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

TL;DR

This paper derives the distribution and asymptotic moments of the maximum queue length during a busy period in an M/M/1 queue, linking it to Lambert series and providing precise asymptotic expansions.

Contribution

It presents a simple derivation of the maximum queue length distribution and connects the asymptotic behavior of its moments to Lambert series, with detailed error estimates.

Findings

Asymptotic expansions for moments with arbitrarily high accuracy.

Connection between queue maximums and Lambert series.

Explicit error bounds for asymptotic approximations.

Abstract

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.