Performance Analysis of Queuing and Computer Networks C9861_FM01_R1.indd 5/6/08 8:52:09 AM CHAPMAN & HALL/CRC COMPUTER and INFORMATION SCIENCE SERIES Series Editor: Sartaj Sahni PUBLISHED TITLES ADVERSARIAL REASONING: COMPUTATIONAL APPROACHES TO READING THE OPPONENT’S MIND Alexander Kott and William M McEneaney DISTRIBUTED SENSOR NETWORKS S Sitharama Iyengar and Richard R Brooks DISTRIBUTED SYSTEMS: AN ALGORITHMIC APPROACH Sukumar Ghosh FUNDEMENTALS OF NATURAL COMPUTING: BASIC CONCEPTS, ALGORITHMS, AND APPLICATIONS Leandro Nunes de Castro HANDBOOK OF ALGORITHMS FOR WIRELESS NETWORKING AND MOBILE COMPUTING Azzedine Boukerche HANDBOOK OF APPROXIMATION ALGORITHMS AND METAHEURISTICS Teofilo F Gonzalez HANDBOOK OF BIOINSPIRED ALGORITHMS AND APPLICATIONS Stephan Olariu and Albert Y Zomaya HANDBOOK OF COMPUTATIONAL MOLECULAR BIOLOGY Srinivas Aluru HANDBOOK OF DATA STRUCTURES AND APPLICATIONS Dinesh P Mehta and Sartaj Sahni HANDBOOK OF DYNAMIC SYSTEM MODELING Paul A Fishwick HANDBOOK OF 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Data Dattatreya, G R Performance analysis of queuing and computer networks / author, G.R Dattatreya p cm (Chapman & hall/CRC computer and information science series) “A CRC title.” Includes bibliographical references and index ISBN 978-1-58488-986-1 (hardback : alk paper) Computer networks Evaluation Network performance (Telecommunication) Queuing theory Telecommunication Traffic I Title TK5105.5956D38 2008 004.6 dc22 2008011866 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com C9861_FM01_R1.indd 5/6/08 8:52:10 AM To my family Contents Introduction 1.1 Background 1.2 Queues in Computers and Computer Networks 1.2.1 Single processor systems 1.2.2 Synchronous multi-processor systems 1.2.3 Distributed operating system 1.2.4 Data communication networks 1.2.4.1 Data transfer in communication networks 1.2.4.2 Organization of a computer network 1.2.5 Queues in data communication networks 1.3 Queuing Models 1.4 Conclusion Characterization of Data Traffic 2.1 Introduction 2.2 The Pareto Random Variable 2.3 The Poisson Random Variable 2.3.1 Derivation of the Poisson pmf 2.3.2 Interarrival times in a Poisson sequence of arrivals 2.3.3 Properties of Poisson streams of arrivals 2.3.3.1 Mean of exponential random variable 2.3.3.2 Mean of the Poisson random variable 2.3.3.3 Variance of the exponential random variable 2.3.3.4 Variance of Poisson random variable 2.3.3.5 The Z transform of a Poisson random variable 2.3.3.6 Memoryless property of the exponential random variable 2.3.3.7 Time for the next arrival 2.3.3.8 Nonnegative, continuous, memoryless random variables 2.3.3.9 Succession of iid exponential interarrival times 2.3.3.10 Merging two independent Poisson streams 2.3.3.11 iid probabilistic routing into a fork 2.4 Simulation 2.4.1 Technique for simulation 2.4.2 Generalized Bernoulli random number 2.4.3 Geometric and modified geometric random numbers 1 2 3 3 13 13 15 22 23 25 26 26 27 28 29 29 30 31 31 31 32 35 37 37 37 39 vii viii 39 40 42 43 46 47 49 52 52 54 56 57 59 The M/M/1/∞ Queue 3.1 Introduction 3.2 Derivation of Equilibrium State Probabilities 3.2.1 Operation in equilibrium 3.2.2 Setting the system to start in equilibrium 3.3 Simple Performance Figures 3.4 Response Time and its Distribution 3.5 More Performance Figures for M/M/1/∞ System 3.6 Waiting Time Distribution 3.7 Departures from Equilibrium M/M/1/∞ System 3.8 Analysis of ON-OFF Model of Packet Departures 3.9 Round Robin Operating System 3.10 Examples 3.11 Analysis of Busy Times 3.11.1 Combinations of arrivals and departures during a busy time period 3.11.2 Density function of busy times 3.11.3 Laplace transform of the busy time 3.12 Forward Data Link Performance and Optimization 3.12.1 Reliable communication over unreliable data links 3.12.2 Problem formulation and solution 3.13 Exercises 63 63 64 70 71 72 76 77 80 81 86 88 94 96 98 99 101 104 104 105 109 State Dependent Markovian Queues 4.1 