Some New Results In Geometric Processes And Stochastic Models

It is well known that Renewal process has many applications in applied proba-bility models. Geometric process, first introduced by Lam in 1987 to 1988, is a gen-eralization of the Renewal process. Many authors have studied the theory, statisticalanalysis and applications of geometric process in reliability and stochastic modelling.In the first chapter, we introduce the concept of geometric process and some fieldsof its applications.In the second chapter of this thesis, we study the solution of the integral equa-tion of the geometric process. We propose some analytic and numerical method forfinding the solution of the integral equation. The analytic solution uses the theoryand method of reproducing kernels. We first give an analytic expression for the so-lution of the integral equation. The analytic solution is a series solution, which hasthe following properties: (1) given the values of the right-hand function of the inte-gral equation, we can construct the solution of the integral equation; (2) the seriessolution is composed by primary functions and after truncation, we can obtain an ap-proximate solution, when the number of the nodes increases, the approximate solutionconverges to the true solution, and with the increasing of the number of nodes, thenorm of error of the approximate solution decreases. The numerical method, in fact,is mechanical quadrature method. Using trapezoidal rule to discretize the integral inthe transformed integral equation, we can easily obtain an recursive formula to com-pute the approximate value of the solution of the integral equation at the nodes in aninterval. Imposing the Euler-Maclaurin expansion of the trapezoidal rule and a gener-alized Gronwall inequality, we prove that our numerical result is of order O(h2). Onemerit of the numerical solution is that we can further consider splitting extrapolationto accelerate convergence. We also compare the numerical results and the results fromsimulation.In the third chapter, we study a new δ-shock model, in which we assume that thesystem will fail if and only if the arriving time of the first shock after replacement orthe inter-arrival time of two successive shocks is less than a specified threshold. Thismodel is called a δ-shock model. Under the assumption that shocks arrive accordingto a renewal process with Weibull, gamma or lognormal distributed inter-arrival time,we find the expression of the long-run average cost per unit time and the analyticexpression for the optimal replacement policy N?. Examples are also given, whichshows that the optimal replacement policy exists and is unique. We also study thesensitivity analysis of the optimal replacement policy and the corresponding long-runaverage cost per unit time with respect to some important parameters in the model.In the forth chapter, we study another stochastic model involving geometricprocess-a deteriorating system with two types of failures. Assume that when the sys-tem fails, it may have two types of failures, one of which can be removed by repair,and the other can only be removed by replacement of the system. Under a generalizedreplacement policy N for the first type of failure, we evaluate the expression for thelong-run average cost per unit time C(N). One can use analytic or numerical methodsto minimize the long-run average cost per unit time for finding the optimal replacementpolicy N?.In the fifth chapter, we study the application of geometric-Process in Time SeriesAnalysis. We use geometric process and its generalization, a threshold geometric pro-cess to fit two famous series in Time Series Analysis, one is the data of Canadian lynx,and the other is the sunspot data. Analysis shows that geometric process and thresholdgeometric process are useful in fitting time series with trend.