, Alfredo Marciello algorithmsiwcsn2006.irmacs.sfu.ca/pdf/ljiljana.trajkovic_edited.pdf · V....
Transcript of , Alfredo Marciello algorithmsiwcsn2006.irmacs.sfu.ca/pdf/ljiljana.trajkovic_edited.pdf · V....
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Discontinuity-induced bifurcations in TCP/RED communication
algorithmsMingjian Liu*, Alfredo Marciello†, Mario di Bernardo†,
and Ljiljana Trajković*
[email protected], [email protected], [email protected], [email protected]
* Simon Fraser University, Vancouver, Canadahttp://www.ensc.sfu.ca/cnl
†University of Naples Federico II, Naples, Italy
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August 18, 2006 IEEE International Workshop on Complex Systems and Networks 2
Roadmap
� Introduction� TCP/RED congestion control algorithms: an overview� Discrete-time dynamical models of TCP Reno with RED:
� modeling assumptions� S-TCP/RED models with:
� one state variable� two state variables
� model validation and modifications� comparison of TCP/RED models
� Bifurcation and chaos phenomena in TCP/RED:� discontinuity-induced bifurcations
� Conclusion and references
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August 18, 2006 IEEE International Workshop on Complex Systems and Networks 3
Roadmap
� Introduction� TCP/RED congestion control algorithms: an overview� Discrete-time dynamical models of TCP Reno with RED:
� modeling assumptions� S-TCP/RED models with:
� one state variable� two state variables
� model validation and modifications� comparison of TCP/RED models
� Bifurcation and chaos phenomena in TCP/RED� discontinuity-induced bifurcations
� Conclusion and references
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August 18, 2006 IEEE International Workshop on Complex Systems and Networks 4
Motivation
� Modeling TCP Reno with RED is important to:� examine the interactions between TCP and RED� understand and predict the dynamical network
behavior� analyze the impact of system parameters� investigate bifurcations and complex behavior
TCP: Transmission Control Protocol
RED: Random Early Detection Gateways for Congestion Avoidance
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August 18, 2006 IEEE International Workshop on Complex Systems and Networks 5
Roadmap
� Introduction� TCP/RED congestion control algorithms: an overview� Discrete-time dynamical models of TCP Reno with RED:
� modeling assumptions� S-TCP/RED models with:
� one state variable� two state variables
� model validation and modifications� comparison of TCP/RED models
� Bifurcation and chaos phenomena in TCP/RED� discontinuity-induced bifurcations
� Conclusion and references
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August 18, 2006 IEEE International Workshop on Complex Systems and Networks 6
TCP
� TCP: Transmission Control Protocol� Fourth layer of the OSI model� Connection oriented, reliable, and byte-stream service� Employs window based flow and congestion control
algorithms
OSI: Open System Interconnection reference model
6 5 4 3 2 1
window size: 5 packets
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August 18, 2006 IEEE International Workshop on Complex Systems and Networks 7
TCP
� Several flavors of TCP:� Tahoe: 4.3 BSD Tahoe (~ 1988)
� slow start, congestion avoidance, and fast retransmit (RFC 793, RFC 2001)
� Reno: 4.3 BSD Reno (~ 1990)� slow start, congestion avoidance, fast retransmit,
and fast recovery (RFC 2001, RFC 2581)� NewReno (~ 1996)
� new fast recovery algorithm (RFC 2582)� SACK (~ 1996, RFC 2018)
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August 18, 2006 IEEE International Workshop on Complex Systems and Networks 8
TCP Reno
SS CAFast retransmit and fast recovery
CA SS
TOSS: Slow StartCA: Congestion AvoidanceTO: Timeout
t
CWND
ssthresh2
ssthresh1
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August 18, 2006 IEEE International Workshop on Complex Systems and Networks 9
TCP Reno: slow start and congestion avoidance
� Slow start:� cwnd = IW (1 or 2 packets)� when cwnd <ssthresh
cwnd = cwnd + 1 for each received ACK� Congestion avoidance:
� when cwnd > ssthreshcwnd = cwnd + 1/cwnd for each ACK
cwnd : congestion window sizeIW : initial window sizessthresh : slow start threshold ACK : acknowledgement RTT : round trip time
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August 18, 2006 IEEE International Workshop on Complex Systems and Networks 10
TCP Reno: fast retransmit and fast recovery
� three duplicate ACKs are received
� retransmit the packet� ssthresh = cwnd/2,
cwnd = ssthresh + 3 packets� cwnd = cwnd + 1, for each
additional duplicate ACK� transmit the new data, if
cwnd allows� cwnd = ssthresh, if ACK for
new data is received
16
17
18
19
20
21
17
ACK 16
ACK 16ACK 16ACK 16ACK 16
TCP sender TCP receiver
three duplicate ACKs
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August 18, 2006 IEEE International Workshop on Complex Systems and Networks 11
TCP Reno: timeout
� TCP maintains a retransmission timer� The duration of the timer is called
retransmission timeout� Timeout occurs when the ACK for the
delivered data is not received before the retransmission timer expires
� TCP sender retransmits the lost packet� ssthresh = cwnd/2
cwnd = 1 or 2 packets
1
1
TOACK1
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August 18, 2006 IEEE International Workshop on Complex Systems and Networks 12
AQM: Active Queue Management
� AQM (RFC 2309):
� reduces bursty packet drops in routers� provides lower-delay interactive service� avoids the “lock-out” problem� reacts to the incipient congestion before buffers
overflow� AQM algorithms:
� RED (RFC 2309)� ARED, CHOKe, BLUE, …
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August 18, 2006 IEEE International Workshop on Complex Systems and Networks 13
RED
� Random Early Detection Gateways for Congestion Avoidance
� Proposed by S. Floyd and V. Jacobson, LBN, 1993:S. Floyd and V. Jacobson, “Random early detection gateways for congestion avoidance,” IEEE/ACM Trans. Networking, vol. 1, no. 4, pp. 397–413, Aug. 1993.
� Main concept: � drop packets before the queue becomes full
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August 18, 2006 IEEE International Workshop on Complex Systems and Networks 14
RED variables and parameters
� Main variables and parameters:� average queue size:� instantaneous queue size:� drop probability:� queue weight:� maximum drop probability:� queue thresholds: and
maxpminq maxq
qw1+kp
1+kq1+kq
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August 18, 2006 IEEE International Workshop on Complex Systems and Networks 15
RED algorithm
Calculate:� average queue size for each packet arrival
� drop probability
11 )1( ++ ⋅+⋅−= kqkqk qwqwq
pk+1
1+kqminq maxq
maxp
1
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August 18, 2006 IEEE International Workshop on Complex Systems and Networks 16
RED algorithm: drop probability
� if
count: number of packets that arrived since the last packet drop
� else if
� else
� Mark or drop the arriving packet with probability
)( minqq <0=ap
1=ap)( maxqq >
ap
minmax
minmax qq
qqppb −−×=
)( maxmin qqq <<
b
ba pcount
pp×−
=1
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August 18, 2006 IEEE International Workshop on Complex Systems and Networks 17
RED algorithm: drop probability
� if
� else if
� else
� mark or drop the arriving packet with probability
)( min1 qqk ≤+
01 =+kp
11 =+kp)( max1 qqk ≥+
1+kp
maxminmax
min11 p
qqqqp k
k −−= +
+
)( max1min qqq k << +
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August 18, 2006 IEEE International Workshop on Complex Systems and Networks 18
Simulation tool: ns-2
� ns-2 is a discrete event network simulatorhttp://www.isi.edu/nsnam/ns
� Supports simulation of TCP, routing, and multicast protocols over wired and wireless networks
� We used ns-2 to validate the proposed S-TCP/RED model
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August 18, 2006 IEEE International Workshop on Complex Systems and Networks 19
Roadmap
� Introduction� TCP/RED congestion control algorithms: an overview� Discrete-time dynamical models of TCP Reno with RED:
� modeling assumptions� S-TCP/RED models with:
� one state variable� two state variables
� model validation and modifications� comparison of TCP/RED models
� Bifurcation and chaos phenomena in TCP/RED� discontinuity-induced bifurcations
� Conclusion and references
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August 18, 2006 IEEE International Workshop on Complex Systems and Networks 20
Modeling methodology
� Categories of TCP models:� averaged and discrete-time models� short-lived and long-lived TCP connections
� S-TCP/RED models:� discrete-time model with a long-lived connection
� State variables:� window size (TCP)� average queue size (RED)
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August 18, 2006 IEEE International Workshop on Complex Systems and Networks 21
S-TCP/RED model
� Key properties of the proposed S-TCP/RED models:� slow start, congestion avoidance, fast retransmit, and
fast recovery (simplified)� Timeout:
J. Padhye, V. Firoiu, and D. F. Towsley, “Modeling TCP Reno performance: a simple model and its empirical validation,”IEEE/ACM Trans. Networking, vol. 8, no. 2, pp. 133–145, Apr. 2000.
