TREC is a joint INRIA-ENS project-team.

TREC is a joint INRIA-ENS project-team. It is focused on the modeling and the control of communication networks. Its methodological activities are combined with projects defined with industrial partners, notably Thomson, Alcatel, France Télécom, Intel and Sprint. The main research directions are :

communication network control: admission control, flow regulation, congestion control, traffic analysis in controlled networks;

modeling and performance analysis of wireless networks (cellular, WLAN's, ad-hoc): coverage and load analysis, power control, evaluation and optimization of the transport capacity, self organization;

stochastic network dynamics, in particular by means of algebraic methods, with a main emphasis on rare events and large network asymptotics;

the development of mathematical tools based on stochastic geometry, random graphs and spatial point processes: Voronoi tessellations, coverage processes, random spatial trees, spectral analysis, Gibbs fields.

Here is the scientific content of each of the four main research directions.

Modeling and control of communication networks. Here we mean control of admission, flow regulation and feedback control
*à la TCP*, the understanding and improvements of which are major challenges within the context of large networks. Our aim is a mathematical representation of the dynamics of the most
commonly used control protocols, from which one could predict and optimize the resulting end user bandwidth sharing and QoS. The design of scalable simulators that could be used for the
dimensioning of large IP networks is a first practical outcome of this line of research. We currently try to use our better understanding of the dynamics of these protocols on such
structures as 1) Internet multicast overlays and 2) TCP proxies (also known under the name of Split TCP or TCP splicing) as used in wireless access networks.

Modeling and performance analysis of wireless networks. The main focus is on the following three classes of wireless networks: cellular networks, mobile ad hoc networks (MANETs) and Wifi mesh networks.

Concerning cellular networks, a new mathematical representation of interferences based on shot-noise has led to a variety of results on coverage and capacity of large CDMA networks when taking into account intercell interferences and power control. The mathematical analysis of the interference and power control problems allowed for the definition of new decentralized admission and congestion control protocols. The interest in these algorithms, besides their potential pertinence for network operators, comes from the fact that they allow for explicit evaluation of several macroscopic characteristics of the network. Our general goal is to propose a strategy for the densification and parameterization of UMTS networks that is optimized for both voice and data traffic.

Using a similar approach, in particular additive and extremal shot-noise processes, we currently investigate also MAC layer scheduling algorithms and power control protocols for MANETs. We concentrate on cross layer optimizations allowing one to maximize the transport capacity for multihop MANETs. A recent example within this class of problems is the concept of opportunistic routing for MANETs that we currently study with the Hipercom project team of Rocquencourt.

We also continue the line of thoughts on the self-organization of Wifi mesh networks. The general problem within this context is to find robust and fully distributed algorithms for the selection of channels by access points and for the association of users to access points. We proposed and analyzed a new class of such algorithms based on Gibbs' sampler.

Theory of network dynamics. TREC strongly contributed to the development of new directions of research in queueing theory. The main directions investigated this year concern extensions of product form theory and of the theory of insensitivity within the context of the new bandwidth sharing paradigms that have been proposed in the literature lately.

TREC is also pursuing the elaboration of a stochastic network calculus, that would allow the analysis of network dynamics by algebraic methods. The mathematical tools are those of discrete event dynamical systems: semi-rings (max, plus) and inf-convolutions, as well as their non linear extensions (topical and non expansive maps, monotone separable framework); the main probabilistic tools within this framework are ergodic theory, asymptotic analysis, Lyapounov exponent analysis, perturbation analysis and large deviations. The main current contributions bear on the analysis of rare events within this framework.

Stochastic geometry and the theory of point processes. The theory of point processes on the real line plays a key role in teletraffic analysis. The main mathematical tools studied within this framework are Palm calculus, stochastic intensity and Gibbs fields. Stochastic geometry is particularly useful in all subdomains of communications where planar or spatial components are present: access networks, local loop, multicast trees, distributed games, hierarchical network architectures, in addition to all the wireless network problems listed above. TREC's favorite tools within this framework are Voronoi tessellations, coverage processes, random spatial trees and percolation. See http://www.di.ens.fr/~trec/sg/.

Depending on the classes of communication networks, we focus on different issues:

Concerning the Internet, we concentrate on Internet probing and on the design of Internet overlay networks;

Concerning operator networks, we work on the control and the optimization of both wireless cellular networks and wireline access networks;

Concerning self-organized networks, we focus on the design of MAC and routing protocols and on the evaluation of the capacity.

We interact on these questions with the following industrial partners: Thomson (self organized networks), Alcatel (wireline access), France Télécom (wireless cellular networks) and Sprint (Internet probing and wireless access).

A software called SERT (Spatial Erlang for Real Time services) was designed by M. Karray for the evaluation of various properties of large cdma networks and in particular the probability that calls are blocked due to the unfeasibility of the power control inherent to cdma. This tool is based on the research conducted with FT R&D described in Section and in particular on the results of , and patent pendings , , . This software is now part of the dimensioning tools used by Orange for its UMTS network.

Operators require a methodology for analysing Internet traffic and control protocols to do resource planning (buffer capacities and bandwidth) capable of handling every possible mix of traffic (voice, video and data) with predefined end to end QoS, as well as overlay networks gathering large collections of interacting users and flows. Several research directions are pursued, ranging from the analysis of transport protocols on a single link to that of large overlay multicast or peer-to-peer networks.

This year, we started studying the case of long lived flows controlled by
*scalable TCP*, a new protocol meant to handle very high speed connections. In
, we extended the Mellin transform approach to
this case and derived the distribution of the throughput.

A general solution for the class of transport equations that arise within the context of both persistent and non persistent TCP flows was derived in , within the framework of a collaboration with K.B. Kim. This class contains two cases of loss point process models: the rate-independent Poisson case where the packet loss rate is independent of the throughput of the flow and the rate-dependent case where the point process of losses has an intensity which is a function of the instantaneous rate. In , we also give a direct proof of the fact that there is a unique density solving the associated differential equation and we provide a closed form expression for this density and for its mean value. The use of this approach for the anaysis of HTTP flows is considered in .

A survey on the applications of this class of transport equations to cross-layer optimization was presented at CISS in March .

For certain parameter settings, we observed numerically that the system may reach one of these two regimes or the other depending only on the initial phasing of flows. The fact that a congestion steady state could be reached in cases where an interactionless regime is also possible may be seen as an analogue of turbulence.

