《MATHEMATICAL METHODS IN THE THEORY OF QUEUEING》求取 ⇩

PART ⅠTHE INCOMING STREAM OF CALLS11

1THEORY OF THE SIMPLE STREAM11

1 Definition and statement of the problem11

2 Elementary solution12

3 Method of differential equations15

4 Intensity of a simple stream17

5 A Stream with a variable parameter18

2GENERAL PROPERTIES OF STATIONARY STREAMS23

6 The stream of demands as a random process23

7 Fundamental propertiea of stationary streams26

8 General form of a stationary stream without after-effects30

3PALM＇S FUNCTIONS37

9 Definition and proof of existence37

10 Palm＇s formulae39

11 The intensity of a stationary stream—Korolyook＇s theorem41

4STREAMS WITH LIMITED AFTER-EFFECTS43

12 Another method of describing a stream of calls43

13 Streams with limited after-effects44

5LIMIT THEOREM49

14 Statement of the problem—Palm＇s theorem49

15 Limiting behaviour of functions51

16 Limit theorem55

PART ⅡSYSTEMS WITH LOSSES57

17 Introductory remarks57

6ERLANG＇S PROBLEM FOR A FINITE COLLECTION60

18 Statement of the problem60

19 Markoff＇s theorem63

20 Erlang＇s formulae and equations65

21 The ergodic theorem69

7ERLANG＇S PROBLEM FOR AN INFINITE COLLECTION73

22 Equations for the generating function73

23 Solution of the problem75

24 Stream with a variable parameter77

25 The infinite collection with an arbitrary distribution function for the lengths of conversation79

8PALM＇S PROBLEM82

26 Statement of the problem82

27 Elementary calculations84

28 Palm＇s fundamental theorem87

29 Deduction from the basic system of equations88

30 Laplace＇s transformation90

31 Determination of functions ψr(t)91

32 Expansion of functions ψr(t) in partial fractions93

33 Conclusion94

PART ⅢSYSTEMS ALLOWING DELAY96

9THE CASE OF AN EXPONENTIAL DISTRIBUTION FOR THE LENGTH OF CONVERSATIONS96

34 Probabilities of different states96

35 The distribution function of the time of waiting99

10SINGLE-LINE SYSTEMS WITH A FIXED LENGTH OF CONVERSATION102

36 Difference-differential equations for the problem102

37 Distribution function of the waiting time104

11GENERAL THEORY OF SINGLE-LINE SYSTEMS106

38 Statement of the problem and definitions106

39 Subsidiary hypotheses107

40 The characteristic function of the time of waiting114

SECTION NOTES AND REFERENCES119

BIBLIOGRAPHY120