1. Estimating Congestion in TCP Traffic Stephan Bohacek and Boris Rozovskii University of Southern California Objective: Develop stochastic model of TCP Necessary ingredients: Models of the network. Specifically packet drop probability and roundtrip time. The model parameters are indicators of congestion.

2. Outline A very brief introduction to TCP Modeling packet drop probability –Modeling roundtrip time –Dynamics of drop probability parameters A diffusion model of roundtrip time –Estimating the model parameters Stochastic models of TCP Results and insights

3. An Introduction to TCP TCP is acknowledgment based. The sender sends a packet of data. When the receiver receives the packet, it responds by sending an acknowledgment packet to the sender. If the sender fails to receive an acknowledgment, it assumes that the packet has been dropped and decreases the sending rate. If the sender receives an acknowledgment, it assumes that the network is not congested and increases the sending rate.

4. The congestion window, X, defines the maximum number unacknowledged packets. When the sender receives an acknowledgment it increases the congestion window by 1/[X t ]. When a packet drop is detected, the sender divides the congestion window in half. Some TCP Details Note: the congestion window is not the sending rate. sending rate = congestion window / roundtrip time.

5. Number of unacknowledged packets equals Cwnd. Send no more packets. How the Congestion Window Increases send pkt A send pkt B send pkt C send pkt D X=4 ack A ack B ack C ack D send pkt E send pkt F send pkt G send pkt H send pkt F X=4.25 X=4.5 X=4.75 X=5 time Packet arrives at receiver. Receiver sends an acknowledgment. roundtrip time

6. Time Series of the Congestion Window (simulation) 5055606570758085 0 5 10 15 20 25 30 35 40 45 50 Seconds Congestion Window linear increase when no drops occur divide by two when a drop is detected

7. Time Series of the Congestion Window (simulation) 50100150200250300350400450500 0 5 10 15 20 25 30 35 40 45 50 Seconds Congestion Window Stochastic model of TCP Stochastic model of packet drops. Stochastic model of the roundtrip time.

8. Drop Models Ott (1997) considered deterministic drops. Padhye (1998) assumed drops to be highly correlated over short time scales, but independent over longer time scales. Altman (2000) assumes drops are bursty. Altman (2000) drop events are modeled as renewal processes with particular examples, deterministic, Poisson, i.i.d., and Markovian. Savari (1999) drop events are modeled as Poisson where the intensity depends on the window size of the TCP protocol.

9. Models for Packet Drop Probability Let S t be the sending rate at time t. Let R t be the roundtrip time experienced by a packet sent at time t. Let  t be the congestion level at time t. Let g be the probability that a packet is dropped general model memoryless depends on roundtrip time only Preliminary work indicates that, for reasonable sending rates, the drop probability mostly depends on roundtrip time. Since drops are rare, it is difficult to collect data for slow sending rates. * *

10. Determining the Conditional Drop Probability system of linear equations observable assume the congestion level, , is constant We assume that given R tk, d k is independent of R tk-1

11. End-to-end model with many queues q1tq1t q2tq2t q n-1 t qntqnt sourcedestination queues queuing delay at time t = D 1 t queuing delay at time t = D n-1 t The k th is sent at time T k propagation delay

12. Modified Diffusion Approximation for a Single Queue

13. Histogram of Observed RTT Increments (real data) Gaussian (RTT 0 =28) Queue empties slowly Queue empties quickly Agrees with queuing theory (Diffusion Approximation)

14. Observed and Smoothed Conditional Drop Probabilities 1618202224262830 -0.005 0 0.005 0.01 0.015 0.02 0.025 Roundtrip Time 1618202224262830 0 0.005 0.01 0.015 0.02 0.025 Roundtrip Time Drop Probability 1618202224262830 0 0.005 0.01 0.015 0.02 0.025 Roundtrip Time observed conditional drop probability smoothed conditional drop probability night 12 noon 3pm 6pm P(Drop | Roundtrip Time ) 9am

15. Drop Model Parameter Variation g(R t,  t ) =  0 (  t ) +  1 (  ) T 1 (R) +  2 (  ) T 2 (R) + …

16. Autocorrelation of the Increments of the Drop Probability Parameters Ii t :=  i (  t+1 ) -  i (  t ) - increments E(I0 t I0 t+  ) E(I1 t I1 t+  ) E(I2 t I2 t+  ) E(I3 t I3 t+  ) It appears that the increments are uncorrelated.