If
Nstochastic dynamical systems are coupled together through the use of a common resource one may describe the entire system via a histogram of all
Nstates. In the mean field limit as
Nthis histogram tends to a deterministic density. The publication
provides a new approach to this mean field
convergence problem and tackles the mathematical parts of mean field convergence. This approach which first allowed us to analyse RED is now being extended Drop Tail through one router.
Recently this methodology was extended to a wide class of problems including HTTP on/off sources modeled by the model in
. This extension as well as the mathematics
of the mean field are parts of the PhD
.

The scalability result found for the one-to-many TCP Overlay was extended to a class of dynamical discrete event systems, described by a system of Uniform Recurrence Equations (UREs).
This class includes distributed protocols deployed on a large population of agents. We established that the speed of directional last-passage percolation is positive, if tasks completion
times admit a finite moment of a certain order, and under simple organization condition. This condition is based on simple scalar products and it was shown to be necessary for a system of
dimension
d= 2. We extended this scalability result to a system organized over a general regular graph, under a more restrictive moment condition. The properties of a
stationary regime for some of these infinite discrete event systems were analyzed using hydrodynamic limits.

These results are included in the thesis .

TCP was mainly developed for wired reliable links, where packet losses occur mostly because of congestion. In wireless networks, performance problems arise as TCP often over-reacts to losses due to radio transmission errors. The split-connection approach is an overlay structure that has has been adopted to cope with this problem. In Split TCP, each TCP connection is split into two halves in order to achieve higher throughput. In the model that we developed with S. Foss we considered two long-lived TCP-Reno flows traversing two links with different medium characteristics in cascade. A buffer at the end of the first link will prevent the loss of packets that cannot be immediately forwarded on the second link by storing them temporarily. The target of our study is the characterization of the TCP throughput on both links as well as the buffer occupancy. We gave a representation of the system in terms of ``piecewise-deterministic Markov process'', by referring the theory of PDPs developed by M.H.A. Davis in the '80s. A PDP is a mixture of deterministic motion and random jumps associated to various configuration of the system. We proved stability by exploiting some sample pathwise continuity and monotonicity properties. We also developed an event-driven simulator in order to perform statistical tests devoted to verify the finiteness/infiniteness of certain moments as suggested by the analysis.

P2P networks provide better scalability for the filesharing applications they underlie. Unlike traditional server-based approach such as FTP, maintaining a constant QoS with a fixed number of servers seems feasible, whatever the number of peers involved. However, a P2P filesharing network sometimes happens to saturate, notably in a semi-P2P filesharing architecture or during flashcrowd phase, and scalability may fail. Even "smart" networks can encounter situations where the whole file but one piece is downloaded, which we call starvation. In , we suggested in collaboration with F. Mattieu a simple and versatile filesharing model. It applies to all pieces-oriented filesharing protocols used in softwares such as MlDonkey or BitTorrent. Simulations of this model show that starvation may occur even during flashcrowds. We propose a theoretical explanation for the so-called starvation phenomenon.

This axis concerns the analysis and the design of wireless access communication networks, in particular cellular networks, wireless LANs, MANETs, sensor networks etc. We are interested both in macroscopic models, which are particularly important for economic planning and in models allowing the definition and the optimization of protocols. Our approach combines several tools, queueing theory, point processes, stochastic geometry, random graphs, mean field techniques.

In a series of previous papers co-authored by Mohamed Karray, most of which were published in IEEE Infocom, we developed some scalable load control schemes for large cellular networks with power control. These schemes are decentralized, in that they can be implemented in such a way that each base station only has to consider the load brought by its own users. Moreover, they allow one to control both elastic (data) and a CBR (voice) traffic: CBR traffic has a predefined bit-rate, and has to be regulated by an appropriate admission policy. Our approach applies to large, multi-cell networks; it is based on the sub-stochasticity condition imposed on the relative path-loss matrix introduced in . When the spectral radius of this matrix is less than one, the global feasibility of the power allocation is guaranteed. In , using the sub-stochasticity condition, we established some extension of the classical Erlang's formula for blocking rates in cellular networks with fixed traffic. Our spatial Erlang formula is used by Orange; it was implemented in the SERT dimensioning tools (cf. Section ).

The results described above are derived under the assumption that each user is served by at most one base station. It is important to analyze networks where the base stations are jointly encoding and decoding signals. This kind of cooperation between base stations is called macrodiversity. On the downlink, mobiles are receiving a signal from several antennas and on the uplink, mobiles are sending a signal which is received by several antennas. The gain obtained with macrodiversity is an open issue and no admission control protocol has been proposed for these networks. We have derived some interesting counter intuitive results on macrodiversity. In particular, we have proved that the number of mobiles receiving a signal from more than two different antennas is bounded above by the number of antennas. We can prove that as the number of users grows large, the load of the network is asymptotically equivalent to the number of users times an explicit coefficient which depends on the positions of the base stations. Moreover, there exists a classical cellular network whose load is asymptotically equivalent to the load in the network in macrodiversity: macrodiversity on the downlink does not drastically improve the feasibility of the power allocation problem. However, on the uplink, macrodiversity has a much deeper impact on the network. We have extended the results of Hanly to infinite networks and proved that if macrodiversity is enforced the feasibility condition depends only on the mean number of users per base station and the average bit rate required by users. In complete opposition with what happens on the downlink, the geometry of the network does not play any role. These results have been published in .

Unlike voice calls that are characterized by their duration, data flows are characterized by their size (in bits). The corresponding traffic intensity, defined as the product of the flow
arrival rate by the mean flow size (in bits/s), may well exceed the cell capacity in the sense that the number of data flows increases continuously. Thus one distinguishes two milestones of
the analytical evaluation of the performance of these networks: identification of its
*stability region*; i.e., the maximum traffic intensity such that the system remains stable, and the evaluation of the
*steady state characteristics*; e.g., the mean throughput.

The notion of
*cell capacity*is defined in
as the maximum traffic intensity such that
the system remains stable. We applied this notion to various technologies including TDMA, CDMA and OFDM, and compared the results to the maximum capacity given by the information
theory.

In
we are interested in the capacity of large
networks, and more precisely in the
*maximum number of bits per unit of surface and unit of time*that the network can handle without saturation. Inspired by previously developed scalable load control schemes and the
analysis of macrodiversity (see Section
) we have established a general framework for the
stability analysis of such cellular data networks.

