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Hybrid Systems Modeling of Communication Networks
João P. Hespanha
University of Californiaat Santa Barbara
Hybrid Control and Switched Systems
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Motivation
Why model network traffic?
• to validate designs through simulation (scalability, performance)• to analyze and design protocols (throughput, fairness, security, etc.)• to tune network parameters (queue sizes, bandwidths, etc.)
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Types of models
Packet-level modeling• tracks individual data packets, as they
travel across the network• ignores the data content of individual
packets• sub-millisecond time accuracy • computationally very intensive
Fluid-based modeling • tracks time/ensemble-average packet
rates across the network• does not explicitly model individual
events (acknowledgments, drops, queues becoming empty, etc.)
• time accuracy of a few seconds for time-average
• only suitable to model many similar flows for ensemble-average
• computationally very efficient (at least for first order statistics)
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Types of models
Hybrid modeling • keeps track of packet rates for each
flow averaged over small time scales• explicitly models some discrete
events (drops, queues becoming empty, etc.)
• time accuracy of a few milliseconds (round-trip time)
• computationally efficient
provide information about both average, peak, and “instantaneous”
resource utilization(queues, bandwidth, etc.)
captures fast dynamicseven for a small number of flow
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Summary
• Modeling 1st pass: Dumbbell topology & simplified TCP
• Modeling 2nd pass: General topology, TCP and UDP models
• Validation
• Simulation complexity
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1st pass – Dumbbell topology
Several flows follow the same path and compete for bandwidth in a single bottleneck link
Prototypical network to study congestion control
single queuerouting is trivial
q( t ) ´ queue size
r1 bps
r2 bps
r3 bps
rate · B bps
queue
f1
f2
f3
f1
f2
f3
B is unknown to the data sources and possibly time-varying
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Queue dynamics
When f rf exceeds B the queue fills and data is lost (drops)
) drop (discrete event – relevant for congestion control)
q( t ) ´ queue size
r1 bps
r2 bps
r3 bps
rate · B bps
queue
f1
f2
f3
f1
f2
f3
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Queue dynamics
Hybrid automaton representation:
q( t ) ´ queue size
r1 bps
r2 bps
r3 bps
rate · B bps
queue
f1
f2
f3
f1
f2
f3
transition enabling condition
exporteddiscrete event
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Window-based rate adjustment
1st packet sent
e.g., wf = 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 f destination f
wf effectively determines the sending rate rf :
round-trip time
t0
t1
t2
t3
0
1
2
wf (window size) ´ number of packets that can remain unacknowledged for by the destination
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Window-based rate adjustment
wf (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 wf too large
queuegets full
longerRTT
ratedecreases
queuegets empty
negative feedback
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TCP congestion avoidance
1. While there are no drops, increase wf by 1 on each RTT (additive increase)
2. When a drop occurs, divide wf by 2 (multiplicative decrease)
(congestion controller constantly probes the network for more bandwidth)
disclaimer: this is a very simplified version of TCP Reno, better models later…
TCP congestion avoidance
additiveincrease
multiplicative increase
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TCP congestion avoidance
1. While there are no drops, increase wf by 1 on each RTT (additive increase)
2. When a drop occurs, divide wf by 2 (multiplicative decrease)
(congestion controller constantly probes the network for more bandwidth)
disclaimer: this is a very simplified version of TCP Reno, better models later…
Queuing model TCP congestion avoidance
drop
RTT
rfadditiveincrease
multiplicative increase
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TCP congestion avoidance
1. While there are no drops, increase wf by 1 on each RTT (additive increase)
2. When a drop occurs, divide wf by 2 (multiplicative decrease)
(congestion controller constantly probes the network for more bandwidth)
disclaimer: this is a very simplified version of TCP Reno, better models later…
TCP + Queuing model
additiveincrease
multiplicative increase
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Linearization of the TCP model
TCP + Queuing model
additiveincrease
multiplicative increase
Time normalization ´ define a new “time” variable by
1 unit of ´ 1 round-trip time
In normalized time, the continuous dynamics become linear
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Impact-map analysis
additive increase
t0 t1 t2 t3
´ continuous state before the kth multiplicative decrease
x1 x2T
state space
x1
x2
impact map
additive increase additive increase
additive increase
