Stability Analysis of TCP/RED Communication Algorithms

Transcription

1 Stability Analysis of TCP/RED Communication Algorithms Ljiljana Trajković Simon Fraser University, Vancouver, Canada

2 Collaborators Mingjian Liu and Hui Zhang School of Engineering Science Simon Fraser University, Vancouver, Canada Alfredo Marciello and Mario di Bernardo University of Naples Federico II, Naples, Italy Xi Chen, Siu-Chung Wong, and Chi K. Tse Department of Electronic and Information Engineering The Hong Kong Polytechnic University, Hong Kong October 3, 2008 IWCSN 2008, ANU, Canberra 2

3 Roadmap Introduction TCP/RED congestion control algorithms: an overview Discrete-time dynamical models of TCP Reno with RED Discontinuity-induced bifurcations in TCP/RED Stability analysis of RED gateway with multiple TCP Reno connections Stability study of TCP/RED system using detrended fluctuation analysis References October 3, 2008 IWCSN 2008, ANU, Canberra 3

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 investigate stability of the TCP/RED system TCP: Transmission Control Protocol RED: Random Early Detection Gateways for Congestion Avoidance October 3, 2008 IWCSN 2008, ANU, Canberra 4

5 Roadmap Introduction TCP/RED congestion control algorithms: an overview Discrete-time dynamical models of TCP Reno with RED Discontinuity-induced bifurcations in TCP/RED Stability analysis of RED gateway with multiple TCP Reno connections Stability study of TCP/RED system using detrended fluctuation analysis References October 3, 2008 IWCSN 2008, ANU, Canberra 5

6 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) October 3, 2008 IWCSN 2008, ANU, Canberra 6

7 TCP Reno CWND SS: Slow Start CA: Congestion Avoidance TO: Timeout TO ssthresh1 ssthresh2 SS CA CA Fast retransmit and fast recovery SS t October 3, 2008 IWCSN 2008, ANU, Canberra 7

8 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 > ssthresh cwnd = cwnd + 1/cwnd for each ACK cwnd : congestion window size IW : initial window size ssthresh : slow start threshold ACK : acknowledgement RTT : round trip time October 3, 2008 IWCSN 2008, ANU, Canberra 8

9 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 three duplicate ACKs TCP sender TCP receiver ACK 16 ACK 16 ACK 16 ACK 16 ACK 16 October 3, 2008 IWCSN 2008, ANU, Canberra 9

10 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 TO 1 1 ACK1 October 3, 2008 IWCSN 2008, ANU, Canberra 10

11 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, October 3, 2008 IWCSN 2008, ANU, Canberra 11

12 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 , Aug Main concept: drop packets before the queue becomes full October 3, 2008 IWCSN 2008, ANU, Canberra 12

13 RED variables and parameters Main variables and parameters: average queue size: instantaneous queue size: drop probability: queue weight: w q p k+1 maximum drop probability: min queue thresholds: and q q k +1 q k+1 q p max max October 3, 2008 IWCSN 2008, ANU, Canberra 13

14 RED algorithm Calculate: average queue size for each packet arrival q k + 1 = ( 1 wq ) qk + wq qk + 1 drop probability p k+1 1 p max qmin qmax q k +1 October 3, 2008 IWCSN 2008, ANU, Canberra 14

15 RED algorithm: drop probability if else if else ( q ) min < qk + 1 < qmax q q p k+ 1 min k+ 1 = pmax qmax qmin ( q k + 1 qmax) p k+1 =1 ( ) min q k + 1 q p k+1 = 0 mark or drop the arriving packet with probability p k +1 October 3, 2008 IWCSN 2008, ANU, Canberra 15

16 Network model TCP senders Bottleneck link TCP receivers Bottleneck gateway October 3, 2008 IWCSN 2008, ANU, Canberra 16

17 TCP window congestion control algorithm Sender sends W packets at a time Window size = W Additive increase (AI): if no loss, window size increases by one per round trip time Multiplicative decrease (MD): on detection of loss, window size decreases by half October 3, 2008 IWCSN 2008, ANU, Canberra 17

18 TCP window congestion control algorithm Sender sends W packets at a time Window size = W Additive increase (AI): if no loss, window size increases by one per round trip time Multiplicative decrease (MD): on detection of loss, window size decreases by half receiver sender W W+1 October 3, 2008 IWCSN 2008, ANU, Canberra 18

19 TCP window congestion control algorithm Sender sends W packets at a time Window size = W Additive increase (AI): if no loss, window size increases by one per round trip time Multiplicative decrease (MD): on detection of loss, window size decreases by half receiver sender W W/2 October 3, 2008 IWCSN 2008, ANU, Canberra 19

