Efficient Multirate Teletraffic Loss Models Beyond Erlang

List of Figures xvii List of Tables xxv Preface xxix Acronyms xxxiii Symbols xxxvii About the Companion Website xxxix Introduction xli I.1 Traffic-load Definition xlii I.2 Traffic Congestion and GoS/QoS xliii I.3 System Capacity xliv I.4 Teletraffic Models xlv I.5 Traffic-load Properties xlviii I.6 Call Arrival Process l I.6.1 Superposition and Decomposition of Poisson Processes lv I.6.2 Poisson Arrivals See Time Averages lvi I.7 Call Service Time lvii I.7.1 Markov Property lvii I.8 Service Systems lix I.9 Little's Law lxi I.10 Other Performance Metrics of Loss Systems lxii I.11 General Examples lxiii I.12 Service-classes - Bandwidth Sharing Policies lxiv I.13 Classification of Teletraffic Loss Models lxx I.14 Teletraffic Models and the Internet lxxi References lxxiv Part I Teletraffic Models of Random Input 1 1 The Erlang Multirate Loss Model 3 1.1 The Erlang Loss Model 3 1.1.1 The Service System 3 1.1.2 Global and Local Balance 5 1.1.3 Call Blocking Probability 8 1.1.4 Other Performance Metrics 11 1.2 The Erlang Multirate Loss Model 13 1.2.1 The Service System 13 1.2.2 The Analytical Model 15 1.3 The Erlang Multirate Loss Model under the BR policy 28 1.3.1 The Service System 28 1.3.2 The Analytical Model 30 1.4 The Erlang Multirate Loss Model under the Threshold Policy 38 1.4.1 The Service System 38 1.4.2 The Analytical Model 40 1.5 The Erlang Multirate Loss Model in a Fixed Routing Network 44 1.5.1 The Service System 44 1.5.2 The Analytical Model 45 1.5.3 CBP Calculation by the RLA Method 49 1.6 Applications 54 1.6.1 The Erlang-B Formula 54 1.6.2 The Erlang-C Formula 55 1.6.3 The Kaufman-Roberts Recursion 56 1.7 Further Reading 58 References 60 2 Multirate Retry Threshold Loss Models 65 2.1 The Single-Retry Model 65 2.1.1 The Service System 65 2.1.2 The Analytical Model 69 2.2 The Single-Retry Model under the BR Policy 72 2.2.1 The Service System 72 2.2.2 The Analytical Model 75 2.3 The Multi-Retry Model 77 2.3.1 The Service System 77 2.3.2 The Analytical Model 83 2.4 The Multi-Retry Model under the BR Policy 86 2.4.1 The Service System 86 2.4.2 The Analytical Model 87 2.5 The Single-Threshold Model 92 2.5.1 The Service System 92 2.5.2 The Analytical Model 96 2.6 The Single-Threshold Model under the BR Policy 99 2.6.1 The Service System 99 2.6.2 The Analytical Model 101 2.7 The Multi-Threshold Model 107 2.7.1 The Service System 107 2.7.2 The Analytical Model 107 2.8 The Multi-Threshold Model under the BR Policy 109 2.8.1 The Service System 109 2.8.2 The Analytical Model 109 2.9 The Connection Dependent Threshold Model 112 2.9.1 The Service System 112 2.9.2 The Analytical Model 114 2.10 The Connection Dependent Threshold Model under the BR Policy 119 2.10.1 The Service System 119 2.10.2 The Analytical Model 119 2.11 Applications 121 2.12 Further Reading 129 References 130 3 Multirate Elastic Adaptive Loss Models 133 3.1 The Elastic Erlang Multirate Loss Model 133 3.1.1 The Service System 133 3.1.2 The Analytical Model 139 3.2 The Elastic Erlang Multirate Loss Model under the BR Policy 146 3.2.1 The Service System 146 3.2.2 The Analytical Model 149 3.3 The Elastic Erlang Multirate Loss Model under the Threshold Policy 152 3.3.1 The Service System 152 3.3.2 The Analytical Model 156 3.4 The Elastic Adaptive Erlang Multirate Loss Model 163 3.4.1 The Service System 163 3.4.2 The Analytical Model 171 3.5 The Elastic Adaptive Erlang Multirate Loss Model under the BR Policy 175 3.5.1 The Service System 175 3.5.2 The Analytical Model 177 3.6 The Elastic Adaptive Erlang Multirate Loss Model under the Threshold Policy 179 3.6.1 The Service System 179 3.6.2 The Analytical Model 182 3.7 Applications 185 3.8 Further Reading 190 References 191 4 Multirate Elastic Adaptive Retry Loss Models 195 4.1 The Elastic Single-Retry Model 195 4.1.1 The Service System 195 4.1.2 The Analytical Model 201 4.2 The Elastic Single-Retry Model under the BR Policy 206 4.2.1 The Service System 206 4.2.2 The Analytical Model 210 4.3 The Elastic Multi-Retry Model 212 4.3.1 The Service System 212 4.3.2 The Analytical Model 218 4.4 The Elastic Multi-Retry Model under the BR Policy 220 4.4.1 The Service System 220 4.4.2 The Analytical Model 223 4.5 The Elastic Adaptive Single-Retry Model 226 4.5.1 The Service System 226 4.5.2 The Analytical Model 233 4.6 The Elastic Adaptive Single-Retry Model under the BR Policy 237 4.6.1 The Service System 237 4.6.2 The Analytical Model 241 4.7 The Elastic Adaptive Multi-Retry Model 243 4.7.1 The Service System 243 4.7.2 The Analytical Model 248 4.8 The Elastic Adaptive Multi-Retry Model under the BR Policy 250 4.8.1 The Service System 250 4.8.2 The Analytical Model 254 4.9 Applications 258 4.10 