Introduction 4.2 Stochastic Processes 4.2.1 Markov process 4.3 Continuous Parameter Markov Chains 4.3.1 Time intervals between state transitions 4.3.2 State transition diagrams 4.3.3 Development of balance equations 115 115 115 117 118 118 118 119 2.5 2.6 2.7 2.8 2.4.4 Exponential random number 2.4.5 Pareto random number Elements of Parameter Estimation 2.5.1 Parameters of Pareto random variable 2.5.2 Properties of estimators Sequences of Random Variables 2.6.1 Certain and almost certain events Elements of Digital Communication and Data Link Performance 2.7.1 The Gaussian noise model 2.7.2 Bit error rate evaluation 2.7.3 Frame error rate evaluation 2.7.4 Data rate optimization Exercises ix 4.3.4 Graphical method to write balance equations Markov Chains for State Dependent Queues 4.4.1 State dependent rates and equilibrium probabilities 4.4.2 General performance figures 4.4.2.1 Throughput 4.4.2.2 Blocking probability 4.4.2.3 Expected fraction of lost jobs 4.4.2.4 Expected number of customers in the system 4.4.2.5 Expected response time 4.5 Intuitive Approach for Time Averages 4.6 Statistical Analysis of Markov Chains’ Sample Functions 4.7 Little’s Result 4.7.1 FIFO case 4.7.2 Non-FIFO case 4.8 Application Systems 4.8.1 Constant rate finite buffer M/M/1/k system 4.8.2 Forward data link with a finite buffer 4.8.3 M/M/∞ or immediate service 4.8.4 Parallel servers 4.8.5 Client-server model 4.9 Medium Access in Local Area Networks 4.9.1 Heavily loaded channel with a contention based transmission protocol 4.9.1.1 Consequences of modeling approximations 4.9.1.2 Analysis steps 4.9.2 A simple contention-free LAN protocol 4.10 Exercises 123 124 124 127 127 127 127 128 128 129 132 141 141 142 143 143 146 147 148 152 160 The M/G/1 Queue 5.1 Introduction 5.2 Imbedded Processes 5.3 Equilibrium and Long Term Operation of M/G/1/∞ Queue 5.3.1 Recurrence equations for state sequence 5.3.2 Analysis of equilibrium operation 5.3.3 Statistical behavior of the discrete parameter sample function 5.3.4 Statistical behavior of the continuous time stochastic process 5.3.5 Poisson arrivals see time averages 5.4 Derivation of the Pollaczek-Khinchin Mean Value Formula 5.4.1 Performance figures 5.5 Application Examples 5.5.1 M/D/1/∞: Constant service time 5.5.2 M/U/1/∞: Uniformly distributed service time 5.5.3 Hypoexponential service time 5.5.4 Hyperexponential service time 179 179 180 181 181 183 4.4 160 161 162 163 170 185 189 190 193 198 198 198 198 199 199 Review of Probability Theory 435 interrupts statistically independently with probabilities of 0.1, 0.4, and 0.6, respectively Their respective interrupt service times are 5, 3, and msecs Unserviced interrupts are lost There is no queuing at all Determine each of the probabilities with which the microprocessor services A, B, or C after it checks Also determine the average time spent on interrupt service during a 10 msec time period The function GX (x) = − FX (x) is called the complementary cumulative distribution function (ccdf) and also by the name of the tail distribution function It is also commonly denoted by Fc (x) when the random variable under consideration is obvious Prove that the expectation of a nonnegative random variable is the integral of its ccdf over the nonnegative real line Prove that the expectation of the sum of two random variables is the sum of their individual expectations Do not assume that the two random variables are independent Do this for the two cases of continuous and discrete random variables Prove that the variance of the sum of two independent random variables is the sum of the variances of the individual random variables From the Parade weekly news magazine Consider a TV game show in which the host shows the participant three identical doors and tells her that behind one of them is a sports car and behind the others are a goat each The participant should try to guess the door with the car The participant picks one The host opens a