Renewal theory arose from the study of 'self-renewing aggregates',
but more recently has developed into the investigation of some
general results in the theory of probability connected with sums of
independent non-negative random variables. These results are applicable
to quite a wide range of practical probability problems. The
object of this monograph is to present the main results in an applied
mathematical way. That is to say, the emphasis is on formulae that
can be used to answer specific problems rather than on proofs of
theorems under conditions of the utmost generality.
The monograph is intended for students and research workers in
statistics and probability theory, and for others, especially those in
operational research, whose work involves the application of probability
theory. To understand the monograph, familiarity with elementary
probability theory is essential. It is also desirable to know
about the simpler properties of the Laplace transform; the relevant
results are, however, reviewed in one of the preliminary sections.
Author(s): D.R. Cox
Edition: 1
Publisher: Methuen & Co.
Year: 1970
Language: English
Pages: x, 142
Preface
CHAPTER 1 Preliminaries
1.1. Introduction
1.2. The distribution of failure-time
1.3. Laplace transforms
1.4. Special distributions
CHAPTER 2 The Fundamental Models
2.1. The ordinary renewal process
2.2. Two other models
2.3. The Poisson process
2.4. Some further examples of renewal processes
2.5. The time up to the rth renewal
CHAPTER 3 The Distribution of the Number of Renewals
3.1. Some general formulae
3.2. The probability generating function
3.3. The asymptotic distribution of N
3.4. The number of renewals in a random time
CHAPTER 4 The Moments of the Number of Renewals
4.1. The renewal function
4.2. The asymptotic form of the renewal function
4.3. A more detailed study of the renewal function
4.4. The renewal density
4.5. The variance of the number of renewals
4.6. The higher moments
CHAPTER 5 Recurrence-Times
5.1. The backward recurrence-time
5.2. The forward recurrence-time
5.3. The limiting distribution of recurrence-time
5.4. An alternative derivation of the limiting distribution
5.5. An application to the number of renewals in an arbitrary interval
CHAPTER 6 The Superposition of Renewal Processes
6.1. Introduction
6.2. The pooled output of several renewal processes
6.3. Some general properties
6.4. The mean time up to the rth renewal
6.5. The interval between successive renewals
6.6. A large number of component processes
CHAPTER 7 Alternating Renewal Processes
7.1. Introduction
7.2. The renewal functions
7.3. The type of component in use at timet
7.4. Equilibrium alternating renewal processes
7.5. The precision of systematic sampling
CHAPTER 8 Cumulative Processes
8.1. Introduction
8.2. Independent increments
8.3. The cumulative process associated with a Poisson process
8.4. The first passage time
8.5. A general limiting result
CHAPTER 9 Some other Generalizations
9.1. Introduction
9.2. Some results based on the laws of large numbers
9.3. Some limiting results based on the relation between Nc and S.
9.4. Some exact results
9.5. Failure-times which may be negative
CHAPTER 10 Probabilistic Models of Failure
10.1. Introduction
10.2. Failures of many types
10.3. Two types offailure
10.4. Dependence of failures on wear
CHAPTER 11 Strategies of Replacement
11.1. Introduction
11.2. Some simple strategies
11.3. A strategy involving idle time
11.4. Strategies involving wear
APPENDIX I Bibliographical Notes
APPENDIX II Exercises and Further Results
APPENDIX III References
Index
