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Question 1 This question is intended...

Question 1

This question is intended to assess your understanding of birth and death

processes.

You should be able to answer this question after working through Part I of

Book 4.

Members of the public visit a charity shop to donate second-hand books

for sale according to a Poisson process at rate λ. Each person brings

N books, where N has a geometric distribution N ∼ G1(p). Independently

of any others, each book is displayed in the shop for a time that is

exponentially distributed with mean 1/ν (every book is sold eventually).

The number of books on the shelves at time t is denoted by X(t), and the

probability generating function of X(t) is denoted by Π(s, t). You may

assume that a partial differential equation for Π(s, t) is

(1 − qs) ∂Π

∂t

= ν(1 − s)(1 − qs) ∂Π

∂s

− λ(1 − s)Π,

where q = 1−p.

(a) Show that two solutions of the auxiliary equations may be written as

c1 = (1−s)e−νt, c2= (1−qs)λ/(qν)Π.

(b) Hence write down the general solution for Π(s, t).

(c) At time 0, all the books are removed from the shop in order to stock

another shop that is about to open, so X(0) = 0. Show that the

probability generating function of X(t) may be written as

Π(s, t) =$ p+q(1 − s)e−νt 1 − qs/(qν)

(d) On average, 20 people bring books to the shop each week, and the

mean number of books per person is 16. Each book remains on the

shelves for three weeks, on average.

(i) Find the value of p, and write down the values of λ and ν.

(ii) Find Π(s) = lim

t→∞

Π(s, t), and hence identify the equilibrium

distribution of the number of books in the shop in this particular

case. You should substitute for p, λ and ν using the values that

you gave in part

(iii) In the long run, what is the mean number of books in the shop at

any time?

http://www.mediafire.com/view/?fpb89dhod87oa28

This question is intended to assess your understanding of birth and death

processes.

You should be able to answer this question after working through Part I of

Book 4.

Members of the public visit a charity shop to donate second-hand books

for sale according to a Poisson process at rate λ. Each person brings

N books, where N has a geometric distribution N ∼ G1(p). Independently

of any others, each book is displayed in the shop for a time that is

exponentially distributed with mean 1/ν (every book is sold eventually).

The number of books on the shelves at time t is denoted by X(t), and the

probability generating function of X(t) is denoted by Π(s, t). You may

assume that a partial differential equation for Π(s, t) is

(1 − qs) ∂Π

∂t

= ν(1 − s)(1 − qs) ∂Π

∂s

− λ(1 − s)Π,

where q = 1−p.

(a) Show that two solutions of the auxiliary equations may be written as

c1 = (1−s)e−νt, c2= (1−qs)λ/(qν)Π.

(b) Hence write down the general solution for Π(s, t).

(c) At time 0, all the books are removed from the shop in order to stock

another shop that is about to open, so X(0) = 0. Show that the

probability generating function of X(t) may be written as

Π(s, t) =$ p+q(1 − s)e−νt 1 − qs/(qν)

(d) On average, 20 people bring books to the shop each week, and the

mean number of books per person is 16. Each book remains on the

shelves for three weeks, on average.

(i) Find the value of p, and write down the values of λ and ν.

(ii) Find Π(s) = lim

t→∞

Π(s, t), and hence identify the equilibrium

distribution of the number of books in the shop in this particular

case. You should substitute for p, λ and ν using the values that

you gave in part

(iii) In the long run, what is the mean number of books in the shop at

any time?

http://www.mediafire.com/view/?fpb89dhod87oa28

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