The M/M/S Queue

This article describes the M/M/s queueing system, or Erlang delay model, which is a stochastic service system with a Poisson arrival process and s (1 ≤ s < ∞) independent and identical servers, each of which serves customers in an exponentially distributed amount of time. Arriving customers join an infinite queue if they find all of the s servers busy. The main purpose of this article is to summarize the characteristics of the M/M/s system, namely the steady-state probability distribution of the number of customers in the system, the steady-state performance parameters, and steady-state delay probability which is computed using the well-known Erlang C formula. As a special case of the M/M/s system, we also discuss the M/M/s/c queue. Computational issues and some structural properties of the Erlang C formula are also discussed.