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April, 1992 A Distributional Form of Little's Law in Heavy Traffic
Wladyslaw Szczotka
Ann. Probab. 20(2): 790-800 (April, 1992). DOI: 10.1214/aop/1176989806


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 $(1 - \rho)(l(\rho) - \lambda_\rho w(\rho)) \rightarrow_p 0$ as $\rho\uparrow 1$ (in heavy traffic). Here we will show under some conditions that $\sqrt{1 - \rho}(l(\rho) - \lambda_\rho w(\rho)) \rightarrow_D bN\sqrt{M}$ as $\rho \uparrow 1$, 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.


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Wladyslaw Szczotka. "A Distributional Form of Little's Law in Heavy Traffic." Ann. Probab. 20 (2) 790 - 800, April, 1992.


Published: April, 1992
First available in Project Euclid: 19 April 2007

zbMATH: 0759.60101
MathSciNet: MR1159574
Digital Object Identifier: 10.1214/aop/1176989806

Primary: 60K25

Keywords: asymptotic stationarity , heavy traffic , invariance principle , Little formula , queue size , single-server queue , waiting time , weak convergence

Rights: Copyright © 1992 Institute of Mathematical Statistics

Vol.20 • No. 2 • April, 1992
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