MA6453 PROBABILITY & QUEUEING THEORY UNIT I · 2018. 7. 23. · 10. A continuous random variable X has the probability function f (x) k(1 x), 2 x 5. Find P(X< 4). Solution: 27 2 1

PANIMALAR INSTITUTE OF TECHNOLOGY DEPARTMENT OF IT

II Year / IV Sem 1

MA6453 PROBABILITY & QUEUEING THEORY

UNIT I

RANDOM VARIABLES AND STANDARD DISTRIBUTIONS

SYLLABUS: Discrete and continuous random variables – Moments – Moment generating functions– Binomial, Poisson, Geometric, Uniform, Exponential, Gamma and Normal distributions.

COURSE OBJECTIVE: Know the characteristics of Probability distributions by identifying thediscrete and continuous random variables. The Binomial distribution is used in quality control ofitem manufactured by a production line when each item is classified as either defective or non-defective. The Poisson distribution applies in its own right where the possible number of discreteoccurrences is much larger than the average number of occurrences in a given interval of time orspace. It is also used as a convenient approximation to the binomial distribution in somecircumstances

PART-A

1. If a random variable X takes the values 1,2,3,4 such that P(X=1)=3P(X=2)=P(X=3)=5P(X=4).Find the probability distribution of X. (Nov. Dec.2012)Solution:

Assume P(X=3) = α. By the given equation,5

)4(3

)2(2

)1( XPXPXP .

For a probability distribution (and mass function) 1)(xP

P(1)+P(2)+P(3)+P(4) =1

61

301

30

611

532

61

6)4(;

61

30)3(;

61

10)2(;

61

15)1( XPXPXPXP

The probability distribution is given by

61

6

61

30

61

10

61

15)(

4321

xp

X

2. Let X be a continuous random variable having the probability density

function

otherwise

xxxf,0

1,2

)( 3Find the distribution function of x.

Solution:

21

21

31

11

12)()(

xxdx

xdxxfxF

xxx


II Year / IV Sem 2

3. A random variable X has the probability density function f(x) given by

otherwise

xecxxfx

,0

0,)( . Find the value of c and CDF of X.

Solution:

111111)( 000

cceexcdxecxdxxf xxx

xxxxxx

xxxx

eexeexdxexdxexcdxxfxF 1)( 0000

4. A continuous random variable X has the probability density function f(x) given

by xcexf x ,)( . Find the value of c and CDF of X.

Solution:

2

11121212

1211)(

0

0

0

ccecdxec

dxecdxecdxxf

xx

xx

xxxx

xxxx

eecdxecdxecdxxfxF

xiCase

2

1)(

0)(

xexeccxcec

xxecxecx

dxxecdxxecdxx

ex

cx

dxxfxF

xiiCase

22

12

0

0

0

0)(

0)(

0,22

1

0,2

1

)(xe

xexF x

x

5. If a random variable has the probability density

otherwise

xexfx

,0

0,2)(2

. Find the

probability that it will take on a value between 1 and 3. Also, find the probability that it willtake on value greater than 0.5.


II Year / IV Sem 3

Solution:

1

5.0

2

5.0

2

5.0

623

1

23

1

23

1

2)()5.0(

2)()31(

eedxedxxfXP

eeedxedxxfXP

xx

xx

6. Is the function defined as follows a density function?

4,0

42,2318

12,0

)(

x

xx

x

xf

Solution:

1

72

2323

18

1)(

4

2

24

2

4

2

xdxxdxxf

Hence it is density function.

7. The cumulative distribution function (CDF) of a random variable X is

0,)1(1)(

xexXFx

. Find the probability density function of X.

Solution:

( ) 0 1 1 , 0x x xf x F x x e e x e x

8. The number of hardware failures of a computer system in a week of operations has thefollowing probability mass function:No of failures : 0 1 2 3 4 5 6Probability : 0.18 0.28 0.25 0.18 0.06 0.04 0.01Find the mean of the number of failures in a week.Solution:

92.1

)01.0)(6()04.0)(5()06.0)(4(

)18.0)(3()25.0)(2()28.0)(1()18.0)(0()()(

xPxXE

9. Given the p.d.f of a continuous r.v X as follows

elsewhere

xxxxf

,0

10),1(6)( .

Find the CDF of X.

Solution:32

0

32

0

2

00

23236_6)_1(6)()( xxxxdxxxdxxxdxxfxFxxxx


II Year / IV Sem 4

10. A continuous random variable X has the probability function 52),1()( xxkxf .

Find P(X<4).Solution:

27

21

2

271

2

1111)(

5

2

25

2

4

2

kk

xkdxxkdxxf

27

16925

25

1

2

1

27

21

27

2)()4(

4

2

24

2

4

2

xdxxdxxfXP

11. Given the p.d.f of a continuous R.V X as follows

elsewhere

xxxf

,0

5.01.025.15.12)(

Find P(0.2 < X < 0.3)Solution:

1875.0

2.02.053.03.0525.1

25.12

5.12)25.15.12()3.02.0(

22

3.0

2.0

23.0

2.0

x

xdxxXP

12. If the MGF of a continuous R.V X is given by t

tM X

3

3 . Find the mean and variance of X.

Solution:

1 2 3

2 2

2 2

3 11 1 ...

3 3 3 3 313

1 1 2( ) 1! , ( ) 2! 2!

3 9 92 1 1

( ) ( )9 9 9

X

t t t tM t

tt

E X coefficient of t is the mean E X coefficient of t

Variance E X E X

13. If the MGF of a discrete R.V X is given by 4

2181

1

tX etM , find the distribution of X.

Solution:

tttt

tttttX

eeee

eCeCeCeCetM

432

4

4

3

3

2

21

4

81

16

81

32

81

24

81

8

81

1

24242424181

121

81

1

By the definition of MGF,

tttttx

X epepepeppxpetM432

)4()3()2()1()0()(


II Year / IV Sem 5

On comparison with above expansion the probability distribution is

81

16

81

32

81

24

81

8

81

1)(

43210

xp

X

14. Find the MGF of the R.V X whose p.d.f is

elsewhere

xxf,0

100,10

1)( . Hence find its mean.

Solution:

5

.....31

100051

1....!3

1000

!2

100101

10

11

10

1

10

1

10

1

2

321010

0

10

0

toftcoefficienMean

tt

ttt

tt

e

t

edxetM

ttxtx

X

15. Given the probability density function2

( ) ,1

kf x x

x

, find k and C.D.F.

Solution:

1

122

1tantan1tan11

1)( 1112

kk

kxkdxx

kdxxf

xxxxdxx

kdxxfxF

xx

111112

cot1

tan2

1tantan

1tan

1

1)()(

16. It has been claimed that in 60 % of all solar heat installation the utility bill is reduced by atleastone-third. Accordingly what are the probabilities that the utility bill will be reduced by atleastone-third in atleast four of five installations?Solution:

Given n=5, p=60 % =0.6 and q=1-p=0.4

337.0

)4.0()6.0(5)4.0()6.0(5

]5[]4[)4(555

5454

4

cc

xpxpxp

17. The no. of monthly breakdowns of a computer is a r.v. having poisson distribution with mean1.8. Find the probability that this computer will function for a month with only one breakdown.Solution:

8.1,!

)(

givenx

exXp

x

, 2975.0!1

)8.1()1(

18.1

e

xp


II Year / IV Sem 6

18. In a company 5 % defective components are produced. What is the probability that atleast 5components are to be examined in order to get 3 defectives?Solution:

To get 3 defectives, 3 or more components must be examined.

p=5 % =0.05, q = 1- p=0.95 and k=success=3

9995.000048.01

)95.0()05.0(3)95.0()05.0(21

)4()3(1

)5(1)5(

,...2,1,,)1()(

132

032

1

cc

xpxp

xpxp

kkkxqpcxxXp kxkk

19. A discrete R.V X has mgf . Find E(x), var(x), and p(x=0).

Solution: Given

We know that mgf of poisson is

Therefore λ = 2. In poisson E(x) = var(x) = λ

2)(var)( xxEMean

!)(

x

exXp

x

1353.0!0

)0( 20

eee

Xp

20. Find the mean and variance of geometric distribution.Solution:

The pmf of Geometric distribution is given by

,.....3,2,1,)( 1 xqpxXp x

( ) ( ) 1 2 3 .....

11 2 3 ..... 1

1 1 1 1 2 1 3 1

1 122 2 1

Mean E x x p x x p q p x q p q q q

p q q p q p p pp

x x

x x


II Year / IV Sem 7

2 1 1 1( ) 1 ( 1)1 1 1

11 1 2 1 3 11(1 1) 2(2 1) 3(3 1) .....

1 12 22 2(3) 3(4) ..... 2 1 3 6 .....

1 1 2 13 32 1 22

x x xE x x x x p q x x p q x p qx x x

p q pq pqp

p pq pq p q qp p

p q p pp p pp

222

22

2

2

22

111

112112)()(

p

q

p

p

pp

ppppppxExEVariance

21. Find the MGF geometric distribution.

Solution: The PMF of geometric distribution is given by

t

ttt

ttt

ttt

x

xt

x

xxt

x

xtx

x

xtxtxtxx

qe

qeqepe

qeqeqeq

p

qeqeqeq

pqe

q

pqe

q

p

qqpeqpexpeeEtMMgf

1

11

...1

...

)()()(

1

2

32

11

1

1

1

1

22. Show that for the uniform distribution axaa

xf ,2

1)( , the mgf about origin is

at

atsinh

.

Solution: Given axaa

xf ,2

1)(

MGF

at

at

atat

eeat

t

e

adxe

adx

aedxxfeeEtM

atat

a

a

txa

a

txa

a

txtxtxx

sinh

sinh22

1

2

1

2

1

2

1

2

1)()(


II Year / IV Sem 8

23. Define exponential density function and find mean and variance of the same.

Solution: The density function of exponential distribution is given by 0,)( xexf x

1

110)00(

)(

22

02

00

xxxx exe

dxexdxexdxxfxxEMean

233

032

2

0

2

0

222

22200)000(

22)(

xxxxx exeex

dxexdxexdxxfxxE

222

2

2

22 11212)(

xExEVariance

24. State memory less property of exponential distribution.Solution:If X is exponentially distributed with parameter , then for any two positive integers ‘s’ and ‘t’

)()/( tXPsXtsXP .

25. A continuous random variable X that can assume any value between 5&2 xx has a

density function given by )1()( xkxf . Find P(X>4). (Nov/ Dec. 2012)

Solution: Since f(x) is a density function,

27

21

2

27

2

42

2

255

21)1(

5

2

25

2

kkk

xxkdxxk

27

11

2

16

2

251

27

2)1(

27

2)4(

5

4

dxxXP

26. Identify the random variable and name the distribution it follows, from following statement:“A realtor claims that only 30% of the houses in a certain neighbourhood appraised at less thanRupees 20 lakhs. A random sample of 10 houses from the neighbourhood is selected andappraised to check the realtor’s claims acceptable are not”. (Nov/ Dec. 2012)


II Year / IV Sem 9

Solution:

X is a random variable that a house is appraised at less than Rs.20 lakhs. And it follows abinomial distribution with n = 10, p=0.30 and q=0.70

27. A coin is tossed 2 times, if ‘X’ denotes the number of heads, find the probability distribution ofX.(Nov./Dec. 2013)

X: No. of heads 0 1 2

P(X=x) 21 1

2 4

2

1

12

2C

2

2

12

2C

28. If the probability that a target is destroyed on any one shot is 0.5, find the probability that itwould be destroyed on 6th attempt. (Nov./Dec.,2013)

5 6( 6) (0.5)P X q p

29. A continuous random variable X has the probability density function given by2( ) (1 ), 2 5f x a x x , Find ‘a’ and ( 4)P X (May/June 2014)

52

2

1(1 ) 1

42a x dx a ,

42

2

1 31( 4) (1 )

42 63P X x dx

30. For a binomial distribution with mean 6 and standard deviation 2 , Find the first two terms of

the distribution. (May/June 2014)

31. From the given data,2 1

9, ,3 3

n p q

I term:9

0

19

3C

; II term:1 8

1

2 19

3 3C

32. Test whether; 1 1

( )0 ;

x xf x

otherwise

probability density function of a continuous random variable

(Nov./Dec.2014) (April/ May 2015)

11 2

1 0

12 2 1

2 2

xx dx

. Therefore it is a continuous random variable.


II Year / IV Sem 10

33. What do you mean by MGF? Why it is called so? (Nov./Dec.2014)

MGF is a moment generating function which generates all the moments about the origin. It can also be

calculated as a coefficient of!

rt

rand also by differentiating the MGF with respect to ‘t’ , r times , i.e.

0

( ) ( )

( )

txX

r

r Xr

t

M t E e

dM t

dt

34. If the density function of a continuous random variable X is given by

; 0 1

; 1 2( )

3 ; 2 3

0 ;

ax x

a xf x

a ax x

otherwise

then find the value of ‘ a ’ (April/ May 2015)

1 2 3

0 1 2

1(3 ) 1

2ax dx a dx a ax dx a

35. Suppose that on an average, in every three pages of a book there is one typographical error. IFthe number of typographical errors on a single page of the book is a Poisson random variable.What is the probability of at least one error on a specific page of the book? (April/ May 2015)

3( 1) 1 ( 1) 1 ( 0) 1 10!

eP X P X P X e

36. What are the limitations of Poisson distribution? (April/ May 2015)

Poisson distribution is a limiting case of binomial distribution. When 0n and p

Instead of Binomial distribution, Poisson distribution will be applied.

37. A continuous random variable X has the probability density function given by2( ) (1 ), 1 5f x a x x , Find ‘a’ and ( 4)P X (Nov./Dec. 2015)

52

1

3(1 ) 1

136a x dx a

42

1

3 18( 4) (1 )

136 34P X x dx


II Year / IV Sem 11

38. What is meant by memory less property? Which discrete distribution follows this property?(Nov./Dec. 2015)

If X is continuously distributed random variable, then for any two positive integers‘s’ and ‘t’)()/( tXPsXtsXP .Geometric distribution follows memory less property.

39. Let X be the random variable which denotes the number of heads in three tosses of a fair coin.Determine the probability mass function of X? (Nov./ Dec. 2015)

X: No. of heads 0 1 2 3

P(X=x) 31 1

2 8

3

1

13

2C

3

2

13

2C

3

3

13

2C

40. A continuous random variable X has a pdf given by 23( ) (2 ), 0 2

4f x x x x .

Find P( X>1) (Nov./Dec. 2015)

22

1

3 1( 1) (2 )

4 2P X x x dx

41. Let X be a discrete random variable with pmf ( ) , 1, 2,3, 410

xP X x x , Compute

( 3)2

XP X and E

(May/ June 2016)

Ans:: 1 2 3 4

1 2 3 4( ) :

10 10 10 10

X

p x

1 2( 3) 0.3

10 10P X

1 1 2 3 3 4 4( ) 1

2 2 2 10 10 2 10 2 10

X xE p x

42. If a random variable X has the moment generating function3

( )3XM t

t

, Compute 2( )E X

(May/ June 2016)1 2 2

22

3 1 2( ) 1 1 ..... 1 ...

3 3 3 3 3 2! 9

( )2!

X

t t t tM t t

t

tCoefficient of E X


II Year / IV Sem 12

2 2( )

9E X

PART-B

1. A random variable X has the following probability distribution:X=x -2 -1 0 1 2 3

P(x) 0.1 k 0.2 2k 0.3 3k

(i) Find k, (ii) Evaluate 2XP and 22 XP , (iii) Find the PDF of X and

(iv) Evaluate the mean of X (AP) (Nov/Dec 2011) (May/ June 2016)

2. A random variable X has the following probability distribution:X=x -2 -1 0 1 2 3

P(x) 0.1 K 0.2 2K 0.3 3K

Find K, 22 xP , mean of X . (AP) (May/Jun 2009)

3. A random variable X has the following probability functionX=x 0 1 2 3 4 5 6 7P(x) 0 k 2 k 2 k 3 k 2k 2 2k 7 2k + k

(i) Find the value of k , (ii) Evaluate 6XP , 6XP ,

(iii) If 2

1 cXP , find the minimum value of c .

