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 remaining fraction $f$ of an item after a certain number of generations. 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 ($n$)

2. the lower bound of the confidence interval, use eight digit precision and scientific notation output²

3. the upper bound of the confidence interval, use eight digit precision and scientific notation output

Documentation

Of course, your documentation should be handed off according to the usual submission requirements.

1. Document the location of your Algorithm 8.1.2 implementation (source and lines) as well as your "Welford object" (if you are using one).
2. Provide a paragraph or two of well written and thoughtful analysis, address the following topics:

   1. Does the simulation, as written, address the fundamental question? If so, what assumptions are being made? If not, what do you believe is not being modeled correctly and how would you implement remedies?

   2. Estimate what fraction of the pop can used by the radio show host in the 90s is still with us today in the use-recycle-use chain. This can be done in one of two ways:

      1. Using some math and the results of small GENS and $r$ SIM runs,

      2. Making some much longer SIM runs with appropriate parameters (beware of overflow and you are going to need a lot of random numbers and an efficiency bump!)

      Either approach yields an equal amount of credit, but you must explain your methods clearly. You are not required to use the graded SIM implementation to support your findings. You may augment your SIM with optional invocation magic (just so long as it still works with grader.sh) or hack up another analysis. Choosing a language or library with an idfBinomial() variate would be very wise!

Hints and SIM Testing

1. 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 the $f$ falls to zero or the required GENS have been simulated: the number of items in the current generation, and of course the current generation $g$. In this case, the remaining fraction of the original item is \[ f=\frac{\mbox{items}}{r^g} \]

   Even with this more efficient approach, we must beware of numerical issues! Depending on the random numbers (of course), you can still overflow a 32-bit integer tracking the number of items currently in the recycling chain. Use a 64-bit integer, or a language with arbitrary precision integers (Python, for instance) for your items variable.

   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. Your SIM can simply "flip" a $Bernoulli(p)$ coin for each item.

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

Second, set the SIMGRADING environmental variable with:

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

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

$ cd ~/sim/alrsim
$ ls SIM
SIM
$ ~/tmp/AlRecycling/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 3 PDF files for your inspection (one $n$ curve comparison, and two confidence interval comparisons). For plots generating CDF comparison curves: the red line is the result of your SIM with $N=200$ data points; the blue dots are the expected results from an $N=500$ trial, and the blue lines are the results of three separate trials with $N=100$ for comparison. Your SIM's red line should be a better approximation of the blue dots than the blue lines.

For plots generating overlapped confidence intervals: the red lines are the $200$ intervals generated by your SIM; the green lines are $500$ intervals from a good implementation, and the blue lines are the results of three separate trials with $N=100$ intervals. Your SIM's red intervals should not be any more varied from the green intervals than the blue lines are.

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

Submit Your Work

Computational Rubric (part 1)

This work is worth 50 points.

Analysis Rubric (part 2)

This work is worth 25 points.