Rare event probabilities in stochastic networks

This paper deals with the estimation of rare event probabilities using importance sampling (IS), where an optimal proposal distribution is computed with the cross-entropy (CE) method. Although, IS optimized with the CE method leads to an efficient reduction of the estimator variance, this approach remains unaffordable for problems where the repeated evaluation of the score function represents a too intensive computational effort. This is often the case for score functions related to the solution of a partial differential equation (PDE) with random inputs. This work proposes to alleviate computation by the parsimonious use of a hierarchy of score function approximations in the CE optimization process. The score function approximation is obtained by selecting the surrogate of lowest dimensionality, whose accuracy guarantees to pass the current CE optimization stage. The selection relies on certified upper bounds on the error norm. An asymptotic analysis provides some theoretical guarantees on the efficiency and convergence of the proposed algorithm. Numerical results demonstrate the gain brought by the method in the context of pollution alerts and a system modeled by a PDE.

We present a novel method, called the transform likelihood ratio (TLR) method, for estimation of rare event probabilities with heavy-tailed distributions. Via a simple transformation (change of variables) technique the TLR method reduces the original rare event probability estimation with heavy tail distributions to an equivalent one with light tail distributions. Once this transformation has been established we estimate the rare event probability via importance sampling, using the classical exponential change of measure or the standard likelihood ratio change of measure. In the latter case the importance sampling distribution is chosen from the same parametric family as the transformed distribution. We estimate the optimal parameter vector of the importance sampling distribution using the cross-entropy method. We prove the polynomial complexity of the TLR method for certain heavy-tailed models and demonstrate numerically its high efficiency for various heavy-tailed models previously thought to be intractable. We also show that the TLR method can be viewed as a universal tool in the sense that not only it provides a unified view for heavy-tailed simulation but also can be efficiently used in simulation with light-tailed distributions. We present extensive simulation results which support the efficiency of the TLR method.

This dissertation focuses on the importance sampling (IS) variance reduction technique in the Monte Carlo simulation of the performance of digital communication systems over a memoryless, additive Gaussian noise channel. The result is a unbiased estimate of the decision error with significantly smaller estimation variance than conventional Monte Carlo simulation provides. Several common digital (n, k) block codes are used to illustrate the importance sampling approach. Bounds on the variance of the error estimate are developed where the channel description is simple, but where the multiple dimension decision regions are potentially complex. Coherent detection is used as a venue for the comparison of these IS techniques with respect to increasing dimension. Bounds based on the union bound are presented for several IS techniques. An overlap stratification variance reduction technique is introduced and extended with the application of IS. A mixed IS biasing function is employed for concentrating more samples in a particular error region and is employed for several of the common lower rate ([special characters omitted] < 1) block codes. For the random phase channel the IS biasing function is developed as a 2n dimensional pdf consisting of n Rayleigh-Rician pdf pairs. The error performance of several common lower rate block codes is presented. An alternate formulation, having the potential advantage of being able to use both mean translation and variance scaling, is presented. This approach is based on a commonly used technique to numerically generate a single Rayleigh-Rician pdf pair.

Rare events are an important limitation of discrete-event simulation in some areas of applications. The existing rare-event simulation (RES) techniques, such as importance sampling (IS), splitting, and RESTART, have been used for RES so far. However, these techniques not only have some limitations, but also are not general or easy to use. In this paper, the partition of the region (PoR) method is used as a variance reduction technique in a new RES technique for usage with a wide range of modeling and simulation languages. Three variants of the proposed technique will be introduced: simple PoR, recursive PoR, and advanced PoR. Simple and recursive PoR methods are powerful techniques that have some important features such as bounded relative error; however, they are restricted to some specific models. Advanced PoR is a more general technique that integrates the PoR method and the IS technique. To evaluate the efficiency of the proposed technique and its variants, the comparative simulation results of some illustrative examples using stochastic activity networks will be presented.

This paper deals with the estimation of rare event probabilities in finite capacity queueing networks. The traditional product form property of Markovian queueing networks usually vanishes when capacity of queues are finite and clients are blocked or rejected. A new efficient simulation method, derived from Propp & Wilson (Propp 1996), perfect simulation, is applied in the finite capacity queue context. An algorithm directly samples states of the network according to the stationary distribution. This method is adapted for simulation of rare events, typically when events are described by an increasing subset of the state space.Provided that events of the network are monotone, monotonicity techniques are used to reduce the sampling time. Moreover, a stopping mechanism has been developed to interrupt the simulation when an increasing set has been reached. Then, for the estimation of a monotonous reward function, the simulation time could be reduced drastically as in (Vincent and Marchand 2004).

In this paper, we develop the use of an adaptive Importance Sampling (IS) technique in neural network training, for applications to detection in communication systems. Some topics are reconsidered, such as modifications of the error probability objective function (Pe), optimal and suboptimal IS probability density functions (biasing density functions), and adaptive importance sampling. A genetic algorithm was used for the neural network training, having utilized an adaptive IS technique for improving Pe estimations in each iteration of the training. Also, some simulation results of the training process are included in this paper.

