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Student
number
Semester 2 Assessment, 2020
School of Mathematics and Statistics
MAST30001 Stochastic Modelling
Reading time: 30 minutes — Writing time: 3 hours — Upload time: 30 minutes
This exam consists of 25 pages (including this page)
Permitted Materials
• This exam and/or an offline electronic PDF reader, one or more copies of the masked exam
template made available earlier, blank loose-leaf paper and a Casio FX-82 calculator.
• One double sided A4 page of notes (handwritten or printed).
Instructions to Students
• There are 7 questions with marks as shown. The total number of marks available is 80.
• Working and/or reasoning must be given to obtain full credit. Clarity, neatness and style
count.
• During writing time you may only interact with the device running the Zoom session with
supervisor permission. The screen of any other device must be visible in Zoom from the
start of the session.
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to a second device, which must then be disconnected from the internet.
• Write your answers in the boxes provided on the exam that you have printed or the masked
exam template that has been previously made available. If you are unable to answer the
whole question in the answer space provided then you can append additional handwritten
solutions to the end after the 25 numbered pages. If you do this you MUST make a note
in the correct answer space or page for the question, warning the marker that you have
appended additional remarks at the end.
• If you have been unable to print the exam and do not have the masked template write
your answers on A4 paper. The first page should contain only your student number, the
subject code and the subject name. Write on one side of each sheet only. Start each
question on a new page and include the question number at the top of each page.
• Assemble all exam pages (or masked template pages) in correct page number order and
the correct way up. Add any extra pages with additional working at the end. Use a mobile
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Gradescope window. Before leaving Zoom supervision, confirm with your Zoom supervisor
that you have Gradescope confirmation of submission.
c©University of Melbourne 2020 Page 1 of 25 pages Can be placed in Baillieu Library
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MAST30001 Stochastic Modelling Semester 2, 2020
Question 1 (16 marks)
A Markov chain (Xn)n≥0 with state space S = {1, 2, 3, 4, 5} has transition matrix
P =
1/2 0 0 0 1/2
0 1/2 1/2 0 0
0 1/5 4/5 0 0
0 1/4 1/4 1/4 1/4
1/3 0 0 0 2/3
.
(a) Assuming X0 is uniformly distributed on the set {2, 3, 5}, find
(i) P(X4 = 3, X2 = 3|X1 = 2), and
(ii) P(X4 = 3, X2 = 3).
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MAST30001 Stochastic Modelling Semester 2, 2020
(b) Write down the communication classes of the chain. For each class, find the period,
determine whether it is essential, and classify it as transient or positive recurrent or null
recurrent.
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MAST30001 Stochastic Modelling Semester 2, 2020
(c) Describe the long run behaviour of the chain (including deriving long run probabilities
where appropriate).
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MAST30001 Stochastic Modelling Semester 2, 2020
(d) Find the probability of reaching state 1 before state 2 given the chain starts in state 4.
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MAST30001 Stochastic Modelling Semester 2, 2020
Question 2 (15 marks)
A train line operates from 5am to midnight, and the times between trains that stop at a certain
station are independent with distribution uniform between 10 and 20 minutes.
(a) Estimate how many trains stop at the station between 5am and 10am, and give a sym-
metric interval around your estimate that has a 95% chance of covering the true number
of trains that stop.
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MAST30001 Stochastic Modelling Semester 2, 2020
(b) You arrive at the station at 7pm. What is a good estimate for the mean of the time
until the next train stops at the station?
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MAST30001 Stochastic Modelling Semester 2, 2020
(c) Now assume that trains still arrive at the station in the same way, but each train doesn’t
stop at the station with probability 1/10, independently between trains. What is a good
estimate of the number of trains that stop at the station between 5am and 10am?
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MAST30001 Stochastic Modelling Semester 2, 2020
Question 3 (11 marks)
Customers arrive to an outback auto repair shop according to a Poisson process with rate 3
per day. The shop has one mechanic who takes an exponential with rate 3 per day amount
of time to repair a car. In addition, if there are no cars in the shop, the mechanic will wait
an exponential rate 1 per day time, and if no car has arrived in that time, the mechanic will
leave the shop and take a nap for an exponential rate 10 per day time. If a car arrives while
the mechanic is waiting to leave to take a nap, they will begin work on the car. Cars that
arrive when the mechanic is working form a queue. When the queue has three cars in it, or the
mechanic is taking a nap, cars arriving for repair will move on to the next repair shop.
