Giovanni Luca Torrisi

Large Deviations of Poisson Cluster Processes

In this paper we prove scalar and sample path large deviation principles for a large class of Poisson cluster processes. As a consequence, we provide a large deviation principle for ergodic Hawkes point processes.

Generalised shot noise Cox processes

We introduce a class of Cox cluster processes called generalised shot noise Cox processes (GSNCPs), which extends the definition of shot noise Cox processes (SNCPs) in two directions: the point process that drives the shot noise is not necessarily Poisson, and the kernel of the shot noise can be random. Thereby, a very large class of models for aggregated or clustered point patterns is obtained. Due to the structure of GSNCPs, a number of useful results can be established. We focus first on deriving summary statistics for GSNCPs and, second, on how to simulate such processes. In particular, results on first- and second-order moment measures, reduced Palm distributions, the J-function, simulation with or without edge effects, and conditional simulation of the intensity function driving a GSNCP are given. Our results are exemplified in important special cases of GSNCPs, and we discuss their relation to the corresponding results for SNCPs.

Stochastic analysis of self-sustainability in peer-assisted VoD systems

We consider a peer-assisted Video-on-demand system, in which video distribution is supported both by peers caching the whole video and by peers concurrently downloading it. We propose a stochastic fluid framework that allows to characterize the additional bandwidth requested from the servers to satisfy all users watching a given video. We obtain analytical upper bounds to the server bandwidth needed in the case in which users download the video content sequentially. We also present a methodology to obtain exact solutions for special cases of peer upload bandwidth distribution. Our bounds permit to tightly characterize the performance of peer-assisted VoD systems as the number of users increases, for both sequential and non-sequential delivery schemes. In particular, we rigorously prove that the simple sequential scheme is asymptotically optimal both in the bandwidth surplus and in the bandwidth deficit mode, and that peer-assisted systems become totally self-sustaining in the surplus mode as the number of users grows large.

Large deviations of the interference in a wireless communication model

Interference from other users limits the capacity, and possibly the connectivity, of wireless networks. In this paper, we consider a simple model of a wireless ad-hoc network where node locations are described by a homogeneous Poisson point process, and node transmission powers are random. We obtain a large deviation principle for the total interference in this model, and relate the speed of this large deviation principle to the tail of the distribution of transmission powers.

Large Deviations of the Interference in the Ginibre Network Model

Under different assumptions on the distribution of the fading random variables, we derive large deviation estimates for the tail of the interference in a wireless network model whose nodes are placed, over a bounded region of the plane, according to the β-Ginibre process, 0 < β ≤ 1. The family of β-Ginibre processes is formed by determinantal point processes, with different degree of repulsiveness. As β → 0, β-Ginibre processes converge in law to a homogeneous Poisson process. In this sense the Poisson network model may be considered as the limiting uncorrelated case of the β-Ginibre network model. Our results indicate the existence of two different regimes. When the fading random variables are bounded or Weibull superexponential, large values of the interference are typically originated by the sum of several equivalent interfering contributions due to nodes in the vicinity of the receiver. In this case, the tail of the interference has, on the log-scale, the same asymptotic behavior for any value of 0 < ...

Lundberg parameters for non standard risk processes

We consider risk processes with delayed claims in a Markovian environment, and we study the asymptotic behaviour of finite and infinite horizon ruin probabilities under the small claim assumption. We also consider multivariate risk processes of the same kind, and we give upper and lower bounds for the Lundberg parameters of the corresponding total reserve. Our results have strong analogies with those one in the paper by Juri (Super modular order and Lundberg exponents, 2002).

Performance analysis of non-stationary peer-assisted VoD systems

We analyze a peer-assisted Video-on-Demand system in which users contribute their upload bandwidth to the redistribution of a video that they are downloading or that they have cached locally. Our target is to characterize the additional bandwidth that servers must supply to immediately satisfy all requests to watch a given video. We develop an approximate fluid model to compute the required server bandwidth in the sequential delivery case. Our approach is able to capture several stochastic effects related to peer churn, upload bandwidth heterogeneity, non-stationary traffic conditions, which have not been documented or analyzed before. We provide an analytical methodology to design efficient peer-assisted VoD systems and optimal resource allocation strategies.

A class of risk processes with reserve-dependent premium rate: sample path large deviations and importance sampling

Let (X(t)) be a risk process with reserve-dependent premium rate, delayed claims and initial capital u. Consider a class of risk processes {(Xε (t)): ε > 0} derived from (X(t)) via scaling in a slow Markov walk sense, and let Ψ_ε(u) be the corresponding ruin probability. In this paper we prove sample path large deviations for (X ε (t)) as ε → 0. As a consequence, we give exact asymptotics for log Ψ_ε(u) and we determine a most likely path leading to ruin. Finally, using importance sampling, we find an asymptotically efficient law for the simulation of Ψ_ε(u).

Density Estimation of Functionals of Spatial Point Processes with Application to Wireless Networks

In this paper we provide a Monte Carlo algorithm for the density estimation of functionals of spatial point processes on Lipschitz domains with random marks, using the Malliavin calculus. Our method allows us to explicitly compute the Malliavin weight and is applied to density estimation of the interference in a wireless ad hoc network model. This extends and makes more precise, and under stronger assumptions, some recent results of Privault and Wei [J. Appl. Probab., 44 (2007), pp. 806–823], who dealt with the particular case of the half-line.

Peer-Assisted VoD Systems: An Efficient Modeling Framework

We analyze a peer-assisted Video-on-Demand (VoD) system in which users contribute their upload bandwidth to the redistribution of a video that they are downloading or that they have cached locally. Our target is to characterize the additional bandwidth that servers must supply to immediately satisfy all requests to watch a given video. We develop an approximate fluid model to compute the required server bandwidth in the sequential delivery case, as well as in controlled nonsequential swarms. Our approach is able to capture several stochastic effects related to peer churn, upload bandwidth heterogeneity, and nonstationary traffic conditions, which have not been documented or analyzed before. Finally, we provide important hints for the design of efficient peer-assisted VoD systems under server capacity constraints.