2003 Fall Queuing Theory Midterm Exam(Time limit： 2 hours)

1. (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%)

Fig.1

1 2 3 4 5

1

1

3

1

3

1

3

1

1

2

1

2

1

2. ( 6%) Identify the following systems in figure 2 and make complete notations. (eg: X / X / X / X / X)

Fig.2

(a) (3%)

(b) (3%)

100

G

G

G

M

2

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

1

4

33

1

3

2

P

1 P-I z

nP

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%)

1 , )(

)(

1 , )()(

)(

000

11

ktPtd

tPd

ktPtPtd

tPdkkkk

k

dtetPsP stkk

)()(0

*

1)(0 tP 0t

)( * sPk

)(tPk ),2,1,0( ii

k

ii

k

ii

k

ssP

0

1

0*

) (

)(

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%)

e e1

/k k

0

0p kp

k 0p

0 0

0 kp

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. (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%)

(5%)

(5%)

Fig.3

)(lim 1 mqm

p/)p1(

)(mqi

.

1

1

1

when ,

nonnullrecurrent,

nullrecurrent,

transient

0 1 2 3

p1

p

p1

p

p1

p

p1

p

p