ON THE RELIABILITY OF AN n-component SYSTEM 1. INTRODUCTION

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1 ON THE RELIABILITY OF AN -COMPONENT SYSTEM Do Rawligs ad Lawrece Sze Uder assumptios compatible with the theory of Markov chais, we use a property of Vadermode matrices to examie the reliability of a -compoet system of productio or service. Keywords Markov chais; Probability theory; Stochastic processes. 1. INTRODUCTION The reliability of a -compoet system of productio or service is compromised whe ay of its compoets are out of service. Viswaadham ad Narahari [6] examied the reliability of such systems i some detail. We cosider a set of assumptios ot pursued i Ref. [6] : Time, measured i discrete uits, is idetified with the set of positive itegers. At time t = 1, all compoets are i service. The probability of a i service (or available) compoet remaiig i service from oe time to the ext is fixed ad deoted by. For coveiece, the complemetary probability 1 is abbreviated as. The probability of a out of service (or uavailable) compoet remaiig out of service from oe momet to the ext is fixed ad deoted by. Welet = 1. The trasitio of ay compoet betwee the states of beig available ad uavailable is idepedet of the other compoets.

2 As the cases = 0 = ad = 1 = are of little iterest, we make the restrictio 0 < + < 2. For = 1, our set-up coicides with the Greewood model (Ref. [3], p. 71) of cotagious disease: The coectio is made by idetifyig a available compoet with a healthy idividual ad a uavailable compoet with a ifected idividual. Our assumptios allow the theory of Markov chais to be brought to bear. We take the umber of compoets available to be the state of our chai. The trasitio probability p i,j of movig from state j to state i i oe uit of time is readily computed. Such a trasitio results whe l of the j available compoets remai i service ad i l of the j uavailable compoets are retured to service. Thus, i ( )( ) j j p l j l +l i j i l i,j = (1) l i l l =0 Amog may results aimed at measurig the reliability of a -compoet system, we preset but three (uder our assumptios, of course): (+q t ) 1 q R1 The expected umber of compoets available at time t is where, for coveiece, q = + 1. R2 The expected time it takes for the system ( ) to crash (that is, 1 q ( ) all compoets are out of service) is + ((/) k k ( 1) k ) qk. 1 q k R3 As time t, the expected fractio of time for which i of the ( ) i compoets are available approaches i. i (1 q) We begi with a brief discussio of a relevat class of matrices. 2. VANDERMONDE MATRICES For coveiece, we set ( ) i ( )( ) a, b j j +l V i j b j l i l d l i,j = a c (2) c, d l i l l =0 We the defie the th Vadermode matrix with parameters a, b, c, ad d to be the array [ ] ( ( )) a b a, b = V i,j c d c, d 0 i,j

3 For = 3, 3 a a 2 b ab 2 b 3 [ ] 2 a b 3a c 2abc + a 2 d b 2 c + 2abd 3b 2 d = 2 c d 3 3ac bc 2 + 2acd 2bcd + ad 2 3bd 2 c 3 c 2 d cd 2 d 3 I Ref. [5], we proved the followig remarkable fact. Theorem 2.1. If a, b,, h are elemets of a field, the [ ] [ ] [( )( )] a b e f a b e f = c d g h c d g h I other words, the product of two Vadermode matrices is Vadermode. Moreover, the matrix of parameters for the product miraculously coicides with the product of the uderlyig two-by-two matrices of parameters! Theorem 2.1 allows us to multiply, ivert, ad diagoalize Vadermode matrices at will. For istace, if ad bc = 0, the Theorem 2.1 implies that [ ] 1 [ ] [ ] d b a b ad bc ad bc 1 d b = = (3) c a c d (ad bc) c a ad bc ad bc The relevace of Vadermode matrices for our (, ) preset itetios should be apparet: I view of (1) ad (2), p i,j = V i,j,. So our trasitio matrix [ ] P = (p i,j ) 0 i,j = (4) is Vadermode. 3. MULTI-STEP TRANSITION PROBABILITIES AND R1 ( t) Determiatio of the probability p i, j of movig from state j to state i i t uits of time is key i deducig the results R1, R2, ad R3. To this ed, we diagoalize our trasitio matrix P i (4) (which, by Theorem 2.1, is o more difficult tha diagoalizig a 2 2 matrix). Let [ ] [ ] Q = ad D = 1 0 q

4 The, Theorem 2.1 ad (3) together imply that Q 1 PQ = D (5) Corollary 3.1. If 0 < + <2, the the probability of movig from state j to state i i t uits of time is give by where j ( )( ) (t) 1 j j +l p i j b j l i l d l i,j = a c (6) (1 q) l i l l =0 a = + q t, b = (1 q t ), c = (1 q t ), ad d = + q t (7) Moreover, if the chai begis i state j, the the state probability geeratig fuctio at time t is (t) i (a + cz) j (b + dz) j p i,j z = (8) (1 q) i=0 ( t) Proof. As the multi-step trasitio probability p i, j is the ijth etry i P t, we apply (5) ad Theorem 2.1 to obtai [ ] 1 a b P t = QD t Q 1 = (1 q) c d where a, b, c, ad d are as i (7). Hece, (6) ow follows from (2). For (8), observe that (6) implies that j ( ) j+l ( ) (t) i 1 j l j ( j) (i l p ) i,j z = b j l (dz) a ( cz) i l (1 q) l i l i=0 l =0 i=l Two applicatios of the biomial theorem the gives (8). Derivatio of R1 is ow a routie matter from (8): Takig j =, differetiatig with respect to z, ad the settig z = 1 does the job. 4. A WAITING TIME DISTRIBUTION AND R2 To verify R2, we first cosider how log it takes for a Markov chai to ( travel from oe state to aother. Let f i, jt) deote the probability that the chai visits state i for the first time at time t give that the process begis ( i state j. For i = j, f i, it) is the probability of returig to state i for the first

