Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOn the long time behavior of the TCP window size process
161
0
0.0
(
0
)
Added by
Djalil Chafai
Publication date
2009
fields
Informatics Engineering
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
The TCP window size process appears in the modeling of the famous Transmission Control Protocol used for data transmission over the Internet. This continuous time Markov process takes its values in $[0,infty)$, is ergodic and irreversible. It belongs to the Additive Increase Multiplicative Decrease class of processes. The sample paths are piecewise linear deterministic and the whole randomness of the dynamics comes from the jump mechanism. Several aspects of this process have already been investigated in the literature. In the present paper, we mainly get quantitative estimates for the convergence to equilibrium, in terms of the $W_1$ Wasserstein coupling distance, for the process and also for its embedded chain.
rate research
Read More
The TCP window size process appears in the modeling of the famous Transmission Control Protocol used for data transmission over the Internet. This continuous time Markov process takes its values in [0, infty), is ergodic and irreversible. The sample paths are piecewise linear deterministic and the whole randomness of the dynamics comes from the jump mechanism. The aim of the present paper is to provide quantitative estimates for the exponential convergence to equilibrium, in terms of the total variation and Wasserstein distances.
We consider the symmetric simple exclusion process with slow boundary first introduced in [Baldasso {it et al.}, Journal of Statistical Physics, 167(5), 2017]. We prove a law of large number for the empirical measure of the process under a longer time scaling instead of the usual diffusive time scaling.
The logistic birth and death process is perhaps the simplest stochastic population model that has both density-dependent reproduction, and a phase transition, and a lot can be learned about the process by studying its extinction time, $tau_n$, as a function of system size $n$. A number of existing results describe the scaling of $tau_n$ as $ntoinfty$, for various choices of reproductive rate $r_n$ and initial population $X_n(0)$ as a function of $n$. We collect and complete this picture, obtaining a complete classification of all sequences $(r_n)$ and $(X_n(0))$ for which there exist rescaling parameters $(s_n)$ and $(t_n)$ such that $(tau_n-t_n)/s_n$ converges in distribution as $ntoinfty$, and identifying the limits in each case.
We consider the problem of the long time dynamics for a diffuse interface model for tumor growth. The model describes the growth of a tumor surrounded by host tissues in the presence of a nutrient and consists in a Cahn-Hilliard-type equation for the tumor phase coupled with a reaction-diffusion equation for the nutrient concentration. We prove that, under physically motivated assumptions on parameters and data, the corresponding initial-boundary value problem generates a dissipative dynamical system that admits the global attractor in a proper phase space.
Aldous [(2007) Preprint] defined a gossip process in which space is a discrete $Ntimes N$ torus, and the state of the process at time $t$ is the set of individuals who know the information. Information spreads from a site to its nearest neighbors at rate 1/4 each and at rate $N^{-alpha}$ to a site chosen at random from the torus. We will be interested in the case in which $alpha<3$, where the long range transmission significantly accelerates the time at which everyone knows the information. We prove three results that precisely describe the spread of information in a slightly simplified model on the real torus. The time until everyone knows the information is asymptotically $T=(2-2alpha/3)N^{alpha/3}log N$. If $rho_s$ is the fraction of the population who know the information at time $s$ and $varepsilon$ is small then, for large $N$, the time until $rho_s$ reaches $varepsilon$ is $T(varepsilon)approx T+N^{alpha/3}log (3varepsilon /M)$, where $M$ is a random variable determined by the early spread of the information. The value of $rho_s$ at time $s=T(1/3)+tN^{alpha/3}$ is almost a deterministic function $h(t)$ which satisfies an odd looking integro-differential equation. The last result confirms a heuristic calculation of Aldous.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
