{r setup, include=FALSE} knitr::opts_chunk$set(echo = TRUE)  {exercise} Use the alternating renewal theorem to show that under mild conditions, we have that the probability that the number of arrrivals for renewal process in the interval$[0,t]$is even goes to$1/2$as$t \to \infty$. Why do we need mild conditions? Why do we need$t \to \infty$.  {exercise} Consider a$M(3)/M(5)/1$queue. Let$t=323\$, so the queue has had a chance to settle in. * By running simulations, plot the probability mass function for the random number of customers in the system. Do you recognize it? Any guesses? * Start keeping track of the departures and assume they proceed into a new shop. By simulations, produce a histogram of the inter-arrivals times of the items entering the new shop. You might be surprised. 
{exercise, name="Little's law"} There are many simple exercises on can do with Little's law. See [here](http://web.eng.ucsd.edu/~massimo/ECE158A/Handouts_files/Little.pdf) 
* Version: r format(Sys.time(), '%d %B %Y') * [Rmd Source](https://tsoo-math.github.io/ucl/QHW7.Rmd)