```{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 $t \to \infty$.    
```

<br>


<br>


```{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.
```  

<br>





```{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)
```


<br>

*  Version: `r format(Sys.time(), '%d %B %Y')`
* [Rmd Source](https://tsoo-math.github.io/ucl/QHW7.Rmd)

