= Heavy-traffic limit of a queue
{title2=$\varepsilon Q_\varepsilon(\lfloor t/\varepsilon^2\rfloor)\Rightarrow\Gamma(\sigma B(t)-dt)$}
For fixed <independent and identically distributed> arrivals of mean $\mu$ and positive finite <variance> $\sigma^2$, put service at $\mu+\varepsilon d$. Rescaling net input by space $\varepsilon$ and time $\varepsilon^{-2}$ gives <Brownian motion> with drift $-d$ by the <functional central limit theorem>. The continuous <Skorokhod reflection map on the half-line> gives the displayed transient limit, with a correspondingly scaled initial workload. A stationary approximation additionally requires justification of limit interchange. When valid, the <stationary law of negatively drifted reflected Brownian motion> predicts $\mathbb P(Q_\varepsilon>b/\varepsilon)\approx e^{-2db/\sigma^2}$.
Back to article page
