✬ ✩ Transport Equations for Internet Transmission Control F. Baccelli INRIA and ENS ICERM, October 17, 2011 ✫ ✪
✬ ✩ 1 Summary Isolated TCP flows – Persistent Flows – On-Off Flows Interaction of Parallel TCP Flows – Persistent Flows – On-Off Flows Interaction of TCP Flows in Series ✫ ✪ F. Baccelli
✬ ✩ 2 Persistent Flows – Dynamics of TCP – Square root formula – Markov analysis – Distributions ✫ ✪ F. Baccelli
✬ ✩ 3 TCP Congestion Control IP Network Source Destination PACKETS ACKS – Error control Each packet received by the destination is acknowledged; – Congestion control The number of unacknowledged pack- ets in transit in the network is limited by the source to a maximal value W called the window. If the Round Trip Time (RTT) is R , the throughput of the connection is X = W R ✫ ✪ F. Baccelli
✬ ✩ 4 Congestion Avoidance Phase of TCP TCP dynamic window size (updated when acks are received) w n +1 = g ( w n , F ( n )) , F ( n ) : feedback signal on the state of congestion, TCP Reno: AIMD � w n � g ( w n , OK ) = w n + 1 every w n acks , g ( w n , LOSS ) = 2 Scalable TCP: MIMD Kelly 03 g ( w n , OK ) = w n + a, g ( w n , LOSS ) = ⌊ w n b ⌋ , 0 < 1 < b. TCP Tahoe, HighSpeed TCP Floyd 03, Fast TCP Low 05 ..... ✫ ✪ F. Baccelli
✬ ✩ 5 Hybrid Model for TCP RENO Congestion Avoidance AI: the window is increased of 1 unit every W ack: – In dt , the number of acks that arrive is X ( t ) dt ; – Hence the window increases of X ( t ) dt/W ( t ) = dt/R . MD: in case of a loss event, the windows is cut by half. Differential equation, with N ( t ) the loss point process: dW ( t ) = dt R − W ( t − ) dX ( t ) = dt R 2 − X ( t − ) N ( dt ) N ( dt ) 2 2 First studied by M. Mathis, J. Semke, J. Mahdavi and T. Ott, 97 (square root formula) ✫ ✪ F. Baccelli
✬ ✩ 6 Loss Point Processes Losses are modeled by two kinds of point processes N ( t ) : 1. rate independent (RI) case: homogeneous Poisson point process with intensity λ 2. rate dependent (RD) case: point process with a stochastic intensity pX ( t ) Rationale 1. RI: losses caused by physical layer events arising on wire- less links (fast fading) or DSL links (impulse noise) 2. RD: PER (packet error rate) due to congestion or trans- mission errors. ✫ ✪ F. Baccelli
✬ ✩ 7 Loss Point Processes ( continued ) �� �� �� �� � � �� �� �� �� � � � � � � � � � � � � �� �� �� �� �� �� �� �� �� �� � � �� �� �� �� �� �� �� �� y y � � �� �� �� �� �� �� �� �� �� �� �� �� � � � � � � �� �� � � �� �� �� �� � � �� �� �� �� � � � � � � � � � � �� �� � � � � �� �� �� � � �� �� � � � � �� � � 0 t 0 t ✫ ✪ F. Baccelli
