System reliability using the survival signature

Transcription

1 System reliability using the survival signature Frank Coolen GDRR Ireland 8-10 July 2013 (GDRR 2013) Survival signature 1 / 31

2 Joint work with: Tahani Coolen-Maturi (Durham University) Ahmad Aboalkhair (Mansoura University, Egypt) Abdullah Al-nefaiee (PhD student) (GDRR 2013) Survival signature 2 / 31

3 System reliability: structure function System with m components: State vector x =(x 1, x 2,...,x m ), with x i = 1 if the ith component functions and x i = 0 if not. Structure function (x) =0 if not. (x) =1 if system functions with state x and Assume: (x) is not decreasing in any of the components of x ( coherent system ) and (0) =0 and (1) =1. (This assumption can be deleted) (GDRR 2013) Survival signature 3 / 31

4 System signature T S > 0: random failure time of the system. T j:m : j-th order statistic of the m random component failure times for j = 1,...,m, with T 1:m apple T 2:m apple...apple T m:m. Assume: component failure times are independent and identically distributed with CDF F(t) (exchangeability can be assumed instead). The system s signature is the m-vector q with j-th component q j = P(T S = T j:m ) so q j is the probability that the system failure occurs at the moment of the j-th component failure. (GDRR 2013) Survival signature 4 / 31

5 The signature provides a qualitative description of the system structure that can be used in reliability quantification. with P(T j:m > t) = P(T S > t) = mx r=m j+1 mx q j P(T j:m > t) j=1 m [1 F(t)] r [F(t)] m r r The system structure is fully taken into account through the signature and is separated from the information about the random failure times of the components. (GDRR 2013) Survival signature 5 / 31

6 Consider two systems, each with m components and all failure times of the 2m components assumed to be iid. Let the signature of system A be q a and of system B be q b, and let their failure times be T a and T b. If mx j=r q a j mx j=r q b j for all r = 1,...,m, then for all t > 0. P(T a > t) P(T b > t) (GDRR 2013) Survival signature 6 / 31

7 Straightforward to extend signature to system with more components (same type) and same system failure time. Two systems with different types of components (one type per system) can be compared by with m b Xm a X P(T a < T b )= qi a qj b P(Ti:m a a < Tj:m b b ) i=1 j=1 P(T a i:m a < T b j:m b )= Z 1 0 f i a(t)p(t b j:m b > t)dt where f i a(t) is the PDF for T a i:m a, which, with PDF f a (t) for the failure time of components in system A, is equal to f i a(t) =f a (t) Xm a r a=m a i+1 ma r a [1 F a (t)] ra 1 [F a (t)] ma ra 1 [r a m a (1 F a (t))] (GDRR 2013) Survival signature 7 / 31

8 Computing the signature is difficult for most practical systems, but only needs to be derived once. One can work with bounds of signature ( imprecise signature ), which can be used to determine if its further computation is needed in case of a specific inference. Unfortunately: Generalizing the signature to systems with multiple types of components (nearly all real-world systems) requires (many) probabilities for orderings of order statistics of different distributions to be computed - this is very difficult. (GDRR 2013) Survival signature 8 / 31

9 Survival signature For systems with one type of component: Let (l), for l = 1,...,m, denote the probability that a system functions given that precisely l of its components function. For coherent systems (l) is an increasing function of l. There are m l state vectors x with P m i=1 x i = l; let S l be the set of these state vectors. Due to the iid assumption for the failure times of the m components, (l) = m 1 X l x2s l (x) We call (l) the (system) survival signature. (GDRR 2013) Survival signature 9 / 31

10 Let C t 2{0, 1,...,m} denote the number of components in the system that function at time t > 0. If the probability distribution of the component failure time has CDF F(t), then for l 2{0, 1,...,m} P(C t = l) = m l [F(t)] m l [1 F(t)] l and P(T S > t) = mx l=0 (l)p(c t = l) The survival signature is easily derived from the signature mx (l) = j=m l+1 so it adops all the nice properties of signatures. q j (GDRR 2013) Survival signature 10 / 31

