Research Article A Formula for the Reliability of a d-dimensional Consecutive-k-out-of-n:F System

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1 International Combinatorics Volume 2015, Article ID , 5 pages Research Article A Formula for the Reliability of a -Dimensional Consecutive-k-out-of-n:F System Simon Cowell Department of Mathematical Sciences, UAE University, Al Ain, Abu Dhabi, UAE Corresponence shoul be aresse to Simon Cowell; scowell@uaeu.ac.ae Receive 17 March 2015; Accepte 21 June 2015 Acaemic Eitor: Jiang Zeng Copyright 2015 Simon Cowell. This is an open access article istribute uner the Creative Commons Attribution License, which permits unrestricte use, istribution, an reprouction in any meium, provie the original work is properly cite. We erive a formula for the reliability of a -imensional consecutive-k-out-of-n:f system, that is, a formula for the probability that an n 1 n array whose entries are (inepenently of each other) 0 with probability p an 1 with probability q=1 poes not inclue a contiguous s 1 s subarray whose every entry is Introuction Consier a pipeline punctuate by several pumping stations. Suppose the pumping stations are reunant in that if one of them fails, then its preecessor will be strong enough to pump the flui past it to the next pumping station. Perhaps each pump will even be able to compensate for the failure of its successor an its successor s successor but perhaps not for the failure of the first 3 pumping stations in the chain of its successors. In this case, the system fails if an only if among the sequence of pumping stations there exists a consecutive run of 3 or more faile pumping stations. This is an example of what is known as a consecutive-k-out-of-n:f system. Suppose that such a system having n noes fails if an only if k consecutive noes fail, an suppose that any given noe in the system works correctly with probability p [0, 1], inepenently of the other noes. Then the probability that any given noe fails is q=1 p.thereliability of the system is the probability R(k,n;q)that the system oes not fail. Then R(k,n;q) = 1 P(k,n;q),where P(k,n;q)is the probability that the sequence of n noes inclues a contiguous interval of k or more faile noes. Here we are using notation as in [1]. As note in [1], the concept of the reliability of a consecutive system was introuce to Engineering by Kontoleon in 1980 [2]. In the following year Chiang an Niu [3]iscusse some applications of consecutive systems. The concept has been generalise in several irections, for instance, to systems eeme to have faile if an only if they inclue k consecutive faile components or f faile components [4], k consecutive components of which at least r have faile [5], an at least m nonoverlapping runs of k consecutive faile components [6]. In 2001 Chang et al. publishe a book on the subject [7]. Four reviews are also available [8 11]. An exact formula for R(k,n;q) first appeare in [12]. e Moivre s approach epens on eriving the generating function (see [13]) for P(k,n;q). This work has been rephrase in a moern style in [14]. Much later a ifferent exact formula for R(k,n;q)using Markov chains was foun [15 17]. Both of these exact formulae for R(k,n;q) are ifficult to use in Engineering, where values of k an n may be very large. Efficient algorithms have been foun [18, 19], but even with moern computing resources, evaluating the formulae takes a lot of time an memory. This problem has been aresse at length in the stuy of upper an lower bouns for R(k,n;q). Such bouns have been foun which are close approximations to the exact value of R(k,n;q)an which are also much easier to evaluate than the exact value of R(k,n;q). In [1, 20, 21] the authors compare many such results from the 1980s to the present ay. Dăuş an Beiu were the first to use e Moivre s original formula to erive upper an lower bouns for R(k,n;q).As shown in [1]thismethocomparesveryfavourablywiththe previous upper an lower bouns, especially for small q, an also with the exact algorithms given in [18, 19] (using their bouns is about 100 times faster than the algorithm from [18] an 1000 times faster than the algorithm from [19]).

