Adaptive state‐ dependent importance sampling simulation of markovian queueing networks

Abstract

In this paper, a method is presentedfor the efficient estimation of rarecvent (buffer overflow) probabilities in queueing networks using importance sampling. Unlike previously proposed change of measures, the one used here is not static, i.e., it depends on the buffer contenrs at each of the network nodes. The 'optimal' statedependent change of measure is determined adaptively during the simulntion. using the cross-entropy method. The adaptive statedependent importance sampling algorithm proposed in this paper yields asymptotically efficient simulation of models for which it is shown (formally or otherwise) that no effective static change of measure exists. Simulation results for queueing models of communication systems are presented to demonstrate the effectiveness of the method.

Alternative proof and interpretations for a recent state-dependent importance sampling scheme

“…It turns out that already for a relatively simple queueing network problem, namely overflow of the total population of two queues in tandem, such a change of measure is not asymptotically optimal; see [3,6]. In several publications [4,10], state-dependent changes of measure were proposed for this two-node tandem queue and experimentally found to be asymptotically optimal; however, for none of them a rigid mathematical optimality proof is available.…”

The Cross-Entropy Method for Network Reliability Estimation

“…The method can also be used for solving optimisation problems (Rubinstein, 1999(Rubinstein, , 2001). The Cross-Entropy method has been successfully applied to a wide range of combinatorial and continuous optimisation problems (Dubin, 2002;Lieber, 1998;Margolin, 2002;Rubinstein, 1999), including problems in reliability theory (Lieber, Rubinstein, and Elmakis, 1997), buffer allocation (Alon et al, 2005), telecommunication systems (de Boer, 2000;de Boer, Kroese, and Rubinstein, 2004;de Boer and Nicola, 2002;de Boer, Nicola, and Rubinstein, 2000), neural computation (Dubin, 2002), control and navigation (Helvik and Wittner, 2001;Wittner and Helvik, 2002), DNA sequence alignment (Keith and Kroese, 2002), scheduling (de Mello and Rubinstein, 2002;Margolin, 2002) and Max-Cut and bipartition problems (Rubinstein, 2002). A short review of the basic ideas behind the Cross-Entropy method is given at the end of this section, but for details we refer to the book on Cross-Entropy , and the tutorial in de Boer et al (2005).…”

The cross-entropy method with patching for rare-event simulation of large Markov chains

“…We like to have fast and efficient algorithms. 4. We do not know in advance the order of magnitude of the target probabilities, thus this makes the stopping criterion in iterative methods uncertain, or maybe unnecessary small.…”

“…Then we apply our patching algorithm but now we do not keep the parameters fixed while iterating subsequent patches. Other attempts of reducing the parameter space in the cross-entropy method has been reported in Boer and Nicola (2002). The authors use also a grouping idea, however they group states along boundary layers that depend on the contents of the queues.…”