Large Deviation Analysis Research Articles

The Shape of the Loss Curve and the Impact of Long-Range Dependence on Network Performance

Empirical studies showed that many types of network traffic exhibit long-range dependence (LRD), i.e., burstiness on a wide variety of time-scales. Given that traffic streams are indeed endowed with LRD properties, a next question is: what is their impact on network performance? To assess this issue, we consider a generic source model: traffic generated by an individual user is modeled as a fluid on/off pattern with generally distributed on- and off-times; LRD traffic is obtained by choosing the on-times heavy-tailed. We focus on an aggregation of many i.i.d. sources, say n, multiplexed on a FIFO queue, with the queueing resources scaled accordingly. Large deviations analysis says that the (steady-state) overflow probability decays exponentially in n; we call the corresponding decay rate, as a function of the buffer size B, the loss curve. To get insight into the influence of the distribution of the on- and off-times, we list the most significant properties of the loss curve. Strikingly, for small B, the decay rate depends on the distributions only through their means. For large B there is no such insensitivity property. In case of heavy-tailed on-times, the decay of the loss probability in the buffer size is slower than exponential; this is in stark contrast with light-tailed on-times, in which case this decay is at least exponential. To assess the sensitivity of the performance metrics to the probabilistic properties of the input, we compute the loss curve for a number of representative examples (voice, video, file transfer, web browsing, etc.), with realistic distributions and parameters. Our conclusions on the impact of LRD on the performance can be summarized as follows: (1) If the maximally tolerable delay is relatively small, there is hardly any difference between heavy-tailed and light-tailed inputs; this gives a theoretical handle on observations that appeared in the literature. Only for very delay tolerant applications the above-mentioned large B results kick in. (2) The level of aggregation is a significant factor. If the ratio between the link rate and the peak rate of a single source is high, a high utilization can be achieved, while at the same time the delay requirements are met; this holds even if the delay requirements are stringent.

Large-deviations analysis of the fluid approximation for a controllable tandem queue

A fluid approximation gives the main term in the asymptotic expression of the value function for a controllable stochastic network. The policies that have the same asymptotic of their value functions as the value function of the optimal policy are called asymptotically optimal policies. We consider the problem of finding from this set of asymptotically optimal policies a best one in the sense that the next term of its asymptotic expression is minimal. The analysis of this problem is closely connected with large-deviations problems for a random walk.

Large Deviation Analysis of Subexponential Waiting Times in a Processor-Sharing Queue

We investigate the distribution of the waiting time V in a stable M/G/1 processor-sharing queue with traffic intensity ρ < 1. When the distribution of a customer service request B belongs to a large class of subexponential distributions with tails heavier than e−√x, it is shown that [Formula: see text] Furthermore, we demonstrate that the preceding relationship does not hold if the service distribution has a lighter tail than e−√x.

Exponentially hard problems are sometimes polynomial, a large deviation analysis of search algorithms for the random satisfiability problem, and its application to stop-and-restart resolutions.

A large deviation analysis of the solving complexity of random 3-satisfiability instances slightly below threshold is presented. While finding a solution for such instances demands an exponential effort with high probability, we show that an exponentially small fraction of resolutions require a computation scaling linearly in the size of the instance only. This exponentially small probability of easy resolutions is analytically calculated, and the corresponding exponent is shown to be smaller (in absolute value) than the growth exponent of the typical resolution time. Our study therefore gives some theoretical basis to heuristic stop-and-restart solving procedures, and suggests a natural cutoff (the size of the instance) for the restart.

