Anna University Coimbatore 2005 B.E Computer Science and Engineering Probability and queueing theory - Question Paper

 

B.E./B.Tech. DEGREE EXAMINATION, APRILA{AY 2005

Fourth Semester

; Computer Science Engineering

I

MA O4O - PROBABILITY AND QUEUEING THEORY

Time : Three hours Maximum : 100 marks

Answer ALL questions.

PART A - (10 x2 = 20 marks)

, rt/1 1. IfA and B areindependent events, prove that A- and B are also independent.

. l

,

2. A random variableXhas the p.d.f. f (x) given by

r, l c x e - * , i f r > 0

: l\3c)=10,

i f r ( 0 .

Find the value of C and C.D.F. ofX.

'

3. If the joint p.d.f. of ( X, Y ) is given by

f ( x , l ) = e - ( ' + ! ) tr)0, y >o

frnd E (XY).

4. Defrne wide sense stationary and strict sense stationary random processes.

I

i

'_ 5. The tangent of the angle between the lines of regression y on r and x on y is 0.6

.1

and o- = I o ,. Find the correlation coeffrcient.

6. Let X(t) beaPoissonprocesswithrate2 . Findcorrelationfunctiono f X ( t ) .

7. Establish the relations among the hazard function, reliability and failure time

densitv function

8.

9.

10.

The one-step transition probability matrix of a Markov chain with states

[o r l

{0,1} is given as

"=l; ;l

(a) Draw a transition diagram (b) Is i t

irreducible Markov chaini

Consider anM/MlL queueing system. If ). = 6 and p = 8, find the probability

of atleast 10 customers in the system.

Consider an M/IVI/C queueing system. Find the probability that an arriving

customer is forced to join the queue.

PARTB-(5x16=80marks)

11. (i) Let the random variable X have the p.d.f.

l1e-'/2,rc>o

f(x)=12-

.0, otherwise.

Find the moment generating function, mean and variance ofX. (6)

(ii) A die is tossed until 6 appears. What is the probability that it must be

tossed more than 4 times. @)

(iii) If X is a uniform random variable in the interval (-2,2) frnd the p.d.f of

Y = x2. ( 6 )

12. (a) (i) The joint p.d.f. of R.V.s X and Yis given by

r,, fs (x +y),0<rc<1,03y<L,x+y<L

f \x ' Y' =

1 o. otherwise.

Find

(1)

(2)

(3)

(ii) The j

\,

the marginal p.d.f. ofX

P (X +Y < I l 2 )

Cou (X ,Y ) .

oint p.d.f. of R.vs X and Y is given by

ff e-,t! e-t;x)0,y>0

f(x,Y)=1,

|.0, otherwise.

P (X >L/Y = y ) .

(8)

Find (8)

Or

(b) (i) The random variables X and Y have joint p.d.f.

(1) Are X and Y independent?

(2) Find the conditional p.d.f. of X given Y. (8)

(ii) Suppose X and Y are two random variables having the joint p.d.f.

f (x'l)=[n*'u-t12

+" ) ' x'Y>o

I o. otherwise.

f , x \

r, . lr" *!,0<r<\,0<y<2

| \x,l)=\ 3

|.0, otherwise.

Find the p.d.f. of z =

13. (a) (i) Consider a random process X (t) defrned by

X (t) =U cost + V sinf , where [.I and V are independent random

variables each of which assumes the values -2 and L with

probabilities 1/3 and 2/3 respectively. Show that X (, ) is widesense

stationary and not strict-sense stationary? (8)

(ii) Derive Poisson process with rate 2 and hence find its mean. Is

Poisson process stationary? Explain. (8)

Or

(b) (i) Show that the inter-arrival time of a Poisson process with intensity

(8)

(6)

(10)

space {0,1} and transition

function.

L4. (a) (i) Consider a Markov chain with state

probabilitmy atrix

"

= [- 1 - 9

-l

lr/2 Lt2 )

(1) Is the state 0 recurrent? Explain?

(2) Is the state 1 transient? Explain.

). obeysan exponential law with mean ].

)

Suppose {N(r):t)0} is a renewal process and F(t) is the

interarrival time distribution. (1) Derive the renewal function and

renewal density (2) obtain the renewal equation for the renewal

(ii) Discuss the reliability under preventive maintenance and

MTTF of a system with preventive maintenance.

Or

3

(8)

derive

(8)

 

15. (a)

(i)

(ii)

(i)

ft) A system has 4 identical components connected in parallel and

shows a system reliability of 0.90. How many more components

should be added in parallel to get a system reliability of 0.99? (10)

Discuss the steady state availabiiity of one-unit system. (6)

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 varidble with mean 8 minutes.

(1) Find the average number of customers Z, in the shop

(2) Find the average time a customer spends in the shop Wt

(3) Find the average number of customers in the queue .Lo

(4) What is the probabiiity that the server is idle.

(ii) Automatic car wash facility operators with only one bay. Cars

arrive according to a Poisson process at the rate 4 cars per hour and

may wait in the facility's parking lot if the bay is busy. If the service

time for all cars is constant and equal to 10 minutes, determine

(1) mean number of customers in the system .Lg

(2) mean number of customers in the queue Zo

(3) mean waiting time of a customer in the system W't

(4) mean waiting time of a customer in the queue V[o . (8)

Or

(b) (i) Find the average number of customers L s in the MA4/1AI

queueing system when ) = /r. (8)

(ii) A car servicing station has 2 bays where service can be offered

simultaneously. Because of space limitation only 4 cars are accepted

for servicing. The arrival process is Poisson with 12 cars per day.

The service time in both the bays is exponentially distributed with

F = 8 car per day per bay. Find the average number of cars in the

service station and the average time a car spends in the system. (8)

(8)