Michele Pagano

1 Michele Pagano – A Survey on TCP Performance Evaluation and Modeling

Department of Information Engineering University of Pisa

Network Telecomunication Research Groupwwwtlc.iet.unipi.it

Michele Pagano and Raffaello Secchi

A Survey on TCP Performance Evaluation and Modeling


Outline

• Fast overview on TCP congestion control mechanisms

• Models of TCP congestion control

• A simple stationary models

• The long-term TCP bandwidth

• TCP in high bandwidth-delay product networks

• TCP interactions with AQM

• Tuning RED parameters through linear control theory


TCP congestion control algorithm

ReceiverSender

ReceiverSender• Key parameters• cwnd

• ssthresh

• Additive-Increase Multiplicative Decrease • TCP increases its cwnd by roughly one MSS every RTT as long as no loss event occurs (linear increase phase or congestion avoidance)

• Slow Start• TCP increases its rate exponentially fast by doubling its value of cwnd every RTT

• Reaction to loss events (triple duplicate ACKs)• Fast Retransmit

• Fast Recovery


Evolution of TCP’s Congestion Window

0

2

4

6

8

10

12

14

16

18

0 2 4 6 8 10 12 14 18 20 22 24 26 28

Time (RTT)

cwnd

(M

SS)

ssthresh = 8

cwnd = 16Loss detected by a timeout

Loss detected by a triple DupACK

cwnd = 14

ssthresh = 7

TCP Reno vs. TCP Tahoe


Models of TCP congestion control

• Single connection models – Assume the knowledge of network characteristics, such as mean RTT

and loss probability, and try to evaluate the performance of TCP connections

– This class can be further divided into models for short-lived and long-lived connections

• Models of interaction with AQM– Derive the performance of TCP and network statistics– Introduce a sub-model of TCP and a sub-model of IP network protocol

and solves through fixed-point procedures

• Models for TCP Network Optimization– Interpret the steady-state behaviour of TCP sources as the solution of a

large optimization problem– An utility function is associated to each source. The aggregate of TCP

sources converges toward a global optimality point


Single source traffic models

• Underlying assumptions:

• Steady state

• The loss rate and RTT are independent from the source

• No ACK loss

• Neglect the slow-start phase

• TCP-Reno model:

• Congestion Avoidance

• Fast Retransmit – Fast Recovery

• Delayed-ACK


W(t)

Wmax/2

Wmax

2maxWb83

1p

Loss

Probability

periodic behaviour of congestion window

Total packets per cycle

2maxmax

maxmax Wb

8

3

2

WbW

2

W

2

1

b · Wmax/2 b · Wmax time (RTT)

2bp

3

RTT

MSSThroughput

3bp

8max WMaximum cwnd

Simple stationary model


• The previous expression does not take into account the timeout mechanisms

• It is an optimistic estimate of the bandwidth of a TCP connection.

– It is accurate in the range of small loss probabilities

– It is not suitable to determine performance of TCP over slow-speed line (few packets in transit)

Simple stationary model


TCP window size evolution

W(t)

t

ni cycles with Additive Increase

(cwnd-cycles)

Ends of Congestion Avoidance phase

(timeout mechanism)

Timeout period

New TCP cycle

Acj Tcj


Throughput estimation

timeoutcycle

cycle

T QT

ABw

2cycle Eb8

3A W

2

WERTTbTcycle

Mean duration of a cwnd-cycle period

Probability that congestion is detected by timeout

Mean duration of a timeout period

Amount of data delivered in a cwnd-cycle period

3bp

8WE


Modeling timeout

• T0 is the initial value of the timeout period• For each unsuccessful retransmission (which happens with probability p) the

timeout period is doubled until a threshold value (64T0) is reached• The retransmit timeout remains constant after 64T0

Exponential BackoffExponential Backoff

T0 2T0 4T0

loss loss

0

6

1k

k

1k

1kk T

p)2(1

2pp1pL

7kforT6k6463

6kforT12L

0

0k

k

Mean duration of a timeout period

loss

t


Fast Retransmit / Fast Recovery

w

t

W(t)

A period of congestion A period of congestion window increasingwindow increasing

•The losses in consecutive RTT are independent

•The losses of packets within the same round are correlated since DropTail discipline induces a bursty dropping behaviour

