1. Loss Model of TCP Presented by: Rajarshi Gupta WebTP Group

2. Paper being Presented z“Stochastic Differential Equation Modeling and Analysis of TCP-Windowsize Behavior” zAuthors yVishal Misra, completing PhD, UMass EE yWei-Bo Gong, faculty, UMass EE yDon Towsley, faculty, UMass CS zPresented at Performance’99, Istanbul, Turkey, October’99 zftp://gaia.cs.umass.edu/pub/Misra99-TCP-Stochastic.ps.gz

3. Key Ideas zConsider network as source of losses and sources as recipient of these signals zModel loss arrival as Poisson process zUse Stochastic Differential Equations + Queuing Theory to estimate Rate zCompare with existing data and analytical model (Padhye)

4. Loss Model zLoss arrival may be modeled by Poisson process zConnection travels through many hops yKhinchine’s Theorem zMany flows at each router yProbabilistic thinning yKallenberg’s Theorem zDenote arrival processes ydN TD : 3-dup-ack loss ydN TO : timeout loss

5. Window Size Model Congestion Avoidance zLet W = window size z zTerms y1st: additive increase y2nd: multiplicative decrease (TD loss) y3rd: window cutback to 1 (TO loss) Slow Start + Cong Avoidance z z T=slow-start threshold z Denote y TD =TD loss arrival rate y TO =TO loss arrival rate

6. Analysis (no slow-start) z z Throughput = Expected window size / RTT z Then, z NOTE: We haven’t considered effect of ymaximum window size (important effect) ytimeout backoff Taking t 

7. Maximum Window Size z We solve for P[W=M] by looking at Window Size Evolution for TCP as the Virtual Waiting Time in an M/G/1 queue with finite buffer Then, z Here y 1 = TO y 1 = TD yK = service rate = 1 / RTT z As before, Throughput = E[W]/RTT M is the maximum window size

8. Timeout Backoff zAfter a timeout, the window does not grow for a period of T0 zNeed to replace indicator function I M with I M I T0 where I T0 is 0 for T0 seconds after a timeout zEvents {W=M} and {W  TO} are mutually exclusive z Two corrections needed for “silent time” yE[W] during active stage only yScaling RTT to account for the silent periods z Throughput = E[W active ] / RTT scaled for active period

9. Comparing with other Models zTo compare with traditional TCP analysis yloss/sec = 1 + 2 ypkts/sec = R yloss/pkt p = ( 1 + 2 )/R zNo timeout is likely in the case of TCP SACK zIn short connections, TO > TD z No TO losses ( 1 =0) z Unlimited window (M  ),

10. Experiments zUsed the datasets collected by Padhye et al for the SIGCOMM’98 paper zUsed the sets of 100  100 sec traces zPlotted against SIGCOMM formula values zOn the plots, the 100 points are arranged in descending order of throughput zFormula works badly for very low throughputs ymultiple timeouts y TO too low since throughput is low yloss rates of 60-80% is non-poisson

11. Results Experiment Sequence Number Throughput Experiment Sequence Number

12. More Results Throughput Experiment Sequence Number

13. Still More Results

14. Poisson ? zFor Poisson Arrivals, need to test for yindependent inter-arrival times yexponential inter-arrival times zNote that exponentiality alone is not enough zUse the following tests yIndependence: Lewis and Robinson test (>90%) yExponentiality: Anderson-Darling test (60-80%) zIndependence property more important than exponential property

15. Poisson Experiments

16. Conclusions zLosses modeled as Poisson arrivals from network zSDE gives closed form solution for comprehensive TCP model zExperimental verification with real data