Section: New Results

Analytic models

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

New books about Markovian models and applications. The book [65] is the french version of the book [66] . Markov chains are a fundamental class of stochastic processes. They are the main modeling tool used in our team. They are widely used to solve problems in a large number of domains such as operations research, computer science, communication networks and manufacturing systems. The success of Markov chains is mainly due to their simplicity of use, the large number of available theoretical results and the quality of algorithms developed for the numerical evaluation of many metrics of interest. The books present the theory of both discrete-time and continuous-time homogeneous Markov chains. They examine the explosion phenomenon, the Kolmogorov equations, the convergence to equilibrium and the passage time distributions to a state and to a subset of states. These results are applied to birth-and-death processes. A detailed study of the uniformization technique by means of Banach algebra results is also developed. This technique is used for the transient analysis of several queuing systems.

Another book entitled “Markov Chains and Dependability Theory” will be published soon by Cambridge University Press (see http://www.amazon.fr/Markov-Chains-Dependability-Theory-Gerardo/dp/1107007577/ ). Dependability metrics are omnipresent in every engineering field, from simple ones through to more complex measures combining performance and dependability aspects of systems. The book presents the mathematical basis of the analysis of these metrics in the most used framework, Markov models, describing both basic results and specialised techniques. It presents both discrete and continuous time Markov chains before focusing on dependability measures, which necessitate the study of Markov chains on subsets of states representing different user satisfaction levels for the modelled system. Topics covered include Markovian state lumping, analysis of sojourns on subset of states of Markov chains, analysis of most dependability metrics, fundamentals of performability analysis, and bounding and simulation techniques designed to evaluate dependability measures. The book is of interest to graduate students and researchers in all areas of engineering where the concepts of lifetime, repair duration, availability, reliability and risk are important.

Fluid models. In [53] and [44] we propose a new way of transporting video flows on a peer-to-peer architecture of the Bit-Torrent type. We analyze the performance obtained by our proposal by means of fluid views of the systems, that is, by representing them using differential equations. In [53] the basic idea is to select the downloading peers according to their progress in the downloading process: a given peer only sends chunks to other peers that are downloading at least roughly in the same “area” of the stream. The system is improved in [44] where the main resource (the available bandwidth) is distributed differently among the peers, giving some kind of priority to those nodes remaining more time connected.

In [39] , we look at the problem of approximating Markovian views of the Machine Repaiman Model where life-times and repair times have Phase-type distributions, by differential equations. The machine population goes to infinity, and we analyze the properties of the limiting differential equation (once the Markovian sequence of models is properly scaled) and their relations with the initial models. In [63] we describe these results and other results concerning the same type of limiting processes, but concerning peer-to-peer networks. We discuss here the convergence aspects; the properties of the fluid models themselves are discussed in the two papers [53] and [44] mentioned before.