(a) Model this system as a continuous time Markov chain and write down its state space
and generator.
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MAST30001 Stochastic Modelling Semester 2, 2020
(b) Determine the stationary distribution of the chain.
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MAST30001 Stochastic Modelling Semester 2, 2020
(c) What is the stationary average number of customers at the auto repair shop (both those
waiting for service and in service)?
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MAST30001 Stochastic Modelling Semester 2, 2020
(d) What proportion of customers are rejected from the system?
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MAST30001 Stochastic Modelling Semester 2, 2020
(e) Given that a customer is not immediately rejected from the system, what is the average
time they spend waiting to be served?
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MAST30001 Stochastic Modelling Semester 2, 2020
Question 4 (17 marks)
Phone calls arrive to a large phone bank according to a Poisson process with rate 2 per minute.
Received calls are answered immediately, and the time a call lasts (rounded up to the nearest
minute), is distributed as a geometric random variable X satisfying
P(X = k) = (2/3)k−1(1/3), k = 1, 2, . . . ,
and the times calls last are independent.
(a) What is the chance that at least 2 calls are received in a 2 minute time interval?
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MAST30001 Stochastic Modelling Semester 2, 2020
(b) What is the chance that at least 2 calls lasting exactly 1 minute are received in a 2
minute time interval?
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(c) Given that 10 calls were received in a given 5 minute time interval, what is the chance
that exactly 3 were received in the first minute of the interval?
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MAST30001 Stochastic Modelling Semester 2, 2020
(d) Given that 10 calls were received in a given 5 minute time interval, what is the chance
that exactly 3 calls lasting exactly 1 minute were received in the first minute of the
interval?
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MAST30001 Stochastic Modelling Semester 2, 2020
(e) Assuming there are no queued calls when the phone bank opens, for each n = 1, 2, . . . ,
find the distribution ofXn equal to the number of calls currently being handled nminutes
after the phone bank opens.
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MAST30001 Stochastic Modelling Semester 2, 2020
(f) Now assume the phone bank also receives calls from an additional independent source
according to a Poisson process with rate 5 per minute, and having the same service
discipline. Now what is the chance that at least 2 calls lasting exactly 1 minute are
received (from either source) in a 2 minute time interval?
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MAST30001 Stochastic Modelling Semester 2, 2020
Question 5 (11 marks)
Let (Bt)t≥0 be a Brownian motion. For any normal probabilities below, you can write your
answer in terms of the standard normal distribution function Φ(x) = (2π)−1/2
∫ x
−∞ e
−t2/2dt.
(a) Find P(B10 ≥ −2|B4 = −1).
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MAST30001 Stochastic Modelling Semester 2, 2020
(b) Find P(B10 ≥ −2|B4 = −1, B2 = 1).
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MAST30001 Stochastic Modelling Semester 2, 2020
(c) Find P(B4 ≥ −2|B10 = −1).
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MAST30001 Stochastic Modelling Semester 2, 2020
(d) Let Z be standard normal random variable that is independent of (Bt)t≥0, and for
0 ≤ t ≤ 1 set
Xt = (Bt − tB1) + tZ.
Show that (Xt)0≤t≤1 is a standard Brownian motion (restricted to the interval [0, 1]).
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MAST30001 Stochastic Modelling Semester 2, 2020
Question 6 (6 marks)
Let K be a positive integer. A Markov chain (Xn)n≥0 with state space S = {0, 1, . . . ,K} has
transition probabilities
pi,i+1 = pi,i−1 = 1/2, 1 ≤ i ≤ K − 1,
and
p0,1 = 1− p0,0 = pK,K−1 = 1− pK,K = α.
Let T = inf
{
n ≥ 1 : Xn ∈ {0,K}
}
. For each i ∈ {0, 1, . . . ,K} find a simple expression in terms
of α,K, and i for
E[T |X0 = i].
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MAST30001 Stochastic Modelling Semester 2, 2020
Question 7 (4 marks)
Let (Xt)t≥0 be a continuous time Markov chain on S = {0, 1, 2, . . .} with generator A = (aij)i,j∈S
satisfying
ai,i+1 = 2
i+1/2, i ≥ 0, ai,i−1 = 2i, i ≥ 1,
and with all other off-diagonal entries zero. Show that (Xt)t≥0 is an explosive continuous time
Markov chain.
End of Exam — Total Available Marks = 80
Page 25 of 25 pages