Share & Care is a nonprofit car share company that rents cars. When a customer makes a reservation, they specify their pick up time and the number of time slots they will hold the vehicle, where each time slot equals 15 minutes. For example, if the pickup time is 1pm, then possible drop off times are 1:15 (1 slot), 1:30 (2 slots), etc. Share&Care charges $5.00 for each time slot in the reservation. To discourage customers from returning the rented cars beyond their drop off time, they charge $20 per time slot used beyond the drop off time. For example, if a customer’s drop off time is 2:30 and he returns the vehicle at 2:47, then he is charged $40 for the 2 time slots he used beyond his reservation (and of course, $5.00 per slot that he reserved).
Larry runs a small business that makes deliveries on Fridays. The number of time slots he needs is well modeled by a Poisson distributed with mean 3. He books his car 2 days in advance, before he knows his needs exactly. (He does this to ensure availability of a car.) Assume on the day of the reservation Larry learns his needs and has little control over the number of slots he needs. For example, if he needs 5 slots but booked 4 slots, then he uses the car for 5 slots for a total charge of 4 x $5.00 + $20 = $40. If he ends up booking the car for more time than he needs, the extra time on the car has no value to him.
a. What is the probability that he needs exactly 2 time slots?
b. Suppose he books the car for 2 time slots. How likely is he to pay $40 or more in late fees?
c. To minimize his rental costs, how many time slots should Larry reserve the car for?
d. Larry has a new plan. Whenever his need for the car exceeds the number of slots he has rented the car for, he returns the car on time to avoid the fine, and uses his bicycle for the remaining deliveries. Unfortunately, bike deliveries take 3 times longer than car deliveries. Suppose he rents the car for 5 slots (75 minutes). If he needs 7 slots, then he uses the car for 5 slots, and the remaining 2 slots are done by bicycle, which take 6 slots (1.5 hours). How much time does he expect to bicycle on average (in terms of slots) given that Larry follows your suggestion in part c?