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Analysis of a splitting estimator for rare event probabilities in Jackson networks. (English) Zbl 1291.60150
Summary: We consider a standard splitting algorithm for the rare-event simulation of overflow probabilities in any subset of stations in a Jackson network at level \(n\), starting at a fixed initial position. It was shown in [T. Dean and P. Dupuis, Stochastic Processes Appl. 119, No. 2, 562–587 (2009; Zbl 1157.60019)] that a subsolution to the Isaacs equation guarantees that a subexponential number of function evaluations (in \(n\)) suffices to estimate such overflow probabilities within a given relative accuracy. Our analysis here shows that in fact \(O(n^{2\beta_V+1})\) function evaluations suffice to achieve a given relative precision, where \(\beta_V\) is the number of bottleneck stations in the subset of stations under consideration in the network. This is the first rigorous analysis that favorably compares splitting against directly computing the overflow probability of interest, which can be evaluated by solving a linear system of equations with \(O(n^d)\) variables.

60J22 Computational methods in Markov chains
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