• A packet is lost with probability p given that the previous was not lost

• All the packets following the first packet lost in a round of packet transmission would be also lost.

w


Probability of timeout

w

k

p11

p1k)A(w,

Probability of having k<w successful transmissions in the penultimate round

Distribution of the number m of packets successfully transmitted in the last round

nmforp1

1nmforp1p)mn,(

n

m

C

otherwisemk,Ckw,Akw,A

3 wif1

(w)Q̂ w

3k

2

0m

2

0k

Probability that the cwnd-cycle ends with a timeout (the sender receives less than three duplicate ACKs)


Probability of timeout

WEQ̂WQEwWPrwQ̂Q1w

•A good numerical approximation of the conditional timeout probability is the limit as p→0 of expression of Q:

w

31,minwQ̂

•This expression is based on the assumption that, when p→0, all packets in a particular round are equally liked to be dropped, with at most one drop per round. In that case, any one of last 3 packets in a round can cause a timeout if dropped

•Finally, the probability of timeout is computed as a function of the mean size of congestion window E[W].

3bp

8WE


Model validation

2

0

max

32p1p8

3bp31,minT

32bp

RTT

1,

RTT

Wmin)( pB

From [PFTK]

Additional term related to the impact of the window limitation


TCP in high bandwidth-delay product networks

• The goal of TCP is to keep outstanding an amount of data equal to the bandwidth-delay product of path.

• Over WANs TCP experiences a round trip delay of the same order of magnitude of buffering delay.

• Keep the pipe full can be difficult if TCP suffers occasional random losses due to:

– transient congestion– lossy link (wireless)– link sharing with uncontrolled load (real-time traffic)

• Performance of TCP-Reno with respect to …– WAN delay-bandwidth product– rate of random losses


The case loss-free path (fluid model)

BcTWmax

•When the size of window exceeds Wmax a buffer overflow occurs and the cwnd is set to Wmax /2

•The cwnd-evolution is governed by following equation

•The ACK reception rate is equal to the link rate c if the bottleneck is congested, otherwise it is equal to the sending rate W/T

1/cτT

The total latency of the path isthe sum of transmission delay and propagation delay

dt

da

W

1

dt

da

da

dW

dt

dW

c} ,T

Wmin{

dt

da

TCP

B c

TCP

bottleneck link


The case loss-free path

W(t)

time

The queueis filling up loss

epoch

The first sub-period of congestion avoidance

pipesizecapacity

linear increase

cT

1T 2T

1N2N

1T

211max

1 2T

2T/TTWdt

T

tWN

/2WcTTT max1

22 TcN

2c

cTWT

22max

2

The second sub-period

of congestion avoidance


The case loss-free path

• The performance of TCP can be expressed as a function of the ratio between the bottleneck buffer size and pipe size– TCP suffers the presence of small buffers

– Larger buffers determine an increase of delays

– To fully exploit the capacity of bottleneck the buffer should be at least equal to pipe size

2

2

21

21

cTB

cTB

1

cTB

1

4

3c

TT

NNB

• The mean throughput of TCP-Reno is then given by:


Random loss scenario

• A packet, successfully delivered at the bottleneck link, can be lost randomly with probability q.

• The evolution of congestion window is determined by the window size w at the beginning of cwnd-cycle (Markov process)

• We introduces two functions:

random loss

wn,W Window size after n successful packet transmissions (w initial window)

wn,T Time required to complete n successful packet transmissions

TCP

B c q

TCP

bottleneck link


Markov chain analysis

0

q1

iw

q

…1 wi-1 wi…wi+1 2wi-1

q1 q1

… … Nw(wi)

0

q1

…1 wi-1 …

q1 q1

0

q1

…1 wi-1 …

q1 q1

2

1w i

2

w i

Since the independent loss model used …q1 q1 q1q1

q qq q q

ii1i

ii1i

w,NTT

w,NW2

1w

The cwnd evolution is expressed through these recursive equations

Once solved the time-homogenous Markov chain, we can evaluate the throughput

i

i

TE

NEBw

… …


General comments

• This analysis can be extended also to other versions of TCP

• Since the analysis is computational expensive, approximated solutions have been proposed (see [LM97]).

• Even small loss leads to a significant throughput deterioration over networks with high bandwidth-delay products.