� Captures the basic RED algorithm
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August 18, 2006 IEEE International Workshop on Complex Systems and Networks 22
Assumptions
� Long-lived TCP connection� Constant propagation delay between the source and the
destination� Constant packet size � ACK packets are never lost� Timeout occurs only due to packet loss� The system is sampled at the end of every RTT interval
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August 18, 2006 IEEE International Workshop on Complex Systems and Networks 23
TCP/RED model simplifications
� Simplified fast recovery
SS CA
Fast retransmit and fast recovery
CA SS
TO
SS: Slow StartCA: Congestion AvoidanceTO: Timeout
cwnd
t
ssthresh 2ssthresh 1
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August 18, 2006 IEEE International Workshop on Complex Systems and Networks 24
TCP Reno: fast recovery
SS CAFast retransmit and fast recovery
CA SS
TOSS: Slow StartCA: Congestion AvoidanceTO: Timeout
t
CWND
ssthresh2
ssthresh1
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August 18, 2006 IEEE International Workshop on Complex Systems and Networks 25
S-TCP/RED model simplifications
� TO = 5 RTTV. Firoiu and M. Borden, “A study of active queue management for congestion control,” in Proc. of IEEE INFOCOM 2000, vol. 3, pp. 1435–1444, Tel-Aviv, Israel, Mar. 2000.
� RED: parameter count is not used
ba pp =
minmax
minmax qq
qqppb −−×=
if )( maxmin qqq <<
b
ba pcount
pp×−
=1
)( maxmin qqq <<if
minmax
minmax qq
qqppa −−×=
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August 18, 2006 IEEE International Workshop on Complex Systems and Networks 26
Network topology
� Components: one source, two routers, and one destination
� The link between routers 1 and 2 is the only bottleneck� RED algorithm is deployed in router 1
Source Router 1 Router 2 Destination
Bottleneck
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August 18, 2006 IEEE International Workshop on Complex Systems and Networks 27
Roadmap
� Introduction� TCP/RED congestion control algorithms: an overview� Discrete-time dynamical models of TCP Reno with RED:
� modeling assumptions� S-TCP/RED models with:
� one state variable� two state variables
� model validation and modifications � comparison of TCP/RED models
� Bifurcation and chaos phenomena in TCP/RED� discontinuity-induced bifurcations
� Conclusion and references
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August 18, 2006 IEEE International Workshop on Complex Systems and Networks 28
TCP/RED models
� S-models: discrete nonlinear dynamical models of TCP Reno with RED
� State variables: � window size� average queue size
� The proposed TCP/RED models are:� simple and intuitively derived� able to capture detailed dynamical behavior of
TCP/RED systems� have been verified via ns-2 simulations
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August 18, 2006 IEEE International Workshop on Complex Systems and Networks 29
S-TCP/RED: parameters and variables
� Variables:w : window size q : average queue sizep : drop probability q : instantaneous queue size
� Parameters:qmax: maximum queue thresholdqmin: minimum queue thresholdpmax: maximum drop probability wq : queue weight d : propagation delayM : packet size C : link capacity
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August 18, 2006 IEEE International Workshop on Complex Systems and Networks 30
Roadmap
� Introduction� TCP/RED congestion control algorithms: an overview� Discrete-time dynamical models of TCP Reno with RED:
� modeling assumptions� S-TCP/RED models with:
� one state variable� two state variables
� model validation and modifications� comparison of TCP/RED models
� Bifurcation and chaos phenomena in TCP/RED� discontinuity-induced bifurcations
� Conclusion and references
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August 18, 2006 IEEE International Workshop on Complex Systems and Networks 31
Simplified S-TCP/RED model
� M-model, a discrete nonlinear dynamical model of TCP Reno with RED:P. Ranjan, E. H. Abed, and R. J. La, “Nonlinear instabilities in TCP-RED,”in Proc. IEEE INFOCOM 2002, New York, NY, USA, June 2002, vol. 1, pp. 249–258 and IEEE/ACM Trans. on Networking, vol. 12, no. 6, pp. 1079–1092, Dec. 2004.
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August 18, 2006 IEEE International Workshop on Complex Systems and Networks 32
Simplified S-TCP/RED model:one state variable
� Variables:� qk+1: average queue size in round k+1� qk: average queue size in round k� wq: queue weight in RED� N: number of TCP connections� K: constant =� pk: drop probability in round k� C: capacity of the link between the two routers� d: round-trip propagation delay� M: packet size� rwnd: receiver's advertised window size
2/3
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August 18, 2006 IEEE International Workshop on Complex Systems and Networks 33
Simplified S-TCP/RED model: packet lost
� Drop probability:
where:: TCP sending rate: the number of incoming packets: the number of outgoing packets
MdC
pNK
CMqd
MCNRTT
RTTpKq
MRTTCNRTTpBqq
k
kk
kkk
kkkkk
⋅−⋅=
⋅+−⋅⋅⋅
+=
⋅−⋅⋅+=
++
+++
)(
)(
11
111
0≠kp
)( kpBNRTTpB kk ⋅⋅ +1)(
MRTTC k 1+⋅
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August 18, 2006 IEEE International Workshop on Complex Systems and Networks 34
Simplified S-TCP/RED model: packet lost
The average queue size is:
hence)0,max()1(1 M
dCpKNwqwqk
qkqk⋅−⋅⋅+⋅−=+
11 )1( ++ ⋅+⋅−= kqkqk qwqwq
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August 18, 2006 IEEE International Workshop on Complex Systems and Networks 35
Simplified S-TCP/RED model: no loss
� Drop probability:
The average queue size is:
hence
MdCNrwnd
CMqd
MCNRTT
RTTrwndq
MRTTCNRTTpBqq
kk
kk
kkkkk
⋅−⋅=
⋅+−⋅⋅+=
⋅−⋅⋅+=
++
+++
)(
)(
11
111
0=kp
)()1(1 MdCNrwndwqwq qkqk
⋅−⋅⋅+⋅−=+
11 )1( ++ ⋅+⋅−= kqkqk qwqwq
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August 18, 2006 IEEE International Workshop on Complex Systems and Networks 36
Simplified S-TCP/RED model
� Dynamical model of TCP/RED:
=⋅−⋅⋅+⋅−
≠⋅−⋅⋅+⋅−=+
0)()1(
0)0,max()1(
1
kqkq
kk
qkq
k
pifM
dCNrwndwqw
pifM
dCpKNwqw
q
![Page 37: , Alfredo Marciello algorithmsiwcsn2006.irmacs.sfu.ca/pdf/ljiljana.trajkovic_edited.pdf · V. Firoiu and M. Borden, “A study of active queue management for congestion control,”](https://reader033.fdocuments.in/reader033/viewer/2022052102/603c2df399693850c041f152/html5/thumbnails/37.jpg)
August 18, 2006 IEEE International Workshop on Complex Systems and Networks 37
Roadmap
� Introduction� TCP/RED congestion control algorithms: an overview� Discrete-time dynamical models of TCP Reno with RED:
� modeling assumptions� S-TCP/RED models with:
� one state variable� two state variables
� model validation and modifications� comparison of TCP/RED models