20 RED algorithm Average queue length: x = (1 α) x + αq α q k k k 1 k : :queue averaging weight 0< <1 : :current queue size α Marking/dropping probability: x 0 0 x < X k min pmax Xmin xk Xmax Xmax Xmin pb = x X p (1 p ) X < x 2X 1 2Xmax < xk B p b p k = 1 c p k max max max max k max Xmax Xmin October 3, 2008 IWCSN 2008, ANU, Canberra 20 X m b k min

21 RED marking/dropping probability no dropping or marking mark with p linearly increasing from Pmax to 1 1 p b Xmin Xmax 2Xmax mark with p linearly increasing from 0 to Pmax drop with p=1 p max X min X max 2X max B x average queue length drop probability p October 3, 2008 IWCSN 2008, ANU, Canberra 21

22 RED gateway: small queue length Send Xmax Xmin Receiver Send Receiver p b Send 1 Receiver p max X min X max 2X max B x October 3, 2008 IWCSN 2008, ANU, Canberra 22

23 RED gateway: small queue length Send Xmax Xmin Receiver Send Receiver p b Send 1 Receiver p max X min X max 2X max B x October 3, 2008 IWCSN 2008, ANU, Canberra 23

24 RED gateway: target queue length Send Xmax Xmin Receiver Send Receiver p b Send 1 Receiver p max X min X max 2X max B x October 3, 2008 IWCSN 2008, ANU, Canberra 24

25 RED gateway: target queue length Send Xmax Xmin Receiver Send Receiver p b Send 1 Receiver p max X min X max 2X max B x October 3, 2008 IWCSN 2008, ANU, Canberra 25

26 RED gateway: large queue length Send Xmax Xmin Receiver Send Receiver p b Send 1 Receiver p max X min X max 2X max B x October 3, 2008 IWCSN 2008, ANU, Canberra 26

27 Roadmap Introduction TCP/RED congestion control algorithms: an overview Discrete-time dynamical models of TCP Reno with RED Discontinuity-induced bifurcations in TCP/RED Stability analysis of RED gateway with multiple TCP Reno connections Stability study of TCP/RED system using detrended fluctuation analysis References October 3, 2008 IWCSN 2008, ANU, Canberra 27

28 Modeling methodology Categories of TCP models: averaged and discrete-time models short-lived and long-lived TCP connections TCP/RED model: discrete-time model with a long-lived connection State variables: window size (TCP) average queue size (RED) October 3, 2008 IWCSN 2008, ANU, Canberra 28

29 TCP/RED model Key properties of the proposed TCP/RED model: 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 , Apr Captures the basic RED algorithm October 3, 2008 IWCSN 2008, ANU, Canberra 29

30 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 October 3, 2008 IWCSN 2008, ANU, Canberra 30

31 TCP/RED model simplifications Simplified fast recovery cwnd SS: Slow Start CA: Congestion Avoidance TO: Timeout Fast retransmit and fast recovery TO ssthresh 2 ssthresh 1 t SS CA CA SS October 3, 2008 IWCSN 2008, ANU, Canberra 31

32 TCP Reno: fast recovery CWND SS: Slow Start CA: Congestion Avoidance TO: Timeout TO ssthresh1 ssthresh2 SS CA CA Fast retransmit and fast recovery SS t October 3, 2008 IWCSN 2008, ANU, Canberra 32

33 TCP/RED model simplifications TO = 5 RTT V. Firoiu and M. Borden, A study of active queue management for congestion control, in Proc. of IEEE INFOCOM 2000, vol. 3, pp , Tel-Aviv, Israel, Mar RED: parameter count is not used if q < q < q ) ( min max if ( q < q < q ) max min p b p a = p max q q max pb = 1 count q q p b min min p a = p b p a = p max q q max q q min min October 3, 2008 IWCSN 2008, ANU, Canberra 33

34 Simple 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 and IEEE/ACM Trans. on Networking, vol. 12, no. 6, pp , Dec One state variable: average queue size The proposed TCP/RED model is: simple and intuitively derived able to capture detailed dynamical behavior of TCP/RED systems has been verified via ns-2 simulations October 3, 2008 IWCSN 2008, ANU, Canberra 34

35 Simple S-TCP/RED model: state variable and parameters Variables: q k+1 : average queue size in round k+1 q k : average queue size in round k w q : queue weight in RED N: number of TCP connections K: constant = 3 / 2 p k : 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 October 3, 2008 IWCSN 2008, ANU, Canberra 35