11 Multiple types of components Let (l 1, l 2,...,l K ), for l k = 0, 1,...,m k, denote the probability that a system functions given that precisely l k of its components of type k function, for each k 2{1, 2,...,K }. There are m k l k state vectors x k with precisely l k of their m k components xi k = 1, so with P m k i=1 x i k = l k. Let S l1,...,l K denote the set of all state vectors for the whole system for which P m k i=1 x i k = l k, k = 1, 2,...,K. Due to the iid assumption for the failure times of the m k components of type k, (l 1,...,l K )= " K Y k=1 mk l k 1 # X x2s l1,...,l K (x) (GDRR 2013) Survival signature 11 / 31

12 Let Ct k 2{0, 1,...,m k } denote the number of components of type k in the system that function at time t > 0. If the probability distribution for the failure time of components of type k is known and has CDF F k (t), then the probability that the system functions at time t > 0 is m 1 X P(T S > t) = m 1 X l 1 =0 m K X l K =0 " l 1 =0 m K X l K =0 (l 1,...,l K ) (l 1,...,l K )P( KY k=1 mk l k K\ {Ct k = l k })= k=1 # [F k (t)] m k l k [1 F k (t)] l k (GDRR 2013) Survival signature 12 / 31

13 Example Figure: System with 2 types of components (GDRR 2013) Survival signature 13 / 31

14 l 1 l 2 (l 1, l 2 ) l 1 l 2 (l 1, l 2 ) / / / / Table: Survival signature of this system (GDRR 2013) Survival signature 14 / 31

15 Comments on survival signature Computation of the survival signature can be a complicated task, particularly for real systems and networks. But it is only required once, and one can work with bounds based on partial knowledge or computation (and can check if further computation is required if there is a specific reliability aim). Louis Aslett (Trinity College Dublin) has created an R-function for it! Package: ReliabilityTheory: Tools for structural reliability analysis Function: computesystemsurvivalsignature (GDRR 2013) Survival signature 15 / 31

16 Recent results Combining survival signatures of two subsystems in series or parallel configuration. For series: Xl 1 Xl K apple S(l 1,...,l K ) =... 1 (l1 1,...,l1 K ) 2 (l 1 l1 1,...,l K lk 1 ) l1 1=0 lk 1 =0 KY m 1 k m 2 1 k mk k=1 l 1 k l k l 1 k l k Effect of component replacement: l k (l 1,...,l k 1, l k, l k+1,...,l K )= m k (l 1...,l k 1, l k 1, l k+1,...,l K, 1)+ m k l k m k (l 1...,l k 1, l k, l k+1,...,l K, 0) (GDRR 2013) Survival signature 16 / 31

17 Imprecise probability Uncertainty quantification mostly by precise probabilities: For event A, a single (classical, precise) probability P(A), typically satisfying Kolmogorov s axioms Classical probability requires a high level of precision and consistency of information, and is often too restrictive to carefully represent the multi-dimensional nature of uncertainty Lower and Upper Probabilities: P(A) and P(A) respectively, with 0 apple P(A) apple P(A) apple 1 and P(A c )=1 P(A) (conjugacy) (GDRR 2013) Survival signature 17 / 31

18 Walley (1991, imprecise probability ) uses a subjective interpretation, Weichselberger (2001, interval probability ) generalizes Kolmogorov s axioms without an explicit interpretation. Introduction to Imprecise Probabilities (Wiley, late 2013) P(A) reflects evidence in favour of A, and 1 against A. P(A) reflects evidence We attempt to base inference on few model assumptions, not sufficiently strong to lead to precise probabilities. The lower and upper probabilities are the maximum lower and minimum upper bounds, respectively, for all precise probabilities that are in agreement with the assumptions made and the data-based inferences following from these assumptions. (GDRR 2013) Survival signature 18 / 31

19 Nonparametric Predictive Inference (NPI): A nonparametric frequentist statistical approach Predictive: inference for one or more future observation(s) Depends on Hill s assumption A (n) (Hill 1968) (GDRR 2013) Survival signature 19 / 31

20 NPI for Bernoulli random quantities n + m exchangeable Bernoulli trials, each success or failure. Yj l : number of successes in trials j to l. R t = {r 1,...,r t }, with 1 apple t apple m + 1 and 0 apple r 1 < r 2 <...<r t apple m, and define s+r 0 s = 0. For s 2{0,...,n} (Coolen, 1998) n + m 1 P(Y n+m n+1 2 R t Y1 n = s) = n tx apple s + rj s + rj 1 n s + m rj s s n s j=1 Lower probability derived via conjugacy property (GDRR 2013) Survival signature 20 / 31

