## Queueing Theory with Applications to Packet TelecommunicationQueueing Theory with Applications to Packet Telecommunication is an efficient introduction to fundamental concepts and principles underlying the behavior of queueing systems and its application to the design of packet-oriented electrical Key features of communication systems, such as correlation in packet arrival processes at IP switches and variability in service rates due to fading wireless links are Queueing Theory with Applications to Packet Telecommunication is intended both for self study and for use as a primary text in graduate courses in queueing theory in electrical engineering, computer science, operations research, and mathematics. Professionals will also find this work invaluable because the author discusses applications such as statistical multiplexing, IP switch design, and wireless communication systems. In addition, numerous modeling issues, such as the suitability of Erlang-k and Pade approximations are addressed. |

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### Contents

TERMINOLOGY AND EXAMPLES | 1 |

11 The Terminology of Queueing Systems | 2 |

12 Examples of Application to System Design | 9 |

122 Multiplexing Packets at a Switch | 11 |

123 CDMABased Cellular Data | 14 |

13 Summary | 17 |

REVIEW OF RANDOM PROCESSES | 19 |

21 Statistical Experiments and Probability | 20 |

44 Supplementary Problems | 156 |

THE BASIC MG1 QUEUEING SYSTEM | 159 |

51 MG1 Transform Equations | 161 |

511 Sojourn Time for MG1 | 165 |

512 Waiting Time for MG1 | 167 |

52 Ergodic Occupancy Distribution for MG1 | 170 |

522 Recursive Approach | 180 |

523 Generalized StateSpace Approach | 183 |

212 Conditioning Experiments | 22 |

22 Random Variables | 27 |

23 Exponential Distribution | 33 |

24 Poisson Process | 39 |

25 Markov Chains | 45 |

ELEMENTARY CTMCBASED QUEUEING MODELS | 57 |

31 MM1 Queueing System | 58 |

312 Stochastic Equilibrium MM1 Distributions | 60 |

313 Busy Period for MM1 Queueing System | 76 |

32 Dynamical Equations for General BirthDeath Process | 81 |

33 TimeDependent State Probabilities for FiniteState Systems | 83 |

331 Classical Approach | 84 |

332 Jensens Method | 88 |

34 Balance Equation Approach for Systems In Equilibrium | 91 |

35 Probability Generating Function Approach | 98 |

36 Supplementary Problems | 101 |

ADVANCED CTMCBASED QUEUEING MODELS | 107 |

41 Networks | 108 |

Fixed Routing | 109 |

412 Arbitrary Open Networks | 110 |

413 Closed Networks of Single Servers | 111 |

42 PhaseDependent Arrivals and Service | 122 |

421 Probability Generating Function Approach | 124 |

422 Matrix Geometric Method | 138 |

423 Rate Matrix Computation via Eigenanalysis | 143 |

424 Generalized StateSpace Methods | 146 |

43 PhaseType Distributions | 152 |

53 Expected Values Via Renewal Theory | 210 |

532 Busy Periods and Alternating Renewal Theory | 216 |

54 Supplementary Problems | 219 |

THE MG1 QUEUEING SYSTEM WITH PRIORITY | 225 |

61 MG1 Under LCFSPR Discipline | 226 |

62 MG1 System with Exceptional First Service | 229 |

63 MG1 under HOL Priority | 236 |

631 Higher Priority Customers | 238 |

632 Lower Priority Customers | 241 |

64 Ergodic Occupancy Probabilities for Priority Queues | 244 |

65 Expected Waiting Times for under HOL Priority | 246 |

651 HOL Discipline | 248 |

652 HOLPR Discipline | 249 |

VECTOR MARKOV CHAIN ANALYSIS | 253 |

71 The MG1 and GM1 Paradigms | 254 |

72 GM1 Solution Methodology | 259 |

73 MG1 Solution Methodology | 261 |

74 An Application to Statistical Multiplexing | 265 |

Complex Boundaries | 278 |

76 Summary | 290 |

77 Supplementary Problems | 294 |

CLOSING REMARKS | 297 |

301 | |

309 | |

About the Author | 315 |