA ( B) = P() + P() + P(3), PA ( ) = P() + P(), PB ( ) = P() + P(3), which shows ha PA ( B) = PA ( ) + PB ( ) P(), as P() = P( A B), P( A B) = P( A) + P( B) P( A B). (.8) The above expression can be exended o n evens E, E,, E n by he following equaion: P( E E... E ) = P( E ) + P( E ) +... + P( E ) n [ PE ( E) + PE ( E ) +... + PE ( E)] 3 n + [ PE ( E E) + PE ( E E ) +... + PE ( E E)] 3 3 4 n n n n+ ( ) ( n ) PE E... E. n n (.9) A B 3 Figure.6 Venn diagram for A B
. Conceps of Probabiliy Theory 3...5 Toal Probabiliy Theorem Le A, A,, A n be n muually exclusive evens forming a sample space S and P (A i ) >, i =,,, n (Figure.7). For an arbirary even B one has B = B S = B ( A A... A ) = ( B A ) ( B A )... ( B A ) where he evens B A, B A,, B A n are muually exclusive: P( B Ai ) = n P ( B) = P( A ) P( B A ). (.) i This is called he oal probabiliy heorem. i i i n A 3 A 4 A 5 S A B B A 6 A A n Figure.7 Sample space conaining n muually exclusive evens...6 Bayes Theorem From he definiions of condiional probabiliy, PA ( B) PA B ( ) =, PB ( ) PA ( B) = PB ( ) PA B ( ) ( a), PA ( B) PB A ( ) =, PA ( ) PA ( B) = PA ( ) PB A ( ) ( b). Equaing boh (a) and (b) we have P ( B) P( A B) = P( A) P( B A). We can obain P(A B) as follows: P( A) P( B A) P( A B) =. (.) P( B)
4 Basic Reliabiliy Mahemaics This is a useful resul ha enables us o solve for P(A B) in erms of P(B A). In general, if P(B) is wrien using he oal probabiliy heorem, we obain he following general resul, which is known as Bayes heorem: P( Ai ) P( B Ai ) P( Ai B) =. (.) P( A ) P( B A ) i i i Bayes heorem presens a way o evaluae poserior probabiliies P(A i B) in erms of prior probabiliies P(A i ) and condiional probabiliies P(B A i ). This is very useful in updaing failure daa as more evidence is available from operaing experience. The basic conceps of probabiliy and saisics are explained in deail in [, ]...3 Random Variables and Probabiliy Disribuions I is imporan o represen he oucome from a random experimen by a simple number. In some cases, descripions of oucomes are sufficien, bu in oher cases, i is useful o associae a number wih each oucome in he sample space. Because he paricular oucome of he experimen is no known in advance, he resuling value of our variable is no known in advance. For his reason, random variables are used o associae a number wih he oucome of a random experimen. A random variable is defined as a funcion ha assigns a real number o each oucome in he sample space of a random experimen. A random variable is denoed by an uppercase leer and he numerical value ha i can ake is represened by a lowercase leer. For example, if X is a random variable represening he number of power ouages in a plan, hen x shows he acual number of ouages i can ake, say,,,,, n. Random variables can be classified ino wo caegories, namely, discree and coninuous random variables. A random variable is said o be discree if is sample space is counable. The number of power ouages in a plan in a specified ime is a discree random variable. If he elemens of he sample space are infinie in number and he sample space is coninuous, he random variable defined over such a sample space is known as a coninuous random variable. If he daa is counable hen i is represened wih a discree random variable, and if he daa is a measurable quaniy hen i is represened wih a coninuous random variable.
. Conceps of Probabiliy Theory 5..3. Discree Probabiliy Disribuion The probabiliy disribuion of a random variable X is a descripion of he probabiliies associaed wih he possible values of X. For a discree random variable, he disribuion is ofen specified by jus a lis of he possible values along wih he probabiliy of each. In some cases, i is convenien o express he probabiliy in erms of a formula. Le X be a discree random variable defined over a sample space S = {x, x, x n }. Probabiliy can be assigned o each value of S. I is usually denoed by f(x). For a discree random variable X, a probabiliy disribuion is a funcion such ha f( x ), n i= i f( x ) =, i f ( x ) = P( X = x ). i i Probabiliy disribuion is also known as probabiliy mass funcion. Some examples are binomial, Poisson, and geomeric disribuions. The graph of a discree probabiliy disribuion looks like a bar char or hisogram. For example, in five flips of a coin, where X represens he number of heads obained, he probabiliy mass funcion is shown in Figure.8. f(x).35.3.5..5..5 3 4 5 x (Number of Heads) Figure.8 A discree probabiliy mass funcion
6 Basic Reliabiliy Mahemaics.8.6 F(x).4. 3 4 5 x Figure.9 A discree CDF The cumulaive disribuion funcion (CDF) of a discree random variable X, denoed as F(x), is F ( x) = P( X x) = f ( x ). xi x i F(x) saisfies he following properies for a discree random variable X: F() x = P( X x) = f( x ), xi x Fx ( ), if x yhen Fx ( ) Fy ( ). i The CDF for he coin-flipping example is given in Figure.9...3. Coninuous Probabiliy Disribuions As he elemens of sample space for a coninuous random variable X are infinie in number, he probabiliy of assuming exacly any of is possible values is zero. Densiy funcions are commonly used in engineering o describe physical sysems. Similarly, a probabiliy densiy funcion (PDF) f(x) can be used o describe he
. Conceps of Probabiliy Theory 7 probabiliy disribuion of a coninuous random variable X. If an inerval is likely o conain a value for X, is probabiliy is large and i corresponds o large values for f(x). The probabiliy ha X is beween a and b is deermined as he inegral of f(x) from a o b. For a coninuous random variable X, a PDF is a funcion such ha f() x, + f() x =, Pa ( X b) = f( xdx. ) b a The CDF of a coninuous random variable X is x F( x) = P( X x) = f (θ ) dθ. (.3) The PDF of a coninuous random variable can be deermined from he CDF by differeniaing. Recall ha he fundamenal heorem of calculus saes ha d x f ( ) d f ( x) dx θ θ =. Now differeniaing F(x) wih respec o x and rearranging for f(x), df( x) f ( x) =. (.4) dx..3.3 Characerisics of Random Variables In order o represen he probabiliy disribuion funcion of a random variable, some characerisic values such as expecaion (mean) and variance are widely used. Expecaion or mean value represens he cenral endency of a disribuion funcion. I is mahemaically expressed as Mean = E() x = x f ( x ) for discree, i i i + = xf ( x) dx for coninuous.
