Student
number
Semester 2 Assessment, 2021
School of Mathematics and Statistics
MAST30001 Stochastic Modelling, Assignment 2
Submission deadline: 8 p.m. Friday 15 October
This assignment consists of 3 pages (including this page)
Instructions to Students
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Writing
• There are 3 questions with marks as shown. The total number of marks available is 34.
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©University of Melbourne 2021 Page 1 of 3 pages Do not place in Baillieu Library
MAST30001 Stochastic Modelling, Assignment 2 Semester 2, 2021
Question 1 (12 marks)
Let Nt be the number of earthquakes observed in New Zealand (NZ) up to time t (time units
in this question are always years). We will assume that (Nt)t≥0 is a Poisson process with rate 5
(per year), and that the magnitude of each earthquake is Exponentially distributed with mean
2, independent of all other earthquakes.
(a) Let A be the event that there are no earthquakes in the second half year. Express A as
an event in terms of (Nt)t≥0 and find P(A).
(b) Let T1 be the time of the first earthquake. Find P(T1 > 1/2|N1 = 2).
(c) What is the distribution of the number of earthquakes in NZ of magnitude 6 or more in
the first 10 years?
(d) Suppose that there are exactly 7 earthquakes in each of the first 3 years. What is the
probability that there are also 7 earthquakes in the 4th year?
(e) Find the distribution function of the magnitude of the largest earthquake in the first year
(if there is no earthquake this is defined to be 0).
(f) Give a reason why modelling earthquake occurrences as a Poisson process may be unre-
alistic.
Question 2 (14 marks)
In a small remote neighbourhood consisting of k = 5 individuals, one individual is infected with
a virus at time 0. Suppose that each pair of individuals in this neighbourhood meets at times
of a Poisson process of rate 1 (independent of other meetings). Meetings between an infected
individual and an uninfected individual result in the uninfected individual being infected with
probability p ∈ (0, 1). Infected individuals remain infected for an Exponential amount of time
with mean µ, independent of everything else. Noone interacts with anyone from outside the
town. Let Nt denote the number of infected individuals at time t (so N0 = 1). For the CTMC
(Nt)t≥0:
(a) Draw the transition diagram.
(b) Find the generator matrix.
(c) Find all stationary distributions.
(d) Find the probability that at some time everyone in the neighbourhood is (simultaneously)
infected if p = µ = 1.
(e) Find the expected time until noone in the neighbourhood is infected by the virus when
µ = p = 1.
(f) How would your answers to (d) and (e) above change as µ↗ ∞ or µ↘ 0?
(g) If µ = p = 1, what happens to the answers to (d) and (e) as k (the number of individuals
in the neighbourhood) goes to infinity? (Note that for (d), the state k that you are trying
to reach is growing as k → ∞, but so is e.g. p1,2)
Page 2 of 3 pages
MAST30001 Stochastic Modelling, Assignment 2 Semester 2, 2021
Question 3 (8 marks)
At a small airport, arriving passengers arrive as a Poisson process of rate 1 per minute and
each such passenger is assigned uniformly at random (independent of the length of the queues)
to one of k servers (that each serve the customers in their queue in the order in which they
arrived). Service times at each queue are Exponential random variables with mean 2.5 minutes
(independent of everything else). Let N
(i)
t denote the number of customers in queue i = 1, . . . , k
at time t, and Nt =
∑k
i=1N
(i)
t .
(a) Is this system an M/M/k queue? Why or why not?
(b) How large does k have to be to ensure that the queuing system is stable (i.e. that
no matter how many customers are currently in the system, with probability 1 it will
eventually become empty)?
(c) Assuming that k is sufficiently large so that the system is stable, find the limiting distri-
bution for Nt as t→ ∞, e.g. find limt→∞ P(Nt = n) for each n = 0, 1, 2, . . . .
End of Assignment — Total Available Marks = 34
Page 3 of 3 pages