Efficient simulation of a tandem Jackson network

The two-node tandem Jackson network serves as a convenient reference model for the analysis and testing of different methodologies and techniques in rare event simulation. In this paper we consider a new approach to efficiently estimate the probability that the content of the second buffer exceeds some high level L before it becomes empty, starting from a given state. The approach is based on a Markov additive process representation of the buffer processes, leading to an exponential change of measure to be used in an importance sampling procedure. Unlike changes of measures proposed and studied in recent literature, the one derived here is a function of the content of the first buffer. We prove that when the first buffer is finite, this method yields asymptotically efficient simulation for any set of arrival and service rates. In fact, the relative error is bounded independent of the level L ; a new result which is not established for any other known method. When the first buffer is infinite, we propose a natural extension of the exponential change of measure for the finite buffer case. In this case, the relative error is shown to be bounded (independent of L ) only when the second server is the bottleneck; a result which is known to hold for some other methods derived through large deviations analysis. When the first server is the bottleneck, experimental results using our method seem to suggest that the relative error is bounded linearly in L .

A large deviations analysis of the transient of a queue with many Markov fluid inputs

This article analyzes the transient buffer content distribution of a queue fed by a large number of Markov fluid sources. We characterize the probability of overflow at time t , given the current buffer level and the number of sources in the on-state. After scaling buffer and bandwidth resources by the number of sources n , we can apply large deviations techniques. The transient overflow probability decays exponentially in n . In case of exponential on/off sources, we derive an expression for the decay rate of the rare event probability under consideration. For general, Markov fluid sources, we present a plausible conjecture. We also provide the "most likely path" from the initial state to overflow (at time t ). Knowledge of the decay rate and the most likely path to overflow leads to (i) approximations of the transient overflow probability, and (ii) efficient simulation methods of the rare event of buffer overflow. The simulation methods, based on importance sampling, give a huge speed-up compared to straightforward simulations. The approximations are of low computational complexity, and accurate, as verified by means of simulation experiments.

Effective bandwidth and associated CAC procedure for traffic streams multiplexed over a VBR link

In this paper we introduce the definition of the effective bandwidth for traffic streams that are multiplexed over a VBR link. We assume an arbitrary traffic mix multiplexed into a FIFO buffer, which has a service rate, regulated by VBR traffic parameters as opposed to the traditional CBR server capacity. In order to analytically obtain the generalized effective bandwidth we define an equivalent queuing model, which reduces the problem of the variable service rate multiplexer to a constant rate server. Following the Large Deviation Analysis of our equivalent queuing model we derive the effective bandwidth in terms of VBR link parameters. One of the appealing features of our method is that any dimensioning, resource allocation schemes that are based on the effective bandwidth concept but originally have been devised for CBR links can be easily extended to be applicable for VBR links as well. For example, one can easily build a Connection Admission Control (CAC) algorithm for a VBR link by reusing the existing methods based on the effective bandwidth concept. In this paper we also give an example for a CAC algorithm and for other practical application scenarios.

Large deviation analysis for multiplexing independent regulated inputs

In this paper, we consider the performance analysis problem for a work conserving link with a large number of independent regulated inputs. For such a problem, we derive simple stochastic bounds under a general traffic constraint for the inputs. The bound for queue length is shown to be a stochastic extension of the deterministic worst case bound and it is asymptotically tighter than the bound in Kesidis and Konstantopoulos [5]. We also test the bound by considering periodic inputs with independent starting phases. Based on importance sampling, we propose a fast simulation algorithm that achieves significant variance reduction. The simulations results are compared with our stochastic bound and the bound [5].

Explicit Solutions for Variational Problems in the Quadrant

We study a variational problem (VP) that is related to semimartingale reflecting Brownian motions (SRBMs). Specifically, this VP appears in the large deviations analysis of the stationary distribution of SRBMs in the d-dimensional orthant Rd+. When de2, we provide an explicit analytical solution to the VP. This solution gives an appealing characterization of the optimal path to a given point in the quadrant and also provides an explicit expression for the optimal value of the VP. For each boundary of the quadrant, we construct a “cone of boundary influence”, which determines the nature of optimal paths in different regions of the quadrant. In addition to providing a complete solution in the 2-dimensional case, our analysis provides several results which may be used in analyzing the VP in higher dimensions and more general state spaces.

