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Hybrid Modeling of
TCP Congestion Control
João P. Hespanha, Stephan Bohacek,
Katia Obraczka, Junsoo Lee
University of
Southern 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
sources
destinations
B
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 is unknown to the data sources and possibly time-varying
Congestion control in data networks
r1 bps
queue (temporary
storage for data)
r2 bps
rate · B bps
r3 bps
q( t ) ´ queue size
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?
Congestion control in data networks
r1 bps
queue (temporary
storage for data)
r2 bps
rate · B bps
r3 bps
q( t ) ´ queue size
When i ri exceeds B the queue fills and data is lost (drops)
) drop (discrete event)
Event-based control:
The sources adjust their rates based on the detection of drops
Window-based rate adjustment
wi (window size) ´ number of packets that can remain
unacknowledged for by the destination
e.g., wi = 3
source i
destination i
1st packet sent
2nd packet sent
3rd packet sent
t0
t2
t0
1st ack. received )
4th packet can be sent
t3
t1
t2
t1
t
1st packet received & ack. sent
2nd packet received & ack. sent
3rd packet received & ack. sent
t
wi effectively determines the sending rate ri :
round-trip time
Window-based rate adjustment
wi (window size) ´ number of packets that can remain
unacknowledged for by the destination
per-packet
transmission time
´ sending rate
total
round-trip
time
queue
gets full
longer
RTT
propagation
time in queue
delay
until transmission
rate
decreases
queue
gets empty
negative feedback
This mechanism is still not sufficient to prevent a catastrophic
collapse of the network if the sources set the wi too large
TCP Reno congestion control
1. While there are no drops, increase wi by 1 on each RTT
2. When a drop occurs, divide wi by 2
(congestion controller constantly probe the network for more bandwidth)
Network/queue dynamics
Reno controllers
drop detected

drop occurs
(one RTT after occurred)

disclaimer: this is a simplified version of Reno that ignores some interesting phenomena…
Switched system model for TCP
queue-not-full
transition enabling
condition
(drop occurs)
queue-full
(drop detected)
state reset
Switched system model for TCP
queue-not-full
s=1
(drop occurs)
queue-full
(drop detected)
s=2
alternatively…
s 2 {1, 2 }
continuous dynamics
discrete dynamics
reset dynamics
Linearization of the TCP model
Time normalization ´ define a new “time” variable t by
1 unit of t ´ 1 round-trip time
In normalized time, the continuous dynamics become linear
queue-not-full
queue-full
Switching-by-switching analysis
x1
t0
t1
queue-not-full
t2
queue
full
x2
T
t3
queue-not-full
t4
queue
full
t5
queue-not-full
t6
queue
full
´ kth time the system enters the queue-not-full mode
x1
x2
impact map
state space
Switching-by-switching analysis
x1
t0
t1
queue-not-full
x2
T
t2
queue
full
t3
queue-not-full
t4
queue
full
t5
queue-not-full
t6
queue
full
´ kth time the system enters the queue-not-full mode
Theorem. The function T is a contraction. In particular,
Therefore
• xk ! x1 as k !1
• x( t ) ! x1 ( t ) as t ! 1
x1 ´ constant
x1(t) ´ periodic limit cycle
NS-2 simulation results
Flow 1
N1
S1
Flow 2
N2
S2
TCP Sources
Bottleneck link
TCP Sinks
Router
R2
Router
R1
20Mbps/20ms
Window and Queue Size (packets)
500
Flow 7
N7
S7
Flow 8
N8
S8
window size w1
window size w2
window size w3
window size w4
window size w5
window size w6
window size w7
window size w8
queue size q
400
300
200
100
0
0
10
20
30
time (seconds)
40
50
Results
t0
t1
queue-not-full
t2
queue
full
t3
queue-not-full
t4
queue
full
Window synchronization:
convergence is exponential, as fast as .5k
Steady-state formulas:
average drop rate
average RTT
average throughput
t5
queue-not-full
t6
queue
full
What next?
Other models for drops:
r1 bps
queue
r2 bps
B bps
r3 bps
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
What next?
Other models for drops:
How many drops?
qmax
q( t )
queue-not-full
Which flows suffer drops?
# of drops ´ squeue-full (inrate  outrate)
queue
full
t
number of packets
that are out for flow i
total number of
packets that are out
This probabilistic hybrid model seems to match well with packet-level
simulations, e.g., with drop-head queuing disciplines. Analysis ???
What next?
More general networks:
qB
flow 1
qC
flow 2
flow 3
qA
qD
What next?
More general networks:
qB
flow 1
qC
flow 2
qA
qD
flow 3
portion of the queue due to flow i
total queue size

drop occurs
outgoing rate of flow i
What next?
More general networks:
qB
flow 1
qC
flow 2
qA
qD
flow 3

drop occurs
What next?
Even multicast (current work)…
qB
flow 1
qC
flow 2
qA
qD

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
xk
xk+1
impact map
state space
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…