8 Basic Reliabiliy Mahemaics A measure of dispersion or variaion of probabiliy disribuion is represened by variance. I is also known as he cenral momen or second momen abou he mean. I is mahemaically expressed as Variance = E x mean = x mean f x (( ) ) ( ) ( ) for discree, x + = ( x mean) f( x) dx for coninuous..3 Reliabiliy and Hazard Funcions Le T be a random variable represening ime o failure of a componen or sysem. Reliabiliy is he probabiliy ha he sysem will perform is expeced job under specified condiions of environmen over a specified period of ime. Mahemaically, reliabiliy can be expressed as he probabiliy ha ime o failure of he componen or sysem is greaer han or equal o a specified period of ime (): R( ) = P( T ). (.5) As reliabiliy denoes failure-free operaion, i can be ermed success probabiliy. Conversely, he probabiliy ha failure occurs before he ime is called failure probabiliy or unreliabiliy. Failure probabiliy can be mahemaically expressed as he probabiliy ha ime o failure occurs before a specified period of ime : R ( ) = P( T < ). (.6) As per he probabiliy erminology, R() variable T: is he same as he CDF of he random F()= R ( ) = P( T < ). (.7) Going by he firs axiom of probabiliy, he probabiliy of he sample space is uniy. The sample space for he coninuous random variable T is from o. Mahemaically, i is expressed as PS ( ) =, P( < T < ) =.
.3 Reliabiliy and Hazard Funcions 9 The sample space can be made ino wo muually exclusive inervals: one is T < and he second is T. Using he hird axiom of probabiliy, we can wrie P( < T < ) =, PT ( < T ) =, PT ( < ) + PT ( ) =. Subsiuing Equaions.5 and.7, we have F ( ) + R( ) =. (.8) As he ime o failure is a coninuous random variable, he probabiliy of T having exacly a precise will be approximaely zero. In his siuaion, i is appropriae o inroduce he probabiliy associaed wih a small range of values ha he random variable can ake on: P( < T < + Δ) = F( + Δ) F( ). The PDF f() for coninuous random variables is defined as P ( < T< +Δ) f() = L Δ Δ F ( +Δ) F ( ) = L Δ Δ df() = d dr() = (from Equaion.8). d From he above derivaion we have an imporan relaion beween R(), F(), and f(): df( ) dr( ) f ( ) = =. (.9) d d Given he PDF, f() (Figure.), hen F() = f() d, R () = f() d. (.)
3 Basic Reliabiliy Mahemaics The condiional probabiliy of a failure in he ime inerval from o ( + Δ) given ha he sysem has survived o ime is ( T + Δ T ) P R( ) R( + Δ) =. R( ) R( ) R( + Δ) Then R( ) Δ is he condiional probabiliy of failure per uni of ime (failure rae). R () R ( +Δ) [ R ( +Δ) R ()] λ() = lim = lim Δ R () Δ Δ Δ R () dr() f () = =. d R () R () (.) λ() is known as he insananeous hazard rae or failure rae funcion. Figure. Probabiliy disribuion funcion Reliabiliy as a funcion of hazard rae funcion can be derived as follows. We have he following relaions from he above expression: dr() λ() =, d R() dr() λ() d =. R ()
.4 Disribuions Used in Reliabiliy and Safey Sudies 3 Inegraing and simplifying, we have R( ) = exp λ ( θ ) dθ. (.).4 Disribuions Used in Reliabiliy and Safey Sudies This secion provides he mos imporan probabiliy disribuions used in reliabiliy and safey sudies. They are grouped ino wo caegories, namely, discree probabiliy disribuions and coninuous probabiliy disribuions..4. Discree Probabiliy Disribuions.4.. Binomial Disribuion Consider a rial in which he only oucome is eiher success or failure. A random variable X wih his rial can have success (X = ) or failure (X = ). The random variable X is said o be a Bernoulli random variable if he probabiliy mass funcion of X is given by PX ( = ) = p, PX ( = ) = p, where p is he probabiliy ha he rial is a success. Suppose now ha n independen rials, each of which resuls in a success wih probabiliy p and in a failure wih probabiliy p, are o be performed. If X represens he number of successes ha occur in he n rials, hen X is said o be a binomial random variable wih parameers n, p. The probabiliy mass funcion of binomial random variable is given by n i n i P( X = i) = c p ( p) i =,,,...,n. (.3) i The probabiliy mass funcion of a binomial random variable wih parameer (,.) is presened in Figure.. The CDF is given by i n j n j ( X i) = c j p ( p). (.4) j = P
3 Basic Reliabiliy Mahemaics f(x).35.3.5..5..5 3 4 5 6 7 8 9 x Figure. Binomial probabiliy mass funcion The mean of he binomial disribuion is calculaed as follows: Ex () = xf() x n n i n i i ci p ( p) i= n n i n i np ci p ( p) i= m m j m j np c j p ( p) j = = = = = np. Similarly, variance can also be derived as Var iance = npq. Example I is known from experience ha 4% of hard disks produced by a compuer manufacurer are defecive. Ou of 5 disks esed, wha is he probabiliy (i) of having zero defecs; (ii) ha all are defecive? Soluion: q = 4% of hard disks produced by a compuer manufacurer are defecive. We know ha p + q =, p = q =.4, p =.96.
.4 Disribuions Used in Reliabiliy and Safey Sudies 33 According o he binomial disribuion, P(X = x) = n cx p x q n x. Now: (i) In he case of zero defecs, i.e., p(x = ), P(X = ) = n cx p x n x q 5 C (.4) (.96) = ( 5 ) =.99. (ii) In he case ha all are defecive, i.e., p(x = 5), P(X = 5) = n cx p x n x q = 5 5 C (.4) 5 (.96) ( 5 5) =.87. Or pu anoher way, P(X = 5) = P(X = ) =.99 =.87. Example To ensure high reliabiliy, riple modular redundancy (which demands ha a leas ou of 3 insrumens be successful) is adoped in insrumenaion sysems of a nuclear power plan. I is known ha he failure probabiliy of each insrumenaion channel from operaing experience is.. Wha is he probabiliy of success of he whole insrumenaion sysem? Soluion: q = failure probabiliy from operaion experience is.. We know ha p = q =. =.99. According o he binomial disribuion, P(X = x) = n p x q n x. cx The sample space is hen developed as in Table.4. Table.4 Calculaions Formula Numerical soluions Value (i) (ii) (iii) (iv) P(X = ) = n p x n x q cx P(X = ) = n p x n x q cx P(X = ) = n p x n x q cx P(X = 3) = n p x n x q cx P() = 3 C (.99) ( 3 ) (.) P() = E 6 P() = 3 C (.99) ( 3 ) (.) P() =.9E 4 P() = 3 C (.99) ( 3 ) (.) P() =.9E P() = 3 C (.99) 3 ( 3 3) (.) 3 P(3) =.97 Now he failure probabiliy is he sum of (i) and (ii), which is obained as.98e 4, and he success probabiliy is he sum of (iii) and (iv), which is obained as.9997.
34 Basic Reliabiliy Mahemaics.4.. Poisson Disribuion The Poisson disribuion is useful o model when he even occurrences are discree and he inerval is coninuous. For a rial o be a Poisson process, i has o saisfy he following condiions: The probabiliy of occurrence of one even in ime Δ is λδ where λ is consan. The probabiliy of more han one occurrence is negligible in inerval Δ. Each occurrence is independen of all oher occurrences. A random variable X is said o have a Poisson disribuion if he probabiliy disribuion is given by λ x e ( λ) f () x = x =,,,... (.5) x! λ is known as he average occurrence rae and x is he number of occurrences of Poisson evens. The CDF is given by x F( x) = f ( X = i). (.6) i= The probabiliy mass funcion and CDF for λ =.5/year and = year are shown in Figure.. Boh he mean and variance of he Poisson disribuion are λ. If he probabiliy of occurrence is near zero and sample size very large, he Poisson disribuion may be used o approximae he binomial disribuion. Example 3 If he rae of failure for an iem is wice a year, wha is he probabiliy ha no failure will happen over a period of years? Soluion: Rae of failure, denoed as λ = /year. Time = years. The Poisson probabiliy mass funcion is expressed as λ x e ( λ) f ( x) =. x! In a case of no failures, x =, which leads o e ( ) f ( X = ) = =.83.!