36 October 3, 2008 IWCSN 2008, ANU, Canberra 36 Simple S-TCP/RED model: case 1 Drop probability: where: : TCP sending rate : the number of incoming packets : the number of outgoing packets M d C p N K C M q d M C N RTT RTT p K q M RTT C N RTT p B q q k k k k k k k k k k k = + + = + = ) ( ) ( p k ) ( k p B N RTT p B k k ) +1 ( M RTT C k 1 +

37 Simple S-TCP/RED model: case 1 The average queue size is: q k + 1 = ( 1 wq ) qk + wq qk+ 1 hence q k + 1 = (1 w q ) q k + w q N max( K p k C d M, 0) October 3, 2008 IWCSN 2008, ANU, Canberra 37

38 October 3, 2008 IWCSN 2008, ANU, Canberra 38 Simple S-TCP/RED model: case 2 Drop probability: The average queue size is: hence M d C N rwnd C M q d M C N RTT RTT rwnd q M RTT C N RTT p B q q k k k k k k k k k = + + = + = ) ( ) ( = 0 p k ) ( ) (1 1 M d C N rwnd w q w q q k q k + = ) 1 ( = k q k q k q w q w q

39 October 3, 2008 IWCSN 2008, ANU, Canberra 39 Simple S-TCP/RED model Dynamical model of TCP/RED: = + + = + 0 ) ( ) (1 0 0), max( ) (1 1 k q k q k k q k q k p if M d C N rwnd w q w p if M d C p K N w q w q

40 Validation: simulation scenario 100 Mbps 0 ms delay source router Mbps 10 ms delay 100 Mbps 0 ms delay router 2 sink 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 October 3, 2008 IWCSN 2008, ANU, Canberra 40

41 RED: default parameters RED parameters: S. Floyd, RED: Discussions of Setting Parameters, Nov. 1997: Queue weight (w q ) Maximum drop probability (p max ) Minimum queue threshold (q min ) Maximum queue threshold (q max ) Packet size (M) (packets) 15 (packets) 4,000 (bytes) October 3, 2008 IWCSN 2008, ANU, Canberra 41

42 TCP/RED model validation Waveforms of the state variable with default parameters: average queue size Validation for various values of the system parameters: queue weight: w q maximum drop probability: p max queue thresholds: q min and q max, q max /q min = 3 October 3, 2008 IWCSN 2008, ANU, Canberra 42

43 Average queue size: waveforms Average queue size (packets) Time (sec) Average queue size (packets) Time (sec) TCP/RED model ns-2 October 3, 2008 IWCSN 2008, ANU, Canberra 43

44 Model validation: w q average queue size during steady state: October 3, 2008 IWCSN 2008, ANU, Canberra 44

45 Model validation: w q Comparison of system variables: Parameter Average RTT (msec) Sending rate (packets/sec) Drop rate (%) weight (w q ) TCP/RED model ns-2 TCP/RED model ns-2 TCP/RED model ns October 3, 2008 IWCSN 2008, ANU, Canberra 45

46 Model validation: p max average queue size during steady state: Average queue size (packets) ns-2 proposed model Pmax October 3, 2008 IWCSN 2008, ANU, Canberra 46

47 Model validation: p max Comparison of system variables: Parameter Average RTT (msec) Sending rate (packets/sec) Drop rate (%) p max TCP/RED model ns-2 TCP/RED model ns-2 TCP/RED model ns October 3, 2008 IWCSN 2008, ANU, Canberra 47

48 Model validation: q min and q max average queue size during steady state: Average queue size (packets) ns-2 proposed Model Minimum queue threshold, qmin (packets) October 3, 2008 IWCSN 2008, ANU, Canberra 48

49 Model validation: q min and q max Comparison of system variables: Average RTT (msec) Sending rate (packets/sec) Drop rate (%) q min (packets) TCP/RED model ns-2 TCP/RED model ns-2 TCP/RED model ns October 3, 2008 IWCSN 2008, ANU, Canberra 49

50 TCP/RED: model evaluation Waveforms of the average queue size: match the ns-2 simulation results Sending rate: reasonable agreement with ns-2 simulation results Average RTT and drop rate: disagreement with ns-2 simulation results October 3, 2008 IWCSN 2008, ANU, Canberra 50

