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 $\mu \phi(X(t))$, where $\phi(x)$ is some bounded function and $\mu>0$. 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 $\phi(x)$ is defined by $\phi(x) = 1-\varepsilon ((x\wedge a/\varepsilon)\vee(-b/\varepsilon))$ 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 $\phi$ is replaced with $\Phi(x)=1-\varepsilon x$ 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 \mu(1-\eps\E(X(t)))

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

La nano-intégration : un besoin d'adaptation des procédés de fabrication actuels

L'arrivée des nanomatériaux parmi les objets du quotidien semble porteuse de fonctions d'usage innovantes, mais pose de nombreux problèmes de mise en œuvre lors de l'intégration de particules nanométriques dans des procédés de fabrication standardisés. Il semble donc impératif d'adapter les procédés de transformation habituels

An identification problem in an urn and ball model with heavy tailed distributions

We consider in this paper an urn and ball problem with replacement, where balls are with different colors and are drawn uniformly from a unique urn. The numbers of balls with a given color are i.i.d. random variables with a heavy tailed probability distribution, for instance a Pareto or a Weibull distribution. We draw a small fraction $p\ll 1$ of the total number of balls. The basic problem addressed in this paper is to know to which extent we can infer the total number of colors and the distribution of the number of balls with a given color. By means of Le Cam's inequality and Chen-Stein method, bounds for the total variation norm between the distribution of the number of balls drawn with a given color and the Poisson distribution with the same mean are obtained. We then show that the distribution of the number of balls drawn with a given color has the same tail as that of the original number of balls. We finally establish explicit bounds between the two distributions when each ball is drawn with fixed probability $p$

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

On the Statistical Characterization of Flows in Internet Traffic with Application to Sampling

A new method of estimating some statistical characteristics of TCP flows in the Internet is developed in this paper. For this purpose, a new set of random variables (referred to as observables) is defined. When dealing with sampled traffic, these observables can easily be computed from sampled data. By adopting a convenient mouse/elephant dichotomy also dependent on traffic, it is shown how these variables give a reliable statistical representation of the number of packets transmitted by large flows during successive time intervals with an appropriate duration. A mathematical framework is developed to estimate the accuracy of the method. As an application, it is shown how one can estimate the number of large TCP flows when only sampled traffic is available. The algorithm proposed is tested against experimental data collected from different types of IP networks

Micro-Nano : des comportements différents, des procédés à adapter

6 pagesL'arrivée de nanomatériaux parmi les différents matériaux utilisés en pyrotechnie pose de nombreux problèmes de mise en œuvre. En effet, leurs propriétés texturales particulières ne permettent pas de les utiliser comme des matériaux classiques. Il est impératif d'adapter les procédés de mise en œuvre habituels. Pour obtenir l'adéquation de ces nouveaux matériaux et de nos procédés, des études spécifiques du comportement des nanomatériaux au cours des procédés de mise en œuvre par coulée ont été réalisés. L'observation des écarts de comportement des nanomatériaux par rapport aux "micro-matériaux" pour une même nature chimique et des conditions identiques de mise en œuvre permet d'établir des lois de comportement spécifiques, fonction de la taille des particules manipulées. Les mentalités, comme les procédés, doivent évoluer et prendre en compte l'élévation de surface spécifique qu'offre le passage à l'échelle nanométrique. Ainsi un chargement moindre en nanomatériaux ne sera pas forcement synonyme de lacunes sur les caractéristiques du produit fini (propriétés mécaniques ou pyrotechniques). Deux questions peuvent alors se poser : faut-il adapter les formulations aux procédés, en prenant en compte les caractéristiques des matières premières ou bien adapter des procédés devant supporter des taux de chargement élevés en nanomatériaux ? Le manque de connaissance et le recul sur le comportement des nanomatériaux sont un frein à leur intégration dans des composés pyrotechniques mais les propriétés exhibées laissent présagées d'un grand intérêt en vue de l'optimisation des propriétés des composés fabriqués

A Flow Time Model for Melt-Cast Insensitive Explosive Process

International audienceDiphasic flows of concentrated suspensions of melt-cast insensitive explosives exhibit specific rheological properties. In order to limit the handling of pyrotechnical products presenting a risk with respect to the mechanical and thermal shocks, a lot of work has been undertaken for many years in the civil engineering sector. The objective of this study is to propose a predictive model of the flow time of a concentrated suspension through a nozzle located at the bottom of a tank. Similar to our industrial process, the suspension is made out of insensitive energetic materials and flows under gravity. Experimental results are compared to three models (Quemada, Krieger-Dougherty, and Mooney) predicting the viscosity μ of a suspension as a function of the solid volume fraction ϕ, the maximum packing density ϕm and the viscosity μ0 of the interstitial liquid. De Larrard's model is used to calculate ϕm. The value of viscosity measured for the pure liquid is close to the one predicted by the Bernoulli theorem, where liquids are considered as incompressible and inviscid. Finally, it was found that the Quemada's model gives a fair agreement between predictions and experiments

Adaptive algorithms for identifying large flows in IP traffic

We propose in this paper an on-line algorithm based on Bloom filters for identifying large flows in IP traffic (a.k.a. elephants). Because of the large number of small flows, hash tables of these algorithms have to be regularly refreshed. Recognizing that the periodic erasure scheme usually used in the technical literature turns out to be quite inefficient when using real traffic traces over a long period of time, we introduce a simple adaptive scheme that closely follows the variations of traffic. When tested against real traffic traces, the proposed on-line algorithm performs well in the sense that the detection ratio of long flows by the algorithm over a long time period is quite high. Beyond the identification of elephants, this same class of algorithms is applied to the closely related problem of detection of anomalies in IP traffic, e.g., SYN flood due for instance to attacks. An algorithm for detecting SYN and volume flood anomalies in Internet traffic is designed. Experiments show that an anomaly is detected in less than one minute and the targeted destinations are identified at the same time