.4 Disribuions Used in Reliabiliy and Safey Sudies 35 f(x).4.35.3.5..5..5 3 4 5 6 7 8 9 x (a) F(x).8.6.4. 3 4 5 x (b) Figure. Probabiliy funcions for Poisson disribuion: (a) probabiliy mass funcion and (b) CDF.4..3 Hypergeomeric Disribuion The hypergeomeric disribuion is closely relaed o he binomial disribuion. In he hypergeomeric disribuion, a random sample of n iems is chosen from a finie populaion of N iems. If N is very large wih respec o n, he binomial disribuion is a good approximaion of he hypergeomeric disribuion. The random variable X denoes x successes in he random sample of size n from populaion N conaining k iems labeled success. The hypergeomeric disribuion probabiliy mass funcion is f(x) = p(x, N, n, k) = K cx N k Cn x N C n, x =,,, 3, 4,, n. (.7)
36 Basic Reliabiliy Mahemaics The mean of he hypergeomeric disribuion is E(x) = nk N. (.8) The variance of he hypergeomeric disribuion is V(x) = nk N K N N n N. (.9).4..4 Geomeric Disribuion In he case of binomial and hypergeomeric disribuions, he number of rials n is fixed and number of successes is a random variable. The geomeric disribuion is used if one is ineresed in number of rials required o obain he firs success. The random variable in a geomeric disribuion is he number of rials required o ge he firs success. The geomeric disribuion probabiliy mass funcion is x f(x) = P(x; p) = p( p), x =,, 3,, n, (.3) where p is he probabiliy of success on a rial. The mean of he geomeric disribuion is E(x) = p. The variable of he geomeric disribuion is p V(x) =. p The geomeric disribuion is he only discree disribuion which exhibis he memoryless propery, as does he exponenial disribuion in he coninuous case.
.4 Disribuions Used in Reliabiliy and Safey Sudies 37.4. Coninuous Probabiliy Disribuions.4.. Exponenial Disribuion The exponenial disribuion is mos widely used disribuion in reliabiliy and risk assessmen. I is he only disribuion having consan hazard rae and is used o model he useful life of many engineering sysems. The exponenial disribuion is closely relaed o he Poisson disribuion, which is discree. If he number of failures per uni ime is a Poisson disribuion hen he ime beween failures follows an exponenial disribuion. The PDF of he exponenial disribuion is f () = λe λ for, = for <. (.3) The exponenial PDFs are shown in Figure.3 for differen values of λ. 3.5 f().5 λ =.5 λ = λ = 3.5 3 4 5 6 Figure.3 Exponenial PDFs The exponenial CDF can be derived from is PDF as follows: λ λ λ e e f ( ) d = λe d = λ = λ = λ λ λ F( ) = e λ. (.3) The reliabiliy funcion is he complemen of he CDF: R λ ( ) = F( ) = e. (.33)
38 Basic Reliabiliy Mahemaics The exponenial reliabiliy funcions are shown in Figure.4 for differen values of λ..8.6 λ =.5 λ = λ = 3 R().4. 3 4 5 6 Figure.4 Exponenial reliabiliy funcions The hazard funcion is he raio of he PDF and is reliabiliy funcion; for he exponenial disribuion i is λ f ( ) λe h( ) = = = λ. (.34) λ R( ) e The exponenial hazard funcion is he consan λ. This is he reason for he memoryless propery for he exponenial disribuion. The memoryless propery means he probabiliy of failure in a specific ime inerval is he same regardless of he saring poin of ha ime inerval. Mean and Variance of Exponenial Disribuion E () = f() = λ e d. λ
.4 Disribuions Used in Reliabiliy and Safey Sudies 39 Using he inegraion by pars formula ( = vdu uv udv ) () e e E d e. λ λ λ λ λ λ λ λ λ λ λ λ λ = = + = = Thus mean ime o failure of he exponenial disribuion is he reciprocal of he failure rae: Variance() = E(T ) (mean), = = ) ( ) ( d e d f T E λ λ. Using he inegraion by pars formula, ( ) ( ) e e ET d e d. λ λ λ λ λ λ λ λ λ = = + Bu he inegral erm in he above expression is E(T), which is equal o /λ; subsiuing he same, = ) ( T E λ λ λ λ = +. Now he variance is λ λ λ = = Variance. (.35) Example 4 The failure ime (T) of an elecronic circui board follows an exponenial disribuion wih failure rae λ = 4 /h. Wha is he probabiliy ha (i) i will fail before h; (ii) i will survive a leas, h; (iii) i will fail beween h and, h? Deermine (iv) he mean ime o failure and (v) he median ime failure also.
4 Basic Reliabiliy Mahemaics Soluion: (i) PT ( < ) = FT ( = ). λ For he exponenial disribuion F( T ) = e and subsiuing λ = 4 /h, λ PT ( < ) = e =. 956. (ii) P ( T > ) = R( T = ). λ For he exponenial disribuion R( T ) = e and subsiuing λ = 4 /h, λ P( T > ) = e =.3678. (iii) P( < T < ) = F() F() = [ R()] F(). From (i), we have F() =.956 and from (ii) we have R() =.3678, P( < T < ) = [.3678].956 =.537. (iv) Mean ime o failure = /λ = / 4 =, h. (v) Median ime o failure denoes he poin where 5% failures have already occurred; mahemaically i is R( T ) =.5 λ e =.5 Applying logarihms on boh sides and solving for, = ln(5). = 69347h.. λ.4.. Normal Disribuion The normal disribuion is he mos imporan and widely used disribuion in he enire field of saisics and probabiliy. I is also known as he Gaussian disribuion and i is he very firs disribuion, inroduced in 733. The normal disribuion ofen occurs in pracical applicaions because he sum of a large number of saisically independen random variables converges o a normal disribuion (known as he cenral limi heorem). The normal disribuion can be used o represen wear-ou regions of a bah-ub curve where faigue and aging can be modeled. I is also used in sress-srengh inerference models in reliabiliy sudies. The PDF of normal disribuions is f μ () = σ e,. (.36) σ π where μ and σ are parameers of he disribuion. The disribuion is bell-shaped and symmerical abou is mean wih he spread of disribuion deermined by σ. I is shown in Figure.5.
