A fractal stochastic point process (FSPP) is used to model molecular evolution in agreement with the relationship between the variance and mean numbers of synonymous and nonsynonymous substitutions in mammals. Like other episodic models such as the doubly stochastic Poisson process, this model accounts for the large variances observed in amino acid substitution rates, but unlike other models, it also accounts for the results of Ohta's (1995) analysis of synonymous and nonsynonymous substitutions in mammalian genes. That analysis yields a power-law increase in the index of dispersion and an inverse power-law decrease in the coefficient of variation with the mean number of substitutions, as predicted by the FSPP model but not by the doubly stochastic Poisson model. This result is compatible with the selection theory of evolution and the nearly-neutral theory of evolution.

This thesis deals with the reliability and maintenance of structures that are damaged by shocks arriving randomly in time. The degradation is modeled as a cumulative stochastic point process. Previous studies mostly adopted expected cost rate criterion for optimizing the maintenance policies, which ignores practical implications of discounting of maintenance cost over the life cycle of the system.Therefore, detailed analysis of expected discounted cost criterion has been done, which provides a more realistic basis for optimizing the maintenance. Examples of maintenance policies combining preventive maintenance with age- based replacement are analyzed. Derivation for general cases involving preventive maintenance damage level have been discussed. Special cases are also considered.

This thesis deals with the reliability and maintenance of structures that are damaged by shocks arriving randomly in time. The degradation is modeled as a cumulative stochastic point process. Previous studies mostly adopted expected cost rate criterion for optimizing the maintenance policies, which ignores practical implications of discounting of maintenance cost over the life cycle of the system.Therefore, detailed analysis of expected discounted cost criterion has been done, which provides a more realistic basis for optimizing the maintenance. Examples of maintenance policies combining preventive maintenance with age- based replacement are analyzed. Derivation for general cases involving preventive maintenance damage level have been discussed. Special cases are also considered.