Use of Monte Carlo when estimating reliability of complex systems - PowerPoint PPT Presentation

Use of Monte Carlo when estimating reliability of complex systems COMPSTAT 2010 : August 27, 2010 Jaromir Antoch, Yves Dutuit, Julie Berthon Charles University Prague, Thales Bordeaux, University Bordeaux 1

• Clusters and scan statistics : simple example • Simulation methods  Monte-Carlo  Petri nets • Markov approach  Simplified Markov chain - one scan window  Simplified Markov chain - double scan window  Complete Markov chain • Simulation results and comparison • Conclusions Charles University Prague, Thales Bordeaux, University Bordeaux 1 1

August 6 August 2 nd Flight 1153 of Tuninter Flight 358 of Air France landed on see close to went out of runways Palermo during landing in Toronto August 14 Flight 522 of Hélios August 23 crashed into a Flight 204 of Trans mountain close to Crached approaching Athens Amazonie August 16 Flight 1153 lof West Caribbean Crached in Venezuela Unreliable series, isn’t it, BUT… Scan statistics allow to evaluate or approximer probability of occurence of such a “cluster” of events. Charles University Prague, Thales Bordeaux, University Bordeaux 1 2

Goal : calculate probability that we will observe a cluster of k or more events in a scanning windows of length w moving during a fixed period of length T .  Any window of length w can constain a cluster Problems  Windows overlap Charles University Prague, Thales Bordeaux, University Bordeaux 1 3

Example:   T one year, i.e. 365 days  λ  or p correspond to 8 events per year (on mean)    (w, k) (10,3) : 3 events in 10 day Solutions • Monte Carlo simulations • direct (implemented using a specific algorithm) • supported by Petri nets • Markov chains Two probability models : • Bernoulli Be( p ) • Poisson λ P( ) 4

Direct Monte-Carlo simulation • Dates of accidents are generated along the considered distribuion to cover given period of observation [0,T[      ε ε ε 0 ... T 1 2 S • List of dates is scanned until the cluster is observed • Counter of clusters - Nb_Cluster – is incremented by 1 We estimate unknown parameter using the quantity Nb_Cluster N N est is number of repetitions of the simulation. Charles University Prague, Thales Bordeaux, University Bordeaux 1 5

Use of Petri nets stimulating Monte Carlo simulation • Counting processes (simple counting medium) • 2 places and 2 transitions • Initialization place 1 is set to one  Nb_Cluster = 0  • Variables ε i (i =1,...,k) indicates dates of k consecutive accidents • Index I allows to calculate continuously time elapsed between eventsl i and (i+k-1) • Nb_Cluster passe à 1 when k accidents appear within a window of lentgh w Charles University Prague, Thales Bordeaux, University Bordeaux 1 6

MARKOV MODELS Scanning observation period Notation N(u,w) X i 0 T i-1 i u u+w 1 2 3 • X i … random variable denoting number of events in interval [i-1,i[ • N(u,w)… random variable counting number of events in window [u,u+w[ • p probability that an event will appear in a subinterval of the length equal t o1  1 with probabilit y p   Bernoulli model X  i  0 with probabilit y q 1 - p Charles University Prague, Thales Bordeaux, University Bordeaux 1 7

FIRST MARKOV MODEL X u+1 “Arrival” of random “Lost” of random variable X u+w+1 variable X u+1 X u+w+1 From window N(u,w) à to window N(u+1,w) independent      N ( u 1 , w ) N ( u , w ) X X   u 1 u w 1 dependent n n        P ( X 0 N ( u , w ) n ) 1 P ( X 1 N ( u , w ) n )   u 1 u 1 w w Charles University Prague, Thales Bordeaux, University Bordeaux 1 8

FIRST MARKOV MODEL States: E 0 , E 1 , E 2 : 0, 1 or 2 events in current window E 3 : 3 events or more in current window Markov chain       p p 1 1       q q 1 1     w w w w   w   w   1 1     p p 1 1 p p       w   w   2 2     p p 1 1     1 1 E 1 E 1 E 2 E 2 E 3 E 3 E 0 E 0 E E q q 0 0 q q 2q 2q w w w w       2p 2p 2 2       q q 1 1     w w w w Probability of one cluster of 3 events or more in a window of size w=10 Charles University Prague, Thales Bordeaux, University Bordeaux 1 9

FIRST MARKOV MODEL   q p 0 0     q p w 1 w 1    q p 0 w w w w    Transition matrix M  2 q 2p w 2    0 q 0   w w w     0 0 0 1 Vector of initial probabilities   w q   w - 1  pq   T T 0 0 X   2 w - 2 p q        w w - 1 2 w - 2   1 q pq p q w w Number of iterations T T T T T T T T T T T T T T T T T T T T 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 3 1 1 2 3 1 1 2 3 1 2 3 1 2 2 2 4 4 4 N=T-w+1 N=T-w+1 Charles University Prague, Thales Bordeaux, University Bordeaux 1 10

FIRST MARKOV MODEL Probability to find cluster consisting of k=3 events or more in window of size w=10 scanning the period of length T=365 is given by product M N X N=356 11

SECOND MARKOV MODEL Problem : Model allows the “ways” which cannot be realized in practice E 0 E 1 E 0 E 0 E 1 E 1 12

SECOND MARKOV MODEL Division of the scanning window into two sub-windows E ’ E 0 E 1 1 13

SECOND MARKOV MODEL either pair (i,j) if i+j<k State is: absorbing state if i+j=k Transition matrix is matrix of size D × D avec D=k(k-1)+1 Transition probabilities and vector of initial probabilities are calculated analogously as before       2   2    w  q 0 q 0 0 0 0      b 0, , p   w      2              2 2 2 2                  w w p q 1 p 0 q 1 0 0       b 0, , p b 1, , p           w w w w       2 2              2 2 2 2 4                      0 q q 1 0 q q 0  w w                  w w w w w b 1, , p b 0, , p         2 2            2 4 2 2                       M 0 p 1 0 q 1 p 1 0 0 X w w                w w w w  b 0, , p b 2, , p                            2 2 2 2 4 2 2 2 2 4                          0 q p 1 q p q 1 1 p 0         w w                 w w w w w w w w         b 1, , p b 1, , p             2 2  2 2 4             0 0 0 0 q 1 q 1 0               w w w w w       b 2, , p b 0, , p             2 4 2 2           0 0 0 p p 1 p 1 1             1 B 2, w, p  w w 14

THIRD MARKOV MODEL “Complete” model X i 0 T i-1 i u u+w 1 2 3 either w-uplet (X 1 , X 2 ,…, X w ) if X 1 + X 2 +…+ X w <k  State is: Or absorbing A if X 1 + X 2 +…+ X w =k   w        A    Space of states is E (X , X ,..., X ) X 0,1 and X k 1 2 w i i    i 1       1      w w w With dimension 1 ...  1 2 k 1 Notation: state (i 1 ,i 2 ,…,i m ) if i 1 =i 2 =…=i m =1 and i l =0 otherwise 15

THIRD MARKOV MODEL Transition matrix Transition of state (i,j) to state (i-1,j-1) with probability q: i j i-1 j-1 Transition of state (i,j) to absorbing state with probability p: i j i-1 j-1 16