multiplicative decrease multiplicative decrease
transition surface
multiplicativedecrease
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Impact-map analysis
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
additive increase
t0 t1 t2 t3
x1 x2T
additive increase additive increase
multiplicative decrease multiplicative decrease
´ continuous state before the kth multiplicative decrease
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NS-2 simulation results
0
100
200
300
400
500
0 10 20 30 40 50
Win
dow
and
Que
ue S
ize
(pac
kets
)
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
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Results
Window synchronization:
convergence is exponential, as fast as .5 k
Steady-state formulas:
average drop rate
average RTT
average throughput (well known TCP-friendly formula)
additive increase
t0 t1 t2 t3
additive increase additive increase
multiplicative decrease multiplicative decrease
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2nd pass – general topology
network dynamics (queuing & routing)
congestion control
server
client
data
acks
A communication network can be viewed as theinterconnection of several blocks with specific dynamics
b) Queuing:
in-queuerate
out-queuerate queue size
c) End2end cong. control
serversending
rateacks
& drops
a) Routing:
in-noderate
out-noderates
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end2end sending rate of flow f
Routing
in-queue rate of flow f
n
f
upstream out-queue rate of flow f
Conservation of flows:
determines the sequence of links followed by each flow
n’
n'
indexes and ’ determined by routing tables
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Routing
Multicast Multi-path routing
n’
n'
n''
”
n1’
n'
2
determines the sequence of links followed by each flow
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Queue dynamics
in-queue rates out-queue rates
…drop rates
Queue dynamics:
link bandwidth
total queue size queue size due to flow f
the packets of each flow are assumed uniformly distributed in the queue
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Queue dynamics
queue not empty/full
queue full
queue empty
same in and out-queue rates
out-queue rates proportional to fraction of
packets in the queue
no drops
drops proportional to fraction in-queue rates
in-queue rates out-queue rates
…drop rates
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Drops events
t0 t2t1
total in-queue ratepacket size
total out-queue rate(link bandwidth)
in-queue rates out-queue rates
…drop rates
When?
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Drops events
Which flows?
t0 t2t1
flow that suffers drop at time tk
(drop tail dropping)
When?
in-queue rates out-queue rates
…drop rates
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Hybrid queue model
-queue-not-full
-queue-full
transition enabling condition
exporteddiscrete event
discrete modes
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Hybrid queue model
Random Early Dropactive queuing
stochastic counter-queue-not-full
-queue-full
discrete modes
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Network dynamic & Congestion control
routing
queue dynamics
sendingrates
drops
out-queuerates
in-queue rates
end2end congestion control
TCP/UDP
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Additive Increase/Multiplicative Decrease
congestion-avoidance
TCP-Reno is based on AIMD but uses other discrete modes to improve performance
set of links transversed by flow f
propagation delays
1. While there are no drops, increase wf by 1 on each RTT (additive increase)
2. When a drop occurs, divide wf by 2 (multiplicative decrease)
(congestion controller constantly probe the network for more bandwidth)
importeddiscrete event
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Slow start
3. Until a drop occurs (or a threshold ssthf is reached), double wf on each RTT4. When a drop occurs, divide wf and the threshold ssthf by 2
cong.-avoid.slow-start
especially important for short-lived flows…
In the beginning, pure AIMD takes a long time to reach an adequate window size
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Fast recovery
5. During retransmission, data is sent at a rate consistent with a window size of wf /2
After a drop is detected, new data should be sent while the dropped one is retransmitted
(consistent with TCP-SACK for multiple consecutive drops)
cong.-avoid.
fast-recovery
slow-start
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3th packet received & ack. sent
1st packet sent2nd packet sent
4th packet sent
Timeouts
6. When a drop is detected through timeout:a. the slow-start threshold ssthf is set equal to half the
window size,b. the window size is reduced to one,c. the controller transitions to slow-start
Typically, drops are detected because one acknowledgment in the sequence is missing.
2nd packet received & ack. sent
4th packet received & ack. sent
source destination
three acks received out of order
drop3th packet sent
drop detected, 1st packet re-sent
When the window size becomes smaller than 4, this mechanism fails and drops must be detected through acknowledgement timeout.
