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

Perfect sampling of Jackson Queueing Networks

We consider open Jackson networks with losses with mixed finite and infinite queues and analyze the efficiency of sampling from their exact stationary distribution. We show that perfect sampling is possible, although the underlying Markov chain may have an infinite state space. The main idea is to use a Jackson network with infinite buffers (that has a product form stationary distribution) to bound the number of initial conditions to be considered in the coupling from the past scheme. We also provide bounds on the sampling time of this new perfect sampling algorithm for acyclic or hyperstable networks. These bounds show that the new algorithm is considerably more efficient than existing perfect samplers even in the case where all queues are finite. We illustrate this efficiency through numerical experiments. We also extend our approach to non-monotone networks such as queueing networks with negative customers.On considère les réseaux de Jackson avec perte comportant des files finies et infinies, et l'on s'intéresse à l'efficacité des techniques d'échantillonnage de leur distribution stationnaire exacte. Nous démontrons que la simulation parfaite est possible même si la chaîne de Markov sous-jacente a un espace d'états potentiellement infini. L'idée principale est d'utiliser un réseau de Jackson aux files infinies (qui admet une distribution de forme-produit) pour borner les conditions initiales à considérer dans l'algorithme de simulation parfaite. Nous donnons également des bornes sur le temps d'échantillonnage de ce nouvel algorithme dans le cas des réseaux acycliques, ainsi que pour des réseaux hyperstables. Ces bornes prouvent que le nouvel algorithme est considérablement plus efficace que les échantillonneurs parfaits acuels, même dans le cas où toutes les files sont finies. Nous illustrons cette efficacité par des expériences numériques. Enfin, nous généralisons notre approche au cas des réseaux non-monotones comme les réseaux aux clients négatifs

Perfect sampling of Jackson Queueing Networks

We consider open Jackson networks with losses with mixed finite and infinite queues and analyze the efficiency of sampling from their exact stationary distribution. We show that perfect sampling is possible, although the underlying Markov chain may have an infinite state space. The main idea is to use a Jackson network with infinite buffers (that has a product form stationary distribution) to bound the number of initial conditions to be considered in the coupling from the past scheme. We also provide bounds on the sampling time of this new perfect sampling algorithm for acyclic or hyperstable networks. These bounds show that the new algorithm is considerably more efficient than existing perfect samplers even in the case where all queues are finite. We illustrate this efficiency through numerical experiments. We also extend our approach to non-monotone networks such as queueing networks with negative customers.On considère les réseaux de Jackson avec perte comportant des files finies et infinies, et l'on s'intéresse à l'efficacité des techniques d'échantillonnage de leur distribution stationnaire exacte. Nous démontrons que la simulation parfaite est possible même si la chaîne de Markov sous-jacente a un espace d'états potentiellement infini. L'idée principale est d'utiliser un réseau de Jackson aux files infinies (qui admet une distribution de forme-produit) pour borner les conditions initiales à considérer dans l'algorithme de simulation parfaite. Nous donnons également des bornes sur le temps d'échantillonnage de ce nouvel algorithme dans le cas des réseaux acycliques, ainsi que pour des réseaux hyperstables. Ces bornes prouvent que le nouvel algorithme est considérablement plus efficace que les échantillonneurs parfaits acuels, même dans le cas où toutes les files sont finies. Nous illustrons cette efficacité par des expériences numériques. Enfin, nous généralisons notre approche au cas des réseaux non-monotones comme les réseaux aux clients négatifs

A perfect sampling algorithm of random walks with forbidden arcs

In this paper we show how to construct an algorithm to sample the stationary distribution of a random walk over a grid with forbidden arcs. This algorithm combines the rejection method and coupling from the past of a set of trajectories of the Markov chain that generalizes the classical sandwich approach. We also provide a complexity analysis of this approach in several cases showing a coupling time that is logarithmic in the size of the grid, when no arc is forbidden, and an experimental study of its performance

