A Study On Aggregated Stochastic Models And Their Related Problems

The performance of system is one of the most important issues in both theory and practice.Therefore,a series of indexes which can describe the system's performance properly and correctly is required to be proposed for evaluating the system's performance.Although it has a lot of indexes for evaluation of system performance such as reliability,availability and safety and so forth,they still cannot meet the variety requirements on the evaluation of system performance.Thus,based on the aggregated stochastic process theory,it is an important work to introduce and derive the pont-wise and interval-wise indexes.The main contents of this dissertation are as follows.Firstly,the aggregated stochastic model with classifications based on sojourn times is developed under an alternative renewal process.Two types of related probability measures,point-wise and interval-wise probabilities,including their concepts and computation formulae,are developed under an alternative renewal process and its derivative aggregated stochastic process.All limits of the introduced new measures are discussed too,and the relationship among these new measures is studied.The corresponding reliability indexes are obtained,when the operating system follows exponential distributions.Secondly,the aggregated Markov repairable model with multi-states is proposed,by describing the specific repairable system with distinct aggregated stochastic process.In this model,the state space is partitioned into two subsets,the working subset W and the failure subset F.The system is regarded as stable,if the state of system enters one subset,either W or F,at any instance and sojourns within the subset exceeding a given nonnegative threshold?.Otherwise,the system is regarded as unstable.Thereafter,the point-wise and interval-wise probabilities indexes are introduced under a Markov process and its derivative aggregated stochastic process,their compution formulae are derived.The research results can be used not only in reliability but it may also be used in finance,economy and other fields.Thirdly,a single-unit repairable model with working and repair time omission is researched under an alternative renewal process,as the working time is shorter than the threshold?₁,which leads some working states are regarded as failure states during the observation.Likewise,some failure states are regarded as working states during the observation,because the repair time is shorter than the threshold?₂.In this model,the point and interval availability indexes are derived under model assumptions.Fourthly,if the sojourn time of the system with each state subjects to two given thresholds,a 3-state stochastic model with state classifications based on sojourn times is studied according to the alternative renewal theory.To be specific,if the sojourn time of the underlying process in some class is greater than the given threshold?₂,then the newly defined process is in state 2;if the sojourn time of the underlying process in some class is less than the given threshold?₁,then the newly defined process is in state 0;if the sojourn time of the underlying process in some class is less than the given threshold value?₂and greater than?₁,then the newly defined process is in state 1.In this model,the two-category indexes are introduced,named as point state and interval state probability measures,respectively.In the meantime,some computation formulae are derived.This dissertation has important theoretical and practical significance.The results of the dissertation promote the theoretical development of the aggregated stochastic processes theory and expand its application in reliability and maintenance modeling.