Section: New Results

Asymptotic Models

The analysis of a set of $n$ stochastic entities interacting with each others can be particularly difficult. The mean field approximation is a very effective technique to characterize the transient probability distribution or steady-state regime of such systems when the number of entities $n$ grows very large. The idea of mean-field approximation is to replace a complex stochastic system by a simpler deterministic dynamical system. This dynamical system is constructed by assuming that the objects are asymptotically independent. Each object is viewed as interacting with an average of the other objects (the mean-field). When each object has a finite or countable state-space, this dynamical system is usually a non-linear ordinary differential equation (ODE). An introduction to these techniques is provided in the book chapter [29].

• Mean-field games model the rational behavior of an infinite number of indistinguishable players in interaction [79]. An important assumption of mean-field games is that, as the number of player is infinite, the decisions of an individual player do not affect the dynamics of the mass. Each player plays against the mass. A mean-field equilibrium corresponds to the case when the optimal decisions of a player coincide with the decisions of the mass. This leads to a simpler computation of the equilibrium.

  It has been shown in [72], [96] that for some games with a finite number of players, the Nash equilibria converge to mean-field equilibria as the number of players tends to infinity. Hence, many authors argue that mean-field games are a good approximation of symmetric stochastic games with a large number of players. The classical argument is that the impact of one player becomes negligible when the number of players goes to infinity. In [17], [36], we show that, in general, this convergence does not hold. We construct an example for which the mean-field limit only describes a sub-set of the limiting equilibria. Each finite-player game has an equilibrium with a good social cost, this is not the case for the limit game.

• Computer system and network performance can be significantly improved by caching frequently used information. When the cache size is limited, the cache replacement algorithm has an important impact on the effectiveness of caching. In [21], [3], [20] we introduce approximations to determine the cache hit probability of two classes of cache replacement algorithms: the recently introduced $h$-LRU and LRU($m$). These approximations only require the requests to be generated according to a general Markovian arrival process (MAP). This includes phase-type renewal processes and the IRM model as special cases. We provide both numerical and theoretical support for the claim that the proposed TTL approximations are asymptotically exact. We further show, by using synthetic and trace-based workloads, that $h$-LRU and LRU($m)$ perform alike, while the latter requires less work when a hit/miss occurs.

• In [16], we consider stochastic models in presence of uncertainty, originating from lack of knowledge of parameters or by unpredictable effects of the environment. We focus on population processes, encompassing a large class of systems, from queueing networks to epidemic spreading. We set up a formal framework for imprecise stochastic processes, where some parameters are allowed to vary in time within a given domain, but with no further constraint. We then consider the limit behaviour of these systems as the population size goes to infinity. We prove that this limit is given by a differential inclusion that can be constructed from the (imprecise) drift. We also we discuss different numerical algorithms to compute bounds of the so-obtained differential inclusions. We are currently working on an implementation of these algorithms in a numerical toolbox.

• In [37], we develop a fluid-limit approach to compute the expected absorbing time $T_n$ of a $n$-dimensional discrete time Markov chain. We show that the random absorbing time $T_n$ is well approximated by a deterministic time $t_n$ that is the first time when a fluid approximation of the chain approaches the absorbing state at a distance $1/n$. We show the applicability of this approach with three different problems: the coupon collector, the erasure channel lifetime and the coupling times of random walks in high dimensional spaces.