Stochastic Differential Equation Modeling and Analysis of TCP-Windowsize Behavior

1. Stochastic Differential Equation Modeling and Analysis of TCP-Windowsize Behavior EE228a Class Presentation Anshi Liang

2. Outline of this presentation • Introduction • Modeling • Analysis • Result • Conclusion

3. Outline of this presentation • Introduction • Modeling • Analysis • Result • Conclusion

4. Introduction • Transmission Control Protocol (TCP) and networking: Used in many applications like HTTP, SMTP, FTP and Telnet Reliability Stability TCP friendly

5. Introduction • Studies of TCP behavior with traditional models: Current models came from a source-centric point of view, assume that packets go out on the network with some loss probability p which may be constant or depend upon factors like current window size etc.

6. Introduction • Study of TCP behavior with a new model: The new model considers the network as the source of losses (congestion) and sources receive these signals (loss indications) as a Possion process with some rate λ; Models the window size of TCP as a fluid, having continuous increments.

7. Introduction • With this model: Builds a formulation of the window size behavior as a Possion Counter driven Stochastic Differential Equation (PCSDE); Analyzes the PCSDE and obtain closed form solutions for TCP throughput; Accounts for the maximum window size limitation for TCP connections.

8. Outline of this presentation • Introduction • Modeling • Analysis • Result • Conclusion

9. Modeling-Loss modeling • TCP implements the additive increase multiplicative decrease scheme. • Window based method, at any time, window size number of data packets are allowed in the network. • Detection of congestion is implicit. • Source centric loss model.

10. Modeling-Loss modeling • Network centric loss model: loss indications arrive at the source from the network at a certain rate, the arrival process is a Possion process.

11. Modeling-Traffic modeling • Continuous increase, represented by dt/RTT. • Triple duplicate ack (TD) losses and time out losses (TO). • Possion process N with rate λ:

12. Modeling-Differential equation for the window size • Let W be the window size: • Slow start behavior is not included (the analysis with slow start is more complicate and does not affect the results significantly, claimed by the authors).

13. Outline of this presentation • Introduction • Modeling • Analysis • Result • Conclusion

14. Analysis-maximum window size not considered • Goal: the expected value of window size and throughput (R): Solve the above for E[W], we get:

15. Analysis-maximum window size not considered • Consider the steady state solution (t∞): • The throughput (R) of the connection is obtained by dividing the expected window size by RTT:

16. Analysis-maximum window size considered • A new equation with maximum window size considered: • Solve the equation we get:

17. Analysis-maximum window size considered • Use the some mathematics technique we can get P[W=M], where λTD=λ2, λTO=λ1 and K is the service rate (1/RTT):

18. Analysis-timeout backoff • The timeout backoff effect was not considered in the previous analysis; • The window size does not grow for a period of T0 seconds, after which it starts growing at the normal rate. Here we use {WєTO} to represent the event of timeout

19. Analysis-timeout backoff • We get: • Where

20. Analysis-comparison • Transform the formula to ones involving a packet loss probability: • If we analyze TCP under the assumption of no timeouts (λ1=0):

21. Analysis-comparison • In addition, many analyses of TCP are done with the assumption that there is no limit on window size, in this case, M∞:  Which is consistent with other research results derived from the source centric model.

22. Outline of this presentation • Introduction • Modeling • Analysis • Result • Conclusion

23. Result • Compare the throughput predicted by the formula with that of actual throughput (as well as throughput predicted by other formulas) • The formula does quite well in regions of moderately low to high throughput. It does not do as well in the case of very low throughput

24. Result

25. Result • Reasons for low performance in the very low throughput case: • TCP goes into multiple timeouts, contradicted with the assumption with only one timeout; • The estimate of λTO is not accurate, since only very few packets get transmitted, there are only a few loss indications, thereby artificially introducing a low loss arrival rate

26. Conclusion • A completely different loss model. • Quite accurate in predicting real life measurements. • Ignore some details like: fast recovery, fast retransmit, slow start; make a fluid approximation of the window size. • The paper is too short, adding more details and steps inside may be helpful.

27. Observation • The paper ignored some details like multiple timeout and slow start but still have a pretty good match in the high throughput case.  These details are significant only when the network is highly congested, which translate to the low throughput case.  So it is not strange to have a good match in the high throughput case. 

28. Thank you very much! Anshi Liang