� Bifurcation and chaos phenomena in TCP/RED� discontinuity-induced bifurcations
� Conclusion and references
![Page 38: , Alfredo Marciello algorithmsiwcsn2006.irmacs.sfu.ca/pdf/ljiljana.trajkovic_edited.pdf · V. Firoiu and M. Borden, “A study of active queue management for congestion control,”](https://reader033.fdocuments.in/reader033/viewer/2022052102/603c2df399693850c041f152/html5/thumbnails/38.jpg)
August 18, 2006 IEEE International Workshop on Complex Systems and Networks 38
TCP/RED model: state variable and parameters
� qk+1: instantaneous queue size in round k+1� qk+1: average queue size in round k+1� Wk+1: current TCP window size in round k+1� wq: queue weight in RED� pk: drop probability in round k� RTTk+1: round-trip time at k+1� C: capacity of the link between the two routers� M: packet size� d: round-trip propagation delay� ssthesh: slow start threshold� rwnd: receiver's advertised window size
![Page 39: , Alfredo Marciello algorithmsiwcsn2006.irmacs.sfu.ca/pdf/ljiljana.trajkovic_edited.pdf · V. Firoiu and M. Borden, “A study of active queue management for congestion control,”](https://reader033.fdocuments.in/reader033/viewer/2022052102/603c2df399693850c041f152/html5/thumbnails/39.jpg)
August 18, 2006 IEEE International Workshop on Complex Systems and Networks 39
S-TCP/RED: discrete-time model for TCP Reno with RED
� Calculate the average queue size:
� the average queue size is updated after each packet arrival
� is updated times in k+1-th round
From (1) and (2):
MdCW
CMqd
MCWqq
k
kkkk
⋅−=
⋅+−+=
+
++
1
11 )(
(1)
11 )1( ++ ⋅+⋅−= kqkqk qwqwq (2)
1+kq 1+kW
)0,max())1(1()1( 1111
MdCWwqwq k
Wqk
Wqk
kk⋅−⋅−−+⋅−= ++
++
![Page 40: , Alfredo Marciello algorithmsiwcsn2006.irmacs.sfu.ca/pdf/ljiljana.trajkovic_edited.pdf · V. Firoiu and M. Borden, “A study of active queue management for congestion control,”](https://reader033.fdocuments.in/reader033/viewer/2022052102/603c2df399693850c041f152/html5/thumbnails/40.jpg)
August 18, 2006 IEEE International Workshop on Complex Systems and Networks 40
S-TCP/RED model: drop probability
� Calculate the drop probability:
−−
≥≤
=+
+
+
+
otherwisepqqqq
qqifqqif
pk
k
k
k
maxminmax
min1
max1
min1
1 10
![Page 41: , Alfredo Marciello algorithmsiwcsn2006.irmacs.sfu.ca/pdf/ljiljana.trajkovic_edited.pdf · V. Firoiu and M. Borden, “A study of active queue management for congestion control,”](https://reader033.fdocuments.in/reader033/viewer/2022052102/603c2df399693850c041f152/html5/thumbnails/41.jpg)
August 18, 2006 IEEE International Workshop on Complex Systems and Networks 41
S-TCP/RED model: three cases
� No packet lost: � slow start� congestion avoidance
� Single packet lost:� fast retransmit� fast recovery
� At least two packets lost:� timeout
![Page 42: , Alfredo Marciello algorithmsiwcsn2006.irmacs.sfu.ca/pdf/ljiljana.trajkovic_edited.pdf · V. Firoiu and M. Borden, “A study of active queue management for congestion control,”](https://reader033.fdocuments.in/reader033/viewer/2022052102/603c2df399693850c041f152/html5/thumbnails/42.jpg)
August 18, 2006 IEEE International Workshop on Complex Systems and Networks 42
S-TCP/RED model: no packet loss
� number of lost packets: � window size:
� where:� Wk+1: window size in round k+1� ssthesh: slow start threshold� rwnd: receiver's advertised window size
≥+<
=+ ssthreshWifrwndWssthreshWifssthreshW
Wkk
kkk ),1min(
),2min(1
5.0<kkWp
![Page 43: , Alfredo Marciello algorithmsiwcsn2006.irmacs.sfu.ca/pdf/ljiljana.trajkovic_edited.pdf · V. Firoiu and M. Borden, “A study of active queue management for congestion control,”](https://reader033.fdocuments.in/reader033/viewer/2022052102/603c2df399693850c041f152/html5/thumbnails/43.jpg)
August 18, 2006 IEEE International Workshop on Complex Systems and Networks 43
S-TCP/RED model: no packet loss
� current queue size:
� where:� RTTk+1: round-trip time at k+1� C: capacity of the link between the two routers� M: packet size� d: round-trip propagation delay
MdCW
CMqd
MCWq
MRTTCWqq
k
kkk
kkkk
⋅−=
+−+=
⋅−+=
+
+
+++
1
1
111
)(
![Page 44: , Alfredo Marciello algorithmsiwcsn2006.irmacs.sfu.ca/pdf/ljiljana.trajkovic_edited.pdf · V. Firoiu and M. Borden, “A study of active queue management for congestion control,”](https://reader033.fdocuments.in/reader033/viewer/2022052102/603c2df399693850c041f152/html5/thumbnails/44.jpg)
August 18, 2006 IEEE International Workshop on Complex Systems and Networks 44
S-TCP/RED model: no packet loss
� average queue size:
� hence:
)0,max())1(1()1( 1111
MdCWwqwq k
Wqk
Wqk
kk⋅−⋅−−+−= ++
++
)0,max()1( 11 MdCWwqwq kqkqk
⋅−⋅+⋅−= ++
![Page 45: , Alfredo Marciello algorithmsiwcsn2006.irmacs.sfu.ca/pdf/ljiljana.trajkovic_edited.pdf · V. Firoiu and M. Borden, “A study of active queue management for congestion control,”](https://reader033.fdocuments.in/reader033/viewer/2022052102/603c2df399693850c041f152/html5/thumbnails/45.jpg)
August 18, 2006 IEEE International Workshop on Complex Systems and Networks 45
S-TCP/RED model: no packet loss
� number of lost packets:
� window size:
� average queue size:
)0,max())1(1()1( 1111
MdCWwqwq k
Wqk
Wqk
kk⋅−⋅−−+−= ++
++
≥+<
=+ ssthreshWifrwndWssthreshWifssthreshW
Wkk
kkk ),1min(
),2min(1
5.0<kkWp
![Page 46: , Alfredo Marciello algorithmsiwcsn2006.irmacs.sfu.ca/pdf/ljiljana.trajkovic_edited.pdf · V. Firoiu and M. Borden, “A study of active queue management for congestion control,”](https://reader033.fdocuments.in/reader033/viewer/2022052102/603c2df399693850c041f152/html5/thumbnails/46.jpg)
August 18, 2006 IEEE International Workshop on Complex Systems and Networks 46
S-TCP/RED model: one packet loss
� number of lost packets:
� window size:
� average queue size:
kk WW21
1 =+
5.15.0 <≤ kkWp
)0,max())1(1()1( 1111
MdCWwqwq k
Wqk
Wqk
kk⋅−⋅−−+−= ++
++
![Page 47: , Alfredo Marciello algorithmsiwcsn2006.irmacs.sfu.ca/pdf/ljiljana.trajkovic_edited.pdf · V. Firoiu and M. Borden, “A study of active queue management for congestion control,”](https://reader033.fdocuments.in/reader033/viewer/2022052102/603c2df399693850c041f152/html5/thumbnails/47.jpg)
August 18, 2006 IEEE International Workshop on Complex Systems and Networks 47
S-TCP/RED model: two packet losses
� number of lost packets:
� window size:
� average queue size:
5.1≥kkWp
01 =+kW
kk qq =+1
![Page 48: , Alfredo Marciello algorithmsiwcsn2006.irmacs.sfu.ca/pdf/ljiljana.trajkovic_edited.pdf · V. Firoiu and M. Borden, “A study of active queue management for congestion control,”](https://reader033.fdocuments.in/reader033/viewer/2022052102/603c2df399693850c041f152/html5/thumbnails/48.jpg)
August 18, 2006 IEEE International Workshop on Complex Systems and Networks 48
Roadmap
� Introduction� TCP/RED congestion control algorithms: an overview� Discrete-time dynamical models of TCP Reno with RED:
� modeling assumptions� S-TCP/RED models with:
� one state variable� two state variables
� model validation and modifications� comparison of TCP/RED models