2 2 International Combinatorics One possible extension to higher imensions is as follows. Given N an n 1,n 2,...,n N, suppose that some system is represente by an n 1 n 2 n array of 0s an 1s. Suppose that each noe in the array is 0 with probability p [0, 1] an is 1 with probability q=1 p, inepenently of the other noes. Suppose that s 1,s 2,...,s N are given, with s r n r for each r, an suppose that the system is eeme to have faile if an only if the -imensional array inclues a contiguous s 1 s 2 s subarray of 1s. Define the reliability of the system as the probability R(s 1,...,s,n 1,...,n ;q)that the system oes not fail, an let P(s 1,...,s,n 1,...,n ;q) = 1 R(s 1,...,s,n 1,...,n ;q),so that P(s 1,...,s,n 1,...,n ;q)isthe probability that the array inclues a contiguous s 1 s 2 s subarray of 1s. Let us survey in chronological orer some recent papers ealing with the higher imensional case. In [22] the2- imensional special case R(s,s,n,n;q)is treate. Upper an lower bouns are foun by reucing to the 1-imensional case. Using empirical approximations to the exact value, R(s,s,n,n;q) is plotte as a function of q for n = 5an s = 2, 3, an 4, for n = 10 an s = 2, 4, 6, an 8, an for n=50 an s=2, 5, 10, an 20. The authors o not attempt to fin an explicit expression for R(s, s, n, n; q). In[23] the 2-imensional case is consiere, for R(s 1,s 2,n 1,n 2 ;q) with (s 1,s 2 ) {1,n 1 } {1,n 2 }. Exact formulae are erive by comparison with the 1-imensional case. Similar results are also given for a cylinrical version of the 2-imensional array. The 2-imensional case R(s,s,n,n;q) of [22] isrevisitein [24]. Again, upper an lower bouns are given, but in the more general case where the probabilities of istinct noes of the system failing are not necessarily equal. For the general 2- imensional case R(s 1,s 2,n 1,n 2 ;q), recursive formulae for the exact reliability have been given by Yamamoto an Miyakawa [25] analsobynoguchietal.[26]. The result presente herein is istinct from their work, in that we give a closeform formula for the exact reliability as a polynomial in q,for a general system, of arbitrary imension. The 3-imensional case is treate in [27], where exact formulae for the reliability aregiveninspecialcasesanalogoustothosestuiein[23]. In the present paper we erive general close-form exact formulae for R(s 1,...,s,n 1,...,n ;q)an for P(s 1,..., s,n 1,...,n ;q),wheretheimension N is arbitrary. As far as the author knows, no such formulae were known before for > 1, even in the case = 2. Inee, in [24] theauthorswrite Itisveryifficult(probablyimpossible)to erive simple explicit formulas for the reliability of a general 2D-consecutive-k-out-n:F system. 2. Results Theorem 1 (main result). Let the terms, s 1,...,s, n 1,...n, p, q, R(s 1,...,s,n 1,...,n ;q) P(s 1,...,s,n 1,...,n ;q) be as in the Introuction, an let Then E= (1) {1,...,n r s r + 1}. (2) R(s 1,...,s,n 1,...,n ;q)=1 + P(s 1,...,s,n 1,...,n ;q)= ( 1) J ( 1) J exp [( 1) J ln ( 1 q ) max (0,s r (max e r min e r ))], (3) exp [( 1) J ln ( 1 q ) max (0,s r (max e r min e r))], (4) where P(A) enotes the power set of the set A, thatis,theset of subsets of A,an0 enotes the empty set. Theorem 1 implies the following combinatorial result. Corollary 2. With all terms as in Theorem 1, thenumber a(s 1,...,s,n 1,...,n ) of faile systems among the 2 n r possible systems is a(s 1,...,s,n 1,...,n )= 2 n r ( 1) J exp [( 1) J ln (2) max (0,s r (max e r min e r))]. (5) Proof. In case p=q=1/2, all possible systems occur with equal probability, so the probability measure on the power setofthesetofallsystemsbecomesthecountingmeasure, ivie by a factor of 2 n r.

