Question 6 Queueing (20 marks) Clients are arriving according to a Poisson process with rate = 40 per hour and are being served by only one person (server) when there are 2 clients or less. Service time follows an exponential distribution with rate = 30 per hour. When there are more than 2 clients in the system, an assistant comes from the back store to help the first person. The service time is then reduced by half. We suppose that the system capacity is equal to 4 clients.

a) Draw the state diagram

b) Write the balance equations

c) Find the numerical value of P0, P1, P2, P3 and P4

d) What fraction of the time both servers are free?

e) What is the expected number of clients in the system?