We consider the M/M/∞ queue with arrival rate λ, service rate μ and traffic intensity ρ = λ/μ. We analyze the first passage distribution of the time the number of customers N(t) reaches the level c, starting from N(0) = m>c. If m = c + 1 we refer to this time period as the congestion period above the level c. We give detailed asymptotic expansions for the distribution of this first passage time for ρ → ∞, various ranges of m and c, and several different time scales. Numerical studies back up the asymptotic results.
