In: Advanced Math
1. Suppose that New York fire department receives an average of 12 requests for fire engines each hour, and that these requests occur according to a Poisson process. Each request causes a fire engine to be unavailable for an average of 12 minutes. To have at least a 90% chance of being able to respond to a request, how many fire engines should the fire department have?
2. Two one-barber shop sit side by side in New York. Each shop can hold a maximum of 5 people, and any potential customer who finds a shop full will not wait for a haircut. Barber 1 charges $18 per haircut and takes an average of 20 minutes to complete a haircut. Barber 2 charges $13 per haircut and takes 15 minutes to complete a haircut. On average 12 potential customers arrive per hour at each barber shop. Of course, a potential customer become an actual customer only if he or she finds that the shop is not full. Assuming that inter-arrival times and haircut times are exponential, which barber will earn more money?