51 Queue size vs. w q p max = 0.1, q min = 5, q max = 15 October 3, 2008 IWCSN 2008, ANU, Canberra 51

52 Average queue size vs. w q p max = 0.1, q min = 5, q max = 15 October 3, 2008 IWCSN 2008, ANU, Canberra 52

53 Queue size vs. p max w qx = 0.04, q min = 5, q max = 15 October 3, 2008 IWCSN 2008, ANU, Canberra 53

54 Average queue size vs. p max w q = 0.04, q min = 5, q max = 15 October 3, 2008 IWCSN 2008, ANU, Canberra 54

55 Queue size vs. q min /q max w q = 0.2, p max = 0.1, q max = 3 q min October 3, 2008 IWCSN 2008, ANU, Canberra 55

56 Average queue size vs. q min /q max w q = 0.2, p max = 0.1, q max = 3 q min October 3, 2008 IWCSN 2008, ANU, Canberra 56

57 S-TCP/RED model S-model: a discrete nonlinear dynamical model of TCP Reno with RED Two state variables: window size average queue size The proposed TCP/RED model is: simple and intuitively derived able to capture detailed dynamical behavior of TCP/RED systems has been verified via ns-2 simulations October 3, 2008 IWCSN 2008, ANU, Canberra 57

58 S-TCP/RED model: state variable and parameters q k+1 : instantaneous queue size in round k+1 q k+1 : average queue size in round k+1 W k+1 : current TCP window size in round k+1 w q : queue weight in RED p k : drop probability in round k RTT k+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 October 3, 2008 IWCSN 2008, ANU, Canberra 58

59 October 3, 2008 IWCSN 2008, ANU, Canberra 59 S-TCP/RED model: no packet loss, details current queue size: M d C W C M q d M C W q M RTT C W q q k k k k k k k k = + + = + = ) ( average queue size:,0) max( ) 1 ( 1 1 M d C W w q w q k q k q k + = + +

60 S-TCP/RED model: no packet loss drop probability: p k W k < 0.5 window size: W k + 1 min(2w = min( Wk average queue size: k, ssthresh) + 1, rwnd) if if W W k k < ssthresh ssthresh q k + 1 = + Wk + 1 Wk+ 1 ( 1 wq ) qk + (1 (1 wq ) ) max( Wk 1 C d M, 0) October 3, 2008 IWCSN 2008, ANU, Canberra 60

61 S-TCP/RED model: one packet loss drop probability: 0.5 p k W < 1.5 k window size: 1 W k +1 = W k 2 average queue size: q k + 1 Wk+ 1 Wk + 1 ( 1 w ) q + (1 (1 w ) ) max( Wk 1 = q k q + C d M, 0) October 3, 2008 IWCSN 2008, ANU, Canberra 61

62 S-TCP/RED model: two packet losses drop probability: p k W k 1.5 window size: W k +1 = 0 average queue size: q k+1 = q k October 3, 2008 IWCSN 2008, ANU, Canberra 62

63 Conclusion We developed discrete-time models for TCP Reno with RED TCP/RED models include: slow start, congestion avoidance, fast retransmit, timeout, elements of fast recovery, and RED Proposed models were validated by comparing its performance with ns-2 simulation results They capture the main features of the dynamical behavior of TCP Reno with RED These models were used to study bifurcation and chaos in TPC/RED systems with a single connection October 3, 2008 IWCSN 2008, ANU, Canberra 63

64 Roadmap Introduction TCP/RED congestion control algorithms: an overview Discrete-time dynamical models of TCP Reno with RED Discontinuity-induced bifurcations in TCP/RED Stability analysis of RED gateway with multiple TCP Reno connections Stability study of TCP/RED system using detrended fluctuation analysis References October 3, 2008 IWCSN 2008, ANU, Canberra 64

65 RED: default parameters RED parameters: S. Floyd, RED: Discussions of Setting Parameters, Nov. 1997: Queue weight (w q ) Maximum drop probability (p max ) Minimum queue threshold (q min ) Maximum queue threshold (q max ) Packet size (M) (packets) 15 (packets) 4,000 (bytes) October 3, 2008 IWCSN 2008, ANU, Canberra 65

66 TCP/RED: bifurcation and chaos Bifurcation diagrams for various values of the system parameters: queue weight: w q maximum drop probability: p max queue thresholds: q min and q max (q max /q min = 3) round-trip propagation delay: d Queue weight (w q ) Maximum drop probability (p max ) Minimum queue threshold (q min ) Maximum queue threshold (q max ) Packet size (M) (packets) 15 (packets) 4,000 (bytes) October 3, 2008 IWCSN 2008, ANU, Canberra 66

67 Average queue size vs. w q p max = 0.1, q min = 5, q max = 15, and sstresh = 80 October 3, 2008 IWCSN 2008, ANU, Canberra 67

68 Average queue size vs. p max w q = 0.01, q min = 5, q max = 15, and ssthresh = 20 October 3, 2008 IWCSN 2008, ANU, Canberra 68

