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

TL;DR

This paper derives exact and computationally efficient formulas for the distribution of the partial busy period in multi-server queues, extending previous results to any number of servers and including asymptotic analyses.

Contribution

It introduces novel spectral and algebraic methods to obtain explicit distribution formulas for the partial busy period in M/M/c queues with any number of servers.

Findings

Provides explicit distribution formulas for any number of servers.

Develops asymptotic results for large server counts and time.

Connects new results with existing literature on queue distributions.

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.