# Question

Consider the inventory example presented in Sec. 29.1 except that demand now has the following probability distribution:

P{D = 0} = 1/4, P{D = 2} = 1/4,

P{D = 1} = 1/2, P{D > 3} = 0.

The ordering policy now is changed to ordering just 2 cameras at the end of the week if none are in stock. As before, no order is placed if there are any cameras in stock. Assume that there is one camera in stock at the time (the end of a week) the policy is instituted.

(a) Construct the (one-step) transition matrix.

(b) Find the probability distribution of the state of this Markov chain n weeks after the new inventory policy is instituted, for n = 2, 5, 10.

(c) Find the µij (the expected first passage time from state i to state j) for all i and j.

(d) Find the steady-state probabilities of the state of this Markov chain.

(e) Assuming that the store pays a storage cost for each camera remaining on the shelf at the end of the week according to the function C(0) = 0, C(1) = $2, and C(2) = $8, find the long-run expected average storage cost per week.

P{D = 0} = 1/4, P{D = 2} = 1/4,

P{D = 1} = 1/2, P{D > 3} = 0.

The ordering policy now is changed to ordering just 2 cameras at the end of the week if none are in stock. As before, no order is placed if there are any cameras in stock. Assume that there is one camera in stock at the time (the end of a week) the policy is instituted.

(a) Construct the (one-step) transition matrix.

(b) Find the probability distribution of the state of this Markov chain n weeks after the new inventory policy is instituted, for n = 2, 5, 10.

(c) Find the µij (the expected first passage time from state i to state j) for all i and j.

(d) Find the steady-state probabilities of the state of this Markov chain.

(e) Assuming that the store pays a storage cost for each camera remaining on the shelf at the end of the week according to the function C(0) = 0, C(1) = $2, and C(2) = $8, find the long-run expected average storage cost per week.

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