Passivity Approach to Dynamic Distributed Optimization for Network Traffic Management

1. Passivity Approach to Dynamic Distributed Optimization for Network Traffic Management John T. Wen, Murat Arcak, Xingzhe Fan Department of Electrical, Computer, & Systems Eng. Rensselaer Polytechnic Institute Troy, NY 12180

2. . . . . Network Flow Control Problem Design source and link control laws to achieve: stability, utilization, fairness, robustness Optimization approach: Kelly, Low, Srikant, … x  N y L Rf N sources L links Forward routing matrix (including delays) Source control link control Adjust sending rate based on congestion indication (AIMD, TCP Reno, Vegas) diagonal diagonal AQM: Provide congestion information (RED, REM, AVQ) Rb p L q N Return routing matrix (including delays)

3. Passivity x u y H A System H is passive if there exists a storage function V(x) 0 such that for some function W(x)  0 If V(x) corresponds to physical energy, then H conserves or dissipates energy. Example: Passive (RLC) circuits, passive structure, etc.

4. Passivity Approach: Primal Kelly’s Primal Controller -(q-q*) x-x* -(p-p*) -p -q x y + + RT R RT s IL R - - p-p* y-y* p y h1 s-1 IL h -(q-q*) + K - -(q-q*) + g1 (D+C(sI-A)-1B) s-1 IN -(U’(x)-U’(x*)) - g1 s-1 IN -(U’(x)-U’(x*))

5. . x=K(U’(x)-q)+x Extension • Passive decomposition is not unique: • For first order source controller, the system between –(p-p*) and (y-y*) is also passive. -(p-p*) -(q-q*) x-x* y-y* + RT R - If U’’<0 uniformly (strictly concave), contains a negative definite term in x-x* ---important for robustness! p-p* y-y* h1 • First order dual: (y-y*) to (p-p* ) is also passive • Implementable using delay and loss

6. . . q q . . p p . . . . = (y-c+ b)+p p = (y-c)+b = (y-c)+b b b = (y-c+ b)+p p Passivity Approach: Dual Low’s Dual Controller + -q x -(q-q*) -U’-1 x-x* + - s-1IL g1-1 - - p y q x p-p* y-y* RT R q-q* x-x* sIL RT R y - c y - c D+C(sI-A)-1B

7. -p -q x y + RT R . - x= K (U’(x)-q))+x . p y p = (y - c)+p Passivity Approach: Primal/Dual Controller • Consequence of passivity of first order source controller and first order link controller: combined dynamic controller is also stable. • Generalizes Hollot/Chait controller and easily extended to Kunniyur/Srikant controller.

8. Simulation: Primal Controller (B1:Passive) (A1:Kelly) .25 sec delay

9. Simulation: Dual Controller (B2:Passive) (A2:Low/Paganini) 1 sec delay

10. Robustness in Time Delay • Passivity approach provides Lyapunov function candidates to compute quantitative trade-offs between disturbance and performance, and stability bounds on delays.

11. Extension to CDMA Power Control • Passivity approach is applicable to other distributed optimization problems: minimize power subject to the signal-to-interference constraint.

12. Extension to Multipath Flow Control • Traffic demand in multipath flow control can be incorporated as additional inequality constraints. Same passivity analysis applicable with demand pricing feedback based on r-Hx. z2=x3+x4+x5≥ r2 x3 x4 x5 x1 x1 x2 z1=x1+x2≥ r1 x2 x5 x3 x4