A DISTRIBUTIONAL FORM OF LITTLES LAW IN HEAVY TRAFFIC

• 成果类型: Article
• 署名作者: SZCZOTKA, W
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/aop/1176989806
• 发表日期: 1992
• 页码: 790-800
• 关键词: stationary representation; Weak Convergence; queues

摘要

Consider a single-server queue with units served in order of arrival for which we can define a stationary distribution (equilibrium distribution) of the vector of the waiting time and the queue size. Denote this vector by (w(rho), l(rho)), where rho < 1 is the traffic intensity in the system when it is in equilibrium and lambda(rho) is the intensity of the arrival stream to this system. Szczotka has shown under some conditions that (1 - rho)(l(rho) - lambda(rho)w(rho)) --> p 0 as p up 1 (in heavy traffic). Here we will show under some conditions that square-root 1 - rho (l(rho) - lambda(rho)W(rho)) --> D bN square-root M as rho up 1, where N and M are mutually independent random variables such that N has the standard normal distribution and M has an exponential distribution while b is a known constant.