In the Fall of 2019, I heard the following radio story: Eliminating Single Use Plastic.... In it we learn that a single use aluminum container is recycled $70\%$ of the time, to which the program host says (surprisingly):¹

• Host:	"There's probably a diet-rite that I was drinking in the early 90s whose aluminum is still in the system, is what you're saying."
  Guest:	"That's correct."

Really?

Conceptual Model

We want to know the fraction remaining of a single use recyclable aluminum container after a certain number of "generations". The probability that a single use item in generation $g$ will be recycled and "live on" to generation $g+1$ is $p$. When an item is recycled, we envision it being divided into $r$ equal parts; each contributing to the production of $r$ new (distinct) items in generation $g+1$. This cycle continues (each item in generation $g+1$ has probability $p$ of continuing on to generation $g+2$ as $r$ new distinct items. The initial generation will be $g=0$, so any item in the use-recycle-use chain at generation $g = 1,2,3,4\ldots$ will have $\frac{1}{r^g}$ of the original ($g=0$) item within it.

Specification Model

The parameters $p$ and $r$ will be provided to your SIM via command line parameters. The confidence intervals will be calculated to the half width $\pm0.1$.

Project Requirements

Your SIM will calculate a confidence interval for the average number of generations it takes for an item's remaining fraction $f$ to fall below some threshold $F$. Your SIM will use Algorithm 8.1.2 for the calculation.

Input

The command line parameters provided to your SIM will be:

Output

You will OUTPUT the following values from SIM (in this order, and according to the course submission requirements):

1. the number of simulations run in order to achieve the confidence interval required
2. the lower bound of the confidence interval
3. the upper bound of the confidence interval

Hints and SIM Testing

1. While a particular call to $Generate()$ in your SIM will return an integer value for $g$, the average and confidence interval bounds will of course be $\Re$ valued numbers.

2. Beware of tracking each individual item at each generation with an object - with $p=0.70$, $r=10$ you will have about $75\times10^6$ individual items in a data structure by the ninth generation. Yikes

   It is better to maintain the following two pieces of information as a "generation" loop plods forward, breaking when $f$ becomes less than $F$, these the number of items in the current generation, and of course the current generation $g$. In this case, the remaining fraction of the item is \[ f=\frac{\mbox{items}}{r^g} \] Ideally, we would use a $Binomial(\mbox{items},p)$ random variate to determine how many progress to the next generation. But this would require a numerical inversion of the incomplete beta function (Appendix D), which is beyond the scope of this course.

   To determine how many items progress to the next generation, your SIM can use one of two approaches:

   1. "flip" a $Bernoulli(p)$ coin for each item,
   2. or accumulate $Geometric(p)$ values until all the coins in a generation have been accounted for.

   Recall that $Geometric(p)$ is a random variate that returns the number of $p$-bias successes before the first failure. So if $x=Geometric(p)$, you've accounted for $x+1$ items in the generation; the $x+1$^th one is a failure and is not recycled!

grader.sh

I am providing to students the same tarball the grader will use for testing your SIM. Here is how to use it:

First, download this tarball to your Mines Linux account ("alamode" machines!) and unroll it in a temporary directory.

$ ls XXX-student.*
XXX-student.tar.bz2
$ mkdir tmp
$ cd tmp
$ tar xjf ../XXX-student.tar.bz2

Second, set the SIMGRADING environmental variable with:

$ source ~khellman/SIMGRADING/setup.sh ~khellman/SIMGRADING

Now go to the directory holding your XxXx SIM and execute the grader.sh script from the XXX-grader.tar.bz2 resource.

$ cd ~/sim/cwalksim
$ ls SIM
SIM
$ ~/tmp/Xyz/grader.sh
:
:
:

You will need to read any messages from the script carefully, and hit ENTER several times throughout its course. This script checks for:

1. missing tracefiles
2. truncated tracefiles

3. and the difference between SIM results and expected results

The latter test generates NNN PDF files for your inspection ($OutputVar1$, $OutputVar2$ curves, and $OutputVar3$ confidence intervals). For plots generating CDF comparison curves: the red line is the result of your SIM with $N=GRADER_RUNS$ data points; the blue dots are the expected results from an $N=GOLD_RUNS$ trial, and the blue lines are the results of three separate trials with $N=EXAMPLE_RUNS$ for comparison. Your SIM's red line should be a better approximation of the blue dots than the blue lines.

For plots generating overlaped confidence intervals: the red lines are the $GRADER_RUNS$ intervals generated by your SIM; the green lines are $GOLD_RUNS$ intervals from a good implementation, and the blue lines are the results of three separate trials with $N=EXAMPLE_RUNS$ intervals. Your SIM's <SpanText(red intervals,color=red) should not be any more varied from the <SpanText(green intervals)> than the <SpanText(blue lines,color=blue) are.

Here is an example of one of the plots generated by grader.sh:

xxx1-alternate-title xxx2-alternate-title

Submit Your Work

Rubric

This work is worth ?? points.