In a joint work
with Mohamed Karray, we systematically
evaluate the performance of the congestion control policies derived from the load control schemes described in Section
. We consider the bit-rate configurations identified by
these schemes as
*feasible sets for some classical, maximal fair resource allocation policies*, and study their performance in the long-term evolution of the system. Specifically, we assume Markovian
arrivals, departures and mobility of customers, which transmit some given data-volumes, as well as some temporal channel variability (fading), and study the
*mean throughput*i.e., the mean bit-rates that the policies offer in different parts of a given cell. Explicit formulas are obtained in the case of
*proportional fair policies*, which may or may-not take advantage of the fading, for
*null*or
*infinitely rapid customer mobility*. This approach applies also to a channel shared by the elastic and a fixed traffic, with predefined customer bit-rates, regulated by the respective
admission policy.

In (joint work with S. Borst) and (joint work with N. Hegde) we examined the potential capacity gains in wireless data networks such as UMTS/HSDPA and CDMA 1xEV-DO from cell coordination which combines inter-cell scheduling and optimal cell selection. The inter-cell scheduling involves coordinating the activity phases of interfering base stations so as to avoid inter-cell interference and boost the transmission rates. The cell selection aims at improving the performance by assigning users to base stations based on load and other relevant considerations in addition to signal strength conditions. We consider a dynamic setting where users come and go over time as governed by the arrival and completion of random finite-size data transfers, and evaluate the capacity gains in terms of the maximum sustainable network throughput for a given spatial traffic pattern. We demonstrate that the relative merits of inter-cell scheduling and cell selection strongly depend on the network topology. In sparse (noise-limited) networks, optimal cell selection achieves substantial capacity gains and equalizes the loads across the various cells, while inter-cell scheduling yields no improvement. In contrast, in dense (interference-limited) networks, both inter-cell scheduling and optimal cell selection produce significant capacity gains, but due to interference, optimal cell selection no longer equalizes the loads and may even lead to strong load imbalances across cells.

Using mean field techniques, we presented in a performance analysis of random back-off algorithms, such as the exponential back-off algorithm, in the case of a finite number of saturated sources. The analysis assumed that all links were interfering with each other. In , , we generalize the results to the case of networks with partial interaction, i.e., to the case where all links do not interfere with each other. To do so, we represent the system as a system of interacting particles with a rapidly varying environment, and develop the mean field analysis of such systems. The results allow us to derive explicit expressions of the throughput of the various links, and we are then able to exactly quantify the well-known problem of unfairness in case of hidden nodes.

Mesh networking is a way to route data, voice and instructions between nodes. Mesh networks differ from cellular networks in that the component parts can all connect to each other via multiple hops, and from mobile ad-hoc networks in that their nodes generally are not mobile.

We consider wireless multihop data networks with random multi-access mechanisms at the MAC layer where the various links shared the same channel. The aim of (joint work with N. Hegde) is to study the performance as perceived by users in a dynamic setting where data flows are generated randomly by users and cease upon completion. This task comprises two major difficulties: first, the behavior of random multi-access algorithms at slot-level in a multi-hop network is even more complex than in the case of a single hop hotspot. Second, in order to study user-level performance, one has to characterize the rate region of the system, defined by the set of rates at which the different active users can generate packets without inducing any instabilities in the network. Since links interact with each other through interference, characterizing the rate region is as difficult as studying the behavior of a set of interacting queues. In addition, the behavior of the congestion control algorithm must be taken into account since it impacts the set of active links and then interference. We propose a model, based on decoupling arguments, that circumvents both difficulties and allows the derivation of explicit expressions for the rate region. Finally, using these expressions we analyze the flow-level performance of the network.

Mesh networking has recently been advocated as an efficient and low-cost approach for providing high speed access to the Internet. Before mesh networks are able to provide high speed access to the Internet, a number of challenges must be addressed including routing, channel and radio interface assignments, radio resource management, and MAC scheduling. Our aim in (joint work with N. Hegde) is to provide guidelines on the design of optimal radio resource management and distributed MAC scheduling algorithms. To this aim, we develop a general analytical model to characterize the performance of such networks. This task comprises many major difficulties: analyzing the behavior of distributed MAC schedulers in multi-hop networks, accounting for the interaction of links through interference and quantifying the impact of underlying congestion control algorithms. We present a unified approach partially based on decoupling arguments to circumvent these difficulties. Specifically, we characterize the rate region of the network, which is the set of rates at which the various data flows can be simultaneously transmitted. Based on the derived results, we can propose a set of provably optimal design rules for mesh multi-channel networks. For example, we investigate how to optimally parameterize the coverage of the RTS/CTS signalling messages in IEE802.11-based networks.

The popularity of IEEE 802.11 WLANs has led to today's dense deployments in urban areas. Such high density leads to sub-optimal performance unless wireless devices interfering in these networks learn how to optimally use and share the spectrum.

We proposed a set of distributed algorithms that allow (i) multiple interfering 802.11 Access Points to select their operating frequency in order to minimize interferences, and (ii) users to choose the Access Point they attach to, in order to get their fair share of the whole network bandwidth. Typical functions (choosing a channel to operate on, choosing an access point to associate with) were shown to be well-addressed in a common optimization framework based on Gibbs' sampler via the minimization of a potential energy function.

This scheme does not require explicit coordination among the wireless devices. For a fixed traffic demand, limited by wireless access, it was shown to achieve a fairness criterion identified in the past as the minimal potential delay . We established the mathematical properties of the proposed algorithms and studied their performance using analytical, event-driven simulations. We discussed implementation requirements and showed that significant benefits can be gained even within incremental deployments and in the presence of non-cooperating wireless clients. We investigated several possibilities for real conditions evaluation.

In addition to the INRIA technical report , two papers, one on self-association and the other on power control in such mesh networks will be presented at IEEE INFOCOM 2007. These papers are the outcome of a collaboration with researchers at INTEL (D. Papagiannaki) and THOMSON (C. Diot, A. Chaintreau and V. Mhatre).

A mobile ad-hoc network (MANET) is made of mobile nodes which are at the same time terminals and routers, connected by wireless links, the union of which forms an arbitrary topology. The nodes are free to move randomly and organize themselves arbitrarily. Important issues in such a scenario are connectivity, medium access (MAC), routing and stability.