69 Average queue size vs. q min /q max w q = 0.01, p max = 0.1, q max = 3 q min, and ssthresh = 20 October 3, 2008 IWCSN 2008, ANU, Canberra 69

70 Average queue size vs. d w q = 0.01, p max = 0.1, q min = 5, q max = 15, and ssthresh = 20 October 3, 2008 IWCSN 2008, ANU, Canberra 70

71 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 October 3, 2008 IWCSN 2008, ANU, Canberra 71

72 Discontinuity-induced bifurcations: classification Standard: SN (smooth saddle-node) PD (smooth period-doubling) C-bifurcations or discontinuity-induced bifurcations PWS maps: border collisions of fixed points PWS flows: discontinuous bifurcations of equilibriums Grazing bifurcations of periodic orbits Sliding bifurcations October 3, 2008 IWCSN 2008, ANU, Canberra 72

73 Border collisions in PWS maps Consider a map of the form: F1 ( xk, p), H( xk) < 0 xk + 1 = F2 ( xk, p), H( xk) > 0 A fixed point is undergoing a border-collision bifurcation at p=0 if: µ (-ε,0) x * S 1 µ (0,ε) x * S 2 µ = 0 x * Σ DF 1 DF 2 on Σ October 3, 2008 IWCSN 2008, ANU, Canberra 73

74 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 October 3, 2008 IWCSN 2008, ANU, Canberra 74

75 Persistence µ < 0 x α x+ cµ, x < 0 β x+ cµ, x > 0 0 µ µ = 0 µ > 0 October 3, 2008 IWCSN 2008, ANU, Canberra 75

76 Non-smooth saddle-node µ < 0 x α x+ cµ, x < 0 β x+ cµ, x > 0 0 µ µ = 0 µ > 0 October 3, 2008 IWCSN 2008, ANU, Canberra 76

77 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 onedimensional 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 October 3, 2008 IWCSN 2008, ANU, Canberra 77

78 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, October 3, 2008 IWCSN 2008, ANU, Canberra 78

79 TCP/RED bifurcations: conclusions Discrete-time one and two-dimensional models capture the main features of the dynamical behavior of TCP/RED communication algorithms These 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 October 3, 2008 IWCSN 2008, ANU, Canberra 79

80 Roadmap Introduction TCP/RED congestion control algorithms: an overview Discrete-time dynamical models of TCP Reno with RED Discontinuity-induced bifurcations in TCP/RED Stability analysis of RED gateway with multiple TCP Reno connections Stability study of TCP/RED system using detrended fluctuation analysis References October 3, 2008 IWCSN 2008, ANU, Canberra 80

81 Fluid-flow model of TCP-RED Window size: Instantaneous queue length: dw() t dt dq() t dt = 1 r() t additive increase wt () 2 wt () = N C rt () incoming traffic wt ( rt ( )) rt ( rt ( )) multiplicative decrease outgoing traffic p( t r( t)) loss arrival rate Round trip time: rt () = qt () C queuing delay + R 0 propagation delay October 3, 2008 IWCSN 2008, ANU, Canberra 81

82 Fluid-flow model of TCP-RED Average queue length: dx dt ( α ) ( x() t q() t ) ln 1 = δ α δ : queue averaging weight : sampling rate ~1/C Marking/dropping probability: xt () X 0 0 xt ( ) < X min pmax Xmin x() t Xmax Xmax Xmin pb () t = xt () X p (1 p ) X < x( t) 2X 1 X max < xt () B max max max max max Xmax Xmin min October 3, 2008 IWCSN 2008, ANU, Canberra 82

83 Marking/dropping probability p b 1 p max X min X max 2X max B x October 3, 2008 IWCSN 2008, ANU, Canberra 83

84 Steady-state solution and target queue length Let N(t) N, C(t) C, and p() t = xt () X X X max min min p max Equilibrium point ( q, r, w, p ) : q 0 = X max + X φ min q0 r0 = + R0 C Cr0 w0 = N X 1 + (1 φ) X p p φ(1 ) X min max 0 = max = X min Cr0 max 2 N 2 October 3, 2008 IWCSN 2008, ANU, Canberra 84

85 Linearization and characteristic equation 2 N 1 rc 0 δwt () = ( δwt () + δwt ( r 2 0)) + ( δqt () δqt ( r 2 0)) + δ pt ( r 2 0) r0 C r0 C 2N N 1 δqt () = δwt () δqt () r0 r0 δ pt () = Cln(1 α)( δ pt () βδqt ()) where: δ w= w w0 δ q = q q0 δ p= p p0 pmax β = X X max min Characteristic equation in Laplace domain: N 2 2N α C α N 2α N N 2 Nα C α β r0 r0 C r0 C r0 r0 r0 r0 C r0 2N λ + + α λ + λ + λ λ = λr0 ( C) ( ) ( ) e 0 where : α = ln(1 α) 1 October 3, 2008 IWCSN 2008, ANU, Canberra 85

