Question

Q1.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}.

.Q2 Let {Xn|n ≥ 0} be as in Q6. Show that Xn+1 = X0 + Zn+1 − Xn m=0 min{0, Xm + Vm+1 − Um+1}, where Zn = Xn m=1 (Vm − Um), n ≥ 1.

Q3 Let {Xn|n ≥ 0} be as in Q7. Show that Xn = max{X0 + Zn, max 1≤m≤n Zn − Zm}, n ≥ 1

4.In a discrete time input-output system, let Xn denote the number of units of an item at end of the nth period, (n − 1, n]. In each period (n − 1, n], the system has an input(arrival) Vn and an output (departure) Un, n ≥ 1. The system disregard any output if its exit makes the state of the system negative. We assume that {(Un, Vn)|n ≥ 1} is an i.i.d. sequence such that Un is independent of Vn for all n and P{U1 = k} = qk, k ≥ 0 and P{V1 = k} = pk, k ≥ 0. Assume that X0 is Z +-valued random variable independent of {(Un, Vn)|n ≥ 1}. Show that {Xn|n ≥ 0} is a Markov chain

Answer 1