Showing papers by "Reuven Y. Rubinstein published in 2002"

Cross-entropy and rare events for maximal cut and partition problems

Abstract: We show how to solve the maximal cut and partition problems using a randomized algorithm based on the cross-entropy method. For the maximal cut problem, the proposed algorithm employs an auxiliary Bernoulli distribution, which transforms the original deterministic network into an associated stochastic one, called the associated stochastic network (ASN). Each iteration of the randomized algorithm for the ASN involves the following two phases:(1) Generation of random cuts using a multidimensional Ber(p) distribution and calculation of the associated cut lengths (objective functions) and some related quantities, such as rare-event probabilities.(2) Updating the parameter vector p on the basis of the data collected in the first phase.We show that the Ber(p) distribution converges in distribution to a degenerated one, Ber(pda), pda = (pd,1,...,pd,n) in the sense that someelements of pda, will be unities and the rest zeros. The unity elements of pda uniquely define a cut which will be taken as the estimate of the maximal cut. A similar approach is used for the partition problem. Supporting numerical results are given as well. Our numerical studies suggest that for the maximal cut and partition problems the proposed algorithm typically has polynomial complexity in the size of the network.

A fast cross-entropy method for estimating buffer overflows in queueing networks

Abstract: In this paper, we propose a fast adaptive importance sampling method for the efficient simulation of buffer overflow probabilities in queueing networks. The method comprises three stages. First, we estimate the minimum cross-entropy tilting parameter for a small buffer level; next, we use this as a starting value for the estimation of the optimal tilting parameter for the actual (large) buffer level. Finally, the tilting parameter just found is used to estimate the overflow probability of interest. We study various properties of the method in more detail for theM/M/1 queue and conjecture that similar properties also hold for quite general queueing networks. Numerical results support this conjecture and demonstrate the high efficiency of the proposed algorithm.

Rare event simulation and combinatorial optimization using cross entropy: estimation of rare event probabilities using cross-entropy

Abstract: This paper deals with estimation of probabilities of rare events in static simulation models using a fast adaptive two-stage procedure based on importance sampling and Kullback-Liebler's cross-entropy (CE). More specifically, at the first stage we estimate the optimal parameter vector in the importance sampling distribution using CE, and at the second stage we estimate the desired rare event probability using importance sampling (likelihood ratios). Some theoretical aspects of the proposed method, including its convergence, are established. The numerical results presented suggest that the method effectively estimates rare event probabilities.

Estimation of rare event probabilities using cross-entropy

Abstract: This paper deals with estimation of probabilities of rare events in static simulation models using a fast adaptive two-stage procedure based on importance sampling and Kullback-Liebler's cross-entropy (CE). More specifically, at the first stage we estimate the optimal parameter vector in the importance sampling distribution using CE, and at the second stage we estimate the desired rare event probability using importance sampling (likelihood ratios). Some theoretical aspects of the proposed method, including its convergence, are established. The numerical results presented suggest that the method effectively estimates rare event probabilities.

Inventory processes: quasi-regenerative property, performance evaluation, and sensitivity estimation via simulation

Abstract: We consider a single-commodity, discrete-time, multiperiod (s, S)-policy inventory model with backlog. The cost function may contain holding, shortage, and fixed ordering costs. Holding and shortage costs may be nonlinear. We show that the resulting inventory process is quasi-regenerative, i.e., admits a cycle decomposition and indicates how to estimate the performance by Monte Carlo simulation. By using a conditioning method, the push-out technique, and the change-of-measure method, estimates of the whole response surface (i.e., the steady-state performance in dependence of the parameters s and S) and its derivatives can be found. Estimates for the optimal (s, S) policy can be calculated then by numerical optimization.

Rare event simulation and combinatorial optimization using cross entropy: estimating buffer overflows in three stages using cross-entropy

Abstract: In this paper we propose a fast adaptive Importance Sampling method for the efficient simulation of buffer overflow probabilities in queueing networks. The method comprises three stages. First we estimate the minimum Cross-Entropy tilting parameter for a small buffer level; next, we use this as a starting value for the estimation of the optimal tilting parameter for the actual (large) buffer level; finally, the tilting parameter just found is used to estimate the overflow probability of interest. We recognize three distinct properties of the method which together explain why the method works well; we conjecture that they hold for quite general queueing networks. Numerical results support this conjecture and demonstrate the high efficiency of the proposed algorithm.