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

Gaussian queues in light and heavy traffic

In this paper we investigate Gaussian queues in the light-traffic and in the heavy-traffic regime. The setting considered is that of a centered Gaussian process $X\equiv\{X(t):t\in\mathbb R\}$ with stationary increments and variance function $\sigma^2_X(\cdot)$, equipped with a deterministic drift $c>0$, reflected at 0: $$Q_X^{(c)}(t)=\sup_{-\infty<s\le t}(X(t)-X(s)-c(t-s)).$$ We study the resulting stationary workload process $Q^{(c)}_X\equiv\{Q_X^{(c)}(t):t\ge0\}$ in the limiting regimes $c\to 0$ (heavy traffic) and $c\to\infty$ (light traffic). The primary contribution is that we show for both limiting regimes that, under mild regularity conditions on the variance function, there exists a normalizing function $\delta(c)$ such that $Q^{(c)}_X(\delta(c)\cdot)/\sigma_X(\delta(c))$ converges to a non-trivial limit in $C[0,\infty)$

Large deviations of Gaussian tandem queues and resulting performance formulae

This paper considers a two-node tandem queue where the cumulative input traffic is modeled as a Gaussian process with stationary increments. By applying (the generalized version of) Schilder's sample-path large-deviations theorem, we derive the many-sources asymptotics of the overflow probabilities in the second queue; `Schilder' reduces this problem into finding the most probable path along which the second queue reaches overflow. The general form of these paths is described by recently obtained results on infinite intersections in Gaussian processes; for the special cases of fractional Brownian motion and integrated Ornstein-Uhlenbeck input, they can be explicitly determined, as well as the corresponding exponential decay rate. As the computation of this decay rate is numerically involved, we introduce an explicit approximation (`rough full-link approximation'). Based on this approximation, we propose performance formulae that could be used, for instance, for network provisioning purposes. Simulation is used to assess the accuracy of the formula

Geometrical bounds on the efficiency of wireless network coding

This paper explores wireless network coding both in case of deterministic and random point patterns. Using the Boolean connectivity model we provide upper bounds for the maximum encoding number, i.e., the number of packets that can be combined such that the corresponding receivers are able to decode. For the models studied, this upper bound is of order √N, where N denotes the (mean) number of neighbours. Our simulations show that the √N law is applicable to small-sized networks as well. Moreover, achievable encoding numbers are provided for grid-like networks where we obtain the multiplicative constants analytically. Building on the above results, we provide an analytic expression for the upper bound of the efficiency of wireless network coding. The conveyed message is that it is favourable to reduce computational complexity by relying only on small encoding numbers, for example, XORing only pairs, as the resulting throughput loss is typically small. © 2013 IFIP