Series Jackson Networks and Noncrossing Probabilities

• 成果类型: Article
• 署名作者: Dieker, A. B.; Warren, J.
• 署名单位: University System of Georgia; Georgia Institute of Technology; University of Warwick
• 刊物名称: MATHEMATICS OF OPERATIONS RESEARCH
• ISSN/ISSBN: 0364-765X
• DOI: 10.1287/moor.1090.0421
• 发表日期: 2010
• 页码: 257-266
• 关键词: random-walks; particles

摘要

This paper studies the queue-length process in series Jackson networks with external input to the first station only. We show that its Markov transition probabilities can be written as a finite sum of noncrossing probabilities, so that questions on time-dependent queueing behavior are translated to questions on noncrossing probabilities. This makes previous work on noncrossing probabilities relevant to queueing systems and allows new queueing results to be established. To illustrate the latter, we prove that the relaxation time (i.e., the reciprocal of the spectral gap) of a positive recurrent system equals the relaxation time of an M/M/1 queue with the same arrival and service rates as the network's bottleneck station. This resolves a conjecture of Blanc that he proved for two queues in series.