Modeling TCP Throughput
A Simple Model and its Empirical Validation
Jitendra Padhye
Victor Firoiu
Don Towsley
Jim Kurose
Presented by Jaebok Kim
Introduction
• Simple analytic characterization of the steady state
throughput
– A stochastic model of TCP congestion control
• Deriving mathematical formulas
– Taking account of not only retransmit but also timeout
Contents
• TCP Congestion Avoidance
• Simplifying assumptions
• Loss indications & triple-duplicate ACKs
• Loss indications & triple-duplicate ACKs, time-outs
• Impact of window limitation & a full model
• Empirical validation
• Conclusion
TCP Congestion Avoidance
• How do we resolve this problem?
TCP Congestion Avoidance
• TCP Reno – a newer version
• Slow Start
– W’ = W + 1 (each ACK arrives)
– Eventually, doubling every RTT
TCP Congestion Avoidance
• Additive Increase
– W’ = W + 1/W (each ACK arrives)
– W’’ = W + 1/B (Second round begins)
• B = n of Acknowledged Packets by 1 ACK (Typically, 2)
• W/B ACKs will arrive & each ACK increase 1/W
TCP Congestion Avoidance
• Multiplicative Decrease (3Duplicate ACKs)
– W’ = W * Md
– Eventually, W’ = W/2
– Don’t go back to Slow Start, but Additive Increase
•
Time Out
– Go back to Slow Start
– W=1
Simplifying assumptions
• No time for Fast Recovery
• No time for Slow Start
• Correlated packets losses in a round
– Drop-tail policy
• At a full buffer, drop all packets arriving late
– But, independent between rounds
• Separated by RTT
• Same implementation of TCP-Reno
rP1
P2
P3
P4
P5
P6
Loss indications & triple-duplicate ACKs
• B – long term steady-state TCP throughput
– Windows increases by 1/b
– Windows decreases by a factor of 2
• P – loss probability
• Get B(p) by utilizing Markov Regenerative Process
– B = E[Y] / E[A]
• Y = N of packets sent in TDPi
• A = duration of the period
• E[ ] = Expected value in MRGP
Loss indications & triple-duplicate ACKs
• Why do we need MRGP?
– A cycle will repeat (TDP1, TDP2, TDP3, so on….)
• Like a sequence of output
– New size of windows depends on only previous one’s
• Markov Chain
– Each loss in rounds is separated by RTT (Independently)
• In statistics, a sequence of random variables is independent and
identically distributed (i.i.d.) if each has the same probability
distribution as the others and all are mutually independent
– Representing steady state model
Loss indications & triple-duplicate ACKs
• Markov Model
– Predict the future through the past
– Based on conditional probability
Future state depends
on only current state,
not the past
Loss indications & triple-duplicate ACKs
• P(Rain, Sunny, Cloudy) = ?
= p(Rain) * p(Sunny|Rain) * p(Cloudy|Sunny)
Loss indications & triple-duplicate ACKS
• How do we predict the weather ?
Loss indications & triple-duplicate ACKs
• MRGP
– I.I.D random variables
Loss indications & triple-duplicate ACKs
• To get B(p) = E[Y]/E[A]
– N of packets, including first lost packet, sent in a TDPi : αi
– The round where a loss occurs : Xi
– Yi = αi + Wi – 1
• Total of Yi packets sent in Xi +1 rounds
– E[Y] = E[α] + E[W] – 1
(2)
Loss indications & triple-duplicate ACKs
• To derive E[α]
– Expected value in random process {αi }i : E[α]
– Based on the assumption
• Lost packets in a round are independent on any packets in other
rounds
• Independent & identically distributed random variables
– P[α = k] equal to p that k-1 packets are acknowledged before
a loss
– By using (2) and (4), we could derive (5) E[Y]
Loss indications & triple-duplicate ACKs
• The increase is linear with slope 1/b
• Yi can be expressed by (10)
• Bi : N of packets sent in the last round
– Bi = Wi / 2
Loss indications & triple-duplicate ACKs
• To derive E[W]
– {Wi}, {Xi} all independent sequence of I.I.D random v
– So, derive (12) from (7),(10) and (5)
– Quadratic equation from (11) & (12)
(1-p)/p + w = b* E[W]/4 (3/2 * E[W] – 1) + E[W]/2
Loss indications & triple-duplicate ACKs
• As we get E[W], we could get E[X] & E[A]
• Eventually, B(p) is derived from E[Y]/E[A]
Loss indications & triple-duplicate ACKs,
Time-outs
• The major reason for window decreases
–
–
–
–
Timeout rather than fast retransmit
Occurring when packets(or ACKs) are lost
After time-out , W’ = 1
The period of time-out will doubles
Loss indications & triple-duplicate ACKs,
Time-outs
• Utilizing MRGP again
–
–
–
–
–
ZTO : duration of a sequence of time-outs
ZTD : time interval b/w 2 consecutive TO sequences
Si = ZiTO + ZiTD
M : N of packets sent during Si
B = E[M] / E[S]
Loss indications & triple-duplicate ACKs,
Time-outs
• How to get B(p) ?
– We’ve already known E[Y], E[A]. So, let’s utilize them
– Ri = N of packets sent during time-out sequence ZTO
• Similar process to get B(p) for TDP
– Getting a full model & an approximate model
Impact of window limitation & a full model
• Keep in mind that limitation of window size
• Windows can’t grow up over Wmax
• Let’s follow the similar process to previous models’
– Unconstrained window size : Wu
– E[Wu] < Wmax
– Wmax approximately equal to E[Wu]
Impact of window limitation & a full model
• A full model
• An approximate model
Empirical validation
• Validating formulae, derived so far, by measurement
– 24 data sets with 1 hour long TCP connection
– Infinite source
X-axis = frequency of loss indication
Y-axis = n of packets sent
TD = only TD intervals
T0 = single TO intervals
T1 = double TO intervals
T2 = Triple TO intervals
TD Only = prediction of TD only model
Full = prediction of full model
Empirical validation
• Analysis of measurement tables
• Overestimation of throughput in
TD Only model
• Full model close to measurement
• Connections suffering from more
time-out rather than 3 duplicate
ACKs
Conclusion
• A simple model of TCP-Reno
– Capturing essence of TCP’s congestion avoidance behavior
• TDP & time-out
– Expressing throughput as a function of loss rate
• Most connections suffered from a considerable number of timeouts
Q&A
• Thank you for listening to my presentation