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Fast recovery, timeouts, drop-detection delay…
TCP SACKversion
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Network dynamic & Congestion control
routing
queue dynamics
sendingrates
drops
out-queuerates
in-queue rates
end2end congestion control
RTTs
see SIGMETRICS paper for on/off TCP & UDP model
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Validation methodology
Compared simulation results from• ns-2 packet-level simulator• hybrid models implemented in Modelica
Plots in the following slides refer to two test topologies
• 10ms propagation delay• drop-tail queuing• 5-500Mbps bottleneck throughput• 0-10% UDP on/off background traffic
• 45,90,135,180ms propagation delays• drop-tail queuing• 5-500Mbps bottleneck throughput• 0-10% UDP on/off background traffic
Y-topologydumbbell
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Simulation traces
• single TCP flow• 5Mbps bottleneck throughput• no background traffic
ns-2
0
140
120
100
80
60
40
20
0 2 4 6 8 10 12 14 16 18 20
cwnd
and
que
ue s
ize
(pac
kets
)
time (seconds)
cwnd of TCP 1queue size
0
140
120
100
80
60
40
20
0 2 4 6 8 10 12 14 16 18 20
cwnd
and
que
ue s
ize
(pac
kets
)
time (seconds)
cwnd of TCP 1queue size
hybrid model
slow-start, fast recovery, and congestion avoidance accurately captured
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Simulation traces
• four competing TCP flow(starting at different times)
• 5Mbps bottleneck throughput• no background traffic
the hybrid model accurately captures flow synchronization
0
140
120
100
80
60
40
20
0 2 4 6 8 10 12 14
16 18 20
cwnd
and
que
ue s
ize
(pac
kets
)
time (seconds)
cwnd size of TCP 2
cwnd size of TCP 3
cwnd size of TCP 4
cwnd size of TCP 1
Queue size of Q1
Queue size of Q2
0
140
120
100
80
60
40
20
0 2 4 6 8 10 12 14 16 18 20
cwnd
and
que
ue s
ize
(pac
kets
)
time (seconds)
cwnd size of TCP 2
cwnd size of TCP 3
cwnd size of TCP 4
cwnd size of TCP 1
Queue size of Q1
Queue size of Q2
ns-2hybrid model
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Simulation traces
CWND size of TCP 1 (Prop=0.045ms)
CWND size of TCP 2 (Prop=0.090ms)
CWND size of TCP 3 (Prop=0.135ms)
CWND size of TCP 4 (Prop=0.180ms)
Queue size of Q1
Queue size of Q3
0
140
120
100
80
60
40
20
0 2 4 6 8 10 12 14 16 18 20
cwnd
and
que
ue s
ize
(pac
kets
)
time (seconds)
CWND size of TCP 1 (Prop=0.045ms)
CWND size of TCP 2 (Prop=0.090ms)
CWND size of TCP 3 (Prop=0.135ms)
CWND size of TCP 4 (Prop=0.180ms)
Queue size of Q1
Queue size of Q3
0
140
120
100
80
60
40
20
0 2 4 6 8 10 12 14 16 18 20
cwnd
and
que
ue s
ize
(pac
kets
)
time (seconds)
ns-2hybrid model
• four competing TCP flow(different propagation delays)
• 5Mbps bottleneck throughput• 10% UDP background traffic
(exp. distributed on-off times)
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Average throughput and RTTs
Thru. 1 Thru. 2 Thru. 3 Thru. 4 RTT1 RTT2 RTT3 RTT4
ns-2 1.873 1.184 .836 .673 .0969 .141 .184 .227
hybrid model 1.824 1.091 .823 .669 .0879 .132 .180 .223
relative error 2.6% 7.9% 1.5% .7% 9.3% 5.9% 3.6% 2.1%
the hybrid model accurately captures TCP unfairness for different propagation delays
• 45,90,135,180ms propagation delays• drop-tail queuing• 5Mbps bottleneck throughput• 10% UDP on/off background traffic
• four competing TCP flow(different propagation delays)
• 5Mbps bottleneck throughput• 10% UDP background traffic
(exp. distributed on-off times)
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Empirical distributions
hybrid model ns-2
the hybrid model captures the whole distribution of congestion windows and queue size
0 10 20 30 40 50 60 700
0.05
0.1
0.15
prob
abil
ity
0 10 20 30 40 50 60 700
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
cwnd & queue sizepr
obab
ilit
y
CWND of TCP1CWND of TCP2CWND of TCP3CWND of TCP4Queue 3
CWND of TCP1CWND of TCP2CWND of TCP3CWND of TCP4Queue 3
cwnd & queue size
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Execution time
0.1
1
10
100
1000
10000
1 10 100 1000
bottleneck bandwidth [Mbps]
execution tim
e for
10m
in
of sim
ula
tion tim
e
[sec]
ns-2
hybrid model
1 flow
3 flows
• ns-2 complexity approximately scales with
• hybrid simulator complexity approximately scales with
number of flows
per-flow throughput
(# packets)
5Mbps
50Mbps
500Mbps
hybrid models are particularly suitable for large, high-bandwidth simulations (satellite, fiber optics, backbone)