[math] \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}[/math]

A discrete time queueing system of capacity [math]n[/math] consists of the person being served and those waiting to be served. The queue length [math]x[/math] is observed each second. If [math]0 \lt x \lt n[/math], then with probability [math]p[/math], the queue size is increased by one by an arrival and, inependently, with probability [math]r[/math], it is decreased by one because the person being served finishes service. If [math]x = 0[/math], only an arrival (with probability [math]p[/math]) is possible. If [math]x= n[/math], an arrival will depart without waiting for service, and so only the departure (with probability [math]r[/math]) of the person being served is possible. Form a Markov chain with states given by the number of customers in the queue. Modify the program FixedVector so that you can input [math]n[/math], [math]p[/math], and [math]r[/math], and the program will construct the transition matrix and compute the fixed vector. The quantity [math]s = p/r[/math] is called the traffic intensity. Describe the differences in the fixed vectors according as [math]s \lt 1[/math], [math]s = 1[/math], or [math]s \gt 1[/math].