Local Stability and Hopf Bifurcation Analysis for Compound TCP
TL;DR: A detailed analytical framework is presented to address the stability of the limit cycles and the type of the Hopf bifurcation by invoking Poincaré normal forms and the center manifold theory and packet-level simulations are conducted to corroborate the analytical insights.
Abstract: We conduct a local stability and Hopf bifurcation analysis for Compound transmission control protocol (TCP), with small Drop-Tail buffers, in three topologies. The first topology consists of two sets of TCP flows having different round-trip times and feeding into a core router. The second topology consists of two distinct sets of TCP flows, regulated by a single-edge router and feeding into a core router. The third topology comprises two distinct sets of TCP flows, regulated by two separate edge routers and feeding into a core router. In each case, we conduct a local stability analysis and obtain conditions on the network and protocol parameters to ensure stability. If these conditions get marginally violated, we show that the underlying systems lose local stability via a Hopf bifurcation. After exhibiting a Hopf, a key concern is to determine the asymptotic orbital stability of the bifurcating limit cycles. We then present a detailed analytical framework to address the stability of the limit cycles and the type of the Hopf bifurcation by invoking Poincare normal forms and the center manifold theory. We finally conduct packet-level simulations to corroborate our analytical insights.
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TL;DR: In this paper, a robust active queue management (RAQM) protocol is proposed to prevent congestion along with being robust against wireless environment issues such as packet error rate and fading.
Abstract: Transmission control protocol, in the transport layer of a network, can usually detect congestion after its occurrence. Therefore, designing a robust active queue management (RAQM) is imperative to prevent congestion along with being robust against wireless environment issues such as packet error rate and fading. Moreover, a communication network suffers from input delay as well as the state delay which is multiplied to the control input signal. The main contribution of the authors’ study is to design a predictive sliding mode control (PSMC) procedure as a RAQM to guarantee the input delay system stability and to regulate the queue length to the desired value. Firstly, a predictor is designed for the original system to obtain an input delay free model. Then, a RAQM is designed based on PSMC for the system with non-singular and singular control gain. The disturbance observer ensures that the estimation error tends to zero. Unlike the prevalent procedures designed for the networks, the proposed method can avoid the singular gain problem in the control design. Furthermore, it can stabilise the system and can prevent congestion with robustness against external disturbances. Finally, the simulation results, obtained from SIMULINK and professional network simulator 2, confirm the analytical consequences.
8 citations
TL;DR: In this article, a performance evaluation of Compound TCP (C-TCP) in a regime where the router buffer sizes are small (i.e., independent of the bandwidth-delay product), and the queue policy is Drop-Tail is presented.
5 citations
Posted Content•
TL;DR: A performance evaluation of Compound TCP in two topologies: a single bottleneck and a multi-bottleneck topology, under different traffic scenarios shows that smaller buffers would also yield fairly good system performance, in terms of throughput and flow completion times.
Abstract: Motivated by recent concerns that queuing delays in the Internet are on the rise, we conduct a performance evaluation of Compound TCP (C-TCP) in two topologies: a single bottleneck and a multi-bottleneck topology, under different traffic scenarios. The first topology consists of a single bottleneck router, and the second consists of two distinct sets of TCP flows, regulated by two edge routers, feeding into a common core router. We focus on some dynamical and statistical properties of the underlying system. From a dynamical perspective, we develop fluid models in a regime wherein the number of flows is large, bandwidth-delay product is high, buffers are dimensioned small (independent of the bandwidth-delay product) and routers deploy a Drop-Tail queue policy. A detailed local stability analysis for these models yields the following key insight: smaller buffers favour stability. Additionally, we highlight that larger buffers, in addition to increasing latency, are prone to inducing limit cycles in the system dynamics, via a Hopf bifurcation. These limit cycles in turn cause synchronisation among the TCP flows, and also result in a loss of link utilisation. For the topologies considered, we also empirically analyse some statistical properties of the bottleneck queues. These statistical analyses serve to validate an important modelling assumption: that in the regime considered, each bottleneck queue may be approximated as either an $M/M/1/B$ or an $M/D/1/B$ queue. This immediately makes the modelling perspective attractive and the analysis tractable. Finally, we show that smaller buffers, in addition to ensuring stability and low latency, would also yield fairly good system performance, in terms of throughput and flow completion times.
5 citations
06 Apr 2019
TL;DR: This case study paper demonstrates the appropriateness and usefulness of the digital bifurcation analysis as a push-button alternative to the classical approaches as traditionally used for analysing the stability of TCP/IP protocols.
Abstract: Digital bifurcation analysis is a new algorithmic method for exploring how the behaviour of a parameter-dependent computer system varies with a change in its parameters and, in particular, for identification of bifurcation points where such variation becomes dramatic. We have developed the method in an analogy with the traditional bifurcation theory and have it successfully applied to models taken from systems biology. In this case study paper, we demonstrate the appropriateness and usefulness of the digital bifurcation analysis as a push-button alternative to the classical approaches as traditionally used for analysing the stability of TCP/IP protocols. We consider two typical examples (congestion control and buffer sizes throughput influence) and show that the method provides the same results as obtained with classical non-automatic analytical and numerical methods.
