Business Simulation Exam
É C O L E D E G E S T I O N T E L F E R S C H O O L O F M A N A G E M E N T
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ADM 3305 – PRACTICE PROBLEMS Version: December 5, 2020
1. The weather can be considered a stochastic process, because it evolves in a probabilistic
manner from one day to the next. Suppose for a certain location that this probabilistic evolution satisfies the following description:
• The probability of rain tomorrow is 0.6 if it is raining today; • The probability of its being clear (no rain) tomorrow is 0.8 if it is clear today.
Use the uniform random numbers below to simulate the evolution of the weather for 10 days, beginning the day after a clear day.
0.3039 0.7914 0.8543 0.6902 0.3004 0.0383 0.3883 0.6052 0.2231 0.4250
2. There are eight employees who wash cars by hand at a car-wash center that opens at 10AM
and closes at 6PM. The arrival of cars to the car wash follows a Poisson process with rate 12/hour. Two people work together on each car, and the time to clean a car by a team of two follows a normal distribution with a mean of 20 minutes and a standard deviation of 2 minutes.
a) The manager would like to have a 10-minute team meeting. What is the probability that
no cars arrive in the next 10 minutes? b) The manager had to delay the meeting to take a phone call for 20 minutes. During this
time, surprisingly no customers arrived to the car wash. If the manager now decides to have the 10-minute meeting, compared to the answer in part a), she faces: (1) an increased chance, (2) a decreased chance, (3) the same chance of a customer arriving during the meeting. Choose one of the options and justify your answer.
3. If a random variable X follows an exponential distribution with parameter λ, and
independently, a random variable Y follows an exponential distribution with parameter μ, what is probability that X < Y?
4. Let X1, X2,…, Xn be independent random variables that are exponentially distributed with
respective parameters λ1, λ2,…, λn. Identify the probability distribution function of the random variable V = min{X1,...,Xn}.
5. Let X1, X2,…, Xn be independent random variables, all exponentially distributed with the same parameter λ. Determine the probability distribution function for the random variable Z = min{ X1,...,Xn }.
6. A flashlight requires two good batteries in order to shine. Suppose, for the sake of this
exercise, that the lifetimes of the batteries in use are independent random variables that are exponentially distributed with parameter λ = 1. Reserve batteries do not deteriorate. You begin with five fresh batteries. On average, how long can you shine your light?
7. Consider a gambling game where you win $1 with probability p = 0.4, and lose $1 with
probability 1 - p = 0.6 on each turn. The game ends when you either accumulate $3 or go broke. You start with $1. Let Xn denote your fortune after t turns of the game.
a) Model {Xn= {X1, X2,...}, the evolution of your gambling fortune, as a Markov chain by
providing its corresponding graphical representation (i.e., states and transition probabilities).
b) Should you play this gambling game? c) Will the game eventually end? d) What is the probability you win $3 or go broke? e) How does everything change with the value of p?
8. A store that stocks a particular laptop model uses the following (s, S) ordering policy; if the
number of laptops in inventory at the beginning of a day is x, then it orders: • 0 if x ≥ s, • S - x if x < s.
The order is immediately filled. The daily demands are independent and equal j with probability aj. All demands that cannot be immediately met are lost. Let Xn denote the inventory level at the end of the n-th day. Model {Xn, n ≥ 1} as a Markov chain by providing its corresponding graphical representation (i.e., states and transition probabilities). Assume that the store set the policy with s = 1 and S = 3.
9. Consider an electronic device. Suppose electric shocks occur according to a Poisson process with rate l = 3/year, and suppose that each electric shock, independently, causes the device to fail with probability p = 0.1. Let N denote the number of electric shocks that it takes for the device to fail and let T denote the time to failure.
a) Estimate the value of N and T by simulating this stochastic process until device failure.
Consider only one replication and use the uniform random numbers below.
0.5118 0.2836 0.8416 0.2951 0.7105 0.2756 0.1561 0.7259 0.8209 0.0857
b) Assume that during the first year five electric shocks occurred. Use the uniform random number above to simulate the time of each electric shock.
