Local bifurcation analysis of some dual congestion control algorithms

Abstract

Abstract-We perform the necessary calculations to determine the stability and asymptotic forms of solutions bifurcating from steady state in a nonlinear delay differential equation with a single discrete delay. The results are used to examine the loss of local stability in a selection of congestion control algorithms employed over a single link. In particular, we analyze the fair and the delay-based dual algorithms. Explicit conditions are derived to ensure the onset of stable limit cycles as these algorithms just lose local stability. Further, we are able to quantify the effect parameters of the system have on the amplitude of the bifurcating periodic solutions.

The modified optimal velocity model: stability analyses and design guidelines

“…To determine if system (2) undergoes a stability loss via a Hopf bifurcation, we follow [37] and introduce an exogenous, non-dimensional parameter κ > 0. A general system of DDEṡ…”

“…Further, we assume that F(S t , µ) is analytic, and that F and L µ depend analytically on the bifurcation parameter µ, for small |µ|. The objective now is to cast (37) in the standard form of an OpDE:…”

A non-equilibrium analysis and control framework for active queue management

“…Papers concerned with deriving mean or limit models [23], [29], [30], obtaining AQM performance with linear control methods [11], [31], or with carrying out describing function based analysis [32] typically assume a random noise. The papers concerned with local instability/bifurcation analysis [22], [33]- [36], and (global) numerical investigations [1], [3], [22] using deterministic fluid models show these oscillations as self-excited. With simple fluid models, analytical methods from bifurcation theory have been used to show that these self-excited oscillations can arise as a result of supercritical Hopf bifurcation [33], [35], [37] and of period doubling and border-collision bifurcations [22].…”

“…The papers concerned with local instability/bifurcation analysis [22], [33]- [36], and (global) numerical investigations [1], [3], [22] using deterministic fluid models show these oscillations as self-excited. With simple fluid models, analytical methods from bifurcation theory have been used to show that these self-excited oscillations can arise as a result of supercritical Hopf bifurcation [33], [35], [37] and of period doubling and border-collision bifurcations [22].…”

“…The stochastic modeling approach we take enables us to carry out: Bifurcation analysis: Although the methods of local bifurcation theory can be used to analyze the instability and (local) onset of bifurcation as in [33], [35], these methods are less relevant to the AQM problem because of the global nature of the queue oscillation and presence of discontinuities such as saturations in these models. For instance, the chaotic oscillations have been typically studied using numerical simulations as in [3], [22].…”