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

Backpressure-based control protocols: design and computational aspects

Congestion control in packet-based networks is often realized by feedback protocols. In this paper we assess their performance under a back-pressure mechanism that has been proposed and standardized for Ethernet metropolitan networks. In such a mechanism the service rate of an upstream queue is reduced when the downstream queue is congested, in order to protect the downstream queue. We study a Markovian model that captures the essentials of the protocol, but at the same time allows for numerical analysis. We first derive explicit results for the stability condition of the model (which turns out to be nontrivial). Then we present logarithmic estimates of the probability of buffer overflow in the second queue, which are subsequentially used when devising an efficient simulation procedure based on importance sampling. We conclude the paper by presenting a number of numerical results, and some general design guidelines

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

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

An efficient multilevel splitting scheme

Rare event analysis has been attracting continuous and growing attention over the past decades. It has many possible applications in different areas, e.g., queueing theory, insurance, engineering etc. As explicit expressions are hard to obtain, and asymptotic approximations often lack error bounds, one often applies simulation methods to obtain performance measures of interest.\ud Obviously, the use of standard Monte Carlo simulation for estimating rare event probabilities has an inherent problem: it is extremely time consuming to obtain reliable estimates since the number of samples needed to obtain an estimate of a certain predefined accuracy is inversely proportional to the probability of interest. Two important techniques to speed up simulations are Importance Sampling (IS) and Multilevel Splitting (MS). \ud IS prescribes to simulate the system under a new probability measure such that the event of interest occurs more frequently, and corrects the simulation output by means of likelihood ratios to retain unbiasedness. The likelihood ratios essentially capture the likelihood of the realization under the old measure with respect to the new measure. The choice of a ‘good’ new measure is rather delicate; in fact only measures that are asymptotically efficient are worthwile to consider. We refer to [3] for more background on IS and its pitfalls. \ud The other technique, multilevel splitting (MS), is conceptually easier, in the sense that one can simulate under the normal probability measure. When a sample path of the process is simulated, this is viewed as the path of a ‘particle’. When the particle approaches the target set to a certain distance, the particle splits into a number of new particles, each of which is then simulated independently of each other. This process may repeat itself several times, hence the term multilevel splitting. Typically, the states where particles should be split are determined by selecting a number of level sets of an importance function f. Every time a particle (sample path) crosses the next level set of the importance function f, it is split. The splitting factor (i.e. the number of particles that replaces the original particle) may depend on the current level.\ud The challenge is to choose an importance function that will ensure that the probability of reaching the target set is roughly the same for all states that belong to the same level. Moreover, choosing the splitting factors appropriately is also important, see [1, 2]. Sample paths will hardly ever end up in the rare set if this factor is too small, while the number of particles (and consequently the simulation effort) will grow fast if this factor is too large. For an overview of the MS method see [5].\ud There are not many examples of asymptotically efficient MS schemes for estimating general types of rare events in the present literature. Most articles deal either with effective heuristics for particular (queueing) models, usually providing good estimates without rigorous analysis, see e.g. [6]; or with restrictive models, see e.g. [2]. The recent work in [1] does enable one to construct an asymptotically efficient MS scheme for estimating the probability of first entrance to a rare set, when the decay rate of the probability is known for all starting states. The authors used control-theoretic techniques to derive and prove their results.\ud In this work we also provide a simple and asymptotically efficient MS scheme for estimating the probability of first entrance to some rare set. The scheme can be seen as part of the class of asymptotically efficient MS schemes developed in [1]. However, since we are only interested in easy-to-implement (but still efficient) schemes, we use a fixed, pre-specified splitting factor R, to be used for all levels. This is in contrast to the setting in [1] where the splitting factor may vary between levels and is usually noninteger (which is then implemented by using a randomization procedure). We accompany the scheme with a proof of its asymptotic efficiency which is relatively easy, in the sense that it only uses probabilistic arguments and some simple bounds, thereby giving insight into why the scheme works so well.\ud The rest of the paper’s structure is as follows. In Section 2 we first describe the model of interest and, after a brief review of the MS method, we provide the MS scheme itself. A sketch of the proof of asymptotic efficiency of the scheme is given in Section 3. Supporting numerical results for a two-node tandem model are presented in Section 4 and compared with results from IS on the same model; in fact it turns out that MS can be a good alternative to IS for certain parameter settings.\u

Rare-event simulation for tandem queues: a simple and efficient importance sampling scheme

This paper focuses on estimating the rare event of overflow in the downstream queue of a Jacksonian two-node tandem queue, relying on importance sampling. It is known that in this setting ‘traditional’ state-independent schemes perform poorly. More sophisticated state-dependent schemes yield asymptotic efficiency. Their drawback, however, is that they require a per-state computation of the new measure, so that it still consumes considerable machine time. The contribution of this paper is a scheme that combines asymptotic efficiency with low complexity. It retains the quality of the original state-dependent scheme, but its implementation is almost as simple as for state-independent analogues

Increasing Detection Performance of Surveillance Sensor Networks

We study a surveillance wireless sensor network (SWSN) comprised of small and low-cost sensors deployed in a region in order to detect objects crossing the field of interest. In the present paper, we address two problems concerning the design and performance of an SWSN: optimal sensor placement and algorithms for object detection in the presence of false alarms. For both problems, we propose explicit decision rules and efficient algorithmic solutions. Further, we provide several numerical examples and present a simulation model that combines our placement and detection methods

Winning rate in the full-information best-choice problem

Following a long-standing suggestion by Gilbert and Mosteller, we derive an explicit formula for the asymptotic winning rate in the full-information best-choice problem