different door that shows a goat The host then gives a chance for the participant to switch the choice to the remaining door The question is, at that time, what are the probabilities of finding the car in each of the two closed doors? This will help the participant to determine if she should switch the doors if given a chance Remember that any time anyone has to choose between two or more identical choices (to the knowledge of the person making the choice), one is picked “at random.” This problem is based on a fictional story titled “The curious prisoner and the warden.” There were three prisoners A, B, and C, in different cells They knew that one of them had already been chosen at random to be executed the next morning Prisoner A went to the warden and said “I know you are not allowed to tell me which one of us will be executed But I know that one of the other two will not be executed Will you please tell me which of the other two will not be executed?” The warden reflected for a moment and obliged the prisoner A with a true answer Did this change A’s calculation of the probability that he would be executed? And according to A’s calculations, what is the probability that the other candidate (other than the survivor as disclosed by the warden) would be executed? 436 Performance Analysis of Queuing and Computer Networks Instead of checking with the warden, let us say that A found a true document that he was not supposed to have access to And let us say that the document implied that a particular prisoner from {B, C} will be alive following the next day Now repeat the same calculations as above and compare Prove the Markov inequality That is, for any nonnegative random variable X, with an expectation η and for any α > 0, show that P [X ≥ α] ≤ η α (A.170) 10 Prove the Chebyshev inequality That is, for any random variable with an expectation of η and variance of σ , and for any > 0, show that P [|X − η| ≥ ] ≤ σ2 (A.171) This is named after Pafnuty Chebyshev (1821–1894), a Russian mathematician Index continuous parameter Markov chain, 119 discrete parameter Markov chain, 232 global, 123, 272, 273, 279, 283 graphical method, 123, 233 local, 123 open network, 270 balking, 153 bank teller, 173 Bayes’ theorem, 417 a posteriori pdf, 420 continuous conditioning, 420 discrete partition, 419 Bayes, T., 417 beacon signal, 168 BER, 52, 54, 105 Bernoulli generalized, 401 Bernoulli, J., 400 Bertrand box paradox, 419 Bertrand, J L F., 419 biased die, 402 binomial distribution, 23 random variable, 34 bit error, rate, 54, 105 blocking probability, 127, 146 broadcast, link, Brown, R., 330 buffer capacity, 124 content, 365 a posteriori density, 420 a posteriori probability, 419 a priori probability, 419 acknowledgment, addressing mode, 223 algorithm convolution, 286, 343 decision, 62 fixed-point, 177 numerical, 320 scheduling, 2, 149 transmission, 206 aperiodic chain, 224 long term behavior, 240 state, 223, 225 approximation hyperexponential, 337 model, arrival, 1, 63 rate state dependent, 124 theorem, 294 time state probability, 190 attenuation, 52 autocorrelation, 329, 332 matrix, 333 sequence, 330 axiom infinite additivity, 50, 398 probability, 397 backbone subnet, balance equation closed network, 283 437 438 Performance Analysis of Queuing and Computer Networks finite, 124 fluid, 364 occupancy, 246 average, 73 closed network, 289 Burke, P J., 85 bursty traffic, 332 bus computer, 262 busy probability, 72 time, 96 M/M/1/∞ queue pdf, 101 parallel server, 150 Buzen, J P., 286 cable channel, 160 call telephone, 171 capacity buffer, 124 carrier sense, 160 CBR, 326 ccdf, 435 cdf, 17, 401 central moment, 410 processing unit, centroid random variable, 407 channel cable, 160 propagation, 147 request, 209 Chapman, S., 220 Chapman-Kolmogorov equations, 220 Chebyshev inequality, 436 Chebyshev, P., 436 classical teletraffic, 329 client-server system, 152 clock cycle, closed queuing network, 9, 268 arrival theorem, 294 