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Functional limit theorems for Nonstationary Stochastic Systems

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Document pages: 191 pages

Abstract: Modern service systems, such as call centers and healthcare systems, are usually large-scale and have non fixed data, including customer arrival rate, service time and staffing. A better understanding of such service systems will help to improve system efficiency, optimize system management and reduce costs. However, due to the nonstationarity, randomness and large scale of this kind of system, accurate analysis is difficult. Stochastic models under heavy traffic conditions help to obtain approximate values and provide insights for operation management. The main work of this paper is to study the stochastic model with nonstationarity and establish the relevant functional limit theorems.t theorems for two-parameter processes describing the system dynamics of non-stationary non-Markovian queueing models. In particular, I have studied i) infinite- server queues with non-stationary arrival and service times (arrival dependent services), ii) infinite-server queues with non-stationary arrival and weakly dependent general service times, and iii) non-Markovian many-server queues with non-stationary arrival and service times.I have also studied a new class of non-stationary shot noise models where the distribution of each shot noise depends on the shot time. Shot noise processes have been extensively studied, and have many applications in physics, insurance risk theory, telecommunications, and service systems. In particular, they include the infinite-server queueing models as special cases. My work weakens the i.i.d. assumption on shot noises in the existing literature and thus provides more flexibilities for modeling non-stationary shot noise phenomena.Non-stationary stochastic models usually suffer from mathematical intractability. The main theoretical difficulty is the lack of appropriate maximal inequalities for general stochastic processes. I have developed maximal inequalities for two-parameter stochastic processes via the chaining method. In addition, sample path properties for Gaussian processes and classical criteria for weak convergence of stochastic processes have been generalized to broader settings.

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