Hybrid Modeling ofTCP Congestion Control

João P. Hespanha, Stephan Bohacek,Katia Obraczka, Junsoo Lee

University ofSouthern California

Background

• TCP/IP– Transmission Control Protocol/Internet Protocol– WWW, Telnet, FTP – UNIX, Windows 98, Windows 2000 all include

TCP/IP– The evolution of TCP/IP is supported by Internet

Engineering Task Force(IETF)– Window based congestion control– If congestion occurs reduce sending rate to half,

otherwise increase window size by 1 for each round trip time

Congestion control in data networks

Congestion control problem:How to adjust the sending rates of the data sources to make sure that the bandwidth B of the bottleneck link is not exceeded?

B

sourcesdestinations

B is unknown to the data sources and possibly time-varying

Congestion control in data networks

q( t ) ´ queue size

r1 bps

r2 bps

r3 bps

rate · B bps

Congestion control problem:How to adjust the sending rates of the data sources to make sure that the bandwidth B of the bottleneck link is not exceeded?

queue (temporary storage for data)

Congestion control in data networks

When i ri exceeds B the queue fills and data is lost (drops)

rate · B bps

) drop (discrete event)

Event-based control:The sources adjust their rates based on the detection of drops

r1 bps

r2 bps

r3 bpsq( t ) ´ queue size

queue (temporary storage for data)

Window-based rate adjustmentwi (window size) ´ number of

packets that can remain unacknowledged for by the destination

1st packet sent

e.g., wi = 3

t

2nd packet sent3rd packet sent 1st packet received & ack. sent

2nd packet received & ack. sent3rd packet received & ack. sent1st ack.

received )4th packet can be sent

t

source i destination i

wi effectively determines the sending rate ri :

round-trip time

t0

t1

t2

t3

0

1

2

Window-based rate adjustmentwi (window size) ´ number of

packets that can remain unacknowledged for by the destination´ sending rate

totalround-trip

time propagationdelay

per-packettransmission time

time in queueuntil transmission

This mechanism is still not sufficient to prevent a catastrophic collapse of the

network if the sources set the wi too large

queuegets full

longerRTT

ratedecreases

queuegets empty

negative feedback

TCP Reno congestion control

1. While there are no drops, increase wi by 1 on each RTT2. When a drop occurs, divide wi by 2

disclaimer: this is a simplified version of Reno that ignores some interesting phenomena…

Network/queue dynamics Reno controllers

drop occurs

drop detected(one RTT after occurred)

(congestion controller constantly probe the network for more bandwidth)

Switched system model for TCPqueue-not-full

queue-full

(drop occurs)

(drop detected)

transition enabling condition

state reset

Switched system model for TCPqueue-not-full

queue-full

(drop occurs)

(drop detected)

2 {1, 2 }

alternatively…continuous dynamics

discrete dynamics

reset dynamics

= 2

= 1

Linearization of the TCP model Time normalization ´ define a new “time” variable by

queue-not-full

queue-full

In normalized time, the continuous dynamics become linear

1 unit of ´ 1 round-trip time

Switching-by-switching analysis

queue-not-full queue full

queue-not-full queue full

queue-not-full queue full

t0 t1 t2 t3 t4 t5 t6

´ kth time the system enters the queue-not-full mode

x1 x2T

state space

x1

x2

impact map

queue-not-full

queue-full

Switching-by-switching analysis

queue-not-full queue full

queue-not-full queue full

queue-not-full queue full

t0 t1 t2 t3 t4 t5 t6

´ kth time the system enters the queue-not-full mode

x1 x2T

Theorem. The function T is a contraction. In particular, Therefore

• xk ! x1 as k !1 x1 ´ constant• x( t ) ! x1 ( t ) as t ! 1 x1(t) ´ periodic limit cycle

NS-2 simulation results

0

100

200

300

400

500

0 10 20 30 40 50

Win

dow

an

d Q

ueu

e S

ize (

pack

ets

)

time (seconds)

window size w1window size w2window size w3window size w4window size w5window size w6window size w7window size w8queue size q

Router R1

Router R2

TCP Sources TCP SinksBottleneck link

20Mbps/20ms

Flow 1

Flow 2

Flow 7

Flow 8

N1

N2

N7

N8

S1

S2

S7

S8

Results

queue-not-full queue full

queue-not-full queue full

queue-not-full queue full

t0 t1 t2 t3 t4 t5 t6

Window synchronization:

convergence is exponential, as fast as .5k

Steady-state formulas:

average drop rate

average RTT

average throughput

What next?

queue-not-full

queue-full

One drop per flow is very specific to this network:•all flows share the same

queue•similar propagation

delays for all flows•constant bit-rate cross

traffic•“drop-tail” queuing

discipline

r1 bps

r2 bps

r3 bps

B bps

queue

Other models for drops:

What next?Other models for drops:

How many drops?

t

q( t )

queue-not-full queue full

qmax

# of drops ´ squeue-full (inrate outrate)

Which flows suffer drops?number of

packets that are out for flow i

total number of packets that are

outThis probabilistic hybrid model seems to match well with packet-level simulations, e.g., with drop-head queuing disciplines. Analysis ???

What next?More general networks:

qA

qC

qD

qB

flow 1

flow 2

flow 3

What next?More general networks:

portion of the queue due to flow i

outgoing rate of flow i

total queue size

drop occurs

qA

qC

qD

qB

flow 1

flow 2

flow 3

What next?More general networks:

qC

qDqA

qB

flow 1

flow 2

flow 3

drop occurs

What next?Even multicast (current work)…

qC

qDqA

qB

flow 1

flow 2

drop occurs

Conclusions

Hybrid systems are promising to model network traffic in the context of congestion control:• retain the low-dimensionality of continuous

approximations to traffic flow•are sufficiently expressive to represent event-

based control mechanisms

Hybrid models are interesting even as a simulation tool for large networks for which packet-by-packet simulations are not feasible

Complex networks will almost certainly require probabilistic hybrid systems

END

Switching-by-switching analysis

state space

xk

xk+1

impact map

queue-not-full

queue-full

Impact maps are difficult to compute because their computation requires:Solving the

differential equations on each

mode(in general only

possible for linear dynamics)

Intersecting the continuous trajectories

with a surface(often transcendental

equations)It is often possible to prove that T is a contraction without an

explicit formula for T…