(iv) (1.5 4.5 / 2)P X X (AP)(April/May 2012)(April/ May 2015)4. If X is a Poisson variate such that 690492 XPXPXP . Find

(i) Mean and 2XE , (ii) .2XP (AP)(April/May 2012)

5. In a continuous distribution, the probability density is given by .20),2()( xxkxxf Find

k, mean, variance and the distribution function. (AP) (May/Jun 2007)

6. The sales of a convenience store on a randomly selected day are X thousand dollars, where X isa random variable with a distribution function of the following form:

otherwise

xxxk

xx

x

xF

1

21)4(

102

00

)(2

2

Suppose that this convenience store’s total sales on any given day are less than $2000.(a) Find the value of k,(b) Let A and B be the events that ‘tomorrow the store’s total sales are between 500 and

1500 dollars, and over 1000 dollars,’ respectively. Find P(A) and P(B).


II Year / IV Sem 13

(c) Are A and B independent events? (U) (Nov/Dec 2007)

7. A random variable X has the pdf

otherwise

xxxf

,0

10,2)( find (a)

2

1XP (b)

2

1

4

1XP (c)

2

14

3

X

XP (AP) (Nov/Dec 2008)

8. If

0,0

0,)(2

2

x

xxexp

x

.Show that p(x) is a pdf and also find F(x) (AP) (May/Jun 2009)

9. If the cumulative distribution function of a RV X is given by

2,0

2,4

1)( 2

x

xxxF

find (a) 3XP (b) 54 XP (c) 3XP (AP) (Apr/May 2008)

10. If the density function of a continuous random variable X is given

by

otherwiseo

xaxa

xa

xax

xf323

21

10,

)(

Find a . Find the cdf of X . (AN) (Nov/Dec 2008)

11. The distribution function of a random variable X is given by 0;)1(1)( xexxF x .

Find the density function, mean and variance of X .(AP) (Nov/Dec 2010)

12. If X is a random variable with a continuous distribution function )(xF , prove that

)(xFY has a uniform distribution in (0, 1). Further if

otherwise

xxXf

0

31),1(2

1)( ,

find the range of Y corresponding to the range 9.21.1 x .(AP)

13. The (DF) cumulative distribution function (cdf) of a random variable X is given by

Find the pdf of X and evaluate 1XP and 431 XP using both pdf and cdf (PDF).

(U)(May/June2007, (Nov/Dec 2011)

3,1

32

1,3

25

31

2

10,

0,0

)(2

2

x

xx

xx

x

xF


II Year / IV Sem 14

14. The probability function of an infinite discrete distribution is given by ;21 jjXP

,...,2,1j . Verify that the total probability is 1 and find the mean and variance of the

distribution. Find also ,evenisXP 5XP and .3bydivisibleisXP (AP) (Nov/Dec

2011)15. Find the moment –generating function of the binominal random variable with parameters m

and p and hence find it mean and variance. (AN) (April/May 2011)(April/ May 2015)

16. Find the moment generating function of an exponential random variable and hence find itsmean and variance.(AN)(April/May 2012)(May/ June 2014)

17. Find the moment generating function of the geometric random variable with the pdf

,....3,2,1,)( 1 xpqxf x and hence obtain its mean and variance. (May/Jun 2007)(April/

May 2015) (AN)18. Derive mean and variance of a Geometric distribution. Also establish the forgetfulness property

of the Geometric distribution. (AN)(April/May 2011)19. Describe the situations in which geometric distribution could be used. Obtain its MGF. (AN)

(April/May 2010)20. By calculating the MGF of Poisson distribution with parameter , prove that the mean and

variance of the Poisson distribution are equal. (AN) (April/May 2010)(May/ June 2014)21. Define Weibull distribution and write its mean and variance. (R)(April/May 2011)22. Define Gamma distribution and find its mean and variance. (AN)(Nov/Dec 2011)

23. If the density function of X equals

00

02

x

xCexf

x

. Find C . What is 2XP . (AP)

(April/May 2010)

24. A discrete RV has moment generating function5

4

3

4

1)(

t

X etM . Find XE , XVar

and 2XP .(AP) (Apr/May 2008)

25. If the moments of a random variable X are defined by ,..3,2,1;6.0 rXE r show that

02,6.01,4.00 XPXPXP . (AN)(Nov/Dec 2008)

26. A coin having probability p of coming up heads is successively flipped until the thr head

appears. Argue that X , the number of flips required will be n , rn with

probability 1 ,1

r n rP X n n C p q n rr

. (AN)(April/May 2010)

27. A coin is tossed until the first head occurs. Assuming that the tosses are independent and theprobability of a head occurring is ‘p’. Find the value of ‘p’ so that the probability that an oddnumber of tosses required is equal to 0.6. Can you find a value of ‘p’ so that the probability is0.5 that an odd number of tosses are required? (U)(Nov/Dec 2010)

28. The time (in hours) required to repair a machine is exponentially distributed with

parameter2

1 . What is the probability that the repair time exceeds h2 ? What is the

conditional probability that a repair takes at least h10 given that its duration exceeds h9 . (AP)


II Year / IV Sem 15

(Nov/Dec 2010)29. Suppose that telephone calls arriving at a particular switchboard follow a Poisson process with

an average of 5 calls coming per minute. What is the probability that up to a minute will elapseunit 2 calls have come in to the switch board? (AP)(April/May 2011)

30. A machine manufacturing screws is known to produce 5% defective. In a random sample of 15screws, what is the probability that there are

(i) exactly 3 defectives(ii) not more than 3 defectives (AP)(Nov/Dec 2008)

31. Out of 800 families with 4 children each, how many families would be expected to have(i) 2 boys and 2 girls

(ii) at least 1 boy(iii) at most 2 girls(iv) Children of both sexes.

Assume equal probabilities for boys and girls. (AP)(May/Jun 2009)

32. The mileage which cars owners get with a certain kind of certain kind of radial tire is a randomvariable having an exponential distribution with mean 40,000 km. Find the probabilities thatone of these tires will last

(i) At least 20,000 km(ii) At most 30,000 km. (AP)(April/May 2015)

33. The number of monthly breakdown of a computer is a random variable having a Poissondistribution with mean equal to 1.8. Find the probability that this computer will function for amonth

(i) without a breakdown(ii) with only one breakdown(iii)With at least one break down (AP) (May/Jun 2007), (Nov. /Dec.2012)

34. Experience has shown that while walking in a certain park, the time X (in minutes) between

seeing two people smoking has a density function of the form

otherwise

xxexf

x

0

0)(

.

(a) Calculate the value of .(b) Find the distribution function of X .(c) What is the probability that Jeff, who has just seen a person smoking, will see another

person smoking in 2 to 5 minutes? In atleast 7 minutes? (AP) (Nov/Dec 2007)35. Let the random variable X follows binomial distribution with parameter n and p. Find

(1) probability mass function of X(2) moment generating function(3) mean and variance of X (AP)(Nov/Dec 2006)

36. The number of personal computers (PC) sold daily at a Computer world is uniformlydistributed with a minimum of 2000 PC and a maximum of 5000 PC. Find

(i) the probability that daily sales will fall between 2,500 and3,000 PC.(ii) What is the probability that Computer world will sell at least 4000 PCs?(iii)What is the probability that Computer world will sell exactly 2500 PCs?


II Year / IV Sem 16

(AP) (Nov/Dec 2006)37. Define the probability density function of normal distribution and standard normal distribution.

Write the important properties of this distribution. (R)(Nov/Dec 2006)38. An electrical firm manufactures light bulbs that have a life, before burn out, that is normally

distributed with mean equal to 800 hours and a standard deviation of 40 hours. Find(i) the probability that a bulb burns more than 834 hours.(ii) the probability that the bulb burns between 778 and 834 hours. (AP) (Nov/Dec

2006)39. The lifetime X in hours of a component is modeled by a Weibull distribution with 2 .

Starting with a large number of components, it is observed that %15 of the components thathave lasted 90 hrs fail before 100 hrs. Find the parameter .(AP)(May/Jun 2007)

40. A die is cast until 6 appear. What is the probability that it must be cast more than 5 times?(AP) (Nov/Dec 2008)

41. Let X be a RV with 1XE and 41 XXE . Find )32(,2

XVarX

Var

(AP) (Apr/May

2008)

42. If X is a continuous RV with pdf

otherwise

xx

xx

xf

0

21)1(2

3

10

)( 2 find the cumulative

distribution function xF of X and use it to find

2

5

2

3XP (AP)(Apr/May 2008).

43. If a RV X has geometric distribution, i.e., 3,2,1,)( 1 xpqxXP x where

1,1 popq show that xXPyXyxXP (AP)(Apr/May 2008)

44. Let the pdf for X be given by

otherwise

xexf

x

0

0,2

1)(

22

.

Find (i)

2

1XP , (ii) Moment generating function for X (iii) XE

(iv) Var( X ) (AP)(Apr/May 2008)

45. If the probability density of X is given by

otherwise

xforxxf

0

10)1(2)(

(i) Show that )2)(1(

2

rrXE r

(ii) Use this result to evaluate 212 xE (AP)(Nov/Dec 2006)

46. A random variable X has density function given by

otherwise

kxkxf0

01

)(


II Year / IV Sem 17

Find mgf, rth moment, mean, variance (AP)(Nov/Dec 2006)

47. Let the random variable X assume the value ‘r’ with the probability law:

,..3,2,1,1 rpqrXP r Find the moment generating function and hence its mean and

variance. (AN)(May/Jun 2006)48. Find the moment generating function of a normal distribution. (AN) (May/Jun 2006)49. The pdf of samples of the amplitude of speech wave forms is found to decay exponentially at

the rate so the following pdf is proposed. xCexf x ,)( . Find the constant C

and also XP and E(X). (AP) (Apr/May 2008)

50. If X is uniformly distributed over 0),,( , find so that

(i) 3

11 XP (ii) 11 XPXP (AP) (Nov/Dec 008)

51. The lifetime of a TV tube (in years) is an exponential random variable with mean 10. What isthe probability that the average lifetime of a random sample of 36 TV tubes is at least 0.5

(AP) (Nov/Dec 2007)52. The atoms of radioactive element are randomly disintegrating. If every gram of this element, on

average, emits 3.9 alpha particles per second, what is the probability that during the nextsecond the number of alpha particles emitted from 1 gram is(i) at most 6 (ii) At least 2(iii) at least 3 and at most 6 (AP)(Nov/Dec 2007)

53. Starting at 5.00am every half hour there is a flight from San Francisco airport to Los AngelesInternational Airport. Suppose that none of these planes is completely sold out and that theyalways have room for passengers. A person who wants to fly to L.A. arrives at the airport at arandom time between 8.45 a.m., and 9.45., am. Find the probability that she waits(i) At most 10 minutes (ii) At least 15 minutes (AP) (Nov/Dec 2007)

54. A man with ‘n’ keys wants to open his door and tries the keys independently and at random.Find the mean and variance of the number of trials required to open the door if unsuccessfulkeys are not eliminated from further selection. (AN) (Nov/Dec 2007)

55. Write the pdf of Gamma distribution. Find the MGF, mean and variance. (May/Jun 2007)56. In a certain city, the daily consumption of electric power in millions of kilowatt hours can be

treated as a random variable having Gamma distribution with parameters2

1 and 3v . If

the power plant of this city has daily capacity of 12 million kilowatt-hours, what is theprobability that this power supply will be inadequate on any given day. (AP)(April/May 2012)

57. A continuous random variable has the pdf 11,)( 4 xkxxf , Find the value of k and also

4

1

2

1XXP (AP) (May/ June 2013)

58. Find the moment generating function of uniform distribution and hence find its mean andvariance. (AN) (May/ June 2013)

59. Find the moment generating function and rth moment for the distribution whose pdf is

.0,)( xKexf x Hence find the mean and variance. (AN) (May/ June 2013)


II Year / IV Sem 18

60. In a large consignment of electric bulb, 10 % are defective. A random sample of 20 is takenfor inspection. Find the probability that (1) all are good bulbs (2) atmost there are 3 defectivebulbs (3) Exactly there are 3 defective bulbs(May/ June 2013) (AP)

61. If a random variable X takes the values 1,2,3,4 such that P(X=1) = 3P(X=2) = P(X=3) =5P(X=4). Find the probability distribution of X. (AP)(Nov. Dec.2012).

62. Find the MGF of the binomial distribution and hence find its mean? (AN)(Nov/ Dec. 2012)

63. If the probability that an applicant for a driver’s license will pass the road test on any trial is0.8, what is the probability that he will finally pass the test (1) on the4th trial (2) in fewer than4 trials? (AP) (Nov./ Dec.2012)

64. Find the MGF of the random variable ‘X’ having the pdf, 0 1

( ) 2 , 1 2

0,

x for x

f x x for x

otherwise

(AP) (Nov./Dec.2013)65. 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 abox fail to meet the guaranteed quality? (AP)(Nov./Dec.2013)

66. 6 dice are thrown 729 times. How many times do you expect atleast three dice to show a fiveor a Six? (AP)(Nov./Dec.2013)

67. If a continuous RV X follows uniform distribution in the interval (0,2) and a continuous RV, Yfollows exponential distribution with parameter such that ( 1) ( 1)P X P Y (AP)

(Nov./Dec.2013)68. Suppose that a trainee soldier shoots a target in an independent fashion. IF the probability that

the target is shot on any one shot is 0.7. (1) What is the probability that the target would behit on tenth attempt? (2) What is the probability that it takes him less than 4 shots? (3) Whatis the probability that it takes him an even number of shots? (AP)(May/ June2014)

69. Trains arrive at a station at 15 minutes intervals starting at 4 a.m. If the passenger arrive at astation at a time that is uniformly distributed between 9.00 and 9.30, find the probability thathe has to wait for the train for (1) less than 6 minutes (2) more than 10 minutes (AP)(May/June 2014)

70. If2

2 , 0( )0 ; 0

x

xe xf xx

then show that ( )f x is a pdf and find F(x) (AP)(Nov./Dec.2014)

71. Find the MGF of a poisson random variable and hence find its mean and variance (AN)(Nov./Dec.2014)

72. A random variable X takes the values -2,-1, 0 and 1 with probabilities1 1 1 1

, ,8 8 4 2

and

respectively. Find and draw the probability distribution function (AP)(Nov./Dec.2014)73. In a normal distribution, 31% of the items are under 45 and 8% are over 64. Find the mean and

variance of the distribution. (AP)(Nov./Dec.2014)


II Year / IV Sem 19

74. The distribution function of a random variable X is given by ( ) 1 (1 ) , 0xF x x e x . Find

the density function, mean and variance of X. (AP) (April/May 2015)75. Messages arrive at a switch board in a poisson manner at an average rate of six per hour. Find

the probability for each of the following events: (1) exactly two messages arrive within onehour.(2) no message arrives within one hour (3) Atleast three messages arrive within one hour

(AP)(April/ May 2015)76. The peak temperature T, as measured in degrees Fahrenheit, on a particular day is the Gaussian

(85,10) random variable. What is P(T>100), P(T<60) and (70 100)P T (AP) (April/ May

2015)

77. The CDF of the random variable of x is given by

0 , 0

1 1( ) , 0

2 21

1,2

x

F x x x

x

Draw the graph of

CDF. Compute1 11( ),4 3 2

P X P X

(AP)(April/ May 2015)

78. Ten percent of the tools produced in a certain manufacturing company turn out to be defective.Find the probability that in a sample of 10 tools chosen at random, exactly 2 will be defectiveby using (1) binomial distribution (2) The Poisson approximation to the binomial distribution.(AP) (Nov/Dec. 2015)

79. The number of typing mistakes that a typist makes on a given page has a Poisson distributionwith a mean of 3 mistakes. What is the probability that she makes (1) exactly 7 mistakes?(2) fewer than 4 mistakes? (3) no mistakes on a given page? (AP) (Nov/Dec. 2015)

80. The lifetime X of particular brand of batteries is exponentially distributed with a mean of 4weeks. Determine (1) the mean and variance of X (2) what is the probability that the battery lifeexceeds 2 weeks? (3) Given that the battery has lasted 6 weeks, what is the probability that itwill last at least another 5 weeks? (AP) (Nov/Dec. 2015)

81. Find the moment generating function of ),( 2N normal random variable and hence determine

the mean and variance. (AN) (Nov/Dec. 2015)82. A component has an exponential time to failure distribution with mean of 10,000 hours.