buffer occupancy, 289 cyclic, 295 marginal state probability, 288 mean value analysis, 293 noncyclic, 296 performance figure, 288 response time, 290 throughput, 289 traffic equation, 283 utilization, 289 cluster packet, 329 cognitive radio, 359 coin toss random experiment, 400 collision, 5, 160, 249 detection, 160 communication data, digital, 52 link, network, complement, 399 complementary cdf, 435 componentwise inequality, 353 composite pdf round robin IAT, 93 computer bus, 262 mouse, conditional expectation, 411 pdf, 405 pmf, 405 probability, 399 confidence interval, 42 constant bit rate, 326 contention, 11, 160 continuous parameter, 116 state, 116 control packet, 167, 168, 207 parameter, 147 control parameter, convergent matrix, 348 Index convolution algorithm, 286, 343 integral, 417 matrix, 288 sum, 416 CPU, 2, 221, 303 crossbar, 256, 263 CSMA/CA wireless LAN, 175 CSMA/CD, 160 cumulative distribution, 17, 401 customer, cycle clock, cyclic network, 295 data analysis, 42 communication, frame, 5, 63 packet, 1, 19, 63 rate optimization, 106 traffic, 13 intensity, 329 datalink, 363, 395 forward, 104 finite buffer, 146 layer, performance, 52 reliable communication, 104 unidirectional, 4, 104, 112 unreliable, 104 decision from noisy data, 53 delay, 13, 76 queue, delta function Dirac, 81 Kronecker, 271 density probability, 403 total, 414 departure, 1, 63 M/M/1/∞, 81, 85 439 time instant, 363 detection error, determinant, 373 diagonal matrix, 348 die biased, 402 differential equation, 67, 119, 365, 368 general solution, 370 initial conditions, 374 linear, 369 ordinary, 369 partial, 367 steady state solution, 67, 120 Dirac delta function, 81 Dirac, P., 81 discrete parameter, 116 random variable, 402 state, 116 distributed operating system, distribution cumulative, 401 probability, 401 DLL, DMA, 221 document size WWW, 330 DOS, drain, 365, 370 fluid, 364 dummy packet, 164 early arrival system, 210 effective load G/M/1/∞ queue, 319, 323 eigenvalue, 373, 387 eigenvector, 373, 387 electromagnetic signal, 52 embedded Markov chain, 180 emit packet, 337 empty system, 72 ensemble, 129 440 Performance Analysis of Queuing and Computer Networks expectation, 128, 129 Markov chain, 129 stochastic process, 115 epoch, 210 equation balance closed network, 283 open network, 270 effective load G/M/1/∞ queue, 317 global balance open network, 272 traffic closed network, 283 open network, 273 equilibrium M/M/1/∞ queue, 67 Markov chain, 120 operation, 70 state probabilities M/M/1/∞, 69 Markov chain, 121, 237 erf, 55 ergodic theorem, 243 ergodicity, 140 Erlang B formula, 151, 177 C formula, 151, 177 Erlang, A K., 151, 177 error bit, detection, function, 55 rate bit, 52, 54 frame, 56 estimation parameter, 42 Pareto, 43 theory, 42 estimator properties, 46 unbiased, 46 event, 397 almost certain, 49 certain, 49 impossible, 399 null, 398 occurrence, 398 execute instruction, 223 expectation, 407 conditional, 411 total, 415 expected number G/M/1/∞ queue, 321 G/M/1/k queue, 322 M/G/1/∞ queue, 197 M/G/1/k queue, 202 M/M/1/∞ queue, 73 expected response time G/M/1/∞ queue, 321 G/M/1/k queue, 322 M/G/1/∞ queue, 198 M/M/1/∞ queue, 77 expected value, 407 experiment random, 397 exponential random variable, 26, 428 variance, 28 service time, 76 FBM, 330, 335 feedback, 270 feedback probability, 283 FER, 56 fetch instruction, 223 FGN, 330, 336 FIFO, 1, 63 finite buffer, 143 G/M/1 queue, 322 M/G/1 queue, 200 M/M/1/ queue, 124 fixed-point algorithm, 177 flip-flop, 221 fluid arrival, 363 buffer, 364 departure, 363 drain, 365 Index flow Little’s result, 377 M/M/1/∞ queue, 387 model, 363 multistate chain, 384 random unit, 382 traffic, 363 two state Markov chain input, 364 transmission, 365 forking probabilistic, 35 forward link, 104 fractal, 335 fractional Brownian motion, 330, 335 Gaussian noise, 330, 336 frame error rate, 56 free server, 8, 72 fricative, 173 G/M/1/∞ queue, 307 effective load, 319 expectation, 321 expected response time, 