1. Long-term averages
A local hospital needs to determine the size and staffing levels for its maternity ward. From historical records, the birth rate for the local community is 5 births per day on average. Women stay in the maternity ward for 2 days on average before going home with their newborn. Every woman will occupy one bed during her stay in the maternity ward. In addition, this maternity ward wishes to maintain an average occupancy no more than 75%.
Q1.1 - Using the above information, provide an estimate of the minimal number of beds needed in this maternity ward?
Please explain your calculation.
Q 1.2 - Suppose that due to space constraints, the ward was able to accommodate 10 maternity beds and, in reality, the average length of stay in the maternity ward was 2.5 days. What is the maximum number of births per day on average to guarantee an average occupancy of no more than 75%?
Please explain your calculation
2. Capacity and Queues
Webflux is an Internet-based DVD rental business specializing in hard-to-find, obscure films. Its operating model is as follows. When a customer finds a film on the Webflux Web site and decides to watch it, she puts it in the virtual shopping cart. If a DVD is available, it is shipped immediately (assume it can be shipped during weekends and holidays, too). If not available, the film remains in the customer's shopping cart until a rented DVD is returned to Webflux, at which point it is shipped to the customer if she is next in line to receive it. Webflux maintains an internal queue for each film and a returned DVD is shipped to the first customer in the queue (first-in, first-out).
Webflux has one copy of the 1990 film Sundown, the vampire in Retreat, starring David Carradine and Bruce Campbell. The average time between requests for the DVD is 8 days, with a coefficient of variation of 1.1. On average, a customer keeps the DVD for 5 days before returning it. It also takes 1 day to ship the DVD to the customer and 1 day to ship it from the customer back to Webflux. The standard deviation of the time between shipping the DVD out from Webflux and receiving it back is 7 days (i.e., it takes on average 7 days to (a) ship it, (b) have it with the customer, and (c) ship it back); hence, the coefficient of variation of this time is 1.
Q2.1 - On average, how many customers are in Webflux's internal queue for Sundown, the vampire in Retreat? Assume customers do not cancel their items in their shopping carts.
Q2.2 - What is the average time that a customer has to wait to receive the Sundown DVD after the request? Recall it takes 1 day for a shipped DVD to arrive at a customer address (i.e., in your answer, you have to include the 1-day shipping time).
Q 2.3 - Webflux is considering acquiring a second copy of the Sundown DVD. If Webflux owns a total of two copies of Sundown DVDs (whether in Webflux's internal stock, in customer's possession or in transit), what will be the new average time that a customer has to wait to receive the Sundown DVD after the request?