In an ongoing work, also with Paul Mühlethaler, we focus on the analysis of routing protocols in multi-hop mobile wireless networks. In particular, we investigate the potential gains of opportunistic routing strategies which take advantage of both time and space diversity compared to classical routing strategies, where packets are routed on a pre-defined route usually obtained by a shortest path routing protocol. In the opportunistic routing scheme we consider, the relay is selected among the nodes having captured the packet transmission (if any) as the node which maximizes the progress of the packet towards the destination. In such a scheme, opportunism consists in taking advantage at each hop of the local pattern of transmission, where locality is understood in both its time and space sense. In our study we use a spatial version of Aloha for the MAC layer, which has been shown to scale well in multi-hop networks and a well established definition for the capture of packets based on the Signal over Interference and Noise Ratio (SINR) model. Our simulation study shows that such an opportunistic scheme very significantly outperforms classical routing schemes. It also shows how to optimally tune the MAC parameters so as to minimize the average number of hops from origin to destination everywhere in the network. This optimization is shown by simulation to be independent of the network density, a property that we back by a mathematical proof based on a scale invariance argument. We submitted our results to a conference organized in 2007.

The stability of decentralized access protocols is an important and difficult problem. In collaboration with Serguei Foss and Vsevolod Shneer we analyze the stability of an Aloha-type
access protocol with a simplified form of spatial interaction
. To the best of our knowledge, no rigorous
results were known in this field. At each time slot, an incoming flow of user arrives in the system and each user in the system tries to emit with a probability which is the inverse of the
number of users in interaction with him (including himself). If a user is the only emitting user in its contention neighborhood, he/she leaves the system. If all users are jointly
interacting, the stability condition for this system is well-known: the maximal arrival intensity is
e^{-1}. Under appropriate technical assumptions, we have proved that the maximal arrival intensity is
e^{-1}where
(0, 1]and
is the mean arrival intensity of users in interaction with a given user. This work paves the way to completely new results in this field.

With advances in coding theory, many techniques have emerged that allow the recovery of the lost information efficiently. Examples of the techniques that require only some fraction of the packets to be received correctly for error-free construction of the complete message include digital fountain and network coding. Moreover, real-time applications can tolerate some loss without much degradation of the quality perceived by the end user. The loss-tolerance can be used to enhance the stability region of the system and also to simplify certain network protocols. In this work in progress, P. Chaporkar and A. Proutière characterize the stability (bounded mean queue length) of such systems and obtain policies that maximize the system throughput (the total number of packets received correctly at the receivers per unit time) while stabilizing the system.

In and in we compare the performance of three usual allocations, namely max-min fairness, proportional fairness and balanced fairness, in a communication network whose resources are shared by a random number of data flows. The model consists of a network of processor-sharing queues. The vector of service rates, which is constrained by some compact, convex capacity set representing the network resources, is a function of the number of customers in each queue. This function determines the way network resources are allocated. We show that this model is representative of a rich class of wired and wireless networks. We give in this general framework the stability condition of max-min fairness, proportional fairness and balanced fairness and compare their performance on a number of toy networks.

In modern computer-science and telecommunication networks, it is advisable to share the resources so that, as for the telephone networks, the performance of these systems does not depend on the fine characteristics of the traffic generated by users. Thus for the data networks, one could identify the allowance of bandwidth ensuring that the average time of transfer of a queue depends only on the intensity of traffic and not on the distribution of the size of queues. This result uses a class of networks of processor-sharing queues for which the property of insensitivity is equivalent to a property of balance known since works of Kelly and Whittle.

We want to identify "the" class of insensitive queueing networks, i.e. to extend the preceding result with disciplines of service more general than processor sharing. A first result indeed shows that a "symmetric" queueing network satisfying the property of balance is insensitive. A more general result shows that this network remains insensitive even if random permutations are allowed at each event, and even if random permutations occur at the times of an independent Poisson process. It remains to show the reciprocal one, i.e. to determine if this class constitutes the set of insensitive queueing networks. The results were presented at the conferences MCQT 06 ( http://www.mat.ucm.es/~mcqt/confe06/conf06.html) and ``Journées MAS de la SMAI, Modèles Spatiaux'' ( http://math.univ-lille1.fr/~mas2006/).

A network belongs to the monotone separable class if its state variables are homogeneous and monotone functions of the epochs of the arrival process. This framework contains several
classical queueing network models, including generalized Jackson networks, max-plus networks, polling systems, multiserver queues, and various classes of stochastic Petri nets. We use
comparison relationships between networks of this class with i.i.d. driving sequences and the
GI/
GI/1/1queue to obtain the tail asymptotics of the stationary maximal dater under light-tailed assumptions for service times
. The exponential rate of decay is given as a
function of a logarithmic moment generating function. We exemplify an explicit computation of this rate for the case of queues in tandem under various stochastic assumptions.

In a paper with Ton Dieker we extended previous results obtained with Serguei Foss in . We studied the stationary solution to a (max, plus)-linear recursion. Under subexponentiality assumptions on the input to the recursion, we obtained the tail asymptotics of certain (max, plus)-linear functionals of this solution. In the event graph setting, two special cases of our results are of particular interest and have already been investigated in the literature. Firstly, the functional may correspond to the end-to-end response time of the event graph. Secondly, for two queues in tandem, the functional may correspond to the sojourn time in the second queue. Our results allow for more general networks; we illustrated this by studying the asymptotics of the resequencing delay due to multi-path routing.

Due to random delays over different paths in a system, the packets or updates may arrive at the receiver in a different order than their chronological order. In such a case, a resequencing buffer at the receiver has to store disordered packets temporarily. In , we analyze both the waiting time of a packet in the resequencing buffer and the size of this resequencing queue. We derive the exact asymptotics for the large deviation of these quantities under heavy-tailed assumptions. In contrast with results obtained for light-tailed distributions, we show that there exists several ``typical paths'' that lead to the large deviation. We derive explicitly these different ``typical paths'' and give heuristic rules for an optimal balancing.

Active probing began by measuring end-to-end path metrics, such as delay and loss, in a direct measurement process which did not require inference of internal network parameters. The field has since progressed to measuring network metrics, from link capacities to available bandwidth and cross traffic itself, which reach deeper and deeper into the network and require increasingly complex inversion methodologies. The paper is an outcome of a collaboration with S. Machiraju, D. Veitch, and J. Bolot. In this paper, we proposed inversion formulas based on queueing theory allowing one to analyze the law of cross traffic in a router from the time series of the end-to-end delays experienced by probes. We also investigated the limitations of such inversion formulas. We used the resulting insight to design practical estimators for cross traffic, which we tested in simulation and validated by using router traces.

