Advanced Theory for Poisson Shock Models

In this chapter, we extend and generalize approaches and results of the previous chapter to various reliability related settings of a more complex nature. We relax some assumptions of the traditional models except the one that defines the underlying shock process as the nonhomogeneous Poisson process (NHPP). Only in the last section, we suggest an alternative to the Poisson process to be called the geometric point process. It is remarkable that although the members of the class of geometric processes do not possess the property of independent increments, some shock models can be effectively described without specifying the corresponding dependence structure. Most of the contents of this chapter is based on our recent work [5–11] and covers various settings that, we believe, are meaningful both from the theoretical and the practical points of view. The chapter is rather technical in nature; however, general descriptions of results are reasonably simple and illustrated by meaningful examples. As the assumption of the NHPP of shocks is adopted, many of the proofs follow the same pattern by using the time-transformation of the NHPP to the HPP (see the derivation of Eq. ( 2.31 )). This technique will be used often in this chapter. Sometimes the corresponding derivations will be reasonably abridged, whereas other proofs will be presented at full length.