(1) The component has already been in operation for its mean life. What is the probability thatit will fail by 15,000 hours? (Nov/Dec. 2015)

(2) At 15,000 hours the component is still in operation. What is the probability that it willoperate for another 5000 hours? (AP) (Nov/Dec. 2015)

83. The average percentage of marks of candidates in an examination is 42 with a standarddeviation of 10. If the minimum mark for pass is 50% and 1000 candidates appear for theexamination, how many candidates can be expected to get the pass mark if the marks follownormal distribution? If it is required, that the double the number of the candidates should pass,what should be the minimum marks for pass? (AP) (Nov/Dec. 2015)


II Year / IV Sem 20

84. A continuous random variable X has the pdf 3( ) , 0xf x kx e x . Find the rth moment about

the origin, moment generating function, mean and variance of X. (AP) (Nov/Dec. 2015)

85. Let X be a continuous random variable with pdf ( ) , 0xf x xe x , find (i) the cumulative

distribution function of X (ii) Moment generating function of X (iii) P(X<2) (iv) E(X) (AP)(May/ June 2016)

86. Let1

3 1( ) , 1, 2,3...

4 4

x

P X x x

be the probability mass function of a random variable

X, Compute (1) P(X>4) (2) P(X >4/ X>2) (3) E(x) (4) Var (X) (AP) (May/ June 2016)

87. Let X be a uniformly distributed random variable over [-5, 5], Determine (1) P(X<=2)

(2) ( 2)P X , (3) Cumulative distribution function of X, (4) Var(X). (AP) (May/ June 2016)

Find the moment generating function of Poisson distribution with parameter and hence prove thatthe mean and variance of the Poisson distribution are equal. (AN) (May/ June 2016) (Nov/Dec.2015)

COURSE OUTCOME: Acquire skills in handling one random variable and functions of randomvariables.

UNIT – IITWO DIMENSIONAL RANDOM VARIABLES

SYLLABUS: Joint distributions – Marginal and conditional distributions – Covariance – Correlationand Linear regression – Transformation of random variables.

COURSE OBJECTIVE: Know the fundamental concepts of joint, marginal and conditionaldistributions to understand covariance and correlation. Have an ability to design a model or a processto meet desired needs within realistic constraints such as environmental conditions

PART–A1. Define joint probability density function of two random variables X andY .

If YX , is a two dimensional continuous random variable such that

dydxyxfdy

yYdy

ydx

xXdx

xP ,22

,22

, then yxf , is called the joint pdf

of YX , , provided yxf , satisfies the following conditions

1,

,0,

R

dydxyxfii

Ryxallforyxfi

2. State the basic properties of joint distribution of YX , where X and Y are random variables.


II Year / IV Sem 21

Statement:

Properties of joint distribution of YX , are

yxfyx

FyxfofcontinuityofspoAtv

caFcbFdaFdbFdYcbXaPiv

cxFdxFdYcxXPiii

yaFybFyYbXaPii

FandxFyFi

,,,int

,,,,,

,,,

,,,

1,,0,

2

3. Can the joint distributions of two random variables X and Y be got if their marginaldistributions are random?Solution:

If the random variables X and Y are independent then the joint distributions of two randomvariables can be got if their marginal distributions are known.

4. Let X and Y be two discrete random variable with joint pmf

otherwise

yxyx

yYxXP,0

2,1;2,1,18

2, . Find the marginal pmf of X and XE .?

Solution:

The joint PMF of YX , is given by

Marginal pmf of X is

9

4

18

8

18

5

18

31 XP ,

9

5

18

10

18

6

18

42 XP

.

9

14

9

10

9

4

9

52

9

41xpxXE

5. Let X and Y be integer valued random variables with

............,2,1,,, 22 mnpqnYmXP nm and 1 qp . Are YandX independent?

Solution:

1 2

1

18

3

18

4

2

18

5

18

6


II Year / IV Sem 22

The marginal PMF of X is

1112

1123212

1

112

1

112

1

22

1.........1

mm

mm

n

nm

n

nm

n

nm

pqqpq

ppqppppq

ppqppqpqxp

The marginal PMF of Y is

1112

1123212

1

112

1

112

1

22

1.........1

nn

nn

m

mn

m

nm

m

nm

pqqpq

ppqppppq

ppqppqpqyp

nYmXPpqpqpqypxp nmnm 2211 .

Therefore YandX are independent random variables.

6. The joint probability density function of the random variable YX , is given by

0,0,,22

yxeyxkyxf yx . Find the value of k. (Nov./Dec.2013)

Solution:Given yxf , is the joint pdf , we have

1, dydxyxf put tx 2

10 0

22

dydxeyxk yx dtdxx 2

10 0

22

dydxeeyxk yx

2

dtdxx

10 0

22

dydxexeyk xy txwhenandtxwhen ,0,0

120 0

2

dydt

eeyk ty

12 0

0

2

dyeeyk ty


II Year / IV Sem 23

1102 0

2

dyeyk y

tywhenandtywhen

dtdyy

dtdyy

typut

,0,02

2

2

414

1104

14

1202 0

kkk

ekdtte

k t .

7. The joint PDF of the random variable YX , is

otherwise

yxyxkyxf

,0

20;20,, .

Find the value of k .


8

11

222

12210202

12

11,

2

0

220

2

0

2

0

2

0

20

2

0

22

0

2

0

ky

yk

dyykdyyk

dyxyx

kdydxyxkdydxyxf

8. The joint pdf of the random variable YX , is

otherwise

yxyxcyxf

,0

20;20,, . Find the

value of c .


2

0

2

0

2

0

22

0

2

0

10212

11, dyycdyx

ycdydxyxcdydxyxf

4

1141

22

2

0

2

cc

yc

9. If two random variables YandX have probability density function yxkyxf 2, for

3020 yandx . Evaluate k .

Solution: Given yxf , is the joint PDF, we have


II Year / IV Sem 24

21

11211912

12

24124

12

2121,

3

0

230

3

0

3

0

20

2

0

23

0

2

0

kkk

yykdyyk

dyxyx

kdydxyxkdydxyxf

10. If the function 10,10,11, yxyxcyxf is to be a density function, find the value

of c.

Solution:

Given yxf , is the joint PDF, we have

414

14

1

2

11

22

1

2

1

122

11

22

11

122

111111,

1

0

210

1

0

1

0

1

0

1

0

210

1

0

210

1

0

1

0

1

0

1

0

cc

cy

yc

dyy

cdyy

yc

dyx

yxyx

xc

dydxxyyxcdydxyxcdydxyxf

Therefore the value of c is 4c

11. Find the marginal density functions of YandX if 10,10,525

2, yxyxyxf .

Solution:

Marginal density of X is

10,15

4

2

52

5

2

252

5

252

5

2,

1

0

210

1

0

xxx

yyxdyyxdyyxfxf X

Marginal density of Y is


II Year / IV Sem 25

10,25

251

5

2

52

25

252

5

2, 1

0

1

0

21

0

yyy

xyx

dxyxdxyxfyfY

12. If YandX have joint pdf

otherwise

yxyxyxf

;0

10,10;, . Check whether YandX are

independent.

Solution:

10,2

1

2,

1

0

210

1

0

xx

yyxdyyxdyyxfxf X

10,2

1

2, 1

0

1

0

21

0

yyxy

xdxyxdxyxfyfY

yxfyxyx

xyyxyfxf YX ,4

1

222

1

2

1.

Therefore YandX are not independent variables.

13. If YandX are random variables having the joint density function

42,20,68

1, yxyxyxf , find 3YXP .

Solution:

3 ,

333 3 21 1 36 68 8 2

02 0 20

P X Y f x y dx dy

yyxyx y dx dy y x dy

3 31 1 1 12 26 3 3 18 9 38 2 8 2

2 2

33 332 3 31 1

18 98 2 3 2 3

2 2 2

1 9 1 118 3 2 9 4 27 8 0 1

8 2 3 6

1 45 19 1 518

8 2 3 6 24

2

32

y y y dy y y y dy

yy yy


II Year / IV Sem 26

14. Let YandX be continuous random variable with joint pdf

10,10,2

3, 22 yxyxyxf XY . Find yxf YX .

Solution:

2

1

2

3

3

1

2

3

32

3

2

3, 221

02

1

0

31

0

22

yyxy

xdxyxdxyxfyfY

3

1

3

1

2

32

3,

2

22

2

22

y

yx

y

yx

yf

yxfyxf

YYX .

15. If the joint pdf of YX , is given by 10;2, yxyxyxf , find XE .

Solution:

8

1

24

3

24

5

3

1

46

5

36

5

23

2322

22,

1

0

41

0

31

0

1

0

3233

2

1

0 0

2

0

3

0

2

1

0 0

21

0 0

yydyyydy

yyy

dyx

yxx

dydxxyxxdydxyxxdydxyxfxXE

yyy

yy

16. Let YandX be random variable with joint density function

otherwise

yxyxyxf XY ,0

10,10;4, . Find YXE .

Solution:

.9

4

3

1

3

4

33

4

3

4

3444,

1

0

31

0

2

1

0

1

0

32

1

0

1

0

221

0

1

0

ydyy

dyx

ydydxyxdydxyxxydydxyxfxyXYE

17. Let YandX be any two random variables and ba , be constants. Prove that

YXabbYXaCov ,cov, Solution: YEXEXYEYXCov ,


II Year / IV Sem 27

YXCovba

YEXEXYEabYEbXEaYXEab

bYEaXEbYaXEYbXaCov

,

,

18. If 32 XY , find YXCov , .Solution:

XVarXEXE

XEXEXEXE

XEXEXXE

XEXEXXE

YEXEXYEYXCov

22

3232

3232

3232

,

22

22

2

19. If 1X has mean 4 and variance 9 while 2X has mean 2 and variance 5 and the two are

independent, find 52 21 XXVar .Solution:

Given 9,4 11 XVarXE

5,2 22 XVarXE

41536594452 2121 XVarXVarXXVar

20. Find the acute angle between the two lines of regression.Solution:

The equations of the regression lines are

2

1

yyy

xrxx

xxx

yryy

Slope of line 1 isx

yrm

1

Slope of line 2 isxr

ym

2

If is the acute angle between the two lines, then


II Year / IV Sem 28

22

2

2

22

2

2

2

2

21

21

1

1

1

1

.1

1tan

yxr

yxr

x

yx

x

y

r

r

x

y

x

y

r

r

xr

y

x

yr

xr

y

x

yr

mm

mm

21. If YandX are random variables such that bXaY where a and b are real constants, show

that the correlation co-efficient YXr , between them has magnitude one.Solution:

Correlation co-efficient YX

YXCovYXr

,

,

222

22

2

,

XaXVaraXEXEa

XEbXEaXEbXEa

bXEaXEXbXaE

bXaEXEbXaXEYEXEXYEYXCov

222 YEYEY

222222 2 bXEabXabXaEbaXEbaXE

222222 22 bXEabXEabXEabXEa

222222XaXVaraXEXEa

Therefore XY a and 1.

,2

XX

X

a

aYXr

Therefore, the correlation co-efficient YXr , between them has magnitude one.

22. If X and Y are two independent random variables with variances 2 and 3, find the variance of3X+4Y (May/ June 2013)

Solution: (3 4 ) 9 ( ) 16 ( ) 9(2) 16(3) 66Var X Y Var X Var Y

23. If the joint pdf of (X,Y) is given by 10,2),( yxyxf , Find E(X) (May/ June 2013)

Solution:

The marginal density function of X is )1(2),()(1

xdyyxfxfx


II Year / IV Sem 29

3

1)1)(2()(()(

1

0

1

0

dxxxdxxfxXE

24. When will the two regression lines be (A) at right angles (b) coincident? (Nov./ Dec. 2012)Solution:If 1r , the regression lines will coincide.If 0r , the regression line will be at right angle to each other.

25. A small college has 90 male and 30 female professors. An ad-hoc committee of 5 is selected atrandom to unite the vision and mission of the college. If X and Y are the number of men andwomen in the committee, respectively. What is the joint probability mass function of X and Y?

(Nov./ Dec. 2012)

X=No. ofmen

Y= No. offemale

Probability

X=5, Y=0 5

5 12

95

C

X=4, Y=1 14

4 12

3

12

95

C

X=3, Y=2 23

3 12

3

12

95

C

X=2, Y=3 32

2 12

3

12

95

C

X=1, Y=4 41

1 12

3

12

95

C

X=0, Y=5 50

0 12

3

12

95

C

26. Find the value of ‘k’ if the joint density function of (X, Y) is given by

( , ) (1 )(1 ), 0 4, 1 5f x y k x y x y (May/ Nov. 2014)4 5

0 1

1(1 )(1 ) 1

32k x y dxdy k

27. Given the joint probability density function of (X,Y) as1

( , ) , 0 2, 0 36

f x y x y

Determine the marginal density. (May/June 2014)


II Year / IV Sem 30

2 3

0 0

1 1 1 1( ) ; ( )

6 3 6 2f y dx f x dy

28. The joint pdf of a two dimensional random variable (X,Y) is given by

, 0 2, 0( , )

0,

ykxe x yf x y

otherwise

Find the value of ‘k. (April / May 2015)(Nov./Dec.2014)

2 2

0 0 0 0

11

2y ykxe dy dx k x dx e dy k

29. In a partially destroyed laboratory record of an analysis of correlation data, the following

results only are legible: Variance of x = 9; Regression equations are8 10 66 0 40 18 214 0X Y and X Y . What are the mean values of X and Y?

(April/ May 2015)

Solving the equations8 10 66 0

40 18 214 0

X Y

X Y

Mean value of X is 13 and the mean value of Y is 17.30. The joint probability mass function of a two dimensional random variable (X,Y) is given by

( , ) (2 3 ); 0,1, 2; 1, 2,3.p x y k x y x y Find the value of k. (April/ May 2015)

XY

0 1 2

1 3k 5k 7k2 6k 8k 10k3 9k 11k 13k

1

72k

31. What do you mean by correlation between two random vairble ? (April/ May 2015)When the random variables X and Y are correlated, the correlation coefficient between X andY lies between -1 and +1If the correlation coefficient is 0, then the random variable are uncorrelated.If the correlation coefficient is -1, the random variables are negatively correlated.If the correlation coefficient is +1, the random variables are positively correlated.

32. Given the two regression lines 3 12 19 3 9 46X Y and Y X , Find the coefficient of

correlation between X and Y ? (Nov./Dec. 2015)

From the first equations1

4yxb

and from the second equation1

3xyb

Then the correlation coefficient between x and y is1

0.0812xy xy yxb b

33. The joint probability density function of bivariate random variable (X,Y) is given by( , ) 4 , 0 1, 0 1f x y xy x y , Find ( 1)P X Y (Nov./Dec. 2015)


II Year / IV Sem 31

1 1

0 0

1( 1) 4

6

x

P X Y xy dy dx

34. Determine the value of the constant ‘c’ if the joint density function of two discrete random

variables X and Y is given by ( , ) , 1, 2,3 1, 2,3p m n cmn m and n (Nov./Dec. 2015)

35. The lines of regression in a bivariate distribution are49

9 7, 43

X Y and Y X , Find the

correlation coefficient? (Nov./Dec. 2015)

36. Comment on the statement: “If COV(X,Y)=0,then X and Y are uncorrelated”. (Nov./ Dec.2014)Since X and Y are uncorrelated, the correlation coefficient between them is zero. Therefore x and Y areindependent random variables.

