Process of successive cleaning

A Poisson stream of particles arrives to a half-line $[0;\infty)$ with rate $\lambda$ and mean density 1. A server moves on a half-line at unit speed to the right, stopping to perform service of every particle encountered. The service times are all taken to be mutually independent and exponentially distributed with mean $\mu$. At the initial moment the server is in zero. We study $Y(T)$ — its position at the moment $T$. The main result is the following:
$$ \lim_{T\to\infty}\frac{Y(T)}{\ln T} =\frac{\mu}{\lambda}\qquad\mboxa.s. $$