1 (5 points). Right after the end of her class at 11:20 am, a student, Lucy, started to walk to a bus stop to catch bus #5 for home which runs not on a schedule but continuously at a rate of 10 buses per hour, tting to an exponential distribution nicely. Lucy arrived at her bus stop at 11:30. (a) Find the probability that the next bus #5 would come within 5 minutes. (b) She had waited 10 minutes and a friend arrived at the bus stop who was also to catch bus #5. Find the probability that they had to wait at least for another 10 minute for bus #5. (c) Bus #8 can also take Lucy home. It runs like an exponential distribution as well but with a mean of 15 minutes between arrivals. Find the probability that Lucy would catch a bus ride home between 11:30 and 11:35. (d) Find the probability that Lucy did not miss any of her ride while she was walking to the bus top. (e) Find the probability that Lucy missed at least one ride while she was walking to the bus top. 2 (5 points). A gas station has 4 gasoline pumps. Cars stop for gasoline according to a Poisson process at a mean rate of 20 per hour. However, if all pumps are being used, the next customer will simply drive to a gas station across the street. The time required to service a car has an exponential distribution with a mean of 5 minutes. (a) What is the state space of this queueing system? (b) Construct the rate diagram for this queueing system. (c) Develop the balance equations. ) (d Solve these equations to nd the steady-state probability distribution of the number of cars at the station. (e) What is the waiting time in the queue, W