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

PART A-(10 X 2=20 marks)

1. Given the probability density function f(x) = k/(1+x

^{2}), -∞ < x < ∞, Find k

2. If the probability is 0.10 that a certain kind of measuring device will show

3. If X has mean 4 and variance 9, while Y has mean -2 and variance 5, and

^{2})

4.Let X and Y be continuous RVs with J.p.d.f

f( x, y) = 2xy+3/2y2, 0

5. Define (a).Markov chain (b).Wide-Sense stationary process.

6. State any two properties of the Poisson process

7. In the usual notation of an M/M/I queuing system, if Î» = 3/hour and

8. Find P(X=c+n) for an M/M/C queuing system.

PART B-(5 X 16=80 marks)

11a (i).The time required to repair a machine is exponentially distributed

(ii). A discrete R.V. X has moment generating function MX(t)= (1/4+3/4e

^{ t})

^{5 }

Find E(X),Var(X) and P(X=2). (8)

(OR)

11.(b).(i).Find the moment generating function of a poisson variable and

(ii). A man draws 3 balls from an urn containing 5 white and and 7

12.(a).(i).(X, Y) is a two dimensional random variable uniformly distributed

(ii). Suppose that orders at a restaurant are i.i.d random variables with

(OR)

^{-(x2+y2) }, x>0 , y>0

^{2}+y

^{2}). (8)

g(x,y) = e

^{-(x+y) }, x>=0, y>=0

then

(1) find the m.p.d.f. of X

(2) find the m.p.d.f of Y.

(3) Are X and Y independent RVs? Explain?

(4) Find P(X > 2, Y<4)

(5) Find P(X>Y). (8)

0 1 0

p = ½ 0 ½

1 0 0

(1).Sketch the transition diagram.

(2).Is the chain irreducible? Explain

(3).Is the chain Ergodic? Explain. (8)

consecutive arrivals is (1) more than 1 minute (2) between 1 and 2

minutes (3) less than 4 minutes. (8)

(OR)

13.(b).(i).Consider a random process X(t) defined by X(t)=Ucost+(V+1)sint,

where U and independent random variables for which

E(U)=E(V)=0;E(U

^{2})=E(V

^{2})=1.

(1).Find the auto-covariance function of X(t)

(2).Is X(t) wide-sense stationary? Explain your answer. (8)

(ii). Ther are 2 white marbles in urn A and 4 red marbles in urn B. At each

step of the process, a marble is selected from each urn and the 2

chain is the number of red balls in A after the interchange. What is

the probability that there are 2 red balls in urn A (i) after 3 steps and

(ii) in the long run? (8)

transmits them over a single transmission line. Suppose that

messages arrives according to a Poisson process at a rate of one

message every 4 milliseconds and suppose that message transmission

times are exponentially distributed with mean 3ms. Find the mean

number of messages in the system and the mean total delay in the

system. What percentage increase in arrival rate results in a doubling

of the above mean total delay? (8)

(ii). Discuss the M/M/1 queuing system finite capacity and obtain its

steady-state probabilities and the mean number of customers in the

system. (8)

(OR)

14.(b).(i).A petrol pump station has 2 pumps. The service times follow the

exponential distribution 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? (8)

(ii) There are 3 typists in an office. Each typist can type an average of 6

letters per hour. If letters arrive for being typed at the rate of 15 letters

per hour, what fraction of time all the typists will be busy? What is

the average number of letters waiting to be typed? (8)

15. (a)(i). Automatic car wash facility operates with only one bay. Cars

arrive according to a Poisson process, with mean of 4 cars per hour

and may wait in the facility parking lot if the bay is busy. If the

service time for the cars is constant and equal to 10 min, determine

(1).mean number of customers in the system, (2).mean number of

customers in the queue (3).mean waiting time in the system (4).mean

waiting time in the queue. (8)

(ii) A repair facility shared by a large number of machines has 2

sequential stations with respective service rates of 2 per hour and 3 per

hour. The cumulative failure rate of all the machines is 1 per hour.

Assuming that the system behavior may be approximated by the

2-stage tandem queue, find

(1) the average repair time including the waiting time.

(2) the probability that both the service stations are idle and

(3) the bottleneck of the repair facility. (8)

(OR)

_{1 }and S

_{2}at a Poisson rate of 35/hour and form a queue at the entrance.

On studying the situation at the centre, they decide to go to either S

_{1 }or

S

_{2 .}The decision making takes on the average 30 seconds in an

exponential fashion. Nearly 55% of the customers go to S

_{1, }that

consists of 3 parallel servers and the rest go to S

_{2, }that consist of 7

parallel servers. The service times at S

_{1, }are exponential with a mean

of 6 minutes and those at S

_{2 }with a mean of 20 minutes. About 2% of

customers, on finishing service at S

_{1}go to S

_{2 }and about 1% of

customers, on finishing service at S

_{2}go to S

_{1}. Find the average queue

sizes in front of each node and the total average time a customer

spends in the service centre.

_{ }(16)

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