The contents of a reservoir depend on the available rainfall in the region and the demands on the water supply. Suppose that a reservoir holds up to 4 units of water (a unit might be a million gallons), and policy for the use of the water for irrigation and drinking water never allows the contents of the reservoir to drop below 1 unit of water. The (rounded) amount of water in the reservoir from week to week seems to follow the transition matrix

(a) Determine and interpret the stable distribution for the matrix.
(b) Suppose that the weekly benefits to recreation in the area around the reservoir are estimated to be $4000 when there is 1 unit in the reservoir, $6000 when there are 2 units in the reservoir, $10,000 when there are 3 units, and $3000 when there are 4 units. Determine the average weekly benefits to be realized in the long run?

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