5 time after t steps. The fudametal relatioship betwee the waitig time ( t) ( t) probabilities f i, j ad the multi-step probabilities p i, j is as follows. Theorem 4.1 (First Etrace Theorem). If we let (t) t M f (t) t i,j (z) = (1 z) p i,j z ad F i,j (z) = i,j z, the M i,j (z) F i,j (z) = (9) 1 z + M i,i (z) for z < 1. Moreover, if ( ) (t) p i = lim p i,j exists, is idepedet of j, ad is positive, (10) t the F i,j is a probability geeratig fuctio, that is, F i,j is left-cotiuous at z = 1 with F i,j (1) = 1. Proof. As i Ref. [3],p.89, t 1 f (t) (t) (k) (t k) i,j = p i,j f i,j p i,i (11) for all t 1. Multiplyig (11) by z t ad summig over t 1 gives a formula equivalet to (9): (t) (t) F i,j (z) = p i,j z t F i,j (z) p i,i z t Next, suppose that (10) holds. For z < 1, observe that ( ) (t) (t 1) t M i,j (z) = p i,j p i,j z (12) ( 0) where p i, j = 0. As the series ( ) (t) (t 1) p i,j p i,j telescopes to p i ( ), it follows from Abel s Theorem that M i,j (z) is leftcotiuous at z = 1 with M i,j (1) = p i ( ). Thus, lim z 1 F i,j (z) = 1. We ow take aim at our secod result R2. First, ote that it is correct whe = 1: I this case, the expected crash time i ifiite. So cosider

6 the case < 1. As we are also assumig that 0 < + < 2, it follows that q < 1. Corollary 3.1 the guaratees that (10) holds for i = 0. Thus, F 0, (z) is a probability geeratig fuctio. Also by Corollary 3.1, ( ) ( ) (t) (1 q t ) (t) + q t p 0, = ad p 0,0 = (13) 1 q 1 q Note that M 0, (1) = /(1 q) = M 0,0 (1) With the aid of (12), (13), the exteded biomial theorem, ad a little fiaglig, we obtai ( ) ( ( ) ) k(t 1) M (1) = + t ( 1) k (q k 0, 1)q 1 q k t 2 ( ) ( ( ) ) q k = 1 + ( 1) k 1 1 q k 1 q k Similarly, M ( ) ( ( ) ) q k 0,0 (1) = 1 (/)k 1 q k 1 q k Fially, applyig logarithmic differetiatio to (9) yields R2: M 0, (1) M 0,0 F (1) 1 0, (1) = M 0, (1) M 0,0 (1) ( ) 1 q ( ) k q = + (( 1) k 1 + (/) k ) k 1 q k I the cotext of -player Russia roulette, the case = 1 of R2 is discussed i Rawligs [4]. Bartholdi [1] cosidered a variatio of R2 i which lamps are tured o ad off accordig to a set of determiistic rules. 5. THE EXPECTED NUMBER OF VISITS AND R3 To get at our fial result R3, we cosider how ofte a give state is visited o a fixed time iterval. This issue is resolved by Theorem 5.1 (a proof of which may be foud i Ref. [2], p. 105). Theorem 5.1. If a Markov chai begis i state j, the the expected umber of (t) times state i is visited o the iterval [1, ] is p t=1 i,j.

7 For R3, we agai assume 0 < + <2 ad that the chai begis i state. If = 1 (so <1), the R3 is easily see to be correct: It s 1 if i = 0 ad0if i > 1. So assume that <1. The, by Corollary 3.1, ( ) i ( ) i i (t) q t t p ) i i, = (1 q 1 + i (1 q) So, relative to the time iterval [1, ], Theorem 5.1 implies that the expected fractio of the time for which our system has i compoets available is ( ) i ( ) i i 1 q t (1 q t ) i 1 + (14) i (1 q) t=1 1 As lim (1 + a kq t ) = 1 whe a 1, a 2,, a are real ad q < 1, t=1 lettig i (14) gives R3. As a exercise, Viswaadham ad Narahari [6] (p. 206) pose the problem of determiig the asymptotic expected fractio of time for which at least m of the compoets are available whe = 3 ad whe uavailable compoets are retured to service accordig to a certai determiistic rule. Uder our assumptios, R3 gives the solutio to their problem for ay o-egative iteger as ( ) i i /(1 q) i=m. i REFERENCES 1. Bartholdi, L. Lamps, factorizatios, ad fiite fields. Amer. Math. Mothly 1999, 107, Bhat, U.N. Elemets of Applied Stochastic Processes; 2d ed.; Joh Wiley: New York, Iosifescu, M. Fiite Markov Processes ad Their Applicatios; Wiley: New York, Rawligs, D. Sequetial searches: Proofreadig, Russia roulette, ad the icomplete q-euleria polyomials revisited. Amer. Math. Mothly 2001, 108, Rawligs, D.; Sze, L. O the metamorphosis of Vadermode s idetity. Math. Mag. 2005, Jue, Viswaadham, N.; Narahari, Y. Performace Modelig of Automated Maufacturig Systems; New Jersey: Pretice Hall, 1992.