Further Reading 258 References 260 5 ON-OFF Multirate Loss Models 263 5.1 The ON-OFF Multirate Loss Model 263 5.1.1 The Service System 263 5.1.2 The Analytical Model 265 5.2 The ON-OFF Multirate Loss Model under the BR Policy 275 5.2.1 The Service System 275 5.2.2 The Analytical Model 276 5.3 The ON-OFF Multirate Loss Model in a Fixed Routing Network 280 5.3.1 The Service System 280 5.3.2 The Analytical Model 280 5.4 Applications 285 5.5 Further Reading 288 References 289 Part II Teletraffic Models of Quasi-Random Input 291 6 The Engset Multirate Loss Model 293 6.1 The Engset Loss Model 293 6.1.1 The Service System 293 6.1.2 The Analytical Model 293 6.2 The Engset Multirate Loss Model 298 6.2.1 The Service System 298 6.2.2 The Analytical Model 300 6.3 The Engset Multirate Loss Model under the BR Policy 308 6.3.1 The Service System 308 6.3.2 The Analytical Model 310 6.4 The Engset Multirate Loss Model under the TH Policy 312 6.4.1 The Service System 312 6.4.2 The Analytical Model 312 6.5 Applications 318 6.6 Further Reading 324 References 327 7 Finite Multirate Retry Threshold Loss Models 331 7.1 The Finite Single-Retry Model 331 7.1.1 The Service System 331 7.1.2 The Analytical Model 333 7.2 The Finite Single-Retry Model under the BR Policy 338 7.2.1 The Service System 338 7.2.2 The Analytical Model 340 7.3 The Finite Multi-Retry Model 342 7.3.1 The Service System 342 7.3.2 The Analytical Model 344 7.4 The Finite Multi-Retry Model under the BR Policy 348 7.4.1 The Service System 348 7.4.2 The Analytical Model 349 7.5 The Finite Single-Threshold Model 353 7.5.1 The Service System 353 7.5.2 The Analytical Model 355 7.6 The Finite Single-Threshold Model under the BR Policy 360 7.6.1 The Service System 360 7.6.2 The Analytical Model 362 7.7 The Finite Multi-Threshold Model 363 7.7.1 The Service System 363 7.7.2 The Analytical Model 364 7.8 The Finite Multi-Threshold Model under the BR Policy 366 7.8.1 The Service System 366 7.8.2 The Analytical Model 366 7.9 The Finite Connection Dependent Threshold Model 367 7.9.1 The Service System 367 7.9.2 The Analytical Model 368 7.10 The Finite Connection Dependent Threshold Model under the BR Policy 373 7.10.1 The Service System 373 7.10.2 The Analytical Model 373 7.11 Applications 374 7.12 Further Reading 374 References 375 8 Finite Multirate Elastic Adaptive Loss Models 377 8.1 The Elastic Engset Multirate Loss Model 377 8.1.1 The Service System 377 8.1.2 The Analytical Model 380 8.2 The Elastic Engset Multirate Loss Model under the BR Policy 383 8.2.1 The Service System 383 8.2.2 The Analytical Model 385 8.3 The Elastic Adaptive Engset Multirate Loss Model 387 8.3.1 The Service System 387 8.3.2 The Analytical Model 389 8.4 The Elastic Adaptive Engset Multirate Loss Model under the BR Policy 392 8.4.1 The Service System 392 8.4.2 The Analytical Model 394 8.5 Applications 402 8.6 Further Reading 405 References 405 9 Finite ON-OFF Multirate Loss Models 407 9.1 The Finite ON-OFF Multirate Loss Model 407 9.1.1 The Service System 407 9.1.2 The AnalyticalModel 408 9.2 Generalization of the f-ON-OFF Model to include Service-classes with a Mixture of a Finite and an Infinite Number of Sources 415 9.3 Applications 416 9.4 Further Reading 422 References 423 Part III Teletraffic Models of Batched Poisson Input 425 10 The Erlang Multirate Loss ModelWith Batched Poisson Arrivals 427 10.1 The Erlang Multirate Loss Model with Batched Poisson Arrivals 427 10.1.1 The Service System 427 10.1.2 The AnalyticalModel 428 10.2 The Erlang Multirate Loss Model with Batched Poisson Arrivals under the BR Policy 435 10.2.1 The Service System 435 10.2.2 The AnalyticalModel 435 10.3 The Erlang Multirate Loss Model with Batched Poisson Arrivals under the Threshold Policy 441 10.3.1 The Service System 441 10.3.2 The Analytical Model 441 10.4 Applications 445 10.5 Further Reading 451 References 451 11 Batched Poisson Multirate Elastic Adaptive Loss Models 455 11.1 The Elastic Erlang Multirate Loss Model with Batched Poisson Arrivals 455 11.1.1 The Service System 455 11.1.2 The Analytical Model 456 11.2 The Elastic Erlang Multirate Loss Model with Batched Poisson Arrivals under the BR Policy 461 11.2.1 The Service System 461 11.2.2 The Analytical Model 463 11.3 The Elastic Adaptive Erlang Multirate Loss Model with Batched Poisson Arrivals 466 11.3.1 The Service System 466 11.3.2 The Analytical Model 467 11.4 The Elastic Adaptive Erlang Multirate Loss Model with Batched Poisson Arrivals under the BR Policy 475 11.4.1 The Service System 475 11.4.2 The Analytical Model 477 11.5 Applications 482 11.6 Further Reading 483 References 485 Appendix A Interdependency of the Teletraffic Models 487 Index 491

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