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 q_ij of the pollution from region i ends up the next day at region j. A fraction q_i =1−Σ_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 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).