Exact Results for the Distribution of the Partial Busy Period for a
Multi-Server Queue
Abstract
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.