A new global variance reduction technique based on pseudo flux method

The authors extend the formulation for both classical Monte Carlo (MC) and importance sampling (IS) techniques to include the bit error rate (BER) and symbol error rate (SER) of multilevel systems based on symbol by symbol observation. A new IS technique and analysis of BER and SER based on the extended formulation are presented. The authors briefly summarize the previous work on BER of binary systems. The extended formulation is then presented. The conventional IS technique for BER of binary systems is shown to be applicable in simulating BER and SER of M-ary systems, and a new improved IS technique is presented. Numerical results for all the methods are provided. >

Modified efficiency importance sampling (MEIS), an importance sampling technique which can be used to yield a significant saving of run time for the assessment of bit error rate (BER) in digital communication systems using computer simulation, is described. The concept of equivalent noise source is introduced to avoid the effect of system memory on the efficiency of importance sampling, and a nonGaussian probability density function (p.d.f.) is selected as the biased equivalent noise to further improve run time. MEIS has been used successfully in the simulation of M-ary quadrature amplitude modulation (MQAM) systems. The MEIS is shown to be effective in the simulation of systems with large memory length, and in mildly nonlinear systems. The experimental results show that MEIS is a great improvement over Monte Carlo (MC) methods and that is even more efficient than the efficient importance sampling (EIS) technique. >

The design of complex systems often requires reliability assessments involving a large number of uncertainties and low probability of failure estimations (in the order of 10−4). Estimating such rare event probabilities with crude Monte Carlo (CMC) is computationally intractable. Specific numerical methods to reduce the computational cost and the variance estimate have been developed such as importance sampling or subset simulation. However, these methods assume that the uncertainties are defined within the probability formalism. Regarding epistemic uncertainties, the interval formalism is particularly adapted when only their definition domain is known. In this paper, a method is derived to assess the reliability of a system with uncertainties described by both probability and interval frameworks. It allows one to determine the bounds of the failure probability and involves a sequential approach using subset simulation, kriging, and an optimization process. To reduce the simulation cost, a refinement strategy of the surrogate model is proposed taking into account the presence of both aleatory and epistemic uncertainties. The method is compared to existing approaches on an analytical example as well as on a launch vehicle fallout zone estimation problem.

Importance sampling is a variance reduction technique for efficient estimation of rare-event probabilities by Monte Carlo. In standard importance sampling schemes, the system is simulated using an a priori fixed change of measure suggested by a large deviation lower bound analysis. Recent work, however, has suggested that such schemes do not work well in many situations. In this paper we consider dynamic importance sampling in the setting of uniformly recurrent Markov chains. By “dynamic” we mean that in the course of a single simulation, the change of measure can depend on the outcome of the simulation up till that time. Based on a control-theoretic approach to large deviations, the existence of asymptotically optimal dynamic schemes is demonstrated in great generality. The implementation of the dynamic schemes is carried out with the help of a limiting Bellman equation. Numerical examples are presented to contrast the dynamic and standard schemes.

A stratified beta-sphere sampling method combined with important sampling and active learning for rare event analysis

Risk analysis using stochastic reliability methods applied to two cases of deterministic water quality models

In order to increase productivity and reduce energy consumption of steelmaking-continuous casting (SCC) production process, especially with complicated technological routes, the cross entropy (CE) method was adopted to optimize the SCC production scheduling (SCCPS) problem. Based on the CE method, a matrix encoding scheme was proposed and a backward decoding method was used to generate a reasonable schedule. To describe the distribution of the solution space, a probability distribution model was built and used to generate individuals. In addition, the probability updating mechanism of the probability distribution model was proposed which helps to find the optimal individual gradually. Because of the poor stability and premature convergence of the standard cross entropy (SCE) algorithm, the improved cross entropy (ICE) algorithm was proposed with the following improvements: individual generation mechanism combined with heuristic rules, retention mechanism of the optimal individual, local search mechanism and dynamic parameters of the algorithm. Simulation experiments validate that the CE method is effective in solving the SCCPS problem with complicated technological routes and the ICE algorithm proposed has superior performance to the SCE algorithm and the genetic algorithm (GA).

Project scheduling is essential to be planned before activities. Standard methods of project scheduling based on precedence constraints scheduling events and the assumption that resources are not unlimited capacity. But in fact the project activity has resource limits. The main objective is to minimize the total duration of the project with precedence constraints and resource constraints for all project activities. Many optimization methods are used to improve the quality of scheduling and speed up the calculation time. This paper proposed the use of Cross Entropy (CE) method to solve resource constraints project scheduling problem, as well as comparing the advantages and disadvantages between the CE methods with Differential Evolution (DE) method. The purpose CE method consist of four critical steps including generating a sample of random solution, calculating the performance according to the specific fitness function, selecting elite sample and finally updating the previous parameters to get a better sample in the next iteration. To speed up the computation time, in this study decrease the number of samples for each iteration in Cross Entropy algorithm. Numerical experiments with several data sets from the Project Scheduling library (PSPLIB) showed that CE can provide the optimal total duration of the project same as the DE with calculation time is faster than DE.