37. The joint pdf of RV (X,Y) is given as1

( , ) , 0 1f x y y xx , Find the marginal pdf of Y.

(May/ June 2016)1 1 1

( ) logy

y

f y dxx y

38. Let X and Y be two independent random variables with Var (X)=9, Var(Y)=3, Find theVar(4X-2Y+6) (May/ June 2016)

(4 2 6) 16 ( ) 4 ( ) 16 cov( , )

16(9) 4(3) 0 144 12 156

Var X Y Var X Var Y X Y

PART-B

1. Let X andY have the joint pdf

Y

X

0 1 2

0 0.1 0.4 0.1

1 0.2 0.2 0

Find

(i) 1YXP

(ii) the probability mass function xXP of the RV X

(iii) 11 XYP

(iv) XYE (AP)(Apr/May 2008)

2. The joint density function of the random variable YX , is given by


II Year / IV Sem 32

elsewhere

xyxxyyxf

,0

0,10,8),(

(i) Find the marginal density ofY

(ii) Conditional density of yYX

(iii)

2

1XP (AP)(May/Jun 2007)

3. The joint probability density function of a two dimensional random variable YX , is given by

10,20;8

),(2

2 yxx

xyyxf

Compute 1)(,,12

1,

2

11,

2

1,1

YXPYXPXYP

Y

XPYPXP .

(AP) (May/Jun 2009) (Nov/Dec 2007)4. Given )(),( yxcxyxf , 20 x , xyx and ‘0’ elsewhere. Evaluate ‘c’ find

xf X and yfY (AP)(Nov/Dec 2010)

5. In producing gallium – arsenide microchips, it is known that the ratio between gallium andarsenide is independent of producing a high percentage of workable water, which are maincomponents of microchips. Let X denote the ratio of gallium to arsenide andY denote thepercentage of workable micro wafers retrieved during a 1 hour period. X andY are independentrandom variables with the joint density being known as

otherwise

yxyx

yxf

0

10,204

)31(),(

2

Show that YEXEXYE . (AP)(Nov/Dec 2006)

6. Given the joint density function

elsewhere

yxyx

yxf

0

10,204

)31(),(

2

, Find the marginal

densities )(,)( yhxg and the conditional density )/( yxf and evaluate

3/1/

2

1

4

1YxP (AP)(Apr/May2011)

7. The joint pmf of YX , is given by .3,2,1;2,1,0),32(),( yxyxkyxp Find all the

marginal and conditional probability distributions. Also find the probability distribution

of YX . (AP)(Nov/Dec 2007) (Nov/Dec 2011)


II Year / IV Sem 33

8. Let X and Y be two random variables having the joint probability function).32(),( yxkyxf where x and y can assume only the integer values 0, 1 and 2. Find all

the marginal and conditional distributions. (AP)(April/May 2012)

9. If the joint density of 21 , XX is given by

otherwise

xxforexxf

xx

0

0,06),( 21

23

21

21

find the

probability density of 21 XXY . (AP)(Nov/Dec 2006)

10. Let X and Y be independent random variables, both uniformly distributed on (0, 1). Calculatethe probability density of YX . (AP)(April/May 2010)

11. Two random variables X and Y have the joint density

function 20,103

),( 2 yxxy

xyxf XY . Find the conditional density functions.

Check whether the conditional density functions are valid. (AP)(Nov/Dec 2006)12. Suppose that X and Y are independent non-negative, continuous random variables having

densities xf X and yfY respectively. Compute YXP (AP)(April/May 2010)

13. The joint density of X and Y is given by

otherwise

yxyeyxf

xy

0

20,02

1),( .

Calculate the conditional density of X given 1Y . (AP)(April/May 2010)14. Determine whether the random variables X and Y are independent , given their joint

probability density function as

otherwise

yxxy

xyxf

0

20,10,3),(

2

(April/May 2011)15. Can YXXY 5.03,8.25 be the estimated regression equations ofY on X and X on

Y respectively? Explain your answer with suitable theoretical arguments.(AP)(Nov/Dec 2007)

16. Two random variables X andY have the following joint probability density function

otherwise

yxyxyxf

0

10,102),(

Find(a) Marginal probability density functions of X andY(b) Conditional density functions

(c) XVar and YVar . (AP)(May/June 2006)

17. Two random variables X andY have the following joint probability density function

elsewhere

yxyxcyxf

0

20,20)4(),( Find YX ,cov . Find the equations of two

lines of regression.(AP)(Apr/May 2012) (Nov/Dec. 2015)


II Year / IV Sem 34

18. If X and Y are independent random variables with pdf’s xe , 0x and ye , 0y

respectively, find the density functions ofYX

XU

and YXV . Are U and

V independent?(AP) (Nov/Dec 2011)19. Let YX , be a two dimensional non negative continuous random variable having the joint

density

elsewhere

yxxyeyxf

yx

0

0,0,4),(

)( 22

Find the density function of 22 YXU

(AP) (May/Jun 2006)20. Two dimensional random variable YX , have the joint probability density function

elsewhere

yxxyyxf

0

10,8),( (i) Find

4

1

2

1YXP , (ii) Find the marginal and

conditional distributions. (iii) Are X and Y independent. (AP)(April/May 2012)

21. If the joint probability density of 21 , XX is given by

elsewhere

xxforexxf

xx

0

0,0),( 21

)(

21

21

. Find the probability of21

1

XX

XY

(AP)(Nov/Dec 2006)22. If X and Y are independent random variables having density functions

00

0,2)(

2

x

andxexf

x

00

0,3)(

3

y

yeyf

y

respectively, find the density

functions YXz . (AP)(Apr/May 2011)23. If the joint distribution function of X andY is given by

otherwise

yxforeeyxF

yx

0

0,0)1)(1(),(

(i) Find the marginal densities of X andY(ii) Are X andY independent

(iii) Find )21,31( YxP (AP)(Nov/Dec 2008)

24. Let X be a random variable with pdf xexf x ,

2

1)( 22

, find the pdf of the RV

2XY . (AP)(Apr/May 2008)25. Compute the co-efficient of correlation between X andY using the following data

(AP)(Nov/Dec 2010)

X 1 3 5 7 8 10

Y 8 12 15 17 18 20

26. If the correlation coefficient is 0, then can we conclude that they are independent? Justify youranswer through an example. What about the converse? (AN)(April/May 2010)


II Year / IV Sem 35

27. Find the coefficient of correlation between industrial production and export using the followingdata: (AP)(Nov/Dec 2008)

Production( X ) 55 56 58 59 60 60 62

Export(Y ) 35 38 37 39 44 43 44

28. Obtain the equations of the regression lines from the following data, using the method of leastsquares. Hence find the coefficient of correlation between X andY . Also estimate the valueofY when 38X and the value of X when 18Y . (AP)(May/Jun2009)(April/May 2015)

X 22 26 29 30 31 33 34 35

Y 20 20 21 29 27 24 27 31

29. Calculate the correlation coefficient for the following data: (AP) (May/Jun 2007) (Nov/Dec2007)

X 65 66 67 67 68 69 70 72

Y 67 68 65 68 72 72 69 71

30. Find the coefficient of correlation and obtain the lines of regression from the data given below:(AP)(Nov/Dec 2006)

X 50 55 50 60 65 65 65 60 60 50

Y 11 14 13 16 16 15 15 14 13 13

31. Find the coefficient of correlation and obtain the lines of regression from the data given below:(AP)(May/Jun 2006)

X 62 64 65 69 70 71 72 74

Y 126 125 139 145 165 152 180 208

32. Let the random variable X have the marginal density2

1

2

1,1)(1 xxf and let the

conditional density ofY be

2

1011

02

1,11

xxyx

xxyxxyf

. Show that the variables are

uncorrelated. (AP)(May/Jun 2006)

33. Let z be a random variable with probability density2

1)( zf in the range 11 z . Let the

random variable zX and the random variable 2zY . Obviously X andY are not

independent since YX 2 . Show, none the less, that X andY are uncorrelated. (AP)(Nov/Dec 2006)


II Year / IV Sem 36

34. Two random variables X andY are defined as 94 XY . Find the correlation coefficientbetween X andY . (AP)(Nov/Dec 2006)

35. If X andY are independent exponential random variables each with parameter 1, find the pdf ofYXU . (AP)(May/Jun 2007)(May/ June 2013) (Nov/Dec 2015)

36. If X is any continuous RV having the pdf

otherwise

xxxf

,0

10,2)( and XeY , find the pdf of

RV Y . (AP)(Apr/May 2008)

37. Find 42 YXP when the joint pdf of X andY is given by

otherwiseo

yxeyxg

yx 0,0,),(

)(

.

Are X andY independent RVs? Explain. And find the pdf of the RVY

XU (AP)

(Apr/May 2008)(May/ June 2016)

38. If the joint pdf of the RVs X andY is given by

otherwise

yxyxf

0

10,1),( find the pdf of

the RVY

XU (AP)(Apr/May 2008)

39. The two lines of regression are02141840

066108

yx

yx

.The variance of X is 9.

Find the mean values of X andY , Correlation coefficient between X andY . (AP)(Nov/Dec

2008)40. For two random variables X andY with the same mean, the two regression equations are

baxy and dcyx . Find the common mean, ratio of the standard deviations and also

show thatc

a

d

b

1

1. (AP)(Nov/Dec2010)

41. The joint probability density function of a two dimensional random variable (X,Y) is

42,20,8

6),(

yx

yxyxf .

Find )3/1()3()3()2()31()1( YXPYXPYXP (AP) (May/ June 2013)

42. The marks obtained by 10 students in Mathematics and Statistics are given below. Find thecorrelation coefficient between the two subjects (AP) (May/ June 2013)

Marks in Mathematics 75 30 60 80 53 35 15 40 38 48

Marks in Statistics 85 45 54 91 58 63 35 43 45 44

43. A distribution with unknown mean has variance equal to 1.5, Use central limit theorem to

find how large a sample should be taken from the distribution in order that the probability willbe atleast 0.95 that the sample mean will be within 0.5 of the population mean. (AP)

(May/ June 2013)


II Year / IV Sem 37

44. Obtain the equation of the lines of regression from the following data: (AP) (Nov./ Dec. 2012)X: 1 2 3 4 5 6 7Y: 9 8 10 12 11 13 14

45. The joint pdf of random variable X and Y is given by

otherwise

yxxyyxf

;0

10,),(

2

(1) Determine the value of (2) Find the marginal probability density function of X and Y(3) Find the conditional pdf f(x/y) (AP) (May/ June 2016)(Nov./ Dec. 2012)

46. The regression equations of X and Y is 010853 xy . If the mean value of Y is 44 and the

variance of X were th16

9of the variance of Y. Find the mean value of X and the correlation

coefficient. (AP) (Nov. / Dec. 2012)47. Let 10021 ,....,, XXX be independent identically distributed random variables with 2

And4

12 . Find )210...192( 10021 XXXP . (AP) (Nov./Dec.2012)

54. Let X and Y be random variables having joint density function

2 23( ), 0 1, 0 1

( , ) 20,

x y x yf x y

elsewhere

Find the correlation coefficient xy (AP)(Nov./Dec.2013)

48. The joint distribution of X and Y is given by ( , ) , 1,2,3, 1,221

x yf x y x y

. Find the

marginal distributions and conditional distributions. (AP) (Nov./Dec.2013) (Nov/Dec. 2015)

49. If the pdf of ‘X’ is ( ) 2 , 0 1f x x x , find the pdf of Y= 3X+1 (AP) (Nov./Dec.2013)

50. The joint probability density function of two random variables X and Y is given b

26( , ) , 0 1, 0 2

7 2

xyf x y x x y

, Find the conditional density function of X

given Y and the conditional density function of Y given X (AP)(May/ June 2014)51. If the independent random variables X and Y have the variances 36 and 16 respectively. Find

the correlation coefficient uvr , where U =X+Y and V= X-Y (AP)(May/ June, 2014)

52. The joint probability density function of two random variables X and Y is2 2( , ) ( ) ( ) , 0 ( , ) 1f x y k x y x y x y , Show that X and Y are uncorrelated but

not independent? (AP)(May/ June 2014)53. Calculate the coefficient of correlation for the following data: (AP)

X: 9 8 7 6 5 4 3 2 1Y: 15 16 14 13 11 12 10 8 9 (Nov./Dec. 2014)


II Year / IV Sem 38

54. IF nXXX ,....,, 21 are Poisson variates with parameter 2 , Use the central limit theorem to

estimate (120 160)nP S where 1 2 ... , 75n nS X X X n (AP)(Nov./Dec.2014)

55. The joint probability density function of a two dimension random variable (X,Y) is given by2

2( , ) ; 0 2; 0 18

xf x y xy x y .Compute

1 1 1( 1), ( ), 1/ , / 1 , ( ) ( 1)

2 2 2P X P Y P X Y P Y X P X Y and P X Y

(AP)(April/ May 2015)56. Find the equation of the regression line Y on X from the following data:

X: 3 5 6 8 9 11Y: 2 3 4 6 5 8 (AP)(April/ May 2015)

57. Assume that the random variable X and Y have the joint

PDF 31( , ) ; 0 2; 0 1

2f x y x y x y Determine if X and Y are independent (April/ May

2015) (AP)

58. The joint pdf of the random variable X and Y is defined as 5( , ) 25 ; 0 0.2, 0yf x y e x y (1) find the marginal PDFs of X and Y (2) what is the covariance of X and Y? (April/ May2015) (AP)

59. Find the constants k such that ( , ) (1 ) , 0 1, 0yf x y k x e x y is the joint pdf of the

continuous random variable (X, Y), Are X and Y independent r.v’s Explain. (May/ June 2016)(AP)

60. The joint distribution of X and Y is given by ( , ) , 1,2, 1,2,3,432

x yf x y x y

. Compute

the covariance of X and Y . (AP)(May/ June 2016)

61. Let the joint pdf of (X,Y) be given as 2( , ) , 0 1f x y Cxy x y , Determine the value of C ,

Find the marginal pdf of X and Y and find the conditional pdf f(x/ y) (AP)(May/ June 2016)

62. If X and Y are independent random variables with pdf’s xe , 0x and ye , 0y respectively,

find the density functions ofYX

XU

. (AP) (Nov/Dec 2011) (Nov/Dec 2015)

63. The probability density function of X and Y is given by

26( , ) ,0 1, 0 2

7 2

xyf x y x x y

(1) compute the marginal density function of X and Y,

(2) Find E(X) and E(Y) (3) Find1 1

, ,0 1, 0 22 2

P X Y x y

. (AP)(Nov/Dec 2015)

64. If X, Y and Z are uncorrelated random variables with zero means and standard deviation 5, 12and 9 respectively and if U = X + Y and V = Y + Z, find the correlation coefficient between Uand V. (AP)(Nov/Dec 2015)

65. If X1, X2,…… Xn are Poisson variates with parameter 2 , use CLT to estimate P(120 < Sn

< 160) where Sn = X1 + X2 +,……+ Xn and n = 75. (AP)Nov/Dec 2015)


II Year / IV Sem 39

COURSE OUTCOME: Acquire skills in handling more than one random variable and correlationbetween the random variables.

UNIT III

CLASSIFICATION OF RANDOM PROCESSES

SYLLABUS: Classification – Stationary process – Markov process - Poisson process – Discreteparameter Markov chain – Chapman Kolmogorov equations – Limiting distributions.

