ISyE 6644 — Fall 2017
Homework #1 Solutions
1. (Deterministic Model.) Suppose you throw a rock off a cliff having height h0 = 500 feet. You’re a strong bloke, so the initial downward velocity is v0 = 100 feet/sec (
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ISyE 6644 — Fall 2017
Homework #1 Solutions
1. (Deterministic Model.) Suppose you throw a rock off a cliff having height h0 = 500 feet. You’re a strong bloke, so the initial downward velocity is v0 = 100 feet/sec (slightly under 70 miles/hr). Further, in this neck of the woods, it turns out there is no friction in the atmosphere — amazing! Now you remember from your Baby Physics class that the height after time t is
When does the rock hit the ground?
2. (Stochastic Model.) Consider a single-server queueing system where the times between customer arrivals are independent, identically distributed Exp(λ = 2/hr) random variables; and the service times are i.i.d. Exp(µ = 3/hr). Unfortunately, if a potential arriving customer sees that the server is occupied, he gets mad and leaves the system. Thus, the system can have either 0 or 1 customer in it at any time. This is what’s known as an M/M/1/1 queue. If P (t) denotes the probability that a customer is being served at time t, trust me that it can be shown that
P (t) = λ + P (0) − λ e−(λ+µ)t.
If the system is empty at time 0, i.e., P (0) = 0, what is the probability that there will be no people in the system at time 20 minutes?
This study source was downloaded by 100000815948112 from CourseHero.com on 06-01-2021 13:37:11 GMT -05:00
3. (History.) Harry Markowitz (one of the big wheels in simulation language development) won his Nobel Prize for portfolio theory in 1990, though the work that earned him the award was conducted much earlier in the 1950s. Who won the 1990 Prize with him? You are allowed to look this one up.
4. (Applications.) Which of the following situations might be good candidates to use simulation? (There may be more than one correct answer.)
(a) We are interested in investing our entire portfolio in fixed-interest U.S. bonds, and we are interested in determining the portfolio’s value in 5 years.
(b) We are interested in investing one half of our portfolio in fixed-interest U.S. bonds and the remaining half in a stock market equity index. We have some information concerning the distribution of stock market returns, but we do not really know what will happen in the market with certainty.
(c) We have a new strategy for baseball batting orders, and we would like to know if this strategy beats other commonly used batting orders (e.g., a fast guy bats first, a big, strong guy bats fourth, etc.). We have information on the performance of the various team members, but there’s a lot of randomness in baseball.
(d) We have an assembly station in which “customers” (for instance, parts to be manufactured) arrive every 5 minutes exactly and are processed in precisely 4 minutes by a single server. We would like to know how many parts the server can produce in a hour.
(e) Consider an assembly station in which parts arrive randomly, with indepen- dent exponential interarrival times). There is a single server who can process the parts in a random amount of time that is normally distributed. Moreover,
This study source was downloaded by 100000815948112 from CourseHero.com on 06-01-2021 13:37:11 GMT -05:00
the server takes random breaks every once in a while. We would like to know how big any line is likely to get.
(f) Suppose we are interested in determining the number of doctors needed on Friday night at a local emergency room. We need to insure that 90% of patients get treatment within one hour.
5. (Baby Examples.) The planet Triskelion has 80-day years.
(a) Suppose there are 2 Triskelions in the room. What’s the probability that they’ll have the same birthday?
(b) Now suppose there are 3 Triskelions in the room. (They’re big, so the room is getting crowded.) What’s the probability that at least two of them have the same birthday?
6. (More Baby Examples.) Inscribe a circle in a unit square and toss n = 1000 random darts at the square.
This study source was downloaded by 100000815948112 from CourseHero.com on 06-01-2021 13:37:11 GMT -05:00
(a) Suppose that 760 of those darts land in the circle. Using the technology developed in class, what is the resulting estimate for π?
(b) What would our estimate be if we let n and we applied the same ratio strategy?
(c) BONUS: With the previous question in mind, suppose that we can somehow toss n random darts into a unit cube. Further, suppose that we’ve inscribed a sphere with radius 1/2 inside the cube. Let pˆn be the proportion of the n darts that actually fall within the sphere. Give a Monte Carlo scheme to estimate π.
7. (Still Another Baby Example.) Suppose customers arrive at a single-server ice cream parlor at times 3, 6, 15, and 17. Further suppose that it takes the server 7, 9, 6, and 10 minutes, respectively, to serve the four customers. When does customer 4 leave the shoppe?
8. (Generating Randomness.) Suppose we are using the (awful) pseudo-random num- ber generator
Xi = (5Xi−1 + 1) mod(8),
This study source was downloaded by 100000815948112 from CourseHero.com on 06-01-2021 13:37:11 GMT -05:00
with starting value (“seed”) X0 = 2. Find the second PRN, U2 = X2/m = X2/8.
9. (Generating Randomness.) Suppose we are using the “desert island” pseudo- random number generator
Xi = 16807 Xi−1 mod(231 − 1),
with seed X0 = 12345678. Find the resulting integer X1. Feel free to use something like Excel if you need to.
10. (Generating Randomness.) Suppose that we generate a pseudo-random number
U = 0.228. Use this to generate an Exponential(λ = 1/3) random variate.
Note: It turns out that
X = −(1/λ)ln(1 − U ) = −3ln(0.772) = 0.776
would also have been an acceptable answer. Can you see why?
