Asymptotic Behavior of Heterogeneous TCP Flows and RED Gateway

1. Asymptotic Behavior of Heterogeneous TCP Flows and RED Gateway Peerapol Tinnakornsrisuphap University of Maryland (joint work with Armand M. Makowski and Richard J. La) IPAM Large-Scale Communication Networks Lake Arrowhead, CA September 30th, 2003

2. Outline • Background, motivation and the approach • Model description • The Law of Large Numbers (or why the deterministic models are justified) • The Central Limit analysis (or what the deterministic models cannot capture) • Conclusion & future work

3. TCP in a nutshell • TCP controls around 90% of Internet traffic • TCP operates in two regimes • Slow-start --- Exponential growth of window • Congestion avoidance --- the additive increase/ multiplicative decrease (AIMD) feedback mechanism

4. Random Early Detection (RED) • RED signals a TCP flow to back off by randomly dropping or marking incoming packets • The probability of dropping/marking is calculated through a function of the exponentially averaged queue size • Typically, the memory of the average is long, hence consecutive incoming packets are marked with almost identical probability

5. Modeling TCP and RED • Difficult in realistic situations • Feedback-based and stochastic system • Explosion of state-space (for large # of flows) • Several layers interacting with each other • Session dynamics • Variable round-trips • Qualitative and quantitative results needed in order to understand and control TCP traffic effectively

6. Typical Modeling Approaches • Too simplistic • Not scalable • Too crude (e.g., ad-hoc approximations) • Enforced average to reduce number of states • Fixed point approximation • Accurate only in certain regimes • Shot noise process (for low congestion networks) models only the slow-start • M/G/1/PS (for heavily congested networks) models only the congestion-avoidance

7. The Approach • We seek limiting behavior of queue dynamics when large number of TCP sessions, say N going to infinity • The session layer and the variable round-trip delay are explicitly incorporated into the model • Complement the control-theoretic viewpoint and yield better understanding of the roles of random round-trip, file sizes, marking mechanisms

8. Why Limit? • Networking problem interesting usually under heavy usage. • Simplification usually occurs in the limit, as the redundant details are removed. • Limit theorems are central to the Theory of Probability and hold under very weak assumptions.

9. Outline • Background, motivation and the approach • Model description • The Law of Large Numbers • The Central Limit analysis • Conclusion & future work

10. Overview of our model • Session Dynamics • Need to keep track of the remaining packets in the session for an active flow • Session arrivals according to Poisson process • TCP Dynamics • Slow-start and Congestion avoidance window recursion • Slow-start is only for recently-active flows which have never experienced congestion

11. Overview of our model (cont’d) • Network Dynamics • Update the queue size according to the capacity and packet arrivals • Mark packets with probability according to the queue size • Round-trip Dynamics • Randomly chosen at the beginning of each session • Update congestion window using the delayed information

12. Interaction between Layers

13. The model (without variable round-trips) Discrete time with slotted time Duration of the slotted time equals to the round-trip Recursive queue dynamics – similar to Lindley’s recursion N sessions Capacity = NC packets/timeslot Infinite Buffer

14. Notation • Wi(N)(t) Congestion window size of Flow i (out of N) at the beginning of the timeslot [t, t+1), its value is an integer between [0, Wmax]. • Q(N)(t) Queue size at the beginning of the timeslot [t, t+1). • f (N): R+’[0,1]Marking probability function.

15. Additional Notation • Xi(N)(t) Remaining workload of the session of user i (out of N) at the beginning of the timeslot [t,t+1) • Si(N)(t) Indicator function indicating whether TCP flow i is in slow-start or congestion avoidance • Ai(N)(t) The amount of traffic user i injected into the network in the timeslot [t,t+1)

16. Session Dynamics

17. TCP Dynamics

18. TCP Dynamics (cont’d)

19. Network Dynamics

20. Outline • Background, motivation and the approach • Model description • The Law of Large Numbers • The Central Limit analysis • Conclusion & future work

21. Assumptions • (A1) There exists a continuous function f : R+’[0,1] such that for each N = 1,2,… • (A2) For each N, the dynamics start with initial conditionsfor i = 1,…,N.