In active probing, PASTA is invoked to justify the sending of probe packets (or trains) at Poisson times in a variety of contexts. However, due to the diversity of aims and analysis techniques used in active probing, the benefits of Poisson based measurement, and the utility and role of PASTA, are unclear. With the same group of authors (S. Machiraju is now at SPRINT), and using a combination of rigorous results, examples and counter-examples, we have shown that PASTA is of very limited use in active probing. In particular, Poisson probes are not unique in their ability to sample without bias. Furthermore, PASTA ignores the issue of estimation variance, and the central need for an inversion phase to estimate the quantity of interest based on what is directly observable. These issues are addressed in where we also give concrete examples of when Poisson probes should not be used, and explain why, and offer initial guidelines on suitable alternative sending processes.

Active probing suffers presently of the ``Bottleneck'' limitation: all characteristics of the path after the bottleneck link are unreachable with current techniques. The botleneck link erases all the later effects. In a joint work with Darryl Veitch, we are currently investigating a new tomography technique, based on the measurements of end-to-end delays, which should allow one to have access to several hidden metrics such as the available bandwidth for every link on the path.

This year our work within this framework is focused on percolation models, on spanning trees, both random and minimal ones, on spectral properties of random matrices associated with random graphs of the Euclidean space.

Connectivity is probably the first issue that has to be addressed when considering large-scale MANETs. The mathematical analysis of this problem involves random graphs associated with various models typically driven by Poisson point processes on the plane. Percolation properties of these graphs (existence of a giant component) are interpreted as an indication that the connectivity of the ad hoc network scales well with the size. Probably the first percolation model explicitly proposed for wireless communication networks, was studied by Gilbert already in 1961 (see ). It is now considered as the classical continuum model in percolation theory and accepted in wireless communications as such, despite the fact that it ignores the interference effect that arises when many transmitters are active at the same time. Recently the use of shot-noise processes allowed us to study the impact of interferences on the connectivity of large-scale ad-hoc networks using percolation theory (see , ). An important observation is that, contrary to Gilbet's model, connectivity is not always improved by densification.

The existing model assumes constant emitted powers and bi-directional connections. Current work in this domain (in TREC) concerns extensions to random (in particular controlled) powers and uni-directional communications.

We have defined and analyzed a new model in geometric probability: the radial spanning tree (RST). The RST is a spanning tree over a locally finite point set. It has a distinctive root taken as the origin and a simple local structure: the ancestor of a vertex is the closest point from the vertex which is closer than the vertex to the origin. When this point set is a Poisson point process on the plane we have performed a careful analysis of its main features. This work is the object of a forthcoming publication in the Annals of Applied Probability.

This work has been extended to the more general framework of the analysis of decentralized spanning trees over Poisson point processes. This type of structures appears naturally in routing issues in wireless networks and in peer to peer networks. This work on trees has led to a research report , a publication in the proceedings of Spaswin and a submission. Such trees are also considered in the thesis .

Freshman calculus tells us how to find a minimum
x_{*}of a smooth function
f(
x): set the derivative
and check
. The related series expansion tells us, for points
xnear to
x_{*}, how the distance
= |
x-
x_{*}|relates to the difference
=
f(
x)-
f(
x_{*})in
f-values:
scales as
^{2}. This
*scaling exponent*2 persists for functions
: if
x_{*}is a local minimum and
(
): = min{
f(
x)-
f(
x_{*}):|
x-
x_{*}| =
), then
(
)scales as
^{2}for a generic smooth function
f.

Combinatorial optimization, exemplified by the
*traveling salesman problem*(TSP), is traditionally viewed as a quite distinct subject, with theoretical analysis focusing on the number of steps that algorithms require to find the
optimal solution. To make a connection with calculus, compare an arbitrary tour
xthrough
npoints with the optimal (minimum-length) tour
x_{*}, by considering the two quantities

where
s(
n)is the length of the minimum length tour. Now define
_{n}(
)to be the minimum value of
_{n}(
x)over all tours
xfor which
_{n}(
x)
. Although the function
_{n}(
)will depend on
nand the problem instance, we anticipate that for typical instances drawn from a suitable probability model it will converge in the
nlimit to some deterministic function
(
). The
*universality*paradigm from statistical physics suggests there might be a scaling exponent
defined by

and that the exponent should be robust under model details.

There is fairly strong evidence that for TSP the scaling exponent is 3. This is based on analytic methods in a
*mean-field*model of interpoint distances (distances between pairs of points are random, independent for different pairs, thus ignoring geometric constraints) and on Monte-Carlo
simulations for random points in 2, 3 and 4 dimensional space. The analytic results build upon a recent probabilistic reinterpretation of the work of Krauth and Mézard establishing the
average length of mean-field TSP tours. But neither part of these TSP assertions is rigorous, and indeed rigorous proofs in
ddimensions seem far out of reach of current methodology. In contrast, for the
*minimum spanning tree*(MST) problem, a standard algorithmically easy problem, a simple heuristic argument strongly suggests that the scaling exponent is 2 for any reasonable probability
model. The goal of the paper
, in collaboration with D. Aldous is to work
through the details of a rigorous proof.

Random matrices have a large field of applications in communication networks and they have been used successfully for example in information theory and network epidemics. We have started
to investigate the potential applications of random matrices in geometric probability. Following the seminal work of
, we have analyzed the spectral properties of
Euclidean random matrices. We draw
npoints in a compact set and the entry
(
i,
j)of such matrices is a function of the distance between points
iand
j. If this function is an indicator function, this matrix is the adjacency matrix of the geometric graph. In
we prove rigorously some of the results stated
in
and we derive new formulas for the spectral
radius of these matrices and for the empirical measure of their eigenvalues. In particular, we relate the spectrum of Euclidean random matrices to the Fourier transform of the function of the
distance between two points.

This work is the first step of an ongoing research project. From the work of , , we may infer a relation between the critical percolation threshold and the limiting empirical measure of the eigenvalues of the Laplacian matrix of a graph. We are currently trying to extend the results to a larger class which include the Laplacian matrix of a geometric graph. The aim is to obtain a new characterization of the percolation threshold based on spectral properties.

