Asymptotic Behavior of the Moments of the Maximum Queue Length During a Busy Period

We give a simple derivation of the distribution of the maximum L of the length of the queue during a busy period for the M/M/1 queue with lambda<1 the ratio between arrival rate and service rate. We observe that the asymptotic behavior of the moments of L is related to that of Lambert series for the generating functions for the sums of powers of divisors of positive integers. We show how to obtain asymptotic expansions for these moments with error terms having order as large a power of 1-lambda as desired.