Analyzing the Busy Period of the M/M/1 Queue via Order Statistics and Record Values

In this study, we assume an M/M/1 queue adequately models some random process of interest. We consider a finite sample of random variables
$X_{1}, X_{2}\ldots, X_{n}$
that represent the queue lengths and response times from busy periods of an M/M/1 queue and compute performance measures using the order statistics and record values. Specifically, given a sequence of
$n$
independent and identically distributed (iid) busy periods, for the largest or
$nth$
busy period, we develop formulas to compute: the maximum expected queue length, an upper bound for the maximum expected response time, and record maximum queue lengths over several congestion periods.