Spectral properties of the tandem Jackson network, seen as a quasi-birth-and-death process

• 成果类型: Article
• 署名作者: Kroese, DP; Scheinhardt, WRW; Taylor, PG
• 署名单位: University of Queensland; University of Twente; Centrum Wiskunde & Informatica (CWI); University of Melbourne
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/105051604000000477
• 发表日期: 2004
• 页码: 2057-2089
• 关键词: markov-chains; stationary distributions; models

摘要

Quasi-birth-and-death (QBD) processes with infinite phase spaces can exhibit unusual and interesting behavior. One of the simplest examples of such a process is the two-node tandem Jackson network, with the phase giving the state of the first queue and the level giving the state of the second queue. In this paper, we undertake an extensive analysis of the properties of this QBD. In particular, we investigate the spectral properties of News's R-matrix and show that the decay rate of the stationary distribution of the level process is not always equal to the convergence norm of R. In fact, we show that we can obtain any decay rate from a certain range by controlling only the transition structure at level zero, which is independent of R. We also consider the sequence of tandem queues that is constructed by restricting the waiting room of the first queue to some finite capacity, and then allowing this capacity to increase to infinity. We show that the decay rates for the finite truncations converge to a value, which is not necessarily the decay rate in the infinite waiting room case. Finally, we show that the probability that the process hits level n before level 0 given that it starts in level 1 decays at a rate which is not necessarily the same as the decay rate for the stationary distribution.