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452 Observed relative error behavior
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In Section 441 we discussed the fact that, in some cases, the simulation technique can degrade when rarity increases, but the numerical values coming from the simulation run hides this phenomenon, leading the user to accept incorrect results We illustrated this by means of an example where rarity is parameterized by , and where in spite of the fact that RE is unbounded as 0, we will necessarily observe that RE suddenly becomes essentially constant, that is, independent of Of course, this is not a systematic fact appearing in these contexts, but it simply underlines the necessity of being careful if we observe this type of behavior Speci cally, as a diagnostic rule, the idea is to simulate (with small sample sizes) the network for different values of larger than in the original problem, that is, to simulate much less rare events, with a small and xed sample size, before running the real simulation if things seem to go well What is the incorrect behavior we try to detect We look to see whether the estimated relative variance seems rst to increase, then suddenly drops and stays xed This is due to the fact that important events (or paths, depending on the context) in terms of contribution to the variance (and to the estimation itself), are not sampled anymore This trend of regular growth and sudden drop is likely to be a good hint of rare event problems An illustration of this was provided by Example 1, Table 41 If we use a sample size n = 1000, ten times smaller than that used in Table 41, we observe the same phenomenon, always coherent with the formulas given in Section 441 We observed the same behavior with different con gurations This type of phenomenon does not appear in the case of the M/M/1/B model presented in Example 3 Increasing B (see Table 42), we observe uctuations of the relative error, but no trend similar to that exhibited before The diagnostic can hardly be conclusive in this model, as it was in the rst example This illustrates that the tests of the section are traditional rejection tests Let us now consider another example [1]: Example 5 Consider the discrete-time Markov chain X given in Figure 46 and de ne = E(K 1 K 1(X(k) = 1) | X(0) = 1) k=1
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Figure 46 A two-state Markov chain We look at the average fraction of interval {1, 2, , K} where the chain is in state 1, starting at state 0 at time 0
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The exact value of is = a 1 (1 a b)K 1 (1 a b) a+b K(a + b)
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Assume nevertheless that we use IS to estimate and let us consider the cases where 099 001 04 06 P = , P = 01 09 05 05 We look for the value of when K = 30 We know that the exact answer is 6713 2 , and, of course, this is easy to estimate with the crude estimator We used the proposed IS scheme for n = 105 samples, changing the seed of the pseudo-random number generator We got the results shown in Table 43 We can observe here that over these six runs, the relative error uctuates without a clear trend, but in ve of the six cases, the exact value is outside the con dence interval (the case where the exact value is in the con dence interval is for seed) If we increase the value of K, increasing the possible number of paths, we get the results given in Table 44 The RE exhibits no trend again, but we know that the estimations are horribly bad, and that the exact value is never inside the obtained con dence intervals In conclusion, for this test, involving checking the behavior of the relative error as a function of rarity, for small sample sizes, we observe good results when rarity is associated with transitions and the state space has a xed topology, and no clear indications when rarity comes from the increasing length of good paths, as in the M/M/1/B case Table 43 Estimating (whose value is 6713 2 ) in the two-state Markov chain of Example 5, with a = 001, b = 01, a = 06, b = 05, K = 30, for different seeds (using drand48( ) under Unix), for n = 105 samples seed
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