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

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

The effective bandwidth problem revisited

The paper studies a single-server queueing system with autonomous service and $\ell$ priority classes. Arrival and departure processes are governed by marked point processes. There are $\ell$ buffers corresponding to priority classes, and upon arrival a unit of the $k$th priority class occupies a place in the $k$th buffer. Let $N^{(k)}$, $k=1,2,...,\ell$ denote the quota for the total $k$th buffer content. The values $N^{(k)}$ are assumed to be large, and queueing systems both with finite and infinite buffers are studied. In the case of a system with finite buffers, the values $N^{(k)}$ characterize buffer capacities. The paper discusses a circle of problems related to optimization of performance measures associated with overflowing the quota of buffer contents in particular buffers models. Our approach to this problem is new, and the presentation of our results is simple and clear for real applications.Comment: 29 pages, 11pt, Final version, that will be published as is in Stochastic Model

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

Proofs for some conjectures of Rajaratnam and Takawira on the peakedness of handoff traffic

The purpose of this paper is to supplement a recent paper by M. Rajaratnam and F. Takawira (see ibid., vol.49, p.817-34, 2000), which deals with a model for the performance analysis of cellular mobile networks. The key performance measure is a second-order characteristic (peakedness) of the traffic stream that serves as a model for handoff traffic. We show that this quantity may be obtained by evaluating an explicit formula rather than by solving a set of equations. This result enables us to verify some conjectures formulated by Rajaratnam and Takawira on the basis of numerical experiments. We also show the uniqueness of the solution to a system of nonlinear equations, required in the performance analysis, as conjectured by Rajaratnam and Takawira

Bounds for expected supremum of fractional Brownian motion with drift

We provide upper and lower bounds for the mean ${\mathscr M}(H)$ of $\sup_{t\geqslant 0} \{B_H(t) - t\}$, with $B_H(\cdot)$ a zero-mean, variance-normalized version of fractional Brownian motion with Hurst parameter $H\in(0,1)$. We find bounds in (semi-)closed-form, distinguishing between $H\in(0,\frac{1}{2}]$ and $H\in[\frac{1}{2},1)$, where in the former regime a numerical procedure is presented that drastically reduces the upper bound. For $H\in(0,\frac{1}{2}]$, the ratio between the upper and lower bound is bounded, whereas for $H\in[\frac{1}{2},1)$ the derived upper and lower bound have a strongly similar shape. We also derive a new upper bound for the mean of $\sup_{t\in[0,1]} B_H(t)$, $H\in(0,\tfrac{1}{2}]$, which is tight around $H=\tfrac{1}{2}$.Comment: 16 pages, 3 figure

Proofs for some conjectures of Rajaratnam and Takawira

The purpose of this note is to supplement a recent paper by Rajaratnam and Takawira ({\it IEEE Trans. Vehicular Technol.} {\bf 49} (2000) 817-834), which deals with a model for the performance analysis of cellular mobile networks. We show that the key performance quantity may be obtained by evaluating an explicit formula rather than by solving a set of equations. This result enables us to verify some conjectures formulated by Rajaratnam and Takawira on the basis of numerical experiments. We also show uniqueness of the solution to a system of nonlinear equations, required in the performance analysis, as conjectured by Rajaratnam and Takawira. \u