On the Wulff crystal in the Ising model

We study the phase separation phenomenon in the Ising model in dimensions $d \geq 3$. To this end we work in a large box with plus boundary conditions and we condition the system to have an excess amount of negative spins so that the empirical magnetization is smaller than the spontaneous magnetization $m^*$. We confirm the prediction of the phenomenological theory by proving that with high probability a single droplet of the minus phase emerges surrounded by the plus phase. Moreover, the rescaled droplet is asymptotically close to a definite deterministic shape, the Wulff crystal, which minimizes the surface free energy. In the course of the proof we establish a surface order large deviation principle for the magnetization. Our results are valid for temperatures $T$ below a limit of slab-thresholds $\hat{T}_c$ conjectured to agree with the critical point $T_c$. Moreover, $T$ should be such that there exist only two extremal translation invariant Gibbs states at that temperature, a property which can fail for at most countably many values and which is conjectured to be true for every $T$. The proofs are based on the Fortuin–Kasteleyn representation of the Ising model along with coarse-graining techniques.To handle the emerging macroscopic objects we employ tools from geometric measure theory which provide an adequate framework for the large deviation analysis. Finally,we propose a heuristic picture that for subcritical temperatures close enough to $T_c$, the dominant minus spin cluster of the Wulff droplet permeates the entire box and has a strictly positive local density everywhere.

A large deviation analysis of errors in measurement based admission control to buffered and bufferless resources

In measurement based admission control, measured traffic parameters are used to determine the maximum number of connections that can be admitted to a resource within a given quality constraint. The assumption that the measured parameters are the true ones can compromise admission controls measured parameters are random quantities, causing additional variability. This paper analyzes the impact of measurement error within the framework of Large Deviation theory. For a class of admission controls, large deviation principles are established for the number of admitted connections, and for the attained overflow rates. These are applied to admission to bufferless resources, and buffered resources in both the many sources and large buffer asymptotic. The sampling properties of effective bandwidths are presented, together with a discussion the impact of the temporal extent of individual samples on estimator variability. Sample correlations are shown to increase estimator variances procedures to make admission control robust with respect to these are described.

The theory of large deviations: from Boltzmann’s 1877 calculation to equilibrium macrostates in 2D turbulence

After presenting some basic ideas in the theory of large deviations, this paper applies the theory to a number of problems in statistical mechanics. These include deriving the form of the Gibbs state for a discrete ideal gas; describing probabilistically the phase transition in the Curie–Weiss model of a ferromagnet; and deriving variational formulas that describe the equilibrium macrostates in models of two-dimensional turbulence. A general approach to the large deviation analysis of models in statistical mechanics is also formulated.

Asymptotically Optimal Importance Sampling and Stratification for Pricing Path‐Dependent Options

This paper develops a variance reduction technique for Monte Carlo simulations of path‐dependent options driven by high‐dimensional Gaussian vectors. The method combines importance sampling based on a change of drift with stratified sampling along a small number of key dimensions. The change of drift is selected through a large deviations analysis and is shown to be optimal in an asymptotic sense. The drift selected has an interpretation as the path of the underlying state variables which maximizes the product of probability and payoff—the most important path. The directions used for stratified sampling are optimal for a quadratic approximation to the integrand or payoff function. Indeed, under differentiability assumptions our importance sampling method eliminates variability due to the linear part of the payoff function, and stratification eliminates much of the variability due to the quadratic part of the payoff. The two parts of the method are linked because the asymptotically optimal drift vector frequently provides a particularly effective direction for stratification. We illustrate the use of the method with path‐dependent options, a stochastic volatility model, and interest rate derivatives. The method reveals novel features of the structure of their payoffs.

