Probabilistic sampling of finite renewal processes

Consider a finite renewal process in the sense that interrenewal times are positive i.i.d. variables and the total number of renewals is a random variable, independent of interrenewal times. A finite point process can be obtained by probabilistic sampling of the finite renewal process, where each renewal is sampled with a fixed probability and independently of other renewals. The problem addressed in this work concerns statistical inference of the original distributions of the total number of renewals and interrenewal times from a sample of i.i.d. finite point processes obtained by sampling finite renewal processes. This problem is motivated by traffic measurements in the Internet in order to characterize flows of packets (which can be seen as finite renewal processes) and where the use of packet sampling is becoming prevalent due to increasing link speeds and limited storage and processing capacities.Comment: Published in at http://dx.doi.org/10.3150/10-BEJ321 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Modelling the Duration of Multihop Paths in Mobile Ad Hoc Networks

Mobile ad hoc networks are characterized by having nodes that are cooperative and communicate without any kind of infrastructure. The mobility and multihop capability of these networks leads the network topology to change rapidly and unpredictably; this aspect must be incorporated in effective models to describe the dynamics of multihop paths.\newline When modeling the duration of multihop paths, a great part of the literature assumes that the links of multihop paths behave independently. This simplifies the modeling and reduces the complexity of computations. However, each link shares a common node with each of its neighbor links, turning the independent link assumption ge-nerally not valid. In this paper, we use a piecewise deterministic Markov model that characterizes the random behaviour of a multihop path not assuming independent links. We obtain the mean path duration of multihop paths and compare the results for the used model with the ones obtained by assuming independent links. Numerical results illustrate that independent link approximation results underestimate the mean path duration, with the most significant differences being observed with low node mobility and higher path durations

Stochastic networks with multiple stable points

This paper analyzes stochastic networks consisting of a set of finite capacity sites where different classes of individuals move according to some routing policy. The associated Markov jump processes are analyzed under a thermodynamic limit regime, that is, when the networks have some symmetry properties and when the number of nodes goes to infinity. An intriguing stability property is proved: under some conditions on the parameters, it is shown that, in the limit, several stable equilibrium points coexist for the empirical distribution. The key ingredient of the proof of this property is a dimension reduction achieved by the introduction of two energy functions and a convenient mapping of their local minima and saddle points. Networks with a unique equilibrium point are also presented.Comment: Published in at http://dx.doi.org/10.1214/009117907000000105 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Analysis of loss networks with routing

This paper analyzes stochastic networks consisting of finite capacity nodes with different classes of requests which move according to some routing policy. The Markov processes describing these networks do not, in general, have reversibility properties, so the explicit expression of their invariant distribution is not known. Kelly's limiting regime is considered: the arrival rates of calls as well as the capacities of the nodes are proportional to a factor going to infinity. It is proved that, in limit, the associated rescaled Markov process converges to a deterministic dynamical system with a unique equilibrium point characterized by a nonstandard fixed point equation.Comment: Published at http://dx.doi.org/10.1214/105051606000000466 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Perturbation Analysis of a Variable M/M/1 Queue: A Probabilistic Approach

Motivated by the problem of the coexistence on transmission links of telecommunication networks of elastic and unresponsive traffic, we study in this paper the impact on the busy period of an M/M/1 queue of a small perturbation in the server rate. The perturbation depends upon an independent stationary process (X(t)) and is quantified by means of a parameter \eps \ll 1. We specifically compute the two first terms of the power series expansion in \eps of the mean value of the busy period duration. This allows us to study the validity of the Reduced Service Rate (RSR) approximation, which consists in comparing the perturbed M/M/1 queue with the M/M/1 queue where the service rate is constant and equal to the mean value of the perturbation. For the first term of the expansion, the two systems are equivalent. For the second term, the situation is more complex and it is shown that the correlations of the environment process (X(t)) play a key role

Small and large scale behavior of moments of poisson cluster processes

Poisson cluster processes are special point processes that find use in modeling Internet traffic, neural spike trains, computer failure times and other real-life phenomena. The focus of this work is on the various moments and cumulants of Poisson cluster processes, and specifically on their behavior at small and large scales. Under suitable assumptions motivated by the multiscale behavior of Internet traffic, it is shown that all these various quantities satisfy scale free (scaling) relations at both small and large scales. Only some of these relations turn out to carry information about salient model parameters of interest, and consequently can be used in the inference of the scaling behavior of Poisson cluster processes. At large scales, the derived results complement those available in the literature on the distributional convergence of normalized Poisson cluster processes, and also bring forward a more practical interpretation of the so-called slow and fast growth regimes. Finally, the results are applied to a real data trace from Internet traffic.NSA grant [H98230-13-1-0220]info:eu-repo/semantics/publishedVersio

Upstream traffic capacity of a WDM EPON under online GATE-driven scheduling

Passive optical networks are increasingly used for access to the Internet and it is important to understand the performance of future long-reach, multi-channel variants. In this paper we discuss requirements on the dynamic bandwidth allocation (DBA) algorithm used to manage the upstream resource in a WDM EPON and propose a simple novel DBA algorithm that is considerably more efficient than classical approaches. We demonstrate that the algorithm emulates a multi-server polling system and derive capacity formulas that are valid for general traffic processes. We evaluate delay performance by simulation demonstrating the superiority of the proposed scheduler. The proposed scheduler offers considerable flexibility and is particularly efficient in long-reach access networks where propagation times are high

On the minimum hop count and connectivity in one-dimensional ad hoc wireless networks

This paper investigates connectivity in one-dimensional ad hoc networks by means of the distribution of the minimum hop count between source and destination nodes. We derive the exact probability distribution of the minimum hop count from the location density of relay nodes in the multihop path selected with the Most Forward within Radius (MFR) scheme. The probability that the source and destination nodes are connected (provided by Ghasemi and Nader-Esfahani [IEEE Commun. Lett. 10(4):251–253, 2006]) can be obtained by summing the probability masses for each possible value of the minimum hop count, which provides new insights to the connectivity probability. Numerical results show the effect of the number of nodes and the transmission range on the minimum hop count