Download MODEL-PQT SET 2

LOYOLA – ICAM
COLLEGE OF ENGINEERING AND TECHNOLOGY (LICET)
Loyola Campus, Nungambakkam , Chennai – 34
Branch: CSE & IT
Semester: IV
Date: 10-04-2015
Time: 10.00 – 01.00 pm (3 Hrs)
MODEL EXAM
PROBABILITY AND QUEUING THEORY (MA2262)
1.
2.
3.
4.
Part-A (Answer all the questions)
(10x2=20)
A continuous random variable X that can assume any value between 𝑥 = 2 and 𝑥 = 5 has a density
function given by 𝑓(𝑥) = 𝑘(1 + 𝑥). Find 𝑃(𝑋 < 4).
If the probability that a target is destroyed on any one shot is 0.5, find the probability that it would be
destroyed on 6th attempt.
The regression equations of X on Y and Y on X are respectively
5𝑥 − 𝑦 = 22 & 64𝑥 − 45𝑦 = 24. Find the means of X and Y.
A small college has 90 male and 30 female professors. An ad-hoc committee of 5 is selected at random
to unite the vision and mission of the college. If X and Y are the number of men and women in the
committee, respectively, what is the joint probability mass function of X and Y?
5
1
5. If the initial state probability distribution of a markov chain is 𝑝(0) = (6 6) and the transition
0 1
probability matrix of the chain is ( 1 1 ), find the probability distribution of the chain after 2 steps.
2
2
6. Define continuous time random process and discrete state random process.
7. Define M/ M/ 2 Queueing model. Discuss why the notation M is used?
8. Define steady state and transient state in Queuing Theory
9. What do you mean by bottleneck of a network?
10. Define series queues.
PART – B (16 X 3 = 48 )
11a. i ) If the random variable X takes the values 1, 2, 3 & 4 such that
2𝑃(𝑋 = 1) = 3𝑃(𝑋 = 2) = 𝑃(𝑋 = 3) = 5𝑃(𝑋 = 4), then find the probability distribution and
cumulative distribution function of X.
ii ) A manufacturer of pins knows that 2% of his products are defective. If he sells pins in boxes of 100
and guarantees that not more than 4 pins will be defective, what is the probability that a box fail to
meet the guaranteed quality?
(Or)
𝑥, 𝑓𝑜𝑟 0 < 𝑥 < 1
b. i) Find the MGF of the random variable X having the pdf 𝑓(𝑥) = {2 − 𝑥, 𝑓𝑜𝑟 1 < 𝑥 < 2}
0,
𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒.
ii) Given that X is distributed Normally, if 𝑃(𝑋 < 45) = 0.31 & 𝑃(𝑋 > 64) = 0.08, find the mean and
standard deviation of the distribution
12a. i) Two dimensional random variables (X, Y) have the joint probability density function
8 xy, 0  x  y  1
1
1
f ( x, y )  
Find 𝑃 (𝑋 < 2 ∩ 𝑌 < 4).
 0, elsewhere
Find the marginal and conditional distributions. Are X and Y independent?
ii) Obtain the equations of the lines of regression from the following data:
X
1
2
3
4
5
6
7
Y
9
8
10 12 11 13 14
(Or)
b. i) If X and Y are independent random variables having density function
2e2 x , x  0
3e3 y , y  0
f ( x)  
& f ( y)  
 0, x  0
 0, y  0 find the density functions of 𝑧 = 𝑋 − 𝑌.
ii) The joint probability mass function of (X, Y) is given by 𝑃 = 𝐾(2𝑥 + 3𝑦), 𝑥 = 0;
𝑦 = 1, 2, 3. Find all the marginal and conditional probability distributions.
13a. i) Show that the random process 𝑋(𝑡) = 𝐴 cos(𝜔0 𝑡 + 𝜃) is wide sense stationary, if A and 𝜔0 are
constants and 𝜃 is uniformly distributed RV in (0, 2𝜋).
ii) Find the limiting state probabilities associated with the following transition probability matrix
0.4 0.5 0.1
[0.3 0.3 0.4].
0.3 0.2 0.5
(Or)
b. i) Suppose that children are born at a Poisson rate of five per day in a certain hospital. What is the
probability that (i) atleast two babies are born during the next six hours. (ii) no babies are born
during the next two days?
ii) Classify the states for the Markov chain with transition probability matrix
𝑃=
0.4 0.5 0.1
[0.3 0.3 0.4].
0.3 0.2 0.5
14a. If people arrive to purchase cinema tickets at the average rate of 6 per minute, it takes an average of 7.5
seconds to purchase a ticket. If a person arrives 2 min before the picture starts and it takes exactly 1.5
min to reach the correct seat after purchasing the ticket,
 Can he expect to be seated for the start of the picture?
 What is the probability that he will be seated for the start of the picture?
 How early must he arrive in order to be 99% sure of being seated for the start of the
picture?
(Or)
b. A super market has two girls running up sales at the counters. If the service time for each customer is
exponential with mean 4 minutes and if people arrive in Poisson fashion at the rate of 10 per hour, find
the following
(i) What is the probability that a customer has to wait for service?
(ii) What is the expected percentage of idle time for each girl?
(iii) What is the expected length of customer’s waiting time?
15a. i) Derive Pollaczek – Khintchine formula of M / G / 1 queue.
0 0.6 0.3
ii) Given (𝑟1 , 𝑟2 , 𝑟3 ) = (1, 4, 3) and 𝑃𝑖𝑗 = [0.1 0 0.3] determine the average arrival rate 𝜆𝑗
0.4 0.4 0
to the node j for j = 1, 2, 3.
(Or)
b. i) Write short notes on the following:
i)
Queue networks
ii)
Series queues
iii)
Open networks
ii) Automatic car wash facility operates with only one bay. Cars arrive according to a Poisson
distribution with a mean 4 cars per hour and may wait in the facility’s parking lot if the bay is busy.
Find the average number of customers in the system if the service time is (i) constant and is equal to 10
minutes (ii) uniformly distributed between 8 and 12 minutes.