Question: On average, how many customers are in GTUX’s internal queue for a tuxedo? Assume customers do not cancel their items in their shopping carts.

Part 1 - Real Options (30 points)
Think of an example of a real option for managing information risk that you observed in practice or your own life. Remember that real options are designed to protect against the negative impacts of some information risk. Please answer the questions below. Note: please do not choose the option to return a product as your example - we discussed it exhaustively in class.

Please describe what the option is, what information risk it is mitigating, who can use the option, and who offers it. For example, Essmart customers have the option to return products they dislike. This option is offered by Essmart and mitigates the risk customers have regarding product quality and usefulness.
How does this option address the information risk you mentioned above? Is the risk shifted to someone else, or does it simply disappear? For example, when Essmart offers customers a return option, the quality and fit-risk are transferred from the customer to Essmart. Conversely, the option to put the Circored plant on "standby" eliminates the downside of price risk.
What are the operating expenses and assets associated with this option? For example, in Essmart's case, offering customers a return option has costs and assets associated with return logistics and refurbishing the returned product.
Part 2 - GTUX: Fancy Tuxedos (35 points)
You and your colleagues expect a return of fancy events at Georgia Tech this Spring. As a result, you launch GTUX, a small company specialized in renting tuxedos to college students.

GTUX’s operating model is as follows. When a customer finds a tuxedo on GTUX’s website and decides to rent it, they put it in a virtual shopping cart. If a tuxedo is available, it is shipped directly to the customer (assume it can be shipped during weekends and holidays too). If not available, the customer enters a queue and waits until a tuxedo is available. When a tuxedo is returned to GTUX, it is shipped to the customer next in line to receive it. GTUX manages this queue such that a returned tuxedo is shipped to the first customer in the queue (which is the customer that has been waiting for the longest).

Since you are currently experimenting with the business model, GTUX only owns one tuxedo. The average time between requests for the tuxedo is 10 days, with a coefficient of variation of 1.

On average, a customer keeps the tuxedo for 6 days before returning it. It also takes 1 day to ship the tuxedo to the customer and 1 day to ship it from the customer back to GTUX. Thus, it takes 8 days on average between GTUX shipping the tuxedo to a customer and receiving it back. The standard deviation of the time between shipping the tuxedo to a customer from GTUX and receiving it back in 8 days (thus the coefficient of variation is 1).

Question 1 (10 points)

What is the average time that a customer has to wait to receive the tuxedo after the request? Recall that it takes 1 day for a shipped tuxedo to arrive at a customer address.

Question 2 (10 points)

On average, how many customers are in GTUX’s internal queue for a tuxedo? Assume customers do not cancel their items in their shopping carts.

Question 3 (15 points)

Thanks to a very successful advertisement campaign featuring Buzz in a tuxedo, the demand for tuxedos has increased. Now the average interarrival time for tuxedo requests at GTUX is 3 days, with a coefficient of variation of 1. Other numbers remain unchanged. To satisfy the increased demand, GTUX is considering acquiring more tuxedos to rent.

How many tuxedos should GTUX own (whether in GTUX’s internal stock, in customer’s possession, or in transition) so that a customer waits, on average, less than 3 days to receive the tuxedo at home? Assume all tuxedos are identical. Recall that it takes 1 day for a shipped tuxedo to arrive at a customer's address.

Part 3 -Covid and Queues (35 points)
Question 1 (10 points)

In the Atlanta Metropolitan Area (which, for this question, we assume has 5 million people), 0.5% of the population are tested every day for Covid-19.

For 80% of patients, the test is negative (i.e., they do not have the virus).
For the remaining 20% of patients who do have the virus:

60% do not need to be hospitalized and recover from their symptoms at home.
40% of patients are admitted to the hospital.
Of the 40% of patients who are hospitalized:

90% do not need to be admitted to intensive care (ICU) and recover in a regular hospital bed.
10% of patients will require ICU admission after first spending 4 days, on average, in a regular hospital bed.
If a patient is admitted to the hospital but not to the ICU, they spend 14 days in the hospital on average.
If a patient is admitted to the ICU, they then spend 7 days there on average. 50% of patients recover in the ICU, while another 50% do not survive.
Those who survive then return to a hospital bed to recover, where they spend a further 5 days on average.
On average, how many Covid-19 patients will be occupying (i) hospital beds and (ii) ICU beds in Atlanta?

Question 2 (5 points)

Assume that those hospitalized above arrive randomly to one of 20 hospitals in the Atlanta Metropolitan Area (i.e., the allocation decision is not made based on how busy each hospital is, but instead it is the result of a “flip of the coin”).

The ICUs at each hospital are all equally sized and have 56 beds each. Assume that the ICUs are only treating Covid-19 patients. The coefficient of variation of arrival times and service times (i.e., CVa and CVs) are both equal to 1.

On average, how long will a patient have to wait before they are admitted to the ICU? Hint: What happens with a queue when the arrival rate is higher than the service rate?

Question 3 (20 points)

The government is very unhappy with the answer to Question 2. It takes two immediate measures. The first measure is expanding ICU capacity. The government works with hospitals to increase the number of ICU beds from 56 to 75 beds in each hospital.

The second measure is to ensure that patients should not have to wait more than 6 hours before being admitted to an ICU bed in a regular hospital if they need to be. To achieve this, the government has mandated a policy under which patients can be transferred between hospitals whenever the next available bed becomes available. This is a “first-in-first-out” policy: whichever patient has been waiting in the queue, the longer will be moved to the next available ICU bed, regardless of which hospital this is in.

To simplify this process and reduce the need to transfer patients across large distances over the city, the government is looking to separate the hospitals into groups (of 2, 3, 4, or 5 hospitals), so that patient transfers only occur within that group. For example, suppose there were only 4 hospitals, Hospitals A, B, C, and D, and these were formed into 2 groups of 2 such that Hospitals A and B are in one group, and Hospitals C and D in the other group. A patient requiring an ICU bed at Hospital A could be transferred to Hospital B, but not to Hospital C or D.

a) What is the smallest number of hospitals in each group that will ensure that, on average, patients should not have to wait more than 6 hours before being admitted to an ICU bed at a regular hospital if they need to? Hint: Interpret an ICU with s beds and patients that spend x days per bed as a multi-server queue with s servers each with a service rate of 1/x.

b) On average, how many patients will be either waiting for or occupying an ICU bed at each regular hospital under this system?