stochastic differential equation modeling and analysis of tcp - windowsize behavior presented by sri...
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![Page 1: Stochastic Differential Equation Modeling and Analysis of TCP - Windowsize Behavior Presented by Sri Hari Krishna Narayanan (Some slides taken from or](https://reader035.vdocuments.mx/reader035/viewer/2022062423/5697bf701a28abf838c7d622/html5/thumbnails/1.jpg)
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)
![Page 2: Stochastic Differential Equation Modeling and Analysis of TCP - Windowsize Behavior Presented by Sri Hari Krishna Narayanan (Some slides taken from or](https://reader035.vdocuments.mx/reader035/viewer/2022062423/5697bf701a28abf838c7d622/html5/thumbnails/2.jpg)
Outline
• Introduction
• TCP window Algorithms
• Poisson counter driven stochastic differential equations
• Expressing windowsize changes
• Results
• Statistical tests
![Page 3: Stochastic Differential Equation Modeling and Analysis of TCP - Windowsize Behavior Presented by Sri Hari Krishna Narayanan (Some slides taken from or](https://reader035.vdocuments.mx/reader035/viewer/2022062423/5697bf701a28abf838c7d622/html5/thumbnails/3.jpg)
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
![Page 4: Stochastic Differential Equation Modeling and Analysis of TCP - Windowsize Behavior Presented by Sri Hari Krishna Narayanan (Some slides taken from or](https://reader035.vdocuments.mx/reader035/viewer/2022062423/5697bf701a28abf838c7d622/html5/thumbnails/4.jpg)
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
![Page 5: Stochastic Differential Equation Modeling and Analysis of TCP - Windowsize Behavior Presented by Sri Hari Krishna Narayanan (Some slides taken from or](https://reader035.vdocuments.mx/reader035/viewer/2022062423/5697bf701a28abf838c7d622/html5/thumbnails/5.jpg)
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
![Page 6: Stochastic Differential Equation Modeling and Analysis of TCP - Windowsize Behavior Presented by Sri Hari Krishna Narayanan (Some slides taken from or](https://reader035.vdocuments.mx/reader035/viewer/2022062423/5697bf701a28abf838c7d622/html5/thumbnails/6.jpg)
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
![Page 7: Stochastic Differential Equation Modeling and Analysis of TCP - Windowsize Behavior Presented by Sri Hari Krishna Narayanan (Some slides taken from or](https://reader035.vdocuments.mx/reader035/viewer/2022062423/5697bf701a28abf838c7d622/html5/thumbnails/7.jpg)
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.
![Page 8: Stochastic Differential Equation Modeling and Analysis of TCP - Windowsize Behavior Presented by Sri Hari Krishna Narayanan (Some slides taken from or](https://reader035.vdocuments.mx/reader035/viewer/2022062423/5697bf701a28abf838c7d622/html5/thumbnails/8.jpg)
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
![Page 9: Stochastic Differential Equation Modeling and Analysis of TCP - Windowsize Behavior Presented by Sri Hari Krishna Narayanan (Some slides taken from or](https://reader035.vdocuments.mx/reader035/viewer/2022062423/5697bf701a28abf838c7d622/html5/thumbnails/9.jpg)
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.
![Page 10: Stochastic Differential Equation Modeling and Analysis of TCP - Windowsize Behavior Presented by Sri Hari Krishna Narayanan (Some slides taken from or](https://reader035.vdocuments.mx/reader035/viewer/2022062423/5697bf701a28abf838c7d622/html5/thumbnails/10.jpg)
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
slide taken from presentation by Vishal Mishra
![Page 11: Stochastic Differential Equation Modeling and Analysis of TCP - Windowsize Behavior Presented by Sri Hari Krishna Narayanan (Some slides taken from or](https://reader035.vdocuments.mx/reader035/viewer/2022062423/5697bf701a28abf838c7d622/html5/thumbnails/11.jpg)
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
slide taken from presentation by Vishal Mishra
![Page 12: Stochastic Differential Equation Modeling and Analysis of TCP - Windowsize Behavior Presented by Sri Hari Krishna Narayanan (Some slides taken from or](https://reader035.vdocuments.mx/reader035/viewer/2022062423/5697bf701a28abf838c7d622/html5/thumbnails/12.jpg)
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
![Page 13: Stochastic Differential Equation Modeling and Analysis of TCP - Windowsize Behavior Presented by Sri Hari Krishna Narayanan (Some slides taken from or](https://reader035.vdocuments.mx/reader035/viewer/2022062423/5697bf701a28abf838c7d622/html5/thumbnails/13.jpg)
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
![Page 14: Stochastic Differential Equation Modeling and Analysis of TCP - Windowsize Behavior Presented by Sri Hari Krishna Narayanan (Some slides taken from or](https://reader035.vdocuments.mx/reader035/viewer/2022062423/5697bf701a28abf838c7d622/html5/thumbnails/14.jpg)
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.
