Problem 1
Arrivals at a Jiffy Lube follow a Poisson process with rate 10/hour. Service time is exponentially distributed with mean 15 minutes per customer. There are 2 servers available. Since there is a competitor across the street, only 80% of arriving customers stay for service if they find one car WAITING in queue for service; only 60% stay if they find two cars waiting in queue for service; only 40% stay if they find three cars waiting in queue for service; none stay for service if they find four cars waiting in queue for service. Let X(t) represent the number of cars in the system at time. Convert all rates to hourly rates.
(a) What is the probability that an arriving customer will be served immediately?
(b) What is the long run expected number of cars in the system?
(c) What is the average amount of time a customer spends at the system?
(d) What is the arrival rate of customers INTO Jiffy Lube?
Problem 2
Cars pass a point on the highway according to a Poisson process at a rate of one car every two minutes. 20% of the cars are Toyotas and 10% are Hondas.
(a) What is the probability that at least one Toyota passes in an hour?
(b) Given that 10 Hondas have passed in 2 hours, what is the expected number of Toyotas to have passed in that time?
(c) Given that 60 cars pass in an hour, what is the probability that exactly 40 of them were neither Toyotas nor Hondas.
Problem 3
Players and spectators enter a ballpark according to independent Poisson processes having respective rates 5 and 20 per hour. Starting at an arbitrary time, compute the probability that at least 3 players arrive before 4 spectators.