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. (Unit demand inventory system) Consider an inventory system in discrete time with the following description....

. (Unit demand inventory system) Consider an inventory system in discrete time with the following description. At the beginning of the period the inventory decreases by one unit if the inventory level at the beginning is positive other the level remains zero till the end of the period. At the end of the period nth period, the inventory is increased by an amount Vn, where {Vn|n ≥ 1} is i.i.d. with P{V1 = i} = pi , i ≥ 0. Let Xn denote the level of the inventory at the beginning(just before the probable inventory decrease) of the period [n, n + 1). Show that {Xn|n ≥ 0} is a Markov chain under the assumption that X0 is Z + valued random variable which is independent of {Vn|n ≥ 1}. Also find its transition matrix.

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