  Exact explicit results are derived for the distribution of the partial busy
period of the M/M/c multi-server queue for a general number of servers. A
rudimentary spectral method leads to a representation that is amenable to
efficient numerical computation across the entire ergodic region. An
alternative algebraic approach yields a representation as a finite sum of
Marcum Q-functions depending on the roots of certain polynomials that are
explicitly determined for an arbitrary number of servers. Asymptotic forms are
derived in the limit of a large number of servers under two scaling regimes,
and also for the large-time limit. Connections are made with previous work. The
present work is the first to offer tangible exact results for the distribution
when the number of servers is greater than two.
