Defective renewal equations

Abstract

Defective renewal equations arise in many different areas of application of applied probability. For example, the equilibrium waiting time distribution in many queueing models and the number of offspring distribution in branching processes satisfy certain defective renewal equations. For a detailed discussion of these applications, see Feller (1971) or Resnick (1992). Defective renewal equations also play an important role in insurance risk theory. Many functionals of interest associated with the time of ruin in the classical risk model are in the form of the solution of a defective renewal equation, as will be seen later. On the other hand, compound geometric distributions are appealing for the reasons that (i) the analytic expression of a compound geometric tail is obtainable for some individual claim amount distributions, (ii) tight upper and lower bounds based on reliability classifications are available for compound geometric tails, and (iii) there exists a satisfactory approximation to compound geometric tails. These desirable properties are discussed in chapters 7 and 8. Motivated by the fact that a compound geometric distribution may be viewed as the solution of a special defective renewal equation, we explore a close connection between defective renewal equations and compound geometric distributions in this chapter. The solution of a defective renewal equation is expressed in terms of a compound geometric distribution. This connection allows for the use of analytic properties of a compound geometric distribution and for the application of exact and approximate results which have been developed for the compound geometric tail in chapters 7 and 8 to the solution of the renewal equation itself.