Applications-Oriented Problems
1. (2 group members) The African Tanker Problem. If you like tricky little problems, then this is the one for you! This famous exercise concerns the simulation of a somewhat complicated port, and is described in Law (2015) (Chapter 2 exercises). You should model this and conduct a little more output analysis than is asked for in the text of the problem.
2
2. (2 members) Go to some intersections and model a couple of traffic lights. Collect some real data. Try to answer as many interesting questions as you can think of. For instance, what would the effect of synchronization be? Or, how do cell phones affect driver behavior? Justify everything you conclude by conducting careful output analysis. This could be a great project!
If a kid gets infected by Tommy, he will then become infectious for 3 days as well, starting on the next day.
(a) What is the distribution of the number of kids that Tommy infects on Day 1?
(b) What is the expected number of kids that Tommy infects on Day 1?
(c) What is the expected number of kids that are infected by Day 2 (you can
count Tommy if you want)?
(d) Simulate the number of kids that are infected on Days 1,2,. . . . Do this many times. What are the (estimated) expected numbers of kids that are infected by Day i, i = 1, 2, . . .? Produce a histogram detailing how long the “epidemic” will last.
Language- and Modeling-Oriented Problems
3
5. (3 members) Learn and implement the self-contained simulation language SIMLIB from Chapter 2 of the Law (2015) text. Although SIMLIB is not the most-efficient simulation language around, it will really teach you how an actual simulation lan- guage is written. In your project write-up, give your comments and opinions on the language, provide an easy user’s guide, and demonstrate how SIMLIB works on one or two example models. Note: I’ll allow you to replicate the functionality of SIMLIB using any language such as C++, Python, etc.
6. (2 members) A different modeling paradigm. Read and comment on Lee Schruben’s fundamental paper, “Simulation Modeling with Event Graphs,” Communications of the ACM, Volume 26, pp. 957–963 (1983). Then do the same for the Sargent (1988) and Som and Sargent (1989) articles cited in the References section of the Law (2015) text. (The latter two papers extend and refine the event graphs technique, and they provide lots of nice examples.)
Programming-Oriented Problems
7. (1 member) Let’s gamble! Implement a blackjack simulation; in fact, simulate a few reasonable strategies. Statistically analyze these strategies by playing lots of independent blackjack games. Then determine which choice maximizes your profit. (Assume that the House plays by the usual rules.) Include all code and commentary. Be creative!
8. (1–2 members) Let’s gamble some more! Same thing as in Problem 7, except for
solitaire. (This isn’t easy, but it’s fun.)
9. (1–2 members) Another game. There are two players, A and B. At the beginning of the game, each starts with 4 coins, and there are 2 coins in the pot. A goes first, then B, then A,. . . . During a particular player’s turn, the player tosses a 6-sided die. If the player rolls a:
• 1, then the player does nothing.
• 2: then the player takes all coins in the pot.
• 3: then the player takes half of the coins in the pot (rounded down).
• 4,5,6: then the player puts a coin in the pot.
A player loses (and the game is over) if they are unable to perform the task (i.e., if they have 0 coins and need to place one in the pot). We define a cycle as A and then B completing their turns. The exception is if a player goes out; that is the
4
final cycle (but it still counts as the last cycle). We are trying to determine the expected number (and maybe even the distribution) of cycles the game will last for. I’m guessing that you can use “first-step” analysis to get the expected value. Simulation seems the easiest thing to do to get the entire distribution.
Theory-Oriented Problems
10. (2 members) Work on generalizations of the “Birthday Problem” that we did in class. E.g., what’s the probability that at least three kids in the room will have the same birthday? What happens when the birth patterns change a bit over the year (e.g., more babies born in the spring)?
11. (2 members) Prepare a tutorial on Monte Carlo integration techniques and give
several examples.
And. . .
12. Any Other Problem That You Choose — as long as you can convince us. Succinctly describe the problem and why you think it would be cool for this class. Needs to be nontrivial, useful, and interesting.