With Giovanni Luca Torrisi we worked on the computation of the derivatives of functionals of Poisson point processes in . A paper has been submitted for publication in Advances in Applied Probability. We have derived Monte Carlo methods to estimate these derivatives. This work is motivated by the sensitivity analysis of well known structures in geometric probability such as the cluster containing the origin in continuum percolation in the subcritical regime or the typical Poisson Voronoi cell. In one dimension we have also applied our results to some stopping times in a Poisson shot-noise. The importance sampling technique allows, through a change of measure, to write this problem in terms of derivatives of functional of Poisson point process. Applications of this work include sensitivity analysis of ruin probability in insurance mathematics and tail probability of waiting time in a M/GI/1 queue.

In another work, we have computed the large deviation principle for the number of points of Poisson cluster process in a large set. The rate function is explicitly computed and we have derived in particular new formulas for the void probability of a large set and for the number of points in a large set for a Hawkes process. This work has been submitted to Stochastic Models.

Explicit performance evaluation of stochastic geometry models of wireless communication networks is almost entirely confined to scenarios with stationary (homogeneous) Poisson repartition of objects. Adopting homogeneous scenarios in communication models, however, is often too simplistic, since it ignores the spatial fluctuations of the traffic. Modeling of inhomogeneity is not an easy task; adequate and tractable non-homogeneous models are yet to be identified.

We considered Poisson-Poisson cluster processes in the context of the SINR coverage process developed in , as they still allow for many explicit formulas and at the same time offer much more flexibility in modeling than the class of homogeneous Poisson processes. This work was reported in .

Mathematical tractability often requires exponential distributions, which are also some kind of simplifying assumption (even if compatible with Rayleigh fading). We also considered phase-type emitted power distributions in the context of the SINR coverage process. This work was reported in .

The collaboration with the new Paris Lab of THOMSON has been developing very fast since its creation. The scientific ties with C. Diot, L. Massoulié and A. Chaintreau are quite strong and materialize into:

joint seminars and reading groups (notably the new Paris-Networking series that we jointly initiated);

joint research actions, particularly on routing in ESS mesh WiFi networks and on CDMA networks;

a grant from Thomson which allows us to invite well known scientists in Communications (like e.g. V. Anantharam from Berkeley);

various ongoing projects of joint proposals in national and European agencies.

several joint papers published this year or to be published soon (including two Infocom 07 papers).

We initiated a new project with the Network Strategy Group of Alcatel Antwerp in continuation of the "End to End" OSC that ended in 2005. The new research directions are pursued in collaboration with Danny de Vleeschauwer and Koen Laevens. They bear on the modeling of the interaction of a large collection of multimedia sources that join and leave and that share an access network. The main objective is the design of optimal choking policies for the transport of layer encoded video in access networks.

The collaboration with France Télécom was structured in two parts:

The PRIMO CRC (Partage de Ressources dans l'Internet et les réseaux MObiles). The CRC, which ended in August 2006 after two years of existence, was focused on teletraffic theory and aimed at extending this theory to the classes of communication networks that are used and designed by operators today. The research activity covered both wireline and wireless networks. The main mathematical tools to be investigated within this context are queueing theory, information theory and stochastic geometry.

Our "Spontaneous Collaboration" with M. Karray on the coverage and capacity of the CDMA/UMTS networks. The current work primarily bears on macrodiversity in UMTS, multisectorial antennas in CDMA, distributed power and admission control for UMTS and associated QoS questions. Three patents were filed on these questions, one by INRIA and two jointly by INRIA and FT. The pertinence of our approach has already been recognized by Orange. This operator uses some of our methods in the program SERT (see Section ) integrated to its dimensioning tools. This year again, the spontaneous collaboration with M. Karray lead to a new joint paper accepted for publication at Infocom 07.

The interaction with the research lab of Sprint (Sprint ATL, in Burlingame, California) is made possible through a research grant. This interaction has been focused on two main topics:

The design of active probing methods for the estimation of internal properties of core or access networks based on end-to-end measurements. In the paper we proposed inversion formulas allowing one the analyze the law of cross traffic in a router from the end-to-end delay of probes. We also investigated the limitations of such inversion formulas. There are several continuations of this line of thoughts currently under investigation.

Cross layer optimization in CDMA cellular networks. In the paper , we gave an explicit formula for the optimal processing gain of a TCP source on the downlink of a CDMA channel. The analysis can be extended to other cross layer optimization problems involving congestion control and physical layers like in particular within the context of adaptive modulation.

Other projects have been started on the architecture of heterogeneous wireless networks. This collaboration is quite fruitful. It lead to two papers this year (including a Sigcomm 06 paper).

The France-Stanford Center has accepted a joint project entitled "Analysis and Design of Next-Generation Wireless Networks" that we submitted jointly with Prof. N. Bambos of Stanford in 2006. This project will be funded in 2006-2007.

TREC obtained in November 2006 the INRIA status of
*Associated Lab*(équipe associée) for the group of Prof. Darryl Veitch of the University of Melbourne for developing our joint research action on active probing.

TREC is a partner in the
*European Network of Excellence (NoE)*called EuroNGI (2004–2006;
http://eurongi.enst.fr/en_accueil.html) led by Groupement des Ecoles des Télécoms (GET). TREC was the coordinator of the INRIA participation to this NoE.

the following scientists gave talks in 2006:

France

Defense of the PhD thesis of Augustin Chaintreau
*ENS Paris*, "Processes of Interaction in Data Networks", January 16,

Romain Brette from
*ENS Paris*talking on "Réseaux de neurones et probabilités", March 8,

Pascal Moyal (CEREMADE, Université Paris Dauphine) "Stationarity of pure delay systems and queues with impatient customers via stochastic recursions", May 5,

Charles Bordenave (ENS) "Navigation on a Poisson point process", May 5,

Glenn Merlet "Théorèmes limites pour des suites récurrentes stochastiques d'applications topicales", May 5,

Fabien Mathieu (France Télécom) "Dynamic Roommates: a Versatile Framework for P2P Networks", May 5,

Defense of the PhD thesis of Charles Bordenave, ENS Paris, "Analyse stochastique des réseaux spatiaux", July 4,

Defense of the PhD thesis of Julien Reynier, ENS Paris, "Modélisation mathématique de TCP à l'aide de méthodes de champ moyen", September 15,

Dao Thi Thu Ha from
*LIAFA-Paris VII*talking on "Les files 0-automatiques", October 20,

Justin Salez from
*ENS Paris*talking on "Convergence locale de graphes aléatoires et propagation de croyances en milieu cyclique", October 20,

Laurent Massoulié from
*Thomson Research Paris,*talking on "Stability of decentralized control mechanisms: from congestion control to peer-to-peer broadcasting", November 3,

Eitan Altman from
*INRIA Sophia Antipolis,*talking on "Potential Games in Wireless Networks", November 13.

