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Modeling, Simulation and Measurements of Queuing Delay under Long-tail Internet Traffic Michele Garetto Politecnico di Torino, Italy Don Towsley University of Massachusetts, Amherst ACM SIGMETRICS 2003

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2 Outline Motivation Numerical example Model TCP model Queue model Comparison with simulation Comparison with measurements

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3 Internet traffic (many flows) ? Motivation The behavior of a queue loaded by Internet traffic is a fundamental problem : Network planning (bandwidth provisioning, buffer dimensioning) Network operation (QoS guarantees, Service Level Agreements) Performance evaluation of transport protocols End-to-End delay measurements Congestion control Active Queue Management

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4 Related work Internet traffic is not Poisson ! (Paxson, Floyd ’94) Long-Range Dependence, Self-similarity, Multi-fractal behaviors (Willinger, Park, Crovella, Bestavros, Taqqu, Bolot, Feldman, Gilbert, Erramilli, Narayan) Measurement-based traffic modeling parameter estimation from real traces (at multiple time-scales) Physical modeling physical explanation based on how traffic is generated (heavy- tailed file sizes, transport protocol) Impact on Queue behavior performance evaluation, network engineering TCP modeling (finite flows) ( Roberts, Savage )

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5 Our contribution We study the case of a high-speed, uncongested queue (infinite buffer space) fed by a large number of finite TCP flows (with general file size distribution, e.g. long-tail) Simple Markovian analysis of TCP behavior Novel queueing system We obtain the entire queue length distribution starting only from physical system parameters (results confirmed by simulation and measurements) We offer original insights into the effect on the network of long-tail file size distributions

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6 TCP flows (Poisson arrival process of new flows ) infinite buffer Numerical example Constant size packets (1000 bytes) Link capacity = 28 Mb/s (3500 pkt/s) Variable RTTs (10-100 ms) Bernoulli losses 1% (artificially introduced in simulation) TCP NewReno, maximum window size 32 80 % Link utization Flow size geometrically distributed (mean 20 packets)

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7 1e-05 0.0001 0.001 0.01 0.1 1 050100150200250 Probability Number of packets in the queue ns sim M/M/1 M/D/1 G/M/1 M [X] /M/1 Numerical example

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8 TCP model Computes the batches to be used into the queue model Batches correspond to the clusters of packets transmitted every RTT by the TCP sources We build a Stochastic Finite State Machine (SFSM) to describe the behavior of all connections sharing the same end-to-end path through the network We assume (for now) that the flow size distribution is geometrically distributed The packet loss probability suffered by the TCP connections is assumed to be known a priori (open loop analysis)

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9 TCP model L2L2 L3L3 L W/2 E2E2 E3E3 E W/2 LWLW EWEW entry point of new connections Linear growth (congestion avoidance) window size Exponential growth (slow start) threshold Fast retransmit timeout

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10 Modeling TCP connections transferring a geometrically distributed amount of data Example: congestion avoidance without losses flow size ~ geom (q) packet loss probability = p( s = 1 - p ) N = residual number of packets to be transferred N ~ geom (q) batch size = i batch size = j L i + 1 LiLi

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11 Extension to long-tail flow length distributions Anja Feldmann and Ward Whitt “Fitting mixtures of exponentials to long-tail distributions to analyze network performance models” In Proceedings of IEEE INFOCOM '97, Kobe, Japan, April 1997 A long-tail distribution can be approximated arbitrarily closely by a mixture of geometric distributions Each geometric component is analyzed separately by an instance of the SFSM Arrival rates of batches of the same size are then added up to obtain the actual batch size distribution

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12 Batch size pdf Geometric decomposition log TCP model TCP model TCP model Flow length pdf fit Batch size pdf

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13 1e-05 0.0001 0.001 0.01 0.1 1 050100150200250 Probability Number of packets in the queue sim M/M/1 M/D/1 G/M/1 M [X] /M/1 Numerical example

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14 The queue length distribution predicted by the M [X] /M/1 model can be further refined considering that packets belonging to a batch do not enter the queue simultaneously, but arrive one after the other We model this fact assuming that packets within a batch are equally spaced over time by a constant amount of time We call this queuing system M S [X] /M/1, a queue with “spread” batch arrivals Queue model

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15 : fluid analogy 3 fluid chunks arriving concurrently

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16 If we rely on the assumption of a poisson arrival of batches, the number of concurrent batches R has the same distribution as the number of customers in an M/G/∞ queue: 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0123456 R = 0.3 = 0.6 = 0.8 = 0.95

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17 An approximate solution of the M S [X] /M/1 model is obtained “modulating” the amplitude of the batches considering the joint effect on the queue produced together with a given number of concurrent batches We reduce the solution of the M S [X] /M/1 model to a standard M [X] /M/1 with a modified batch size distribution X We can consider an arbitrary batch size distribution and use well-known results of the M [X] /M/1 model (closed form expressions)