86 Characteristic equation With Padé(1,1) approximation: we obtain: e λr r0 λ 2 r0 λ 2 where: λ λ λ λ a1 + a2 + a3 + a4 = 0 3 a1 = ( α1) C Κ 6N 2 3α 1 2 a2 ( ) C 3 2 = Κ + Κ Κ 4N 6Nα1 2α1 2α1N a3 = ( + ) C Κ Κ Κ ΦΚ 1 α1 4 a4 = 4 Nα1( ) C 4 Κ ΦΚ K = Cr0 Φ = Xmax Xmin October 3, 2008 IWCSN 2008, ANU, Canberra 86 3

87 Stability conditions for TCP/RED The Routh-Hurwitz stability criterion provides the necessary and sufficient stability conditions for the approximated system in terms of network parameters (N, α, K, Φ): 4N 6Nα1 2α1N 3 6N 2 ( 2 α 2 1+ )[( α1)( + 3 α 2 1) Κ Κ Φ Κ Κ Κ 4N 6Nα 2α 2α > Κ Κ Κ ΦΚ Κ Κ Φ ( )] 4 Nα 3 2 1( α1) ( ) 0 October 3, 2008 IWCSN 2008, ANU, Canberra 87

88 Stable region (K, N, α) Unstable Stable Φ = 256 packets October 3, 2008 IWCSN 2008, ANU, Canberra 88

89 Stable region (K, N, α) Φ = 128 packets Φ = 256 packets Φ = 384 packets October 3, 2008 IWCSN 2008, ANU, Canberra 89

90 Stable and unstable waveforms of queue length: ns-2 simulations (1) Stable Unstable K = packets (r 0 = 64 ms), Φ = 256 packets, α = 0.01 K = packets (r 0 = 73 ms), Φ = 256 packets, α = 0.01 October 3, 2008 IWCSN 2008, ANU, Canberra 90

91 Stable and unstable waveforms of queue length: ns-2 simulations (2) Stable Unstable K = packets (r 0 = 64 ms), Φ = 256 packets, α = 0.01 K = packets (r 0 = 75 ms), Φ = 256 packets, α = 0.01 October 3, 2008 IWCSN 2008, ANU, Canberra 91

92 Comparison of simulation methods and approximations ns ns-2 simulation S Full fluid flow simulation Stable Unstable SL SL1 SL2 A1 A2 Linearized fluid flow simulation Linearized fluid flow simulation with Padé(0,1) approximation Linearized fluid flow simulation with Padé(1,1) approximation Analytical closed form solution with Padé(0,1) approximation Analytical closed form solution with Padé(1,1) approximation Φ = 256 packets and α = October 3, 2008 IWCSN 2008, ANU, Canberra 92

93 Comparisons: K = Cr o Capacity vs. Round Trip Time: N = 500, 600, and 800 Φ = 800 packets q 0 = 500 packets α = October 3, 2008 IWCSN 2008, ANU, Canberra 93

94 Comparisons: varying α Φ = 256 packets and q 0 = 384 packets October 3, 2008 IWCSN 2008, ANU, Canberra 94

95 Comparisons: varying N ns-2 results vs. analytical solution for Φ = 256 packets and q 0 = 384 packets SIMULINK results vs. analytical solution for Φ = 256 packets and q 0 = 384 packets October 3, 2008 IWCSN 2008, ANU, Canberra 95

96 Comparisons: varying Φ N = 256, α = and q 0 = 384 packets October 3, 2008 IWCSN 2008, ANU, Canberra 96

97 Appendix 1 October 3, 2008 IWCSN 2008, ANU, Canberra 97

98 Stable queue length waveforms: Simulink K = packets (r 0 = 51 ms), Φ = 256 packets, α = 0.01 October 3, 2008 IWCSN 2008, ANU, Canberra 98

99 Stable queue length: Simulink K = packets (r 0 = 51 ms), Φ = 256 packets, α = 0.01 October 3, 2008 IWCSN 2008, ANU, Canberra 99

100 Unstable queue length waveforms: Simulink K = packets (r 0 = 58 ms), Φ = 256 packets, α = 0.01 October 3, 2008 IWCSN 2008, ANU, Canberra 100

