Thesis (M.Sc.) -- University of Adelaide, Dept. of Applied Mathematics, 1978.

Thesis (Ph. D.)--University of North Texas, May, 2007. / Title from title page display. Includes bibliographical references.

This research models the behavior of a multi-task system with respect to time. The type of multi-task system considered here is one in which not all system components are required to perform each task. Each component may, however, be used for more than one task. Also, it is possible that some of the components may be required for every task that the system performs. The components that are required for a subset of the tasks are considered to be intermittently demanded components and those components required for every task are continuously demanded components. This modeling approach assumes that the system is subject to a Modified Age Replacement Policy (MARP). With a MARP the intermittently demanded components are maintained during their idle periods and the continuously demanded components are replaced according to their age replacement times. A renewal theory approach is used to develop an availability expression for the multi-task system. Past research has focused on systems consisting of continuously demanded components and typically does not distinguish between elapsed clock time and elapsed operating time in the renewal density function expressions. This research recognizes that the operational age of an intermittently demanded component is different than the chronological age of the component. The renewal density function and availability measures are modeled using joint density functions defined on both clock time and operating time. The expressions are evaluated numerically using a multidimensional numerical integration routine. The results show logical behavior of the joint density functions used to define the availability measure. The availability measure also displays behavior consistent with the definition of a multi-task system. This model is an important development in the need for stochastic models of highly complex systems. The model is also a first step in defining performance measures for systems composed of both intermittently demanded components and continuously demanded components. / Ph. D.

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The search for a satisfactory model for complexity, meant as an intermediate condition between total order and total disorder, is still subject of debate in the scientific community. In this dissertation the emergence of non-Poisson renewal processes in several complex systems is investigated. After reviewing the basics of renewal theory, another popular approach to complexity, called modulation, is introduced. I show how these two different approaches, given a suitable choice of the parameter involved, can generate the same macroscopic outcome, namely an inverse power law distribution density of events occurrence. To solve this ambiguity, a numerical instrument, based on the theoretical analysis of the aging properties of renewal systems, is introduced. The application of this method, called renewal aging experiment, allows us to distinguish if a time series has been generated by a renewal or a modulation process. This method of analysis is then applied to several physical systems, from blinking quantum dots, to the human brain activity, to seismic fluctuations. Theoretical conclusions about the underlying nature of the considered complex systems are drawn.

Renewal theory began development in the early 1940s, as the need for it in the industrial engineering sub-discipline operations research had risen. In time, the theory found applications in many stochastic processes. In this thesis I investigated the effect of seasonal effects on Poisson and non-Poisson renewal processes in the form of perturbations. It was determined that the statistical analysis methods developed at UNT Center for Nonlinear Science can be used to detect the effects of seasonality on the data obtained from Poisson/non-Poisson renewal systems. It is proved that a perturbed Poisson process can serve as a paradigmatic model for a case where seasonality is correlated to the noise and that diffusion entropy method can be utilized in revealing this relation. A renewal model making a connection with the stochastic resonance phenomena is used to analyze a previous neurological experiment, and it was shown that under the effect of a nonlinear perturbation, a non-Poisson system statistics may make a transition and end up in the of Poisson basin of statistics. I determine that nonlinear perturbation of the power index for a complex system will lead to a change in the complexity characteristics of the system, i.e., the system will reach a new form of complexity.

Thesis (Ph. D.)--University of North Texas, May, 2008. / Title from title page display. Includes bibliographical references.

Thesis (Ph. D.)--Ohio State University, 2006. / Title from first page of PDF file. Includes bibliographical references (p. 152-153).

Many social, biological and technological systems can be viewed as complex networks with a large number of interacting components. However despite recent advancements in network theory, a satisfactory description of dynamic processes arising in such cooperative systems is a subject of ongoing research. In this dissertation the emergence of dynamical complexity in networks of interacting stochastic oscillators is investigated. In particular I demonstrate that networks of two and three state stochastic oscillators present a second-order phase transition with respect to the strength of coupling between individual units. I show that at the critical point fluctuations of the global order parameter are characterized by an inverse-power law distribution and I assess their renewal properties. Additionally, I study the effect that different types of perturbation have on dynamical properties of the model. I discuss the relevance of those observations for the transmission of information between complex systems.

Same types of systems with the same age usually have different amounts of cumulated usage. These systems when in operation usually have different performance and effectiveness. In this case the existing models of the univariate measures of system effectiveness are inadequate and incomplete. For example, the univariate availability measures for these same-aged systems are all the same even though with different amounts of usage. This is the motivation for this research to pursue a bivariate approach in reliability and maintenance modeling. This research presents a framework for bivariate modeling of a single-unit system. Five key efforts are identified and organized as: (i) bivariate failure modeling, (ii) bivariate renewal modeling, (iii) bivariate corrective maintenance (CM) modeling, (iv) bivariate preventive maintenance (PM) modeling, and (v) bivariate availability modeling. The results provide a foundation for further study of bivariate and multivariate models. For bivariate failure modeling, several bivariate failure models are constructed to represent the possible correlation structures of the two system aging variables, time and usage. The behavior of these models is examined under various correlation structures. The developed models are used to analyze example maintenance problems. Models for bivariate renewal, bivariate CM, and bivariate PM are derived based on the constructed bivariate failure models and the developed bivariate renewal theory. For bivariate CM modeling, corrective maintenance is modeled as an alternating bivariate renewal process or simply an ordinary bivariate renewal process. For bivariate PM modeling, PM models are examined under a bivariate age replacement preventive maintenance policy. The Laplace transforms of the renewal functions (and densities) for these models are obtained. Definitions for bivariate availability functions are developed. Based on the derived CM and PM models, the Laplace transforms for their corresponding bivariate availability models are constructed. The idea of the quality of availability measure is also defined in terms of bivariate availability models. The most significant observation is that this framework provides a new way to study the reliability and maintenance of equipment for which univariate measures are incomplete. Therefore, a new area of reliability research is identified. The definitions offered may be modified and the approach to model formulation presented may be used to define other models. / Ph. D.