17. Transition Probability for Drop Model Parameter It appears that the transition probability is spatially homogenous.

18.  controls the rate at which the transition probability converges to the stationary distribution. A Simple Stochastic Model of the Roundtrip Time CIR Model (short-term interest rates) Note that the stationary distribution does not depend on . forward equation Define R t to be the queuing delay. Roundtrip time – propagation delay mean reverting mean = /  similar to gamma distribution

19. Estimation of, , and  (many approaches) likelihood ratio maximum likelihood estimates note the change in representation

20. Estimation of, , and  Select a set of observations {R tk } where t k << t k+1 Hence, {R tk } are approximately i.i.d. the maximum likelihood estimates satisfy mean of R t mean of log( R t ) under the independence assumption

21. Estimating Diffusion

22. Estimation of, , and  The transition probability is known Computationally difficult  and  are found as before and

23. Roundtrip Time Quasi-stationary Distribution Night Midday = 2  = 0.17  = 0.38 50556065707580859095100 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 ns-2 simulation RTT P(RTT) Observed RTT Dist Fitted RTT Dist RTT P(RTT)

24. Basic Stochastic Model of TCP N t is a Cox process with intensity n(R t, X t,  t ) n(R t, X t,  t ) = X t / R t · g(R t, X t,  t ) Sending Rate = X t / R t = Packets per RTT / Seconds per RTT = Packets / Second Drop Probability = g(R t, X t,  t ) X t – congestion window R t + T prop – roundtrip time drop probability

25. Stochastic Model of TCP with fast recovery After a drop is detected, the congestion window, X, is halved and remains constant until the dropped packet is resent and acknowledged. A drop is detected at t=0, so W 0 =R 0 W t > 0 for 0 < t < R 0 W Ro = 0 and the congestion window continues to increase until the next drop occurs. fast recovery

26. Stochastic Model of TCP with fast recovery and time-out When a time-out occurs, the congestion window is set to 1.

27. The Forward Equation No Time-outs Stationary environment (model parameters are constant) Assumptions:

28. Midday RTT and Drop Data = 4,  = 0.85,  = 0.5 02468101214 0 0.05 0.1 0.15 0.2 0.25 RTT Prob(RTT) Fitted Measured 02468101214 0 0.005 0.01 0.015 0.02 0.025 0.03 RTT drop prob Fitted Measured Drop Probability Roundtrip Time Stationary Distribution

29. ns-2 (network simulator) topology Source12345 Destination ABDC STUV Competing TCP Sources

30. ns-2 Simulation 203040506070 0 5 10 15 20 25 30 X - Congestion Window R - Roundtrip Time Marginal Density p(X,R) = 2,  = 0.17,  = 0.34 05101520253035404550 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 RTT P(RTT) Fitted Measured Drop Probability Roundtrip Time Stationary Distribution 01020304050 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 current Simulated Marginal Density p(X)

31. The Dependence on Drop Probability Drops are modeled as a Cox process with intensity n(R t, X t,  t ) = X t / R t · g(R t, X t,  t ) Scale

32. Dependence on  051015202530 0 0.02 0.04 0.06 0.08 0.1 0.12 X - Congestion Window p  = 0.069  = 0.138  = 0.275  = 0.413  = 0.551  = 0.688 15202530 0 5 10 15 20 25 X - Congestion Window R - Roundtrip Time Marginal Density p(X,R)  = 0.07 recall that  controls the rate at which the transition probability converges to the stationary distribution

33. 051015202530 0 0.02 0.04 0.06 0.08 0.1 0.12 Conditional Probability Density p(X | RTT) Sending Rate p compatible sending rate Application to Variable Bit Rate Video Transmission TCP Friendly – Send data at a rate similar to the rate that TCP would. When the video image changes quickly, the bit rate increases and the sending rate must also increase. The “compatibility” with TCP’s sending rate can the judged by examining the probability density function of the congestion window.

34. Future Work Better models: dynamic models that depend on the past sending rate. –For example: Suppose that the sending rate is initially high. Then other TCP flows should decrease their sending rate. If the sending rate suddenly decreases, then there is temporarily extra capacity and there should be few drops (maybe). Time-out and slow start. Doubly stochastic processes: Allow the parameters to vary with time. More accurate roundtrip time models Experimental verification More data, do general models for drop probability exist?

35. Conclusions System theoretical (input/output) view point to the Internet is valid. –Drop probability models –Roundtrip time models (queuing theory works!) Stochastic models –seem to accurately predict the TCP in complex networks. –give useful insight into the performance of TCP (e.g. the dependence on  ) –might be for other types of congestion control (e.g. VBR video)