.4 Disribuions Used in Reliabiliy and Safey Sudies 4.8.6 μ = ; σ = μ = ; σ =.5 μ = ; σ =.5 f().4. -5-4 -3 - - 3 4 5 Figure.5 Normal PDFs The normal disribuion is no a rue reliabiliy disribuion since he random variable ranges from o +. Bu if he mean μ is posiive and is larger han σ by severalfold, he probabiliy ha random variable T akes negaive values can be negligible and he normal can herefore be a reasonable approximaion o a failure process. The normal reliabiliy funcion and CDF are R( ) = e σ π μ σ d, (.37) F( ) = e σ π μ σ d. (.38) As here is no closed-form soluion o hese inegrals, he reliabiliy and CDF are ofen expressed as a funcion of he sandard normal disribuion (μ = and σ = ) (Figure.6). Transformaion o he sandard normal disribuion is achieved wih he expression μ z =, σ
4 Basic Reliabiliy Mahemaics The CDF of z is given by z z φ ( z) = e dz. (.39) π.8 μ = ; σ = μ = ; σ =.5 μ = ; σ =.5.6.4 F(). -5-4 -3 - - 3 4 5 Figure.6 Normal CDFs Table A. (see Appendix) provides he cumulaive probabiliy of he sandard normal disribuion. This can be used o find he cumulaive probabiliy of any normal disribuion. However, hese ables are becoming unnecessary, as elecronic spreadshees (e.g., Microsof Excel) have buil-in saisic funcions. 8 μ = ; σ = μ = ; σ =.5 μ = ; σ =.5 6 4 8 6 4 H() -5-3 - 3 5 Figure.7 Normal hazard rae funcions
.4 Disribuions Used in Reliabiliy and Safey Sudies 43 The hazard funcion can expressed as f ( ) f ( ) h( ) = =. (.4) R( ) Φ( z) The hazard funcion is an increasing funcion, as shown in Figure.7. This feaure makes i suiable o model aging componens. Example 5 Failure imes are recorded from he life esing of an engineering componen as 85, 89, 9, 955, 98, 5, 36, 47, 65, and. Assuming a normal disribuion, calculae he insananeous failure rae a h. Soluion: Given daa are n =, N =. Mean = x xi 9889 = = = 988. 9. n Now, he sample sandard deviaion (σ ) is σ n n n xi ( x ) i= i i= = = nn ( ) 84. 8455. (Calculaions are given in Table.5.) The insananeous failure rae is given by he hazard funcion, and is esablished by f ( ) f () φ( z).4669 h( ) = = = = =.4. R( ) R() Φ( z).55 Table.5 Calculaions x i x i 85 75 89 79 9 8484 955 95 98 964 5 565 36 7396 47 969 65 345 544 x i = 9889 x i
44 Basic Reliabiliy Mahemaics.4..3 Lognormal Disribuion A coninuous posiive random variable T has a lognormal disribuion if is naural logarihm is normally disribued. The lognormal disribuion can be used o model he cycles o failure for meals, he life of ransisors and bearings, and repair imes. I appears ofen in acceleraed life esing as well as when a large number of saisically independen random variables are muliplied. The lognormal PDF is f σ () = e, > σ π ln μ, (.4) where μ and σ are known as he locaion parameer and shape parameer, respecively. The shape of disribuion changes wih differen values of σ as shown in Figure.8..4E-.E-.E- μ = ; σ =.5 μ = ; σ = μ = ; σ = 3 8.E- f() 6.E- 4.E-.E-.E+ 3 4 5 Figure.8 Lognormal PDFs The lognormal reliabiliy funcion and CDF are ln μ R( ) = Φ, (.4) σ ln μ F( ) = Φ. (.43) σ
.4 Disribuions Used in Reliabiliy and Safey Sudies 45 Lognormal failure disribuion funcions and lognormal hazard funcions are shown in Figures.9 and.. The mean of he lognormal disribuion is σ μ + E ( ) = e (.44) and he variance is (μ+ σ ( ) ) σ V = e ( e ). (.45).8.6 μ = ; σ =.5 μ = ; σ = μ = ; σ = 3 F().4. 3 4 5 Figure.9 Lognormal CDFs.3.5..5 H(). μ = ; σ =.5 μ = ; σ = μ = ; σ = 3.5 3 4 5 Figure. Lognormal hazard funcions
46 Basic Reliabiliy Mahemaics Example 6 Deermine he mean and variance of ime o failure for a sysem having lognormally disribued failure ime wih µ = 5 years and σ =.8. Soluion: The mean of he lognormal disribuion is σ μ + E ( ) = e. 8. 5+ = e = 4. 3839. The variance of he lognormal disribuion is (μ+ σ ( ) ) σ V = e ( e ), ( + (8) ) 8 = e. ( e. ), = 37448. 49..4..4 Weibull Disribuion The Weibull disribuion was inroduced in 933 by P. Rosin and E. Rammler [3]. I has a wide range of applicaions in reliabiliy calculaions due o is flexibiliy in modeling differen disribuion shapes. I can be used o model ime o failure of lamps, relays, capaciors, germanium ransisors, ball bearings, auomobile ires, and cerain moors. In addiion o being he mos useful disribuion funcion in reliabiliy analysis, i is also useful in classifying failure ypes, roubleshooing, scheduling prevenive mainenance, and inspecion aciviies. The Weibull PDF is β β α β f ( ) = e, >, (.46) α α.e-.5e- β =.5 β = β = β=.5 β=3.6 f().e- 5.E-.E+ 5 5 Figure. Weibull PDF
.4 Disribuions Used in Reliabiliy and Safey Sudies 47 where α and β are known as he scale parameer (or characerisic life) and he shape parameer, respecively. An imporan propery of he Weibull disribuion is as β increases, he mean of he disribuion approaches α and he variance approaches zero. Is effec on he shape of he disribuion can be seen in Figure. wih differen values of β (α = is assumed in all he cases). I is ineresing o see from Figure. ha all are equal o or approximaely maching wih several oher disribuions. Due o his flexibiliy, he Weibull disribuion provides a good model for much of he failure daa found in pracice. Table.6 summarizes his behavior. Table.6 Disribuions wih differen values of β β Remarks Idenical o exponenial Idenical o Rayleigh.5 Approximaes lognormal 3.6 Approximaes normal Weibull reliabiliy and CDF funcions are R ) β α ( = e, (.47) β α F ( ) = e. (.48) Reliabiliy funcions wih differen values of β are shown in Figure...E+ 8.E- 6.E- R() 4.E- β =.5 β = β = β=.5 β=3.6.e-.e+ 5 5 Figure. Weibull reliabiliy funcions
48 Basic Reliabiliy Mahemaics.E+ 8.E- 6.E- H() 4.E- β =.5 β = β = β=.5 β=3.6.e-.e+ 5 5 Figure.3 Weibull hazard funcions The Weibull hazard funcion is β β H ( ) =. (.49) β α The effecs of β on he hazard funcion are demonsraed in Figure.3. All hree regions of bah-ub curve can be represened by varying he β value: β < resuls in decreasing failure rae (burn-in period). β = resuls in consan failure rae (useful life period). β > resuls in increasing failure rae (wear-ou period). The mean value of he Weibull disribuion can be derived as follows: β β β α Mean= f () d = e d. α α Le β x =, α β dx = α α Now Mean = e y β dy. β Since = α x, d.
