A Markovian analysis of additive-increase multiplicative-decrease algorithms

Abstract

The additive-increase multiplicative-decrease (AIMD) schemes designed to control congestion in communication networks are investigated from a probabilistic point of view. Functional limit theorems for a general class of Markov processes that describe these algorithms are obtained. The asymptotic behaviour of the corresponding invariant measures is described in terms of the limiting Markov processes. For some special important cases, including TCP congestion avoidance, an important autoregressive property is proved. As a consequence, the explicit expression of the related invariant probabilities is derived. The transient behaviour of these algorithms is also analysed.

TCP and iso-stationary transformations

“…in which form the solution to (22) was derived in [8,11]. We now employ our transformations to obtain the stationary distribution for the general (α, β)-case.…”

“…Ott et al [21] use a space transformation to solve the idealized TCP case for "packet time" (α < 1, β = 0, Q = c) and use a time transformation to obtain the limiting stationary distribution for "clock time" (α = 0, β = 1, Q = c). Dumas et al [8] consider the case α = 0, β > −1, and present the stationary distribution for β = 1. 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.…”

An extension of the square root law of TCP

“…The constant K depends on the version of TCP, but also on the models. Several different approaches have been used in this line of work; for example direct modeling and performance analysis [2,3], simulation [7], approximations [21] and probabilistic scaling [5,11,19,20]. In this work, we combine probabilistic scaling and performance analysis methods to analyze a class of congestion avoidance algorithms which we (following [3]) call general increase multiplicative decrease (GIMD) algorithms.…”

Steady state and scaling limit for a traffic congestion model

“…2.1), proving a full hydrodynamic limit for the empirical measure (5.1) in Theorem 5.1. The result relates partially to the limiting process from equation (2.1) in [19] (α = constant, n = 1, non-interactive dynamics) and the convergence to the process given by equation (2.7) quoted in [14] from [8] (α(x) = x, n = 1, non-interactive dynamics). Recurrence issues and the ergodic properties of the one-particle scaling limit process with rate α(x) are analyzed in greater detail in [13].…”