10. Consider a sequence of items from a production process, with each item being graded as
good or defective. Suppose that a good item is followed by another good item with probability 0.99 and is followed by a defective item with probability 0.01. Similarly, a defective item is followed by another defective item with probability 0.12 and is followed by a good item with probability 0.88.
a) If the first item is good, what is the probability that the first defective item to appear is
the fifth item? b) What is the probability that the fourth item is defective given that the first item is
defective? c) Let Xn denote the quality of the n-th item with Xn = 0 meaning “good” and Xn = 1 meaning
“defective”. Model Xn, the quality of the item produced, as a Markov chain by providing its corresponding graphical representation (i.e., states and transition probabilities).
d) In the long run, what is the probability that an item produced by this process is defective? e) What would be the answers to parts a) and b) based on one replication of the simulation
of the production process. Use the uniform random numbers in Problem 9.
11. A time-shared computer system has three terminals that are attached to a central processing unit (CPU) that can simultaneously handle at most two active users. If a person logs on and requests service when two other users are active, then the request is held in a buffer until it can receive service. Assume that the time between two consecutive service requests is exponentially distributed with mean 5 minutes and that an active session takes a time that is exponentially distributed with mean 4 minutes. Let X(t) be the total number of requests that are either active or in the buffer at time t.
a) Model the evolution of X(t), the total number of requests that are either active or in the
buffer, as a continuous time Markov chain by providing the corresponding graphical representation.
b) Simulate the evolution of the total number of requests that are either active or in the buffer using the uniform random numbers provided in Problem 1. Assume that no request is active or in the buffer at the beginning of the simulation.
c) Determine the long run probability that the computer is fully loaded. 12. Consider the following trunk reservation problem for a link in a communication network. The
link has 4 circuits (or frequencies), which is the capacity of the link. Transmitting a message at any time occupies or uses one circuit. Hence, at any time, the link can transmit a maximum of 4 messages. Two types of messages, called Type 1 and Type 2, arrive at the link requesting a frequency for transmission. Their arrivals follow Poisson processes with rates 15 and 20 messages per hour, respectively. The transmission times for both types of messages are independent and have the same exponential probability distribution with mean 4 minutes. The admission of a message upon its arrival follows the following trunk reservation rule:
• A Type 1 arrival is admitted as long as there is at least one circuit available; • A Type 2 arrival is admitted only when there are (strictly) more than 2 circuits available.
This limit is known as the reservation level for Type 1 traffic.
A message that is not admitted is said to be blocked. Blocked messages are lost permanently. a) Model the evolution of the number of circuits being used over time as a continuous time
Markov chain by providing its corresponding graphical representation. Feel free to use any of the alterative characterizations discussed in class.
b) Determine the stationary (i.e., steady state) probabilities for this stochastic process. c) In the communication industry, the most important measure for the quality of service is
the blocking probability, i.e., the probability that an arriving message is blocked (and hence not transmitted). Based on the values obtained in part b), determine the blocking probability for each type of traffic.
d) Simulate the number of circuits being used over time until you run out of Uniform(0,1) random numbers. Perform only one replication and start your simulation with no circuits in use. Please use the following random numbers:
1) 0.7696 2) 0.1349 3) 0.6783 4) 0.4355 5) 0.2989 6) 0.8081 7) 0.0685 8) 0.0207 9) 0.1851 10) 0.3491
13. An airline reservation system has two computers, only one of which is in operation at any
given time. A computer may break down on any given day with probability p = 0.2. There is a single repair facility that takes 2 days to restore a computer to normal. The facility is such that only one computer at a time can be dealt with. Model the system as a discrete-time Markov chain by taking as states the pairs (x, y), where x is the number of machines in operating condition at the end of a day and y is 1 if a day’s labor has been expended on a machine not yet repaired and 0 otherwise.
In the long run, what is the probability that at least one computer is operating? What fraction of time is the repair facility idle?