Stochastic Hybrid Systems: Application to Communication Networks

We propose a model for Stochastic Hybrid Systems (SHSs) where transitions between discrete modes are triggered by stochastic events much like transitions between states of a continuous-time Markov chains. However, the rate at which transitions occur is allowed to depend both on the continuous and the discrete states of the SHS. Based on results available for Piecewise-Deterministic Markov Process (PDPs), we provide a formula for the extended generator of the SHS, which can be used to compute expectations and the overall distribution of the state. As an application, we construct a stochastic model for on-off TCP flows that considers both the congestion-avoidance and slow-start modes and takes directly into account the distribution of the number of bytes transmitted. Using the tools derived for SHSs, we model the dynamics of the moments of the sending rate by an infinite system of ODEs, which can be truncated to obtain an approximate finite-dimensional model.

[1]  Karl Henrik Johansson,et al.  Dynamical properties of hybrid automata , 2003, IEEE Trans. Autom. Control..

[2]  M.H.A. Davis,et al.  Markov Models & Optimization , 1993 .

[3]  Stephan Bohacek,et al.  A stochastic model of TCP and fair video transmission , 2003, IEEE INFOCOM 2003. Twenty-second Annual Joint Conference of the IEEE Computer and Communications Societies (IEEE Cat. No.03CH37428).

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

[5]  V. Borkar,et al.  A unified framework for hybrid control : b background, model, and theory , 1994 .

[6]  R. Srikant,et al.  Analysis and design of an adaptive virtual queue (AVQ) algorithm for active queue management , 2001, SIGCOMM '01.

[7]  S. Sastry,et al.  Towars a Theory of Stochastic Hybrid Systems , 2000, HSCC.

[8]  Marta Z. Kwiatkowska,et al.  Automatic verification of real-time systems with discrete probability distributions , 1999, Theor. Comput. Sci..

[9]  Alain Bensoussan,et al.  Impulse Control and Quasi-Variational Inequalities , 1984 .

[10]  L. Tavernini.  Differential automata and their discrete simulators , 1987 .

[11]  E. Boukas,et al.  Mean square stochastic stability of linear time-delay system with Markovian jumping parameters , 1998, IEEE Trans. Autom. Control..

[12]  Biplab Sikdar,et al.  TCP Reno with random losses: latency, throughput and sensitivity analysis , 2001, Conference Proceedings of the 2001 IEEE International Performance, Computing, and Communications Conference (Cat. No.01CH37210).

[13]  Mariëlle Stoelinga,et al.  An Introduction to Probabilistic Automata , 2002, Bull. EATCS.

[14]  John Lygeros,et al.  Towars a Theory of Stochastic Hybrid Systems , 2000, HSCC.

[15]  Vishal Misray,et al.  Stochastic Differential Equation Modeling and Analysis of TCP-Windowsize Behavior , 2005 .

[16]  Mark H. Davis.  Markov Models and Optimization , 1995 .

[17]  George J. Pappas,et al.  Hybrid Systems: Computation and Control: 7th International Workshop, Hscc 2004, Philadelphia, Pa, Usa, March 2004: Proceedings (Lecture Notes in Computer Science, 2993) , 2004 .

[18]  John Guckenheimer,et al.  A Dynamical Simulation Facility for Hybrid Systems , 1993, Hybrid Systems.

[19]  Joao P. Hespanha,et al.  Analysis of a TCP hybrid model , 2002 .

[20]  Roberto Segala,et al.  Modeling and verification of randomized distributed real-time systems , 1996 .

[21]  Fernando Paganini,et al.  Internet congestion control , 2002 .

[22]  Roger W. Brockett,et al.  Hybrid Models for Motion Control Systems , 1993 .

[23]  Luca de Alfaro,et al.  Stochastic Transition Systems , 1998, CONCUR.

[24]  Biplab Sikdar,et al.  Analytic models for the latency and steady-state throughput of TCP tahoe, Reno, and SACK , 2003, TNET.

[25]  M. Egerstedt,et al.  On the regularization of Zeno hybrid automata , 1999 .

[26]  João Pedro Hespanha,et al.  Hybrid Modeling of TCP Congestion Control , 2001, HSCC.

[27]  Jerzy A. Filar,et al.  Control of singularly perturbed hybrid stochastic systems , 2001, IEEE Trans. Autom. Control..

[28]  João Pedro Hespanha.  Polynomial Stochastic Hybrid Systems , 2005, HSCC.

[29]  Donald F. Towsley,et al.  Modeling TCP Reno performance: a simple model and its empirical validation , 2000, TNET.

[30]  Paul Barford,et al.  Changes in Web Client Access Patterns , 1998, The Web Conference.

[31]  M. Fragoso,et al.  Stability Results for Discrete-Time Linear Systems with Markovian Jumping Parameters , 1993 .

[32]  Yuguang Fang,et al.  Stabilization of continuous-time jump linear systems , 2002, IEEE Trans. Autom. Control..

[33]  S. Floyd,et al.  Tcp-friendly unicast rate-based flow control , 1997 .

[34]  John Lygeros,et al.  Controllers for reachability specifications for hybrid systems , 1999, Autom..

[35]  Karl Henrik Johansson,et al.  Dynamical Systems Revisited: Hybrid Systems with Zeno Executions , 2000, HSCC.

[36]  J. Hespanha,et al.  Stochastic models for chemically reacting systems using polynomial stochastic hybrid systems , 2005 .

[37]  Nancy A. Lynch,et al.  Compositionality for Probabilistic Automata , 2003, CONCUR.

[38]  Johannes Schumacher,et al.  An Introduction to Hybrid Dynamical Systems, Springer Lecture Notes in Control and Information Sciences 251 , 1999 .

[39]  João Pedro Hespanha,et al.  A hybrid systems modeling framework for fast and accurate simulation of data communication networks , 2003, SIGMETRICS '03.

[40]  R. Srikant,et al.  How good are deterministic fluid models of Internet congestion control? , 2002, Proceedings.Twenty-First Annual Joint Conference of the IEEE Computer and Communications Societies.

[41]  A. Hassibi,et al.  Control with random communication delays via a discrete-time jump system approach , 2000, Proceedings of the 2000 American Control Conference. ACC (IEEE Cat. No.00CH36334).

[42]  Steven H. Low,et al.  A duality model of TCP and queue management algorithms , 2003, TNET.

[43]  A. Shiryaev,et al.  Limit Theorems for Stochastic Processes , 1987 .

[44]  Matthew Mathis,et al.  The macroscopic behavior of the TCP congestion avoidance algorithm , 1997, CCRV.

[45]  Joao P. Hespanha,et al.  Stochastic Hybrid Systems: Application to Communication Networks , 2004, HSCC.

[46]  Martin F. Arlitt,et al.  Workload characterization of a Web proxy in a cable modem environment , 1999, PERV.

[47]  M. K. Ghosh,et al.  Ergodic Control of Switching Diffusions , 1997 .

[48]  John Lygeros,et al.  Stochastic Hybrid Models: An Overview , 2003, ADHS.

[49]  Thomas A. Henzinger,et al.  Hybrid Systems: Computation and Control , 1998, Lecture Notes in Computer Science.

[50]  Panos J. Antsaklis,et al.  Hybrid System Modeling and Autonomous Control Systems , 1992, Hybrid Systems.

[51]  R. Srikant,et al.  Robustness of real and virtual queue-based active queue management schemes , 2005, IEEE/ACM Transactions on Networking.