MATH268: Mock Exam
Dr Yi Zhang
All the problems in the mock exam are seen either in tutorials or in assignments.
Do not expect any problem in the mock exam to be present in the final/resit exam.
This is just to give you the style of the eaxm (four questions). In the final/resit
exam, there can be questions from other blocks. (There is no question from Block
5 about time series forecasting.)
Exercise 1 The government of a fictitious country is attempting to determine whether tourists
should be allowed to enter the country. The government is concerned with a contagious disease.
Assume that the decision will be made on a financial basis. Each tourist who is allowed into the
country and has the disease costs the country £100,000, and each tourist who enters and does
not have the disease will contribute £10,000 to the national economy. Assume that 10% of all
potential tourists have the disease. The government may admit the tourist, reject the tourist,
or test the tourist for the disease before determining whether he/she should be admitted.
If costs £100 to test a person for the disease; the test result is either positive or negative.
If the test result is positive, the person definitely has the disease. However, 20% of the people
who do have the disease tests negative. A person who does not have the disease will always test
negative.
The government’s goal is to maximize the expected economic benefit (net profit). Suggests
with justifications the optimal policy of the government. What is the expected economic benefit
(per person) under this policy?
Exercise 2 Consider the M/M/1 queueing system with arrival rate λ > 0 and service rate
µ > 0.
(a) Compute the expected number of arrivals during a service time (also called service period).
(b) Compute the probability that no customers arrive during a service period.
Exercise 3 Fast food restaurant Mark has two windows: one serving drinks and the other
one serving snacks. In front of each of the two windows, one queue is built up by arriving
customers. For each queue, the inter-arrival times are independently exponentially distributed
with parameter 40 (i.e., the customers for snacks and those for drinks arrive according to two
independent Poisson processes both with parameter 40). At each window, there is one server.
Suppose the service times are independently exponentially distributed with parameter 60. The
time unit is fixed here and below.
(a) Calculate the mean number of customers waiting in lines in the steady state, altogether.
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(b) Suppose now there is only one window, in front of which there is only one queue being
built up. At the window we put two servers who can provide both drinks and snacks, and
their service times are independently exponentially distributed with parameter 60. What
is the mean number of waiting customers in the steady state?
(c) Based on parts (a) and (b), which arrangement is more reasonable if we want to minimize
the expected number of enqueued customers (excluding those receiving service)?
Exercise 4 Suppose it is known how to simulate a random variable X with the cumulative
distribtuion function F defined by F (x) = P (X ≤ x) for all x ∈ (−∞,∞). Based on this fact,
suggest how to simulate a random variable Y whose cumulative distribution function G is given
by G(x) = F (x)n, where n ≥ 2 is an integer.
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