.4 Disribuions Used in Reliabiliy and Safey Sudies 49 β x Mean = α ( x) e dx where Γ ( x) is known as he gamma funcion: x y Γ ( x) = y e dy. Similarly variance can be derived as = α Γ + β, (.5) = Γ + Γ σ α +. (.5) β β Example 7 The failure ime of a componen follows a Weibull disribuion wih shape parameer β =.5 and scale parameer =, h. When should he componen be replaced if he minimum recurring reliabiliy for he componen is.95? Soluion: Subsiuing ino he Weibull reliabiliy funcion gives β ( ) α R () = e,.5.5 ( ) ( ) 95. = e = e. 95. Taking naural logarihms on boh sides,.5 ln = ( ).95. Taking logs on boh sides, log.593.899 =.5log = log.5.85996 = log log log.85996 = log = 38.38 h..4..5 Gamma Disribuion As he name suggess, he gamma disribuion derives is name from he wellknown gamma funcion. I is similar o he Weibull disribuion, where by varying he parameer of he disribuion a wide range of oher disribuions can be derived. The gamma disribuion is ofen used o model lifeimes of sysems. If an even akes place afer n exponenially disribued evens ake place sequenially,
5 Basic Reliabiliy Mahemaics he resuling random variable follows a gamma disribuion. Examples of is applicaion include he ime o failure for a sysem consising of n independen componens, wih n componens being sandby componens; ime beween mainenance acions for a sysem ha requires mainenance afer a fixed number of uses; ime o failure of sysem which fails afer n shocks. The gamma PDF is β =Γ =, Γ α α β f ;, e, () ( αβ) where Γ ( α ) = ( α ) α x x e dx. (.5) where α and β are parameers of disribuion. The PDF wih parameer β = is known as a sandardized gamma densiy funcion. By changing he parameerα, differen well-known disribuions can be generaed as shown in Figure.4 and Table.7. The CDF of random variable T having a gamma disribuion wih parameers α and β is given by F () = P( T < ) β = Γ α ( α ) e α β d. (.53) The gamma CDF in general does no have a closed-form soluion. However, ables are available giving he values of CDFs having a sandard gamma disribuion funcion...8 α =.5 α = α = α = 3 f().4 3 4 5 Figure.4 Gamma PDFs
.4 Disribuions Used in Reliabiliy and Safey Sudies 5 Table.7 Disribuion wih differen values of α α Disribuion α = Exponenial α = ineger Erlangian α = Chi-square α > Normal The mean of he gamma disribuion is α E ( T ) = (.54) β and he variance is α V ( T ) =. (.55) β For ineger values of α, he gamma PDF is known as he Erlangian PDF..4..6 Erlangian Disribuion The Erlangian disribuion is a special case of he gamma disribuion where α is an ineger. In his case PDF is expressed as f () = α β α ( α )! e β. (.56) The Erlangian reliabiliy funcion is e α β R. (.57) k () = k =! k β The hazard funcion is α h () =. (.58) k α α Γ β β k! ( ) α k =
5 Basic Reliabiliy Mahemaics.5.5 α =.5 α = α = 3 h().5 3 4 5 Figure.5 Erlangian hazard funcions By changing he value of α, all hree phases of bah-ub curves can be seleced (Figure.5). If α <, failure rae is decreasing, α =, failure rae is consan, and α >, failure rae is increasing..4..7 Chi-square Disribuion A special case of he gamma disribuion, wih α = and β = /ν, is he chisquare (χ ) disribuion, which is used as a deerminan of goodness of fi and confidence limis. The chi-square PDF is χ (x, v) = f(x) = v / x Γ( v / ) ( v / ) x / e, x >. (.59) The shape of he chi-square disribuion is shown in Figure.6. The mean of he chi-square disribuion is E(x) = ν and he variance is V(x) = ν. If x, x,, x n are independen, sandard normally disribued variables, hen he sum of squares of a random variable (i.e., X + X + + X ν ) has a chisquare disribuion wih ν degrees of freedom.
.4 Disribuions Used in Reliabiliy and Safey Sudies 53.5..5..5.3 3 5 7 9 3 5 x f(x) v= v=6 v= Figure.6 Chi-square PDF I is ineresing o noe ha he sum of wo or more independen chi-square variables is also a chi-square variable wih degrees of freedom equal o he sum of degrees of freedom for he individual variable. As ν becomes large, he chisquare disribuion approaches normal wih mean ν and variance ν..4..8 F-disribuion If χ and χ are independen chi-square random variables wih v and v degrees of freedom, hen he random variable F defined by F= v v χ χ (.6) is said o have an F-disribuion wih v and v degrees of freedom. The PDF of random variable F is given by f(f)=, + Γ Γ + Γ + v v v v v F v F v v v v v v F >. (.6)
54 Basic Reliabiliy Mahemaics f(f) v=8,v=6 v= 6,v=8 v= 5,v=5 3 4 5 F Figure.7 F PDFs wih differen v and v Figure.7 shows F PDFs wih differen v and v. The values of he F-disribuion are available from ables. If f α (v, v ) represens he area under he F PDF, wih degrees of freedom v and v, o he righ of α, hen F (v,v ) = α. (.6) F α ( v, v ) I is ineresing o observe ha if s and s are he variance of independen random samples of size n and n drawn from a normal populaion wih variance of σ and σ respecively, hen he saisic s / σ F = s / σ σ s = σ s has an F-disribuion wih v = n and v = n degrees of freedom. (.63).4..9 -disribuion If Z is a normally disribued random variable and he independen random variable χ follows a chi-square disribuion wih v degrees of freedom hen he random variable defined by = z χ / v (.64) is said o have a -disribuion wih v degrees of freedom.
.4 Disribuions Used in Reliabiliy and Safey Sudies 55 Table.8 Summary of applicaion areas Disribuion Poisson Binomial Exponenial Weibull Lognormal Normal Gamma Chi-square F Areas of applicaion in reliabiliy sudies To model occurrence raes such as failures per hour or defecs per iem (defecs per compuer chip or defecs per auomobile) To model K ou of M or voing redundancy such as riple modular redundancies in conrol and insrumenaion To model useful life of many iems Life disribuion of complex non-repairable sysems β > ofen occurs in applicaions as failure ime of componens subjeced o wear-ou and/or faigue (lamps, relays, mechanical componens) Scheduling inspecion and prevenive mainenance aciviies To model he cycles o failure for meals, he life of ransisors, he life of bearings. Size disribuion of pipe breaks To model repair ime Prior parameer disribuion in Bayesian analysis Modeling buildup of olerances Load-resisance analysis (sress-srengh inerference) Life disribuion of high-sress componens To model ime o failure of sysem wih sandby unis To model ime beween mainenance acions Prior parameer disribuion in Bayesian analysis Couning he number of failures in an inerval Applicaions involving goodness of fi and confidence limis To make inferences abou variances and o consruc confidence limis To draw inferences concerning means and o consruc confidence inervals for means when he variance is unknown The PDF of is given by f() = v + Γ + Γ( v/) Πv v ( v ) +,. (.65)
56 Basic Reliabiliy Mahemaics Like he sandard normal densiy, he -densiy is symmerical abou zero. In addiion, as v becomes larger, i becomes more and more like he sandard normal densiy. Furher, E() = and v() = v/(v ) for v >..4.3 Summary The summary of applicaions of he various disribuions is described in Table.8..5 Failure Daa Analysis The credibiliy of any reliabiliy/safey sudy depends upon he qualiy of he daa used. This secion deals wih he reamen of failure daa and subsequen usage in reliabiliy/safey sudies. The derivaion of sysem reliabiliy models and various reliabiliy measures is an applicaion of probabiliy heory, whereas he analysis of failure daa is primarily an applicaion of saisics. The objecive of failure daa analysis is o obain reliabiliy and hazard rae funcions. This is achieved by wo general approaches. The firs is deriving empirical reliabiliy and hazard funcions direcly from failure daa. These mehods are known as nonparameric mehods or empirical mehods. The second approach is o idenify an approximae heoreical disribuion, esimae he parameer(s) of disribuion, and perform a goodness-of-fi es. This approach is known as he parameric mehod. Boh approaches are explained in his secion..5. Nonparameric Mehods In hese mehods empirical reliabiliy disribuions are direcly derived from he failure daa. The sources of failure daa are generally from () operaional or field experience and/or () failures generaed from reliabiliy esing. Nonparameric mehods are useful for preliminary daa analysis o selec appropriae heoreical disribuion. They also find applicaion when no parameric disribuion adequaely fis he failure daa. Consider life ess on a cerain uni under exacly he same environmen condiions wih number of unis N ensuring ha failures of he individual unis are independen and do no affec each oher. A some predeermined inervals of ime, he number of failed unis is observed. I is assumed ha es is carried ou unil all he unis have failed. Le us now analyze he informaion colleced hrough his es.
