This is an overview of a relatively recent application of long-range dependence (LRD) to the area of communication networks, in particular to problems concerned with the dynamic nature of packet flows in high-speed data networks such as the Internet. We demonstrate that this new application area offers unique opportunities for significantly advancing our understanding of LRD and related phenomena. These advances are made possible by moving beyond the conventional approaches associated with the wide-spread "black-box" perspective of traditional time series analysis and exploiting instead the physical mechanisms that exist in the networking context and that are intimately tied to the observed characteristics of measured network traffic. In order to describe this complexity we provide a basic understanding of the design, architecture and operations of data networks, including a description of the TCP/IP protocols used in today's Internet. LRD is observed in the large scale behavior of the data traffic and we provide a physical explanation for its presence. LRD tends to be caused by user and application characteristics and has little to do with the network itself. The network affects mostly small time scales, and this is why a rudimentary understanding of the main protocols is important. We illustrate why multifractals may be relevant for describing some aspects of the highly irregular traffic behavior over small time scales. We distinguish between a time-domain and wavelet-domain approach to analyzing the small time scale dynamics and discuss why the wavelet-domain approach appears to be better suited than the time-domain approach for identifying features in measured traffic (e.g., relatively regular traffic patterns over certain time scales) that have a direct networking interpretation (e....

The theory of large deviations provides a simple unified basis for statistical mechanics, information theory and queueing theory. The objective of this paper is to use large deviation theory and the Laplace method of integration to provide an simple intuitive overview of the recently developed theory of effective bandwidth for high speed digital networks, especially ATM networks. This includes (i) identification of the appropriate energy function, entropy function and effective bandwidth function of a source, (ii) the calculus of the effective bandwidth functions, (iii) bandwidth allocation and buffer management, (iv) traffic descriptors, and (v) envelope processes and conjugate processes for fast simulations and bounds.

We consider a multiclass multiplexer with support for multiple service classes, and dedicated buffers for each service class. Under specific scheduling policies for sharing bandwidth among these classes, we seek the asymptotic (as the buffer size goes to infinity) tail of the buffer overflow probability for each dedicated buffer. We assume dependent arrival and service processes as is usually the case in models of bursty traffic. 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 introduce a novel optimal control approach to address these problems. In particular, we relate the lower bound derivation to a deterministic optimal control problem, which we explicitly solve. Optimal state trajectories of the control problem correspond to typical congestion scenarios. We explicitly and in detail characterize the most likely modes of overflow. We specialize our results to the ...

This paper describes an efficient technique for estimating, via simulation, the probability of buffer overows in a queueing model that arises in the analysis of ATM (Asynchronous Transfer Mode) communication switches. There are multiple streams of (autocorrelated) traffic feeding the switch that has a buffer of finite capacity. Each stream is designated as either being of high or low priority. When the queue length reaches a certain threshold, only high priority packets are admitted to the switch's buffer. The problem is to estimate the loss rate of high priority packets. An asymptotically optimal importance sampling approach is developed for this rare event simulation problem. In this approach, the importance sampling is done in two distinct phases. In the first phase, an importance sampling change of measure is used to bring the queue length up to the threshold at which low priority packets get rejected. In the second phase, a different importance sampling change of measure is used to move the queue length from the threshold to the buffer capacity.

In an earlier paper [MR] the authors introduced the inverse measure y (dt) of a given measure (dt) on [0; 1] and presented the `inversion formula' f y (ff) = fff(1=ff) which was argued to link the respective multifractal spectra of and y . A second paper [RM2] established the formula under the assumption that and y are continuous measures. Here, we investigate the general case which reveals telling details of interest to the full understanding of multifractals. Subjecting self-similar measures to the operation 7! y creates a new class of discontinuous multifractals. Calculating explicitly we find that the inversion formula holds only for the `fine multifractal spectra' and not for the `coarse' ones. As a consequence, the multifractal formalism fails for this class of measures. A natural explanation is found when drawing parallels to equilibrium measures. In the context of our work it becomes natural to consider the degenerate Holder exponents 0 and 1.

Abstract—The performance of Neyman–Pearson detection of correlated random signals using noisy observations is considered. Using the large deviations principle, the performance is analyzed via the error exponent for the miss probability with a fixed false-alarm probability. Using the state-space structure of the signal and observation model, a closed-form expression for the error exponent is derived using the innovations approach, and the connection between the asymptotic behavior of the optimal detector and that of the Kalman filter is established. The properties of the error exponent are investigated for the scalar case. It is shown that the error exponent has distinct characteristics with respect to correlation strength: for signal-to-noise ratio (SNR) I, the error exponent is monotonically decreasing as the correlation becomes strong whereas for SNR I there is an optimal correlation that maximizes the error exponent for a given SNR. Index Terms—Autoregressive process, correlated signal, error exponent, Gauss–Markov model, Neyman–Pearson detection. I.

New results on the speed of spread of the one-dimensional spatial branching process are described. Generalizations to the multitype case and to d dimensions are discussed. The relationship of the results with deterministic theory is also indicated. Finally the theory developed is used to re-prove smoothly (and improve slightly) results on certain data-storage algorithms arising in computer science.

The tail behaviour of the limit of the normalized population size in the simple supercritical branching process, W, is studied. Most of the results concern those cases when a tail of the distribution function of W decays exponentially quickly. In essence, knowledge of the behaviour of transforms can be combined with some `large deviation' theory to get detailed information on the oscillation of the distribution function of W near zero or at infinity. In particular we show how an old result of Harris (1948) on the asymptotics of the moment generating function of W translates to tail behaviour.

this paper either #(x) = p -1 |x| p , for some p > 0 or #(x) = e x 1. We also see that under certain conditions, the rate function in the LDP for some certain stochastic processes has the form I(z) = # # # # # # # # M #(z # (t)) dt, if z(0) = 0 and z is absolutely continuous otherwise

We consider the problem of Quality of Service (QoS) provisioning in modern highspeed, multimedia, communication networks. We quantify QoS by the probabilities of loss and excessive delay of an arbitrary packet, and introduce the model of a multiclass node (switch) which provides network access to users that may belong to multiple service classes. We treat such a node as a stochastic system which we analyze and control. In particular, we develop an analytical approach to estimate both the delay and the buffer overflow probability per service class, based on ideas from large deviations and optimal control. We exploit these performance analysis results by devising a call admission control algorithm which can provide per class QoS guarantees. We compare the proposed approach to alternative worst-case and effective bandwidth-based schemes and argue that it leads to increased efficiency. Finally, we discuss extensions to the network case in order to provide end-to-end QoS guarantees. Key w...