Summary: 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: \(\lim_{T \to\infty} {Y(T)\over\ln T} ={\mu \over\lambda} \text{ a.s}\).