4 citations
Cites background from "Local Stability and Hopf Bifurcatio..."
...Bifurcation analysis of TCP under various congestion control mechanisms have been studied by several authors [16,25,30,32,40,41]....
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TL;DR: This article at first proposes a predictor to eliminate the input delay and then designs a compensated system to deal with the input signal constraint, and the chattering phenomena is eliminated based on the adaptive laws designed for the controller parameters.
Abstract: This article studies congestion control of wireless access networks. In a wireless access network, it is necessary to design a robust active queue management (RAQM) technique to control congestion occurrence and to make the network robust simultaneously against some issues in the wireless link aspects, such as packet error rate (PER) and fading effects. On the other hand, a network is often described by a nonlinear model, in which the delayed packet drop probability is assumed as the input signal. So in this article, a RAQM technique is proposed based on an adaptive fractional order predictive sliding mode control (AFOPSMC) method, through which the following are achieved: (1) the stability of the nonlinear system with input delay is assured, (2) the congestion occurrence is prevented by controlling the queue measurement to the desired value, (3) the robustness against the external disturbances is achieved, and (4) the input signal obtained by the controller is limited between 0 and 1. In contrast to most recently published papers where input delay is ignored in the system description and input saturation is achieved through designing parameters, this article at first proposes a predictor to eliminate the input delay and then designs a compensated system to deal with the input signal constraint. Furthermore, the chattering phenomena, as a commonly caused issue in the sliding mode control, is eliminated based on the adaptive laws designed for the controller parameters. The theoretical results are validated and compared with some other related protocols through numerical simulations using Simulink and professional network simulator 2 (NS2).
1 citations
References
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Book•
01 Jan 1966
TL;DR: In this paper, the Riesz representation theorem is used to describe the regularity properties of Borel measures and their relation to the Radon-Nikodym theorem of continuous functions.
Abstract: Preface Prologue: The Exponential Function Chapter 1: Abstract Integration Set-theoretic notations and terminology The concept of measurability Simple functions Elementary properties of measures Arithmetic in [0, ] Integration of positive functions Integration of complex functions The role played by sets of measure zero Exercises Chapter 2: Positive Borel Measures Vector spaces Topological preliminaries The Riesz representation theorem Regularity properties of Borel measures Lebesgue measure Continuity properties of measurable functions Exercises Chapter 3: Lp-Spaces Convex functions and inequalities The Lp-spaces Approximation by continuous functions Exercises Chapter 4: Elementary Hilbert Space Theory Inner products and linear functionals Orthonormal sets Trigonometric series Exercises Chapter 5: Examples of Banach Space Techniques Banach spaces Consequences of Baire's theorem Fourier series of continuous functions Fourier coefficients of L1-functions The Hahn-Banach theorem An abstract approach to the Poisson integral Exercises Chapter 6: Complex Measures Total variation Absolute continuity Consequences of the Radon-Nikodym theorem Bounded linear functionals on Lp The Riesz representation theorem Exercises Chapter 7: Differentiation Derivatives of measures The fundamental theorem of Calculus Differentiable transformations Exercises Chapter 8: Integration on Product Spaces Measurability on cartesian products Product measures The Fubini theorem Completion of product measures Convolutions Distribution functions Exercises Chapter 9: Fourier Transforms Formal properties The inversion theorem The Plancherel theorem The Banach algebra L1 Exercises Chapter 10: Elementary Properties of Holomorphic Functions Complex differentiation Integration over paths The local Cauchy theorem The power series representation The open mapping theorem The global Cauchy theorem The calculus of residues Exercises Chapter 11: Harmonic Functions The Cauchy-Riemann equations The Poisson integral The mean value property Boundary behavior of Poisson integrals Representation theorems Exercises Chapter 12: The Maximum Modulus Principle Introduction The Schwarz lemma The Phragmen-Lindelof method An interpolation theorem A converse of the maximum modulus theorem Exercises Chapter 13: Approximation by Rational Functions Preparation Runge's theorem The Mittag-Leffler theorem Simply connected regions Exercises Chapter 14: Conformal Mapping Preservation of angles Linear fractional transformations Normal families The Riemann mapping theorem The class L Continuity at the boundary Conformal mapping of an annulus Exercises Chapter 15: Zeros of Holomorphic Functions Infinite Products The Weierstrass factorization theorem An interpolation problem Jensen's formula Blaschke products The Muntz-Szas theorem Exercises Chapter 16: Analytic Continuation Regular points and singular points Continuation along curves The monodromy theorem Construction of a modular function The Picard theorem Exercises Chapter 17: Hp-Spaces Subharmonic functions The spaces Hp and N The theorem of F. and M. Riesz Factorization theorems The shift operator Conjugate functions Exercises Chapter 18: Elementary Theory of Banach Algebras Introduction The invertible elements Ideals and homomorphisms Applications Exercises Chapter 19: Holomorphic Fourier Transforms Introduction Two theorems of Paley and Wiener Quasi-analytic classes The Denjoy-Carleman theorem Exercises Chapter 20: Uniform Approximation by Polynomials Introduction Some lemmas Mergelyan's theorem Exercises Appendix: Hausdorff's Maximality Theorem Notes and Comments Bibliography List of Special Symbols Index
9,642 citations
Book•
01 Jan 1995
TL;DR: One-Parameter Bifurcations of Equilibria in continuous-time systems and fixed points in Discrete-Time Dynamical Systems have been studied in this paper, where they have been used for topological equivalence and structural stability of dynamical systems.
