You work at Marriott and are tasked with figuring out how often a given hotel is vacant. Suppose that the number of guests checking into the hotel is Poisson distributed with mean 1. Also, suppose that the number of days a guest stays in the hotel is geometrically distributed with parameter 0.8. Thus, a guest who spent the previous night in the hotel will, independently of how long they have already spent in the hotel, check out the next day with probability 0.8.

You work at Marriott and are tasked with figuring out how often a given hotel is vacant. Suppose that the number of guests checking into the hotel is Poisson distributed with mean 1. Also, suppose that the number of days a guest stays in the hotel is geometrically distributed with parameter 0.8. Thus, a guest who spent the previous night in the hotel will, independently of how long they have already spent in the hotel, check out the next day with probability 0.8. Part A (15 POINTS) Suppose that guests are independent of each other. It is also given that the maximum capacity of the hotel is 4. 1. Model the number of guests checked into the hotel as a Markov Chain. (3 points) 2. What is the probability that the hotel is empty in 5 days time, regardless of the number of occu- pied rooms today? The initial probability distribution is assumed to be (0.2, 0.3, 0.2, 0.2,0.1) for being in states {0,1,2,3,4} respectively. (3 points) 3. The hotel shuts down if it remains vacant for 5 consecutive days. Compute the probability that hotel shuts down in the next four days given the hotel is empty today but had 1 guest yesterday. (4 points) 4. The hotel shuts down if the hotel remains vacant for 5 consecutive days. Compute the probability that hotel shuts down over the next 12 days given the hotel has 2 guests today. (5 points) Solution Part B (10 POINTS) Now assume that the maximum capacity of the hotel is x. Given the hotel is at maximum capacity, derive an expression for the expected number of consecutive days that the hotel will remain at its maximum capacity, as a function of x (maximum capacity). Plot the computed expression as a function of x. Solution: You work at Marriott and are tasked with figuring out how often a given hotel is vacant. Suppose that the number of guests checking into the hotel is Poisson distributed with mean 1. Also, suppose that the number of days a guest stays in the hotel is geometrically distributed with parameter 0.8. Thus, a guest who spent the previous night in the hotel will, independently of how long they have already spent in the hotel, check out the next day with probability 0.8. Part A (15 POINTS) Suppose that guests are independent of each other. It is also given that the maximum capacity of the hotel is 4. 1. Model the number of guests checked into the hotel as a Markov Chain. (3 points) 2. What is the probability that the hotel is empty in 5 days time, regardless of the number of occu- pied rooms today? The initial probability distribution is assumed to be (0.2, 0.3, 0.2, 0.2,0.1) for being in states {0,1,2,3,4} respectively. (3 points) 3. The hotel shuts down if it remains vacant for 5 consecutive days. Compute the probability that hotel shuts down in the next four days given the hotel is empty today but had 1 guest yesterday. (4 points) 4. The hotel shuts down if the hotel remains vacant for 5 consecutive days. Compute the probability that hotel shuts down over the next 12 days given the hotel has 2 guests today. (5 points) Solution Part B (10 POINTS) Now assume that the maximum capacity of the hotel is x. Given the hotel is at maximum capacity, derive an expression for the expected number of consecutive days that the hotel will remain at its maximum capacity, as a function of x (maximum capacity). Plot the computed expression as a function of x. Solution