## Section: New Results

### Performance evaluation

Participants : Laura Aspirot, Raymond Marie, Gerardo Rubino, Bruno Sericola.

We maintain a set of activities about the evaluation of the performance of specific systems, and about the development of techniques for performing these evaluations.

In [16] , we proposed in a fully decentralized algorithm to provide each node with a value reflecting its connectivity quality. Comparing these values between nodes, enables to have a local approximation of a global characteristic of the graph. Our algorithm relies on an anonymous probe visiting the network in a unbiased random fashion. Each node records the time elapsed between visits of the probe which is called the return time of the random walk. Computing the standard deviation of such return times enables to approximate the conductance of the graph. Typically, this information may be used by nodes to assess their position, and therefore the fact that they are critical, in a graph exhibiting low conductance. This work is a collaboration with the Inria team-project Asap.

In [27] , we expose a methodology to analyze maximum level and hitting probabilities in a Markov driven fluid queue for various initial condition scenarios and in both cases of infinite and finite buffers. Step by step we build up our argument that finally leads to matrix differential Riccati equations for which there exists a unique solution. The power of the methodology resides in the simple probabilistic argument used that permits to obtain analytic solutions of these differential equations. We illustrate our results by a fluid model that we exactly solve.

We continue our collaboration with the Inria team-projects Adept and Ipso. In [32] , we present an analytic study of the impact of churn in cluster-based overlay networks and we accurately predict the frequency at which the topology of the overlay changes according to the number of join/leave operations.

It is well-known that peer-to-peer overlays networks can only survive Byzantine attacks if malicious nodes are not able to predict what will be the topology of the network for a given sequence of join and leave operations. In [76] , we investigate adversarial strategies by following specific games. Our analysis demonstrates first that an adversary can very quickly subvert DHT-based overlays by simply never triggering leave operations. We then show that when all nodes (honest and malicious ones) are imposed on a limited lifetime, the system eventually reaches a stationary regime where the ratio of polluted clusters is bounded, independently from the initial amount of corruption in the system. These results have been obtained using Markov models. In [77] , we consider the behavior of a stochastic system composed of several identically distributed, but non independent, discrete-time absorbing Markov chains competing at each instant for a transition. The competition consists in determining at each instant, using a given probability distribution, the only Markov chain allowed to make a transition. We analyze the first time at which one of the Markov chains reaches its absorbing state. We obtain its distribution and its expectation and we propose an algorithm to compute these quantities. We also exhibit the asymptotic behavior of the system when the number of Markov chains goes to infinity. Actually, this problem comes from the analysis of large-scale distributed systems and we show how our results apply to this domain.

In [49] we analyzed the idea of a peer-to-peer (P2P) system where some of the arriving peers are given priority over the remaining ones, based on the fact that they trend (statistically) to stay connected longer. We modeled this situation and explored the quantitative properties of the resulting system by simulation and by using deterministic models (ODEs). In the literature, this last approach is mathematically explored under the general label of mean field techniques. They are used to analyze the way, in some cases, a large Markovian model converges toward a deterministic differential one. We are exploring these convergence aspects in the case of models, focusing first on the case of P2P networks. Some preliminary results were presented in [57] and in [67] . In particular, we obtained convergence results allowing to find specific performance metrics defined in the initial large Markov model by working with the (compact) deterministic one.

Concerning the use of large Markov models and the way of analyzing
them using different techniques (simulation, numerical methods,
bounding procedures), we started to present the characteristics of a
tool we are developing, called AF ([68] ). The tool allows
to describe the Markov model using a general language (`C` in
our implementation) and it has been deigned to provide different
facilities to researchers aiming at developing analysis techniques,
belonging to the three categories described below.

In [68] , we explore the concept of power of a queueing model proposed by Kleinrock in the 80s. Kleinrock's idea was to build a metric combining two “competing” ones, the mean trhoughput and the mean respnse time, for the system in equilibrium. The power is defined as the ratio of normalized versions of those metrics. We discuss different ways of adapting this concept to more general queueing systems such as queueing networks.

The paper [45] has been published under the scientific sponsoring of IFIP WG 6.3 and 7.3. Because the elements of modeling in general and of performance evaluation of discrete event systems (DES) in particular have undergone a tremendous transformation during these last four decades, the aim of this paper is to look back over all this evolution, trying to retain some particular experiences from the past together with some suggestions to preserve the quality of the expertise of the community are proposed.

Finally, we managed the publication of [75] , together with two colleagues from Turin (S. Donatelli) and from Montréal (P. Panangaden), where the general topic is the performance evaluation of computer and communication systems.