22. The Weak Law of Large Numbers Assume (A1)-(A2). Then, for each t=0,1,… there exist a (non-random) constant q(t) and rvs (W(t),X(t),S(t)) such that • The convergence takes place;

23. WLLNs (cont’d) • For any integer I = 1,2,…, the rvs {(Wi(N)(t),Xi(N)(t),Si(N)(t)), i=1,…,I} becomes asymptotically independent as N becomes large • For any bounded function g: IN+3’IR, • Simplified recursions for q(t) and (W(t),X(t),S(t)), t = 0,1,… (closely related to the single flow model)

24. Implications:The weak law of large numbers • Queue dynamics can be approximated by a simple recursion, i.e.,The deterministic recursion for q(t) does not depend on N, hence the model is scalable. It is also more accurate as N becomes large. • The dependency between each session becomes negligible as N becomes large, i.e., “RED breaks global synchronization when the number of sessions is large”

25. Comparison to other models Assume • (A3) The marking function is monotonically increasing with Then • For C!1,the model is similar to the time-reversed shot-noise model, i.e., the slow-start mechanism dominates • For C¼0, the queueing behavior approaches that of M/G/1/PS model

26. A Steady-State Analysis • (A4) for some non-random q* and rvs W*, X*, S*which immediately implies the steady-state marking probability f(q*). • (A5) Average workload is large comparing to the average window size, i.e.,

27. Steady-state Analysis (cont’d) • Simple relationship between model parameters and the queue length • Numerical examples show that the formula provides a reasonable approximation at the steady-state

28. Model w/ variable round-trips • Duration of the slotted time equals to the greatest common divisor of the round-trips • Congestion window update uses information which is delayed by a round-trip • Connection transmits once per round-trip (all packets aggregated into one timeslot) • Bursty behavior --- ACK compression • Provide upper bound on the fluctuation • Have minimal effect with queue averaging

29. With variable round-trips… • Similar WLLNs can be derived • The steady-state analysis suggests that only the mean round-trip delay affects the mean steady-state queue level • BUT the magnitude of queue fluctuation will be larger for random round-trips with higher variance • This will be established rigorously in the CLT

30. Outline • Background, motivation and the approach • Model description • The Law of Large Numbers • The Central Limit analysis • Conclusion & future work

31. N1/2 L(t) Nq(t) A Central Limit Analysis • Since we have ,does there exist a rv L(t) s.t. • Sharper approximation in the formon the queue distribution.

32. A Central Limit Theorem • Assume (A1)-(A2) to hold with f being continuously differentiable. Set Then, for each t = 0,1,…, there exists a R2-valued rv L(t) s.t.

33. CLT (cont’d) Moreover, the distributional recurrenceholds, where K(t) is the residual capacity per user in the limit.

34. f(x) 1 x maxthresh By product of the theorem.. for some constant c(t) and rv x(t).The magnitude of the queue fluctuation increases as f’ increases.This agrees with observation in [Firoiu, INFOCOM’00].

35. Why ? • The only feedback information from RED is f(N)(Q(N)(t)) • This feedback info also fluctuates around its limiting mean f(q(t)) • The Delta Method: Iff is continuously differentiable, thenimplies

36. Sources of queue fluctuations • Fluctuation in feedback information (through the delta method) • Limited granularity in feedback info (through binary marking scheme in ECN) • Properly scaled and centered file sizes and round-trips – also converges weakly to Gaussian rvs • Overall fluctuation is well approximated by a Guassian rv – combined 1-3 through the protocol structure

37. Discussion • Two types of fluctuations • Deterministic – Oscillation predictable by control-theoretic models • Random – Established in the CLT analysis • How to distinguish between the two? • How to reduce the random component in the fluctuation? • Select f(.) carefully • Increase the number of ECN bits

38. Effects of f(x) ‘Shallow’ f(x) ‘Steep’ f(x)

39. Effects of variable round-trips Uniform round-trip Bimodal round-trip

40. Conclusions • A large number of TCP flows leads to a simpler modeland a better understanding of the dynamics • Deterministic models are fine for the average behavior, but the CLT yields a more precise dynamics • RED breaks global synchronization and provides a control of TCP traffic, but the feedback function has to be chosen with care • Steady-state behavior is simple to analyze. TCP flows are “decoupled” from the network in the asymptotic regime

41. Future work • Selection of f(x) • Low queueing delay needs a steep function • Small fluctuation needs a flat function • Optimization problem • Extension to other AQMs and congestion control mechanisms • Provide a simple, unified framework to analyze and compare different AQMs • Multiple Bottlenecks