321 imbedded Markov chain, 307 G/M/1/k queue, 322 expected number, 322 expected response time, 322 throughput, 322 Gauss, C F., 52 Gaussian, 52 noise, 105 general service time, 179 solution differential equation, 370 Geom/Geom/m/k queue, 253 geometric random variable, 403 global balance equation, 123 open network, 272 Gordon, W J., 282 441 handshake, hardware control, 221 heavy load, 262 tailed distribution, 337 Homeland Security, 305 homogeneous Markov chain, 118, 218 host machine, housekeeping operation, 2, 156 Hurst parameter, 16, 323, 335 Hurst, H E., 16 hyperexponential approximation, 337 pdf, 93, 179 I/O, 2, 303 IAT, 6, 13, 63 exponential, 50 Poisson arrivals, 50 idle, 72 IDT, 82 iid, 13, 417 imbedded Markov chain, 180 G/M/1/∞ queue, 307 immediate service, 147 impossible event, 399 increment, 334 FBM, 336 stationary, 335 independence mutual, 399 statistical, 399 inequality componentwise, 353 loose, 348, 354 strict, 354 infinite additivity axiom, 398 inhomogeneous Markov chain, 219 initial condition, 374 input/output, 2, 203 instruction 442 Performance Analysis of Queuing and Computer Networks addressing mode, 223 execute, 223 fetch, 223 register mode, 223 intensity data traffic, 329 interarrival time, 6, 13, 22 memoryless, 63 Poisson, 25 interdeparture time, 85 interrupt, 434 interval estimate, 42 irreducible, 220 IRS, 304 Jackson, J R., 273 job, 2, 67 payload, 156 joint density, 406 Markov chain source and queue, 344 probability, 32, 406 Kendall’s notation, Kendall, D G., 7, 96, 197 Khinchin, A Y., 197 Kolmogorov, A N., 220 Kronecker delta function, 271 Kronecker, L., 271 L’Hospital, G F A., 25 LAN, contention-free M/M/1/∞ approximation, 168 protocol contention-free, 163, 206 slotted, 212, 266 Laplace transform, 81, 428 exponential pdf, 26 Laplace, P.-S., 428 late arrival system, 258 Lavenberg, S S., 293 layer, datalink, medium access, network, transport, leaky bucket, 364 LHS, 426 LIFO, line of sight, 104 link, broadcast, data, forward, unidirectional, 4, 104, 112 unreliable, 59 Little’s result, 129, 141 FIFO, 141 fluid flow system, 377 non-FIFO, 142 Little, J D C., 129 load normalized, 69, 195 local area network, balance equation, 123 long range dependence, 329 long term behavior aperiodic chain, 240 long-term operation M/G/1/∞, 181 loose inequality, 348 LOS, 104 lost customer, 127, 145 LRD, 329 M/G/1 queue finite buffer, 200 M/G/1/∞ queue Pareto service time, 200 sample function continuous time, 189 imbedded Markov chain, 185 sluggish os, 206 stability, 184 state recursion, 181 Index state transition diagram, 182 stochastic process, 180 M/M/∞ queue, 147 M/M/1/∞ queue, 63 busy time Laplace transform, 102 pdf, 101 departure, 81, 85 expected number, 73 performance figures, 72 response time, 76 round robin, 88 stability, 69 state, 64 MAC, saturated LAN, 160 machine sequential, 221 Maclaurin series, 102 Maclaurin, C., 102 Mandelbrot, 335 marginal density, 406 probability, 412 state probability closed network, 288 Markov chain, 115, 117, 364 continuous parameter, 117, 244 applications, 143 balance equation, 119 equilibrium probabilities, 121 sample function, 132 transition rate, 118 discrete parameter, 181, 216 applications, 249 balance equations, 232 Chapman-Kolmogorov equations, 220 equilibrium, 231 homogeneous, 218 inhomogeneous, 218 irreducible, 220 null recurrent, 229 positive recurrent, 229 recurrent, 229 443 recurrent non-null, 229 stable, 232 time averages, 239 transient, 229 transition probability, 211 unstable, 232 ergodicity, 140 fluid flow input, 364 homogeneous, 118 imbedded, 180 irreducible, 119 joint source and queue, 344 state dependent queue, 124 state transition, 118 unstable, 126 Markov inequality, 436 Markov modulated Poisson process, 330 Markov process, 117 Markov, A A., 117 mathematical statistics, 42 Matlab, 147 matrix convergent, 348 convolution, 288 eigenvalue, 373 eigenvector, 373 maximization throughput, 106, 163 mean, 19, 407 mean value analysis, 293 mean