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.

Tail asymptotics for exponential functionals of Lévy processes

“…Z ¼ R 1 0 e ÀX ðtÞ dt. Such random variables appear in several applications, like mathematical finance (Z as the present value of a perpetuity [11], Asian options and COGARCH process [24]), Additive Increase Multiplicative Decrease (AIMD) algorithms [12,21], order-picking strategies in carousel systems [26], mathematical physics, and more. A recent monograph devoted to such exponential functionals is [32].…”

Flow Control as a Stochastic Optimal Control Problem with Incomplete Information

“…Most works considering optimization and modeling of transmission protocols disregard this dependence and assume that the loss process intensity is constant [5,6]; perhaps, only in [19] the authors investigate the asymptotic of a piecewise deterministic Markov process with state-dependent loss process under small intensity of losses. Most works considering optimization and modeling of transmission protocols disregard this dependence and assume that the loss process intensity is constant [5,6]; perhaps, only in [19] the authors investigate the asymptotic of a piecewise deterministic Markov process with state-dependent loss process under small intensity of losses.…”

A survey on performance analysis of warehouse carousel systems

“…For instance, they are relevant to the analysis of randomized algorithms (F lajolet , 2004) and in mathematical finance (B ertoin and Y or , 2005). In D umas et al. (2002) and G uillemin et al.…”

“…The functional I ( q ) has been intensively studied in the literature. Its density was obtained independently in D umas , G uillemin and R obert (2002) and B ertoin , B iane and Yor (2004), and in L itvak and A dan (2001) for q =1/2. C armona , P etit and Yor (1997) derived a density of for a large class of functions , in particular, for h ( n )= q n .…”