## Section: New Results

### Performance evaluation

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

An important problem arising when dimensioning a P2P system is to understand the evolution of the peers' population with time. The number of units being usually large, the standard stochastic models used to represent this kind of system (e.g. a Markovian stochastic process) are difficult to use in practice. Instead, it is popular today to move to deterministic continuous-state (fluid) models whose dynamics is governed by differential equations. It is then of interest to analyze the conditions under which the latter are the limit, in some sense, of the former. We started to develop this program in [36] by focusing on some popular models of P2P systems, and analyzed when and how the deterministic model is the limit of the stochastic one when the number of peers goes to infinity.

In [60] , we continued to 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 throughput and the mean response 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. In this research line, [60] opens the way for a definition of efficiency, which is currently analyzed in the team.

In [30] , we expose a clear 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. We illustrate our results by a comprehensive fluid model that we solve exactly.

In [65] , we analyze the transient behavior of a fluid queue driven by a general ergodic birth and death process using spectral theory in the Laplace transform domain. These results are applied to the stationary regime and to the busy period analysis of that fluid queue.

Finally, in [71] we present a global view of the performance evaluation area in computer and communication systems, an extended and reviewed version of a talk given in 2010.