Question 11.2.21: (Roberts7) A city is divided into 3 areas 1, 2, and 3. It is......
(Roberts7) A city is divided into 3 areas 1, 2, and 3. It is estimated that amounts u1, u2, and u3 of pollution are emitted each day from these three areas. A fraction qij of the pollution from region i ends up the next day at region j. A fraction qi =1−Σj qij > 0 goes into the atmosphere and escapes. Let wi(n) be the amount of pollution in area i after n days.
(a) Show that w(n)= u + uQ + ··· + uQn−1.
(b) Show that w(n)→ w, and show how to compute w from u.
(c) The government wants to limit pollution levels to a prescribed level by prescribing w. Show how to determine the levels of pollution u which would result in a prescribed limiting value w.
7F. Roberts, Discrete Mathematical Models (Englewood Cliffs, NJ: Prentice Hall, 1976).
The problem should assume that a fraction
q_i= 1-\sum\limits_{j}{q_{ij}}\gt 0
of the pollution goes into the atmosphere and escapes.
(a) We note that u gives the amount of pollution in each city from today’s emission, uQ the amount that comes from yesterday’s emission, uQ2 from two days ago, etc. Thus
wn = u + uQ + · · · uQn−1.
(b) Form a Markov chain with Q-matrix Q and with one absorbing state to which the process moves with probability qi when in state i. Then
I + Q + Q2+ · · · + Qn−1→ N ,
so
w(n)→ w = uN .
(c) If we are given w as a goal, then we can achieve this by solving w = Nu for u, obtaining
u = w(I − Q) .