POSITIVE RECURRENCE OF PIECEWISE ORNSTEIN-UHLENBECK PROCESSES AND COMMON QUADRATIC LYAPUNOV FUNCTIONS

• 成果类型: Article
• 署名作者: Dieker, A. B.; Gao, Xuefeng
• 署名单位: University System of Georgia; Georgia Institute of Technology
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/12-AAP870
• 发表日期: 2013
• 页码: 1291-1317
• 关键词: stability; matrices; criteria; systems; LIMITS; QUEUE

摘要

We study the positive recurrence of piecewise Ornstein-Uhlenbeck (OU) diffusion processes, which arise from many-server queueing systems with phase-type service requirements. These diffusion processes exhibit different behavior in two regions of the state space, corresponding to overload (service demand exceeds capacity) and underload (service capacity exceeds demand). The two regimes cause standard techniques for proving positive recurrence to fail. Using and extending the framework of common quadratic Lyapunov functions from the theory of control, we construct Lyapunov functions for the diffusion approximations corresponding to systems with and without abandonment. With these Lyapunov functions, we prove that piecewise OU processes have a unique stationary distribution.