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18 M [X] /M/1 : istantaneous batch arrivals 01ii+1i+2i+3i+4i+5i+6i+7 …… 3 3 3 R 0 … … 01ii+1i+2i+3i+4i+5i+6i+7 M S [X] /M/1 : “modulated” batch arrivals 3 R 1 3 1/2 R 2 …

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19 Comparison of queue models 0.0001 0.001 0.01 0.1 1 020406080100120140 Probability Number of packets in the queue sim M [X] / M / 1 M [X] / E 10 / 1 S M [X] / M / 1 S 0.01 0.1 012345678910

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20 simulation (95% conf) (relative error) 0.3 0.917 (± 0.015) 2.293 ( 150% ) 0.997 ( 8.7% ) 0.939 ( 2.4% ) 0.6 4.950 (± 0.128) 8.025 ( 62.1% ) 5.527 ( 11.6% ) 5.122 ( 3.5% ) 0.8 16.696 (± 0.698) 21.401 ( 28.2% ) 18.420 ( 10.3% ) 16.979 ( 1.7% ) 0.9 41.091 (± 2.887) 48.145 ( 17.2% ) 45.617 ( 11.0% ) 41.819 ( 1.8% ) Comparison of queue models Access link scenario - average number of packets in the queue

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21 Ns simulations We studied the sensitivity of various parameter on the resulting queue length distribution: link utilization, link capacity, round trip times, loss probability, loss correlation, maximum window size, flow length distribution, access links capacity, network topology, arrival process of new connections The model is robust on a wide range of operating conditions, but has some limitations (due to the assumption of a Poisson arrival of batches) The accuracy of the model improves increasing the number of active flows (large bandwidth-delay product)

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22 Example of “long-tail” distribution 1e-07 1e-06 1e-05 0.0001 0.001 0.01 0.1 1 10 100100010000 Probability Flow length (packets) = 0.89 - mean 10 = 0.1 - mean 100 = 0.01 - mean 1000 overall distribution overall mean = 28.9 Mixture of 3 geometric components

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23 Impact of “long-tail” distribution 1e-05 0.0001 0.001 0.01 0.1 050100150200250300350400 p = 0.01 sim - 28 Mbps mod sim - 155 Mbps 1e-05 0.0001 0.001 0.01 0.1 050100150200250300350400 Probability Number of packets in the queue p = 0.1 sim mod sim - 155 Mbps sim - RTTs x 10

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24 1e-05 0.0001 0.001 0.01 0.1 050100150200250300350400 Probability Number of packets in the queue sim - S = 10 sim - S = 20 sim - S = 40 sim - S = 80 sim - S = 320 sim - S = 1000 Effect of file size (single geometric component) [ link utilization = 0.8 – p = 0.01 ]

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25 Impact of link speed on additional correlations (not modeled) 1e-06 1e-05 0.0001 0.001 0.01 0.1 0100200300400500600700800900 Probability Number of packets in the queue sim - 10 Mbps sim - 28 Mbps sim - 155 Mbps sim - 622 Mbps mod zero packet loss probability link utilization = 80% average flow length = 320 packets

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26 0.01 0.1 020406080100120 Probability Number of packets in the queue sim Finite buffer size ( drop-tail ) Buffer size = 128 pkts Link utilization = 90% Average flow length = 60 packets mod Loss event probability obtained by sim

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27 Measurement setup Internet Turin Polytechnic LAN Probe packets One-way delay samples obtained sending a Poisson stream of probe TCP packets 20 packets/s for 10 minutes ~ 12000 samples per trace No synchronization between source and receiver (clock skew removal) Assumption: the variable part of the delay completely due to queueing delay on the access link Active measurements Passive measurements MRTG: average link utilization during an interval of 10 minutes TSTAT: post-processing of the trace of all packets traversing the link model parameters: - Flow length distribution - Average packet size and MSS - Maximum TCP window size - Loss event probability

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28 Flow length distribution Fitted mixture of 7 geometric distributions

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29 Model / simulation / measurements 69 % link utilization

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30 Model / simulation / measurements 82 % link utilization 0.0001 0.001 0.01 0.1 1 05101520253035404550 Probability queuing delay (ms) simulation (28Mbps) simulation (155 Mbps) Poisson model measurement (28Mbps)

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31 Conclusions Traditional Markovian analysis can be successfully applied to model the behavior of a queue loaded by Internet traffic A Poisson batch arrival process captures most of the traffic correlations on a variety of network scenarios Additional correlations (not primarily due to long- tail flow length distributions) vanish increasing the number of active flows (i.e. on high speed links)

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