Exam (elaborations)

ESE503 - Simulation Modeling & Analysis (Homework #2) -University of Pennsylvania

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ESE 503

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ESE 503

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ESE503 - Simulation Modeling & Analysis (Homework #2)

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.

––––––––––––––––––––––––––––––––––––

Your Name _____________________

Problem Electronic Paper
Number Submission Submission Both Optional Comments
1
2
3
4
5
6
SP#1 X Must be Done
––––––––––––––––––––––––––––––––––––

,––––––––––––––––––––––––––––––––––––

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.

––––––––––––––––––––––––––––––––––––

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.

––––––––––––––––––––––––––––––––––––

2

, ––––––––––––––––––––––––––––––––––––

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.

––––––––––––––––––––––––––––––––––––

3

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Exam (elaborations)

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