[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]

Consider the simple queueing process of Example~\ref{exam

5.21}. Suppose that you watch the size of the queue. If there are </math>j[math] people in the queue the next time the queue size changes it will either decrease to [/math]j - 1[math] or increase to [/math]j + 1[math]. Use the result of Exercise~[[exercise:8b126d52c7 |Exercise]] to show that the probability that the queue size decreases to [/math]j - 1[math] is [/math]\mu/(\mu + \lambda)[math] and the probability that it increases to [/math]j + 1[math] is [/math]\lambda/(\mu + \lambda)[math]. When the queue size is 0 it can only increase to~1. Write a program to simulate the queue size. Use this simulation to help formulate a conjecture containing conditions on [/math]\mu[math]~and~[/math]\lambda<math> that will ensure that the queue will have times when it is empty.