Promo is a wedding organizer that is evaluating the installation of portable toilets at an outdoor venue where a wedding is going to take place. Location of a group of toilets is titles Area X. There will be one single line/queue to access all the portable toilets in this area and only one person can use a toilet at a time. Promo estimates that there will be, on average, one arrival to the toilets every 15 seconds with a standard deviation of the time between arrivals of 15 seconds. The average time a person spends using the toilet is 3.12 minutes with standard deviation of 2 minutes.
(a) What is the minimum number of portable toilets that Promo must install to ensure that the queue length will not continue to grow without limit (i.e., so that the line/queue will be stable)? Suppose Promo installs 13 portable toilets. On average, what fraction of the toilets will be occupied during the wedding?
(b) Given 13 toilets, what is the average wait time to use a toilet?
(c) There is another toilet area located at a little distance: Area Y. All the data for Area Y is exactly the same as Area X. If Area Y toilets are moved to be adjacent to Area X toilets and if a common line/queue is maintained for all the toilets, what will be the average wait time to use a toilet?