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Posted By: ADITYA Member Level: Gold Posted Date: 24 May 2008
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2006 Anna University B.E Computer Science PROBABILITY AND QUEUEING THEORY Question paper
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B.E. DEGREE EXAMINATION.
Fourth Semester
Computer Science and Engineering
MA 040 — PROBABILITY AND QUEUEING THEORY
Time : Three hours Maximum : 100 marks
Answer ALL questions.
PART A — (10 ? 2 = 20 marks)
1. If A and B are independent events, prove that and B are also independent events.
2. Find the cumulative distribution function corresponding to the p.d.f. , for .
3. The joint p.d.f. of a bivariate R.V. is given by
Find .
4. State the central limit theorem for independent and identically distributed random variables.
5. Describe Bernoulli process.
6. Show that if a random process is wide sense stationary, then it must also be covariance stationary.
7. Consider a Markov chain with two states and transition probability matrix . Find the stationary probabilities of the chain.
8. Establish the relation between hazard function and reliability function.
9. Consider an M/M/1 queueing system. Find the probability of finding at least ‘n’ customers in the system.
10. Consider an M/M/C queueing system. Find the probability that an arriving customer is forced to join the queue.
PART B — (5 ? 16 = 80 marks)
11. (i) Derive the distribution of Poisson process and hence find its mean and variance.
(ii) Let be a renewal process with distribution . Prove that the probability is and the expected number of renewals by , where is the n–fold convolution of with itself. Find also and for .
12. (a) (i) A box contains 5 red and 4 white balls. Two balls are drawn
successively from the box without replacement and it is noted that
the second one is white. What is the probability that the first is also
white?
(ii) Let . Find the p.d.f. of Y if X is a uniform random variable over .
(iii) Let X be a uniform random variable over . Determine the moment Generating function of X and hence find variance of X.
Or
(b) (i) Determine the binomial distribution for which the mean is 4 and
the variance 3.
(ii) If , find , where X is a random variable.
(iii) Let . Find the p.d.f. of Y if X is a uniform random variable over .
13. (a) (i) If X and Y have the joint p.d.f.
find and .
(ii) Suppose the joint probability mass function of a bivariate random variable is given by
(1) Are X and Y independent?
(2) Are X and Y uncorrelated?
Or
(b) (i) Consider . Show that if X and Y are independent Poisson
Random variables with parameter and respectively, then z is
also a Poisson random variable with parameter .
(ii) Let X and Y be independent standard normal random variables. Find the p.d.f. of .
14. (a) (i) Consider a Markov chain with state space and transition
probability matrix .
(1) Draw a transition diagram
(2) Show that state 0 is recurrent
(3) Show that state 1 is transient
(4) Is the state 1 periodic? If so, what is the period?
(5) Is the chain irreducible?
(6) Is the chain ergodic? Explain.
(ii) Distinguish the terms reliability and availability.
(iii) Find the MTTF for constant hazard function.
Or
(b) (i) Describe the two–unit parallel system with repair.
(ii) Describe the preventive maintenance of the system and hence find MTTSF.
15. (a) (i) Customers arrive at a watch repair shop according to a Poisson
process at a rate of one per every 10 minutes, and the service time
is an exponential random variable with mean 8 minutes. Find the
average number of customer , the average waiting time a
customer spends in the shop and the average time a customer
spends in the waiting for service .
(ii) Determine the steady–state probabilities for M/M/C queueing system.
Or
(b) (i) Cars arrive at a derive–in restaurant with mean arrival rate of 24 cars per hour and the service rate of the cars is 20 cars per hour. The arrival rate and the service rate follow Poisson distribution. The number of parking space for cars is only 4. Find (1) the mean number of cars in the system (2) , the mean waiting time in the system.
(ii) Determine the steady–state probabilities of the system size for the M/M/ queueing system and hence find mean number of customers in the system and the mean waiting time in the system.
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