• TCP performance is strongly dependent on the parameter q(cT)2

and decreases sharply as this parameter increases– “too early” drops in the TCP cycle induce the over-reaction

• Random losses should be avoided– flow isolation– link layer protocols


Interaction between TCP and AQM

• Fluid model:– The congestion window is a continuous variable– A continuous flow of data

• Interaction between TCP-Reno and AQM mechanism

• Fixed-Point approach

TCPTCP

NetworkNetwork

load ploss RTT


• RED (Random Early Detection): implicit congestion avoidance mechanism

• RED discards packets randomly in order to:

– Prevent the incipient congestion by reacting earlier

– Avoid the synchronization between sources

– Mechanism of Dropping/Marking based on the mean queue length

– Moving Average Algorithm used to smooth the instantaneous queue size

x

p(x)

1

TmaxTmin

Pmax

Mean queue size

Pro

ba

bili

ty o

f D

rop

pin

g

Active Queue Management: RED


Active Queue Management: RED

Moving average filter

Sampled data system

kTqdτebkTxeT1kxT1k

kT

τkTaaT

kTqαkTxα1T1kx

tqT

α1lntx

T

α1ln

dt

dx

t

x(t)

Instantaneous queue length

Mean queue length


Model of the network

•The network is modelled as a set of L links with capacities cl l = 1,2, … , L and the links are shared by a set of S sources indexed by s = 1,2, … , S each using a subset Ls of links

• Basic quantities

routing matrix

otherwise

LlA s

ls ,0

if,1

congestion window associatedwith each TCP source

)(tWs

)()( tqtp llprobability of drop and instantaneous length associated with each link



Parameters related to the s-th TCP connection

sLl l

lss c

tqτtRTT

• Round trip time

• End-to-end dropping probability

Ll

lls

L

1lllss (t)pA(t))pA(11(t)p̂

s is the round trip propagation delay

since we are considering AQM/RED, we may reasonably assume that drops at different queue are independent



lc

S

1s s

slsltq

l

(t)RTT

(t)WAc1

dt

(t)dql

S

1s s

sls )t(RTT

(t)WA

• Differential version of the Lyndley equation

• Mean transient behaviour (by approximating the expectation of both sides):

S

1s s

slsl}tE{ql (t)}E{RTT

(t)}E{WAc1(t)}E{q

dt

dl

Parameters related to dynamic of the l-th queue

tlqIncoming traffic Outgoing traffic


Model of the source

tdN2

(t)W

tRTT

dt(t)dW s

s

ss

t

W(t)

loss events

Additive Increase Multiplicative Decrease

(t)λs

• Again, taking the expectation

(t)dtλ2

(t)}E{W

}tE{RTT

dt(t)}dE{W s

s

ss

• Packet losses at flow s are modelled by a Poisson process with time varying rate

• Ni(t): number of losses suffered by flow i

• t: point of time when the flow detects losses

• Evolution of cwnd:


Model of the source

• In proportional marking schemes the dropping rate is proportional to the share of the connection

(t)RTT

(t)W(t)p

s

sl expected value for

drop rate at link l Ls

• Actually, drops occur at the node about a round trip time before they can detected by the sender (the latency of feedback is important in a control system since it impacts on stability)

•This equation governs the evolution of congestion window of s-th connection

)τ(tp̂)τ(tRTT

)τ(tW

2

(t)W

(t)RTT

1

dt

(t)Wdss

s

sss

s

s


Stochastic differential equations system

ss

ss

sss

s

sτtp̂

τtRTT

τtW

2

tW

)(RTT

1

dt

tWd

t

sl

Ls ss

ss

l0tql

τtRTT

τtWc1

dt

tqd

tqT

α1lntx

T

α1ln

dt

xdll

l

• 2L+S coupled equations in the unknowns (x,q,W) that can be solved numerically

L}{1,2,...,l

S}{1,2,...,s

The time needed to solve the system is several order of magnitude less than that needed for the simulation of the same network scenario

L}{1,2,...,l

TCP

RED

Lindley


Linearized analysis of TCP with AQM

• Goal: linearization of the previous set of equations in the case of single bottleneck link topology

• The linearized system is suitable to be studied through the classic tools of linear control theory.

