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Performance models of TCP can simulate (ns-2) + faithful to operation of TCP - expensive, time consuming deterministic approximations + quick - ignore some TCP details, steady state fluid models + transient behavior - ignore some TCP details TCP behavior TCP runs at end-hosts congestion control: decrease sending rate when loss detected, increase when no loss routers discard, mark packets when congestion occurs interaction between end systems (TCP) and routers? want to understand congested router drops packets (quantify) this interaction Generic TCP behavior window algorithm (window W) up to W packets in network return of ACK allows sender to send another packet cumulative ACKS increase window by one per RTT W <- W +1/W per ACK W <- W +1 per RTT seeks available network bandwidth receiver W sender Generic TCP behavior window algorithm (window W) increase window by one per RTT W <- W +1/W per ACK loss indication of congestion decrease window by half on detection of loss, (triple duplicate ACKs), W <- W/2 receiver TD sender Generic TCP Behavior window algorithm (window W) increase window by one per RTT W <- W +1/W per ACK halve window on detection of loss, W <- W/2 timeouts due to lack of ACKs -> window reduced to one, W <- 1 receiver sender TO Generic TCP Behavior window algorithm (window W) increase window by one per RTT (or one over window per ACK, W <- W +1/W) halve window on detection of loss, W <- W/2 timeouts due to lack of ACKs, W <- 1 successive timeout intervals grow exponentially long up to six times TCP throughput/loss relationship loss occurs Idealized model: W W is maximum supportable window size (then loss TCP occurs) window TCP window starts at W/2 size grows to W, then halves, W/2 then grows to W, then halves… one window worth of packets each RTT to find: throughput as time (rtt) function of loss, RTT TCP throughput/loss relationship # packets sent per “period” = W TCP window size W/2 period time (rtt) TCP throughput/loss relationship # packets sent per “period” = W W W /2 W W + + 1 + ... + W ( + n) 2 2 n 0 2 TCP W W W /2 window + 1 + n size 2 2 n 0 W/2 W W W / 2(W / 2 + 1) + 1 + period 2 2 2 3 3 W2 + W 8 4 3 2 time (rtt) W 8 TCP throughput/loss relationship 3 2 # packets sent per “period” W 8 W 1 packet lost per “period” implies: TCP window 8 8 ploss or: W size 3W 2 3 ploss W/2 period 3 packets B avg._thrup W ut 4 rtt 1.22 packets B avg._thrup ut ploss rtt time (rtt) B throughput formula can be extended to model timeouts and slow start (PFTK) Recall RED queue management dropping/marking packets depends on average queue length -> p = p(x) 1 Marking probability p More generally: active queue management pmax (AQM) 0 tmin tmax 2tmax Average queue length x Bottleneck behavior bottleneck router: capacity fully utilized all interfering sessions see same loss prob. do all sessions see same thruput? i Bi (RTTi ,p) = C C - router bandwidth Bi - throughput of flow i Single bottleneck: infinite flows N infinite TCP sessions two way propagation delay Ai, i = 1,…,N throughput Bi(p,RTTi) one bottleneck router RED queue management • avg. queue length x ; dropping probability p(x) to discover Bi: TCP sessions’ throughput, router behavior, e.g., drop prob. avg. queue len. Model and solution model p = p(x) (AQM) RTTi = Ai + x /C (round trip time) i Bi (x) = C, for i =1 ,…,N i B (p , RTT i) = C, for i =1 ,…,N solve a fixed point problem for x unique solution provided B is monotonic and continuous on x resulting x can be used to obtain RTTi and p Model versus simulation: single bottleneck, infinite flows • fixed router capacity 4 Mbps and RED parameters • 10-120 TCP flows • two-way prop. delay 20+2i ms, i = 1,…,N throughput router loss Bottleneck principle: a qualitative result Bnew(p) new/improved, Bnew(p) thruput TCP, BTCP(p) BTCP(p) Bnew(p) > BTCP(p) p Sharing bottleneck with TCP C Nnew NTCP Nnew Bnew(p) + NTCP Bni(p) = C Bnew(p) > BTCP(p) a win! friendly? p Replacing