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now ready to give the estimator proposed in [11]:
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Z(G) = H + (1 H )
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(V = h)Z(Gh )
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Let us denote by v a sample from the distribution of V Then a sample Z (k) (G) of Z(G) can be deduced from a sample Z (k) (Gv ) of Z(Gv ) by Z (k) (G) = H + (1 pH )Z (k) (Gv ) If s and t are merged into a nal single node, the unreliability of Gv is equal to 0 and then Z (k) (G) = H + (1 pH ) 0 = H If s and t are not connected, the unreliability of Gv is equal to 1 and then Z (k) (G) = H + (1 pH ) 1 = 1 Otherwise, we have found an st-cut in Gv and we proceed again as before The main interest of this procedure is that Gv is smaller than G, and sometimes much smaller, because of the series parallel simpli cations, deletions and contractions performed A function which returns a trial of Z(G) can be summarized as follows: TRIAL-RVR(G, K) 1 Check end recursion condition: If |K| = 1 return(0) If G is not K-connected return(1) 2 Construct sp-red(G) by applying series parallel reductions to G 3 Find a K-cut in sp-red(G): = {l1 , , lH } 4 Compute the probability H that all links in are down 5 Compute the probability mass function distribution of the random variable V 6 Generate a trial v of V 7 Construct the network Gv = sp-red((G l1 l2 lv 1 ) lv ) 8 Recursive step: return( H + (1 H ) Trial-RVR(Gv , Kv )) The memory space complexity of the function TRIAL-RVR(G, K) is of order O(|E|(|E| + |V|)) and time complexity is, in the worst case, of order O(|E|(|K|2 |E||V|2 )) The worst case corresponds to a version using a maximal ow procedure in order to select a K-cut at step 3 of the above algorithm By calling the function TRIAL-RVR(G, K) N times, we obtain N independent trials Z (k) (G) of Z(G), 1 k N The sample mean Z(G) of these trials leads to an estimate of Q = 1 R and the variance is estimated by VRV R 1 = N (N 1)
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Z(G) Z (k) (G)
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MONTE CARLO TECHNIQUES IN STATIC MODELS
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In [12] it is shown how the computational complexity of the RVR method can be improved by generating the N samples simultaneously, thus avoiding replicating a large part of the computations, and [13] discusses the sensitivity of the RVR s accuracy to the strategy of choosing cuts Instead of using cuts to recursively change the original problem into a smaller one, the method in [10] exploits paths and that in [14] exploits both paths and cuts leading to a more interesting behavior than for versions based on only paths or cuts
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To the best of our knowledge, the CE-MP method [37] which uses the cross-entropy technique to further improve the performance of the merge process method [29] and the RVR technique which exploits series parallel reductions and a minimum cost st-cut strategy [13], where each link l has value ln(ql ), are the most suitable procedures published in the literature to compute network reliability in a rare event context In this section we present some numerical illustrations of these methods For the examples, we consider highly reliable grid topologies G3 and G6 (see Figure 71), where links are assigned equal unreliability q = 10 3 or q = 10 6 as in [37] and K is the set of the four corner nodes For those networks exact values of Q = 1 R are tabulated in column 3 of Table 71 Each exact unreliability Q is used in the computation of the relative error parameter which helps to appreciate the quality of the estimates produced by the two estimators considered Tables 72 and 73 show that both methods lead to small relative errors and the RVR method offers the most accurate estimates In the general case, we do not know the exact values Then, the best estimator in terms of accuracy is the one with smallest variance for a xed sample size N , leading to smallest lengths of con dence intervals Column 6 of Table 73 shows that the RVR method signi cantly reduces the variance with respect to the CE-MP method, and the best gains are obtained for highly reliable cases To illustrate the behavior of the RVR method on dense networks, let us now consider the evaluation of complete topologies for which we calculate the exact
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