Problem 3, Page 953
New York City produces 1,000 tons of air pollution per day, Jersey City 100 tons, and Newark 50 tons. Each day, ⅓ of New York’s pollution is blown to Newark, ⅓ dissipates, and ⅓ remains in New York. Each day, ⅓ of Jersey City’s pollution is blown to New York, ⅓ stays in Jersey City, and ⅓ is blown to Newark. Each day, ⅓ of Newark’s pollution stays in Newark, and the rest is blown to Jersey City. On a typical day, which city will be the most polluted?
Transition Matrix:
|
New York(NY)
|
Newark (NW)
|
Dissipate
|
Jersey (J)
|
New York(NY)
|
1/3
|
1/3
|
1/3
|
0
|
Newark (NW)
|
0
|
1/3
|
0
|
2/3
|
Dissipate
|
0
|
0
|
1
|
0
|
Jersey (J)
|
1/3
|
1/3
|
0
|
1/3
|
1000-2/3NY+1/3J = 0
50-2/3NW+1/3NY+1/3J = 0
100-2/3J+2/3NW = 0 +
1150 = 1/3NY
NY= 3450
1000-2/3(3450)+1/3J = 0
J = 3900
300-2(3900)+2NW = 0
NW = 3750
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