3 International Combinatorics 3 Setting =1, n 1 =n,ans 1 = 2, 3, an 4 in Corollary 2 yiels the sequences [28], [29], an [30], respectively. Setting =2, n 1 =n 2 =n,ans 1 =s 2 = 2inCorollary2 yiels the sequence [31], whose complementary sequence is [32]. Proof of Theorem 1. Call a system e an elementary failure case if an only if it inclues a contiguous s 1 s subarray of 1s, an its other elements are all 0. We ientify the set of all elementary failure cases e with the set E, accoring to the rule e (e 1,...,e ) (6) if an only if the (i 1,...,i )th element of e is 1 precisely when, for all r {1,...,}, e r i r e r +s r 1. (7) For example, in the two-imensional case, we ientify an elementary failure case with the inices (e 1,e 2 ) of the top left corner of its contiguous subrectangle of 1s. Denote by G the simple acyclic irecte graph whose vertices are all the possible systems an in which system b is a irect successor of system a if an only if b is obtaine from a by changing a single 0 element to 1. Then the faile systems f in G are precisely those vertices having some elementary failure case e Eas an ancestor (incluing the possibility that f=e). Therefore if F enotes the set of all faile systems f (in terms of probability, F is the event that the system fails), then F= the set of escenents of e, (8) e E where by b is a escenent of a wemeantoincluethe possibility that b=a.then where by the union of several n 1 n systems we mean the system whose generic element is 1 if an only if the corresponing element in at least one of those systems is also 1. Thus P(s 1,...,s,n 1,...,n ;q)= ( 1) J +1 P( the set of escenents of e) = ( 1) J +1 P(the set of escenents of e). However, for any vertex a of G, P (the set of escenents of a) = P ({b}) b is a escenent of a = p n 1n 2 n k q k, b is a escenent of a (12) (13) where k=k(b)isthenumberoffailenoesinthesystemb. Gathering like powers of q in the above sum, we have P (the set of escenents of a) = n 1 n 2 n k=k(a) ( n 1n 2 n k(a) )p n 1n 2 n k q k k k(a) P(s 1,...,s,n 1,...,n ;q)=p(f) =P( the set of escenents of e) e E = ( 1) J +1 (9) =q k(a) n 1 n 2 n k=k(a) =q k(a) n 1 n 2 n k(a) j=0 ( n 1n 2 n k(a) )p n 1n 2 n k q k k(a) k k(a) =q k(a) (p + q) n 1n 2 n k(a) =q k(a). ( n 1n 2 n k(a) )p n 1n 2 n k(a) j q j j (14) P( the set of escenents of e), by the Inclusion-Exclusion Principle for the probability measure on the power set of the set of all possible systems. But the set of escenents of e = the set of escenents of hc (J), (10) where by hc(k) we mean the highest common escenent of any given set K of vertices in G. That is, in the partial orer on the vertices of G efine by a bprecisely when a is an ancestor of b,hc(k) is the least upper boun of K.Moreover, hc (J) = e, (11) That is, for any vertex a of G, P (the set of escenents of a) =q k(a), (15) where k(a) enotes the number of faile noes in the system a. Therefore from (12)we obtain P(s 1,...s,n 1,...n ;q)= ( 1) J +1 P(the set of escenents of e) = ( 1) J +1 q k( e). (16) The next task is to compute the number k( e) of 1s in the system e,wherej is some nonempty subset of

4 4 International Combinatorics 1.0 on the power set of the set of elements of a system, to obtain q k ( e) = ( 1) J +1 k ( e). (17) Each elementary failure case consists entirely of 0s except for a contiguous s 1 s subarray of 1s. Therefore e also consists entirely of 0s except for a (possibly empty) contiguous subarray of imensions t 1 t,where,for each r {1,...}, t r = max (0,(min (e r +s r 1) max e r + 1)) = max (0,s r (max e r min e r )). (18) The volume of that contiguous t 1 t subarray is equal to k( e);thatis, R(1, 2; q) R(1, 2, 2, 3; q) R(1, 2, 3, 2, 3, 4; q) Figure 1 the set E of elementary failure cases e.weusetheinclusion- Exclusion Principle again, this time for the counting measure k( e) = = t r max (0,s r (max e r min e r )). Consiering (16), (17), an(19), we have (19) P(s 1,...,s,n 1,...,n ;q)= ( 1) J +1 q k( e) = ( 1) J +1 exp [ ln (q) [ ( 1) J +1 k( e)] ] = ( 1) J +1 exp [ ln (q) [ = ( 1) J ( 1) J +1 max (0,s r (max e r min e r ))] ] exp [( 1) J ln ( 1 q ) max (0,s r (max e r min e r ))], (20) which proves (4).Equation(3) follows immeiately. As an example, we compute the reliabilities of 3 consecutive systems, having imensions 1, 2, an 3: + 4q 18 8q 19 8q 20 12q q 22 8q 23 +q 24. (21) R(1, 2;q)=1 2q+q 2, R(1, 2, 2, 3;q)=1 4q 2 + 2q 3 + 4q 4 4q 5 +q 6, R(1, 2, 3, 2, 3, 4;q)=1 8q 6 + 4q 8 + 4q 9 + 4q 10 8q q 12 16q 14 16q 15 12q q 17 See also Figure Conclusion We have provie a novel formula for the reliability of a general -imensional consecutive-k-out-of-n:f system, as an exact polynomial in q. We believe ours to be the first general exact close-form formula publishe for the reliabilityin imensions. This answers an open question in Engineering.

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