arXiv subjects

Christine Fricker

Publications and source records attributed to Christine Fricker.

25 records · Page 2Linked to original sources

arXiv2009

Analysis of a Bloom Filter Algorithm via the Supermarket Model

This paper deals with the problem of identifying elephants in the Internet Traffic. The aim is to analyze a new adaptive algorithm based on a Bloom Filter. This algorithm uses a so-called min-rule which can be described as in the supermarket model. This model consists of joining the shortest queue among d queues selected at random in a large number of m queues. In case of equality, one of the shortest queues is chosen at random. An analysis of a simplified model gives an insight of the error generated by the algorithm on the estimation of the number of the elephants. The main conclusion is that, as m gets large, there is a deterministic limit for the empirical distribution of the filter counters. Limit theorems are proved and the limit is identified. It depends on key parameters. The condition for the algorithm to perform well is discussed. Theoretical results are validated by experiments on a traffic trace from France Telecom and by simulations.

cs.DM↗

arXiv2008

Perturbation analysis of an M/M/1 queue in a diffusion random environment

We study in this paper an $M/M/1$ queue whose server rate depends upon the state of an independent Ornstein-Uhlenbeck diffusion process $(X(t))$ so that its value at time $t$ is $μϕ(X(t))$, where $ϕ(x)$ is some bounded function and $μ>0$. We first establish the differential system for the conditional probability density functions of the couple $(L(t),X(t))$ in the stationary regime, where $L(t)$ is the number of customers in the system at time $t$. By assuming that $ϕ(x)$ is defined by $ϕ(x) = 1-\varepsilon ((x\wedge a/\varepsilon)\vee(-b/\varepsilon))$ for some positive real numbers $a$, $b$ and $\varepsilon$, we show that the above differential system has a unique solution under some condition on $a$ and $b$. We then show that this solution is close, in some appropriate sense, to the solution to the differential system obtained when $ϕ$ is replaced with $Φ(x)=1-\varepsilon x$ for sufficiently small $\varepsilon$. We finally perform a perturbation analysis of this latter solution for small $\varepsilon$. This allows us to check at the first order the validity of the so-called reduced service rate approximation, stating that everything happens as if the server rate were constant and equal to $μ(1-\eps\E(X(t)))$.

cs.NI↗

arXiv2008

Inference of Flow Statistics via Packet Sampling in the Internet

We show in this note that by deterministic packet sampling, the tail of the distribution of the original flow size can be obtained by rescaling that of the sampled flow size. To recover information on the flow size distribution lost through packet sampling, we propose some heuristics based on measurements from different backbone IP networks. These heuristic arguments allow us to recover the complete flow size distribution.

cs.NI↗

arXiv2007

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.

math.PR↗

arXiv2007

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.

cs.NI↗

arXiv2006

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.

cs.NI↗

arXiv2006

Integration of streaming services and TCP data transmission in the Internet

We study in this paper the integration of elastic and streaming traffic on a same link in an IP network. We are specifically interested in the computation of the mean bit rate obtained by a data transfer. For this purpose, we consider that the bit rate offered by streaming traffic is low, of the order of magnitude of a small parameter \eps \ll 1 and related to an auxiliary stationary Markovian process (X(t)). Under the assumption that data transfers are exponentially distributed, arrive according to a Poisson process, and share the available bandwidth according to the ideal processor sharing discipline, we derive the mean bit rate of a data transfer as a power series expansion in \eps. Since the system can be described by means of an M/M/1 queue with a time-varying server rate, which depends upon the parameter \eps and process (X(t)), the key issue is to compute an expansion of the area swept under the occupation process of this queue in a busy period. We obtain closed formulas for the power series expansion in \eps of the mean bit rate, which allow us to verify the validity of the so-called reduced service rate at the first order. The second order term yields more insight into the negative impact of the variability of streaming flows.

cs.NI↗