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54. K. Park and W. Wang. QoS-sensitive transport of real-time MPEG video using adaptive forward error correction. In Proc. IEEE Multimedia Systems '99, pp. 426 432, 1999. 55. K. Park. Warp control: a dynamically stable congestion protocol and its analysis. In Proc. ACM SIGCOMM '93, pp. 137 147, 1993. 56. M. Parulekar and A. Makowski. Tail probabilities for a multiplexer with self-similar traf c. In Proc. IEEE INFOCOM '96, pp. 1452 1459, 1996. 57. V. Paxson and S. Floyd. Wide-area traf c: the failure of Poisson modeling. In Proc. ACM SIGCOMM '94, pp. 257 268, 1994. 58. S. Plotkin. Competitive routing of virtual circuits in ATM networks. IEEE J. Select. Areas Commun., 13(6):1128 1136, 1995. 59. M. L. Puterman. Markov Decision Processes: Discrete Stochastic Dynamic Programming. Wiley, New York, 1994. 60. B. Ryu and A. Elwalid. The importance of long-range dependence of VBR video traf c in ATM traf c engineering: myths and realities. In Proc. ACM SIGCOMM '96, pp. 3 14, 1996. 61. P. Shahabuddin. Rare event simulation in stochastic models. In Proc. 1995 Winter Simulation Conference, pp. 178 185, 1995. 62. A. Shaikh, J. Rexford, and K. Shin. Load-sensitive routing of long-lived IP ows. In Proc. ACM SIGCOMM '99, 1999. 63. S. Shenker. A theoretical analysis of feedback ow control. In Proc. ACM SIGCOMM '90, pp. 156 165, 1990. 64. A. Shwartz and A. Weiss. Large Deviations for Performance Analysis. Chapman and Hall, London, 1995. 65. S. Stidham. Optimal control of admission to a queueing system. IEEE Trans. Automat. Contr., AC-30(8):705 713, 1985. 66. B. Tsybakov and N. D. Georganas. Self-similar traf c and upper bounds to buffer over ow in an ATM queue. Perf. Eval., 36(1):57 80, 1998. 67. T. Tuan and K. Park. Multiple time scale congestion control for self-similar network traf c. Perf. Eval., 36:359 386, 1999. 68. T. Tuan and K. Park. Performance evaluation of multiple time scale TCP under self-similar traf c conditions. Technical Report CSD-TR-99-040, Department of Computer Sciences, Purdue University, 1999. 69. T. Tuan and K. Park. Multiple time scale redundancy control for QoS-sensitive transport of real-time traf c. To appear in Proc. IEEE INFOCOM '00, 2000. 70. W. Willinger, M. Taqqu, R. Sherman, and D. Wilson. Self-similarity through highvariability: statistical analysis of Ethernet LAN traf c at the source level. In Proc. ACM SIGCOMM '95, pp. 100 113, 1995. 71. C. Yang and A. Reddy. A taxonomy for congestion control algorithms in packet switching networks. IEEE Network, 9:34 45, 1995.