� Bifurcation and chaos phenomena in TCP/RED� discontinuity-induced bifurcations
� Conclusion and references
![Page 49: , Alfredo Marciello algorithmsiwcsn2006.irmacs.sfu.ca/pdf/ljiljana.trajkovic_edited.pdf · V. Firoiu and M. Borden, “A study of active queue management for congestion control,”](https://reader033.fdocuments.in/reader033/viewer/2022052102/603c2df399693850c041f152/html5/thumbnails/49.jpg)
August 18, 2006 IEEE International Workshop on Complex Systems and Networks 49
Simulation scenario
1.54 Mbps
10 ms delaysource router 1 router 2 sink
100 Mbps
0 ms delay
100 Mbps
0 ms delay
� source to router1:
� link capacity: 100 Mbps with 0 ms delay
� router 1 to router 2: the only bottleneck in the network
� link capacity: 1.54 Mbps with 10 ms delay
� router 2 to sink:
� link capacity: 100 Mbps with 0 ms delay
![Page 50: , Alfredo Marciello algorithmsiwcsn2006.irmacs.sfu.ca/pdf/ljiljana.trajkovic_edited.pdf · V. Firoiu and M. Borden, “A study of active queue management for congestion control,”](https://reader033.fdocuments.in/reader033/viewer/2022052102/603c2df399693850c041f152/html5/thumbnails/50.jpg)
August 18, 2006 IEEE International Workshop on Complex Systems and Networks 50
RED: default parameters
15 (packets)Maximum queue threshold (qmax)
4,000 (bytes)Packet size (M)
5 (packets)Minimum queue threshold (qmin)
0.1Maximum drop probability (pmax )0.002Queue weight (wq )
� RED parameters:
S. Floyd, “RED: Discussions of Setting Parameters,” Nov. 1997: http://www.icir.org/floyd/REDparameters.txt
![Page 51: , Alfredo Marciello algorithmsiwcsn2006.irmacs.sfu.ca/pdf/ljiljana.trajkovic_edited.pdf · V. Firoiu and M. Borden, “A study of active queue management for congestion control,”](https://reader033.fdocuments.in/reader033/viewer/2022052102/603c2df399693850c041f152/html5/thumbnails/51.jpg)
August 18, 2006 IEEE International Workshop on Complex Systems and Networks 51
S-TCP/RED model validation
� Waveforms of the state variables with default parameters:� window size� average queue size
� Validation for various values of the system parameters:� queue weight: wq� maximum drop probability: pmax� queue thresholds: qmin and qmax , qmax/qmin = 3
![Page 52: , Alfredo Marciello algorithmsiwcsn2006.irmacs.sfu.ca/pdf/ljiljana.trajkovic_edited.pdf · V. Firoiu and M. Borden, “A study of active queue management for congestion control,”](https://reader033.fdocuments.in/reader033/viewer/2022052102/603c2df399693850c041f152/html5/thumbnails/52.jpg)
August 18, 2006 IEEE International Workshop on Complex Systems and Networks 52
0 10 20 30 40 50 60 70 800
5
10
15
20
25
30
35
Time (sec)
Win
dow
siz
e (p
acke
ts)
Window size: waveforms
40 41 42 43 44 45 46 47 48 49 500
5
10
15
20
25
30
35
Time (sec)
Win
dow
siz
e (p
acke
ts)
40 41 42 43 44 45 46 47 48 49 500
5
10
15
20
25
30
35
Time (sec)
Win
dow
siz
e (p
acke
ts)
S-TCP/RED model ns-2
0 10 20 30 40 50 60 70 800
5
10
15
20
25
30
35
Time (sec)
Win
dow
siz
e (p
acke
ts)
![Page 53: , Alfredo Marciello algorithmsiwcsn2006.irmacs.sfu.ca/pdf/ljiljana.trajkovic_edited.pdf · V. Firoiu and M. Borden, “A study of active queue management for congestion control,”](https://reader033.fdocuments.in/reader033/viewer/2022052102/603c2df399693850c041f152/html5/thumbnails/53.jpg)
August 18, 2006 IEEE International Workshop on Complex Systems and Networks 53
0 10 20 30 40 50 60 70 800
1
23
45
6
78
910
Time (sec)
Ave
rage
que
ue s
ize
(pac
kets
)
40 41 42 43 44 45 46 47 48 49 505
5.5
6
6.5
7
7.5
8
Time (sec)
Ave
rage
que
ue s
ize
(pac
kets
)
0 10 20 30 40 50 60 70 800
1
23
45
6
78
910
Time (sec)
Ave
rage
que
ue s
ize
(pac
kets
)
Average queue size: waveforms
40 41 42 43 44 45 46 47 48 49 505
5.5
6
6.5
7
7.5
8
Time (sec)
Ave
rage
que
ue s
ize
(pac
kets
)
S-TCP/RED model ns-2
![Page 54: , Alfredo Marciello algorithmsiwcsn2006.irmacs.sfu.ca/pdf/ljiljana.trajkovic_edited.pdf · V. Firoiu and M. Borden, “A study of active queue management for congestion control,”](https://reader033.fdocuments.in/reader033/viewer/2022052102/603c2df399693850c041f152/html5/thumbnails/54.jpg)
August 18, 2006 IEEE International Workshop on Complex Systems and Networks 54
0 10 20 30 40 50 60 70 800
5
10
15
20
25
30
Time (sec)
Win
dow
siz
e (p
acke
t)
0 10 20 30 40 50 60 70 800
5
10
15
20
25
30
Time (sec)
Win
dow
siz
e (p
acke
ts)
Model validation: wq
� wq =[0.001, 0.01], with other parameters default
� window size: wq = 0.006
S-TCP/RED model ns-2
![Page 55: , Alfredo Marciello algorithmsiwcsn2006.irmacs.sfu.ca/pdf/ljiljana.trajkovic_edited.pdf · V. Firoiu and M. Borden, “A study of active queue management for congestion control,”](https://reader033.fdocuments.in/reader033/viewer/2022052102/603c2df399693850c041f152/html5/thumbnails/55.jpg)
August 18, 2006 IEEE International Workshop on Complex Systems and Networks 55
1 2 3 4 5 6 7 8 9 10x 10-3
0
2
4
6
8
10
Wq
Ave
rage
que
ue s
ize
(pac
kets
)S-RED modelns-2
Model validation: wq
� average queue size during steady state:
![Page 56: , Alfredo Marciello algorithmsiwcsn2006.irmacs.sfu.ca/pdf/ljiljana.trajkovic_edited.pdf · V. Firoiu and M. Borden, “A study of active queue management for congestion control,”](https://reader033.fdocuments.in/reader033/viewer/2022052102/603c2df399693850c041f152/html5/thumbnails/56.jpg)
August 18, 2006 IEEE International Workshop on Complex Systems and Networks 56
Model validation: wq
� Comparison of system variables:
11.720.550.610.083384.70385.028.9635.738.90.010
11.110.550.610.109384.68385.108.9035.839.00.008
7.910.560.600.093384.73385.088.9335.839.00.006
6.120.560.590.083384.79385.118.8036.239.40.004
2.560.550.560.056384.77384.9810.8336.039.90.002
1.290.540.550.073384.71384.9911.6336.140.30.001
Δ(%)ns-2S-RED modelΔ(%)ns-2S-RED
modelΔ(%)ns-2S-RED modelweight (wq)
Drop rate (%)Sending rate(packets/sec)Average RTT (msec)Parameters
![Page 57: , Alfredo Marciello algorithmsiwcsn2006.irmacs.sfu.ca/pdf/ljiljana.trajkovic_edited.pdf · V. Firoiu and M. Borden, “A study of active queue management for congestion control,”](https://reader033.fdocuments.in/reader033/viewer/2022052102/603c2df399693850c041f152/html5/thumbnails/57.jpg)
August 18, 2006 IEEE International Workshop on Complex Systems and Networks 57
0 10 20 30 40 50 60 70 800
5
10
15
20
25
30
Time (sec)
Win
dow
siz
e (p
acke
ts)
0 10 20 30 40 50 60 70 800
5
10
15
20
25
30
Time (sec)
Win
dow
siz
e (p
acke
ts)
Model validation: pmax
� pmax = [0.05, 0.95], with other parameters default
� window size: waveforms, pmax = 0.5
S-TCP/RED ns-2
![Page 58: , Alfredo Marciello algorithmsiwcsn2006.irmacs.sfu.ca/pdf/ljiljana.trajkovic_edited.pdf · V. Firoiu and M. Borden, “A study of active queue management for congestion control,”](https://reader033.fdocuments.in/reader033/viewer/2022052102/603c2df399693850c041f152/html5/thumbnails/58.jpg)
August 18, 2006 IEEE International Workshop on Complex Systems and Networks 58
