A Mean Field Approach for Optimization in Particles Systems and Applications

This paper investigates the limit behavior of Markov Decision Processes (MDPs) made of independent particles evolving in a common environment, when the number of particles goes to infinity. In the finite horizon case or with a discounted cost and an infinite horizon, we show that when the number of particles becomes large, the optimal cost of the system converges almost surely to the optimal cost of a discrete deterministic system (the ``optimal mean field''). Convergence also holds for optimal policies. We further provide insights on the speed of convergence by proving several central limits theorems for the cost and the state of the Markov decision process with explicit formulas for the variance of the limit Gaussian laws. Then, our framework is applied to a brokering problem in grid computing. The optimal policy for the limit deterministic system is computed explicitly. Several simulations with growing numbers of processors are reported. They compare the performance of the optimal policy of the limit system used in the finite case with classical policies (such as Join the Shortest Queue) by measuring its asymptotic gain as well as the threshold above which it starts outperforming classical policies

A stochastic approximation algorithm for stochastic semidefinite programming

Motivated by applications to multi-antenna wireless networks, we propose a distributed and asynchronous algorithm for stochastic semidefinite programming. This algorithm is a stochastic approximation of a continous- time matrix exponential scheme regularized by the addition of an entropy-like term to the problem's objective function. We show that the resulting algorithm converges almost surely to an $\varepsilon$-approximation of the optimal solution requiring only an unbiased estimate of the gradient of the problem's stochastic objective. When applied to throughput maximization in wireless multiple-input and multiple-output (MIMO) systems, the proposed algorithm retains its convergence properties under a wide array of mobility impediments such as user update asynchronicities, random delays and/or ergodically changing channels. Our theoretical analysis is complemented by extensive numerical simulations which illustrate the robustness and scalability of the proposed method in realistic network conditions.Comment: 25 pages, 4 figure

Distributing Labels on Infinite Trees

Sturmian words are infinite binary words with many equivalent definitions: They have a minimal factor complexity among all aperiodic sequences; they are balanced sequences (the labels 0 and 1 are as evenly distributed as possible) and they can be constructed using a mechanical definition. All this properties make them good candidates for being extremal points in scheduling problems over two processors. In this paper, we consider the problem of generalizing Sturmian words to trees. The problem is to evenly distribute labels 0 and 1 over infinite trees. We show that (strongly) balanced trees exist and can also be constructed using a mechanical process as long as the tree is irrational. Such trees also have a minimal factor complexity. Therefore they bring the hope that extremal scheduling properties of Sturmian words can be extended to such trees, as least partially. Such possible extensions are illustrated by one such example.Comment: 30 pages, use pgf/tik

Penalty-regulated dynamics and robust learning procedures in games

Starting from a heuristic learning scheme for N-person games, we derive a new class of continuous-time learning dynamics consisting of a replicator-like drift adjusted by a penalty term that renders the boundary of the game's strategy space repelling. These penalty-regulated dynamics are equivalent to players keeping an exponentially discounted aggregate of their on-going payoffs and then using a smooth best response to pick an action based on these performance scores. Owing to this inherent duality, the proposed dynamics satisfy a variant of the folk theorem of evolutionary game theory and they converge to (arbitrarily precise) approximations of Nash equilibria in potential games. Motivated by applications to traffic engineering, we exploit this duality further to design a discrete-time, payoff-based learning algorithm which retains these convergence properties and only requires players to observe their in-game payoffs: moreover, the algorithm remains robust in the presence of stochastic perturbations and observation errors, and it does not require any synchronization between players.Comment: 33 pages, 3 figure

Distributed Adaptive Routing in Communication Networks

In this report, we present a new adaptive multi-flow routing algorithm to select end- to-end paths in packet-switched networks. This algorithm provides provable optimality guarantees in the following game theoretic sense: The network configuration converges to a configuration arbitrarily close to a pure Nash equilibrium. In this context, a Nash equilibrium is a configuration in which no flow can improve its end-to-end delay by changing its network path. This algorithm has several robustness properties making it suitable for real-life usage: it is robust to measurement errors, outdated information and clocks desynchronization. Furthermore, it is only based on local information and only takes local decisions, making it suitable for a distributed implementation. Our SDN-based proof-of-concept is built as an Openflow controller. We set up an emulation platform based on Mininet to test the behavior of our proof-of-concept implementation in several scenarios. Although real-world conditions do not conform exactly to the theoretical model, all experiments exhibit satisfying behavior, in accordance with the theoretical predictions

Distributed Adaptive Routing in Communication Networks

In this report, we present a new adaptive multi-flow routing algorithm to select end- to-end paths in packet-switched networks. This algorithm provides provable optimality guarantees in the following game theoretic sense: The network configuration converges to a configuration arbitrarily close to a pure Nash equilibrium. In this context, a Nash equilibrium is a configuration in which no flow can improve its end-to-end delay by changing its network path. This algorithm has several robustness properties making it suitable for real-life usage: it is robust to measurement errors, outdated information and clocks desynchronization. Furthermore, it is only based on local information and only takes local decisions, making it suitable for a distributed implementation. Our SDN-based proof-of-concept is built as an Openflow controller. We set up an emulation platform based on Mininet to test the behavior of our proof-of-concept implementation in several scenarios. Although real-world conditions do not conform exactly to the theoretical model, all experiments exhibit satisfying behavior, in accordance with the theoretical predictions

Coupling from the past in hybrid models for file sharing peer to peer systems

International audienceIn this paper we show how file sharing peer to peer systems can be modeled by hybrid systems with a continuous part corresponding to a fluid limit of files and a discrete part corresponding to customers. Then we show that this hybrid system is amenable to perfect simulations (i.e. simulations providing samples of the system states which distributions have no bias from the asymptotic distribution of the system). An experimental study is carried to show the respective influence that the different parameters (such as time-to-live, rate of requests, connection time) play on the behavior of large peer to peer systems, and also to show the effectiveness of this approach for numerical solutions of stochastic hybrid systems

Optimal Routing in Deterministic Queues in Tandem

In this paper we address the problem of routing a stream of customers in two parallel networks of queues in tandem with deterministic service times in order to minimize the average response time of the whole system. We show that the optimal routing is a Sturmian word which density depends on the decomposition in continuous fraction of the maximum service time on each route. In order to do this we study the output process of deterministic queues when the input process is Sturmian

Maximizing the Robustness of TDMA Networks with Applications to TTP/C

In this study we show how one can use Fault-Tolerant Units (FTU) in an optimal way to make a TDMA network robust to bursty random perturbations. We consider two possible objectives. If one wants to minimize the probability of losing all replicas of a given message, then the optimal policy is to spread the replicas over time. This is proved using convexity properties of the loss probability. On the contrary if one wants to minimize the probability of losing at least one replica, then the optimal solution is to group all replicas together. This is proved by using majorization techniques. Finally we show how these ideas can be adapted for the TTP/C protocol