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Self-Similar Network Traf c and Performance Evaluation Edited by Kihong Park, Walter Willinger Copyright 2000 John Wiley & Sons, Inc. ISBNs: 0-471-31974-0 (Hardback); 0-471-20644-X (Electronic)
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INDEX
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Acivity period, 185, 262 Admission control, 291, 371, 408, 412, 423, 466, 543, 548 549 Aggregate process, 18, 49, 74, 195 Allan variance, 75 Asynchronous transfer mode (ATM), 6, 369, 405, 407, 481, 488, 522 Autocorrelation function, 5, 20, 356 Autocovariance function, 17, 441 Bandwidth, 9 10, 15, 106, 329, 340, 356 357, 362, 405, 422 423, 426, 499 Benes method, 112 ``Black box'' models, 13, 171, 533 Blocking, 371, 416 417 Buffer capacity, 9 10, 15 16, 306, 356, 362, 532 nite, 9, 253, 308, 323, 482, 532, 539, 541 ineffectiveness, 293 over ow, 178, 183, 325, 361, 448, 459 sizing, 106, 218, 220, 532 Bufferless, 406, 408, 422, 540 543 Burstiness, 5, 193, 249, 362, 368 369, 450, 448 Busy period, 110, 309, 404 Cantor set, 2 one-dimensional, 3 two-dimensional, 2 Cantor traf c, 3 Cascade, 4, 519, 524, 534 conservative cascade, 519, 523 Central limit theorem (CLT), 94, 101, 117 118, 423, 426, 496, 525, 541 Chapman Kolmogorov equation, 320, 332 Chernoff bound, 212, 243, 435, 516 Coef cient of concentration, 145 Conditional mean, 424, 427, 451, 474, 547, 550 Conditional probability density, 453, 455, 466 Congestion, 409, 414, 424, 435 Congestion control, 10, 448, 457, 458, 465, 508, 511, 514, 523, 533, 547 Connection admission control, see Admission control arrival, 367, 509 duration, 10 11, 22, 509, 511, 533, 549 lifetime, see Connection, duration Control closed-loop, 402, 408, 534 open-loop, 402, 406 407, 534, 548 549 Convergence, 92, 95, 121, 132, 320, 467, 474, 532, 537 Cumulative process, 4, 17, 24 Delay, see Performance measure Delay-bandwidth product, 11, 16, 458, 478, 533, 545 Distribution a-stable distribution, 90, 94 exponential, 23, 359, 381, 439, 538 Beta, 292 Gamma, 292 Gaussian, 20, 24, 101 102, 116, 511 513 heavy-tailed (long-tailed), 8, 21, 25 26, 89 90, 126, 143, 171, 217, 251, 354, 510 integrated tail, 233, 252 light-tailed, 21, 134, 182 lognormal, 22, 235, 239, 381, 520, 523 negative-binomial, 187 Pareto, generalized, 9, 22, 90, 126, 181, 216, 235, 240, 298, 326, 439, 510 stable, 90, 94 subexponential, 217, 237, 251 Weibull, 9, 22, 234, 239, 298, 381, 384 Effective (equivalent) bandwidth, 10, 230, 291, 416, 484, 495, 498 Elapsed holding time (age), 424 Elastic traf c, 404, 407 408 Erlang's formula, 416
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INDEX Level process, 422 423 Lindley's recursion, 148, 231, 252 Local area network (LAN), 6, 143, 510, 512 Log scale diagram, 56 Long-range dependence (LRD), 6, 8, 15, 20, 40 41, 48, 53, 68, 101, 115, 143, 171, 176, 340, 351, 361, 448, 472, 510, 538, 543, 547 Loss rate (behavior), see Performance measure Markovian model, 1, 7 Markov chain, 303, 433 Markov decision process (MDP), 548 Markovian ( uid) on=off process, 262, 322 Markov modulated G=G=1, 253 Markov process, 323, 422 semi-Markov process, 422 Mandelbrot, B. B., 2, 8 Maximum likelihood estimator (MLE), 381 Mellin transform, 271 Mobility, 535 Monofractal, 69, 514, 534 Multifractal formalism, 54, 514, 534 multifractional Brownian motion, 54 process, 54, 68 scaling, 7, 12, 514, 522, 534 spectrum (Hausdorff), 54 55, 517, 525 Multiple time scale congestion control, 10, 448, 457 traf c control, 16, 478, 