• The linear analysis gives us many suggestions on the way to modify the algorithm in order to achieve stability and robustness

BOTTLENECK

losseslosses


Linear analysis: the single link case

• Let us consider N identical TCP Reno flows (with the same RTT) sharing a common link with capacity C.

CNR(t)

W(t)(t)q

τ)p(tτ)2R(t

τ)W(tW(t)

R(t)

1(t)W

C

q(t)τR(t)

• We have assumed that the server is always transmitting packets (bottleneck)

• Common value of RTT:


Block diagram

+

X

(.)

1

RTT tW tW

2

1

(.)

1

RTT X

(.)

1

RTT

tp LOWPASS

N +

dx

dpxK )(

TCPTCP

tq

K

REDRED

tq

CCongested

Router

Congested Router

tq

tq tq

-

controller


Small signal analysis

• Goals of an AQM (RED) controller

• Stable closed-loop system

• Acceptable transient response

• Insensitivity to variations of model parameters

• Insensitivity to disturbance factors (short lived flows)

• Strategy

• Linearization around the operating point (W0, q0, p0)

• Input: Loss probability

• Output: Queue size

• Design of RED using dominant pole compensation


Small signal analysis

N

CRW

pW

00

00

2

01

02

1

0

0

00

20

0

NR

W

pR

W

R

• Operating point derivative equal to zero:

δq(t)R

1δW(t)

R

N(t)qδ

τ)δp(t2N

CRtδWτtδW

CR

N(t)Wδ

00

2

20

20

• Difference variables linearization in a neighbourhood of the operating point

tδW2


Laplace representation

2NCR

s1

4NCR

sP 20

3

330

tcp

sR1

NsP

0queue

• The static gain of plant is

• proportional to RTT and capacity

• inversely proportional to the number of active flows

• A small number of TCP flows lead to an oscillatory response

• An increase in the round trip time reduces the controllability of the system

• High speed links are difficult to control

• Representation of the system in the Laplace domain:


Small signal model

s/β1

1KsCred

minmax

max

tt

pK

T

α)ln(1β

controllergain

• RED acts as a proportional controller

controllertime-constant

• Internet routers typically implement a drop tail policy in the queues (ON-OFF control strategy) strong oscillation in queue size, with the alternation of emptiness and buffer overflow

• RED should reduce the extent of variations in queue length

• Trade-off between acceptable queuing delay and link utilization.

sP sτe sCred++

-


RED Design

• In choosing the parameters of RED controller (K,β), it is necessary to introduce some bounds on the number of TCP sessions and on RTT:

max0min RR,NN

• Basic Result: Under previous constraints, if K and β satisfy the following condition:

1β

ω

2N

CRK2

g

2min

3max

the system converges exponentially fast to the equilibrium,for whatever initial condition.

Queuemin

TCPming ω,ωmin

10

1ω where

dominant pole

compensation


Designing AQM/RED

Bode Plots

Amplitude

Phase -900

-1800

QueueωTCPωgω

phase

margin

ω

ω

Usually the dynamics of the queue are faster than those of TCP


Conclusions

• Summary of analytical modelling for the performance evaluation of Internet congestion control

• Bandwidth achieved by a TCP connection in response to network conditions

– These models are also useful in asymptotic conditions with many sources

• Interaction between TCP and AQM (RED) schemes

– Qualitative understanding of TCP transient behaviour.

– Powerful tools of linear control theory

– Selection of the network parameters leading to stable and robust working conditions


A few references

[PFTK98] J. Padhye, V. Firoiu, D. Towsley and J. Kurose, “Modeling TCP Throughput: A Simple Model and its Empirical Validation”, In SIGCOMM, 1998.

[LM97] T. Lackshman and U. Madhow, “The performance of TCP/IP for networks with high bandwidth-delay products and random loss”, In Transaction on Networking, 1997

[VGT99] V. Misra, W. Gong, D. Towsley, “Stochastic Differential Equation Modeling And Analysis of TCP-Windowsize Behavior”, In PERFORMANCE, Istanbul, Turkey, 1999.

[HMTG01] C. Hollot, V. Misra, D. Towsley and W. Gong. “A ControlTheoretic Analysis of RED”, In INFOCOMM 2001

Michele Pagano – A Survey on TCP Performance Evaluation and Modeling 1 Department of Information Engineering University of Pisa Network Telecomunication.

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