State-dependent importance sampling for a Jackson tandem network

This paper considers importance sampling as a tool for rare-event simulation. The focus is on estimating the probability of overflow in the downstream queue of a Jacksonian two-node tandem queue – it is known that in this setting ‘traditional’ state-independent importance-sampling distributions perform poorly. We therefore concentrate on developing a state-dependent change of measure, that we prove to be asymptotically efficient.\ud More specific contributions are the following. (i) We concentrate on the probability of the second queue exceeding a certain predefined threshold before the system empties. Importantly, we identify an asymptotically efficient importance-sampling distribution for any initial state of the system. (ii) The choice of the importance-sampling distribution is backed up by appealing heuristics that are rooted in large-deviations theory. (iii) Our method for proving asymptotic efficiency is substantially more straightforward than some that have been used earlier. The paper is concluded by simulation experiments that show a considerable speed up

State-dependent Importance Sampling for a Slow-down Tandem Queue

In this paper we investigate an advanced variant of the classical (Jackson) tandem queue, viz. a two-node system with server slow-down. The slow-down mechanism has the primary objective to protect the downstream queue from frequent overflows, and it does so by reducing the service speed of the upstream queue as soon as the number of jobs in the downstream queue reaches some pre-specified threshold. To assess the efficacy of such a policy, techniques are needed for evaluating overflow metrics of the second queue. We focus on the estimation of the probability of the following rare event: overflow in the downstream queue before exhausting the system, starting from any given state in the state space.\ud Due to the rarity of the event under consideration, naive, direct Monte Carlo simulation is often infeasible. We therefore rely on the application of importance sampling to obtain variance reduction. The principal contribution of this paper is that we construct an importance sampling scheme that is asymptotically efficient. In more detail, the paper addresses the following issues. (i) We rely on powerful heuristics to identify the exponential decay rate of the probability under consideration, and verify this result by applying sample-path large deviations techniques. (2) Immediately from these heuristics, we develop a proposal for a change of measure to be used in importance sampling. (3) We prove that the resulting algorithm is asymptotically efficient, which effectively means that the number of runs required to obtain an estimate with fixed precision grows subexponentially in the buffer size. We stress that our method to prove asymptotic efficiency is substantially shorter and more straightforward than those usually provided in the literature. Also our setting is more general than the situations analyzed so far, as we allow the process to start off at any state of the state space, and in addition we do not impose any conditions on the values of the arrival rate and service rates, as long as the underlying queueing system is stable

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

Simple and efficient importance sampling scheme for a tandem queue with server slow-down

This paper considers importance sampling as a tool for rare-event simulation. The system at hand is a so-called tandem queue with slow-down, which essentially means that the server of the first queue (or: upstreanm queue) switches to a lower speed when the second queue (downstream queue) exceeds some threshold. The goal is to assess to what extent such a policy succeeds in protecting the first queue, and therefore we focus on estimating the probability of overflow in the downstream queue.\ud It is known that in this setting importance sampling with traditional state-independent distributions performs poorly. More sophisticated state-dependent schemes can be shown to be asymptotically efficient, but their implementation may be problematic, as for each state the new measure has to be computed. This paper presents an algorithm that is considerably simpler than the fully state-dependent scheme; it requires low computational effort, but still has high efficiency

Dynamic importance sampling for queueing networks

Importance sampling is a technique that is commonly used to speed up Monte Carlo simulation of rare events. However, little is known regarding the design of efficient importance sampling algorithms in the context of queueing networks. The standard approach, which simulates the system using an a priori fixed change of measure suggested by large deviation analysis, has been shown to fail in even the simplest network setting (e.g., a two-node tandem network). Exploiting connections between importance sampling, differential games, and classical subsolutions of the corresponding Isaacs equation, we show how to design and analyze simple and efficient dynamic importance sampling schemes for general classes of networks. The models used to illustrate the approach include $d$-node tandem Jackson networks and a two-node network with feedback, and the rare events studied are those of large queueing backlogs, including total population overflow and the overflow of individual buffers.Comment: Published in at http://dx.doi.org/10.1214/105051607000000122 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

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

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

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

Efficient Heuristics for the Simulation of Buffer Overflow in Series and Parallel Queueing Networks

Many of recent studies have proved the tail equivalence result for Egalitarian Processor Sharing system: [EQUATION], where B (resp. V) is service requirement (resp. sojourn time) of a customer; for PS, g = 1 - ρ. In this paper, we consider time-shared systems in which the server capacity is shared by existing customers in proportion to (dynamic) weights assigned to customers. We consider two systems, 1) in which the weight of a customer depends on it Age (attained service), and 2) in which the weight depends on the residual processing time (RPT). We allow for a parameterized family of weight functions such that the weight associated with a customer that has received a service (or, has a RPT) of x units is ω(x) = xα for some -∞ < α < ∞. We then study the sojourn time of a customer under such scheduling discipline and provide conditions on α for tail equivalence to hold true, and also give the value of g as a function of α