A combined splitting—cross entropy method for rare-event probability estimation of queueing networks

Abstract

We present a fast algorithm for the efficient estimation of rare-event (buffer overflow) probabilities in queueing networks. Our algorithm presents a combined version of two well known methods: the splitting and the cross-entropy (CE) method. We call the new method SPLITCE. In this method, the optimal change of measure (importance sampling) is determined adaptively by using the CE method. Simulation results for a single queue and queueing networks of the ATM-type are presented. Our numerical results demonstrate higher efficiency of the proposed method as compared to the original splitting and CE methods. In particular, for a single server queue example we demonstrate numerically that both the splitting and the SPLITCE methods can handle our buffer overflow example problems with both light and heavy tails efficiently. Further research must show the full potential of the proposed method.

The Splitting Method for Decision Making

“…Although there is a vast literature on the splitting method (see Botev and Kroese, 2008;Garvels, 2000;Garvels and Rubinstein, 2002;Lagnoux-Renaudie, 2010;Melas, 1997;L'Ecuyer et al, 2007;Rubinstein, 2010), we follow Rubinstein (2010) and present some background on the splitting method, also called in Rubinstein (2010) the cloning method. The main idea is to design a sequential sampling plan, with a view of decomposing a "difficult" counting problem defined on some set * into a number of "easy" ones associated with a sequence of related sets 0 1 m and such that m = * .…”

On the Use of Smoothing to Improve the Performance of the Splitting Method

“…For relevant references on the splitting method see [2], [4], [5], [7], [8], [9], [10], [11], which contain extensive valuable material as well as a detailed list of references. Recently, the connection between splitting for Markovian processes and interacting particle methods based on the Feynman-Kac model with a rigorous framework for mathematical analysis has been established in Del…”

“…For the sake of clarity, we will focus first on g ε as per (8). With this function, for a given ε, we can split the region explored by the Gibbs sampler in several small (sub) hypercubes or hyperrectangles, as shown schematically in Figure 1.…”

Counting with Combined Splitting and Capture-Recapture Methods

“…Darwin's Finch Data from Yuguo Chen, Persi Diaconis, Susan P. Holmes, and Jun S. Liu: m = 12, n = 17 with row and columns sums r = (14, 13, 14, 10, 12, 2, 10, 1, 10, 11, 6, 2), c = (3, 3, 10, 9,9,7,8,9,7,8,2,9,3,6,8,2,2).…”