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
2
4
6
8
10
Pmax
Ave
rage
que
ue s
ize
(pac
kets
)S-RED modelns-2
Model validation: pmax
� average queue size during steady state:
![Page 59: , Alfredo Marciello algorithmsiwcsn2006.irmacs.sfu.ca/pdf/ljiljana.trajkovic_edited.pdf · V. Firoiu and M. Borden, “A study of active queue management for congestion control,”](https://reader033.fdocuments.in/reader033/viewer/2022052102/603c2df399693850c041f152/html5/thumbnails/59.jpg)
August 18, 2006 IEEE International Workshop on Complex Systems and Networks 59
Model validation: pmax
� Comparison of system variables:
14.370.650.747.60357.55384.63-0.8535.134.80.75
19.090.610.731.48379.37384.983.8034.035.30.50
11.280.590.650.05384.73384.935.8034.536.50.25
2.560.550.560.06384.77384.9810.8336.039.90.10
-11.760.510.450.11384.70385.1316.2738.144.30.05
Δ(%)ns-2S-RED model Δ(%)ns-2S-RED
modelΔ(%)ns-2S-RED modelpmax
Drop rate (%)Sending rate (packets/sec)Average RTT (msec)
![Page 60: , Alfredo Marciello algorithmsiwcsn2006.irmacs.sfu.ca/pdf/ljiljana.trajkovic_edited.pdf · V. Firoiu and M. Borden, “A study of active queue management for congestion control,”](https://reader033.fdocuments.in/reader033/viewer/2022052102/603c2df399693850c041f152/html5/thumbnails/60.jpg)
August 18, 2006 IEEE International Workshop on Complex Systems and Networks 60
0 10 20 30 40 50 60 70 800
5
10
15
20
25
30
35
40
Time (sec)
Win
dow
siz
e (p
acke
ts)
0 10 20 30 40 50 60 70 800
5
10
15
20
25
30
35
40
Time (sec)
win
dow
siz
e (p
acke
ts)
� qmin= [1, 20] packets, qmax/qmin = 3, with other parameters default� window size: waveforms, qmin = 10 packets
S-TCP/RED ns-2
Model validation: qmin and qmax
![Page 61: , Alfredo Marciello algorithmsiwcsn2006.irmacs.sfu.ca/pdf/ljiljana.trajkovic_edited.pdf · V. Firoiu and M. Borden, “A study of active queue management for congestion control,”](https://reader033.fdocuments.in/reader033/viewer/2022052102/603c2df399693850c041f152/html5/thumbnails/61.jpg)
August 18, 2006 IEEE International Workshop on Complex Systems and Networks 61
0 2 4 6 8 10 12 14 16 18 200
5
10
15
20
25
qmin (packets)
Ave
rage
que
ue s
ize
(pac
kets
)S-RED modelns-2
Model validation: qmin and qmax
� average queue size during steady state:
![Page 62: , Alfredo Marciello algorithmsiwcsn2006.irmacs.sfu.ca/pdf/ljiljana.trajkovic_edited.pdf · V. Firoiu and M. Borden, “A study of active queue management for congestion control,”](https://reader033.fdocuments.in/reader033/viewer/2022052102/603c2df399693850c041f152/html5/thumbnails/62.jpg)
August 18, 2006 IEEE International Workshop on Complex Systems and Networks 62
Model validation: qmin and qmax
� Comparison of system variables:
-5.660.160.150.09384.95385.308.3673.079.120
-10.710.220.200.06384.83385.0612.2760.367.715
-6.340.330.310.06384.85385.1013.7248.154.710
2.560.550.560.06384.77384.9810.8336.039.95
10.010.710.780.20382.44383.227.431.133.43
Δ(%) ns-2S-RED modelΔ(%)ns-2S-RED
modelΔ(%)ns-2S-RED model
qmin(packets)
Drop rate (%)Sending rate (packets/sec)Average RTT (msec)
![Page 63: , Alfredo Marciello algorithmsiwcsn2006.irmacs.sfu.ca/pdf/ljiljana.trajkovic_edited.pdf · V. Firoiu and M. Borden, “A study of active queue management for congestion control,”](https://reader033.fdocuments.in/reader033/viewer/2022052102/603c2df399693850c041f152/html5/thumbnails/63.jpg)
August 18, 2006 IEEE International Workshop on Complex Systems and Networks 63
S-TCP/RED: model evaluation
� Waveforms of the window size:� match the ns-2 simulation results
� The average queue size: � mismatch, but similar trend
� System variables RTT, sending rate, and drop rate:� reasonable agreement with ns-2 simulation results,
depending on the system parameters
![Page 64: , Alfredo Marciello algorithmsiwcsn2006.irmacs.sfu.ca/pdf/ljiljana.trajkovic_edited.pdf · V. Firoiu and M. Borden, “A study of active queue management for congestion control,”](https://reader033.fdocuments.in/reader033/viewer/2022052102/603c2df399693850c041f152/html5/thumbnails/64.jpg)
August 18, 2006 IEEE International Workshop on Complex Systems and Networks 64
Roadmap
� Introduction� TCP/RED congestion control algorithms: an overview� Discrete-time dynamical models of TCP Reno with RED:
� modeling assumptions� S-TCP/RED models with:
� one state variable� two state variables
� model validation and modifications� comparison of TCP/RED models
� Bifurcation and chaos phenomena in TCP/RED� discontinuity-induced bifurcations
� Conclusion and references
![Page 65: , Alfredo Marciello algorithmsiwcsn2006.irmacs.sfu.ca/pdf/ljiljana.trajkovic_edited.pdf · V. Firoiu and M. Borden, “A study of active queue management for congestion control,”](https://reader033.fdocuments.in/reader033/viewer/2022052102/603c2df399693850c041f152/html5/thumbnails/65.jpg)
August 18, 2006 IEEE International Workshop on Complex Systems and Networks 65
S-TCP/RED: modification
� The difference in the average queue size between S-TCP/RED model and ns-2 is due to the simplification of :
� Modification of : )1( >⋅= αα ba pp
p
ap
ba pp =
minmax
minmax qq
qqppb −−×=
if )( maxmin qqq <<
b
ba pcount
pp×−
=1
)( maxmin qqq <<if
minmax
minmax qq
qqppa −−×=
![Page 66: , Alfredo Marciello algorithmsiwcsn2006.irmacs.sfu.ca/pdf/ljiljana.trajkovic_edited.pdf · V. Firoiu and M. Borden, “A study of active queue management for congestion control,”](https://reader033.fdocuments.in/reader033/viewer/2022052102/603c2df399693850c041f152/html5/thumbnails/66.jpg)
August 18, 2006 IEEE International Workshop on Complex Systems and Networks 66
Modified S-TCP/RED model
� Modification: 8.1=α
1 2 3 4 5 6 7 8 9 10
x 10-3
0
2
4
6
8
10
wq
Ave
rage
que
ue s
ize
(pac
kets
)
S-RED modelS-RED modified modelns-2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
2
4
6
8
10
pmax
Ave
rage
que
ue s
ize
(pac
kets
)
S-RED modelS-RED modified modelns-2
0 2 4 6 8 10 12 14 16 18 200
5
10
15
20
25
qmin
Ave
rage
que
ue s
ize
(pac
kets
)
S-RED modelS-RED modified modelns-2
![Page 67: , Alfredo Marciello algorithmsiwcsn2006.irmacs.sfu.ca/pdf/ljiljana.trajkovic_edited.pdf · V. Firoiu and M. Borden, “A study of active queue management for congestion control,”](https://reader033.fdocuments.in/reader033/viewer/2022052102/603c2df399693850c041f152/html5/thumbnails/67.jpg)
August 18, 2006 IEEE International Workshop on Complex Systems and Networks 67
Roadmap
� Introduction� TCP/RED congestion control algorithms: an overview� Discrete-time dynamical models of TCP Reno with RED:
� modeling assumptions� S-TCP/RED models with:
� one state variable� two state variables
� model validation and modifications� comparison of TCP/RED models
� Bifurcation and chaos phenomena in TCP/RED� discontinuity-induced bifurcations
� Conclusion and references
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August 18, 2006 IEEE International Workshop on Complex Systems and Networks 68