21 Combining survival signature with NPI System with one type of components: C t is the number out of m future components, so components in the system, that function at time t. In the test of n components, exchangeable with these m components, s t functioned at time t. where P(T S > t) = mx l=0 (l)d(c t = l) D(C t = l) = P(C t apple l) P(C t apple l 1) n + m 1 apple st 1 + l n st + m l = n s t 1 n s t (GDRR 2013) Survival signature 21 / 31

22 System with K types of components (independent): where m 1 X P(T S > t) = l 1 =0 m K X l K =0 KY (l 1,, l K ) D(Ct k = l k ) k=1 D(Ct k = l k ) = P(Ct k apple l k ) P(Ct k apple l k 1) " nk + m 1 s k = k t 1 + l k nk s k t + m k l # k st k 1 n k The corresponding upper probabilities are found similarly, using instead D(C k t = l k )=P(C k t apple l k ) P(C k t apple l k 1) n k s k t (GDRR 2013) Survival signature 22 / 31

23 Example Figure: System with 2 types of components (GDRR 2013) Survival signature 23 / 31

24 Assume n 1 = n 2 = 2 components of each type tested, ordering of their failure times: t1 2 < t1 1 < t2 2 < t1 2 (tk j the j-th ordered failure time of components of type k). t1 2 < t1 1 < t2 2 < t1 2 t 2 P(T S > t) P(T S > t) [0, t1 2) (t1 2, t1 1 ) (t1 1, t2 2 ) (t2 2, t1 2 ) , 1) (t 1 2 Table: S TS (t) and S TS (t) (GDRR 2013) Survival signature 24 / 31

25 While for test failure times ordering t 1 1 < t2 1 < t1 2 < t2 2 : t1 1 < t2 1 < t1 2 < t2 2 t 2 P(T S > t) P(T S > t) [0, t1 1) (t1 1, t2 1 ) (t1 2, t1 2 ) (t 1, t2 2 ) (t2 2, 1) Table: S TS (t) and S TS (t) (GDRR 2013) Survival signature 25 / 31

26 Challenges Computation of survival signatures: link to fault trees, binary decision diagrams, etc Learning survival signature from system-level data (Louis Aslett has presented this for the signature) Use for reliability of (large) networks Applications! (GDRR 2013) Survival signature 26 / 31

27 References Signatures Samaniego FJ (2007) System Signatures and their Applications in Engineering Reliability. Springer. Eryilmaz S (2010) Review of recent advances in reliability of consecutive k-out-of-n and related systems. Journal of Risk and Reliability (GDRR 2013) Survival signature 27 / 31

28 Imprecise Probability and Reliability Coolen FPA, Troffaes MC, Augustin T (2011) Imprecise probability. International Encyclopedia of Statistical Science, M Lovric (Ed.). Springer Coolen FPA, Utkin LV (2011) Imprecise reliability. International Encyclopedia of Statistical Science, M Lovric (Ed.). Springer (GDRR 2013) Survival signature 28 / 31

29 NPI Coolen FPA (2011) Nonparametric predictive inference. International Encyclopedia of Statistical Science, M Lovric (Ed.). Springer Coolen FPA (1998) Low structure imprecise predictive inference for Bayes problem. Statistics & Probability Letters Hill BM (1968) Posterior distribution of percentiles: Bayes theorem for sampling from a population. Journal of the American Statistical Association (GDRR 2013) Survival signature 29 / 31

30 NPI with Signatures Coolen FPA, Al-nefaiee AH (2012) Nonparametric predictive inference for failure times of systems with exchangeable components. Journal of Risk and Reliability Al-nefaiee AH, Coolen FPA. Nonparametric predictive inference for system failure time based on bounds for the signature. Journal of Risk and Reliability to appear. (GDRR 2013) Survival signature 30 / 31

31 Survival Signatures Coolen FPA, Coolen-Maturi T (2012) On generalizing the signature to systems with multiple types of components. Complex Systems and Dependability, W Zamojski et al. (Eds.), Springer Coolen FPA, Coolen-Maturi T, Al-nefaiee AH, Aboalkhair AM (2013) Recent advances in system reliability using the survival signature. Proceedings 20th Advances in Risk and Reliability Technology Symposium 2013, L Jackson, J Andrews (Eds.) (GDRR 2013) Survival signature 31 / 31