Table of Contents
Preface
Part I The Incoming Stream of Calls
1 Theory of the Simple Stream
1 Definition and statement of the problem 11
2 Elementary solution 12
3 Method of differential equations 15
4 Intensity of a simple stream 17
5 A stream with a variable parameter 18
2 General Properties of Stationary Streams
6 The stream of demands as a random process 23
7 Fundamental properties of stationary streams 26
8 General form of a stationary stream without after-effects 30
3 Palm's Functions
9 Definition and proof of existence 37
10 Palm's formulae 39
11 The intensity of a stationary stream - Korolyook's theorem 41
4 Streams with Limited After-Effects
12 Another method of describing a stream of calls 43
13 Streams with limited after-effects 44
5 Limit Theorem
14 Statement of the problem-Palm's theorem 49
15 Limiting behaviour of functions 51
16 Limit theorem 55
Part II Systems with Losses
17 Introductory remarks 57
6 Erlang's Problem for a Finite Collection
18 Statement of the problem 60
19 Markoff's theorem 63
20 Erlang's formulae and equations 65
21 The ergodic theorem 69
7 Erlang's Problem for an Infinite Collection
22 Equations for the generating function 73
23 Solution of the problem 75
24 Stream with a variable parameter 77
25 The infinite collection with an arbitrary distribution function for the lengths of conversation 79
8 Palm's Problem
26 Statement of the problem 82
27 Elementary calculations 84
28 Palm's fundamental theorem 87
29 Deduction from the basic system of equations 88
30 Laplace's transformation 90
31 Determination of functions Ψ r(t) 91
32 Expansion of functions Ψr(t) in partial fractions 93
33 Conclusion 94
Part III Systems Allowing Delay
9 The Case of an Exponential Distribution for the Length of Conversations
34 Probabilities of different states 96
35 The distribution function of the time of waiting 99
10 Single-Line Systems with a Fixed Length of Conversation
36 Difference-differential equations for the problem 102
37 Distribution function of the waiting time 104
11 General Theory of Single-Line Systems
38 Statement of the problem and definitions 106
39 Subsidiary hypotheses 107
40 The characteristic function of the time of waiting 114
Section Notes and References 119
Bibliography 120