.5 Failure Daa Analysis 57 From he classical definiion of probabiliy, he probabiliy of occurrence of an even A can be expressed as follows: PA ( ) n s = = s n N n + n s f, (.66) where n s is he number of favorable oucomes, n f is he number of unfavorable oucomes, and N is he oal number of rials = n s + n f. When N unis are esed, le us assume ha n s () unis survive he life es afer ime and ha n f () unis have failed over he ime. Using he above equaion, he reliabiliy of such a uni can be expressed as R () n () ns(). (.67) N n () + n () s = = s f This definiion of reliabiliy assumes ha he es is conduced over a large number of idenical unis. The unreliabiliy Q() of he uni is he probabiliy of failure over ime, equivalen o he CDF and is given by F(), n () f Q () F () =. (.68) N We know ha he derivaive of he CDF of a coninuous random variable gives he PDF. In reliabiliy sudies, he failure densiy funcion f() associaed wih failure ime of a uni can be defined as follows: df() dq() dnf nf( +Δ) nf( ) f() = = = lim d d N d N Δ Δ. (.69) The hazard rae can be derived from Equaion. by subsiuing f() and R() as expressed below ( ) ( ) h () = lim n. (.7) nf +Δ nf s () Δ Δ Equaions.67,.69, and.7 can be used for compuing reliabiliy, failure densiy, and hazard funcions from he given failure daa. The preliminary informaion on he underlying failure model can be obained if we plo he failure densiy, hazard rae, and reliabiliy funcions agains ime. We can define piecewise coninuous funcions for hese hree characerisics by selecing some small ime inerval Δ. This discreizaion evenually in he limi-
58 Basic Reliabiliy Mahemaics ing condiions i.e., Δ or when he daa is large, would approach he coninuous funcion analysis. The number of inervals can be decided based on he range of daa and accuracy desired. Bu he higher he number of inervals, he beer would be he accuracy of resuls. However, he compuaional effor increases considerably if we choose a large number of inervals, bu here exis an opimum number of inervals given by Surges [4], which can be used o analyze he daa. If n is he opimum number of inervals and N is he oal number of failures, hen n = + 33log. ( N). (.7) Example 8 To ensure proper illuminaion in conrol rooms, higher reliabiliy of elecric lamps is necessary. Le us consider ha he failure imes (in hours) of a populaion of 3 elecric lamps from a conrol room are given in Table.9. Calculae failure densiy, reliabiliy, and hazard funcions. Soluion: Table.9 Failure daa Lamp Failure ime Lamp Failure ime Lamp Failure ime 3 4 5 6 7 8 9 5495.5 887.7 539.66 53. 8887 435.6 99.33 376.4 55.56 55.75 3 4 5 6 7 8 9 35.4 6893.8 853.83 3458.4 77.78 34.6 866.69 63.47 395.6 97.4 3 4 5 6 7 8 9 3 437. 933.79 485.66 458. 653.43 8367.9 9.4 3576.97 843.38 4653.99 The opimum number of inervals as per Surge s formula (Equaion.7) wih N = 3 is n = + 3.3 log(3) = 5.87. In order o group he failure imes under various inervals, he daa is arranged in increasing order. Table. shows he daa wih ascending order of failure imes. The minimum and maximum of failure ime is 99.33 and 8,887 respecively.
.5 Failure Daa Analysis 59 Table. Daa in ascending order Bulb Failure ime Bulb Failure ime Bulb Failure ime 3 4 5 6 7 8 9 99.33 34.6 539.66 55.75 866.69 97.4 933.79 485.66 843.38 853.83 3 4 5 6 7 8 9 9.4 53. 435.6 395.6 3458.4 35.4 376.4 437. 458. 4653.99 3 4 5 6 7 8 9 3 5495.5 63.47 653.43 6893.8 77.78 8367.9 887.7 55.56 3576.97 8887 8887 99.33 Inerval size = Δ i = = 33.7 35. 6 We can now develop a able showing he inervals and corresponding values of R(), F(), f(), and h() respecively. The following noaion is used. The summary of calculaions is shown in Table.. n s ( i ) number of survivors a he beginning of he inerval; n f ( i ) number of failures during ih inerval. The plos of f() and h() are shown in Figures.8 and.9, whereas he plos of R() and F() are given in Figure.3. Table. Calculaions Inerval n s ( i ) n f ( i ) R( i ) F( i ) n ( ) f i f( i ) = NΔ i nf ( i ) h ( i ) = n ( ) Δ 35 3 4.48E 4.48E 4 35 6 7.53.47 7.4E 5.38E 4 63 63 9 6.3.7 6.35E 5.E 4 945 945 3..9.6E 5.5E 4 6 6.66.934.6E 5.58E 4 575 575 89.33.967.6E 5 3.7E 4 s i i
6 Basic Reliabiliy Mahemaics.6E-4.4E-4.E-4.E-4 f() 8.E-5 6.E-5 4.E-5.E-5.E+ 35 63 945 6 575 89 5 Figure.8 Failure densiy funcion 3.5E-4 3.E-4.5E-4 h().e-4.5e-4.e-4 5.E-5.E+ 35 63 945 6 575 89 5 Figure.9 Hazard rae funcion.8 R(), F().6.4 R() F(). 35 63 945 6 575 89 5 Time Figure.3 Reliabiliy funcion/cdf
.5 Failure Daa Analysis 6.5. Parameric Mehods The preceding secion discussed mehods for deriving empirical disribuions direcly from failure daa. The second, and usually preferred, mehod is o fi a heoreical disribuion, such as he exponenial, Weibull, or normal disribuions. As heoreical disribuions are characerized wih parameers, hese mehods are known as parameric mehods. Nonparameric mehods have cerain pracical limiaions compared wih parameric mehods. As nonparameric mehods are based on sample daa, informaion beyond he range of daa canno be provided. Exrapolaion beyond he censored daa is possible wih a heoreical disribuion. This is significan in reliabiliy/safey sudies as he ails of he disribuion arac more aenion. The main concern is deermining he probabilisic naure of he underlying failure process. The available failure daa may be simply a subse of he populaion of failure imes. Esablishing he disribuion he sample came from and no he sample iself is he focus. The failure process is ofen a resul of some physical phenomena ha can be associaed wih a paricular disribuion. Handling a heoreical model is easy in performing complex analysis. In he parameric approach, fiing a heoreical disribuion consiss of he following hree seps:. idenifying candidae disribuions;. esimaing he parameers of disribuions; 3. performing a goodness-of-fi es. All hese seps are explained in he following secions..5.. Idenifying Candidae Disribuions In he earlier secion on nonparameric mehods, we have seen how one can obain empirical disribuions or hisograms from he basic failure daa. This exercise helps one o guess a failure disribuion ha can be possibly employed o model he failure daa. Bu nohing has been said abou an appropriae choice of he disribuion. Probabiliy plos provide a mehod of evaluaing he fi of a se of daa o a disribuion. A probabiliy plo is a graph in which he scales have been changed in such a manner ha he CDF associaed wih a given family of disribuions, when represened graphically on ha plo, becomes a sraigh line. Since sraigh lines are easily idenifiable, a probabiliy plo provides a beer visual es of a disribuion han comparison of a hisogram wih a PDF. Probabiliy plos provide a quick mehod o analyze, inerpre, and esimae he parameers associaed wih a model.
