# MA2262- PROBABILITY AND QUEUEING THEORY ANNA UNIVERSITY PREVIOUS YEAR QUESTION PAPER DOWNLOAD

Fourth Semester

Computer Science and Engineering

MA2262- PROBABILITY AND QUEUEING THEORY

(Regulation 2008)

(Common to Information Technology)

PART A- (10 X 2= 20 marks)

1. Obtain the mean for a Geometric random variable.

2. What is meant by memoryless property? Which continuous distribution follows this property?

3. Give a real life example each for positive correlation and negative correlation.

4. State central limit theorem for independent and identically distributed (iid) random variables.

5. Is a Poisson process a continuous time Markov chain? Justify your answer.

6. Consider the Markov chain consisting of the three states 0, 1, 2 and transition probability matrix P= |1/2 1/2 0 |

|1/2 1/4 1/4|

|0 1/3 2/3| it irreducible? Justify.

7. Suppose that customers arrive at a Poisson rate of one per every 12 minutes and that the service time is exponential at a rate of one service per 8 minutes. What is the average number of customers in the system?

8. Define M/M/2 queuing model. Why the notation M is used?

9. Distinguish between open and closed networks.

10. M/G/1 queuing system is markovian. Comment on this statement.

11. (a) (i) By calculating the moment generating function of Poisson distribution with parameter ¥, prove that the mean and variance of the Poisson distribution are equal.

(ii) If the density function of X equals f(x) = {Ce-2x , 0 < x < ¥

0 , x<0 , find c. What is P[X>2] ?

(Or)

(b) (i) Describe the situations in which geometric distributions could be used. Obtain its moment

generating function.

(ii) A coin having probability p of coming up heads is successively flipped until the rth head appears. Argue that X, the number of flips required will be n, n>= r with probability

P[X = n] = (n-1)

(r-1)pr qn-r n>=r

12. (a) (i) Suppose that X and Y are independent non negative continuous random variables having

densities fx(x) and fy(y) respectively. Compute P[X < Y].

(ii) The joint density of X and Y is given by f(x, y) = {1/2ye-xy , 0< x < ¥, 0< y <2.

0 , otherwise

Calculate the conditional density of X given Y = 1.

(Or)

(b) (i) If the correlation coefficient is 0, then can we conclude that they are independent? Justify your answer, through an example. What about the converse?

(ii) Let X and Y be independent random variables both uniformly distributed on (0, 1). Calculate the probability density of X + Y.

13. (a) (i) Let the Markov Chain consisting of the states 0, 1, 2, 3 have the transition probability matrix.

|0 0 1/2 1/2|

|1 0 0 0 |

|0 1 0 0 |

|0 1 0 0 |

Determine which states are transient and which are recurrent by defining transient and recurrent states.

(ii) Suppose that whether or not it rains today depends on previous weather condition through the last two days. Show how this system may be analyzed by using a Markov chain. How many states are needed?

(Or)

(b) (i) Derive Chapman - Kolmogorov equations.

(ii) Three out of every four trucks on the road are followed by a car, while only one out of every five cars is followed by a truck. What fraction of vehicles on the road are trucks?

14. (a) Define birth and death process. Obtain its steady state probabilities. How it could be used to find the steady state solution for the M/M/1 model? Why is it called geometric?

(Or)

(b) Calculate any four measures of effectiveness of M/M/1 queueing model.

15. (a) Derive Pollaczek- Khintchine formula.

(Or)

(b) Explain how queuing theory could be used to study computer networks.

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