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Question 2 (10 points) A retail store manages the inventory of washing machines as follows. When the number of washing machines in stock decreases to a fixed number k = 3, an order is placed with the manufacturer for m = 2 new washing machines. It takes a random amount of time for the order to be delivered to the retailer. If the inventory is at most k = 3 when an order is delivered (including the newly delivered order), another order for m = 2 items is placed immediately. Suppose the delivery times are iid Exp(6) and that the demand for the washing machines occurs according to a PP(5). Let X(t) be the number of machines in stock at time t (measured in weeks unit). Thus {X(t),t > 0} is a Continues Time Markov Chain (CTMC). Note that the maximum number of washing machines in stock is K = k+m = 3+2 = 5, which happens if the order is delivered before the next demand occurs. The state space is thus S = {0, 1, 2, 3, 4, 5}. The rate matrix is == R= 0 0 6 0 0 0 5 0 0 6 0 0 0 5 0 0 6 0 0 0 5 0 0 6 0 0 0 5 0 0 0 0 0 0 5 0 1. Compute the transition probability matrix P(t) at t = 0.2 (~ 1.4 days). Hint: Use R or other software to find the transition matrix. See, also, Example 7 in slides weeks9-12. 2. If the number of machines in stock at time t = 0 was 3, what is the probability that the retail store will have 2 machines in stock at t = 0.2. 3. If the number of machines in stock at time t = 0 was 3, what is the expected number of machines in stock at t = 0.2. R-code for Step 3 in previous example: t 0} is a Continues Time Markov Chain (CTMC). Note that the maximum number of washing machines in stock is K = k+m = 3+2 = 5, which happens if the order is delivered before the next demand occurs. The state space is thus S = {0, 1, 2, 3, 4, 5}. The rate matrix is == R= 0 0 6 0 0 0 5 0 0 6 0 0 0 5 0 0 6 0 0 0 5 0 0 6 0 0 0 5 0 0 0 0 0 0 5 0 1. Compute the transition probability matrix P(t) at t = 0.2 (~ 1.4 days). Hint: Use R or other software to find the transition matrix. See, also, Example 7 in slides weeks9-12. 2. If the number of machines in stock at time t = 0 was 3, what is the probability that the retail store will have 2 machines in stock at t = 0.2. 3. If the number of machines in stock at time t = 0 was 3, what is the expected number of machines in stock at t = 0.2. R-code for Step 3 in previous example: t

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