6 Basic Reliabiliy Mahemaics Probabiliy plos may also be used when he sample size is oo small o consruc hisograms and may be used wih incomplee daa. The approach o probabiliy plos is o fi a linear regression line of he form menioned below o a se of ransformed daa: y = mx + c. (.7) The naure of he ransform will depend on he disribuion under consideraion. If he daa of failure imes fi he assumed disribuion, he ransformed daa will graph as a sraigh line. λ Consider an exponenial disribuion whose CDF is F( ) = e ; rearranging λ o F( ) = e, and aking he naural logarihm of boh sides, λ ln( F ( )) = ln( e ), ln( F ( )) = λ, ln( ) = λ. F ( ) Comparing i wih Equaion.7: y = mx + c, we have y = ln( ), F ( ) m = λ; x = ; c =. Now if y is ploed on he ordinae, he plo would be a sraigh line wih a slope of λ. The failure daa is generally available in erms of he failure imes of n iems ha have failed during a es conduced on he original populaion of N iems. Since F() is no available, we can make use of E[F( i )]: n i E[ F( )] =. (.73) i i= N + Example 9 Table. gives a chronological sequence of he grid supply ouages a a process plan. Using a probabiliy ploing mehod, idenify he possible disribuions. Soluion: Table.3 gives he summary of calculaions for x and y coordinaes. These are ploed in Figure.3.
.5 Failure Daa Analysis 63 Table. Class IV power failure occurrence ime since..998 Failure number Dae/ime Time o failure (days) Time beween failure (days).4.998/4:35 7.6.998/:3 68 67 3 4.7.998/9:9 5 37 4 3.8.999/:3 59 385 5 7.8.999 64 4 6..999 7 7 7.. 763 4 8.5./5:38 88 9 9 7../5:56 6 79 4.5. 5 9 3.7./9:45 3 5.7./8:5 674 374 3 9.5.3/8:43 976 3 4 8..5 94 964 5.5.6/: 365 5 6 7.5.7/: 3445 38 7.6.7/6:3 346 6 Table.3 Time beween failure (TBF) values for ouage of class IV (for Weibull ploing) i Failure number TBF, (days) F() = (i.3)/(n +.4) y = ln(ln(/r()) x = ln() 5 4.43 3.968.63957 7 6.977.7488.77589 3 3 37.557.789 3.698 4 7 4.644.4398 3.73767 5 5.75.556 3.93 6 67.37586.94 4.4693 7.38557.78 4.65 8 6 7.4459.5376 4.7674 9 8 9.5.3665 4.7793 5 5.55747.46 4.8834 9 79.64943.467 5.87386 9.6744.9754 5.474 3 3 3.79885.6993 5.77 4 374.787356.43753 5.9456 5 6 38.84488.635 5.947 6 4 385.999.8448 5.95343 7 4 964.95977.675 6.879
64 Basic Reliabiliy Mahemaics ln(ln(/r()) - - -3 4 6 8 y =.996x - 5.748-4 -5-6 ln Figure.3 Weibull ploing for he daa The plo is approximaed o a sraigh line: y =.996x 5.748. The shape parameer α =.996; scale parameer, β = e 5.748 = 94.4 days. As he shape parameer is close o uniy, he daa fis an exponenial disribuion. Table.4 summarizes (x, y) coordinaes of various disribuions used in probabiliy ploing. Table.4 Coordinaes of disribuions for probabiliy ploing Disribuion (x, y) y =mx+c Exponenial λ F( ) = e m = λ, ln ˆ c = F( ) Weibull m = α ( β ln, ln ln α F( ) = e ) ˆ c = ln( / β ) F( ) Normal μ F ( ) = Φ σ (, [ F( ) ]) Φ m = σ μ c = σ
.5 Failure Daa Analysis 65.5.. Esimaing he Parameers of Disribuion The preceding secion on probabiliy ploing focused on he idenificaion of a disribuion for a se of daa. Specificaion of parameers for he idenified disribuion is he nex sep. The esimaion of parameers of he disribuion by probabiliy ploing is no considered o be he bes mehod. This is especially rue in cerain goodness-of-fi ess ha are based on he maximum likelihood esimaor (MLE) for he disribuion parameers. There are many crieria based on which an esimaor can be compued, viz., leas-squares esimaion and MLE. The MLE provides maximum flexibiliy and is widely used. Maximum Likelihood Esimaes Le he failure imes,,, n represen observed daa from a populaion disribuion, whose PDF is f( θ,..., θ k ), where θ i is he parameer of he disribuion. Then he problem is o find he likelihood funcion given by n... θk ) = f ( i θ... k ) i= L( θ θ. (.74) The objecive is o find he values of he esimaors of θ,, θ k ha render he likelihood funcion as large as possible for given values of,,, n. As he likelihood funcion is in he muliplicaive form, i is o maximize log(l) insead of L, bu hese wo are idenical, since maximizing L is equivalen o maximizing log(l). By aking parial derivaes of he equaion wih respec o θ,, θ k and seing hese equal o zero, he necessary condiions for finding MLEs can be obained: ln L( θ... θk ) =, i =,,, k. (.75) θ i Exponenial MLE The likelihood funcion for a single-parameer exponenial disribuion whose PDF is f ( ) = λe λ is given by j λ n n j = = e λ λ... λ ) ( λ )( λe )...( λe ) = λ e. (.76) L λ ( n n
66 Basic Reliabiliy Mahemaics Taking logarihms, we have n ln L(,,..., λ ) = nln λ λ. (.77) n j j = Parially differeniaing Equaion.77 wih respec o λ and equaing o zero, we have n ˆλ, (.78) = n j j = where ˆλ is he MLE of λ. Inerval Esimaion The poin esimaes would provide he bes esimae of he parameer, whereas he inerval esimaion would offer he bounds wihin which he parameer would lie. In oher words, i provides he confidence inerval for he parameer. A confidence inerval gives a range of values among which we have a high degree of confidence ha he disribuion parameer is included. Since here is always an uncerainy associaed in his parameer esimaion, i is essenial o find he upper and lower confidence limis of hese wo parameers. Upper and Lower Confidence of he Failure Rae The chi-square disribuion is used o find he upper and lower confidence limis of he mean ime o failure. The chi-square equaions are given as follows: θ θ where: T, (.79) χ LC r, α / T. (.8) χ UC r, α / θ LC, θ UC = lower and upper confidence limis of mean ime o failure; r = observed number of failures; T = operaing ime; α = level of significance.