533, 545 Multiplexing, 7, 10, 257, 261, 406 gain (ef ciency), 134, 290, 423, 533 statistical, 10, 406, 408 Multiplicative process (cascade), 2, 4, 55, 519 Multiresolution analysis (MRA), 44 Multiscale diagram, 70 Multiservice, 401, 536 Network control, see Control delay, see Performance measure performance, 350, 362, 448, 527, 532 topology, 360, 536 traf c, 372 Noncooperative, 409, 536 Numerical approximation (evaluation, results), 338, 411, 417, 436, 439 440, 498 1=f-type process, 52, 82 On=off model, 8, 14, 24, 350 On=off model (period, source), 3, 8, 14, 24, 40, 101, 115, 143, 172, 194, 274, 333, 422, 450, 513
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Estimation, 6, 12 13, 29, 56, 68, 356, 377, 381, 449, 453, 461, 547 Extremal process, 180 Fairness, 409, 411, 475 Fano factor, 76 Feedback congestion control, 16, 350, 537, 544, 549 control, 364, 543, 546 loop, 10, 459, 533, 545 File size, 8, 14, 90, 349, 352, 354, 358 Flow, see Connection Fluid model=queueing system, 115, 144, 148, 257 in nite-source, 184 multisource, 183 single-source, 176, 258 Forward error correction (FEC), 10, 478, 532, 541, 545 Fractal, 2, 507, 511, 534 fractal (Hausdorff) dimension, 53 point process, 76 Fractional Brownian motion (FBM), 24, 26, 48, 78, 103, 116, 133 Fractional Brownian storage, 101 Fractional Brownian traf c, 102 Fractional Gaussian noise (FGN), 14, 24, 49, 512 Gaussian process, 101, 112, 116, 511 Girsanov formula, 112 Heavy load condition, 116 Heavy-tailed, see Distribution Heavy traf c limit, 101, 145 Hurst=self-similarity parameter, 7, 18, 48, 60, 101, 133, 193, 218, 287, 356 357, 359, 377, 513 Identi cation, see Estimation Impact, see Performance, impact Increment process, 4, 17 Index of dispersion of counts (IDC), 193, 216 In nite variance, 21, 89, 150, 171, 410, 511 Intermediate regular varying, 260 Intermittency, 513 Internet, 367, 369, 375, 402, 414, 508 509, 531 532, 539 Internet protocol (IP), 6, 12, 22, 367, 522 Laplace transform, 126, 271, 424, 427 Large deviations, 2, 101, 106, 217, 423 Large deviation principle (LDP), 108, 225, 260, 435, 516, 518, 540 Law of large numbers, 211, 257, 423, 516 Leaky bucket, 405, 407 Level duration, 439
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INDEX Optimistic traf c control, 549 Packet scheduling, see Service discipline train, 3, 24, 351 Pareto, see Distribution Partition function, 55, 517 Performance analysis, 2, 14 15, 320, 532 effect, see Performance, impact evaluation, 538, 541 impact, 14, 125, 128, 166, 249, 340, 472, 541, 543 544 Performance measure delay, 10, 362, 408, 410, 493, 501 rst-order, 16, 532, 538, 541 jitter, 10, 484, 493, 501, 532, 541 loss rate, 9, 286, 323, 329, 335, 406 second-order, 16, 532, 538, 541 Physical modeling, 7, 13, 532 533 Poisson, 1, 76, 119, 146, 172, 184, 193 194, 198, 215, 375, 379, 395, 524, 548 Polaczek-Khintchine formula, 147 Predictability, see Prediction Prediction, 10 11, 17, 22 23, 433, 448, 451, 455, 457, 466, 471, 546 547 Pricing, 414, 466 Proactivity, 11, 17, 533 Protocol stack (hierarchy), 8, 12, 350, 361, 508, 522 Quality of service (QoS), 2, 11, 16, 319, 362, 401, 403, 414, 448, 481, 532, 541 542, 545 Queueing model GI=G=1, 144 145 GI=GI=1, 252 M=G=1, 144, 409 410, 417 M=G=1, 25, 116, 184, 215, 218, 253, 314 Queue length distribution, 9, 40, 105, 204 asymptotic tail behavior, 9, 106, 116, 216, 258, 263, 540 exponential, 9, 229 polynomial, 9, 210 subexponential, 9, 147 Weibullian, 109, 116, 130 bounds, 106, 123, 210, 225, 227, 236 Rare event simulation, 539 Real-time, 10, 16, 404, 423, 541 542, 545 Recovery time, 424, 435 436, 439 Reich's formula, 103 Regularly varying (at in nity), 144 Relevant=critical time scale, 310, 313 Reliability, 350, 362, 542 Renewal process, 118, 174, 350, 509, 549
