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Modeling TCP Throughput Jitendra Padhye Victor Firoiu Don Towsley Jim Kurose Presented by Jaebok Kim A Simple Model and its Empirical Validation

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Introduction Simple analytic characterization of the steady state throughput –A stochastic model of TCP congestion control Deriving mathematical formulas –Taking account of not only retransmit but also timeout

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Contents TCP Congestion Avoidance Simplifying assumptions Loss indications & triple-duplicate ACKs Loss indications & triple-duplicate ACKs, time-outs Impact of window limitation & a full model Empirical validation Conclusion

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TCP Congestion Avoidance How do we resolve this problem?

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TCP Congestion Avoidance TCP Reno – a newer version Slow Start –W’ = W + 1 (each ACK arrives) –Eventually, doubling every RTT

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TCP Congestion Avoidance Additive Increase –W’ = W + 1/W (each ACK arrives) –W’’ = W + 1/B (Second round begins) B = n of Acknowledged Packets by 1 ACK (Typically, 2) W/B ACKs will arrive & each ACK increase 1/W

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TCP Congestion Avoidance Multiplicative Decrease (3Duplicate ACKs) –W’ = W * Md –Eventually, W’ = W/2 –Don’t go back to Slow Start, but Additive Increase Time Out –Go back to Slow Start –W = 1

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Simplifying assumptions No time for Fast Recovery No time for Slow Start Correlated packets losses in a round –Drop-tail policy At a full buffer, drop all packets arriving late –But, independent between rounds Separated by RTT Same implementation of TCP-Reno P1P2P4P3P5P6 r r

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Loss indications & triple-duplicate ACKs B – long term steady-state TCP throughput –Windows increases by 1/b –Windows decreases by a factor of 2 P – loss probability Get B(p) by utilizing Markov Regenerative Process –B = E[Y] / E[A] Y = N of packets sent in TDP i A = duration of the period E[ ] = Expected value in MRGP

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Loss indications & triple-duplicate ACKs Why do we need MRGP? –A cycle will repeat (TDP1, TDP2, TDP3, so on….) Like a sequence of output –New size of windows depends on only previous one’s Markov Chain –Each loss in rounds is separated by RTT (Independently) In statistics, a sequence of random variables is independent and identically distributed (i.i.d.) if each has the same probability distribution as the others and all are mutually independent –Representing steady state model

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Loss indications & triple-duplicate ACKs Markov Model –Predict the future through the past –Based on conditional probability Future state depends on only current state, not the past

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Loss indications & triple-duplicate ACKs P(Rain, Sunny, Cloudy) = ? = p(Rain) * p(Sunny|Rain) * p(Cloudy|Sunny)

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Loss indications & triple-duplicate ACKS How do we predict the weather ?

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Loss indications & triple-duplicate ACKs MRGP –I.I.D random variables

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Loss indications & triple-duplicate ACKs To get B(p) = E[Y]/E[A] –N of packets, including first lost packet, sent in a TDP i : α i –The round where a loss occurs : X i –Y i = α i + W i – 1 Total of Y i packets sent in X i +1 rounds –E[Y] = E[α] + E[W] – 1(2)

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Loss indications & triple-duplicate ACKs To derive E[α] –Expected value in random process {α i } i : E[α] –Based on the assumption Lost packets in a round are independent on any packets in other rounds Independent & identically distributed random variables –P[α = k] equal to p that k-1 packets are acknowledged before a loss –By using (2) and (4), we could derive (5) E[Y]

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Loss indications & triple-duplicate ACKs The increase is linear with slope 1/b Y i can be expressed by (10) B i : N of packets sent in the last round –B i = W i / 2

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Loss indications & triple-duplicate ACKs To derive E[W] –{Wi}, {Xi} all independent sequence of I.I.D random v –So, derive (12) from (7),(10) and (5) –Quadratic equation from (11) & (12) (1-p)/p + w = b* E[W]/4 (3/2 * E[W] – 1) + E[W]/2

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Loss indications & triple-duplicate ACKs As we get E[W], we could get E[X] & E[A] Eventually, B(p) is derived from E[Y]/E[A]

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Loss indications & triple-duplicate ACKs, Time-outs The major reason for window decreases –Timeout rather than fast retransmit –Occurring when packets(or ACKs) are lost –After time-out, W’ = 1 –The period of time-out will doubles

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Loss indications & triple-duplicate ACKs, Time-outs Utilizing MRGP again –Z TO : duration of a sequence of time-outs –Z TD : time interval b/w 2 consecutive TO sequences –S i = Z i TO + Z i TD –M : N of packets sent during S i –B = E[M] / E[S]

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Loss indications & triple-duplicate ACKs, Time-outs How to get B(p) ? –We’ve already known E[Y], E[A]. So, let’s utilize them –Ri = N of packets sent during time-out sequence Z TO Similar process to get B(p) for TDP –Getting a full model & an approximate model

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Impact of window limitation & a full model Keep in mind that limitation of window size Windows can’t grow up over W max Let’s follow the similar process to previous models’ –Unconstrained window size : Wu –E[Wu] < W max –W max approximately equal to E[Wu]

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Impact of window limitation & a full model A full model An approximate model

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Empirical validation Validating formulae, derived so far, by measurement –24 data sets with 1 hour long TCP connection –Infinite source X-axis = frequency of loss indication Y-axis = n of packets sent TD = only TD intervals T0 = single TO intervals T1 = double TO intervals T2 = Triple TO intervals TD Only = prediction of TD only model Full = prediction of full model

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Empirical validation Analysis of measurement tables Overestimation of throughput in TD Only model Full model close to measurement Connections suffering from more time-out rather than 3 duplicate ACKs

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Conclusion A simple model of TCP-Reno –Capturing essence of TCP’s congestion avoidance behavior TDP & time-out –Expressing throughput as a function of loss rate Most connections suffered from a considerable number of time-outs

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Q&A Thank you for listening to my presentation

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