Command-based importance sampling for statistical model checking

Vehicle routing problem belongs to classical Combination Optimization hard problem and has been extensively studied by many researchers. Vehicle Routing Problem with Weight Coefficients and Stochastic Demands (WVRPSD) is established and a new implementation for importance sampling method to solve this model is proposed in this paper, in which the classical exponential change of measure is adopted to construct the family of importance sampling distributions, and the optimal importance sampling distribution is obtained by minimizing the variance of importance sampling estimator. Numerical experiments have been conducted and the results indicate that the method can effectively solve this problem.

We review two types of adaptive Monte Carlo methods for rare event simulations. These methods are based on importance sampling. The first approach selects importance sampling distributions by minimizing the variance of importance sampling estimator. The second approach selects importance sampling distributions by minimizing the cross entropy to the optimal importance sampling distribution. We also review the basic concepts of importance sampling in the rare event simulation context. To make the basic concepts concrete, we introduce these ideas via the study of rare events of M/M/1 queues.

Simulation is widely used to estimate losses due to default and other credit events in financial portfolios. The accurate measurement of credit risk can be modeled as a rare event simulation problem. While Monte Carlo simulation is time-consuming for rare events, importance sampling techniques can effectively reduce the simulation time, thus improving simulation efficiency. This chapter proposes a new importance sampling method to estimate rare event probability in simulation models. The optimal importance sampling distributions are derived in terms of expectation in the normal copula model developed in finance. In the normal copula model, dependency is introduced through a set of common factors of multiple obligors. The intriguing dependence between defaults of multiple obligors imposes hurdles in simulation. The simulated results demonstrate the effectiveness of the proposed approach to solving the portfolio credit risk problem.

Simulation-based validation methods exhibit properties which should allow them to contribute to the release of highly automated driving functions (HADF). Monte-Carlo (MC) simulations make it possible to generate statistical statements about a HADF. However, the fact that accidents happen very rarely in the real world can strongly enlarge the number of MC simulation runs which are needed to obtain statistically stable results. In the past, importance sampling (IS) was used to reduce these simulation costs. This paper considers some of the still existent limitations of IS: The definition of proper safety metrics needed to find good IS distributions and the application of IS to high dimensional parameter spaces. The approach in this paper was able to reduce an IS optimization of 2700 dimensions to 17 optimizations of 9 dimensions. The number of MC runs could be reduced by a factor of 350.

It is well known that in estimating performance measures associated with a stochastic system a good importance sampling distribution (IS) can give orders of magnitude of variance reduction while a bad one may lead to large, even infinite, variance. In this paper we study how this sensitivity of the estimator variance to the importance sampling change of measure may be "dampened" by combining importance sampling with stochastic approximation based temporal difference (TD) method. We consider a finite state space discrete time Markov chain (DTMC) with one-step transition rewards and an absorbing set of states and focus on estimating the cumulative expected reward to absorption starting from any state. In this setting we develop sufficient conditions under which the estimate resulting from the combined approach has a mean square error that asymptotically equals zero even when the estimate formed by using only importance sampling change of measure has infinite variance. In particular, we consider the problem of estimating the small buffer overflow probability in a queuing network, where the change of measure suggested in literature is shown to have infinite variance under certain parameters and where the appropriate combination of IS and TD method can be empirically seen to have a much faster convergence rate compared to naive simulation.

Balanced likelihood ratio importance sampling methods were originally developed for the analysis of fault-tolerant systems. The paper provides a basis for adapting this approach to analyze the rare event probability that total system size reaches a bound before returning to zero in tandem Jackson networks. An optimal importance sampling distribution for the single server case is derived through direct application of the balanced likelihood ratio approach. The generalization of this approach to larger systems is explored via a two-node tandem Jackson network. A general heuristic approach is outlined along with certain open questions whose answers could lead to a more robust solution. Asymptotic characteristics of the proposed importance sampling approach for the two-node network are discussed. Bounded relative error is only possible under certain conditions. Numerical results illustrate the benefits of the approach.