✬ ✩ 8 The Square Root Formula in 3 Lines - RI case If there exists a stationary regime with X integrable, the Rate Conservation Principle gives R 2 = λ 1 2 E 0 N [ X (0 − )] with E 0 N the Palm probability of N ; Pasta implies E 0 N [ X (0 − )] = E [ X (0)] , which gives 2 E [ X (0)] = λR 2 The packet loss probability p is such that p E [ X (0)] = λ ; Hence � 2 E [ X (0)] = pR 2 . ✫ ✪ F. Baccelli
✬ ✩ 9 RD Case If there exists a stationary regime where N has intensity λ and X is integrable, the Rate Conservation Principle gives R 2 = λ 1 2 E 0 N [ X (0 − )] N [ X (0 − )] = E [ X (0) pX (0) From Papangelou’s theorem E 0 λ ] so that 2 E [ X (0) 2 ] = pR 2 No simple identification of the mean TCP throughput. ✫ ✪ F. Baccelli
✬ ✩ 10 Markov Analysis X ( t ) is a Markov Process – with continuous time – with continuous state space It falls in the Piecewise Deterministic Process framework of Davis. The embedded chain (at ”discontinuities”) is geometrically ergodic. ✫ ✪ F. Baccelli
✬ ✩ 11 Analytical Results: Distributions - RI Case For all u > 0 for all continuity point of X ( t ) , X u ( t ) = uX u − 1 ( t ) ˙ ˙ X ( t ) = uX u − 1 ( t ) /R 2 so that � � � t � t uX u − 1 ( v ) 1 − 1 X u ( t ) = X u (0) + X u ( v − ) N ( dv ) dv − R 2 2 u 0 0 Thus � � � t � t M ( t ) = X u ( t ) − X u (0) − u 1 − 1 X u − 1 ( v ) dv + λ X u ( v − ) dv R 2 2 u 0 0 is a martingale s.t. M (0) = 0 so that whenever moments are finite � � ∂t E [ X u ( t )] = u ∂ 1 − 1 R 2 E [ X u − 1 ( t )] − λ E [ X u ( t )] . 2 u ✫ ✪ F. Baccelli
✬ ✩ 12 Analytical Results: Distributions - RI Case ( continued ) Mellin transforms of the density of X at time t : � ∞ z u f ( t, z ) dz = � E [ X u ( t )] = f t ( u + 1) . 0 Functional equation: � � ∂ f t ( u + 1) = u 1 − 1 � R 2 � � f t ( u ) − λ f t ( u + 1) 2 u ∂t PDE ∂f ( z, t ) + 1 ∂f ( z, t ) + λ ( f ( z, t ) − 2 f (2 z, t )) = 0 R 2 ∂t ∂x ✫ ✪ F. Baccelli
✬ ✩ 13 Analytical Results: Distributions - RI Case ( continued ) Stationary ODE: d f ( z ) + ξ ( f ( z ) − 2 f (2 z )) = 0 dz with ξ = λR 2 . Stationary functional equation: � � 1 − 1 u � � f ( u ) = ξ f ( u + 1) 2 u � f ( u ) = g ( u )Γ( u ) ξ − u . Then � g ( u ) = g ( u + 1)(1 − 2 − u ) , (1 − 2 − u − k ) , i.e. g ( u ) = g ( ∞ ) k ≥ 0 ✫ ✪ F. Baccelli
✬ ✩ 14 Analytical Results: Distributions - RI Case ( continued ) Theorem The unique stationary distribution solution of this functional equation has for Mellin transform f ( u ) = φ Γ( u ) ξ − u � � (1 − 2 − u − k ) k ≥ 0 �� � − 1 . with ξ = λR 2 and φ = ξ k ≥ 1 (1 − 2 − k ) The associated probability density is � b n e − ( ξ 2 n ) z f ( z ) = φ n ≥ 0 n � 2 with b 0 = 1 and b n = ( − 1) n (2 k − 1) . k =1 ✫ ✪ F. Baccelli
