POKHARA UNIVERSITY
Level: Bachelor
Programme: BE
Semester – Fall
Course: Probability and Queuing Theory
Year
: 2010
Full Marks : 100
Pass Mark : 45
Time
: 3 hrs
Candidates are required to give their answers in their own words as far
as practicable.
The figures in the margin indicate full marks.
Attempt all the questions.
1. a.
b.
2. a.
Among the 24 invoices prepared by a billing department, 4 7
contain errors while the others do not. If we randomly check 2 of
these invoices, what is the probability that:
i. both will contain errors
ii. neither will contain error
Show that for 1,000,000 flips of a balanced coin the probability is 8
at least 0.99 that the proportion of heads will fall between 0.495
and 0.505.
Suppose that the duration of telephone calls follows a distribution 7
with probability density function given by
f ( x)  1 / 5e  x / 5 , x  0
b.
3.
a.
Find the probability that
i.
P(X>5)
ii.
P(3<X<6)
iii.
P(X<6|X>3)
Let X and Y have the joint probability mass function
(x + y)/21, x = 1, 2, 3 and y = 1, 2.
i.
Find marginal probability of X and Y.
ii.
Find conditional probability distribution of X given
Y= y.
iii.
Are X and Y independent?
The probability of a man hitting a target is 0.4. If he fires 5
times, what is the probability of his hitting the target at least
twice? How many times must he fire so that the probability of
his hitting the target at least once is greater than 2/3?
OR
1
8
7
b.
Light bulbs are installed successively into a socket. Assume that
each has a mean life of 2 months with a standard deviation of
1/4 month.
Find the probability that 40 bulbs last at least 7
years. Also find how many bulbs should be bought so that
8
one can 95% sure that supply will last 5 years.
Let X is a random variable with mean of 11 and variance of 9.
Using Chebyshev’s inequality, find the following:
i. A lower bound for P(6<X<16).
ii. The value of c such that P(|X-11|≥ c)≤0.09.
4.
a.
b.
5.
a.
b.
6. a.
Given the transition probability matrix of a Markov chain 7
below; obtain the matrix of n step transition probabilities and
then find the two step transition probabilities.
Cars arrive at a petrol station having only one petrol unit in
Poisson fashion with an average of 10 cars per hour. The service
time is distributed exponentially with a mean of 3 minutes. Find
(i) the average number of cars in the system, (ii) average time a
car spends in the queue, (iii) average queue length, (iv) the
probability that there will be two cars in the system.
In a railway marshalling yard, trains arrive at a rate of 30 per
day. Assume that the inter arrival times follow an exponential
distribution and the service time distribution is also exponential
distribution with an average of 36 minutes. If the line capacity
of the yard is 9 trains, calculate (i) the probability that the yard
is empty, (ii) the average queue length.
The arrival of large jobs at a computer center forms a Poisson
process with rate 2 per hour. The service time of such jobs are
exponentially distributed with mean 20 minutes. Only 4 large
jobs can be accommodated in the system at a time.
i.
Determine the probability that a large job will be turned
away because of lack of storage.
ii.
What is the probability that the arriving customer has to
wait?
A barber shop has two barbers and three chairs for customers.
Assume that the customers arrive in a Poisson fashion at a rate of
5 per hour and that each barber serves a customer, the service time
being exponentially distributed with a mean of 15 minutes. If a
customer arrives and if there are no empty chairs in the shop, he
2
8
7
8
7
will leave.
i. What is the probability that the shop is empty?
8
ii. What is the expected number of customer in the shop?
b.
In a factory, the customers have to pass through three counters.
The customers buy coupons at the first counter, select and collect
the snacks at the second counter and collect tea at third counter.
The server at each counter takes on an average 1.5 minutes
although the distribution of service time is approximately Poisson
at an average rate of 6 per hour. Find the average time a customer
spends waiting in cafeteria. Also find the most probable time
spent in getting the service.
OR
b.
Define MM/M/s:(N,FIFO) queue and give its operating
characteristics.
7. Write short notes on (Any Two):
2×5
a. Bayes Rule
b. Probability distribution function
c. Random Variable
3