COURSE OBJECTIVE: Ability to analyze the relation between random input and output signalsusing the basics of random process and its characteristics and to solve problems and model situationsusing techniques of Markov process. Have an ability to design a model or a process to meet desiredneeds within realistic constraints such as environmental conditions

PART A

1. State the four types of stochastic processes.The four types of stochastic processes are

1. Discrete random sequence2. Continuous random sequence3. Discrete random process4. Continuous random process

2. Give an example for a continuous time random process.Solution:If )(tX represents the maximum temperature at a place in the interval (0, t), )(tX is a

continuous random process.

3. Define a stationary process.Solution:If certain probability distribution or averages do not depend on t, then the random process

)(tX is called a stationary process.

4. Give an example for a stationary process.A Bernoulli process is a stationary process.

5. Give an example of stationary process and justify your claim.Solution:A Bernoulli process is a stationary stochastic process as the joint probability distributions areindependent of time.

6. Define strict sense and wide sense stationary process. (May/ June 2013)(May/June ,2014)


II Year / IV Sem 40

Solution:A random process is called a strict sense stationary process or strongly stationary process if allits finite dimensional distributions are invariant under translation of time parameter.

A random process )(tX with finite first and second order moments is called a weakly

stationary process or covariance stationary process or wide-sense stationary process if its meanis a constant and the auto correlation depends only on the time difference. i.e, if )]([ tXE

and )()]()([ zRztXtXE

7. Give an example for strict sense stationary process.Bernoulli’s process is an example for strict sense stationary random process.

8. Prove that a first order stationary random process has a constant mean.Proof:

)()( htxftxf as the process is stationary.

)()()()]([

)()()]([

htdhtxfhtxhtXE

dttxftxtXE

( )

( ) ( )

( )

[ ( )] [ ( )]

Put t h u d t h du

x u f x u du

E X u

E X t h E X t

Therefore, )]([ tXE is independent of t.

)]([ tXE is a constant.

9. What is a Markov process?Solution:Markov process is one in which the future value is independent of the past values, given thepresent value.

10. Give an example of a Markov process.Solution:Poisson process is a Markov process. Therefore, number of arrivals in (0,t) is a Poisson processand hence a Markov process.

11. Define Markov chain and one – step transition probability.Solution:


II Year / IV Sem 41

If

110022

11,......,,

nn

nnnn

nn

nnaX

aXPaXaXaXaXPn then the

process nX , n=0,1,2,…. is called a Markov chain.

The conditional probability

in

jn

aXaX

P1

is called the one step transition probability

from state ia to state ja at the thn step.

12. Describe a random walk process. Is it a Markov process?Solution:Suppose a person tosses a fair coin every T seconds and instantly after each toss, he moves adistance d to the right if heads show and to the left if tails show. )(nTX is the position of the

person after n tosses

Then the process )(nTX is a random walk process. The random walk process is a Markov

process.

13. Define Poisson process.Solution:If )(tX represents the number of occurrences of a certain event in (0,t), then the discrete

process )(tX is called the Poisson process.

14. What is homogeneous Poisson process?Solution:

The probability law for the Poisson process is

,.....2,1,0,!

)()( n

n

tentXP

nt

when is

a constant, the Poisson process is called a homogeneous Poisson process.

15. State the postulates of Poisson process.The postulates of Poisson process are

(i) )(1),(0 tOttttinoccurenceP

(ii) )(),(2 tOtttinoccurencesmoreorP (iii) )(tX is independent of the number of occurrences of the event in any interval prior

to and after the interval (0,t).(iv) The probability that the event occurs a specified number of times in

),( 00 ttt depends only on t , but not on 0t

16. State any two properties of Poisson process.

(i) The Poisson process is a Markov process.(ii) Sum of two independent Poisson processes is a Poisson process.


II Year / IV Sem 42

(iii) Difference of two independent Poisson processes is not a Poisson process.

17. Prove that the sum of two independent Poisson processes is also Poisson.Solution:

Let )()()( 21 tXtXtX

nn

t

nn

t

nnnnn

t

n

r

rnrnrt

n

r

rnrnrrtt

rntn

r

rt

n

r

n

te

n

te

nCnCn

te

tn

nCe

rnr

ttee

rn

te

r

te

rntXPrtXP

ntXtXPntXP

)(!

)(!

.........!

!

)!(!

)!(

)(

!

)(

)()(

)()()(

21)(

21)(

121

2221

1212

)(

021

)(

0

21

2

0

1

20

1

21

21

21

21

21

21

21

=!

))(( 21)( 21

n

te

nt

Therefore, )()( 21 tXtX is a Poisson process with parameter t)( 21 .

18. Prove that the difference of two independent Poisson processes is not a Poisson process.Solution:

Let )()()( 21 tXtXtX

2( ) ( ) ( ) ( ) ( ) ( )1 2 1 2 1 1 2E X t E X t X t E X t E X t t t t

221

2 )()()( tXtXEtXE

19. Let )(tX be a Poisson process with rate . Find )()( tXtXE (or) Derive the auto

correlation of the poisson process (May/ June 2016)Solution:


II Year / IV Sem 43

( ) ( ) ( ) ( ) ( ) ( )

2( ) ( ) ( ) ( )

E X t X t E X t X t X t X t

E X t X t X t E X t

2 2( ) ( ) ( ) ( ) ( ) ( ) ( )

2 2 2 2 2 2 ( )

E X t E X t X t E X t E X t E X E X t

t t t t t t t t t

)()( tXtXE ttt )(2

20. For a Poisson process with parameter and for ts show that

nkt

s

t

snCntNksNP

knk

k ,...2,1,0,1)()(

Proof:

( ) ( )( ) ( )

( ) ( )

( )( ) ( ( ))( ) ! ( )!

( ) ( )!

( )!!( )!

P N s k N t n P N s k N t s n kP N s k N t n

P N t n P N t n

s k t s n ke s e t sP N s k P N t s n k k n k

t nP N t n e tn

s k k t s n k n ke s e e t snk n k e

1 1

1

t n ntn k n ks sn k n k k n ks t s t t

t tnC nCk kn n nt tk n ks snCk tkt

( ) ( ) 1 , 0,1,2,....k n ks sP N s k N t n nC k nk t t

21. For the sine wave process ;,cos)( ttYtX constant, the amplitude Y is a

random variable with uniform distribution in the interval 0 to 1. Check whether the process isstationary or not.Solution:Given Y is a random variable with uniform distribution in the interval 0 to 1, then

101)( yyfY

Also given tYtX cos)(

0 1 1( ) cos cos 1 cos cos2 2

E X t E Y t tE t t


II Year / IV Sem 44

Since the mean is time dependent, the process is not stationary.

22. If the TPM of a Markov chain is

21

21

10, find the steady state distribution of the chain.

(May/ June 2013).

Solution:

3

23

1

12

12

110

2

1

21

2121

23. If N(t) is the poisson process, then what can you say about the time we will wait for the firstevent to occur? And the time we will wait for the nth event to occur? (May/ June 2013)Solution:

,.....2,1,0,!

)()( n

n

tentNP

nt

Where N(t) is the number of occurrences in (0, t)

24. Is poisson process stationary? Justify? (May/ June 2013) (April/ May 2015)

Solution:

It is a non-stationary process. ttXE ))(( is the mean of the poisson process, which depends

on time.

25. Prove that the first order stationary random process has a constant mean.(Nov./Dec. 2013)

( ( )) ( , )

( , ) [ ( )]

E X t x f x t dx

x f x t dx E X t

26. Prove that Poisson process is a Markov process. (Nov./Dec.2013)


II Year / IV Sem 45

3 23 2

3 3 2 2 1 1

3 3 2 2 1 1

2 2 1 1

( )3 2

3 2

3 3 2 2

( ) / ( ) , ( )

( ) , ( ) , ( )

( ) , ( )

( )

( )!

( ) / ( )

n nt t

P X t n X t n X t n

P X t n X t n X t n

P X t n X t n

e t t

n n

P X t n X t n

27. A gambler has Rs.2. He bets Re.1 at a time and wins Re.1 with probability ½. He stops playingif he loses Rs.2 or wins Rs. 4. What is the transition probability matrix of the related markovchain? (May/ June 2014)

1 12 2

1 12 2

1 12 2

1 12 2

1 12 2

1 0 0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0 0 1

28. What is a random process? When do you say a random process is a random variable? (April/May 2015)

A random process is a collection of random variables indexed by the time set T , i.e. ( , )X t s ,

When t is fixed, X(t,s) will become X(s) – a random variable.

29. A radioactive source emits particles at a rate of 5 per min. in accordance with Poisson process.Each particle emitted has a probability 0.6 of being recorded. Find the probability that 10particles are recorded in 4 min period. (Nov./ Dec. 2014)

(5(0.6)(4) 10

12 10

(5(0.6)(4))( (4) 10)

10!

(12)

10!

eP X

e

30. Check whether the Markov chain with transition probability matrix

0 1 0

1 102 20 1 0

P

is

irreducible or not? (Nov./Dec.2014)


II Year / IV Sem 46

Given TPM is an irreducible matrix, because each state is reached from the remaining states insome non-zero number of steps.

31. The random process X(t) is given by ( ) cos(2 ), 0X t Y t t Where Y is a random variable

with E(Y)=1. Is the process X(t) stationary? (May/ June 2016)

( ) cos(2 ), 0

( ( )) ( ) cos(2 ) cos(2 )

' '

X t Y t t

E X t E Y t t

which depends on t

X(t) is not a stationary process

32. When a Markov chain is called homogeneous? (Nov./Dec. 2015)

33. Consider a random process ;,cos)( tttX where is a constant and is uniform

variable in 0,2

, show that X(t) is not WSS? (Nov./Dec. 2015)

34. Consider a markov chain with state {0, 1, 2} and TPM

1 10 2 21 102 21 0 0

P

, Draw the

transition diagram. (Nov. Dec. 2015)

PART-B

1. If customers arrive at a counter in accordance with a Poisson process with a mean rate of 2/min,find the probability that the interval between 2 consecutive arrivals is more than 1 min, between1 and 2 mins and 4 mins or less. (AP) (Nov/Dec 2010, 2011)

2. At the receiver of an AM radio, the received signal contains a cosine carrier signal at the carrier

frequency 0 with a random phase that is uniformly distributed over )2,0( . The received

carrier signal is )cos()( tAtX o . Show that the process is second order stationary. (AP)

(May/Jun 2007)3. Show that the random process )cos()( 0 tAtX is wide sense stationary if A and 0 are

constants and is uniformly distributed random variable in )2,0( . (Nov/Dec 2007,2011)

(Nov./Dec.2015) (AP)

3

1 2


II Year / IV Sem 47

4. Show that random process ttBtAtX ,sincos)( is a wide sense stationary

process where A and B are independent random variables each of which has a value -2 withprobability 1/3 and a value 1 with probability 2/3. (April/May2011)(April/May2015)(Nov./Dec.2015) (AP)

5. Given a RV with density )(f and another RV uniformly distributed in ),( and

independent of and )cos()( tatX , prove that )(tX is a WSS process. (AP)

(May/Jun 2006) (Nov/Dec 2008)6. Consider a random process )(tX defined by tVtUtX sincos)( where U and V are

independent random variables for which 1)()(;0)()( 22 VEUEVEUE

(1) Find the auto covariance function tX

(2) Is tX wide sense stationary? Explain your answer. (U)(Apr/May 2008)

7. A stochastic process is described by tBtAtX cossin)( where A and B are independent

random variables with zero means and equal standard deviation show that the process isstationary of the second order. (AP)(Nov/Dec 2006)

8. Suppose that a mouse is moving inside the maze shown in the adjacent figure from one cell toanother, in search of food. When at a cell, the mouse will move to one of the adjoining cells

randomly, for 0n , nX be the cell number the mouse will visit after having changed cells

‘n’ times. Is ,...1,0; nX n a Markov chain? If so, write its state space and transition

probability matrix. (AP)1 4 7

2 5 8

3 6 9

9. A raining process is considered as two states Markov Chain. If it rains, it is considered to bestate 0 and if it does not rain, the chain is in state 1. The transition probability of the Markov

chain is defined as

8.02.0

4.06.0P Find the probability that it will rain for 3 days from today

assuming that it will rain after 3 days. Assume the initial probabilities of state 0 and state 1 as0.4 and 0.6 respectively. (AP)(Nov/Dec 2006)

10. Let the Markov chain consisting of the states 0,1,2,3 have the tpm

0010

0010

00012

1

2

100

. Determine

which states are transient and which are recurrent by defining transient and recurrent states.(AP) (April/May 2010)


II Year / IV Sem 48

11. The following is the transition probability matrix of a Markov chain with state space

4,3,2,1,0 . Specify the classes are transient and which are recurrent. Give reasons.

3/203/100

04/3004/1

2/102/100

03/103/13/1

05/3005/2

. (AP) (Nov/Dec 2010)

12. Suppose that whether or not it rains today depends on previous weather conditions through thelasts two days. Show how this system may be analyzed using a Markov chain. How manystats are needed? (AP)(April/May 2010)

13. A raining process is considered as two states Markov Chain. If it rains, it is considered to bestate 0 and if it does not rain, the chain is in state 1. The transition probability of the Markov

chain is defined as

8.02.0

4.06.0P Find the probability that it will rain for 3 days from today

assuming that it is raining today. Find also the unconditional probability that it will rain afterthree days with the initial probabilities of state 0 and state 1 as 0.4 and 0.6 respectively

(AP) (May/Jun 2006)14. A person owning a scooter has the option to switch over to scooter, bike or car next time with

the probability of (0.3, 0.5, 0.2). If the transition probability matrix is

5.025.025.0

3.05.02.0

3.03.04.0

.

What are the probabilities vehicles related to his fourth purchase? (AP)(Nov/Dec 2006)15. Find the limiting-state probabilities associated with the following transition probability matrix

5.02.03.0

4.03.03.0

1.05.04.0

(AP)(April/May 2011)

16. An engineer analyzing a series of digital signals generated by a testing system observes thatonly 1 out of 15 highly distorted signals follows a highly distorted signal, with norecognizable signal between, whereas 20 out of 23 recognizable signals follow recognizablesignals, with not highly distorted signal between. Given that only highly distorted signals arenot recognizable, find the TPM and fraction of signals that are highly distorted. (AP)(May/Jun 2009) (Nov/Dec 2007) (Nov/Dec 2010)(Nov./Dec.2014)(April/ May 2015)

17. A salesman territory consists of three cities A, B and C. He never sells in the same city onsuccessive days. If he sells in city-A, then the next day he sells in city-B. However if he sellsin either city-B or city-C, the next day he is twice as likely to sell in city-A as in the othercity. In the long run how often does he sell in each of the cities? (AP)(April/May 2012)(Nov/Dec.2013)

18. Derive Chapman-Kolmogorov equations. (AN)(April/May 2010)(April/ May 2015)


II Year / IV Sem 49

19. Derive probability distribution of poisson process and hence find its auto correlation function.(AN)(April/May 2011)

20. Show that the difference of two independent poisson processes is not a poisson process. (AN)(April/May 2011)(May/ June 2013)

21. Define Poisson process and derive the Poisson probability law. (AN)(Nov/Dec 2011)22. Three out of every four trucks on the road are followed by a car, while only one out of every

five cars is followed by a truck. What fraction of vehicles on the road are trucks? (AP)(April/May2010)

23. The process )(tX whose probability distribution is given by

.0,1

,...2,1,)1(

)(

)(1

1

nat

at

nat

at

ntXPn

n

Show that )(tX is not stationary. (May/Jun2006)

(Apr/May 2012)(Nov./Dec.2013) (Nov./Dec.2015) (AP)(1) The transition probability matrix of a Markov chain ....3,2,1, nX n having 3 states

1, 2 and 3 is

3.04.03.0

2.02.06.0

4.05.01.0

and the initial distribution is 1.0,2.0,7.0 . Find 32 XP ,

3 2 1 02, 3, 3 2P X X X X . (Nov/Dec2008)(Nov./Dec.2013)(May/June2014)

(Nov./Dec.2015) (AP)24. Assume that a computer system is in any one of the three states: busy, idle and under repair

respectively denoted by 0, 1, 2. Observing its state at 2 pm each day, we get the transition

probability matrix as

4.006.0

1.08.01.0

2.02.06.0

P . Find out the 3rd step transition probability matrix.