This study source was downloaded by 100000815948112 from CourseHero.com on 06-01-2021 13:37:11 GMT -05:00
ISyE 6644 — Fall 2017
Homework #1 Solutions
ISyE 6644 — Fall 2017
Homework #1 Solutions
1. (Deterministic Model.) Suppose you throw a rock off a cliff having height h0 = 500 feet. You’re a strong bloke, so the initial downward velocity is v0 = 100 feet/sec (slightly under 70 miles/hr). Further, in this neck of the woods, it turns out there is no friction in the atmosphere — amazing! Now you remember from your Baby Physics class that the height after time t is
When does the rock hit the ground?
2. (Stochastic Model.) Consider a single-server queueing system where the times between customer arrivals are independent, identically distributed Exp(λ = 2/hr) random variables; and the service times are i.i.d. Exp(µ = 3/hr). Unfortunately, if a potential arriving customer sees that the server is occupied, he gets mad and leaves the system. Thus, the system can have either 0 or 1 customer in it at any time. This is what’s known as an M/M/1/1 queue. If P (t) denotes the probability that a customer is being served at time t, trust me that it can be shown that
P (t) = λ + P (0) − λ e−(λ+µ)t.
If the system is empty at time 0, i.e., P (0) = 0, what is the probability that there will be no people in the system at time 20 minutes?
This study source was downloaded by 100000815948112 from CourseHero.com on 06-01-2021 13:37:11 GMT -05:00
3. (History.) Harry Markowitz (one of the big wheels in simulation language development) won his Nobel Prize for portfolio theory in 1990, though the work that earned him the award was conducted much earlier in the 1950s. Who won the 1990 Prize with him? You are allowed to look this one up.
4. (Applications.) Which of the following situations might be good candidates to use simulation? (There may be more than one correct answer.)
(a) We are interested in investing our entire portfolio in fixed-interest U.S. bonds, and we are interested in determining the portfolio’s value in 5 years.
(b) We are interested in investing one half of our portfolio in fixed-interest U.S. bonds and the remaining half in a stock market equity index. We have some information concerning the distribution of stock market returns, but we do not really know what will happen in the market with certainty.
(c) We have a new strategy for baseball batting orders, and we would like to know if this strategy beats other commonly used batting orders (e.g., a fast guy bats first, a big, strong guy bats fourth, etc.). We have information on the performance of the various team members, but there’s a lot of randomness in baseball.
(d) We have an assembly station in which “customers” (for instance, parts to be manufactured) arrive every 5 minutes exactly and are processed in precisely 4 minutes by a single server. We would like to know how many parts the server can produce in a hour.
(e) Consider an assembly station in which parts arrive randomly, with indepen- dent exponential interarrival times). There is a single server who can process the parts in a random amount of time that is normally distributed. Moreover,
This study source was downloaded by 100000815948112 from CourseHero.com on 06-01-2021 13:37:11 GMT -05:00
the server takes random breaks every once in a while. We would like to know how big any line is likely to get.
(f) Suppose we are interested in determining the number of doctors needed on Friday night at a local emergency room. We need to insure that 90% of patients get treatment within one hour.
5. (Baby Examples.) The planet Triskelion has 80-day years.
(a) Suppose there are 2 Triskelions in the room. What’s the probability that they’ll have the same birthday?
(b) Now suppose there are 3 Triskelions in the room. (They’re big, so the room is getting crowded.) What’s the probability that at least two of them have the same birthday?
6. (More Baby Examples.) Inscribe a circle in a unit square and toss n = 1000 random darts at the square.
This study source was downloaded by 100000815948112 from CourseHero.com on 06-01-2021 13:37:11 GMT -05:00
(a) Suppose that 760 of those darts land in the circle. Using the technology developed in class, what is the resulting estimate for π?
(b) What would our estimate be if we let n and we applied the same ratio strategy?
(c) BONUS: With the previous question in mind, suppose that we can somehow toss n random darts into a unit cube. Further, suppose that we’ve inscribed a sphere with radius 1/2 inside the cube. Let pˆn be the proportion of the n darts that actually fall within the sphere. Give a Monte Carlo scheme to estimate π.
7. (Still Another Baby Example.) Suppose customers arrive at a single-server ice cream parlor at times 3, 6, 15, and 17. Further suppose that it takes the server 7, 9, 6, and 10 minutes, respectively, to serve the four customers. When does customer 4 leave the shoppe?
8. (Generating Randomness.) Suppose we are using the (awful) pseudo-random num- ber generator
Xi = (5Xi−1 + 1) mod(8),
This study source was downloaded by 100000815948112 from CourseHero.com on 06-01-2021 13:37:11 GMT -05:00
with starting value (“seed”) X0 = 2. Find the second PRN, U2 = X2/m = X2/8.
9. (Generating Randomness.) Suppose we are using the “desert island” pseudo- random number generator
Xi = 16807 Xi−1 mod(231 − 1),
with seed X0 = 12345678. Find the resulting integer X1. Feel free to use something like Excel if you need to.
10. (Generating Randomness.) Suppose that we generate a pseudo-random number
U = 0.228. Use this to generate an Exponential(λ = 1/3) random variate.
Note: It turns out that
X = −(1/λ)ln(1 − U ) = −3ln(0.772) = 0.776
would also have been an acceptable answer. Can you see why?
This study source was downloaded by 100000815948112 from CourseHero.com on 06-01-2021 13:37:11 GMT -05:00
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