PROBLEM 3 (25 POINTS): MODELING COVID-19 VACCINE INVENTORY The first COVID-19 vaccines are now available, heralding the 'beginning of the end' of the pandemic. However, these vaccines are expensive and they have a short shelf-life. As a result Purdue University Student Health (PUSH) can only keep a small inventory of at most 10 doses available. On the other hand, PUSH also intends to vaccinate any individual who wishes to be inoculated. Given these condi- tions, PUSH intends to use an inventory restocking policy, wherein new vaccines are ordered once the inventory falls to a fixed level; let's call this s = 2. Assume that the policy is to "order up" such that the refreshed inventory level is $ = 10. Suppose that the demand for vaccines is geometrically distributed with mean A = 2 per day. Assume that it takes one day to fulfill the re-order request. If demand ex- ceeds the available doses, then that demand is 'lost." Assume that a patient only needs one-dose of the vaccine. 1. Construct a Markov chain model of the inventory. 2. If the inventory starts at 8, what is the probability that the vaccine doses will run out in 4 days? 3. What is the long-run probability that the inventory empties before the next order turns up? 4. What is the average time to return to full inventory? 5. What is the average time to run out of inventory for the first time, starting with a full inventory? 6. How many times is PUSH expected to order new inventory over the next 5 days? 7. What is value of s PUSH should use to ensure that the probability of running out of doses is at most 0.05? Solution