A Class Of Mean Field Interaction Models for Computer and Communication Systems

Abstract

We consider models of N interacting objects, where the interaction is via a common resource and the distribution of states of all objects. We consider the case where the number of transitions per time slot per object vanishes as N grows. We show that, under mild assumptions and for large N , the occupancy measure converges, in probability and in mean square over any finite horizon, to a deterministic dynamical system. Our method of proof is inspired by stochastic approximation algorithms. The convergence results allow us to derive properties valid in the stationary regime. We use this to develop a critique of the fixed point method sometimes used in conjunction with the decoupling assumption.

The Mean Drift: Tailoring the Mean Field Theory of Markov Processes for Real-World Applications

“…In this section, we present the stochastic model of a system and its mean field approximations. For the most part, our notation agrees with [2]. A list of objects appearing in the mathematical discussions that follow are given in Table 1.…”

“…We then use the mean drift to construct a new set of ordinary differential equations which address the analysis of population processes with an arbitrary size.Population processes are stochastic models of systems which consist of a number of similar agents (or particles) [23]. When the impact of each agent on the behaviour of the system is similar to other agents, it is said that the population process is a mean field interaction model [2]. It is possible to apply a symmetric reduction on the state space of these types of processes and gain some efficiency in their analysis.…”

“…In the field of communication engineering and starting with Bianchi's analysis of the IEEE 802.11 DCF protocol [4,5], much research has focused on discussing the validity of the so-called decoupling assumption in this analysis [7,33,29,11]. Several general frameworks have also been proposed which target the analysis of computer and communication systems [2,25]. In the field of computer science the initial application of the approximations was intuitively motivated by methods such as fluid and diffusion approximations of queueing networks [19].…”

“…In the current text, we use only the most essential and trivial concepts to build the mean field approximations. Several decisions were made to achieve this goal: the choice of discrete-time Markov processes to specify the agent behaviour, a detailed discussion on time, the discussion on the derivation of transition maps, and the presentation of the convergence result in two clear steps (following [2]). In our opinion, this attitude is essential in promoting a correct understanding of the theory.…”

Discrete mean field games: Existence of equilibria and convergence

“…We claim that the model with explicit interactions covers several natural phenomena such as information/infection propagation or resource congestion where the cost but also the state dynamics of a player depend on the state of the all the others. This type of behavior is classical in systems with a large number of interacting objects [6] and cannot be handled using previous mean field game models. For instance, in the classical SIR (Susceptible, Infected, Recovered) infection model [39], the rate of infection of one individual depends on the proportion of individuals already infected.…”