[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]

In the queueing problem of Exercise Exercise,

let [math]S[/math] be the total service time required by a customer and [math]T[/math] the time between arrivals of the customers.

• Show that [math]P(S = j) = (1 - r)^{j - 1}r[/math] and [math]P(T = j) = (1 - p)^{j - 1}p[/math], for [math]j \gt 0[/math].
• Show that [math]E(S) = 1/r[/math] and [math]E(T) = 1/p[/math].
• Interpret the conditions [math]s \lt 1[/math], [math]s = 1[/math] and [math]s \gt 1[/math] in terms of these expected values.