Theory and MethodologyThe sensitivity analysis of binary networks via simulation
Introduction
Consider an undirected network G=(V, E), where V is the node set and E={1,2,…,n} is the arc set. Every arc is randomly associated with two states, namely the operated state or the failed state. All arcs are stochastically independent. Let pi denote the probability that arc i operates for i=1,2,…,n and the parameter set be . The system reliability g is defined as the probability that the whole network operates, which is a function of all pi,i=1,2,…,n.
The system analysts usually need to evaluate how the system reliability g changes when the underlying parameter set is changed to . A quantitative measure of such perturbation is 100%, which is the (increasing or decreasing) relative change in total reliability in percent.
This paper presents a simulation method for the estimation of . We first use the score function method (SF) to estimate in Section 2and then we propose another approach in Section 3. A comprehensive survey of SF can be found in 1, 2. In Section 4, the proposed method is examined by a simulation experiment. Finally, just for reference, a summary of notations is listed in Section 5.
Section snippets
The score function method
Let Xi represent the state of arc i, whereand let be the state vector. The structure function Φ of an undirected network G is an indicator function such that
The system G is assumed to be a coherent system ([3], p. 6) throughout this paper. Now suppose that the parameter set is changed from to . The corresponding reliability will then change from to and the
The conditional simulation method
This section proposes another simulation method, named the conditional simulation (CS) method in this paper. Let Di(xi)=(xi−pi)/pi(1−pi) and let be the expectation when the underlying parameter set is . We rewrite , as the following lemmas.
Lemma 3.
Lemma 4.where is the mean conditioned on .
Substituting , into Eq. (1), we have
Simulation experiment and discussion
This section performs a simulation experiment to compare the performance of the score function method and the conditional simulation method. The reliability system is arbitrarily chosen to be a bridge network, as shown in Fig. 1. The parameter set is =(0.55, 0.9, 0.7, 0.6, 0.8), which is then adjusted to =(0.5, 0.91, 0.7, 0.61, 0.81). Now, suppose we use SF and CS to estimate the ratio of change . SF and CS use 200 replications to obtain an estimate of respectively. The 200
Notations and symbols
V set of nodes E set of arcs, E ={1, 2,..., n} G an undirected network, G=(V, E) Xi the random state of arc i. P(Xi=1)=pi and P(Xi=0)=1 − pi the parameters set, associated with E the original parameters set the perturbed parameter set the system reliability given the parameter set the structure function of G given . the mean conditioned on Φ(X)=1 SF score function method CS conditional simulation method
Acknowledgements
We are pleased to thank the editors and referees for their suggestions and proof reading. The research of this paper is supported by National Science Council in Taiwan, ROC. (N.S.C. 87-2213-E-251-004)
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