A population-based optimization algorithm combining the support vector machine (SVM) and importance sampling (IS) is proposed to achieve a global solution to optimal reliability design. The proposed approach is a greedy algorithm that starts with an initial population. At each iteration, the population is divided into feasible/infeasible individuals by the given constraints. After that, feasible individuals are classified as superior/inferior individuals in terms of their fitness. Then, SVM is utilized to construct the classifier dividing feasible/infeasible domains and that separating superior/inferior individuals, respectively. A quasi-optimal IS distribution is constructed by leveraging the established classifiers, on which a new population is generated to update the optimal solution. The iteration is repeatedly executed until the preset stopping condition is satisfied. The merits of the proposed approach are that the utilization of SVM avoids repeatedly invoking the reliability function (objective) and constraint functions. When the actual function is very complicated, this can significantly reduce the computational burden. In addition, IS fully excavates the feasible domain so that the produced offspring cover almost the entire feasible domain, and thus perfectly escapes local optima. The presented examples showcase the promise of the proposed algorithm.

In this paper vehicle routing problem with stochastic demands and simultaneous delivery and pickup is developed and analyzed, which is an important expansion of classical Vehicle Routing Problem(VRP).An effective algorithm based on Important Sampling is designed to solve the model. The optimal importance sampling distribution function was obtained by making use of the expection constructed by likelihood ratio. Numerical experiments have been conducted and the results indicate that the method can effectively solve this problem.

Rare event simulation for highly dependable systems with fast repairs

We propose to perform statistical probabilistic model checking by using perfect simulation in order to verify steady-state and time unbounded until formulas over Markov chains. The model checking of probabilistic models by statistical methods has received increased attention in the last years since it provides an interesting alternative to numerical model checking which is poorly scalable with the increasing model size. In previous statistical model checking works, unbounded until formulas could not be efficiently verified, and steady-state formulas had not been considered due to the burn-in time problem to detect the steady-state. Perfect simulation is an extension of Markov Chain Monte Carlo (MCMC) methods that allows us to obtain exact steady-state samples of the underlying Markov chain, and thus it avoids the burn-in time problem to detect the steady-state. Therefore we suggest to verify time unbounded until and steady-state dependability properties for large Markov chains through statistical model checking by combining perfect simulation and statistical hypothesis testing.

Importance sampling is a change-of-measure technique for speeding up the simulation of rare events in stochastic systems. In this paper the authors establish a number of properties characterizing optimal importance sampling measures for Markovian systems. The authors show how these properties may be used to compute the optimal measure and give specific results for a tandem queueing system. The authors' approach has no apparent computational advantage over other direct methods, but it suggests a number of heuristic approximations which may lead to computationally attractive methods.

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.

In this paper, we consider a statistical problem of learning a linear model from noisy samples. Existing work has focused on approximating the least squares solution by using leverage-based scores as an importance sampling distribution. However, no finite sample statistical guarantees and no computationally efficient optimal sampling strategies have been proposed. To evaluate the statistical properties of different sampling strategies, we propose a simple yet effective estimator, which is easy for theoretical analysis and is useful in multitask linear regression. We derive the exact mean square error of the proposed estimator for any given sampling scores. Based on minimizing the mean square error, we propose the optimal sampling scores for both estimator and predictor, and show that they are influenced by the noise-to-signal ratio. Numerical simulations match the theoretical analysis well.

This paper surveys efficient techniques for estimating, via simulation, the probabilities of certain rare events in queueing and reliability models. The rare events of interest are long waiting times or buffer overflows in queueing systems, and system failure events in reliability models of highly dependable computing systems. The general approach to speeding up such simulations is to accelerate the occurrence of the rare events by using importance sampling. In importance sampling, the system is simulated using a new set of input probability distributions, and unbiased estimates are recovered by multiplying the simulation output by a likelihood ratio. Our focus is on describing asymptotically optimal importance sampling techniques. Using asymptotically optimal importance sampling, the number of samples required to get accurate estimates grows slowly compared to the rate at which the probability of the rare event approaches zero. In practice, this means that run lengths can be reduced by many orders of magnitude, compared to standard simulation. In certain cases, asymptotically optimal importance sampling results in estimates having bounded relative error. With bounded relative error, only a fixed number of samples are required to get accurate estimates, no matter how rare the event of interest is. The queueing systems studied include simple queues (e.g., GI/GI/1), Jackson networks, discrete time queues with multiple autocorrelated arrival processes that arise in the analysis of Asynchronous Transfer Mode communications switches, and tree structured networks of such switches. Both Markovian and non-Markovian reliability models are treated.

In this paper, the optimal importance sampling, which is a variance reduction technique in Monte Carlo estimation, is theoretically studied in the case of electronic circuits. To the best of our knowledge, this problem has not previously been studied in electronic circuits. In this study, the theoretical basis of the optimal importance sampling is developed in the case where no approximate model of the circuit responses is given. Simulation results show good agreement with the theoretical basis.