FUNCTIONAL LIMIT THEOREMS FOR NON-STATIONARY STOCHASTIC SYSTEMS
Open Access
- Author:
- Zhou, Yuhang
- Graduate Program:
- Industrial Engineering
- Degree:
- Doctor of Philosophy
- Document Type:
- Dissertation
- Date of Defense:
- February 15, 2018
- Committee Members:
- Guodong Pang, Dissertation Advisor/Co-Advisor
Guodong Pang, Committee Chair/Co-Chair
Vinayak V Shanbhag, Committee Member
Xingyuan Fang, Committee Member
David Russell Hunter, Outside Member - Keywords:
- Functional limit theorems
Maximal inequalities
Infinite-server queue
Non-stationarity - Abstract:
- Modern service systems, such as call centers and health care systems, are typically of large scale and have non-stationary data, including customer arrival rates, service times and staffing. A better understanding of such service systems will help improving system efficiency, optimizing system managements, and reducing costs. However, exact analysis is hard because of the non-stationarity, randomness and large-scale of such systems. Stochastic models in the heavy-traffic regime turn out to be useful to obtain approximations and provide insights for operational managements. My primary work in this dissertation is to study stochastic models with non-stationarity and to establish associated functional limit theorems. I have established a novel and powerful framework for proving functional limit 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.