101 Unstable queue length: Simulink K = packets (r 0 = 58 ms), Φ = 256 packets, α = 0.01 October 3, 2008 IWCSN 2008, ANU, Canberra 101

102 Appendix 2 October 3, 2008 IWCSN 2008, ANU, Canberra 102

103 Stable and unstable queue length waveform: ns-2 (2) Stable or not? K = packets (r 0 = 155 ms), Φ = 256 packets, α = FFT of average queue for first 100 seconds K = packets (r 0 = 155 ms), Φ = 256 packets, α = October 3, 2008 IWCSN 2008, ANU, Canberra 103

104 Stable and unstable queue length waveform: ns-2 (2) Unstable K = packets (r 0 = 155 ms), Φ = 256 packets, α = FFT of average queue for first 100 seconds K = packets (r 0 = 155 ms), Φ = 256 packets, α = October 3, 2008 IWCSN 2008, ANU, Canberra 104

105 Stable and unstable queue length waveform: ns-2 (2) Stable or not? K = packets (r 0 = 145 ms), Φ = 256 packets, α = FFT of average queue for first 100 seconds K = packets (r 0 = 145 ms), Φ = 256 packets, α = October 3, 2008 IWCSN 2008, ANU, Canberra 105

106 Stable and unstable queue length waveform: ns-2 (2) Stable K = packets (r 0 = 145 ms), Φ = 256 packets, α = FFT of average queue for first 100 seconds K = packets (r 0 = 145 ms), Φ = 256 packets, α = October 3, 2008 IWCSN 2008, ANU, Canberra 106

107 Stability analysis: conclusions We have derived stability boundaries of the RED scheme based on an analytical closed-form solution using the Padé(1,1) linearized fluid flow models Very good match between the stability boundaries found from the Padé(1,1) linearized fluid flow model and ns-2 simulations has been achieved The model (verified by ns-2) can be used to predict dynamical behavior of the system October 3, 2008 IWCSN 2008, ANU, Canberra 107

108 Roadmap Introduction TCP/RED congestion control algorithms: an overview Discrete-time dynamical models of TCP Reno with RED Discontinuity-induced bifurcations in TCP/RED Stability analysis of RED gateway with multiple TCP Reno connections Stability study of TCP/RED system using detrended fluctuation analysis Conclusions and References October 3, 2008 IWCSN 2008, ANU, Canberra 108

109 Detrended fluctuation analysis (DFA) DFA method was first proposed for determining the statistical self-affinity of a signal: *C-K Peng, SV Buldyrev, S Havlin, M Simons, HE Stanley, and AL Goldberger, Mosaic organization of DNA nucleotides, Phys. Rev. E 49: , DFA method has been applied in the analysis of DNA nucleotides, fractals, electrocardiogram (ECG/ EKG), electroencephalography (EEG), climate, and stock market. October 3, 2008 IWCSN 2008, ANU, Canberra 109

110 Detrended fluctuation analysis (DFA) DFA method: permits the detection of intrinsic self-similarity embedded in a seemingly nonstationary time series it avoids the spurious detection of apparent self-similarity, which may be an artifact of extrinsic trends Note: A stationary time series is characterized by its mean, standard deviation, higher moments, and correlation functions being invariant under time translation. Signals that are not stationary are nonstationary. October 3, 2008 IWCSN 2008, ANU, Canberra 110

111 DFA computation Consider signal s(i) of length of L Step 1: integrate s(i) and obtain i yi () = [ s( j) s] j= 1 s 1 L L j = 1 [()] s j Step 2: divide y(i) into boxes of equal length of l Step 3: subtracting the local trend y l (i) for box of length l to detrended y(i): Y() l y() i y () i l Step 4: for each box l, the characteristic size of fluctuation for the integrated and detrended time series is calculated by: L 1 2 Fl () [ Y( j)] L j = 1 Repeat Steps 2-4 for several length of l (different scales) October 3, 2008 IWCSN 2008, ANU, Canberra 111 l l

112 DFA method 1. Reconstruction of the signal 2. Splitting of the record into segments of scale length l 3. Regression in each segment 4 Calculation of variance Y(j) in each segment and then taking averaging to obtain F(l) October 3, 2008 IWCSN 2008, ANU, Canberra 112

113 DFA method (cont.) Scale invariant signals with power-law correlations between the scaling function F(l) and the scale l : Fl () l β, l β represents the degree of the correlation October 3, 2008 IWCSN 2008, ANU, Canberra 113

114 Non stationarity and instability in TCP/RED system DFA method helps quantitatively describe the level of the instability of TCP/RED systems October 3, 2008 IWCSN 2008, ANU, Canberra 114