Large deviations analysis of the generalized processor sharing policy

In this paper we consider a stochastic server (modeling a multiclass communication switch) fed by a set of parallel buffers. The dynamics of the system evolve in discrete-time and the generalized processor sharing (GPS) scheduling policy of l25r is implemented. The arrival process in each buffer is an arbitrary, and possibly autocorrelated, stochastic process. We obtain a large deviations asymptotic for the buffer overflow probability at each buffer. In the standard large deviations methodology, we provide a lower and a matching (up to first degree in the exponent) upper bound on the buffer overflow probabilities. We view the problem of finding a most likely sample path that leads to an overflow as an optimal control problem. Using ideas from convex optimization we analytically solve the control problem to obtain both the asymptotic exponent of the overflow probability and a characterization of most likely modes of overflow. These results have important implications for traffic management of high-speed networks. They extend the deterministic, worst-case analysis of l25r to the case where a detailed statistical model of the input traffic is available and can be used as a basis for an admission control mechanism.

Rare event analysis of the state frequencies of a large number of Markov chains

This paper focuses on the situation of a large number n of i.i.d. Markov chains. The fractions of these n Markov chains being in any specific state are called state frequencies (or:‘empirical distribution’). We provide a large deviations analysis of the probability of these state frequencies reaching a distribution β from a given initial distribution α in a fixed amount of time t. This probability is characterizered asymptotically in the number of Markov chains. Apart from these asymptotics, it is also shown how the rare event occurs: the ‘most likely path’ from α to β in t units of time is explicitly calculated. It is argued that the analysis provides useful insight into the likelihood of extreme fluctuations of the input rate of an ATM switching element or an IP router. A number of numerical examples are included

A Skorokhod Problem formulation and large deviation analysis of a processor sharing model

Generalized processor sharing has been proposed as a policy for distributing processing in a fair manner between different data classes in high-speed networks. In this paper we show how recent results on the Skorokhod Problem can be used to construct and analyze the mapping that takes the input processes into the buffer content. More precisely, we show how to represent the map in terms of a Skorokhod Problem, and from this infer that the mapping is well defined (existence and uniqueness) and well behaved (Lipschitz continuity). As an elementary application we present some large deviation estimates for a many data source model.

Large-deviation analysis of Burgers turbulence with white-noise initial data

The statistics of the solution to the inviscid Burgers equation are investigated when the initial velocity is a white-noise function. Explicit representations for the probability distribution functions (PDFs) of the velocity and the distance between two successive shocks are found in terms of the solution to a PDE system. By using these expressions and large-deviation theory, tight bounds are found for the tails of these distributions and for that of the shock-strength distribution. These bounds give the exact rates of decay for the tails. © 1998 John Wiley & Sons, Inc.

Counterexamples in importance sampling for large deviations probabilities

A guiding principle in the efficient estimation of rare-event probabilities by Monte Carlo is that importance sampling based on the change of measure suggested by a large deviations analysis can reduce variance by many orders of magnitude. In a variety of settings, this approach has led to estimators that are optimal in an asymptotic sense. We give examples, however, in which importance sampling estimators based on a large deviations change of measure have provably poor performance. The estimators can have variance that decreases at a slower rate than a naive estimator, variance that increases with the rarity of the event, and even infinite variance. For each example, we provide an alternative estimator with provably efficient performance. A common feature of our examples is that they allow more than one way for a rare event to occur; our alternative estimators give explicit weight to lower probability paths neglected by leading-term asymptotics.

Large deviation analysis of the single server queue

We establish the large deviation principle (LDP) for the virtual waiting time and queue length processes in the GI/GI/1 queue. The rate functions are found explicitly. As an application, we obtain the logarithmic asymptotics of the probabilities that the virtual waiting time and queue length exceed high levels at large times. Additional new results deal with the LDP for renewal processes and with the derivation of ‘unconditional’ LDPs for ‘conditional ones’. Our approach applies in large deviations ideas and methods of weak convergence theory.

Large deviation analysis of the single server queue

Large deviation analysis of the single server queue