Summary: Exam Queueing Theory -- May 21, 2007
This exam consists of four problems and one sheet of formulas
Please put your name, university and student number on each sheet of paper
Motivate your answers
1. Cars arrive at a small gas station to refuel according to a Poisson process
with rate 30 per hour, and have an exponential service time distribution with
mean 4 minutes. Since there are four gas pumps available, four cars can refuel
simultaneously, but unfortunately there is no room for cars to wait. Hence, if
a car arrives when all pumps are busy, the driver leaves immediately. For each
customer that is served, an average profit is made of 7 euros.
(a) Determine the probability that an arriving car is not refueled.
(b) What is the long-run expected profit per day (consisting of eight hours)?
Some day, the manager has the opportunity to buy an adjacent parking lot,
so there is room to wait for all cars that arrive when the four pumps are busy.
Suppose all drivers decide to wait instead of leaving when this happens.
(c) Determine the probability that a car has to wait.
(d) What is the long-run expected profit per day in this case?
(e) Indicate shortly if the answers for (a), (b), (c) and (d) increase, decrease,
or remain the same when the refueling times have a different distribution
with equal mean 4 minutes, but smaller variance.