TCP with TCP-new N Bnew(pnew) = C C vs N N BTCP(pTCP) = C pnew > pTCP a loss! pTCP pnew simple model for TCP B c , c ≈ 1.2 T p bottleneck principle multiple bottlenecks fluid models Multiple Bottleneck: infinite flows N TCP flows throughputs B = <Bi (Ri,pi)> V congested AQM routers capacities C = <Cv > avg. queue lengths x = <xv > discard prob. p = <pv (xv )> bottleneck router model i Bi (x ) = Cv , v =1,…,V V equations, V unknowns Results: multiple bottleneck, infinite flows • tandem network core, 5 - 10 routers • 2-way propagation delay 20-120 ms • bandwidth, 2-6 Mbps throughput • PFTK model error • throughput < 10% • loss rate < 10% • avg. queue length < 15% • similar results for cyclic networks router loss Comments what about UDP / non-TCP flows? If there are “non-responsive” flows, just decrease bottleneck capacity by non-responsive flow rate what about short lived flows? Hard (some work in sigcomm 2001 – massoulie) note: throughout, assumption that time to send packets in window is less that RTT Dynamic (transient) analysis of TCP fluids model TCP traffic as fluid describe behavior of flows and queues using Ordinary Differential Equations solve resulting ODEs numerically Loss Model AQM Router B(t) Packet Drop/Mark p(t) Sender Receiver Round Trip Delay (t) Loss Rate as seen by Sender: l(t) = B(t-t)*p(t-t) A Single Congested Router focus on single bottlenecked router TCP flow i capacity {C (packets/sec) } queue length q(t) discard prob. p(t) N TCP flows thru router window sizes Wi(t) round trip time AQM router Ri(t) = Ai+q(t)/C C, p throughputs Bi (t) = Wi(t)/Ri(t) Adding RED to the model RED: Marking/dropping based on average queue length x(t) 1 Marking probability p pmax tmin tmax 2tmax Average queue length x - q(t) - x(t) x(t): smoothed, time averaged q(t) t -> System of Differential Equations Timeouts and slow start ignored Window Size: dWk 1 - p ( x (t - t k )) - Wk (t ) Wk (t - t k ) p ( x (t - t k )) dt Rk (t ) 2 Rk (t - t k ) Additive Mult. Loss arrival increase decrease rate dq Wk (t ) Queue length: -C 1{q > 0} + k dt Rk (t ) Outgoing Incoming traffic traffic System of Differential Equations (cont.) Average smoothed dx ln(1 - a ) ln a queue length: x (t ) - q (t ) dt d d Where a = averaging parameter of RED(wth) d = sampling interval ~ 1/C dp dp dx Loss probability: dt dx dt Where dp is obtained from the marking profile dx N+2 coupled equations N flows dWi dt f1 p, Ri , Wi ), i 1, , N Wi(t) = Window size of flow i Ri(t) = RTT of flow i dp dt f 3 q ) dq dt f 2 Wi ) p(t) = Drop probability q(t) = queue length Equations solved numerically using MATLAB Steady slate behavior let t → ∞ dWk 0, p(t ) p, W (t ) W , Rk (t ) Rk dt this yields 1 - p Wk Wk 2(1 - p) 0 - p or Wk Rk 2 Rk p the throughput is 2(1 - p) 2 Bk for small p Rk p Rk p A Queue is not a Network Network - set of AQM routers, V sequence Vi for session i Round trip time - aggregate delay Ri(t) = Ai + vVi qv(t) Loss/marking probability - cumulative prob pi (t) = 1-Pv Vi (1 - pv(qv(t))) Link bandwidth constraints Queue equations How well does it work? OC-12 – OC-48 links RED with target delay 5msec OC-12 2600 TCP flows OC-48 decrease to 1300 at 30 sec. 2600 j 2600 j 1300 j increase to 2600 at 90 sec. t=30 t=90 simulation fluid model instantaneous delay time (sec) Good queue length match average window size simulation fluid model window size simulation fluid model time (sec) time (sec) matches average window size Scaling Properties OC-12 OC-12 j OC-48 OC-48 j 2600 1300 2600 2600 j 2600 j 1300 j t=30 t=90 t=30 t=90 Wk(t) = Wj k(t) qv(t) = qjv(t)/100 Summary: TCP flows as fluids What have we seen? model TCP as constant rate fluid flows rate sensitive to congestion via: capacities C = <Cv > avg. queue lengths x = <xv > discard prob. p = <pv (xv )> dynamic (transient) behavior of TCP modeled as system of differential equations ability to predict performance of system of TCP flows using fluid models

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