![Page 15: Stochastic Differential Equation Modeling and Analysis of TCP - Windowsize Behavior Presented by Sri Hari Krishna Narayanan (Some slides taken from or](https://reader035.vdocuments.mx/reader035/viewer/2022062423/5697bf701a28abf838c7d622/html5/thumbnails/15.jpg)
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.
![Page 16: Stochastic Differential Equation Modeling and Analysis of TCP - Windowsize Behavior Presented by Sri Hari Krishna Narayanan (Some slides taken from or](https://reader035.vdocuments.mx/reader035/viewer/2022062423/5697bf701a28abf838c7d622/html5/thumbnails/16.jpg)
SDE Graphical Representation
Time
Cha
ngin
g W
indo
w s
ize
W
1/RTT
(-W/2)dNTD +(1-W)dNTO
![Page 17: Stochastic Differential Equation Modeling and Analysis of TCP - Windowsize Behavior Presented by Sri Hari Krishna Narayanan (Some slides taken from or](https://reader035.vdocuments.mx/reader035/viewer/2022062423/5697bf701a28abf838c7d622/html5/thumbnails/17.jpg)
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 )
![Page 18: Stochastic Differential Equation Modeling and Analysis of TCP - Windowsize Behavior Presented by Sri Hari Krishna Narayanan (Some slides taken from or](https://reader035.vdocuments.mx/reader035/viewer/2022062423/5697bf701a28abf838c7d622/html5/thumbnails/18.jpg)
Windowsize at steady state
Time
Cha
ngin
g W
indo
w s
ize
W
1/RTT
(-W/2)dNTD +(1-W)dNTO
![Page 19: Stochastic Differential Equation Modeling and Analysis of TCP - Windowsize Behavior Presented by Sri Hari Krishna Narayanan (Some slides taken from or](https://reader035.vdocuments.mx/reader035/viewer/2022062423/5697bf701a28abf838c7d622/html5/thumbnails/19.jpg)
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 )
![Page 20: Stochastic Differential Equation Modeling and Analysis of TCP - Windowsize Behavior Presented by Sri Hari Krishna Narayanan (Some slides taken from or](https://reader035.vdocuments.mx/reader035/viewer/2022062423/5697bf701a28abf838c7d622/html5/thumbnails/20.jpg)
Windowsize at steady state with maximum window size
Time
Cha
ngin
g W
indo
w s
ize
W
1/RTT
(-W/2)dNTD +(1-W)dNTO
M
![Page 21: Stochastic Differential Equation Modeling and Analysis of TCP - Windowsize Behavior Presented by Sri Hari Krishna Narayanan (Some slides taken from or](https://reader035.vdocuments.mx/reader035/viewer/2022062423/5697bf701a28abf838c7d622/html5/thumbnails/21.jpg)
Other TCP features
• Slowstart– Considered unimportant by authors
• Timeout backoff– Modeled similarly to the maximum window
![Page 22: Stochastic Differential Equation Modeling and Analysis of TCP - Windowsize Behavior Presented by Sri Hari Krishna Narayanan (Some slides taken from or](https://reader035.vdocuments.mx/reader035/viewer/2022062423/5697bf701a28abf838c7d622/html5/thumbnails/22.jpg)
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
![Page 23: Stochastic Differential Equation Modeling and Analysis of TCP - Windowsize Behavior Presented by Sri Hari Krishna Narayanan (Some slides taken from or](https://reader035.vdocuments.mx/reader035/viewer/2022062423/5697bf701a28abf838c7d622/html5/thumbnails/23.jpg)
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
![Page 24: Stochastic Differential Equation Modeling and Analysis of TCP - Windowsize Behavior Presented by Sri Hari Krishna Narayanan (Some slides taken from or](https://reader035.vdocuments.mx/reader035/viewer/2022062423/5697bf701a28abf838c7d622/html5/thumbnails/24.jpg)
Results 1