115 Self-similarity in TCP/RED system Unstable Stable October 3, 2008 IWCSN 2008, ANU, Canberra 115

116 DFA results: instability October 3, 2008 IWCSN 2008, ANU, Canberra 116

117 Interpretation of DFA: the waveform viewpoint October 3, 2008 IWCSN 2008, ANU, Canberra 117

118 DFA exponents: sine and saw tooth October 3, 2008 IWCSN 2008, ANU, Canberra 118

119 DFA method: conclusions Stability of the queue length has been explored using the DFA method The degree of instability can be described by DFA exponent, which varies with the relative stability of RED gateway We provided an interpretation of the relationship between the DFA exponent and the stability of RED system The DFA exponent may used as indicator for TCP/RED system and thus control the stability of the system October 3, 2008 IWCSN 2008, ANU, Canberra 119

120 Roadmap Introduction TCP/RED congestion control algorithms: an overview Discrete-time dynamical models of TCP Reno with RED Discontinuity-induced bifurcations in TCP/RED Stability analysis of RED gateway with multiple TCP Reno connections Stability study of TCP/RED system using detrended fluctuation analysis References October 3, 2008 IWCSN 2008, ANU, Canberra 120

121 References: TCP/RED and the model [1] V. Jacobson, Congestion avoidance and control, ACM Computer Communication Review, vol. 18, no. 4, pp , Aug [2] M. Allman, V. Paxson, and W. Steven, TCP congestion control, IETF Request for Comments, RFC 2581, Apr [3] S. Floyd and V. Jacobson, Random early detection gateways for congestion avoidance, IEEE/ACM Trans. Networking, vol. 1, no. 4, pp , Aug [4] V. Firoiu and M. Borden, A study of active queue management for congestion control, in Proc. IEEE INFOCOM 2000, Tel-Aviv, Israel, Mar. 2000, vol. 3, pp [5] J. Padhye, V. Firoiu, and D. F. Towsley, Modeling TCP Reno performance: a sample model and its empirical validation, IEEE/ACM Trans. Networking, vol. 8, no. 2, pp , Apr [6] Khalifa and Lj. Trajkovic, An overview and comparison of analytical TCP models, in Proc. IEEE Int. Symp. Circuits and Systems, Vancouver, BC, Canada, May 2004, vol. V, pp October 3, 2008 IWCSN 2008, ANU, Canberra 121

122 References: bifurcations [7] C. J. Budd and P. Piiroinen, Corner bifurcations in nonsmoothly forced impact oscillators, to appear in Physica D, [8] P. Jain and S. Banerjee, Border-collision bifurcations in one-dimensional discontinuous maps, Int. Journal Bifurcation and Chaos, vol. 13. pp , [9] 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, [10] P. Ranjan and E. H. Abed, Bifurcation analysis of TCP-RED dynamics, in Proc. ACC, Anchorage, AK, USA, May 2002, pp [11] 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 [12] R. J. La, Instability of a tandem network and its propagation under RED, IEEE Trans. Automatic Control, vol. 49, no. 6, pp , June [13] P. Ranjan, E. H. Abed, and R. J. La, "Nonlinear instabilities in TCP-RED, IEEE/ACM Trans. on Networking, vol. 12, no. 6, pp , Dec October 3, 2008 IWCSN 2008, ANU, Canberra 122

123 References: M. Liu, H. Zhang, and Lj. Trajkovic, Stroboscopic model and bifurcations in TCP/RED, in Proc. IEEE Int. Symp. Circuits and Systems, Kobe, Japan, May 2005, pp H. Zhang, M. Liu, V. Vukadinovic, and Lj. Trajkovic, Modeling TCP/RED: a dynamical approach, Complex Dynamics in Communication Networks, Springer Verlag, Series: Understanding Complex Systems, 2005, pp M. Liu, A. Marciello, M. di Bernardo, and Lj. Trajkovic, Discontinuity-induced bifurcations in TCP/RED communication algorithms, Proc. IEEE Int. Symp. Circuits and Systems, Kos, Greece, May 2006, pp X. Chen, S.-C. Wong, C. K. Tse, and Lj. Trajkovic, Stability analysis of RED gateway with multiple TCP Reno connections, Proc. IEEE Int. Symp. Circuits and Systems, New Orleans, LA, May 2007, pp X. Chen, S.-C. Wong, C. K. Tse, and Lj. Trajkovic, Stability study of the TCP-RED system using detrended fluctuation analysis, Proc. IEEE Int. Symp. Circuits and Systems, Seattle, WA, May 2008, pp October 3, 2008 IWCSN 2008, ANU, Canberra 123