Fabrice Guillemin

A Markovian analysis of additive-increase multiplicative-decrease algorithms

The additive-increase multiplicative-decrease (AIMD) schemes designed to control congestion in communication networks are investigated from a probabilistic point of view. Functional limit theorems for a general class of Markov processes that describe these algorithms are obtained. The asymptotic behaviour of the corresponding invariant measures is described in terms of the limiting Markov processes. For some special important cases, including TCP congestion avoidance, an important autoregressive property is proved. As a consequence, the explicit expression of the related invariant probabilities is derived. The transient behaviour of these algorithms is also analysed.

Jitter in ATM networks and its impact on peak rate enforcement

Abstract Cells arriving to an ATM network experience random delays due to queueing in upstream multiplexing stages, notably in customer premises. This is the phenomenon of jitter and the aim of the present paper is to study its influence on peak rate enforcement. We first introduce some general characterizations of jitter and then, describe two models of jittered flows based on simple queueing systems. We discuss the objectives of peak rate enforcement and study the impact of jitter on the dimensioning of jumping window and leaky bucket mechanisms. A useful synthetic characterization of jitter appears to be a remote quantile of the cell delay distribution expressed in units of the initial inter-cell interval.

Tail asymptotics for processor-sharing queues

The basic queueing system considered in this paper is the M/G/1 processor-sharing queue with or without impatience and with finite or infinite capacity. Under some mild assumptions, a criterion for the validity of the reduced-service-rate approximation is established when service times are heavy tailed. This result is applied to various models based on M/G/1 processor-sharing queues.

The formal theory of birth-and-death processes, lattice path combinatorics and continued fractions

Classic works of Karlin and McGregor and Jones and Magnus have established a general correspondence between continuous-time birth-and-death processes and continued fractions of the Stieltjes-Jacobi type together with their associated orthogonal polynomials. This fundamental correspondence is revisited here in the light of the basic relation between weighted lattice paths and continued fractions otherwise known from combinatorial theory. Given that sample paths of the embedded Markov chain of a birth-and-death process are lattice paths, Laplace transforms of a number of transient characteristics can be obtained systematically in terms of a fundamental continued fraction and its family of convergent polynomials. Applications include the analysis of evolutions in a strip, upcrossing and downcrossing times under flooring and ceiling conditions, as well as time, area, or number of transitions while a geometric condition is satisfied.

Experimental analysis of the impact of peer‐to‐peer applications on traffic in commercial IP networks

To evaluate the impact of peer-to-peer (P2P) applications on traffic in wide area networks, we analyze measurements from a high speed IP backbone link carrying TCP traffic towards several ADSL areas. The first observations are that the prevalent part of traffic is due to P2P applications (almost 80% of total traffic) and that the usage of network becomes symmetric in the sense that customers are not only clients but also servers. This latter point is observed by the significant proportion of long flows mainly composed of ACK segments. When analyzing the bit rate created by long flows, it turns out that the TCP connections due to P2P applications have a rather small bit rate and that there is no evidence for long range dependence. These facts are intimately related to the way P2P protocols are running. We separately analyze signaling traffic and data traffic. It turns out that by adopting a suitable level of aggregation, global traffic can be described by means of usual tele-traffic models based on M/G/∞ queues with Weibullian service times. Copyright

Analysis of ADSL traffic on an IP backbone link

Measurements from an Internet backbone link carrying TCP traffic towards different ADSL areas are analyzed. For traffic analysis, we adopt a flow based approach and the popular mice/elephants dichotomy, where mice refer to short traffic transfers and elephants to long transfers. The originality of the reported experimental data, when compared with previous measurements from very high speed backbone links, is that the commercial traffic includes a significant part generated by peer-to-peer applications. This kind of traffic exhibits some remarkable properties in terms of mice and elephants, as we describe. It turns out that by adopting a suitable level of aggregation, the bit rate of mice can be described by means of a Gaussian process. The bit rate of elephants is smoother than that of mice and can also be well approximated by a Gaussian process.