Comparison: S-TCP/RED vs. M-model
� M-model:A discrete nonlinear dynamical model of TCP Renowith RED proposed by a research group from University of Maryland:P. Ranjan, E. H. Abed, and R. J. La, “Nonlinear instabilities in TCP-RED,” in Proc. IEEE INFOCOM 2002, New York, NY, USA, June 2002, vol. 1, pp. 249–258 and IEEE/ACM Trans. on Networking, vol. 12, no. 6, pp. 1079–1092, Dec. 2004.
� One state variable: average queue size
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August 18, 2006 IEEE International Workshop on Complex Systems and Networks 69
0 10 20 30 40 50 60 70 800
12
3
45
6
7
89
10
Time (sec)
Ave
rage
que
ue s
ize
(pac
kets
)
0 10 20 30 40 50 60 70 800
12
3
45
6
7
89
10
Time (sec)
Ave
rage
que
ue s
ize
(pac
kets
)
0 10 20 30 40 50 60 70 800123456789
10
Time (sec)
Ave
rage
que
ue s
ize
(pac
kets
)
� The waveform of the average queue size with default RED parameters:
S-TCP/RED M-model ns-2
Model comparison: default parameters
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August 18, 2006 IEEE International Workshop on Complex Systems and Networks 70
1 2 3 4 5 6 7 8 9 10x 10-3
0
2
4
6
8
10
Wq
Ave
rage
que
ue s
ize
(pac
kets
)
S-RED modelM-modelns-2
Model comparisons: wq
� wq = [0.001, 0.01], with other parameters default� average queue size during steady state:
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August 18, 2006 IEEE International Workshop on Complex Systems and Networks 71
Model comparisons: wq
� system variables:
RTT sending rate drop rate
1 2 3 4 5 6 7 8
x 10-3
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
Wq
RTT
(sec
)
S-RED modelM-Modelns-2
1 2 3 4 5 6 7 8
x 10-3
300
320
340
360
380
400
Wq
Sen
ding
rate
(pac
kets
/sec
)
S-RED modelM-Modelns-2
1 2 3 4 5 6 7 8
x 10-3
0
0.2
0.4
0.6
0.8
1
Wq
Dro
p ra
te (%
)
S-RED modelM-Modelns-2
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August 18, 2006 IEEE International Workshop on Complex Systems and Networks 72
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
2
4
6
8
10
Pmax
Ave
rage
que
ue s
ize
(pac
kets
)
S-RED modelM-modelns-2
Model comparisons: pmax
� pmax= [0.05, 0.95], with other parameters default� average queue size during steady state:
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August 18, 2006 IEEE International Workshop on Complex Systems and Networks 73
Model comparisons: pmax
� system variables:
RTT sending rate drop rate
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80
0.01
0.02
0.03
0.04
0.05
Pmax
RTT
(sec
)
S-RED modelM-Modelns-2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8100
150
200
250
300
350
400
450
Pmax
Sen
ding
rate
(pac
kets
/sec
)
S-RED modelM-Modelns-2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Pmax
Dro
p ra
te (%
)
S-RED modelM-Modelns-2
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August 18, 2006 IEEE International Workshop on Complex Systems and Networks 74
0 2 4 6 8 10 12 14 16 18 200
5
10
15
20
25
qmin (packets)
Ave
rage
que
ue s
ize
(pac
kets
)
S-RED modelM-modelns-2
Model comparisons: qmin and qmax
� qmin= [1, 20] packets, qmax/qmin =3, with other parameters default
� average queue size during steady state:
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August 18, 2006 IEEE International Workshop on Complex Systems and Networks 75
Model comparisons: qmin and qmax
� system variables:
RTT sending rate drop rate
2 4 6 8 10 12 14 16 18 200
0.010.020.030.040.050.060.070.080.090.1
0.110.12
qmin
RTT
(sec
)
S-RED modelM-Modelns-2
2 4 6 8 10 12 14 16 18 200
0.10.20.30.40.50.60.70.80.9
1
qmin
Dro
p ra
te (%
)
S-RED modelM-Modelns-2
2 4 6 8 10 12 14 16 18 20200220240260280300320340360380400
qmin
Sen
ding
rate
(pac
kets
/sec
)
S-RED modelM-Modelns-2
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August 18, 2006 IEEE International Workshop on Complex Systems and Networks 76
Model comparison: summary
� S-TCP/RED model captures dynamical details of TCP/RED� RTT, sending rate, and drop rate: S-TCP/RED model, in
general, matches the ns-2 simulation results better than the M-model
� M-model: average queue size� constant during steady-state� matches better the ns-2 simulation results
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August 18, 2006 IEEE International Workshop on Complex Systems and Networks 77
Roadmap
� Introduction� TCP/RED congestion control algorithms: an overview� Discrete-time dynamical models of TCP Reno with RED:
� modeling assumptions� S-TCP/RED models with:
� one state variable� two state variables
� model validation and modifications� comparison of TCP/RED models
� Bifurcation and chaos phenomena in TCP/RED� discontinuity-induced bifurcations
� Conclusion and references
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August 18, 2006 IEEE International Workshop on Complex Systems and Networks 78
TCP/RED: bifurcation and chaos
� Bifurcation diagrams for various values of the system parameters:
� queue weight: wq
� maximum drop probability: pmax
� queue thresholds: qmin and qmax (qmax/qmin = 3)� round-trip propagation delay: d
15 (packets)Maximum queue threshold (qmax)
4,000 (bytes)Packet size (M)
5 (packets)Minimum queue threshold (qmin)
0.1Maximum drop probability (pmax )
0.002Queue weight (wq )
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August 18, 2006 IEEE International Workshop on Complex Systems and Networks 79
Queue size vs. wq
� pmax = 0.1, qmin = 5, qmax = 15
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August 18, 2006 IEEE International Workshop on Complex Systems and Networks 80
Average queue size vs. wq
� pmax = 0.1, qmin = 5, qmax = 15
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August 18, 2006 IEEE International Workshop on Complex Systems and Networks 81
Queue size vs. pmax
� wqx = 0.04, qmin = 5, qmax = 15
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August 18, 2006 IEEE International Workshop on Complex Systems and Networks 82
Average queue size vs. pmax
� wq = 0.04, qmin = 5, qmax = 15
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August 18, 2006 IEEE International Workshop on Complex Systems and Networks 83
Queue size vs. qmin/qmax
� wq = 0.2, pmax = 0.1, qmax= 3 × qmin
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August 18, 2006 IEEE International Workshop on Complex Systems and Networks 84
Average queue size vs. qmin/qmax
� wq = 0.2, pmax = 0.1, qmax= 3 × qmin
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August 18, 2006 IEEE International Workshop on Complex Systems and Networks 85
Roadmap
� Introduction� TCP/RED congestion control algorithms: an overview� Discrete-time dynamical models of TCP Reno with RED:
� modeling assumptions� S-TCP/RED models with:
� one state variable� two state variables
� model validation and modifications� comparison of TCP/RED models
