Convergence of Discrete Models of TCP Congestion Avoidance. Olga I. - PowerPoint PPT Presentation

Convergence of Discrete Models of TCP Congestion Avoidance. Olga I. Bogoiavlenskaia PetrSU, Department of Computer Science olbgvl@cs.karelia.ru

TCP Congestion Control • The paradigm of Distributed Control in Packet Switching Network • Transmission Control Program, 1974. • Congestion collapse • Variance not important yet 1

TCP Congestion Control Development • Jitter sensitive applications • TCP vs UDP • High BDP links utilization vs Congestion Control • Best effort vs QoS guarantees 2

Variety of Experimental Versions • TCP CUBIC - cubical growth period. RTT independent • High Speed TCP (HSTCP), S. Floyd 2003. Congestion Avoidance co- eff. of linear growth and multiplicative decrease are convex functions of current window size • Scalable TCP (STCP) T. Kelly, 2003. Decreases time of data recovery • TCP Hybla 2003-04. Developed for satellite links. Scales throughput to mimic NewReno and utilize link at the same time. • TCP-YeAH 3

Two mainstream modeling approaches to TCP begavior t Figure 1: The step-wise random process of the congestion window size. 4

Two mainstream modeling approaches to TCP begavior Figure 2: The piecewise linear random process of the congestion window size. 5

Definitions Let’s t n — denote ends of TCP rounds [ t n − 1 , t n ] is RTT and ξ n = t n − t n − 1 is RTT length. Let’s w ( t ) — denotes cwnd. We define stepwise process { w ( t ) } such that  � w ( t n ) � , if during [ t n − 1 , t n ]   α  w ( t n + 0) = TCP lost data,   w ( t n ) + 1 , if all data delivered.  Between moments t n the process { w ( t ) } stays constant. 6

Definitions Lets { X ( t ) } t> 0 takes values from R + and for the intervals [ θ n , θ n +1 ) n = 0 , 1 , . . . grows linearly with the speed b = E [ ξ n ] − 1 , . . X ( t ) = X ( t 0 ) + bt, ∀ [ t 0 , t ] ⊂ [ θ n , θ n +1 ) . At random moments { θ n } n ≥ 0 the process { X ( t ) } t> 0 makes a jump X ( θ n + 0) = X ( θ n ) /α, α > 1 . We assume that the sequence { θ n } n ≥ 0 forms poisson flow with param- eter 0 < λ < ∞ . 7

Convergence Theorem We propose following transformation of coordinates for the process { w ( t ) } t = ns w = ⌊ nX ⌋ . (1) Lets consider following sequence of stepwise processes w n ( s ) = w ( ns ) : λ n = λ/n. We denote w n (0) = ⌊ nx 0 ⌋ X (0) = x 0 . Theorem 1 ∀ s takes place w ( ns ) lim = X ( s ) (2) n n →∞ � − 1 � ∞ � by distribution. And b = xdR ( x ) . 0 8

Proof Let us consider a growth period of X ( s ) . For the interval [ s 1 , s 2 ] , ⊂ [ θ n , θ n +1 ] it takes place X ( s ) = X ( s 1 ) + b ( s − s 1 ) . Let’s denote u n ( s, s 1 ) the number of the moments t m , happened in the interval [ ns 1 , ns ] . The sequence { t k } ∞ m =1 makes renewal process and according to Smith theorem E [ u n ( s, s 1 )] lim n ( s − s 1 ) = b (3) n →∞ Then according to Chebyshev inequality u n ( s, s 1 ) lim = b ( s − s 1 ) (4) n n →∞ by probability. 9

Proof Now denote w n,k = w n ( τ k − 0) and j n,k = u n ( σ n,k , σ n,k − 1 ) , where τ k = nσ n,k . � w n,k − 1 + j n,k = w n,k − 1 � w n,k = − γ n,k + j n,k , (5) α α where 0 ≤ γ n,k < 1 . The interval τ k − τ k − 1 = n ( σ n,k − σ n,k − 1 ) = π k + δ k , where η k = π k /n, has distribution F η k ( x ) = 1 − e − λs , and r.v. δ k /n converges by probability to zero. Henceforth n →∞ ( σ n,k − σ n,k − 1 ) = η k . lim (6) by distribution. 10

Proof Let nθ ′ n — the moment of the last jump of w n ( s ) before moment ns , ν n — its number and w n (0) = j n 0 . then   ν n ν n j n,ν n − i γ n,ν n − i w n ( s ) = 1  + u n ( s − θ ′ � � − n ) . (7)  α i α i α i =0 i =0 From (6) one infers that ν n → ν, by probability and ν satisfies Poisson distribution. n → θ ′ by distribution too, and θ ′ is the moment of the last Also θ ′ jump of the process X ( s ) before time moment s . Hence ν w ( ns ) = b η ν − i α i + b ( s − θ ′ ) = X ( s ) � lim (8) n α n →∞ i =0 by distribution, where η 0 = x 0 . 11

Conclusion • The Development of Congestion Control schemes and two main ap- proaches to its modelling are considered. • The sequence of the stepwise AIMD models is built. • For the sequence the convergence theorem is proved. • Further development: investigate the speed of the convergence. 12