Jitter and bandwidth enforcement

Cells arriving at an ATM (asynchronous transfer mode) network experience random delays due to queuing in upstream multiplexing stages. The alteration of the initial periodic nature of a constant bit rate cell stream due to such delays is the phenomenon of jitter. An assumption of Markovian dependence between the delays of successive cells makes it possible to calculate various quantities of interest for characterizing the jitter. This assumption is appropriate for two ATM multiplex models, and it was possible to investigate the effects of different source and multiplex parameters on the degree of jitter introduced. Jumping window and leaky bucket mechanisms have been dimensioned to enforce the peak rate of jittered streams. The leaky bucket is seen to be considerably more responsive than the jumping window. A synthetic characterization of jitter consists in the remote delay quantile divided by the period of the considered stream. This is sufficient to dimension the leaky bucket for one of the considered multiplex models and supports the heuristic approach of G. Niestegge (1990).<<ETX>>

Analysis of the M / M /1 Queue with Processor Sharing via Spectral Theory

We show in this paper that the computation of the distribution of the sojourn time of an arbitrary customer in a M/M/1 with the processor sharing discipline (abbreviated to M/M/1 PS queue) can be formulated as a spectral problem for a self-adjoint operator. This approach allows us to improve the existing results for this queue in two directions. First, the orthogonal structure underlying the M/M/1 PS queue is revealed. Second, an integral representation of the distribution of the sojourn time of a customer entering the system while there are n customers in service is obtained.

Impact of peer-to-peer applications on wide area network traffic: an experimental approach

To evaluate the impact of peer-to-peer applications on traffic in wide area IP networks, we analyze measurements from a high speed backbone link carrying TCP traffic towards several ADSL areas. The first observations are that the prevalent part of traffic is due to peer-to-peer applications (almost 80% of total traffic) and that the usage of network becomes symmetric in the sense that customers are not only clients but also servers. This latter point is observed by the significant proportion of long flows mainly composed of ACK segments. When analyzing the bit rate created by long flows, it turns out that those TCP connections due to peer-to-peer applications have a rather small bit rate and that there is no evidence for long range dependence. These facts are intimately related to the way peer-to-peer protocols are running.

A flow-based approach to modeling ADSL traffic on an IP backbone link

Measurements from an Internet backbone link carryingtcp traffic towards differentadsl areas are analyzed in this paper. For traffic analysis, we adopt a flow-based approach and the popular mice/elephants dichotomy. The originality of the experimental data reported in this paper, when compared with previous measurements from very high speed backbone links, is that commercial traffic comprises a significant part due to peer-to-peer applications. This kind of traffic exhibits some remarkable properties in terms of mice, elephants and bit rates, which are thoroughly described in this paper. Global traffic is actually decomposed into several flow components on the basis of the mice/elephants dichotomy. Mice due p2p protocols and mice due to classical Internet applications such ashttp, ftp, etc. are analyzed separately. It turns out that by adopting a suitable level of aggregation, global traffic can be described by means of usual tele-traffic models based on M/G/∞ queues with Weibullian service times. The blobal bit rate can then be approximated by the superposition of Gaussian processes perturbed by a white noise.RésuméOn analyse dans cet article des mesures effectuées sur un lien d’un réseau dorsalip transportant du trafictcp vers différentes plaquesadsl. Pour le dépouillement des données, on adopte une approche par flots et la fameuse dichotomie souris/éléphants. L’originalité des mesures analysées ici, en comparaison avec des mesures publiées antérieurement dans la littérature technique, réside dans le fait qu’une grande partie du trafic est engendrée par des applications pair à pair (peer-to-peer, p2p). Ce type de trafic présente des propriétés remarquables en termes de débit et de caractéristiques des souris et des éléphants, qui sont étudiées en détail. Le trafic global est décomposé en plusieurs composantes sur la base des souris et des éléphants. Les souris engendrées par les protocoles p2p et celles par les applications classiques de l’Internet commehttp, ftp, etc. sont examinées séparément. Il s’avère qu’en adoptant un certain niveau d’agrégation des flots, le trafic global peut être décrit à l’aide de modèles classiques du télétrafic, à savoir des files d’attente M/G/∞ avec des temps de service distribués suivant des lois de Weibull. Le débit global peut alors être représenté par la superposition de processus gaussiens perturbés par un bruit blanc.