� Bifurcation and chaos phenomena in TCP/RED� discontinuity-induced bifurcations
� Conclusion and references
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August 18, 2006 IEEE International Workshop on Complex Systems and Networks 86
TCP/RED: bifurcation and chaos
� Bifurcation diagrams for various values of the system parameters:
� queue weight: wq
� maximum drop probability: pmax
� queue thresholds: qmin and qmax (qmax/qmin = 3)� round-trip propagation delay: d
15 (packets)Maximum queue threshold (qmax)
4,000 (bytes)Packet size (M)
5 (packets)Minimum queue threshold (qmin)
0.1Maximum drop probability (pmax )
0.002Queue weight (wq )
![Page 87: , Alfredo Marciello algorithmsiwcsn2006.irmacs.sfu.ca/pdf/ljiljana.trajkovic_edited.pdf · V. Firoiu and M. Borden, “A study of active queue management for congestion control,”](https://reader033.fdocuments.in/reader033/viewer/2022052102/603c2df399693850c041f152/html5/thumbnails/87.jpg)
August 18, 2006 IEEE International Workshop on Complex Systems and Networks 87
Average queue size vs. wq
� pmax = 0.1, qmin = 5, qmax = 15, and sstresh = 80
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August 18, 2006 IEEE International Workshop on Complex Systems and Networks 88
Average queue size vs. pmax
� wq = 0.01, qmin = 5, qmax = 15, and ssthresh = 20
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August 18, 2006 IEEE International Workshop on Complex Systems and Networks 89
Average queue size vs. qmin/qmax
� wq = 0.01, pmax = 0.1, qmax= 3 qmin, and ssthresh = 20
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August 18, 2006 IEEE International Workshop on Complex Systems and Networks 90
Average queue size vs. d
� wq = 0.01, pmax = 0.1, qmin= 5, qmax = 15, and ssthresh = 20
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August 18, 2006 IEEE International Workshop on Complex Systems and Networks 91
An analytical explanation
� Nonsmooth systems may exhibit discontinuity-induced bifurcations: a class of bifurcations unique to their nonsmooth nature
� These phenomena occur when a fixed point, cycle, or aperiodic attractor interacts nontrivially with one of the phase space boundaries where the system is discontinuous
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August 18, 2006 IEEE International Workshop on Complex Systems and Networks 92
� Standard: � SN (smooth saddle-node)� PD (smooth period-doubling)
� C-bifurcations or DIBs� PWS maps: border collisions of fixed points � PWS flows: discontinuous bifurcations of equilibriums� Grazing bifurcations of periodic orbits� Sliding bifurcations
Discontinuity-induced bifurcations: classification
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August 18, 2006 IEEE International Workshop on Complex Systems and Networks 93
Border collisions in PWS maps
� Consider a map of the form:
� A fixed point is undergoing a border-collision bifurcation at p=0 if:
• µ ∈ (-ε,0) ⇒ x* ∈ S1
• µ ∈ (0,ε) ⇒ x* ∈ S2
• µ = 0 ⇒ x* ∈ Σ
• DF1 ≠ DF2 on Σ
11
2
( , ), ( ) 0( , ), ( ) 0
k kk
k k
F x p H xx
F x p H x+
<= >
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August 18, 2006 IEEE International Workshop on Complex Systems and Networks 94
Classifying border collisions
� Several scenarios are possible when a border-collision occurs
� They can be classified by observing the map eigenvalues on both sides of the boundary
� The phenomenon can be illustrated by a very simple 1D map where the eigenvalues are the slopes of the map on both sides of the boundary
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August 18, 2006 IEEE International Workshop on Complex Systems and Networks 95
Persistence
µ < 0
µ = 0 µ > 0
0 µ
, 0, 0
x c xx
x c xα µβ µ
+ <→ + >
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August 18, 2006 IEEE International Workshop on Complex Systems and Networks 96
Non-smooth saddle-node
µ < 0
µ = 0 µ > 0
0 µ
, 0, 0
x c xx
x c xα µβ µ
+ <→ + >
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August 18, 2006 IEEE International Workshop on Complex Systems and Networks 97
Border-collisions in the TCP/RED model
� The analysis has focussed mostly on continuous maps� Recently proposed: further bifurcations are possible when
the map is piecewise with a gap� Complete classification method is available only for the
one-dimensional case� The TCP/RED case is a 2D map with a gap: its dynamics
resemble closely those observed in very different systems: the impact oscillator considered by Budd and Piiroinen, 2006
� They might be explained in terms of border-collision bifurcations of 2D discontinuous maps
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August 18, 2006 IEEE International Workshop on Complex Systems and Networks 98
Numerical evidence
Cascades of corner-impact bifurcations in a forced impact oscillator show a striking resemblance to the phenomena detected in the TCP/RED model. They were explained in terms of border-collisions of local maps with a gap.
C. J. Budd and P. Piiroinen, “Corner bifurcations in nonsmoothly forced impact oscillators,” to appear in Physica D, 2005.
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August 18, 2006 IEEE International Workshop on Complex Systems and Networks 99
Roadmap
� Introduction� TCP/RED congestion control algorithms: an overview� Discrete-time dynamical models of TCP Reno with RED:
� modeling assumptions� S-TCP/RED models with:
� one state variable� two state variables
� model validation and modifications� comparison of TCP/RED models
� Bifurcation and chaos phenomena in TCP/RED� discontinuity-induced bifurcations
� Conclusion and references
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August 18, 2006 IEEE International Workshop on Complex Systems and Networks 100
Conclusions
� We developed two discrete-time models for TCP Renowith RED
� S-TCP/RED models include:� slow start, congestion avoidance, fast retransmit,
timeout, elements of fast recovery, and RED� Proposed models were validated by comparing their
performance to ns-2 simulations� They capture the main features of the dynamical
behavior of TCP/RED communication algorithms
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August 18, 2006 IEEE International Workshop on Complex Systems and Networks 101
Conclusions
� S-TCP/RED models were used to study bifurcations and chaos in TPC/RED systems with a single connection
� Bifurcations diagrams were characterized by period-adding cascades and devil staircases
� The observed behavior can be explained in terms of a novel class of piecewise-smooth maps with a gap
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August 18, 2006 IEEE International Workshop on Complex Systems and Networks 102
References: TCP and RED
[1] M. Allman, V. Paxson, and W. Steven, “TCP congestion control,” IETF Request for Comments (RFC) 2581, Apr. 1999.