Determine the limiting possibilities. (AP)(May/Jun 2007)25. Define stationary transition probabilities. Derive the Chapman- Kolmogorov equations for

discrete time Markov chain. (AN)(Nov/Dec 2007)26. On a given day, a retired English professor Dr. Charles Fish, amuses himself with only one of

the following activities: reading (activity 1), gardening (activity 2), or working on his book

about a river valley (activity 3). For 31 i let iX n if Dr. Fish devotes day ‘n’ to activity

i. Suppose that ....3,2,1, nX n is a Markov chain and depending on which of these

activities on the next day is given by TPM

35.040.025.0

50.010.040.0

45.025.030.0

. Find the proportion of days

Dr. Fish devotes to each activity. (AP)(Nov/Dec 2007)27. Suppose that customers arrive at a bank according to a Poisson process with a mean rate of 3

per minute; find the probability that during a time interval of 2 minutes


II Year / IV Sem 50

(1) exactly 4 customers arrive (2) more than 4 customers arrive (3) Fewer than 4customers arrive (AP) (May/Jun 2006)(Nov./Dec.2013)(April/ May 2015) (Nov./Dec.2015)

28. Queries presented in a computer database are following a Poisson process of rate 6 queriesper minute. An experiment consists of monitoring the database for m minutes and recordingN(m) the number of queries presented.

a) What is the probability that there are no queries in a one minute interval?b) What is the probability that exactly 6 queries arrive in a one minute interval?c) What is the probability of less than 3 queries arriving in a half minute interval?

(AP)(May/Jun 2007).29. Obtain the steady state or long run probabilities for the population size of a birth death process.

(AN)(May/Jun 2007)30. Discuss the pure birth process and hence obtain its probabilities, mean and variance.

(AN)(Apr/May 2008)31. Write a short note on recurrent state, transient state, ergodic state. (R)(Nov/Dec 2008)

32. Let )(tX be a Poisson process with arrival rate . Find stsXtXE 2)()( . (AN)

(Apr/May 2008)33. Let ,...3,2,1; nX n be a Markov chain on the space 3,2,1S with one step transition

probability matrix

0012

10

2

1010

P

(a) Sketch the transition diagram.(b) Is the chain irreducible? Explain(c) Is the chain ergodic? Explain. (AP)(Apr/May 2008)(May/ June 2013)

34. A man either drives a car or catches a train to go to office each day. He never goes 2 days in arow by train but if he drives one day, then the next day he is just as likely to drive again as he isto travel by train. Now suppose that on the first day of the week, the man tossed a fair die anddrove to work iff a 6 appeared. Find (1) the probability that he takes a train on the 3rd day, (2)the probability that he drives to work in the long run. (Nov/Dec 2008) (Nov/Dec 2011)(Nov./Dec.2015) (AP)

35. Show that the process tBtAtX sincos)( is wide sense stationary, if

0)()()(,0)()( 22 ABEandBEAEBEAE where A and B are random variables.(AP)(May/ June 2013)(April/May 2015)

36. Prove that the poisson process is a Markov process. (AN)(May/ June 2013)

37. A fair die is tossed repeatedly. The maximum of the first ‘n’ outcomes is denoted by nX . Is

,....2,1, nX n a Markov chain why or why not? IF it is a Markov chain, calculate its

transition probability matrix. Specify the classes. (AP) (Nov./Dec.2012)(April/May 2015)

38. An observer at a lake notices that when fish are caught, only 1 out of 9 trout is caught afteranother trout, with no other fish between whereas 10 out of 11 non-trout, with no other fishbetween whereas 10 out of 11 non-trout are caught following non-trout, with no trout


II Year / IV Sem 51

between. Assuming that all fish are equally likely to be caught, what fraction of fish in thelake is trout? (AP)(Nov./Dec.2012)(April/ May 2015)

39. The following is the transition probability matrix of a Markov chain with state space

5,4,3,2,1 . Specify the classes are transient and which are recurrent. Give reasons. (AP)

5/305/200

01000

2/102/100

03/203/10

10000

. (Nov/Dec 2012)

40. For an English course, there are four popular textbooks dominating the market. The englushdepartment of an institution allows its faculty to teach only from these 4 text books. Eachyear, prof. Rose Mary O donoghue adopts the same book she was using the previous yearwith probability 0.64. The probabilities of her changing to any of the other 3 books are equal.Find the proportion of years Prof. O’ Donoghue uses each book. (AP)(Nov./ Dec.2012)

41. Explain the steady state probabilities of birth-death process. Also draw the transition graph?(U)(Nov./Dec. 2012)

42. Show that the sum of two independent poisson process with parameter 1 2and is also a

poisson process (U)(May/ June 2014)

43. Find the limiting-state probabilities associated with the following transition probability matrix

0.5 0.4 0.1

0.3 0.4 0.3

0.2 0.3 0.5

45. A soft water plant works properly most of the time. After a day in which the plant is working,the plant is working the next day with probability 0.95. Otherwise a day or repair followed bya day of testing is required to restore the plant to working status. Draw the state transitiondiagram for the status of the plant. Write down the TPM and Classify the status of theprocess. (AP)(Nov./Dec.2014)

46. Suppose that children are born at a poisson rate of five per day in a certain hospital. What isthe probability that (1) atleast two babies are born during the next six hours. (2) no babies areborn during the next two days? (AP)(Nov./Dec.2014)

47. If {N1(t)}and {N2(t)}are two independent Poisson process with parameter 1 and

2 respectively, show that 1 1 2( ( ) / ( ) ( ) ) k n knP N t k N t N t n p q

k

, where

1

1 2

p

and 1

1 2

q

. (AN)(Nov./Dec.2015)


II Year / IV Sem 52

48. Consider a random process 0( ) ( ) cos( )Y t X t w t where X(t) is WSS process, is a

uniformly distributed r.v. over , w0 is a constant. It is assumed that X(t) and are

independent. Show that Y(t) is WSS? (AP)(May/ June 2016)

49. Consider the Markov chain Xn, n=0,1,2.. having state space and TPM P=0.4 0.6

0.8 0.2

(1)

Draw the transition diagram (2) IS the chain irreducible? (3) Is the state -1 ergodic? Explain(4) IS the chain ergodic? Explain (AN)(May/ June 2016)

50. Let X(t) and Y(t) be two independent poisson process with parameters 1 2and resp. Find

(1) P(X(t)+Y(t)) = n, n=0,1,2,… (2) P(X(t)-Y(t)) = n, n=0, 1,-1,2,-2,…. (AN) (May/ June2016)

51. Let ,...3,2,1; nX n be a Markov chain on the space 3,2,1S with one step transition

probability matrix

0012

10

2

1010

P with initial state probability

distribution 0( ) , 1, 2,33

iP X i i . Find (1) 3 2 1 0( 2, 1, 2 / 1)P X X X X

(2) 3 2 1 0( 2, 1, 2 , 1)P X X X X , (3) 2 0( 2 / 2)P X X ,

(4) Invariant probabilities of the Markov chain. (AP)(May/ June 2016)

COURSE OUTCOME: Able to characterize phenomena which evolve with respect to time inprobabilistic manner. Choose an appropriate method to solve a practical problem.

UNIT IV

MARKOVIAN QUEUEING MODELS

COURSE OBJECTIVE: Ability to analyze basic properties of Markov chains and their applicationsin modeling queuing systems and acquire skills in analyzing queueing models. Have an ability todesign a model or a process to meet desired needs within realistic constraints such as environmentalconditions

FORMULAS

MODEL-I: )/(:)1//( FIFOMM

1. Steady state probability 0p

10p


II Year / IV Sem 53

1, 0

nn

n pp

2. Average no. of customers in the system sL

sL

3. Average number of customers in the queue ( qL )

2

L Lq s

4. Average waiting time of a customer in the system sW

1ss

LW

5. Average waiting time of a customer in the system sW

2q

q

LW

6. Probability that the number of customers in the system exceeds ‘k’

1

)(

k

kNp

7. Average number wL of customers in the non-empty queue ( wL )

wL

8. Average waiting time of customers in the queue ,if he has to wait.

1

0qq WWE

9. Probability density function (pdf) of the waiting time in the system

( )( ) ( ) wf w e

10. Probability that the waiting time of a customer in the system exceeds ‘t’


II Year / IV Sem 54

0,1

0,)(

)(

)(

w

we

wg

w

NOTE: qW does not follow an exponential distribution.

MODEL-II: )/(:)//( FIFOcMM

1.

snpss

snp

pn

sn

n

n

,!

1

,

0

0

2. 02

1

1!

1p

s

ssL

s

q

3.

02

1

1!

1p

s

ssLL

s

qs

4.0

1 1 1

! 21

LW p

s s

s

s

ss

5. 02

1!

11p

s

ss

LW

s

qq

6.

1

0

0

1!!

1

1s

ns

s

n

n

ssn

p

MODEL-III )/(:)1//( FIFOMM


II Year / IV Sem 55

1.1

,1

1n

np if

k

1 ,1np if

k

0

1

11

p ifk

01 ,

1p if

k

2. . s inL The average no of customer the systems

,2kL ifs

( 1)

,

1

1

1

kk

L ifs k

3. L Lq s where is the effective arrival rate )1( 0p

4. s

s

LW

5. q

q

LW

MODEL-IV )/(:)//( FIFOksMM

1.

1,

!

1,

0 ,

0

0!

p n sn

p p s n ks

n k

n

n

n

n ss


II Year / IV Sem 56

2.

1 ( )(1 )

2! 10L p k s

s

s k s k sq

where

3.1

( ) 10 0

s kL L where s s n p ps q n n

n n

4. LsWs

5. q

q

LW where is the effective arrival rate

6. 1

1! !

0 1

0

p

n ss

s k n sn s

n sn n s

PART-A

1. In a given FCFSMM //1// queue 6.0 , what is the probability that the queue contains 5or more customers?

Solution: 55

5

)6.0()5(

np

2. What is the probability that a customer has to wait more than 15 min. to get his servicecompleted in )/(:)1//( FIFOMM queue system if hr/10&.min/6 ? (Nov/ Dec.2012)

Solution: )4

1(min)15( hrwpwp ss , parameterwithonentialisws exp

14

)610(

4

)(

41

)(

41

)(

0

)(

eeee

de

3. A duplicating machine maintained for office use is operated by an office assistant. If jobs arriveat a rate of 5 per hour and the time to complete each job varies according to an exponentialdistribution with mean 6 min., find the percentage of idle time of the machine in a day.(Assume that jobs arrive according to a poisson process)

Solution: )/(:)1//( FIFOMM model

hrhr /10min6

1,/5


II Year / IV Sem 57

5.010

5110

p

50 % of time the machine is idle.

4. For )/(:)1//( FIFOMM model write down the little’s formula.

Solution: ,L Ls q ,W Ws q L ws s , W Ws q

5. Consider an (M/M/1) queueing system. If 6 and 6 find the probability of atleast 10customers in the system.

Solution: 101010

75.08

6)10(

np

6. What are the basic characteristics of Queueing process?

Solution:

1. Arrival time pattern (distribution) (a)2. Service time pattern(distribution) (b)3. No. of services (c)4. Capacity of the system (d)5. Service discipline (Queue discipline) (e)

7. Write the Kendall’s notation and explain.Solution:

)/(:)//( edcba is the Kendall’s notation a , b , c , d , e are explained as above in Q.no.(6)8. Derive the average no. of customers in the system for )/(:)1//( FIFOMM

Solution:

0 1 2 302 3

0 01

2

0

0 2

0. 1. 2. 3. .....

2 3 ....

1 2 3 ....

1

1

11 .

1

n

n

L n p p p p ps nn

n p p

p

p

2

9. What is the probability that an arrival to an infinite capacity 3 server poisson queue with

9

1

3

20 pand

c

enters the service without waiting?


II Year / IV Sem 58

Solution:Arriving customer shall enter the system without waiting if number of customer in thesystem Number of server (=3)

0

2

00

210

2

1

1

1

)3(

ppp

pppnp

0!

1p

np

n

n

Given 323

2

33

2 cand

c

9

5

9

14

2

1

9

1.2

9

1)3( np

10. Consider an M/M/C queueing system. Find the probability that an arriving customer isforced to join the queue.

Solution: Arriving customers has to join the queue if

Number of customers in the system number of server = c

132

0

0

1

0

3

0

2

00

1210

)!1(

1.....

!3

1

!2

1

!1

111

)!1(

1.....

!3

1

!2

1

!1

11

....1)(

c

c

c

cp

pc

pppp

ppppcnp

11. What is the probability that an arrival to an infinite capacity 3 server poisson Queueing

system with9

12 0 pand

enters the service without waiting?

Solution: )3//( MM model

Customers enter the system without waiting if less than 3 customers in the system2

1 1 1 2 1 1 5( 3) 40 1 2 0 0 01! 2 9 9 2 9 9

p n p p p p p p

.

12. What is the effective arrival rate for )/4/1//( FCFSMM queueing model?

Solution: )1( 0p

13. For )/(:)//( FIFONcMM model , write down the formula for(a) Average no. of customers in the system (b) average waiting time in the system


II Year / IV Sem 59

Solution:

(a) L Ls q

1 ( )(1 )

0 2! 1

1( 1)

0

sk s k swhere L p k sq

s

ss s p andn sn

(b) LsWs

14. What are the characteristics of a queuing system? (May/ June 2013)

Solution:The basic characteristics of a queueing system are

(a) Arrival pattern (b) Service pattern (C) Number of servers

(b) Number of servers

(c) System Capacity

(d) Queue discipline

15. What is the probability that a customer has to wait more than 15 minutes to get his servicecompleted in a M/M/1 queueing system, if 6 per hour and 10 per hour?

(May/ June 2013)Solution:

Probability that the waiting time in the queue exceeds t te )(

Probability that the waiting time exceeds 15 minutes 276015)610( 10254.510

6

10

6 ee

16. Give a real life situation in which (a) customers are considered for service with last in firstout queue discipline (b) a system with infinite number of servers. (Nov. / Dec. 2012)

a. Inventory systemsb. A super market bill counter, Telephone exchange long distance operators, Petrol pump

17. Consider a random queue with two independent Markovian servers. The situation at server 1is just as in M/M/1 model. What will be the type of queue in server 2? Why?(Nov/ Dec.2012)

Solution:


II Year / IV Sem 60

The type of queue in server 2 is FIFO with infinite capacity.

18. Define Markovian Queueing models. (Nov./Dec.2013)

Queueing models in which both inter arrival time and service time which are exponentiallydistributed are called Markovian Queueing models.

19. Suppose that customers arrive at a poisson rate of one per every 12 minutes and that theservice time is exponential at a rate of one service per 8 minutes. (a) what is the averagenumber of customers in the system? (b) What is the average time of a customer spends in thesystem? (Nov./Dec.2013)

1 1/ min; / min

12 8

2

3

The average number of customers in the system is 21sL

The average time the customer spends in the system1

24mins sW L

20. A supermarket has a single cashier. During peak hours, customers arrive at a rate of 20 perhour. The average number of customers that can be serviced by the cashier is 24 per hour.Calculate the probability that the cashier is idle. (May,/June, 2014)

20

24 , 0

20 11 0.1667

24 6P

21. State the steady state probabilities of the finite source queueing model represented by (M/M/R):(GD/K/K) (May/ June ,2014)

01

0

0

0

1

1 1 1! ! !

1,

!

1,

!

0 ,

n s n sc k

n n s

n

n

n n s

P

n s n s

P n sn

P P s n ks s

n k


II Year / IV Sem 61

22. State the relationship between expected number of customers in the queue and in the system(Nov./Dec.2014)

Expected number of customer in the system is Ls

Expected number of customer in the queue is Lq

s qL L

23. What is the steady state condition for M/M/C queueing model? (Nov./Dec.2014)(May/ June2016)

For Multi server , infinite capacity model:

snpss

snp

pn

sn

n

n

,!

