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MAST30001 Stochastic Modelling – 2019

Assignment 2

If you haven’t already, please complete the Plagiarism Declaration Form (available through

the LMS) before submitting this assignment.

Don’t forget to staple your solutions (note that there are no publicly available staplers

in Peter Hall Building), and to put your name, student ID, tutorial time and day, and the

subject name and code on the first page (not doing so will forfeit marks).

The submission deadline is 4:15pm on Friday, 25 October, 2019, in the appropriate

assignment box in Peter Hall Building (near Wilson Lab).

There are 2 questions, both of which will be marked. No marks will be given for answers

without clear and concise explanations. Clarity, neatness, and style count.

1. Let 쒐N(t)t≥0

be a Poisson process with rate 1, and for ε > 0, let X(ε) 1

, X(ε) 2, . . . bei.i.d. with densityC1 ε y1ey, y > ε,

where the normalising constant is

Cε = Z ∞ε x1ex

dx.

Finally, for t > 0, define the time-scaled compound Poisson processZ(ε) t = NX(tCε) j=1

X(ε) j .

(a) Show that for any ε > 0,Cε ≤ ε1eε,

and that for any 0 < ε < 1,e1

log(1/ε) ≤ Cε.

(b) Show that the number of jumps N(tCε) of the process (Z(ε) s )s≥0 up to time t

converges to infinity in probability as ε → 0+. (That is, show that for all n ∈ N, P(N(sCε) ≥ n) → 1 as ε → 0+.)

(c) Show that the Laplace transform of Z(ε) t : Lε(θ) := EeθZ(ε) t

, θ ≥ 0,

converges pointwise as ε → 0+, and identify the limit as the Laplace transform

of a well-known distribution.

(d) Explain in one or two sentences how the number of jumps can go to infinity, but

the distribution of Z(ε) t

can converge.

In fact, the whole process (Z(ε) t )t≥0 converges to a process having independent increments and marginals given by part (c). The limit is a non-decreasing pure jump

process, with the times of the jumps dense in the positive line.

2. A certain queuing system has two types of customers and two types of servers. Type A

customers arrive according to a Poisson process with rate 3, and, independently,

Type B customers arrive according to a Poisson process with rate 2. If Server A

is free, then an arriving Type A customer begins service with Server A. If Server A

is busy but Server B is free, then an arriving Type A customer will begin service

with Server B. If an arriving Type A customer finds both servers busy, they will leave

the system. If Server B is free, then an arriving Type B customer will be served by

Server B, and otherwise will leave the system. Server A takes an exponential rate 2

time to finish a service, Server B takes an exponential rate 1 time to finish a service,

and all service times are independent and independent of arrivals.

(a) Model the system as a four state Markov chain and write down its generator.

(b) Find the stationary distribution of the Markov chain.

(c) What is the stationary average number of customers in the system?

(d) Given a customer is not immediately rejected from the system, what is the average

time they spend in the system?

(e) What is the long-run proportion of time is there a Type A customer being served

by Server B?

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