.5 Failure Daa Analysis 67 The mean ime represens he mean ime beween failure or mean ime o failure. When he failure model follows an exponenial disribuion, he failure rae can be expressed as λ =. θ Thus, he inverse of θ LC and θ UC will be he maximum and minimum possible value of he failure rae, i.e., he upper and lower confidence limi of he failure rae. Upper and Lower Confidence Limi of he Demand Failure Probabiliy In he case of demand failure probabiliy, he F-disribuion is used o derive he upper and he lower confidence limi: P LC = r r+ D r+ F D r +, r, (.8) ( ) ( ) 95. P UC where: = ( r+ ) F95. ( r +, D r), (.8) ( ) ( ) D r+ r+ F r +, D r 95. P LC, P UC = lower and upper confidence limis for demand failure probabiliies; r = number of failures; D = number of demands; F.95 = 95% confidence limi for variables from F-disribuion Table A.4. Example Esimae he poin and 9% confidence inerval for he daa given in he previous example on grid ouage in a process plan. Soluion: Toal number of ouages = 7; oal period = years; mean failure rae = 7/ =.7/year =.94 4 /h. The represenaion of lower (5%) and upper (95%) limis of he chi-square (χ ) disribuion for failure-erminaed ess is as follows: χ α / ; γ χ α / ; γ λ. (.83) T T
68 Basic Reliabiliy Mahemaics For he case under consideraion: α = 9 = %; n = 7; degrees of freedom = γ = n = 7; T = years. χ χ λ. 5; 7. 95; + 7. Obaining he respecive values from he χ ables Table A.3,.77 λ.55. The mean value of grid ouage frequency is.7/year (.94 4 /h) wih lower and upper limi of.77/year (.3 4 /h) and.55/year (.9 4 /h), respecively..5..3 Goodness-of-fi Tess The las sep in he selecion of a parameric disribuion is o perform a saisical es for goodness of fi. Goodness-of-fi ess have he purpose o verify agreemen of observed daa wih a posulaed model. A ypical example is as follows. Given,,, n as n independen observaions of a random variable (failure ime), a rule is asked o es he null hypohesis: H. The disribuion funcion of is he specified disribuion. H. The disribuion funcion of is no he specified disribuion. The es consiss of calculaing a saisic based on he sample of failure imes. This saisic is hen compared wih a criical value obained from a able of such values. Generally, if he es saisic is less han he criical value, he null hypohesis (H ) is acceped, oherwise he alernaive hypohesis (H ) is acceped. The criical value depends on he level of significance of he es and he sample size. The level of significance is he probabiliy of erroneously rejecing he null hypohesis in favor of he alernaive hypohesis. A number of mehods are available o es how closely a se of daa fis an assumed disribuion. For some disribuion funcions used in reliabiliy heory, paricular procedures have been developed, ofen wih differen alernaive hypoheses H and invesigaion of he corresponding es power. Among he disribuionfree procedures, chi-square (χ ) is frequenly used in pracical applicaions o solve he goodness-of-fi problems.
Exercise Problems 69 The Chi-square Goodness-of-fi Tes The χ es is applicable o any assumed disribuion provided ha a reasonably large number of daa poins are available. The assumpion for he χ goodness-offi ess is ha, if a sample is divided ino n cells (i.e., we have ν degrees of freedom where ν = n ), hen he values wihin each cell would be normally disribued abou he expeced value, if he assumed disribuion is correc, i.e., if x i and E i are he observed and expeced values for cell i: n ( xi Ei ) χ =. (.84) E i= i If we obain a very low χ (e.g., less han he h percenile), i suggess ha he daa corresponds more closely o he proposed disribuion. Higher values of χ cas doub on he null hypohesis. The null hypohesis is usually rejeced when he value of χ falls ouside he 9h percenile. If χ is below his value, here is insufficien informaion o rejec he hypohesis ha he daa come from he supposed disribuion. For furher reading on reamen of saisical daa for reliabiliy analysis, ineresed readers may refer o Ebeling [5] and Misra [6]. Exercise Problems. A coninuous random variable T is said o have an exponenial disribuion wih parameer λ; if he PDF is given by f ( ) = λe λ, calculae he mean and variance of T.. Given he following PDF for he random variable ime o failure of a circui breaker, wha is he reliabiliy for a 5 h operaing life? β β α β f ( ) = e wih α = h and β =.5. α α 5 3. Given he hazard rae funcion λ ( ) =, deermine R() and f() a = 5 h. 4. The diameer of a bearing manufacured by a company under he specified supply condiions has a normal disribuion wih a mean of mm and sandard deviaion of. mm. (i) Find he probabiliy ha a bearing has a diameer beween. and 9.8 mm. (ii) Find he diameers, such ha % of he bearings have diameers below he value.
7 Basic Reliabiliy Mahemaics 5. While esing inegraed circuis (ICs) manufacured by a company, i was found ha 5% are defecive. (i) Wha is he probabiliy ha ou of 5 ICs esed more han are defecive? (ii) Wha is he probabiliy ha exacly are defecive? 6. If he rae of failure for a power supply occurs a a rae of once a year, wha is he probabiliy ha five failures will happen over a period of year? 7. Given he following failure imes, esimae R(), F(), f(), and λ():.84, 58.4,.4, 63.4, 4.89, 63.56, 383., 334.34, 4.76, 83.4, 68.35, 95.68, 3.7, 98.76, 756.86, 7.39, 3., 3., 7.38, 7.. 8. Using he daa given in problem 7, idenify he possible disribuion wih he help of a probabiliy ploing mehod. References. Ross SM (987) Inroducion o probabiliy and saisics for engineers and scieniss. John Wiley & Sons, New York. Mongomery DC, Runger GC (999) Applied saisics and probabiliy for engineers. John Wiley & Sons, New York 3. Weibull W (95) A saisical disribuion of wide applicabiliy. Journal of Applied Mechanics 8:93 97 4. Surges HA (976) The choice of class inerval. Journal of American Saisics Associaion :65 66 5. Ebeling CE (997) An inroducion o reliabiliy and mainainabiliy engineering. Taa McGraw-Hill, New Delhi 6. Misra KB (99) Reliabiliy analysis and predicion. Elsevier, Amserdam
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