Stochastic Fluid Model for P2P Caching Evaluation

In this paper we propose a stochastic fluid model to analyze the performance of Squirrel: a P2P cooperative Web cache. This work provides a scalable and insightful extension of our previous analysis of Squirrel. This new model provides a closed-form expression for the hit probability when documents are equally popular. Realistic object popularity is also addressed through a clustering approximation. The accuracy of this model is validated by a comparison with discrete-event simulations. Our model allows us to study the impact of various parameters on the performance of the Squirrel system. In particular, we emphasize the importance of taking object popularity into account. We also investigate the utility of clients announcing their departure on the resulting hit probability

A perfect sampling algorithm of random walks with forbidden arcs

International audienceIn this paper we show how to construct an algorithm to sample the stationary distribution of a random walk over ${1,...,N}^d$ with forbidden arcs. This algorithm combines the rejection method and coupling from the past of a set of trajectories of the Markov chain that generalizes the classical sandwich approach. We also provide a complexity analysis of this approach in several cases showing a coupling time in $O(N^2 d log d )$ when no arc is forbidden and an experimental study of its performance

A perfect sampling algorithm of random walks with forbidden arcs

In this paper we show how to construct an algorithm to sample the stationary distribution of a random walk over a grid with forbidden arcs. This algorithm combines the rejection method and coupling from the past of a set of trajectories of the Markov chain that generalizes the classical sandwich approach. We also provide a complexity analysis of this approach in several cases showing a coupling time that is logarithmic in the size of the grid, when no arc is forbidden, and an experimental study of its performance

Modèles fluides pour l'analyse des systèmes de distribution de contenu

Content distribution systems (CDS) such as web caches and file sharing systems are large-scale distributed systems that may serve hundreds of thousands of users. These highly dynamic systems exhibit a very large state space which makes them difficult to analyze with classical tools such as Markovian models or simulation. In this thesis we propose macroscopic fluid models to reduce the complexity of these systems. We show that these simple models provide accurate and insightful results on the performance of CDS.In a first part we propose a generic fluid model for distributed caching systems. The idea is to replace cached documents with fluid that increase with unsatisfied requests. Caches may go up and down according to a birth-death process. We apply this model to study two caching systems: cache clusters and a P2P cooperative cache system called Squirrel. We derive an efficient and accurate expression of their hit probabilities and show how the model identifies the key tradeoffs of these systems. We also propose a multiclass approximation for taking into account document popularity.In the second part of the thesis we consider file sharing systems such as BitTorrent. We propose a two-class fluid model which replaces downloaders with fluid. This simple deterministic model may reflect the problem of service differentiation or bandwidth diversity for instance. We provide a closed-form expression of the average download time for each class under the worst-case assumption that users leave the system immediately after completing their download. We also show how to allocate peers bandwidth between classes to achieve service differentiation.Les systèmes de distribution de contenu comme les caches web et les réseaux d'échanges de fichiers doivent pouvoir servir une population de clients à la fois très grande (centaines de milliers) et fortement dynamique (temps de connexion très courts). Ces caractéristiques rendent leur analyse très coûteuse par les approches traditionnelles comme les modèles markoviens ou la simulation. Dans cette thèse nous proposons des modèles fluides simples permettant de s'affranchir de l'une des dimensions du problème. Dans la première partie, nous développons un modèle stochastique fluide pour les systèmes de caches distribués. Les documents stockés sont modélisés par un fluide augmentant avec les requêtes insatisfaites. Nous appliquons ce modèle aux "clusters" de caches et à Squirrel, un système de cache pair-à-pair. Dans les deux cas notre modèle permet de calculer efficacement et avec précision la probabilité de hit, et de mettre en évidence les paramètres clés de ces systèmes. Nous proposons également une approximation multi classes pour modéliser la popularité des documents.Dans la seconde partie de cette thèse nous étudions BitTorrent, un système d'échange de fichiers Pair-à-pair. Nous proposons un modèle fluide multi classes qui remplace les usagers par un fluide. Nous considérons deux classes d'usagers pour modéliser les différences de débits d'accès ou de qualité de service. Nous obtenons une formule close pour le temps de téléchargement dans chaque classe. Nous montrons également comment allouer la bande passante a chaque classe pour offrir un service différencié