Covariances of linear stochastic differential equations for analyzing computer networks

Analyses of dynamic systems with random oscillations need to calculate the system covariance matrix, but this is not easy even in the linear case if the random term is not a Gaussian white noise. A universal method is developed here to handle both Gaussian and compound Poisson white noise. The quadratic variations are analyzed to transform the problem into a Lyapunov matrix differential equation. Explicit formulas are then derived by vectorization. These formulas are applied to a simple model of flows and queuing in a computer network. A stability analysis of the mean value illustrates the effects of oscillations in a real system. The relationships between the oscillations and the parameters are clearly presented to improve designs of real systems.

[1]  Victor Firoiu,et al.  A study of active queue management for congestion control , 2000, Proceedings IEEE INFOCOM 2000. Conference on Computer Communications. Nineteenth Annual Joint Conference of the IEEE Computer and Communications Societies (Cat. No.00CH37064).

[2]  Raj Jain,et al.  Analysis of the Increase and Decrease Algorithms for Congestion Avoidance in Computer Networks , 1989, Comput. Networks.

[3]  Donald F. Towsley,et al.  A control theoretic analysis of RED , 2001, Proceedings IEEE INFOCOM 2001. Conference on Computer Communications. Twentieth Annual Joint Conference of the IEEE Computer and Communications Society (Cat. No.01CH37213).

[4]  M. Grigoriu.  The Itô and Stratonovich integrals for stochastic differential equations with poisson white noise , 1998 .

[5]  Vishal Misra,et al.  Fluid-based analysis of a network of AQM routers supporting TCP flows with an application to RED , 2000, SIGCOMM.

[6]  L. Rogers.  Stochastic differential equations and diffusion processes: Nobuyuki Ikeda and Shinzo Watanabe North-Holland, Amsterdam, 1981, xiv + 464 pages, Dfl.175.00 , 1982 .

[7]  QUTdN QeO,et al.  Random early detection gateways for congestion avoidance , 1993, TNET.

[9]  Fernando Paganini,et al.  Dynamics of TCP/RED and a scalable control , 2002, Proceedings.Twenty-First Annual Joint Conference of the IEEE Computer and Communications Societies.

[10]  Hua Fan,et al.  Using SDE to achieve the stable and statistical analyses for TCP/RED flows , 2010, 2010 8th World Congress on Intelligent Control and Automation.

[11]  Jae Chung,et al.  Analysis of active queue management , 2003, Second IEEE International Symposium on Network Computing and Applications, 2003. NCA 2003..

[12]  G. Kallianpur.  Stochastic differential equations and diffusion processes , 1981 .

[13]  Mohsen Razzaghi,et al.  On the solution of the covariance matrix differential equation for singular systems , 1998, Int. J. Comput. Math..

[14]  Donald F. Towsley,et al.  Analysis and design of controllers for AQM routers supporting TCP flows , 2002, IEEE Trans. Autom. Control..

[15]  T. Bhaskar Reddy,et al.  Performance Comparison of Active Queue Management Techniques , 2008 .