A stochastic epidemiological model and a deterministic limit for BitTorrent-like peer-to-peer file-sharing networks

In this paper, we propose a stochastic model for a file-sharing peer-to-peer network which resembles the popular BitTorrent system: large files are split into chunks and a peer can download or swap from another peer only one chunk at a time. We prove that the fluid limits of a scaled Markov model of this system are of the coagulation form, special cases of which are well-known epidemiological (SIR) models. In addition, Lyapunov stability and settling-time results are explored. We derive conditions under which the BitTorrent incentives under consideration result in shorter mean file-acquisition times for peers compared to client-server (single chunk) systems. Finally, a diffusion approximation is given and some open questions are discussed.Comment: 25 pages, 6 figure

Convergence to Fleming-Viot processes in the weak atomic topology

AbstractStochastic models for gene frequencies can be viewed as probability-measure-valued processes. Fleming and Viot introduced a class of processes that arise as limits of genetic models as the population size and the number of possible genetic types tend to infinity. In general, the topology on the process values in which these limits exist is the topology of weak convergence; however, convergence in the weak topology is not strong enough for many genetic applications. A new topology on the space of finite measures is introduced in which convergence implies convergence of the sizes and locations of atoms, and conditions are given under which genetic models converge in this topology. As an application, Kingman's Poisson-Dirichlet limit is extended to models with selection

Perturbation of strong Feller semigroups and well-posedness of semilinear stochastic equations on Banach spaces

We prove a Miyadera-Voigt type perturbation theorem for strong Feller semigroups. Using this result, we prove well-posedness of the semilinear stochastic equation dX(t) = [AX(t) + F(X(t))]dt + GdW_H(t) on a separable Banach space E, assuming that F is bounded and measurable and that the associated linear equation, i.e. the equation with F = 0, is well-posed and its transition semigroup is strongly Feller and satisfies an appropriate gradient estimate. We also study existence and uniqueness of invariant measures for the associated transition semigroup.Comment: Revision based on the referee's comment

On Optimal Harvesting in Stochastic Environments: Optimal Policies in a Relaxed Model

This paper examines the objective of optimally harvesting a single species in a stochastic environment. This problem has previously been analyzed in Alvarez (2000) using dynamic programming techniques and, due to the natural payoff structure of the price rate function (the price decreases as the population increases), no optimal harvesting policy exists. This paper establishes a relaxed formulation of the harvesting model in such a manner that existence of an optimal relaxed harvesting policy can not only be proven but also identified. The analysis embeds the harvesting problem in an infinite-dimensional linear program over a space of occupation measures in which the initial position enters as a parameter and then analyzes an auxiliary problem having fewer constraints. In this manner upper bounds are determined for the optimal value (with the given initial position); these bounds depend on the relation of the initial population size to a specific target size. The more interesting case occurs when the initial population exceeds this target size; a new argument is required to obtain a sharp upper bound. Though the initial population size only enters as a parameter, the value is determined in a closed-form functional expression of this parameter.Comment: Key Words: Singular stochastic control, linear programming, relaxed contro

Feller property and infinitesimal generator of the exploration process

We consider the exploration process associated to the continuous random tree (CRT) built using a Levy process with no negative jumps. This process has been studied by Duquesne, Le Gall and Le Jan. This measure-valued Markov process is a useful tool to study CRT as well as super-Brownian motion with general branching mechanism. In this paper we prove this process is Feller, and we compute its infinitesimal generator on exponential functionals and give the corresponding martingale

