Exercise 1 : Phones in the Lebanese International University are connected to the university central switchboard, which has 120 external lines to the local telephone exchange. The voice traffic generated by its employees in a typical working day is shown as: Calculate the following: a) The total traffic offered to the PABX. b) The overall mean holding time of the incoming traffic. c) The overall mean holding time of the outgoing traffic. Easy Exercise 2 : Customers arrive at a gas station consisting of 2 pumps P1 and P2 with respect to a Poisson process of intensity λ. If the 2 pumps are free, the customer will go directly to P1. If P1 is busy, customer will go to P2. If both pumps are busy, customers will form a queue at the entrance. When the queue is full, any new customer will go his way and try later to enter the gas station. A customer in P2 has to wait P1 to become free in order to leave the station. In case the customer in P2 finishes before the customer in P1, both will leave the station as soon as P1 becomes free. In this case, if there is a customer in the queue, he/she can go directly to P1. In case P1 is free and P2 is busy, no customers can use P1. The service time is exponentially distributed with parameter µ. The following figure illustrates the gas station system. We will model this system as a Markov process X = (X 1 , X 2 , X 3 ). X 1 represents the number of customers in P1, X 2 represents the number of customers in P2, and X 3 represents the number of clients in the queue. Hence, X i is either 0 or 1, for all i = 1, 2, 3. a) Determine all the possible states of the system b) Determine the infinitesimal generator matrix ~ Q c) Is the chain irreducible? Justify your answer d) Determine the equations that characterize the stationary probability e) Determine the percentage of customers that cannot enter the gas station Queue P2 P1