Europe

Frank Kelly from
*Cambridge University, UK*talking on "Stability, routing and congestion control", January 16,

Serguei Foss from
*Heriot-Watt University, Edinburgh, UK*talking "On the stability of a queueing system with uncountably branching fluid limits" and on "Stability of a network of multi-access
broadcast channels with spatial interaction", March 16,

Artyom Sapozhnikov from
*University College Cork, Ireland*talking on "Connectivity in subcritical continuum percolation", May 15,

Henri Koskinen from
*Helsinki University of Technology, Finland*talking on "Geometric studies in wireless multihop networks", July 5,

Serguei Foss from
*Heriot-Watt University, Edinburgh, UK*talking on "Lower bounds and equivalences for randomly stopped sume", November 29.

America, Asia, Australia

Prasanna Chaporkar
*University of Pennsylvania, Philadelphia, PA, USA / ENS - Paris*talking on "MAC Layer Multicast: Theory and approaches", January 4, 2006,

Nicholas Bambos from
*EECS Stanford, USA*talking on "Power control for wireless networks, an overview and some new directions", February 2,

Yogeshwaran Dhandapani from
*Indian Institute of Science, India,*talking on "Coverage Processes and Applications to Target Tracking in Sensor Networks", May 15,

Wojciech Szpankowski from
*Purdue University, Indiana, USA*talking on "Analytic algorithmics, combinatorics, and information theory", June 2,

David McDonald from
*University of Ottawa, Canada,*talking on "Large deviations of multitype queues", June 26,

Darryl Veith from
*University of Melbourne, Australia*talking on "Generalizing Fractional Brownian Motion in discrete time: a surprising self-similarity", July 12,

Anatolii Puhalskii from
*Colorado University, Denver and Institute for Problems in Information Transmission, Moscow,*talking on "Some Asymptotics for the Erdos-Renyi Random Graph", November 22.

TREC animates the project-team seminar: http://www.di.ens.fr/~trec/trec-eng.html

TREC is a founding member of and participates to Paris-Networking ( http://www.paris-networking.org/), a virtual community of researchers in networking who work in or around Paris (or visit Paris).

M. Lelarge animates the project-team reading group.

B. Błaszczyszyn maintains a web-page on stochastic geometry for communications http://www.di.ens.fr/ trec/sg

B. Błaszczyszynis a member of the organizing committee of the Scientific Colloquium of INRIA Rocquencourt
*Le modéle et l'algorithme*(
http://www-rocq.inria.fr/fr/actualites/modeleetalgorithme/).

F. Baccelli collaborates with G. Giraudon on the follow up of the interactions between INRIA and FT R&D.

P. Brémaud is a member of the editorial board of the following journals:
*Journal of Applied Probability, Advances in Applied Probability, Journal of Applied Mathematics and Stochastic Analysis*;

F. Baccelli is a member of the editorial board of the following journals:
*QUESTA, Journal of Discrete Event Dynamical Systems, Mathematical Methods of Operations Research, Advances in Applied Probability*.

Course on queueing theory and network performance evaluation by T. Bonald and A. Proutière (18H).

Graduate Course on point processes, stochastic geometry and random graphs (program ``Master de Sciences et Technologies''), by F. Baccelli, B. Blaszczyszyn and L. Massoulié (45h).

"Travaux dirigés" on arithmetics by M.A. Tran (39h).

Course on Information Theory of P. Brémaud and C. Bordenave (36h).

Undergraduate course (master level, MMFAI ) of F. Baccelli, T. Bonald, A. Proutière, B. Radunović and M.A. Tran on Communication Networks (48H).

Undergraduate course (master level) of F. Baccelli, P. Brémaud and M. Lelarge on applied probability (48h).

Course on PDE's for flow control by F. Baccelli (30h).

Member of the thesis committee of A. Chaintreau (ENS and Corps des Télécoms, January 16), C. Bordenave (Ecole Polytechnique and Corps des Télécoms, July 4) and J. Reynier (ENS and Corps des Télécoms, September 15).

Member of the program committee of the following conferences: Spaswin 06 ( http://icawww1.epfl.ch/spaswin/), Infocom 06 ( http://www.ieee-infocom.org/2006), Sigmetics/Performance 06 ( http://www.cs.wm.edu/sigm06/), WiOpt 06 ( http://www.wiopt.org/wiopt06/).

Presentation at the following conferences:

Workshop on Information Theory and its Applications, La Jolla, USA, February 2006 ( http://ita.ucsd.edu/events/1/workshops/showall/),

CISS, Princeton, March 2006 ( http://conf.ee.princeton.edu/ciss/),

SPASWIN 2006, Boston, USA, April 2006 ( http://icawww1.epfl.ch/spaswin/).

INFOCOM, Barcelona, Spain, April 2006 ( http://www.ieee-infocom.org/2006),

Conference on Stochastic Networks, Urbana-Champaign, USA, June 2006 ( http://www.ifp.uiuc.edu/~srikant/stochnet.htm),

7th IEEE International Workshop on Signal Processing Advances for Wireless Communications (SPAWC 06), Cannes, France, July 2006; keynote lecture ( http://spawc2006.eurecom.fr/),

Sigcomm Conference, Pisa, Italy, September 2006 ( http://www.acm.org/sigs/ sigcomm/sigcomm2006/),

Second International Conference on Performance Evaluation Methodologies and Tools (VALUETOOLS 06), Pisa, Italy, October 2006 ( http://www.valuetools.org/).

Organization of a session on self organized networks at
*Grand colloque STIC*, Lyon, France, November 2006 (
http://www.rntl.org/colloqueTIC2006/index.html),

Presentation at the following seminars:

TCOM, Philadelphia, USA, March 2006,

Académie des Sciences public session, Nice, France, May 2006.

Series of lectures at the 28th Finnish Summer School on Probability Theory, Nagu, Finland, with a course on Stochastic Geometry and Wireless Network Modeling, June 2006 ( http://www.abo.fi/fak/mnf/mate/nagu06/).

