A local store uses an (s, S) inventory policy for a particular product. Every Friday evening after the store closes, the inventory level is checked. If the stock level for the product is greater than s, no action is taken. Otherwise, if the stock on hand is less than or equal to s, an amount S-s is procured over the weekend and is available when the store re-opens on Monday morning. Let {Xn, n = 1, 2, …} be the stock on hand when the inventory is checked on Friday evening of week n and let {Zn, n = 1, 2, …} be the demand for product during week n. Then for any ω ∈ Ω,
Xn+1(ω) = Xn(ω) - Zn+1(ω) if s < Xn(ω), Zn+1(ω) ≤ Xn(ω)
Xn+1(ω) = S - Zn+1(ω) if Xn(ω) ≤ Zn(ω) ≤ S
Xn+1(ω) = 0 otherwise.
Let s = 3, S = 5, X0 = 0 and assume customers arrive at the store according to a Poisson process with rate λ = 4 per week. (Each customer wants one unit of product.)
A) Show that {Xn, n = 1, 2, ... } is a Markov chain. What is its state space?
B) Write the one-step transition matrix for {Xn, n = 1, 2, ... }.
Present your matrix in terms of the variables pk = P{ exactly k customers arrive during the week }, and rk = P{ k or more customers arrive during the week }. What are the formulas for computing pk and rk?
C) If the starting inventory on Monday morning is 4 units, what is the probability that a shortage occurs by Friday afternoon? (A shortage occurs if a customer arrives and there are no units available.)
D) Write the form for the one-step transition matrix if the value of s were not known, i. e., with s as a variable.