1. 2003 Fall Queuing Theory Midterm Exam(Time limit：2 hours)
(10%) For the Markov Chain in figure 1, which states are: (a) recurrent ? (2%)
(b) transient ? (2%)
(c) aperiodic ? (2%)
(d) periodic ? (2%)
(e) Is this chain reducible ? Why or why not? (2%) 1 2 3 4 5
Fig.1

2. 2. ( 6%) Identify the following systems in figure 2 and make complete notations. (eg: X / X / X / X / X)
(a) (3%) G G
100 G
(b) (3%) M
Fig.2

3. (15%) Consider that the discrete-state,discrete-time Markov chain transition probability matrix is given by .
(a) Find the stationary state probability vector . (5%)
(b) Find (5%)
(c) Find the general form for (5%)

4. (14%) Given the differential-difference equations:
Define the Laplace transform
For the initial condition we assume for Transform the differential-difference equations to obtain a set of linear difference equations in
(a) Show that the solution to the set of equations is: (10%)
(b) From (a), find for the case (4%)

5. (15%) Consider an M/M/1 system with parameters , in which customers are impatient. Specifically, upon arrival, customers estimate their queuing time and then join the queue with probability or leave with probability The estimate is when the new arrival finds in the system. Assume
(a) In terms of , find the equilibrium probabilities of finding in the system. Give an expression for in terms of the system parameters. (5%)
(b) For , under what conditions will the equilibrium solution hold? (5%)
(c) For ,find explicitly and find the average number in the system. (5%)

6. (15%) Consider an M/M/1 queuing system, the arrival rate is and the service rate is :
(a) What three properties would make a Markov chain ergodic? (6%)
(b) Prove that the limiting distribution exists only when (4%)
(c) When , argue that the M/M/1 queuing system is ergodic. (5%)

7. (a) Derive using notations provided above. (10%)
(25%) Consider a discrete-time birth-death chain as shown in figure 3. The death rate is p and the birth rate is (1-p). The ratio between birth rate and death rate is
(a) Derive using notations provided above. (10%)
(Hint: is the probability that the chain starts at state i and visits state 0 before it visits state m.)
(b) Show with derivation that this system is:
(5%) 1 2 3
Fig.3