Greed Considered Harmful

Transcription

1 Greed Considered Harmful Nonlinear (in)stabilities in network resource allocation Priya Ranjan Indo-US workshop 2009

2 Outline Background Model & Motivation Main results Fixed delays Single-user, single-link case Feedback based on instantaneous rate Feedback based on averaged rate State-dependent time-varying delay single-link case Distributed delays-closed form solution for Gamma Kernel Discrete delays with feedback based on average Conclusions Future work

3 Motivation- Kelly s rate-based primal algorithm Rate-based flow control : each resource charges a price per unit flow (derivative of cost) users update their rates according to where

4 Background of rate control in the Internet

5 Background of rate control in the Internet Packet A A A P P P A P P P A P A Router Acknowledgment Question : How do we share resources fairly and efficiently?

6 Background of rate control in the Internet Network users have a great deal of freedom as to how they can share the available bandwidth in the network The increasing complexity and size of the Internet renders a centralized rate allocation impractical distributed algorithm is desired Two classes of flow/congestion control mechanisms rate-based : directly controls the transmission rate based on feedback (example: PFCC) window-based : controls the congestion window size to adjust the transmission rate and backlog (example: Transmission Control Protocol (TCP))

7 Model Optimization Framework (Kelly 1997) Network with a set of users (I) and a set of links (J) User receives a utility when it gets a rate - increasing and strictly concave

8 Model - Example

9 Kelly s Model Optimization framework impractical for a centralized system to compute and allocate the user rates

10 Two parts of original model

11 Model Kelly 1997 One can always find vectors and such that 1) solves 2) solves for all 3) 4) is the unique solution to At the equilibrium between NETWORK and USER problems the system optimum is achieved

12 Motivation- Kelly s rate-based primal algorithm Rate-based flow control : each resource charges a price per unit flow (derivative of cost) users update their rates according to where

13 Motivation- Kelly s rate-based algorithm Theorem (Kelly et al. 98): User rates converge to a point that maximizes the following expression Proposed rate-based (primal) algorithm solves a relaxation of the SYSTEM(U, A, C) problem But, wait! Where is the delay?!?!?!

14 Motivation - Kelly s rate-based algorithm Delays between network elements ignored in Kelly s original formulation! Especially important in wireless environment, (e.g., multihop wireless networks and satellite networks)

15 Motivation Delays in a network vary widely and are hard to predict Uncertainty in delays (e.g., multi-hop wireless networks) Large variation (e.g., wireline vs. satellite connections) Robustness is critical in a large, evolving system Stability of the system independent of the delays or gain parameters (κ) of the algorithm => Improves the portability of the protocol from one technology to another (e.g., cellular to satellite to military networks)

16 Model with Fixed DELAYS Delays between network elements (FIXED for now)

17 Single-User, Single-Link Case End user algorithm (no forward delay) Normalize time by T

18 Single-User, Single-Link Case End user algorithm (no forward delay) Normalize time by T

19 Single-User, Single-Link Case Buffer Forward Delay D1 Server Sender Reverse Delay D2 Generic Network Feedback Model

20 Single-User, Single-Link Case Define Change of coordinates (1)

21 Single-User, Single-Link Case

22 Single-User, Single-Link Case: Motivated by Ivanov Underlying market structure

23 Invariance and Stability

24 Example The utility function given by found to be useful for modeling TCP utility - Price elasticity of demand -Users with smaller a are GREEDIER

25 Example Price function given by Marking function imitating an M/M/1 queue

26 Discrete-time Map Numerical Simulation: 2 homogeneous users a=3, b=5, and C=5, Rate control algorithm is unstable since 6/3=2>1. First Return Map Second Return Map Xn+1=F(xn) 45o line Xn

27 Instability: effect of greed

28 Instability: effect of delay Numerical Simulation: 2 homogeneous users a=3, b=5, and C=5, Unstable since 6/3=2>1.T=1

29 Instability: effect of delay Numerical Simulation: 2 homogeneous users a=3, b=5, and C=5, Unstable since 6/3=2>1.T=10

30 Instability: effect of delay Numerical Simulation: 2 homogeneous users a=3, b=5, and C=5, Unstable since 6/3=2>1.T=50

31 Instability: Period Doubling, What happens if a < b+1?

32 Instability: Period Doubling What happens if a < b+1?

33 Instability: Period Doubling Numerical Simulation: 2 homogeneous users a=3, b=5, and C=5, Rate control algorithm is unstable since 6/3=2>1. First Return Map Second Return Map Xn+1=F(xn) 45o line Xn

