We consider an $M/M/infty$ queue with servers ranked as ${1, 2,
3,\dots}$. The Poisson arrival stream has rate $\lambda$ and each server works
at rate $\mu$. A new arrival takes the lowest ranked available server.We let
$S$ be the set of occupied servers and $|S|$ is the number of elements of
$S$.We study the distribution of max $(S)$ in the asymptotic limit of $\rho =
\lambda/\mu \to \infty$.Setting $P(m) = \Pr[max(S) > m]$ we find that the
asymptotic structure of the problem is different according as $m = O(1)$ or $m
\to \infty$, at the same rate as $\rho$. For the latter it is furthermore
necessary to distinguish the cases $m/\rho < 1, m/\rho \approx 1$ and
$m/\rho > 1$.We also estimate the average amount of wasted stor- age space,
which is defined by $E(max(S)) - \rho$. This is the average number of idle
servers that are ranked below the maximum occupied one.We also relate our
results to those obtained by probabilistic approaches. Numerical studies
demonstrate the accuracy of the asymptotic results.