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Anna University Coimbatore 2011 B.E Computer Science and Engineering Probability and Queuing theory - exam paper

Wednesday, 16 January 2013 05:00Web

MODEL EXAMINATION

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)

Obtain the mean for Geometric random variable.
If the range of X is the set and for all x, determine the mean and variance of the random variable.
A continuous random variable X has the pdf . Find .
Every week the average number of wrong number phone calls received by a certain mail order house is seven. What is the probability that they will receive two wrong calls tomorrow?
Give a real life example for each positive and negative correlation.
State Central limit theorem.
Examine whether the Poisson process is stationary or not?
The regression equation of X on Y and Y on X are . Find the means of X and Y.
Consider the Markov chain consisting of three states 0,1,2 and tpm:is it irreducible? Justify.
Define Markov process.
What is Stationary process.
State the four types of stochastic processes.
Define M/M/2 queuing model. Why the notation M is used?
In the usual notation of M/M/1 queueing system if , find the average number of customers in the system.
What are the basic characteristics of Queueing process.
For , write down Littles formula.
Distinguish between open and closed Jackson network.
Write Pollaczek- Khintchine formula.
Consider a service facility with two sequential stations with respective service rates of 3/min and 4/min. The arrival rate is 2/min. What is the average service time of the system, if the system could be approximated by a two stage tandem queue.
What is the total queue length in D/D/1?

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

a) By calculating the moment generating function of Poisson distribution with parameter ,

prove that the mean and variance of the Poisson process are equal. (8 Marks)

b) If the density function of X equals , find C. What is ?

( 4 Marks)

i)and ii)(6 Marks)

b) The two lines of regression are . The variance of X is 9. Find i) the mean

values of X and Y. ii) Correlation coefficient between X and Y. (6 Marks)

. Find Var(X), Var(Y) and Co-variance between X and Y. (6 Marks)

b) If X and Y are independent Random variables uniformly distributed . Obtain the

distribution of XY. (6 Marks)

b) A petrol station has 2 pumps. The service times follows the exponential with mean of 4 minutes

and cars arrive for service is a Poisson process at the rate of 10 cars per hour. Find the probability

that a customer has to wait for service. What is the probability that the pumps remain idle?(4 marks)

A TVS company in Madurai containing a repair section shared by a large number of machines has 2 sequential stations with respective service rates of 3 per hour and 4 per hour. The cumulative failure rate of all machines is 1 per hour. Assuming that the system behavior can be approximated by a 2 stage tandem queue, find

i) The probability that both the service stations are idle.

ii) The average repair time including waiting time and

iii) The bottle neck of the repair facility.

a) Compute the coefficient of correlation between X and Y using the following data:

X	1	3	5	7	8	10
Y	8	12	15	17	18	20

(6 Marks)

b) The transition probability matrix of a Markov chain having 3 states

1,2&3 is .And the initial distribution is Find

(6 Marks)

x:	-2	-1	0	1	2	3
P(x):	0.1	k	0.2	2k	0.3	3k

Find i) the value of k, ii) evaluate ,

iii) find the cumulative distribution of X. and iv) evaluate the mean of X.

Customers arrive at a one man barber shop according to a Poisson process with a mean inter arrival time of 12 min. The customers spend an average of 10 minutes in the chair. i) what is the expected number of customers in the shop and in the queue? ii) How much time can a customer expect to spend in the shop ? iii) Find the average time spent in the queue.

( 0 Votes )

Earning: Approval pending.