ENM 307 SAMPLE MIDTERM QUESTIONS
1. Consider the following experiment. A fair coin is tossed, if the result is Heads, a sixsided die is rolled, otherwise a four-sided die is rolled. Both dice are unbiased. Let X be
the number that shows up on the die..
a. Find the value of E[X] and Var[X].
b. Use the uniform random numbers in the table below to generate the value of X for 10
times.(Hint: First calculate the probability of X taking the values 1 to 6 for the above
experiment and then use the random numbers below to generate the realizations of X)
Stream
1
2
3
4
5
6
7
8
9
10
0.49
0.27
0.94
0.49
0.72
0.09
0.98
0.62
0.73
0.36
2. The weekly demand for an slow-moving item has a Binomial distribution with
parameters n=3 and p=1/2. A simulation experiment will be performed to find the
optimal ordering quantities. Use the random numbers below to generate 5 demand
realizations from this demand distribution. Note that for a random variable X with a
binomial distribution, the probability that P(X=x) can be calculated as:
𝑛
𝑃(𝑋 = 𝑥) = ( ) 𝑝 𝑥 (1 − 𝑝)𝑛−𝑥
𝑥
Random number
1
2
3
4
5
0.159
0.232
0.098
0.582
0.987
3. Consider a single server queueing system. Arrivals take place every four minutes (first
arrival takes place at t=0). Service times are random and take the value of 1 with
probability 1/2 and 5 with probability ½. Assume that the following service time
sequence is generated: S1=5, S2=5, S3=5, S4=1, S5=5. Simulate for T=20 minutes and
estimate the average number of customers in the system and the average waiting time
of a customer.
4. Consider a two-teller bank where each teller has its separate queue. Customers are
allowed to jockey between queues at departure instances. Note that a customer only
switches queues at departure instances and if the other queue is shorter. New arrivals
join the shorter queue and choose queue 1 if both queues are of equal length. Let n1(t)
and n2(t) denote the number of customers in the system in queues 1 and 2 at time t. The
following customer inter-arrival times (Ai) and service times for tellers 1 (S1i) and
tellers 2 (S2i) have been generated: A1 = 1.8, A2 = 3.2, A3 = 1.1, A4 = 5.2, S11 = 2.1,
S12 = 1.6, S13 = 5.2, S14 = 0.8 and S21 = 4.9, S22 = 2.7, S23 = 4.3, S24 = 2.3.
a. Assume that n1(0) = 2, n2(0) = 3 and that both servers have just started processing
at time 0. Use the above inter-arrival times and server times to simulate the system
for 10 minutes using the below table:
b. Find the time average number of total customers over 10 minutes.