Asymptotic expansions for the congestion period for the M/M/∞ queue

We consider the M/M/infinity queue with arrival rate lambda, service rate mu and traffic intensity rho=lambda/mu. 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 rho --> infinity, various ranges of m and c, and several different time scales. Numerical studies back up the asymptotic results.