stochastic differential equation modeling and analysis of tcp - windowsize behavior presented by sri...

Stochastic Differential Equation Modeling and Analysis of TCP -

Windowsize BehaviorPresented 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


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



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


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 pi

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

€

dw =dt

R−

w

2dN


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) sourceof R and

Solution: Express R and as functions of W (and N, number of flows)

R


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))dNi

• dW = (dt /RTT) + (-W/2)dNTD +(1-W)dNTO

• First term indicates the additive increase of the TCP window

• Second and Third represent the multiplicative decrease.

SDE Graphical Representation

Time

Cha

ngin

g W

indo

w s

ize

W

1/RTT

(-W/2)dNTD +(1-W)dNTO

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

Cha

ngin

g W

indo

w s

ize

W

1/RTT


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] =(2TO2 + TO + TO TD + TO /RTT +2/ RTT2 +2 /RTT )

(1/RTT+1)(2M TO + MTD +2 /RTT )

Windowsize at steady state with maximum window size

Time

Cha

ngin

g W

indo

w s

ize

W

1/RTT


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] =(2TO2 + TO + TO TD + TO /RTT +2/ RTT2 +2 /RTT )

(1/RTT+1)(2M TO + MTD +2 /RTT )– P[W=M] = (2/ RTT2 +2 /RTT )

(1/RTT+1)(MTD +2 /RTT )

= (2/ RTT) (MTD +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

*www.public.iastate.edu/~wqmeeker/ stat533stuff/psnups/chapter16_psnup.pdf

**http://www.itl.nist.gov/div898/handbook/eda/section3/eda35e.htm

Scatter plot of statistic

slide based on presentation by Vishal Mishra

Experiment 1


Experiment 2


Experiment 3


Experiment 4


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

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