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Reproducing kernel Hilbert space, 107 Resource provisioning, 9 10, 16, 32, 532 533, 548 Resetting effect, 308 Riemann boundary value problem, 159 Round-trip time (RTT), 11, 16, 545 Sampling, 22, 92, 537 538, 548 Scalability, 405 Scaling, 39, 48, 59, 61, 104, 116, 513, 522, 525, 534 Scene length=change, 249, 296 297 Scenic model, 305 Scheduling, see Service discipline Schilder's theorem, 107 Selective aggressiveness control (SAC), 448, 457, 460, 545 Self-similarity, 2, 69, 89, 103, 193 deterministic, 4 distributional continuous-time (H-ss, H-sssi), 19, 48, 103 discrete-time, 20 second-order asymptotic, 18, 197, 224, 510 exact, 18, 196, 223 stochastic, 4 Service discipline earliest deadline rst (EDF), 544 rst-come- rst-served (FCFS), 144, 410 rst-in- rst-out (FIFO), see FCFS generalized processor sharing (GPS), 149 last-come- rst-served preemptive resume (LCFS-PR), 144 last-in- rst-out (LIFO), 496, 499 processor sharing (PS), 144, 410 shortest job rst (SJF), 545 shortest remaining processing time (SRPT), 410 411 Session, see Connection Short-range dependence, 12, 20, 53, 56, 222, 287, 351, 448, 473, 534, 543 Simulation, 29, 90, 93, 181, 264, 282, 293, 304, 314, 353, 394, 463, 498, 538 539 Singularity exponent, 514 Small buffer capacity=large bandwidth, 9, 363, 533, 541 Spectral density, 21, 48, 196 a-stable Levy motion, 163 a-stable process, 56 Stability condition, 176, 221 Stationarity=non-stationarity, 13, 40, 61 second-order, 17 strict, 17 trends and LRD, 72
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INDEX Utilization, 406, 418, 482 UNIX le systems, 8, 352 processes, 11, 92, 545 User datagram protocol (UDP), 8, 350, 353, 463 Vanishing moments, 46 Variable bit rate (VBR) video, 7, 143, 285, 481, 542 Virtual circuit (VC), 481, 483 Waiting time distribution, 144, 166, 252 asymptotic tail behavior, 144 Wavelets, 7, 13, 41, 377, 515, 521 continuous wavelet transform (CWT), 42 Daubechies, 43 discrete wavelet transform (DWT), 44 Haar, 43, 515 Meyer, 81 Web, see World Wide Web Whittle estimator, 68, 377 Wide area network (WAN), 6, 16, 143, 369, 509, 522 Workload generation, 536 537 model, 12, 512, 524, 526, 532, 535 process, 145, 161, 525 Workload-sensitive traf c control, 533, 547 World Wide Web (WWW), 6, 8, 14, 189, 352, 370, 535
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Statistical multiplexing, see Multiplexing Steady-state, 90, 95, 336, 341, 423, 435, 540 Stretching-in-space, 361 Stretching-in-time, 361 Structural causality, 8, 350, 450, 464 Structure function, 55, 517 Synthesis (synthetic), 77, 91, 536 537 Throughput, 5, 355, 406, 418, 455, 459, 469, 473 Time scale, 5, 10, 423, 448, 455, 508, 525, 533 534, 543, 545 Time series models ARMA=ARIMA, 171, 482 DAR, 289 FARIMA, 21, 66, 68, 314, 482 GBAR, 291 Trade-off (delay-packet loss), 362 Traf c aggregation, 22, 26, 405 control, 10, 14, 32, 401, 533, 545, 549 engineering, see Traf c, control invariant, 12 measurement, 6, 12 model, 13, 320 shaping, 350 351, 364, 536 Transient analysis, 320, 332, 424 425 Transmission control protocol (TCP), 7 9, 16, 22, 26, 350, 353, 357, 368, 370, 374, 460, 508, 522 523, 526, 534, 537, 544 545, 548 Transport layer, 350, 361, 465, 537
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