34 Instability: Period Doubling What happens if a < b+1?

35 Global Bifurcation: Slowly Oscillating Periodic Periodic Orbit

36 TCP-RED Dynamics: Firiou s Model 2001

37 TCP-RED Dynamics: First Order Nonlinear Map Essentially nonlinear first order discrete time dynamical system Inverse square root nonlinearity comes from TCP transfer function which has an inverse square root-type dependence on drop prob. Model as a self clocking system. Piecewise smooth (with respect to state) map description comes from buffer being bounded between 0 and B. Different increasing and decreasing segments hint at chaos like behaviour. Map is smooth with respect to exponentially averaging parameter w and other system/red parameters

38 TCP-RED Dynamics: Discrete Time Map First and second return maps and their intersection with 45o degree line show the existence of a fixed point and period two orbit

39 TCP-RED-Gentle_ Dynamics: First Order Nonlinear Map First and second return maps for TCP and RED GENTLE_

40 TCP-RED Model: PDB and Border Collision

41 General Case Not Quite Yet! General network with homogeneous delay Normalize time by T

42 General Case Homogeneous Delay Matrix form

43 General Case Homogeneous Delay Discrete time map Establish the correspondence between the stability of the above discrete time map and the delay-differential equations Underlying market mechanism

44 General Case Homogeneous Delay

45 General Case Homogeneous Delay

46 General Case Homogeneous Delay

47 General Case Homogeneous Delay

48 Example Utility functions: Price functions:

49 Example Define

50 General Case (Finally!!) Devices driven by local oscillators with fixed frequencies

51 Fairness User 3 C1 =1 User 1 C2 =1 User 2 Acrobat Document

52 Outline Background Model & Motivation Main results Single-user, single-link case Multiple links case Fairness Conclusions

53 Distributed delay case: Physical Motivation

54 Distributed delay case: Imagine a strip with a continnum of sensors (e.g. smart dust, amorphous computing etc.) Feedback weighted by distributed delay

55 Distributed delay case: Gamma Function Used in Chemostat modeling

56 Distributed Delay: Linear Chain Technique

57 Similar to Hale and Ivanov s Result (1993,JMAA)

58 Hale and Ivanov s Result (1993, JMAA) LPF Xm F( ). LPF X1 Time Delay Stability if xn+1=f(xn) LPF X0

59 Delay Differetial Equation as a Discrete Time Map

60 Invariance and Stability Result

61 Stochastic stability in the presence of perturbations

62 Stochastic stability in the presence of perturbations: Results

63 Conclusions Discrete time maps is an efficient tool for DDEs Robust Stability derived using invariance principle Useful for a information rich-world where feedback may be based on data-fused from many sources Modeling distributed delay is achieved using Gamma Kernel The stability of a system with arbitrary delays can be studied by that of a corresponding discrete time system Much easier to analyze and simulate There is a close relationship between the responsiveness of the price functions and the elasticity of user demand

64 Distributed Boundary Tracking-Some Fun Stuff

65

66 Journal Publications: [1] P. Ranjan, E. H. Abed and R.J. La, Nonlinear instabilities in TCP-RED, IEEE/ ACM Transactions on Networking, Vol. 12, No. 6, Dec. 2004, pp [2] P. Ranjan, R.J. La and E.H. Abed, Global stability conditions for rate control with an arbitrary communication delay, IEEE/ ACM Transactions on Networking, Vol. 14, No. 1, Feb. 2006, pp [3] R.J. La and P. Ranjan, Asymptotic Stability of a Rate Control System with Communication Delays, Accepted for publication as a technical note to IEEE Transactions on Automatic Control, To appear in Oct [4] R.J. La and P. Ranjan, Global Stability of the Dual Algorithm with Arbitrary Network Delays, Accepted for publication in SIAM Journal on Applied Mathematics, July [4] R.J. La and P. Ranjan, Stability of a rate control system with averaged feedback and network delay, Elsevier Automatica, Vol. 42, No. 10, October 2006., pp Many conference publications on this topic

67 People Researching on Global Bifucations in DDE: Nussbaum, Mallet-Paret, Walther, Ivanov etc. Thanks for your patience!!