Performance Analysis of Queuing and Computer Networks

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ISBN: 9781584889861 | 1584889861
Cover: Hardcover
Copyright: 6/9/2008

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Queues are an important topic due to their usefulness in data communication network performance. Exploring queues from a mathematical standpoint, Elements of Queues and Performance Analysis of Computer Networks discusses the design, operation, and interconnections of data communications while providing the required mathematical skills and background. The book includes numerous applications and examples of packet error rate variations, cyclic window constraints, and finite memory buffers as well as mathematical derivations and MATLAB® code. Topics covered include stochastic processes, queuing systems, the G/M/I queue, and the M/M/I queue.

Introduction	p. 1
Background	p. 1
Queues in Computers and Computer Networks	p. 2
Single processor systems	p. 2
Synchronous multi-processor systems	p. 3
Distributed operating system	p. 3
Data communication networks	p. 3
Data transfer in communication networks	p. 3
Organization of a computer network	p. 4
Queues in data communication networks	p. 5
Queuing Models	p. 6
Conclusion	p. 9
Characterization of Data Traffic	p. 13
Introduction	p. 13
The Pareto Random Variable	p. 15
The Poisson Random Variable	p. 22
Derivation of the Poisson pmf	p. 23
Interarrival times in a Poisson sequence of arrivals	p. 25
Properties of Poisson streams of arrivals	p. 26
Mean of exponential random variable	p. 26
Mean of the Poisson random variable	p. 27
Variance of the exponential random variable	p. 28
Variance of Poisson random variable	p. 29
The Z transform of a Poisson random variable	p. 29
Memoryless property of the exponential random variable	p. 30
Time for the next arrival	p. 31
Nonnegative, continuous, memoryless random variables	p. 31
Succession of iid exponential interarrival times	p. 31
Merging two independent Poisson streams	p. 32
iid probabilistic routing into a fork	p. 35
Simulation	p. 37
Technique for simulation	p. 37
Generalized Bernoulli random number	p. 37
Geometric and modified geometric random numbers	p. 39
Exponential random number	p. 39
Pareto random number	p. 40
Elements of Parameter Estimation	p. 42
Parameters of Pareto random variable	p. 43
Properties of estimators	p. 46
Sequences of Random Variables	p. 47
Certain and almost certain events	p. 49
Elements of Digital Communication and Data Link Performance	p. 52
The Gaussian noise model	p. 52
Bit error rate evaluation	p. 54
Frame error rate evaluation	p. 56
Data rate optimization	p. 57
Exercises	p. 59
The M/M/1/[infinity] Queue	p. 63
Introduction	p. 63
Derivation of Equilibrium State Probabilities	p. 64
Operation in equilibrium	p. 70
Setting the system to start in equilibrium	p. 71
Simple Performance Figures	p. 72
Response Time and its Distribution	p. 76
More Performance Figures for M/M/1/[infinity] System	p. 77
Waiting Time Distribution	p. 80
Departures from Equilibrium M/M/1/[infinity] System	p. 81
Analysis of ON-OFF Model of Packet Departures	p. 86
Round Robin Operating System	p. 88
Examples	p. 94
Analysis of Busy Times	p. 96
Combinations of arrivals and departures during a busy time period	p. 98
Density function of busy times	p. 99
Laplace transform of the busy time	p. 101
Forward Data Link Performance and Optimization	p. 104
Reliable communication over unreliable data links	p. 104
Problem formulation and solution	p. 105
Exercises	p. 109
State Dependent Markovian Queues	p. 115
Introduction	p. 115
Stochastic Processes	p. 115
Markov process	p. 117
Continuous Parameter Markov Chains	p. 118
Time intervals between state transitions	p. 118
State transition diagrams	p. 118
Development of balance equations	p. 119
Graphical method to write balance equations	p. 123
Markov Chains for State Dependent Queues	p. 124
State dependent rates and equilibrium probabilities	p. 124
General performance figures	p. 127
Throughput	p. 127
Blocking probability	p. 127
Expected fraction of lost jobs	p. 127
Expected number of customers in the system	p. 128
Expected response time	p. 128
Intuitive Approach for Time Averages	p. 129
Statistical Analysis of Markov Chains' Sample Functions	p. 132
Little's Result	p. 141
FIFO case	p. 141
Non-FIFO case	p. 142
Application Systems	p. 143
Constant rate finite buffer M/M/1/k system	p. 143
Forward data link with a finite buffer	p. 146
M/M/[infinity] or immediate service	p. 147
Parallel servers	p. 148
Client-server model	p. 152
Medium Access in Local Area Networks	p. 160
Heavily loaded channel with a contention based transmission protocol	p. 160
Consequences of modeling approximations	p. 161
Analysis steps	p. 162
A simple contention-free LAN protocol	p. 163
Exercises	p. 170
The M/G/1 Queue	p. 179
Introduction	p. 179
Imbedded Processes	p. 180
Equilibrium and Long Term Operation of M/G/1/[infinity] Queue	p. 181
Recurrence equations for state sequence	p. 181
Analysis of equilibrium operation	p. 183
Statistical behavior of the discrete parameter sample function	p. 185
Statistical behavior of the continuous time stochastic process	p. 189