Multi-Level Equilibrium Signaling for Molecular Communication

International audienceTwo key challenges in diffusion-based molecular communication are low data rates and accounting for the geometry of the fluid medium in the form of obstacles and the boundary. To reduce the need for the receiver to have knowledge of the geometry of the medium, binary equilibrium signaling has recently been proposed for molecular communication with a passive receiver in bounded channels. In this approach, reversible chemical reactions are introduced at the transmitter and the receiver in order for the system to converge to a known equilibrium state. This provides a means of designing simple detection rules that only depend on the transmitted signal and the volume of the bounded fluid medium. In this paper, we introduce multi-level equilibrium signaling, which allows for higher data rates via higher order modulation. We show that for a wide range of conditions, with appropriate receiver optimization, multi-level equilibrium signaling can outperform conventional concentration shift keying schemes. As such, our approach provides a basis to improve data rates in molecular communications without the need to increase the complexity of the system by exploiting techniques such as multiple information-carrying molecules

Infinite-dimensional diffusions as limits of random walks on partitions

The present paper originated from our previous study of the problem of harmonic analysis on the infinite symmetric group. This problem leads to a family {P_z} of probability measures, the z-measures, which depend on the complex parameter z. The z-measures live on the Thoma simplex, an infinite-dimensional compact space which is a kind of dual object to the infinite symmetric group. The aim of the paper is to introduce stochastic dynamics related to the z-measures. Namely, we construct a family of diffusion processes in the Toma simplex indexed by the same parameter z. Our diffusions are obtained from certain Markov chains on partitions of natural numbers n in a scaling limit as n goes to infinity. These Markov chains arise in a natural way, due to the approximation of the infinite symmetric group by the increasing chain of the finite symmetric groups. Each z-measure P_z serves as a unique invariant distribution for the corresponding diffusion process, and the process is ergodic with respect to P_z. Moreover, P_z is a symmetrizing measure, so that the process is reversible. We describe the spectrum of its generator and compute the associated (pre)Dirichlet form.Comment: AMSTex, 33 pages. Version 2: minor changes, typos corrected, to appear in Prob. Theor. Rel. Field

A Molecular Communication Scheme to Estimate the State of Biochemical Processes on a Lab-on-a-Chip

International audienceA key application of advanced spectroscopy methods is to estimate equilibrium states of biochemical processes in situ and in vivo. Nevertheless, an often present difficulty is the requirement that the biochemical process and its environment (such as the substrate) satisfy special conditions. One means of resolving this issue is to communicate information about the equilibrium states of the biochemical process to another location, supported via microfluidic channles within a lab-on-a-chip. In this paper, we develop a signaling strategy and estimation algorithms for equilibrium states of a biochemical process. For a toggle-switch circuit model important in cellular differentiation studies, we study via simulation the tradeoff between the rate of obtaining spectroscopy measurements and the estimation error, providing insights into requirements of spectroscopy devices for high-throughput biological assays. CCS CONCEPTS • Applied computing → Health care information systems; • Computing methodologies → Model verification and validation; Mixture modeling

Spontaneous Resonances and the Coherent States of the Queuing Networks

We present an example of a highly connected closed network of servers, where the time correlations do not go to zero in the infinite volume limit. This phenomenon is similar to the continuous symmetry breaking at low temperatures in statistical mechanics. The role of the inverse temperature is played by the average load.Comment: 3 figures added, small correction

Large Deviations Principle for a Large Class of One-Dimensional Markov Processes

We study the large deviations principle for one dimensional, continuous, homogeneous, strong Markov processes that do not necessarily behave locally as a Wiener process. Any strong Markov process $X_{t}$ in $\mathbb{R}$ that is continuous with probability one, under some minimal regularity conditions, is governed by a generalized elliptic operator $D_{v}D_{u}$, where $v$ and $u$ are two strictly increasing functions, $v$ is right continuous and $u$ is continuous. In this paper, we study large deviations principle for Markov processes whose infinitesimal generator is $\epsilon D_{v}D_{u}$ where $0<\epsilon\ll 1$. This result generalizes the classical large deviations results for a large class of one dimensional "classical" stochastic processes. Moreover, we consider reaction-diffusion equations governed by a generalized operator $D_{v}D_{u}$. We apply our results to the problem of wave front propagation for these type of reaction-diffusion equations.Comment: 23 page