1

,

0

0

1

0

0

1!!

1

1s

ns

s

n

n

ssn

p

For multi server, finite capacity model:

1,

!

1,

0 ,

0

0!

p n sn

p p s n ks

n k

n

n

n

n ss

1

1! !

0 1

0

p

n ss

s k n sn s

n sn n s

24. What do the letters in the symbolic representation ( / / )( / )a b c d e of a queueing model

represent (April/ May 2015)

Arrival time pattern (distribution) (a)Service time pattern(distribution) (b)


II Year / IV Sem 62

No. of services (c)Capacity of the system (d)Service discipline (Queue discipline) (e)

25. What do you mean by balking and reneging? (April/ May 2015)(May/ June 2016)

Balking: A customer may decide to wait no matter how long the queue becomes, or if hequeue is too long to suit him, may decide not to enter it. If a customer decides not to enter thequeue upon arrive, he is said to have balked.

Reneging: Sometimes a customer may enter the queue, but after he may decide to leave thequeue due to impatience, In this case he is said to have reneged.

26. Draw the state transition rate diagram of a M/M/C queueing model (April/ May 2015)

27. What is the probability that a customer has to wait more than 15 min. to get his servicecompleted in a M/M/1 queueing system, if 6 per hour and 10 per hour? (April/ May

2015)

( )

1510 6

160

( )

15( )

60

tP W t e

P W e e

28. What effect does doubling and have on Ls and Ws for an

)/(:)1//( FIFOMM queueing model? (Nov./Dec.2015)

29. Write the steady state probabilities for the )//(:)//( KKGDRMM , KR queueing model.

(Nov./Dec.2015)

30. Which queue is called to be the queue with discouragement? (Nov./Dec.2015)

31. What is the effective arrival rate for )/4(:)1//( FCFSMM queueing model?

(Nov./Dec.2015)

PART-B

1. There are three typists in an office. Each typist can type an average of 6 letters per hour. Ifletters arrive for being typed at the rate of 15 letters per hour

(a) What is the probability that no letters are there in the system?(b) What is the probability that all the typists are busy? (AP)(May/Jun 2007)

S1 S2

1 2


II Year / IV Sem 63

2. Explain an M/M/1, finite capacity queuing model and obtain expressions for the steady stateprobabilities for the system size. Find also the mean number of customers in the system.

(AN) (May/Jun 2007)(Apr/May 2008)

3. For the steady state M/M/1 queueing model prove that 0PPn

n

(AN)(Nov/Dec 2007)

4. Define Kendall’s notation. What are the assumptions made for the simplest queueing model?(R)

5. Calculate any four measures of effectiveness of M/M/1 queueing system. (AN)(April/May2010)

6. Derive the formula for the average number of customers in the queue and the probability thatan arrival has to wait for (M/M/C) with infinite capacity. Also derive for the same model theaverage waiting time of a customer in the queue as well as in the system. (AN)(May/Jun2006)

7. Define birth and death process. Obtain its steady state probabilities. How it could be used tofind the steady state solution for the M/M/1 queueing system. Why is it called geometric?

(AN)(April/May 2010)(April/ May 2015)8. On every Sunday morning, a Dental hospital renders free dental service to patients. As per

hospital rules, 3 dentists who are equally qualified and experienced will be on duty then. Ittakes on an average 10 minutes for a patient to get treatment and the actual time taken isknown to vary approximately exponentially around this average. The patients arriveaccording to the Poisson distribution with an average of 12 per hour. The hospitalmanagement wants to investigate the following:

(1) The expected number of patients waiting in the queue(2) The average time that a patient spends at the hospital. (AP)(Nov/Dec 2007)

9. A concentrator receives messages from a group of terminals and transmits them over a singletransmission line. Suppose that messages arrive according to a Poisson process at a rate ofone message every 4 milliseconds and suppose that message transmission times areexponentially distributed with mean 3 minutes. Find the number of messages in the systemand the mean total delay in the system. What percentage increase in arrival rate results in adoubling of the above mean total delay? (AP)(Apr/May 2008)

10. A duplicating machine maintained for office use is operated by an office assistant who earnsRs.5 per hour. The time to complete each job varies according to an exponential distributionwith mean 6 minutes. Assume a Poisson input with an average arrival rate of 5 jobs per hour.If an 8 – hrs day is used as a base, determine

(a) The percentage idle time of the machine.(b) The average time a job is in the system.(c) The average earning per day of the assistant. (AP)(Nov/Dec 2008)

11. Patients arrive at a clinic according to Poisson distribution at a rate of 30 patients per hour. Thewaiting room does not accommodate more than 14 patients. Examination time per patient isexponential with mean rate of 20 per hour.

(1) What is the effective arrival rate?


II Year / IV Sem 64

(2) What is the probability that an arriving patient will not wait?(3) What is the expected waiting time until a patient is discharged from the clinic?

(AP)(May/Jun 2007) (Nov/Dec 2015)12. There are three 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(a) What fraction of the time all the typists will be busy?(b) What is the average number of letters waiting to be typed?(c) What is the average time a letter has to spend for waiting for being typed?(d) What is the probability that a letter will take longer than 20 minutes waiting to be

typed and being typed?(AP) (May/Jun 2009)(Nov/Dec 2010,2011)(Apr/May 2012)(May/ June 2013)

13. Self-service system is followed in a super market at a metropolis. The customer arrivals occuraccording to a Poisson distribution with an average of 12 per hour. The hospital managementwants to investigate the following:(a) Find the expected number of customers in the system.(b) What is the percentage of time that the facility is idle? (AP)(Nov/Dec 2007)

14. A supermarket has two girls attending to sales at the counters. If the service time for eachcustomer is exponential with mean 4 minutes and if people arrive in Poisson fashion at therate of 10 per hour,

(a) What is the probability that a customer has to wait for service?(b) What is the expected percentage of idle time for each girl? (AP)(Nov/Dec 2008)(c) What is the expected length of customer’s waiting time?

(AP)(May /June 2012)(April/ May 2015)15. A T.V. repairman finds that the time spent on his job has an exponential distribution with

mean 30 minutes. If he repair sets in the order in which they came in and if the arrival of setsis approximately Poisson with an average rate of 10 per 8 hour day. What is the repairman’sexpected idle time each day? How many jobs are ahead of average set just brought?(May/June 2012) (Nov./Dec.2013) (Nov./Dec.2015) (AP)

16. Customers arrive at one window drive-in bank according to Poisson distribution with mean10 per hour. Service time per customer is exponential with mean 5 minutes. The space is frontof window, including that for the serviced car can accommodate a maximum of three cars.Others cars can wait outside this space.(i) What is the probability that an arriving customer can drive directly to the space in

front of the window?(ii) What is the probability that an arriving customer will have to wait outside the

indicated space?(iii) How long is an arriving customer expected to wait before being served?

(AP)(April/May 2011)17. Customers arrive at a one man barber shop according to a Poisson process with a mean inter

arrival time of 12 minutes. Customers spend an average of 10 minutes in the barber’s chair(a) What is the expected number of customers in the barber shop and in the queue?(b) How much time can a customer expect to spend in the barber’s shop?(c) What is the average time customer spends in the queue?


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(d) What is the probability that the waiting time in the system is greater than 30 minutes?(AP)(May/June 2009)

18. Customers arrive at a one man barber shop according to a Poisson process with a mean interarrival time of 12 minutes. Customers spend an average of 10 minutes in the barber’s chair(a) What is the expected number of customers in the barber shop and in the queue?(b) What is the probability that more than 3 customers are in the system? (AP) (Nov/Dec2008)

19. Find the mean number of customers in the queue, system, average waiting time in the queueand system of M/M/1 queueing model. (AN)(Nov/Dec 2011)

20. Show that for the )//(:)1//( FCFSMM , the distribution of waiting time in the system is

0,)()()( ttetw (AN)(April/May 2011)

21. Find the steady state solution for the multi server M/M/C model and hence find L9, W9, Wsand Ls by using Little’s formula. (AN)(April/May 2011)

22. If people arrive to purchase cinema tickets at the average rate of 6 per minute, it takes anaverage of 7.5 seconds to purchase a ticket. If a person arrives 2 min before the picture startsand it takes exactly 1.5 min to reach the correct seat after purchasing the ticket.(i) Can he expect to be seated for the start of the picture?(ii) What is the probability that he will be seated for the start of the picture?(iii) How early must he arrive in order to be 99% sure of being seated for the start of thepicture?

(AP)(Nov/Dec2010)(Nov./Dec. 2014)23. Trains arrive at the yard every 15 minutes and the service time is 33 minutes. If the line

capacity of the yard is limited to 5 trains, find the probability that the yard is empty and theaverage number of trains in the system, given that the inter arrival time and service time arefollowing exponential distribution. (AP)(April/May 2012)

24. Arrivals at a telephone booth are considered to be Poisson with an average time of 12 minutesbetween one arrival and the next. The length of a phone call is distributed exponentially withmean 4 minutes

(a) What is the average number of customers in the system?(b) What fraction of the day will the phone be in use?(c) What is the probability that an arriving customer will have to wait? (AP)(May/Jun

2007)25. Arrivals at a telephone booth are considered to be Poisson with an average time of 12 minutes

between two consecutive calls arrival. The length of a phone call is distributed exponentiallywith mean 4 minutes(a) Determine the probability that a person arriving at the booth will have ot wait.(b) Find the average queue length that is formed from time to time.(c) The telephone company will install a second booth when convinced that an arrival

would be expected to wait at least 5 minutes for the phone. Find the increase in flowsof arrivals which will justify a second booth.

(d) What is the probability that an arrival will have to wait for more than 15 min beforethe phone is free? (AP)(Nov/Dec 2006)


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26. A petrol pump has 2 pumps. The service times follow the exponential distribution with meanof 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 thepumps remain idle? (AP) (Apr/May 2008)

27. Automatic car wash facility operates with only one bay. Cars arrive according to a Poissonprocess, with mean of 4 cars per hour and may wait in the facility’s parking lot if the bay isbusy. If the service time for all cars is constant and equal to 10 minutes, determine

(a) Mean number of customers in the system Ls(b) Mean number of customers in the queue(c) Mean waiting time in the system.(d) Mean waiting time in the queue. (AP) (Apr/May 2008)(May/ June 2013)

26. Derive (1) Ls, average number of customers in the system (2) Lq, the average number ofcustomers in the queue for the queuing model )/(:)1//( FIFONMM .(May/ June

2013)(May/ June 2014) (AP)27. Customers arrive a t a one man barber shop according to a poisson process with a mean inter

arrival time of 20 minutes. Customers spend an average of 15 minutes in the barber chair. Theservice time is exponentially distributed. If an hour is used as a unit of time, then

(1) What is the probability that a customer need not wait for a hair cut?(2) What is the expected number of customer in the barber shop and in the queue?(3) How much time can a customer expect to spend in the barber shop?(4) Find the average time that a customer spends in the queue?(5) Estimate the fraction of the day that a customer will be idle?(6) What is the probability that there will be 6 or more customers?(7) Estimate the percentage of customers who have to wait prior to getting into the

barber’s chair? (May/Jun 2006)(April/ May 2015) (May/ June 2013) (AP)28. At a port there are 6 unloading berths and 4 unloading crews/ when all the berths are full,

arriving ships are diverted to an overflow facility 20kms down the river. Tankers arriveaccording to poisson process with a mean of 1 every 2 hrs. It takes for an unloading crew, onthe average 10 hrs to unload a tanker, the unloading time following an exponentialdistribution. Find (i) how many tankers are at the port on the average? (ii) How long does atanker spend at the port on the average? (iii) What is the average arrival rate at the overflowfacility? (AP)(Nov./Dec.2012)

29. Consider a single server Queueing system with poisson input, exponential service times.Suppose the mean arrival rate is 3 calling units per hour, the expected service time is 0.25hours and the maximum permissible number calling units in the system is two. Find thesteady state probability distribution of the number of calling units in the system and theexpected number of calling units in the system. (AP)(Nov./Dec.2013)

30. A telephone exchange has two long distance operators. It is observed that, during the peakload long distance calls arrive in a poisson fashion at an average rate of 15 per hour. Thelength of service on these calls is approximately exponential distributed with mean length 5


II Year / IV Sem 67

minutes. Find (a) the probability a subscriber will have to wait for long distance call duringthe peak hours of the day. (b) If the subscriber will wait and are served in turn, what is theexpected waiting time . (AP)(Nov./Dec.2013)

31. Customers arrive at a sales counter manned by a single person according to a poisson processwith a mean rate of 20 per hour. The time required to serve a customer has an exponentialdistribution with a mean of 100 seconds. Find the average waiting time of a customer.(Nov./Dec.2013)(AP)

32. Derive the governing equation for the (M/M/1) : (GD / N/ ) queueing model and henceobtain the expression for the steady state probabilities and the average number of customersin the system (AN)(May/June2014)

33. Four counters are being run on the frontiers of the country to check the passports of thetourists. The tourists choose a counter at random. If the arrival at the frontier is poisson atthe rate and the service time is exponential with parameter /2, find the average queuelength at each counter. (AP)(May/ June 2014)

34. Customers arrive at the express checkout lane in a supermarket in a poisson process with arate of 15 per hour. The time to check out a customer is an exponential random variable withmean of 2 minutes. Find the average number of customers present. What is the expectedwaiting time for a customer in the system? (AP)(May/ June 2014)

35. The local one person barber shop can accommodate a maximum of 5 people at a time 4waiting and 1 getting hair cut). Customers arrive according to a Poisson distribution withmean 5 per hour. The barber cuts hair at an average rate of 4/hr (exponential service time) (i)What percentage of time is the barber idle?(ii)What fraction of the potential customers areturned away? (iii) What is the expected number of cutomers waiting for a hair-cut? (iv) howmuch time can a customer expect to spend in the barber shop? (AP)(Nov./Dec.2014)

36. A tax consulting firm has 3 counter in its office to receive people who have problemsconcerning their income, wealth and sales taxes. On the averages 48 persons arrive in an 8 hrday. Each tax advisor spends 15 min. on the average upon arrival. IF the arrivals are poissondistributed and service times are according to exponential distribution, find (i) the averagenumber of customers in the system (ii) the average number of customers waiting to beserviced (iii) the average time a customer spends in the system?(AP)(April/ May 2015)

37. A small mail-order business has one telephone line and a facility for call waiting for twoadditional customers. Orders arrive at the rate of one per minute and each order requires 2min. and 30 seconds to take down the particulars. What is the expected number of callswaiting in the queue? What is the mean waiting time in the queue? (AP) (April/ May 2015)

38. An airport has a single runway.. Airplanes have been found to arrive at the rate of 15 perhour. It is estimated that each landing takes 3 minutes. Assuming a poisson process forarrivals and an exponential distribution for landing time. Find the expected number ofairplanes waiting to land, expected waiting time. What is the probability that the waiting willbe more than 5 minutes? (AP)(April/ May 2015)

39. Customers arrive at a watch repair shop according to a poisson process at a rate of 1 per every10 minutes, and the service time is an exponential random variable with mean 8 minutes.Compute (1) The mean number of customers Ls in the system (2) The mean waiting time Ws


II Year / IV Sem 68

of a customer spends in the system, (3) the mean waiting Wq of a customer spends in thequeue and (4) the probability that the server is idle (AP)(May/ June 2016)

40. A petrol pump has 4 pumps. The service times follow the exponential distribution with meanof 6 minutes and cars arrive for service is a Poisson process at the rate of 30 cars per hour.Find the probability that no car is in the system? Find the probability that a customer has towait for service. Find the mean waiting time in the system? (AP)(May/ June 2016)

41. A one person barber shop has 6 chairs to accommodate people waiting for a haircut,. Assumethat customers who arrive when all the 6 chairs are full leave without entering the barbershop. Customers arrive at the rate of 3 per hour and spend an average of 15 minutes in thebarber’s chair. Compute P0, Lq, P7 and Ws (AP)(May/ June 2016)

42. Consider a single server queue where the arrivals are poisson with rate 10 / hour . The

service distribution is exponential with 5 / hour . Suppose that customers balk at joining

the queue when it is too long. Specifically, when there are n in the system, an arriving

customer joins the queue with probability1

1n . Determine the steady state probability that

there are ‘n’ customers in the system. (AP)(May/June 2016)43. The engineers have two terminals available to aid their calculations. The average computing

job requires 20 minutes of terminal time and each engineer requires some computations onein half an hour. Assume that these are distributed according to an exponential distribution. Ifthe terminals can accommodate only 6 engineers in the waiting space find the expectednumber of engineers in the computing center. (AP)(Nov./Dec.2015)

44. Find the system size probabilities for an / / : / /M M C FIFO queueing system understeady state conditions. Also obtain the expression for average number of customers in thesystem. (Nov./Dec.2015)(AN)In a production shop of a company, the breakdown of the machines is found to be Poissonwith average rate of 3 machines per hour. Breakdown time at one machine costs Rs. 40 perhour to the company. There are two choices before the company for hiring the repairman.One of the repairmen is slow but cheap, the other fast but expensive. The slow repairmandemands Rs. 20 per hour and will repair the broken down machines exponentially at the rateof 4 per hour. The fast repairman demands Rs. 30 per hour and will repair the machinesexponentially at an average rate of 6 per hour. Which repairman should the company hire?

