Grinstead-Snell-Probability · Problem 11.2.21

Markov Chains

Mathematics · absorbing-markov-chains

Problem

(Roberts) A city is divided into 3 areas 1, 2, and 3. It is estimated that amounts

u_1

u_2

, and

u_3

of pollution are emitted each day from these three areas. A fraction

q_{ij}

of the pollution from region

i

ends up the next day at region

j

. A fraction

q_i = 1 - \sum_j q_{ij} > 0

goes into the atmosphere and escapes. Let

w_i^{(n)}

be the amount of pollution in area

i

after

n

days. (a) Show that

\mathbf{w}^{(n)} = \mathbf{u} + \mathbf{u}\mathbf{Q} + \cdots + \mathbf{u}\mathbf{Q}^{n-1}

. (b) Show that

\mathbf{w}^{(n)} \to \mathbf{w}

, and show how to compute

\mathbf{w}

from

\mathbf{u}

. (c) The government wants to limit pollution levels to a prescribed level by prescribing

\mathbf{w}

. Show how to determine the levels of pollution

\mathbf{u}

which would result in a prescribed limiting value

\mathbf{w}