Moran Storage Model. A reservoir with capacity c receives inputs V 1 , V 2 ,... in discrete time periods, where V n are i.i.d. nonnegative integervalued random variables. In a period when the...

Moran Storage Model. A reservoir with capacity c receives inputs V₁, V₂,... in discrete time periods, where V_n
are i.i.d. nonnegative integervalued random variables. In a period when the reservoir has room for y additional units (the reservoir level is c − y), and v units of input occur, then (v − y)+ units of the input are discarded. In addition, the reservoir releases a non-random quantity of u units of water in each time period provided the reservoir level exceeds u, otherwise it releases all the water in the dam. Let
_Xn
denote the amount of water in the reservoir at (the end of) time period n. Justify that
_Xn
is a reflected random walk Markov chain and express its transition probabilities as a function of the distribution q_n
= P{V₁
= n}.