Abstract:The cross-entropy method is a recent versatile Monte Carlo technique. This article provides a brief introduction to the cross-entropy method and discusses how it can be used for rare-event probability estimation and for solving combinatorial, continuous, constrained and noisy optimization problems. A comprehensive list of references on cross-entropy methods and applications is included.Keywords: cross-entropy, Kullback-Leibler divergence, rare events, importance sampling, stochastic search.The cross-entropy (CE) method is a recent generic Monte Carlo technique for solving complicated simulation and optimization problems. The approach was introduced by R.Y. Rubinstein in [41,42], extending his earlier work on variance minimization methods for rare-event probability estimation [40].The CE method can be applied to two types of problem:, where X is a random variable or vector taking values in some set X and H is function on X . An important special case is the estimation of a probability = P(S(X) γ), where S is another function on X .2. Optimization: Optimize (that is, maximize or minimize) S(x) over all x ∈ X , where S is some objective function on X . S can be either a known or a noisy function. In the latter case the objective function needs to be estimated, e.g., via simulation.In the estimation setting, the CE method can be viewed as an adaptive importance sampling procedure that uses the cross-entropy or Kullback-Leibler divergence as a measure of closeness between two sampling distributions, as is explained further in Section 1. In the optimization setting, the optimization problem is first translated into a rare-event estimation problem, and then the CE method for estimation is used as an adaptive algorithm to locate the optimum, as is explained further in Section 2.An easy tutorial on the CE method is given in [15]. A more comprehensive treatment can be found in [45]; see also [46, Chapter 8]. The CE method homepage can be found at www.cemethod.org .The CE method has been successfully applied to a diverse range of estimation and optimization problems, including buffer allocation [1], queueing models of telecommunication systems [14,16], optimal control of HIV/AIDS spread [48,49], signal detection [30], combinatorial auctions [9], DNA sequence alignment [24,38], scheduling and vehicle routing [3,8,11,20,23,53], neural and reinforcement learning [31,32,34,52,54], project management [12], rare-event simulation with light-and heavy-tail distributions [2,10,21,28], clustering analysis [4,5,29]. Applications to classical combinatorial optimization problems including the max-cut, traveling salesman, and Hamiltonian cycle 1

We investigate the simulation of overflow of the total population of a Markovian two-node tandem queue model during a busy cycle, using importance sampling with a state-independent change of measure. We show that the only such change of measure that may possibly result in asymptotically efficient simulation for large overflow levels is exchanging the arrival rate with the smallest service rate. For this change of measure, we classify the model's parameter space into regions of asymptotic efficiency, exponential growth of the relative error, and infinite variance, using both analytical and numerical techniques.

A wide variety of voltage mixers and samplers are implemented with similar circuits employing switches, resistors, and capacitors. Restrictions on duty cycle, bandwidth, or output frequency are commonly used to obtain an analytical expression for the response of these circuits. This paper derives unified expressions without these restrictions. To this end, the circuits are decomposed into a polyphase multipath combination of single-ended or differential switched-series-kernels. Linear periodically timevariant network theory is used to find the harmonic transfer functions of the kernels and the effect of polyphase multipath combining. From the resulting transfer functions, the conversion gain, output noise, and noise figure can be calculated for arbitrary duty cycle, bandwidth, and output frequency. Applied to a circuit, the equations provide a mathematical basis for a clear distinction between a "mixing" and a "sampling" operating region while also covering the design space "in between." Circuit simulations and a comparison with mixers published in literature are performed to support the analysis.

Recently, a state-dependent change of measure for simulating overflows in the two-node tandem queue was proposed by Dupuis et al. (Ann. Appl. Probab. 17(4):1306-1346, 2007, together with a proof of its asymptotic optimality. In the present paper, we present an alternative, shorter and simpler proof. As a side result, we obtain interpretations for several of the quantities involved in the change of measure in terms of likelihood ratios.

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.