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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. [ABSTRACT FROM AUTHOR]

We consider the M/G/1 queue with an arrival rate λ that depends weakly upon time, as λ = λ(ε t) where ε is a small parameter. In the asymptotic limit ε → 0, we construct approximations to the probability p n( t)that η customers are present at time t. We show that the asymptotics are different for several ranges of the (slow) time scale Τ= ε t. We employ singular perturbation techniques and relate the various time scales by asymptotic matching. [ABSTRACT FROM AUTHOR]