A Distributional Form of Little's Law in Heavy Traffic

Abstract

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.