Importance Functions for RESTART Simulation of General Jackson Networks

RESTART is an accelerated simulation technique that allows the evaluation of extremely low probabilities. In this method a number of simulation retrials are performed when the process enters regions of the state space where the chance of occurrence of the rare event is higher. These regions are defined by means of a function of the system state called the importance function. Guidelines for obtaining suitable importance functions and formulas for the importance function of two-stage networks were provided in previous papers. In this paper, we obtain effective importance functions for RESTART simulation of Jackson networks where the rare set is defined as the number of customers in a particular (‘target’) node exceeding a predefined threshold. Although some rough approximations and assumptions are used to derive the formulas of the importance functions, they are good enough to estimate accurately very low probabilities for different network topologies within short computational time

Analysis of a Splitting Estimator for Rare Event Probabilities in Jackson Networks

We consider a standard splitting algorithm for the rare-event simulation of overflow probabilities in any subset of stations in a Jackson network at level n, starting at a fixed initial position. It was shown in DeanDup09 that a subsolution to the Isaacs equation guarantees that a subexponential number of function evaluations (in n) suffice to estimate such overflow probabilities within a given relative accuracy. Our analysis here shows that in fact O(n^{2{\beta}+1}) function evaluations suffice to achieve a given relative precision, where {\beta} is the number of bottleneck stations in the network. This is the first rigorous analysis that allows to favorably compare splitting against directly computing the overflow probability of interest, which can be evaluated by solving a linear system of equations with O(n^{d}) variables.Comment: 23 page

Importance Sampling Simulation of Population Overflow in Two-node Tandem Networks

In this paper we consider the application of importance sampling in simulations of Markovian tandem networks in order to estimate the probability of rare events, such as network population overflow. We propose a heuristic methodology to obtain a good approximation to the 'optimal' state-dependent change of measure (importance sampling distribution). Extensive experimental results on 2-node tandem networks are very encouraging, yielding asymptotically efficient estimates (with bounded relative error) where no other state-independent importance sampling techniques are known to be efficient The methodology avoids the costly optimization involved in other recently proposed approaches to approximate the 'optimal' state-dependent change of measure. Moreover, the insight drawn from the heuristic promises its applicability to larger networks and more general topologies

Adaptive Importance Sampling Simulation of Queueing Networks

In this paper, a method is presented for the efficient estimation of rare-event (overflow) probabilities in Jackson queueing networks using importance sampling. The method differs in two ways from methods discussed in most earlier literature: the change of measure is state-dependent, i.e., it is a function of the content of the buffers, and the change of measure is determined using a cross-entropy-based adaptive procedure. This method yields asymptotically efficient estimation of overflow probabilities of queueing models for which it has been shown that methods using a stateindependent change of measure are not asymptotically efficient. Numerical results demonstrating the effectiveness of the method are presented as well

Variance Reduction Techniques in Monte Carlo Methods

Monte Carlo methods are simulation algorithms to estimate a numerical quantity in a statistical model of a real system. These algorithms are executed by computer programs. Variance reduction techniques (VRT) are needed, even though computer speed has been increasing dramatically, ever since the introduction of computers. This increased computer power has stimulated simulation analysts to develop ever more realistic models, so that the net result has not been faster execution of simulation experiments; e.g., some modern simulation models need hours or days for a single ’run’ (one replication of one scenario or combination of simulation input values). Moreover there are some simulation models that represent rare events which have extremely small probabilities of occurrence), so even modern computer would take ’for ever’ (centuries) to execute a single run - were it not that special VRT can reduce theses excessively long runtimes to practical magnitudes.common random numbers;antithetic random numbers;importance sampling;control variates;conditioning;stratied sampling;splitting;quasi Monte Carlo

Splitting for Rare Event Simulation: A Large Deviation Approach to Design and Analysis

Particle splitting methods are considered for the estimation of rare events. The probability of interest is that a Markov process first enters a set $B$ before another set $A$, and it is assumed that this probability satisfies a large deviation scaling. A notion of subsolution is defined for the related calculus of variations problem, and two main results are proved under mild conditions. The first is that the number of particles generated by the algorithm grows subexponentially if and only if a certain scalar multiple of the importance function is a subsolution. The second is that, under the same condition, the variance of the algorithm is characterized (asymptotically) in terms of the subsolution. The design of asymptotically optimal schemes is discussed, and numerical examples are presented.Comment: Submitted to Stochastic Processes and their Application