[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]

People arrive at a queue according to the following scheme: During each

minute of time either 0 or 1 person arrives. The probability that 1 person arrives is [math]p[/math] and that no person arrives is [math]q = 1 - p[/math]. Let [math]C_r[/math] be the number of customers arriving in the first [math]r[/math] minutes. Consider a Bernoulli trials process with a success if a person arrives in a unit time and failure if no person arrives in a unit time. Let [math]T_r[/math] be the number of failures before the [math]r[/math]th success.

• What is the distribution for [math]T_r[/math]?
• What is the distribution for [math]C_r[/math]?
• Find the mean and variance for the number of customers arriving in the first [math]r[/math] minutes.