Poisson arrivals see time averages	p. 190
Derivation of the Pollaczek-Khinchin Mean Value Formula	p. 193
Performance figures	p. 198
Application Examples	p. 198
M/D/1/[infinity]: Constant service time	p. 198
M/U/1/[infinity]: Uniformly distributed service time	p. 198
Hypoexponential service time	p. 199
Hyperexponential service time	p. 199
Special Cases	p. 200
Pareto service times with infinite variance	p. 200
Finite buffer M/G/1 system	p. 200
Exercises	p. 202
Discrete Time Queues	p. 209
Introduction	p. 209
Timing and Synchronization	p. 209
State Transitions and Their Probabilities	p. 211
Discrete Parameter Markov Chains	p. 216
Homogeneous Markov chains	p. 218
Chapman-Kolmogorov equations	p. 220
Irreducible Markov chains	p. 220
Classification of States	p. 223
Aperiodic states	p. 223
Transient and recurrent states	p. 226
Analysis of Equilibrium Markov Chains	p. 231
Balance equations	p. 232
Time averages	p. 239
Long term behavior of aperiodic chains	p. 240
Continuous parameter Markov chains	p. 244
Performance Evaluation of Discrete Time Queues	p. 245
Throughput	p. 245
Buffer occupancy	p. 246
Response time	p. 247
Relationship between [pi subscript c] and [pi subscript e]	p. 248
Applications	p. 249
The general Geom/Geom/m/k queue	p. 253
Transition probabilities	p. 253
Equilibrium state probabilities	p. 254
Slotted crossbar	p. 256
Late arrival systems	p. 258
Conclusion	p. 259
Exercises	p. 259
Continuous Time Queuing Networks	p. 267
Introduction	p. 267
Model and Notation for Open Networks	p. 268
Global Balance Equations	p. 270
Traffic Equations	p. 273
The Product Form Solution	p. 276
Validity of Product Form Solution	p. 278
Development of Product Form Solution for Closed Networks	p. 282
Convolution Algorithm	p. 286
Performance Figures from the g(n, m) Matrix	p. 288
Marginal state probabilities	p. 288
Average number in a station	p. 289
Throughput in a station	p. 289
Utilization in a station	p. 289
Expected response time in a station	p. 290
Mean Value Analysis	p. 293
Arrival theorem	p. 294
Cyclic network	p. 295
MVA for cyclic queues	p. 295
Noncyclic closed networks	p. 296
MVA for noncyclic networks	p. 298
Conclusion	p. 301
Exercises	p. 301
The G/M/1 Queue	p. 307
Introduction	p. 307
The Imbedded Markov Chain for G/M/1/[infinity] Queue	p. 307
Analysis of the Parameter [alpha]	p. 313
Stability criterion in terms of the parameters of the queue	p. 317
Determination of [alpha]	p. 319
Performance Figures in G/M/1/[infinity] Queue	p. 321
Expected response time	p. 321
Expected number in the system	p. 321
Finite Buffer G/M/1/k Queue	p. 322
Pareto Arrivals in a G/M/1/[infinity] Queue	p. 323
Exercises	p. 326
Queues with Bursty, MMPP, and Self-Similar Traffic	p. 329
Introduction	p. 329
Distinction between Smooth and Bursty Traffic	p. 331
Self-Similar Processes	p. 334
Fractional Brownian motion	p. 335
Discrete time fractional Gaussian noise and its properties	p. 336
Problems in generation of pure FBM	p. 337
Hyperexponential Approximation to Shifted Pareto Interarrival Times	p. 337
Characterization of Merged Packet Sources	p. 339
Product Form Solution for the Traffic Source Markov Chain	p. 340
Evaluation of h, the Constant in the Product Form Solution	p. 343
Joint Markov Chain for the Traffic Source and Queue Length	p. 344
Evaluation of Equilibrium State Probabilities	p. 348
Analysis of the sequence R[subscript (n)]	p. 351
Queues with MMPP Traffic and Their Performance	p. 355
Performance Figures	p. 357
Conclusion	p. 357
Exercises	p. 358
Analysis of Fluid Flow Models	p. 363
Introduction	p. 363
Leaky Bucket with Two State ON-OFF Input	p. 364
Development of differential equations for buffer content	p. 365
Stability condition	p. 376
Little's Result for Fluid Flow Systems	p. 377
Output Process of Buffer Fed by Two State ON-OFF chain	p. 382
General Fluid Flow Model and its Analysis	p. 384
Leaky Bucket Fed by M/M/1/[infinity] Queue Output	p. 387
Exercises	p. 394
Review of Probability Theory	p. 397
Random Experiment	p. 397
Axioms of Probability	p. 397
Some useful results	p. 398
Conditional probability and statistical independence	p. 399
Random Variable	p. 400
Cumulative distribution function	p. 401
Discrete random variables and the probability mass function	p. 402
Continuous random variables and the probability density function	p. 403
Mixed random variables	p. 404
Conditional pmf and Conditional pdf	p. 405
Expectation, Variance, and Moments	p. 407
Conditional expectation	p. 411
Theorems Connecting Conditional and Marginal Functions	p. 412
Sums of Random Variables	p. 415
Sum of two discrete random variables	p. 415
Sum of two continuous random variables	p. 416
Bayes' Theorem	p. 417
Function of a Random Variable	p. 421
Discrete function of a random variable	p. 421
Discrete function of a discrete random variable	p. 421
Discrete function of a continuous random variable	p. 422
Strictly monotonically increasing function	p. 422
Strictly monotonically decreasing function	p. 423
The general case of a function of a random variable	p. 423
The Laplace Transform L	p. 428
The Z Transform	p. 430
Exercises	p. 434
Index	p. 436
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