Equilibria of a class of transport equations arising in congestion control

Abstract

This paper studies a class of transport equations arising from stochastic models in congestion control. This class contains two cases of loss models as particular cases: the rate-independent case where the packet loss rate is independent of the throughput of the flow and the rate-dependent case where it depends on it. This class of equations covers both the case of persistent and of non-persistent flows. For the first time, we give a direct proof of the fact that there is a unique density solving the associated differential equation. This density and its mean value are provided as closed form expressions.

TCP and iso-stationary transformations

“…Altman et al [1] consider the case α < 1 and give an explicit analysis of β = 0, 1 using rather general mappings that involve both space and time transformations. Baccelli et al [3] consider a more general class of models for non-persistent flows, that includes our model for the case α = 0, β ≤ 0. They show that results for the throughput for β > 0 follow from the case β = 0 by applying appropriate substitutions to a differential equation.…”

A stochastic model for the throughput of non-persistent TCP flows

“…The proof of the last theorem is based on a direct analysis of the differential equation (29) and can be found in [5]. It is omitted due to the lack of space.…”

Critical Scale for a Continuous AIMD Model

“…This simple dynamics is the scaled version [9,15,11,6] of an additive increase multiplicative decrease (AIMD) process modeling the traffic flow in internet congestion control [2,7,8,1,11,3]. For a recent approach, a primer on the model and biography we mention [3].…”

“…Explicit calculations for power law function α(x) with p ≥ 0 are done in [1]. Due to homogeneity, the transformation y(t) = A(x(t)) brings down the invariant measure to the case α = constant.…”