✬ ✩ 15 Distributions - RD Formal proof of PDE by the same martingale approach: ∂f ∂t ( z, t ) + 1 ∂f ∂z ( z, t ) = p (4 zf (2 z, t ) − zf ( z, t )) , z ≥ 0 , R 2 – mass leaves the interval [ z, z + dz ] at rate pzf ( z, t ) dz approx- imately. – mass enters this interval because of losses among through- puts in the interval [2 z, 2( z + dz )] at rate p 2 zf (2 z, t ) · 2 dz Functional equation for the stationary Mellin transform of f : � 1 − 2 − u � u � f ( u ) = ξ � f ( u + 2) . with ξ = pR 2 . ✫ ✪ F. Baccelli
✬ ✩ 16 Distributions - RD ( continued ) Theorem The unique density satisfying the ODE is � ξ 2 4 n � � z 2 − f ( z ) = 2 φ a n e n ≥ 0 with − 1 � 2 � 1 n 2 � � √ π 4 (1 − 2 − 2 k +1 ) a n = ( − 1) n φ = and (4 k − 1) . ξ k ≥ 1 k =1 Its Mellin transform is � � 2 � u � u ∞ � � 1 − 2 − u − 2 k � 2 � f ( u ) = φ Γ Π ∞ ( u ) , with Π( u ) = 2 ξ k =0 Its mean is � � 2 1 Π(1) ∼ 1 . 309 Π(2) E [ X (0)] = R √ p. pR 2 π ✫ ✪ F. Baccelli
✬ ✩ 17 Scalable TCP Distributions - RD SDE: (with α = a R ): dX ( t ) = αX ( t ) dt − (1 − b ) X ( t ) N ( dt ) PDE: � 1 � ∂f ( x, t ) + αx∂f ( x, t ) = − pxf ( x, t ) − αf ( x, t ) + 1 b 2 pxf bx, t , ∂t ∂x ODE � 1 � αxd f ( x ) = − pxf ( x ) − αf ( x ) + 1 b 2 pxf bx . dx ✫ ✪ F. Baccelli
✬ ✩ 18 Scalable TCP Distributions - RD ( continued ) Mean throughput: Ott 05, Altman 05 a E [ X ] = − pR log b. Mellin of stationary distribution: � α � u − 1 � � 1 − b u + k − 1 � ˆ f ( u ) = ΨΓ( u − 1) . p k ≥ 0 Distribution: � f ( x ) = Ψ1 n ) x , c n e − ( pR a ( 1 b ) x n ≥ 0 where − 1 n � � b − 1 � 1 − b k +1 � 1 c n = ( − 1) n and Ψ = . (1) ( b − k − 1) log 1 b k =1 k ≥ 0 ✫ ✪ F. Baccelli
✬ ✩ 19 Comparison RENO–Scalable TCP −3 x 10 3 AIMD AIMD MIMD MIMD 0.016 2.5 0.014 2 0.012 Stationary pdf Stationary pdf 0.01 1.5 0.008 1 0.006 0.004 0.5 0.002 0 0 200 400 600 800 1000 1200 1400 1600 1800 2000 0 100 200 300 400 500 600 700 800 Throughput X [pkts] Jump size [pkts] Left: throughput density; Right: jump size density Parameters: p = 0 . 001 , R = 100 ms , a = 0 . 01 , b = 0 . 875 Jump size: for all bounded measurable φ ( · ) : � ∞ E 0 [ φ ( X − )] = E [ φ ( X ) pX ] 1 = φ ( x ) f ( x ) xdx, p E [ X ] E [ X ] 0 ✫ ✪ F. Baccelli
✬ ✩ 20 ON-OFF Flows On-Off TCP flow – PDE for the exponential Model – Moments – Distributions ✫ ✪ F. Baccelli
✬ ✩ 21 Dynamics RD model with packet loss probability p and RTT R . The flow alternates between document downloads and think times, inducing an ON/OFF flow structure – Document sizes F i are i.i.d. with mean 1 /µ – Think times T i are i.i.d. with mean 1 /β Motivation: HTTP 1.1 where the files successively down- loaded by a flow use the same TCP-Reno connection: – Slow Start jump approximation: jumps J i are i.i.d. ✫ ✪ F. Baccelli
✬ ✩ 22 X(t): throughput of the flow Slope 2 1/R J2 J1 F1 F2 T1 �� �� �� � �� t N ( dt ): packet loss point process, with a stochastic intensity pX ( t ). ✫ ✪ F. Baccelli
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