# IEE 475 Arizona State University Hand Simulation and Graph Based Modeling Worksheet

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IEE 475 – Simulating Stochastic Systems
Ira A. Fulton Schools of Engineering
School of Computing, Informatics, and Decision Systems Engineering (CIDSE)
Spring 2019
Instructor: Giulia Pedrielli (giulia.pedrielli@asu.edu)
Assignment 1: Hand Simulation and Graph Based Modeling
Problem 1
You are given the following differential equation:
0
f (x) = 3f (x)
f (0) = 3
Use the linear approximation method with h = 0.01 to simulate 6 steps of the function f (x). Compare the
values with the exact solution of the differential equation and plot a graph for both the approximate function
and the true function for the values of x calculated.
1
Problem 2
Now Take the same function and compute
Z
5
f (x)dx
I=
3
Using Monte carlo Integration. Try with N = 5, 10, 20. What do you observe? Report the detail of all the
steps performed for the integration. For all the cases show the 95% Confidence interval for the estimator of I.
Report the details for the calculation of the mean and the variance of the estimator.
2
Problem 3
Consider the following function:
f (x) = (x + 1)2 , with 1 ≤ x ≤ 3
R3
Compute the integral I = 1 f (x)dx with exact methods and then use the montecarlo integration method with
the same values of N as before. As for the previous problem, report the confidence interval for the integral
estimator with 95% confidence.
3
Problem 4
A football season is either good, reasonable, or bad for the LA Rams. The owner of the team produces ice-cream
besides managing the NFL business and the profit of the two businesses highly depends on the season being
good or bad, as shown in Table 1. Of all seasons, 20% are bad, 25% are reasonable, and 55% are good. Let F
and I be the following random variables:
• F= profit earned by the owner for football business during a season;
• I= profit earned by the owner for ice cream business during a season.
Find Cov (F, I). Also, find the mean and the variance of the total profit the owner earns during the season.
Table 1: Profits for Gotham City
Type of Summer
Reasonable
Good
FBall Profit
(\$1,000)
\$2,000
\$5,000
4
Ice Cream Profit
\$4,500
\$2,000
(\$1,500)
Problem 5
A supermarket has two queues and each has a finite capacity Qi , i = 1, 2. Customers arrive to the supermarket
with Exponential interarrival time with mean 1/λ. If a customer finds both queues full, then s/he goes back in
the grocery department and will come back
with exponential time with mean 1/`. Both servers have LogNormal
Processing time with parameters µ, σ 2 .
What is the state of the system? What are the possible events? Answer these questions and draw the ERG.
5
Problem 6
Take the system before but assume there is a single queue with maximum capacity Q, and the customers that
find the queue full leave the supermarket.
What is the state of the system? What are the possible events? Answer these questions and draw the ERG.
6