[2] B. Barden et al., “Recommendations on queue management and congestion avoidance in the Internet,” IETF Request for Comments (RFC) 2309, Apr. 1998.
[3] R. Braden, “Requirements for Internet hosts: communication layers,” IETF Request for Comments (RFC) 1122, Oct. 1989.
[4] C. Casetti, M. Gerla, S. Lee, S. Mascolo, and M. Sanadidi, “TCP with fasterrecovery,” in Proc. MILCOM 2000, Los Angeles, CA, USA, Oct. 2000, vol. 1, pp. 320–324.
[5] K. Fall and S. Floyd, “Simulation-based comparison of Tahoe, Reno, and SACK TCP,” ACM Communication Review, vol. 26, no. 3, pp. 5–21, July 1996.
[6] V. Jacobson, “Congestion avoidance and control,” ACM Computer Communi-cation Review, vol. 18, no. 4, pp. 314–329, Aug. 1988.
[7] V. Jacobson, “Modified TCP congestion avoidance algorithm,”ftp://ftp.isi.edu/end2end/end2end-interest-1990.mail, Apr. 1990.
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References: TCP and RED
[8] J. Padhye and S. Floyd, “On inferring TCP behavior,” in Proc. ACM SIGCOMM 2001, San Diego, CA, USA, Aug. 2001, pp. 287–298.
[9] V. Paxson and M. Allman, “Computing TCP's retransmission timer,” IETF Request for Comments (RFC) 2988, Nov. 2000.
[10] J. Postel, “Transmission control protocol,” IETF Request for Comments (RFC) 793, Sept. 1981.
[11] W. R. Stevens, TCP/IP Illustrated, Volume 1: The protocols. New York, NY: Addison-Wesley, 1994.
[12] S. Floyd, “RED: discussions of setting parameters,” Nov. 1997: http://www.icir.org/floyd/REDparameters.txt.
[13] S. Floyd and V. Jacobson, “Random early detection gateways for congestion avoidance,” IEEE/ACM Trans. Networking, vol. 1, no. 4, pp. 397–413, Aug. 1993.
[14] S. Floyd, R. Gummadi, and S. Shenker, “Adaptive RED: an algorithm for increasing the robustness of RED’s active queue management,” Aug. 2001: http://www.icir.org/floyd/papers/.
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August 18, 2006 IEEE International Workshop on Complex Systems and Networks 104
References: TCP/RED models
[15] N. Cardwell, S. Savage, and T. Anderson, “Modeling TCP latency,” in Proc. IEEE INFOCOM 2000, Tel Aviv, Israel, Mar. 2000, vol. 3, pp. 1742–1751.
[16] V. Firoiu and M. Borden, “A study of active queue management forcongestion control,” in Proc. IEEE INFOCOM 2000, Tel-Aviv, Israel, Mar.2000, vol. 3, pp. 1435–1444.
[17] J. Padhye, V. Firoiu, and D. F. Towsley, “Modeling TCP Reno performance: a simple model and its empirical validation,” IEEE/ACM Trans. Networking, vol. 8, no. 2, pp. 133–145, Apr. 2000.
[18] C. V. Hollot, V. Misra, D. Towsley, and W. B. Gong, “A control theoretic analysis of RED,” in Proc. IEEE INFOCOM 2001, Anchorage, AK, Apr. 2001, vol. 3, pp. 1510–1519.
[19] C. V. Hollot, V. Misra, D. Towsley, and W. B. Gong, “Analysis and design of controllers for AQM routers supporting TCP flows,” IEEE Trans. on Automatic Control, vol. 47, no. 6, pp. 945–959, June 2002.
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August 18, 2006 IEEE International Workshop on Complex Systems and Networks 105
References: TCP/RED models
[20] P. Ranjan and E. H. Abed, “Bifurcation analysis of TCP-RED dynamics,” in Proc. ACC , Anchorage, AK, USA, May 2002, vol. 1, pp. 2443–2448.
[21] P. Ranjan, E. H. Abed, and R. J. La, “Nonlinear instabilities in TCP- RED,” in Proc. IEEE INFOCOM 2002, New York, NY, USA, June 2002, vol. 1, pp. 249–258.
[22] R. J. La, P. Ranjan, and E. H. Abed, “Nonlinearity of TCP and instability with RED,”in Proc. SPIE ITCom, Boston, MA, USA, July 2002, pp. 283–294.
[23] P. Ranjan, R. J. La, and E. H. Abed, “Bifurcations of TCP and UDP traces under RED,” in Proc. 10th Mediterranean Conference on Control and Automation (MED) 2002, Lisbon, Portugal, July 2002.
[24] P. Ranjan and E. H. Abed, “Chaotic behavior in TCP-RED,” in Proc. 41st IEEE Conference on Decision and Control, Las Vegas, NE, Dec. 2002, pp. 540–542.
[25] P. Ranjan, E. H. Abed, and R. J. La “Communication delay and instability in rate-controlled networks,” in Proc. 42nd Conference on Decision and Control, Maui, HI, Dec. 2003.
[26] R. J. La, “Instability of a tandem network and its propagation under RED,” IEEE Trans. Automatic Control, vol. 49, no. 6, pp. 1006–1011, June 2004.
[27] P. Ranjan, E. H. Abed, and R. J. La, "Nonlinear instabilities in TCP-RED, IEEE/ACM Trans. on Networking, vol. 12, no. 6, pp. 1079–1092, Dec. 2004.
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August 18, 2006 IEEE International Workshop on Complex Systems and Networks 106
References: TCP/RED models
[28] I. Khalifa and Lj. Trajković, “An overview and comparison of analytical TCP models,” in Proc. IEEE International Symposium on Circuits and Systems, Vancouver, BC, Canada, May 2004, vol. V, pp. 469–472.
[29] M. Liu, H. Zhang, and Lj. Trajković, “Stroboscopic model and bifurcations in TCP/RED,” in Proc. IEEE International Symposium on Circuits and Systems, Kobe, Japan, May 2005, pp. 2060–2063.
[30] H. Zhang, M. Liu, V. Vukadinović, and Lj. Trajković, “Modeling TCP/RED: a dynamical approach,” Complex Dynamics in Communication Networks, Springer Verlag, Series: Understanding Complex Systems, 2005, pp. 251–278.
[31] M. Liu, A. Marciello, M. di Bernardo, and Lj. Trajković, “Continuity-induced bifurcations in TCP/RED communication algorithms,” in Proc. IEEE International Symposium on Circuits and Systems, Kos, Greece, May 2006, pp. 2629–2632.
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August 18, 2006 IEEE International Workshop on Complex Systems and Networks 107
References: bifurcations
[32] C. J. Budd and P. Piiroinen, “Corner bifurcations in nonsmoothly forced impact oscillators,” to appear in Physica D, 2005.
[33] P. Jain and S. Banerjee, “Border-collision bifurcations in one-dimensional discontinuous maps,” Int. Journal Bifurcation and Chaos, vol. 13. pp. 3341–3351, 2003.
[34] S. J. Hogan, L. Higham, and T. C. L. Griffin, “Dynamics of a piecewise linear map with a gap,'' to appear in Proc. Royal Society, London, 2005.
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August 18, 2006 IEEE International Workshop on Complex Systems and Networks 108
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