Abstract: Introduction to Dynamical Systems * Topological Equivalence, Bifurcations, and Structural Stability of Dynamical Systems * One-Parameter Bifurcations of Equilibria in Continuous-Time Systems * One-Parameter Bifurcations of Fixed Points in Discrete-Time Systems * Bifurcations of Equilibria and Periodic Orbits in n-Dimensional Systems * Bifurcations of Orbits Homoclinic and Heteroclinic to Hyperbolic Equilibria * Other One-Parameter Bifurcations in Continuous-Time Systems * Two-Parameter Bifurcations of Equilibria in Continuous-Time Dynamical Systems * Two-Parameter Bifurcations of Fixed Points in Discrete-Time Dynamical Systems * Numerical Analysis of Bifurcations * A: Basic Notions from Algebra, Analysis, and Geometry * References * Index.
5,062 citations
TL;DR: The CUBIC protocol modifies the linear window growth function of existing TCP standards to be a cubic function in order to improve the scalability of TCP over fast and long distance networks.
Abstract: CUBIC is a congestion control protocol for TCP (transmission control protocol) and the current default TCP algorithm in Linux. The protocol modifies the linear window growth function of existing TCP standards to be a cubic function in order to improve the scalability of TCP over fast and long distance networks. It also achieves more equitable bandwidth allocations among flows with different RTTs (round trip times) by making the window growth to be independent of RTT -- thus those flows grow their congestion window at the same rate. During steady state, CUBIC increases the window size aggressively when the window is far from the saturation point, and the slowly when it is close to the saturation point. This feature allows CUBIC to be very scalable when the bandwidth and delay product of the network is large, and at the same time, be highly stable and also fair to standard TCP flows. The implementation of CUBIC in Linux has gone through several upgrades. This paper documents its design, implementation, performance and evolution as the default TCP algorithm of Linux.
2,088 citations
"Local Stability and Hopf Bifurcatio..." refers background in this paper
...A major portion of Internet traffic is controlled by the TCP [8], [14]....
[...]
TL;DR: A simple analytic characterization of the steady-state send rate as a function of loss rate and round trip time for a bulk transfer TCP flow is developed and is able to more accurately predict TCP send rate and is accurate over a wider range of loss rates.
Abstract: The steady-state performance of a bulk transfer TCP flow (i.e., a flow with a large amount of data to send, such as FTP transfers) may be characterized by the send rate, which is the amount of data sent by the sender in unit time. In this paper we develop a simple analytic characterization of the steady-state send rate as a function of loss rate and round trip time (RTT) for a bulk transfer TCP flow. Unlike the models of Lakshman and Madhow (see IEE/ACM Trans. Networking, vol.5, p.336-50, 1997), Mahdavi and Floyd (1997), Mathis, Semke, Mahdavi and Ott (see Comput. Commun. Rev., vol.27, no.3, 1997) and by by Ott et al., our model captures not only the behavior of the fast retransmit mechanism but also the effect of the time-out mechanism. Our measurements suggest that this latter behavior is important from a modeling perspective, as almost all of our TCP traces contained more time-out events than fast retransmit events. Our measurements demonstrate that our model is able to more accurately predict TCP send rate and is accurate over a wider range of loss rates. We also present a simple extension of our model to compute the throughput of a bulk transfer TCP flow, which is defined as the amount of data received by the receiver in unit time.
1,192 citations
"Local Stability and Hopf Bifurcatio..." refers background in this paper
...A major portion of Internet traffic is controlled by the TCP [8], [14]....
[...]
TL;DR: A new active queue management scheme, random exponential marking (REM), is described that aims to achieve both high utilization and negligible loss and delay in a simple and scalable manner and presents simulation results of its performance in wireline and wireless networks.
Abstract: We describe a new active queue management scheme, random exponential marking (REM), that aims to achieve both high utilization and negligible loss and delay in a simple and scalable manner. The key idea is to decouple the congestion measure from the performance measure such as loss, queue length, or delay. While the congestion measure indicates excess demand for bandwidth and must track the number of users, the performance measure should be stabilized around their targets independent of the number of users. We explain the design rationale behind REM and present simulation results of its performance in wireline and wireless networks.
1,041 citations