Member of the thesis committee of Nathalie Mitton (Université Lyon 1)

Reviewer of the thesis of Henri Koskinen, (Helsinki University of Technology),

Member of the program committee of Infocom 07 ( http://www.ieee-infocom.org/2007),

Co-organizer of the session ``Géométrie aléatoire et applications aux réseaux'' during ``Journées MAS de la SMAI, Modèles Spatiaux'',
*Polytechnique de Lille*, September 2006, (
http://math.univ-lille1.fr/~mas2006/)

Presentations at the following conferences:

``The World a Jigsaw: Tessellations in the Sciences'', University of Leiden and Lorentz Center, Netherlands, March 2006; invited talk ( http://www.lc.leidenuniv.nl/lc/web/2006/169/info.php3?wsid=169),

9th Conference on Probability, Bȩdlewo, Poland, May 2006 ( http://www.impan.gov.pl/~prob2006/),

``Journées MAS de la SMAI, Modèles Spatiaux'', Polytechnique de Lille, France, September 2006 ( http://math.univ-lille1.fr/~mas2006/),

Workshop on ``Applied Probability and Stochastic Geometry'', University of Wroclaw, Poland, October 2006 ( http://www.math.uni.wroc.pl/news_events/ events.php?eventid=04102006workshop&eventnr=14).

Presentation at the seminar ``Groupe de Travail ``Probabilités'' '', Université Paris 5, December 2006 ( http://www.math-info.univ-paris5.fr/~dhersin/GTProbaP5/GTProbaP5.html).

Member of the program committee of Sigmetrics/Performance 2006 ( http://www.cs.wm.edu/sigm06/), RAWNET 2006 ( http://www.rawnet.org/2006/index.html).

Presentations at CISS, Princeton, March 2006 ( http://conf.ee.princeton.edu/ciss/).

Presentation at the following conferences:

ALEA 2006, Marseille-Luminy, France, March 2006 ( http://www-rocq.inria.fr/ ~robert/Conf/ALEA06/).

SPASWIN 2006, Boston, USA, April 2006 ( http://icawww1.epfl.ch/spaswin/).

Colloquium "Jeunes Probabilistes et Statisticiens", Aussois, France, April 2006,

Conference on "Limit Theorems in Probability Theory and Their Applications", Novosibirsk, Russia, August 2006 ( http://math.nsc.ru/LBRT/v1/conf2006),

``Journées MAS de la SMAI, Modèles Spatiaux'', Polytechnique de Lille, France, September 2006 ( http://math.univ-lille1.fr/~mas2006/).

Participation at WiOpt, Boston, USA, April, 2006 ( http://www.wiopt.org/wiopt06/),

Presentation at the following seminars:

École Nationale Supérieure des Télécommunications, Paris, May, 2006 ( http://www.tsi.enst.fr/~cappe/sta/eve.html).

École Normale Supérieure, Séminaire TREC, Paris, May, 2006 ( http://www.di.ens.fr/~trec/sem-trec_fr2005-2006.html).

Presentation at the conference ACM Sigmetrics/Performance MAMA Workshop, Saint-Malo, June, 2006. ( http://www.cs.wm.edu/sigm06/MAMAindex.html),

Participation at EuroNGI Summerschool on Network Strategy, Design and Dimensioning, Laredo, Spain, September 2006 ( http://www.tlmat.unican.es/eurongi/summerschool2006/).

Received the prize for his PhD thesis
granted by the
*Société des Personnels Enseignants et Chercheurs en Informatique de France*(SPECIF), it will be handled during the SPECIF congress in January 2007 (
http://www.labri.fr/manifestation/specif2007/).

Presentation at the seminar École Normale Suérieure, Séminaire TREC, Paris, January, 2006 ( http://www.di.ens.fr/~trec/sem-trec_fr2005-2006.html).

Presentation at the following seminars:

``Avant-Première Infocom'' Paris Networking, April 2006 ( http://www.paris-networking.org/),

Network modeling day in ENS, May 2006 ( http://www.di.ens.fr/~trec/sem-trec_fr2005-2006.html),

Statistical Laboratory, University of Cambridge,

EECS Stanford University,

Heriot Watt University, Edinburgh,

University of California, Berkeley,

University of Massachusetts, Amherst.

Co-chair of IEEE RAWNET ( http://www.rawnet.org/2006/index.html), the second workshop on Resource Allocation in Wireless Networks, in conjunction with WiOpt 2006 ( http://www.wiopt.org/wiopt06/), Boston, April 2006.

Member of the TPC of the following conferences: IEEE Infocom 2006-2007 ( http://www.ieee-infocom.org/2006) ( http://www.ieee-infocom.org/2007)

Presentation at the following conferences:

CISS, Princeton, March 2006 ( http://conf.ee.princeton.edu/ciss/),

AINA, Vienna, April 2006; Key note speaker ( http://www.takilab.k.dendai.ac.jp/ conf/aina/2006/).

Presentation at the following seminars:

Princeton, March, 2006,

Microsoft Research, Cambridge, June 2006,

Heriot-Watt University, Edinburgh, June 2006,

Bell Labs, August 2006,

UPenn, July 2006,

Sprint Labs, August 2006,

University of Ottawa, October 2006.

Presentation at the following seminars:

EPFL, Lausanne, October 2006,

Microsoft Research, Cambridge, UK, May 2006,

Intel Research, Cambridge, UK, June 2006.

Participations at the following conferences:

ICIW, French Caribbean, February 2006; work presented by Fabien Mathieu ( http://www.iaria.org/conferences/ICIW06.html).

Presentation at the following seminars:

Thomson Paris Research Lab, Boulogne, June 2006 ( http://www.thomson.net/ EN/Home/Technology/research_activities/paris_lab_overview.htm) peer to peer systems and marriage theory.

France Télécom R&D Issy, July, 2006: mean field model for RED.

Motorola Lab Saclay, August 2006: mean field model for RED.

Paris Networking, ENS Paris, September 2006; thesis defense ( http://www.paris-networking.org/).

Presentations at the following conferences:

Second Madrid Conference on Queueing Theory (MCQT 06), Madrid, Spain, July 2006 ( http://www.mat.ucm.es/~mcqt/confe06/conf06.html),

``Journées MAS de la SMAI, Modèles Spatiaux'', Polytechnique de Lille, France, September 2006 ( http://math.univ-lille1.fr/~mas2006/).