![Page 25: Stochastic Differential Equation Modeling and Analysis of TCP - Windowsize Behavior Presented by Sri Hari Krishna Narayanan (Some slides taken from or](https://reader035.vdocuments.mx/reader035/viewer/2022062423/5697bf701a28abf838c7d622/html5/thumbnails/25.jpg)
Results 2
![Page 26: Stochastic Differential Equation Modeling and Analysis of TCP - Windowsize Behavior Presented by Sri Hari Krishna Narayanan (Some slides taken from or](https://reader035.vdocuments.mx/reader035/viewer/2022062423/5697bf701a28abf838c7d622/html5/thumbnails/26.jpg)
Results 3
![Page 27: Stochastic Differential Equation Modeling and Analysis of TCP - Windowsize Behavior Presented by Sri Hari Krishna Narayanan (Some slides taken from or](https://reader035.vdocuments.mx/reader035/viewer/2022062423/5697bf701a28abf838c7d622/html5/thumbnails/27.jpg)
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?
![Page 28: Stochastic Differential Equation Modeling and Analysis of TCP - Windowsize Behavior Presented by Sri Hari Krishna Narayanan (Some slides taken from or](https://reader035.vdocuments.mx/reader035/viewer/2022062423/5697bf701a28abf838c7d622/html5/thumbnails/28.jpg)
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
![Page 29: Stochastic Differential Equation Modeling and Analysis of TCP - Windowsize Behavior Presented by Sri Hari Krishna Narayanan (Some slides taken from or](https://reader035.vdocuments.mx/reader035/viewer/2022062423/5697bf701a28abf838c7d622/html5/thumbnails/29.jpg)
Scatter plot of statistic
slide based on presentation by Vishal Mishra
![Page 30: Stochastic Differential Equation Modeling and Analysis of TCP - Windowsize Behavior Presented by Sri Hari Krishna Narayanan (Some slides taken from or](https://reader035.vdocuments.mx/reader035/viewer/2022062423/5697bf701a28abf838c7d622/html5/thumbnails/30.jpg)
Experiment 1
slide taken from presentation by Vishal Mishra
![Page 31: Stochastic Differential Equation Modeling and Analysis of TCP - Windowsize Behavior Presented by Sri Hari Krishna Narayanan (Some slides taken from or](https://reader035.vdocuments.mx/reader035/viewer/2022062423/5697bf701a28abf838c7d622/html5/thumbnails/31.jpg)
Experiment 2
slide taken from presentation by Vishal Mishra
![Page 32: Stochastic Differential Equation Modeling and Analysis of TCP - Windowsize Behavior Presented by Sri Hari Krishna Narayanan (Some slides taken from or](https://reader035.vdocuments.mx/reader035/viewer/2022062423/5697bf701a28abf838c7d622/html5/thumbnails/32.jpg)
Experiment 3
slide taken from presentation by Vishal Mishra
![Page 33: Stochastic Differential Equation Modeling and Analysis of TCP - Windowsize Behavior Presented by Sri Hari Krishna Narayanan (Some slides taken from or](https://reader035.vdocuments.mx/reader035/viewer/2022062423/5697bf701a28abf838c7d622/html5/thumbnails/33.jpg)
Experiment 4
slide taken from presentation by Vishal Mishra
![Page 34: Stochastic Differential Equation Modeling and Analysis of TCP - Windowsize Behavior Presented by Sri Hari Krishna Narayanan (Some slides taken from or](https://reader035.vdocuments.mx/reader035/viewer/2022062423/5697bf701a28abf838c7d622/html5/thumbnails/34.jpg)
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.
![Page 35: Stochastic Differential Equation Modeling and Analysis of TCP - Windowsize Behavior Presented by Sri Hari Krishna Narayanan (Some slides taken from or](https://reader035.vdocuments.mx/reader035/viewer/2022062423/5697bf701a28abf838c7d622/html5/thumbnails/35.jpg)
Thank you