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 = (X1, X2, X3). X1 represents the number of customers in P1, X2

represents the number of customers in P2, and X3 represents the number of clients in the queue. Hence, Xi is either 0 or 1, for all i = 1, 2, 3.

a) Determine all the possible states of the systemb) Determine the infinitesimal generator matrix ~Qc) Is the chain irreducible? Justify your answerd) Determine the equations that characterize the stationary probabilitye) Determine the percentage of customers that cannot enter the gas stationf) Determine the percentage of customers that cannot enter the gas station due to the current system design

(i.e. customers cannot be served when P2 is busy even though P1 is free)

Queue P2 P1

Exercise 3:Consider the discrete-time Markov chain whose state transition diagram is given in the figure bellow.

a) Find the probability transition matrix ~P .b) Find ~P5.c) Find the equilibrium state probability vector ~π .

Easy.

Exercise 4:Consider a pure-birth process where λk=kλ for ¿0 ,1 ,2 ,…. . Assuming that at time 0, the process starts in state 1. Find the time-dependent expression for this process (that means the probability of being in a state i at time t).

See solution IN TEXTBOOK PAGE 99

Exercise 5:Consider the following homogeneous Markov chain with 2 states A and B. The transition matrix is given by:

P=(1/2 1 /21/2 1 /2)

We are interested in the first time the sequence ABA occurs in the chain. Thus, we formed the process Y n= (Xn , Xn+1 , Xn+2 ).

a) Prove that Y is a homogeneous Markov chain, and give the form of the transition matrix. b) Is Y irreducible? Aperiodic? Justify your answers.c) Calculate the stationary probability of the chain Y.

Hint: you can assign numbers for the states as in the following sequence: AAA: 1, AAB: 2, ….d) Calculate the average elapsed time (at steady state) between 2 occurrences for the sequence ABA.e) Calculate the average elapsed time (at steady state) spent in state ABA.

Solution: