Anna University Coimbatore 2011 B.E Computer Science and Engineering Probability and Queuing theory - Question Paper

MODEL EXAMINATION

PROBABILITY & QUEUEING THEORY

Year/Semester & Branch: II / IV / CSE & IT

Max. Marks: 100 Time: 180 min

PART-A Answer ALL Questions (20 X 2 = 40)

1. A continuous random variable X has probability density function given by . Find K such that .
2. The number of hardware failures of a computer system in a week of operation has the following probability mass function :

No. of failures 0 1 2 3 4 5 6

Probability 0.18 0.28 0.25 0.18 0.06 0.04 0.01

3. The moment generating function of X is given by. Find.
4. Derive mean and variance of Binomial distribution using moment generating function.
5. The joint probability density function of a bivariate random variable (X, Y) is given by. Find.

6.      Find the marginal density function of X and Y if

7. Define Covariance.
8. The two equations of the variables X and Y are and. Find the correlation coefficient between X and Y.
9. Define discrete random process with an example.
10. If the transition probability matrix is, find the limiting distribution of the chain.
11. What will be the superposition of independent Poisson processes with respective average rates?
12. Give an example of Markov process.
13. Define Kendals notation.
14. What is the probability that a customer has to wait more than 15 minutes to get his service completed inqueue system if per hour and per hour ?
15. What is the effective arrival rate for queueing model when and.
16. Draw the state transition diagram for queueing model.
17. Define flow balance equation of open Jackson network.
18. What are the characteristics of Queueing network?
19. Write a note on Closed Jacksons network.
20. Define series queues with blocking.

PART-B Answer ANY FIVE Questions (5 X 12 = 60)

21. A random variable X has the following probability distribution

. Find (a) the value of k,

(b) Evaluate (c) If find the minimum value of C.

(d) Evaluate (e) Find

 

22. a) Suppose the joint probability density function is given by . Obtain the marginal probability density of X and Y. Hence find. (6 Marks)

b) Given the joint probability density function of (X,Y) as .

Find the marginal and conditional PDF of X and Y. Are X and Y independent? (6 Marks)

23. Given a random variable Y with characteristic function and a random process defined by , show that is stationary in the wide sense of .
24. a) The process whose probability distribution under certain conditions is given by . Show that it is evolutionary. (7 marks)

b) Prove that the sum of the two independent Poisson process is also a Poisson process. (5 marks)

25. a) Derive birth and death process (6 Marks)

b) Trains arrive at the yard every 15 minutes and the service time is 33 minutes. If the line capacity at the yard is limited to 4 trains, find (i) the probability that the yard is empty.

(ii) Average number of trains in the system. (6 Marks)

26. In a bookshop there are 2 sections, one for Engineering books and the other section for Mathematics books. There is only one salesman in each section. Customers from outside arrive at the Engineering book section at a Poisson rate of 4 per hour and the Mathematics book section at a Poisson rate of 3 per hour. The service rates of the Engineering book section and Mathematics book section are 8 and 10 per hour respectively. A customer after service at Engineering book section is equally likely to go to the Mathematics book sections or to leave the book shop. However a customer after completion of service at Mathematics book section will go to the Engineering book section with probability 1/3 and will leave the bookshop otherwise. Find the (i) Joint steady state probability that there are 3 customers in the engineering book section and 2 in the Mathematics book section (ii) Average number of customers in the bookshop. (iii) Average waiting time of a customer in the bookshop.
27. A car wash facility operates with only one bay. Cars arrive according to a Poisson distribution with a mean of 4 cars per hour and may wait in the facilities parking lot if the bay is busy. The parking lot is large enough to accommodate any number of cars. Find the average number of cars waiting in the parking lot if the time for washing and cleaning a car follows (i)Uniform distribution between 8 and 12 minutes .(ii) a normal distribution with mean 12 minutes and standard deviation 3 minutes. (iii) a discrete distribution with values equal to 4,8 and 15 minutes and corresponding probabilities0.2,0.6 and 0.2.
28. Two random variables X and Y have the following joint probability density function .Find and and also find covariance and correlation between X and Y.