Stochastic Differential Equation Modeling and Analysis of TCP - Windowsize Behavior Presented by Sri Hari Krishna Narayanan (Some slides taken from or based on presentations by Vishal Mishra)

Outline Introduction TCP window Algorithms Poisson counter driven stochastic differential equations Expressing windowsize changes Results Statistical tests

Introduction This work is directly related to Ross’ presentation last week. The authors propose a new model which is simpler and work with the same data as the previous paper to obtain similar results. TCP is the protocol of choice for communication for many applications. Modeling TCP is hence important. Other applications may use other protocols –TCP friendliness TCP shares the bandwidth fairly amongst hosts competing for network bandwidth

TCP Congestion Control: window algorithm Window: can send W packets at a time increase window by one per RTT if no loss, W <- W+1 each RTT decrease window by half on detection of loss W  W/2 slide taken from presentation by Vishal Mishra

TCP Congestion Control: window algorithm Window: can send W packets increase window by one per RTT if no loss, W <- W+1 each RTT decrease window by half on detection of loss W  W/2 sender receiver W slide taken from presentation by Vishal Mishra

TCP Congestion Control: window algorithm Window: can send W packets increase window by one per RTT if no loss, W <- W+1 each RTT decrease window by half on detection of loss W  W/2 sender receiver W slide taken from presentation by Vishal Mishra

TCP loss indications at the source There are two kinds –Time Outs(TO) –Triple Acknowledgements (TD) Effects on the TCP windowsize –TO causes windowsize to become 1 –TD causes windowsize to halve When there is no packet loss, the windowsize increases.

Other models Model TCP from the point of view of the source –Packets that the source injects into the network. Each packet has an associated loss probability. p – Identical for each packet –Can be dependent on factors such as the current windowsize

This model Models losses in a network centric way The network is the source of the congestion –Not the packets? Losses are events that arrive at the source –Arrivals are then modeled using statistical analysis –In this case arrivals are modeled as a Poisson process.

SDE based model Sender Loss Probability p i Traditional, Source centric loss model Sender Loss Indications arrival rate New, Network centric loss model Loss model enabled casting of TCP behavior as a Stochastic Differential Equation, roughly slide taken from presentation by Vishal Mishra

Refinement of SDE model W(t) = f(,R) Window Size is a function of loss rate (  and round trip time (R) R Network Network is a (blackbox) source of R and Solution: Express R and as functions of W (and N, number of flows) R slide taken from presentation by Vishal Mishra

Poisson Process What is it? –Process with exponential arrival times –Arrivals are independent of each other –Can be used to model natural occurrences Spotting fish in the ocean Occurrence of soft errors

Traffic model The increase in windowsize –Rises by 1 for every round trip time (RTT) Instead of step increase, the increase is considered to be continuous and represented as dt/RTT Falls by half for TD Falls to 1 for a TO

Poisson counter Poisson process N with arrival rate dN ={ 1 at Poisson arrival { 0 elsewhere E[dN] = dt This basically means that for poisson loss events in time dt, there will be spikes.

Poisson Counter Driven Stochastic differential equations (SDE) Dx = f(x(t))dt+∑gi(x(t))dN i dW = (dt /RTT) + (-W/2)dN TD +(1-W)dN TO First term indicates the additive increase of the TCP window Second and Third represent the multiplicative decrease.

SDE Graphical Representation Time Changing Window size W 1/RTT (-W/2)dN TD +(1-W)dN TO

What to do with the SDE There is a lot of mathematics possible This mathematics evaluates the expected value of the windowsize and the throughput of the network at steady state. E[W] =(1/RTT + TO ) /( TD /2 + TO ) R =(1/RTT)*E[W] =(1/RTT)(1/RTT + TO ) /( TD /2 + TO )

Windowsize at steady state Time Changing Window size W 1/RTT (-W/2)dN TD +(1-W)dN TO

Maximum windowsize considerations Restricts the maximum value of the windowsize to M. E[W] =((1- P[W=M]) /RTT + TO ) /( TD /2 + TO ) What does this mean –The continuous function rises as long as its value is not M. In that case it remains constant. After some mathematics, –P[W=M] =(2 TO2 + TO + TO TD + TO /RTT +2/ RTT 2 +2 /RTT ) (1/RTT+1)(2M TO + M TD +2 /RTT )

Windowsize at steady state with maximum window size Time Changing Window size W 1/RTT (-W/2)dN TD +(1-W)dN TO M

Other TCP features Slowstart –Considered unimportant by authors Timeout backoff –Modeled similarly to the maximum window

Comparison with other models This model can be transformed into one involving packet loss –Loss/sec = TO + TD –Packets/sec = R –Loss/packet = (Loss/sec) / (Packets/sec) = ( TO + TD ) /R

Comparison with other models This model can be transformed into one involving no timeouts – TO = 0, no arrival of timeouts –Earlier computation of E[W] changes –P[W=M] =(2 TO2 + TO + TO TD + TO /RTT +2/ RTT 2 +2 /RTT ) (1/RTT+1)(2M TO + M TD +2 /RTT ) –P[W=M] = (2/ RTT 2 +2 /RTT ) (1/RTT+1)(M TD +2 /RTT ) = (2/ RTT) (M TD +2 /RTT ) Similar changes can be made to account for no maximum window size

Results 1

Results 2

Results 3

Results -Analysis Closely mirrors earlier work –Except at low thoughput This represente very high loss zone (60-80%) –Does not really matter –Does not consider 1 hour traces at all So why use this model at all? –Simpler mathematics and analysis So how do we get this simple analytical model?

Trace analysis Loss inter arrival events tested for Independence –Lewis and Robinson test for renewal hypothesis –A sequence of recurrences T1,T2,... is a renewal process if the time between recurrences τj = Tj −j−, j =1, 2,... (T0 = 0) are independent and identically distributed. * Exponentiality –Anderson-Darling test The Anderson-Darling test is used to test if a sample of data came from a population with a specific distribution….. The Anderson-Darling test is an alternative to the chi-square and Kolmogorov-Smirnov goodness-of-fit tests.** slide based on presentation by Vishal Mishra * stat533stuff/psnups/chapter16_psnup.pdf **

Scatter plot of statistic slide based on presentation by Vishal Mishra

Experiment 1 slide taken from presentation by Vishal Mishra

Experiment 2 slide taken from presentation by Vishal Mishra

Experiment 3 slide taken from presentation by Vishal Mishra

Experiment 4 slide taken from presentation by Vishal Mishra

So are there any more magic fits and tests? Definitely there are more traces that can fit Poisson distribution. Motivating Example –Soft errors Cosmic particles hit the chip to cause bit flips The existence of these particles can be modeled using a Poisson process. What about other distributions? –Definitely, there may be other distributions and related mathematics.

Thank you