Zhikui Wang, Fernando Paganini

1. UCLA Bounded States and Global Stability of Congestion Control with Time-Delays Zhikui Wang, Fernando Paganini Electrical Engineering Department, UCLA Sep. 30, 2003

2. Outline • Congestion Controls and Local Stability • Global Performance of the “Dual” Control with Delays • Ultimately Bounded States • Asymptotical Stability • Exponential Stability • Global Performance of the “Primal-Dual” Control • Ultimately Bounded States • Time-scale Analysis • Summary

3. “Dual” Control: Local Scalable Stability Theorem: Suppose the matrix is of full row rank, and Then link and source control laws that linearize around the equilibrium as shown above give a locally stable system for arbitrary delays and link capacities. [ Paganini, Doyle, Low, CDC’01] Sources : Links :

4. “Primal-Dual”: Scalable Stability and Free utility Theorem: [ Paganini, Wang, Low, Doyle, Infocom’02] Suppose is of full row rank, and for each source, then the “primal-daul” control is locally stable for a small depending only on . Sources : Slower tracking of fairness by small Faster control of utilization At equilibrium: Links : Maximizing local benefit

5. Outline • Congestion Controls and Local Stability • Global Performance of the “Dual” Control with Delays • Ultimately Bounded States • Asymptotical Stability • Exponential Stability • Global Performance of the “Primal-Dual” Control • Ultimately Bounded States • Time-scale Analysis • Summary

6. “Dual” Control: Global stability by Small-Gain Theorem: [ Wang, Paganini, CDC’02 ] The “Dual” system with single flow and single link is globally asymptotically stable if . -

7. “Dual” Control: Ultimately Bounded States Proposition: [ Wang, Paganini, CDC’02 ] For the “dual” system with single link, the price p(t) is ultimately upper bounded by where (I) (II) (III)

8. “Dual” Control: Asymptotical Stability (I) Theorem: The “Dual” system is globally asymptotically stable if

9. “Dual” Control: Asymptotical Stability (2) when Contractive Mapping

10. “Dual” Control: Asymptotical Stability (3) Corollary: The “Dual” system with single flow and single link is globally asymptotically stable if . Specially, • scalable w.r.t. delay and capacity • globally asymptotically stable if [wright, 1955] Special Case: Single-Flow Single-link

11. “Dual” Control: Exponential Stability (1) (2) (3) [Deb, CDC’02] (4) Theorem: [ Wang, Paganini, CDC’02 ] For the configuration of figure 1, there exists a positive constant such that for , the origin of the system Is exponentially stable.

12. Outline • Congestion Controls and Local Stability • Global Performance of the “Dual” Control with Delays • Ultimately Bounded States • Asymptotical Stability • Exponential Stability • Global Performance of the “Primal-Dual” Control • Ultimately Bounded States • Time-scale Analysis • Summary

13. s.t. Proposition: The rate x(t) is ultimately bounded, independent of the local stability condition, I.E., there exists such that Proposition: The states is ultimately bounded , independent of the local stability condition, if “Primal-Dual” : Global upper bound of Single-source & Single-link: [ Wang, Paganini, CDC’03 ]

14. “Primal-Dual” : Two time-scales --- Simulation Single-Flow Single-link Quasi-steady states

15. (I) Boundary-Layer system exponentially stable. (II) Reduced system: exponentially stable uniformly with . “Primal-Dual”: Time-scale analysis Claim: The system with bounded states are exponentially stable by taking small enough and .

16. Summary • Global Performance of the “Dual” Control with Delays • Stability by Small-Gain theorem • Ultimately Bounded States independent of Stability • Globally Asymptotical Stability • Globally Exponential Stability • Global Performance of the “Primal-Dual” Control • Ultimately Bounded States • Time-scale Analysis • Future work?