Large deviations for acyclic networks of queues with correlated Gaussian inputs

We consider an acyclic network of single-server queues with heterogeneous processing rates. It is assumed that each queue is fed by the superposition of a large number of i.i.d. Gaussian processes with stationary increments and positive drifts, which can be correlated across different queues. The flow of work departing from each server is split deterministically and routed to its neighbors according to a fixed routing matrix, with a fraction of it leaving the network altogether. We study the exponential decay rate of the probability that the steady-state queue length at any given node in the network is above any fixed threshold, also referred to as the "overflow probability". In particular, we first leverage Schilder's sample-path large deviations theorem to obtain a general lower bound for the limit of this exponential decay rate, as the number of Gaussian processes goes to infinity. Then, we show that this lower bound is tight under additional technical conditions. Finally, we show that if the input processes to the different queues are non-negatively correlated, non short-range dependent fractional Brownian motions, and if the processing rates are large enough, then the asymptotic exponential decay rates of the queues coincide with the ones of isolated queues with appropriate Gaussian inputs

Large deviations of infinite intersections of events in Gaussian processes

The large deviations principle for Gaussian measures in Banach space is given by the generalized Schilder's theorem. After assigning a norm ||f|| to paths f in the reproducing kernel Hilbert space of the underlying Gaussian process, the probability of an event A can be studied by minimizing the norm over all paths in A. The minimizing path f*, if it exists, is called the most probable path and it determines the corresponding exponential decay rate. The main objective of our paper is to identify the most probable path for the class of sets A that are such that the minimization is over a closed convex set in an infinite-dimensional Hilbert space. The `smoothness' (i.e., mean-square differentiability) of the Gaussian process involved has a crucial impact on the structure of the solution. Notably, as an example of a non-smooth process, we analyze the special case of fractional Brownian motion, and the set A consisting of paths f at or above the line t in [0,1]. For H>1/2, we prove that there is an s such that

Statistical Service Guarantees for Traffic Scheduling in High-Speed Data Networks

School of Electrical and Computer Engineerin

Asymptotic analysis of Lévy-driven tandem queues

We analyze tail asymptotics of a two-node tandem queue with spectrally-positive Lévy input. A first focus lies in the tail probabilities of the type P(Q 1>α x,Q 2>(1−α)x), for α∈(0,1) and x large, and Q i denoting the steady-state workload in the ith queue. In case of light-tailed input, our analysis heavily uses the joint Laplace transform of the stationary buffer contents of the first and second queue; the logarithmic asymptotics can be expressed as the solution to a convex programming problem. In case of heavy-tailed input we rely on sample-path methods to derive the exact asymptotics. Then we specialize in the tail asymptotics of the downstream queue, again in case of both light-tailed and heavy-tailed Lévy inputs. It is also indicated how the results can be extended to tandem queues with more than two nodes

Self-similar traffic and network dynamics

Copyright © 2002 IEEEOne of the most significant findings of traffic measurement studies over the last decade has been the observed self-similarity in packet network traffic. Subsequent research has focused on the origins of this self-similarity, and the network engineering significance of this phenomenon. This paper reviews what is currently known about network traffic self-similarity and its significance. We then consider a matter of current research, namely, the manner in which network dynamics (specifically, the dynamics of transmission control protocol (TCP), the predominant transport protocol used in today's Internet) can affect the observed self-similarity. To this end, we first discuss some of the pitfalls associated with applying traditional performance evaluation techniques to highly-interacting, large-scale networks such as the Internet. We then present one promising approach based on chaotic maps to capture and model the dynamics of TCP-type feedback control in such networks. Not only can appropriately chosen chaotic map models capture a range of realistic source characteristics, but by coupling these to network state equations, one can study the effects of network dynamics on the observed scaling behavior. We consider several aspects of TCP feedback, and illustrate by examples that while TCP-type feedback can modify the self-similar scaling behavior of network traffic, it neither generates it nor eliminates it.Ashok Erramilli, Matthew Roughan, Darryl Veitch and Walter Willinge