Big Queues |
Contents
The Single Server Queue | 1 |
11 The SingleServer Queueing Model | 3 |
12 OneDimensional Large Deviations | 6 |
13 Application to Queues with Large Buffers | 9 |
14 Application to Queues with Many Sources | 15 |
Large Deviations in Euclidean Spaces | 23 |
22 Principle of the Largest Term | 25 |
23 Large Deviations Principle | 26 |
66 Queueing Delay | 126 |
67 Departure Process | 128 |
68 Mean Rate of Departures | 130 |
69 QuasiReversibility | 137 |
610 Scaling Properties of Networks | 144 |
611 Statistical Inference for the TailBehaviour of Queues | 146 |
ManyFlows Scalings | 151 |
72 Topology for Sample Paths | 152 |
24 Cumulant Generating Functions | 27 |
25 Convex Duality | 29 |
26 Cramers Theorem | 32 |
27 Sanovs Theorem for Finite Alphabets | 38 |
28 A Generalisation of Cramers Theorem | 41 |
More on the Single Server Queue | 47 |
32 Queues with Many Sources and PowerLaw Source Scalings | 52 |
33 Queues with Large Buffers and PowerLaw Source Scalings | 55 |
Introduction to Abstract Large Deviations | 57 |
42 Definition of LDP | 59 |
43 The Contraction Principle | 63 |
44 Other Useful LDP Results | 67 |
Continuous Queueing Maps | 77 |
Queues with Large Buffers | 78 |
53 The Continuous Mapping Approach | 80 |
54 Continuous Functions | 81 |
55 Some Convenient Notation | 83 |
56 Queues with Infinite Buffers | 84 |
57 Queues with Finite Buffers | 88 |
58 Queueing Delay | 92 |
59 Priority Queues | 94 |
510 Processor Sharing | 95 |
511 Departures from a Queue | 98 |
512 Conclusion | 103 |
LargeBuffer Scalings | 105 |
62 Large Deviations for Partial Sums Processes | 107 |
63 Linear Geodesies | 117 |
64 Queues with Infinite Buffers | 120 |
65 Queues with Finite Buffers | 125 |
73 The Sample Path LDP | 155 |
74 Example Sample Path LDPs | 162 |
75 Applying the Contraction Principle | 165 |
76 Queues with Infinite Buffers | 166 |
77 Queues with Finite Buffers | 170 |
78 Overflow and Underflow | 171 |
79 Paths to Overflow | 173 |
710 Priority Queues | 176 |
711 Departures from a Queue | 177 |
Long Range Dependence | 183 |
82 Implications for Queues | 185 |
for Fractional Brownian Motion | 187 |
84 Scaling Properties | 190 |
85 How Does Long Range Dependence Arise? | 195 |
86 Philosophical Difficulties with LRD Modelling | 197 |
Moderate Deviations Scalings | 199 |
92 Traffic Processes | 202 |
93 Queue Scalings | 203 |
94 Shared Buffers | 205 |
95 Mixed Limits | 208 |
Interpretations | 211 |
102 Numerical Estimates | 218 |
103 A Global Approximation | 226 |
104 Scaling Laws | 230 |
105 Types of Traffic | 232 |
| 239 | |
| 249 | |
| 251 | |
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Common terms and phrases
absolutely continuous arrival process Chapter closed sets compact continuous functions contraction principle convex conjugate Cramér's theorem cumulant generating function define departure process differentiable discrete-time distribution effective bandwidth effective domain estimate Example exponential flows fractional Brownian motion Gaussian Hurstiness independent copies inf I(x inf sup infimum infinite buffer infinite horizons input interval large deviations principle large deviations theory largest term Lemma Let Xn lim inf lim sup limit linear geodesics log P(QN long-range dependence many-flows Markov mean rate moderate deviations principle moment generating function neighbourhood optimal overflow P(Xn probability proof q+Ct queue fed queue length queue size function random variables rate function recursion result sample path LDP satisfies a large satisfies a sample satisfies an LDP Section service rate single-server queue sup log Suppose supremum t>to traffic uniform convergence upper bound Xn satisfies
Popular passages
Page 240 - C.-S. Chang. Stability, queue length, and delay of deterministic and stochastic queueing networks.