exercise:23290014a1: Difference between revisions
(Created page with "<div class="d-none"><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></div> 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...") |
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Consider the simple queueing process of [[guide:D26a5cb8f7#exam 5.21|Example]]. 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: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 | |||
5.21 | |||
< | |||
to < | |||
show that the probability that the queue size decreases to < | |||
< | |||
\lambda)<math> and the probability that it increases to < | |||
\lambda)<math>. When the queue size is 0 it can only increase to | |||
simulate the queue size. Use this simulation to help formulate a conjecture | |||
containing conditions on < | |||
times when it is empty. | times when it is empty. | ||
Latest revision as of 01:21, 14 June 2024
Consider the simple queueing process of Example. 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 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.