ESE503 - Simulation Modeling & Analysis (Homework #2)
Spring Semester, 2018 M. Carchidi
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Instructions:
1.) Do the following six (6) Problems and Simulation Problem #1.
2.) Please indicat...
Spring Semester, 2018 M. Carchidi
––––––––––––––––––––––––––––––––––––
Instructions:
1.) Do the following six (6) Problems and Simulation Problem #1.
2.) Please indicate in the table below whether the problem is submit-
ted electronically (via Canvas) or in hard copy (or both).
3.) Indicate any optional comments that might assist us in interpret-
ing your submission.
4.) For electronic submissions please create a separate file for each
problem. Also, use the following naming format for all electronic
submissions: Example : Your Name HW2 Problem1.xls.
5.) Submit this cover sheet in order with you assignment.
6.) Homework is due on 02/15/18. No late submissions allowed.
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Your Name _____________________
Problem Electronic Paper
Number Submission Submission Both Optional Comments
1
2
3
4
5
6
SP#1 X Must be Done
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Problem #1 (20 points) - Computing Probabilities
a.) (13 points) Suppose that the coordinates of a point (X, Y ) are such that
X ∼ U[0, 1) and Y ∼ U[0, 1). Compute the probability that the distance
from this point to the origin is less than or equal to L. You should consider
the two cases when
√
0≤L≤1 and when 1 ≤ L ≤ 2.
and you should check your results using a Monte-Carlo simulation of 3000
trials for various values of L.
b.) (7 points) Compute the average distance a point will be from the origin
and you should check your results using a Monte-Carlo simulation of 3000
trials.
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Problem #2 (15 points) - Computing Probabilities
Consider the quadratic equation
Ax2 + Bx + C = 0
where A, B and C are independent standard uniform random variables so
that
A ∼ U [0, 1) , B ∼ U[0, 1) and C ∼ U[0, 1).
Compute the probability that the solution to the above equation yields
real results for x and you should check your results using a Monte-Carlo
simulation of 3000 trials.
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2
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Problem #3 (20 points) - A Reliability Problem
A certain component of a machine has a lifetime that is exponentially dis-
tributed with a mean of μ = 10 hours. When this component fails, the
machine is stopped and the component must be replaced. Suppose that
the replacement time is uniformly distributed between a = 25 and b = 35
minutes. In addition, company policy is to replace the component after
T = 15 hours of operation even if it has not failed.
a.) (5 points) Compute (in terms of μ and T ) the average time (in hours) that
a component is used.
b.) (5 points) Compute (in terms of μ, a, b, N and T ) the time for N compo-
nents to be replaced.
c.) (5 points) Compute (in terms of μ, a, b and T ) the average percent of the
time the machine is idle.
d.) (5 points) Compute (in terms of a, b, μ, n and T ) the number of compo-
nents that must be replaced during the first n days of operation.
You should check your analytical results in this problem using the simulation
you constructed for Problem #3 of Homework #1.
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3
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