value formula M/G/1/∞ queue, 193 medium access, 160 layer, memory access, 223 memoryless, 13, 30 merged hyperexponential IAT product form solution, 340 packet source, 339 merging, 32 Poisson streams, 32 states, 260 444 Performance Analysis of Queuing and Computer Networks metric performance, minimal nonnegative solution, 355, 356 mixed random variable, 404 mixture random variable, 206 MMPP, 329, 330, 355 model, approximation, queuing, modified geometric pmf, 39, 72 random variable, 72 moment, 45, 410 central, 410 mouse computer, ms, 19 msec, 19 multiprogramming, mutually exclusive, 398 independent, 399 MVA, 286, 293 network closed, 268 communication, layer, wireless, Neuts, M., 355, 356 Newell, G F., 282 next arrival time for, 31 noise, 14 Gaussian, 52 noncyclic network, 296 nonlinear optimization, 147 nonnegative matrix, 348 normalized load, 69 notation Kendall, open network, 268 null event, 398 recurrent chain, 229 numerical solution, 325, 338 occurrence event, 398 OFF time packet train, 329 ON time packet train, 329 ON-OFF packet train, 19 open queuing network, 9, 267 global balance equation, 272 notation, 268 product form solution, 276 routing probability, 269 traffic equation, 273 operating system, sluggish, 156 operation equilibrium, 70 house keeping, housekeeping, 2, 156 steady state, 70 optimization channel sensing rate, 163 data rate, 57, 106 nonlinear, 147 transmission probabilities, 166 OS, outcome random experiment, 397 output process fluid flow system, 382 packet cluster, 329 data, 1, 19 emission, 337 source merged, 339 train, 19, 86 OFF time, 329 ON time, 329 ON-OFF, 19 transmission, 104, 146 Index paradox Bertrand’s box, 419 parallel server, 148 parameter continuous, 116 discrete, 116, 181 estimation, 42 Hurst, 16, 335 Pareto, 330 IAT, 323 pdf, 16, 323 random variable, 13, 15 parameter estimation, 43 shifted, 59, 337 Pareto service time M/G/1/∞ queue, 200 Pareto, V F D., 15 partial differential equation, 367 partition, 419 PASTA, 190, 201 payload jobs, 156 pdf, 13, 403 conditional, 405 heavy tailed, 337 marginal, 406 Pareto, 16 total, 414 uniform, 198 performance figure, 72 closed network, 288 state dependent queue, 127 metric, queue, period chain, 230 periodic chain, 225 state, 223, 224 periodicity state, 224 piecemeal service, piggyback, 104 pmf, 402 conditional, 405 445 Poisson, 23 point estimate, 42 Poisson arrivals, 63 interarrival times, 25 random variable, 22 mean, 26 variance, 29 Poisson, S D., 13 Pollaczek, F., 197 Pollaczek-Khinchin formula, 193 positive matrix, 348 posterior probability, 53, 419 preemption, 60 prior probability, 419 probabilistic routing, 35 probability a posteriori, 53 a priori, 419 axioms, 397 conditional, 399 density function, 13, 403 feedback, 283 marginal, 412 mass function, 402 Poisson, 23 packet drop, 146 routing, 269 total, 412 product form solution, 273 closed network, 284 merged hyperexponential IAT, 340 open networks, 278 propagation channel, 147 queue, closed network, 282 discrete time, 209 G/M/1/∞, 307 heavily loaded, 262 M/G/1, 179 finite buffer, 200 M/M/∞, 147 M/M/1/∞, 63 446 Performance Analysis of Queuing and Computer Networks MMPP traffic source, 355 notation Kendall, queuing delay, model, network, closed, open, queuing network closed continuous time, 282 product form solution, 284 traffic equation, 283 Markovian closed, 282 continuous time, 270 open, 267 RAM, 221 random experiment, 397 outcome, 397 fluid unit, 382 process, 115 variable, 400 centroid, 407 continuous, 403 discrete, 402 expectation, 407 exponential, 26, 428 function, 421 geometric, 403 memoryless, 30 mixed, 404 mixture, 206 moment, 410 Pareto, 13, 15 Poisson, 22 sequence, 47 sum, 415 rate arrival, 28 reachable, 119, 121 state, 233 real time system, 170 recurrent chain, 229 state, 226 recursion equation M/G/1/∞ queue, 181 register shift, 210 Reiser, M., 293 relationship Markov chains slot edge and center, 248 request channel, 209 retransmission, 105 service, 3, 152 response time, 6, 247 closed network, 290 M/M/1/∞ queue, 76 round robin, 92 retransmission, request, 105 RHS, 417 round robin, IAT composite, 93 M/M/1/∞ queue, 88 routing probabilistic, 35 sample function, 115 continuous parameter, 189 Markov chain, 132 mean, 46 space, 397 schedule transmission, 164 scheduling algorithm, sec, 58 second, 58 self-similar process, 329, 334 self-similarity, 335, 357 sensor network, 265, 303 sequential machine, 221 Index server free, probability, 72 parallel, 148 service, constant time, 198 FIFO, immediate, 147 LIFO, piecemeal, random order, rate state dependent, 124 request, 152 round robin, time, exponential, 76 memoryless, 63 service time general, 179 hyperexponential, 199 hypoexponential, 179, 199 memoryless, 64 shift register, 210 shifted Pareto, 337 simulation, 37 technique, 37 exponential, 39 generalized Bernoulli, 37 geometric, 39 Pareto, 40 singularity, 415 slot, center, 210 edge, 209 just before, 210 soon after, 210 silent, 259 talk, 259 sluggish os, 156 hyperexponential service, 206 M/G/1/∞ queue, 206 smooth traffic, 331 sojourn, 296 time, 6, 297 447 solution numerical, 325 unique, 231 split probabilistic, 35 splitting state, 260 stability fluid flow system, 376 M/G/1/∞ queue, 184 M/M/1/∞ queue, 69 Markov chain, 122 stack, state, 116 classification, 223 continuous, 116 dependent queue, 124 discrete, 116 M/M/1/∞ queue, 64 merging, 260 periodic, 224 probability arrival time, 190 reachable, 233 recurrent, 226 splitting, 260 transient, 226 transition, 118, 211 diagram, 118 Markov chain, 118 rate, 155 vector, 269 state transition diagram M/G/1/∞ queue, 180–182 M/G/1/k queue, 200 statistic, 7, 42 statistical average, 12 independence, 399 steady state operation, 70 solution differential equation, 67 stochastic chain, 217 448 Performance Analysis of Queuing and Computer Networks process, 217 M/G/1/∞ queue, 180 stochastic process, 115 Markov, 117 sample function, 132 continuous parameter, 189 subnet, synchronization, 209 Takacs, L., 96 Taylor, B., 71 TCP, 61 connection, 61 telephone call, 1, 171 traffic, 151, 177 throughput, 104, 127, 145, 245 closed network, 289 G/M/1/k queue, 322 maximization, 106, 163 time average, 239 intuitive approach, 129 sharing, timer, 88 total density, 414 expectation, 415 probability, 412 TPDU, traffic bursty, 332 data, 13 equation, 273 closed network, 283 open network, 273 fluid flow, 363 increment, 334 MMPP, 355 smooth, 331 telephone, 151, 177 video, 330 WWW, 330 train packet, 86 transform Z, 430 Laplace, 428 transformation monotonic random variable, 43 transient chain, 229 state, 226 transition probability, 211 rate, 118, 155 transmission, packet, 104, 146 probabilities optimization, 166 schedule, 164 transmitter, 212, 218, 249 wireless, 175 transport connection, 394 layer, transreceiver, 105 unidirectional link, 4, 104, 112 uniform distribution, 14, 198, 404 uniprocessor, 88 unique solution, 121, 231 balance equations, 237 unreliable link, 59 unstable Markov chain, 126 utilization, 7, 69 closed network, 289 vacant position, 245 variable bit rate, 330 variance, 18, 411 unbounded, 335 VBR, 330 vector state, 269 video traffic, 330 voice over IP, 207 VOIP, 207 Index vowel, 173 waiting time, 6, 80 WAN router, 112 wireless communication, 175 LAN CSMA/CA, 175 network, wrt, 71 WWW document size, 330 traffic, 330 Z transform, 430 Poisson pmf, 29 449 ... Plaza and Chein-I Chang PERFORMANCE ANALYSIS OF QUEUING AND COMPUTER NETWORKS G R Dattatreya THE PRACTICAL HANDBOOK OF INTERNET COMPUTING Munindar P Singh SCALABLE AND SECURE INTERNET SERVICES AND. .. HIGH PERFORMANCE COMPUTER ARCHITECTURES David Kaeli and Pen-Chung Yew C9861_FM01_R1.indd 5/6/08 8:52:10 AM Performance Analysis of Queuing and Computer Networks G R Dattatreya University of Texas... the router on the other side of the forward link, and Performance Analysis of Queuing and Computer Networks • frame boundary bits to determine the start and end of a frame The resulting data