COURSE OUTCOME: Acquire knowledge to characterize a model based on the Markovianqueuing models in different situation

UNIT V

NON-MARKOVIAN QUEUES

SYLLABUS: Finite source models - M/G/1 queue – PollaczekKhinchin formula - M/D/1 andM/EK/1 as special cases – Series queues – Open Jackson networks.


II Year / IV Sem 69

COURSE OBJECTIVE: Analyze probability and stochastic models which evolve with respect totime in a probabilistic manner .Have an ability to design a model or a process to meet desired needswithin realistic constraints such as environmental conditions

PART-A

1. Explain few non-Markovian queues.Solution:Some of the queueing systems don’t have Markov property either in arrival or service patternand known as non-Markovian queues.

Some of the queues are M/Ek/1 – having erlang service time distributions M/D/1-constantservice time distributions.

2. State the P-K transform equation. (May/ June 2013)Solution:

zLz

zLzzG

B

B

1

111

3. Write the P-K mean value formula.Solution:

12

1 22BC

NE

4. Write the formula for number of customers in the non-Markovian queue with constant servicetime.

12

2

SL

5. State the assumptions to derive P-K formula for non-Markovian queues.Solution:

a) the server is not idle whenever a job is waiting for service.b) the scheduling discipline does not base job sequencing on any priori information on job

execution timesc) the scheduling is no preemptive that is once a job is scheduled for service it is allowed to

complete without interruption.

6. Define open and closed queuing network.(May/ June 2013)(May/ June 2014)Solution:An open queuing network is characterized by one or more sources of job arrivals andcorrespondingly one or more sinks that absorb jobs departing from the network. In a closednetwork jobs neither enter nor depart from the network.


II Year / IV Sem 70

7. Write the balance equation for a two-stage tandem network.Solution: 1010110010110 ,11,1,1, kkpkkpkkpkkpp

8. Draw a two-stage tandem network.

9. Write the solution for balance equation for a two-stage tandem network.

Solution: 10110010 11, kkkkp

10. Define bottleneck of a system.Solution:As the arrival rate increases in a network of queues the node with larger with traffic intensity

will become instability and hence the node with largest traffic intensity is called ‘bottleneck’of the system.

11. A two stage tandem network is such that the average service time of node 1 is 1 hour and theaverage service time for node 2 is 2 hour and the arrival rate is 0.5 per hour. Find thebottleneck of the system.

Solution: 0.5, 1, 2, 0.5, 0.251 2 1 2

Node 1 is bottleneck of the system.

12. A two stage tandem network is such that the average service time of node 1 is 1 hour and theaverage service time for node 2 is 2 hour and the arrival rate is 0.5 per hour.Solution:

0.5, 1, 2, 0.5, 0.251 2 1 2

30,0 1 11 2 8

p

13. Write the properties of Jackson network.

a. Arrivals from the “outside” to node I follow a Poisson process with mean rate i.b. Service times at each channel at node I are independent and exponentially distributed with

parameter i.c. The probability that a customer who has completed service at node i will go to next node j

is rij and rij indicates the probability that a customer will leave the system from node i.

14. State Jackson theorem for an open network. (Nov./ Dec. 2012)(Nov/Dec.2014)(April/ May2015)

0 1

0 1


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In an open Jackson network, customers arrive at each station (node) both from outside thesystem and from other stations. The customers may visit the various stations in any order andmay skip some stations. Each station has an infinite queue capacity and may have multiple

servers. A customer leaving station i goes to station j with probability ijp . So, the

probability that customer leaves the system after service at station ‘i’ is given

k

jijio pp

1

1

Let j deonote the total arrival rate of customers to the station j . Then j can be got as

the solution of 1,....2,1,1

jj

jm

iijijj c

wherekjpa

15. Define series queues. (Nov./Dec.2013)

A series queue is one in which customers may arrive from outside the system at any node andmay leave the system from any node.

16. Define open network. (Nov./Dec.2013)

A network of K service facilities (or) nodes is called an open network if it satisfies thefollowing characteristics:

(a) Arrivals from outside to node ‘i’ followa poisson process with mean rate ir and join the

queue at ‘i’ and wait for his turn turn for service

(b) Service times at the channels at node ‘i’ are distributed with parameter i

(c) Once a customer gets the service completed at node ‘i’ he joins the queue at node ‘j’with probability ijp

17. State P-K formula for the average number in the system in a M/G/1 queueing model and

hence derive the same when the service time is constant with mean1

(May/ June 2014)

12

2

SL

18. What do you mean by Ek in the M/Ek/1 queueing model? (April/ May 2015)

Ek denotes the Erlang distributed service time pattern with the mean service time of


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19. What do the letter in the symbolic representation M/G/1 of a queueing model represent Thearrivals are Markovian and the service pattern is generally distributed (UD, ED, GD, RD, ED)with single server. (April/ May 2015)

20. Write the expression for the traffic equation of the open Jackson queueing network (May/June 2016)

Let i denote the total arrival rate of customers to the station and it can be got as the solution

of1

n

j i i ijj

a p

21. An M/D/1 queue has an arrival rate of 10 customers per second and a service rate of 20customers per second. Compute the mean number of customers in the system. (May/ June2016)

M/D/1 queue is with constant service time where var( ) 0T

2 11 34 0.7512(1 ) 2 42( )2

sL

=1.

22. Derive the Pollaczek - Kintchine formula for the average number in the system when the

service time is constant with mean 1 . (Nov./Dec.2015)

23. Distinguish between and open and closed queueing network. (Nov./Dec.2015)

24. Pollaczek - Kintchine formula. (Nov./Dec.2015)

25. What do you mean by bottle neck of a network? (Nov./Dec.2015)

PART-B

1. Derive the Pollaczek-Kninchine formula for M/G/1 queue.(Nov/Dec 2007, 2008, 2010)(April/May 2010)(Nov/ Dec.2012)(Nov/Dec.2014) (April/ May 2015) (Nov./Dec.2015) (AN)

2. Discuss M/G/1 queueing model and derive Pollaczek-Kninchine formula(Nov/Dec 2011) (AN)3. Derive the P-K formula for the (M/G/1): )//( GD queueing model and hence deduce that

with the constant service time the P-K formula reduces to)1(2

2

sL where)(

1

TE

and

. (April/May 2012)(Nov./Dec.2013)(May/ June 2016) (AN)

4. In a heavy machine shop, the overhead crane is 75% utilized. Time study observations gave theaverage slinging time as 10.5 minutes with a standard deviation of 8.8 minutes. What is the


II Year / IV Sem 73

average calling rate for the services of the crane and what is the average delay in gettingservice? If the average service time is cut to 8.0 minutes with a standard deviation of 6.0minutes, how much reduction will occur on average in the delay of getting served? (AP)

5. A one-man barber shop takes exactly 25 minutes to complete one hair-cut. If customers arriveat the barber shop in a Poisson fashion at an average rate of one every 40 minutes, how long onthe average a customer in the spends in the shop. Also find the average time a customer mustwait for service? (AP)(Nov./Dec.2013) (Nov./Dec.2015)

6. A patient who goes to a single doctor clinic for a general check up has to go through 4 phases.The doctor takes on the average 4 minutes for each phase of the check up and the time taken foreach phase is exponentially distributed. If the arrivals of the patients at the clinic areapproximately Poisson at the average rate of 3 per hour, what is the average time spent by apatient (i) in the examination (ii) waiting in the clinic? (AP)

7. A car wash facility operates with only one bay. Cars arrive according to a Poisson fashion witha mean of 4 cars per hour and may wait in the facility’s parking lot if the bay is busy. Theparking lot is large enough to accommodate any number of cars. Find the average time a carspends in the facility, if the time for washing and cleaning a car(1) is constant of 10 minutes (2)is uniformly distributed between 8 and 12 minutes. (AP)(May/ June 2014)

8. A repair facility by a large number of machines has two sequential stations with respective ratesone per hour and two per hour. The cumulative failure rate of all the machines is 0.5 per hour.Assuming that the system behavior may be approximated by the two-stage tandem queue,determine the average repair time; determine the average number of customers in both stationsand the probability that both service stations are idle. (AP) (May/ June 2016) (Nov./Dec.2015)

9. An average of 120 minutes arrive each hour (inter-arrival times are exponential) at thecontroller office to get their hall tickets. To complete the process, a candidate must passthrough three counters. Each counter consists of a single server, service times at each counter 1,20 seconds; counter 2, 15 seconds and counter 3, 12 seconds. On the average how manystudents will be present in the controller’s office. (AP) (April/May 2012)(May/ June 2014)

10. A car wash facility operates with only one bay. Cars arrive according to a Poisson fashion witha mean of 4 cars per hour and may wait in the facility’s parking lot if the bay is busy. Theparking lot is large enough to accommodate any number of cars. Find the average number ofcars waiting in the parking lot, if the time for washing and cleaning a car follows a discretedistribution with values equal to 4,8,15 minutes and corresponding probabilities 0.2, 0.6 and0.2. (AP)

11. For a open queueing network with three nodes 1, 2 and 3, let customers arrive from outside the

system to node j according to a Poisson input process with parameters jr and let ijP denote the

proportion of customers departing from facility i to facility j. Given )3,4,1(),,( 321 rrr and

04.04.0

3.001.0

3.06.00

ijP determine the average arrival rate j to the node j for j= 1, 2, 3.

V (AP)(Apr/May 2012)


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12. Automatic car wash facility operates with only one bay. Cars arrive according to a Poissondistribution with a mean of 4 cars per hour and may wait in the facility’s parking lot if the bayis busy. The parking lot is large enough to accommodate any number of cars. If the service timefor all cars is constant and equal to 10 minutes, determine

(1) mean number of customers in the system Ls

(2) mean number of customers in the queue Lq

(3) mean waiting time of a customer in the system Ws

(4) mean waiting time of a customer in the queue Wq. (AP)(April/May2012)(May/June2013)

13. In a bookshop, there are two sections, one for text books and the other for note books.Customers from outside arrive at the text book section at a Poisson rate of 4 per hour and atthe note book section at a Poisson rate of 3 per hour. The service rates of the text book andnote book sections are respectively 8 and 10 per hour. A customer upon completion of serviceat text book section is equally likely to go to the note book section or to leave the bookshop,whereas a customer upon completion of service at note book section will go to the text book

section with probability 13

and will leave the bookshop otherwise. Find the joint steady state

probability that there are 4 customers in the text book section and 2 customers in the notebook section. Find also the average number of customers in the bookshop and the averagewaiting time of a customer in the shop. Assume that there is only one salesman in eachsection. (AP)

14. In an ophthalmic clinic, there are two sections- one section for assessing the powerapproximately and the other for final assessment and prescription of glasses. Patients arrive atthe clinic in a Poisson fashion at the rate of 3 per hour. The assistant in the first section takesnearly 15 minutes per patient and the doctor in the second section takes nearly 6 minutes perpatient. If the service times in the two sections are approximately exponential, find theprobability that there are 3 patients in the first section and 2 patients in the second section.

15. Derive the expected steady state system size for the single server queues with poisson inputand General service. (AN)(April/May 2011)

16. Write short notes on: (i) Series Queues (ii) Open and Closed Queue Networks (April/May2011) (April/ May 2015) (U)

17. Explain how queueing theory could be used to study computer networks.(U)(April/May2010)

18. Discuss open and closed networks. (U)(Nov/Dec 2011)19. Write short notes on the following: (U) (Nov/Dec 2010)(Nov./Dec.2014)

(i) Queue networks(ii) Series queues(iii) Open networks(iv) Closed networks

20. Consider a system of two servers where customers from outside the system arrive at server 1 ata poisson rate 4 and at server 2 at a poisson rate 5. The service rates for server 1 and 2 are 8 and10 respectively. A customer upon completion of service at server 1 is likely to go to server 2 orleave the system; whereas a departure from server 2 will go to 25 % of the time to server 1 and


II Year / IV Sem 75

will depart the system otherwise / determine the limiting probabilities Ls and Ws?(May/ June2013) (Nov./Dec.2015) (AP)

21. Consider a two stage random queue with external arrival rate to node ‘0’. Let 10 and be

the service rates of the exponential servers at node ‘0’ and ‘1’ respectively. Arrival process ispoisson model this system using a Markov chain and obtains the balance equations.

22. Consider two servers. An average of 8 customers per hour arrive from outside at server 1 andan average of 17 customers per hour arrive from outside at a server 2. Inter arrival times areexponential. Server 1 can serve at an exponential rate of 20 customers per hour and server 2 canserve at an exponential rate of 30 customers per hour. After completing service at station 1, halfthe customers leave the system and half go to server 2. After completing service at station 2, ¾of the customer complete service and ¼ return to server 1. Find the expected number ofcustomers at each server. Find the average time a customer spends in the system.(Nov. /Dec.2012) (AP)

23. A car wash facility operates with only one bay. Cars arrive according to a poisson distributionwith mean of 4cars per hour and may wait in the facility’s parking lot if the bay is busy. Theparking lot is large enough to accommodate any number of cars. If the service time for a car hasuniform distribution between 8 and 12 minutes. Find (a) The average number of cars waiting inthe parking lot and (b) the average waiting time of a car in the parking lot. (Nov./Dec.2013)(AP)

24. There are two salesmen in a ration shop one incharge of billing and receiving payment and theother incharge of weighing and delivering the items. Due to limited availability of space, onlyone customer is allowed to enter the shop, that too when the billing clerk is free. The customerwho has finished his billing job has to wait there until the delivery section becomes free. If thecustomers arrive in accordance with a poisson process at rate 1 and the service times of twoclerks are independent and have exponential rate of 3 and 2 find (1) the proportion of customerswho enter the ration shop (2) the average number of customers in the shop (3) the averageamount of time that an entering customer spends in the shop. (Nov./Dec.2013) (AP)

25. Consider a open queueing network with parameter values shown below: (AP)

Facility jjS j ja 1i 2i 3i

1j 1 10 1 0 0.1 0.4

2j 2 10 4 0.6 0 0.4

3j 1 10 3 0.3 0.3 0

(i) Find the steady state distribution of the number of customers at facility 1, facility2,andfacility3.

(ii) Find the expected total number of customers in the system.(iii)Find the expected total waiting time for a customer May/June2014)(Nov./Dec.2015)

26. A repair facility shared by a large number of machines has 2 series stations with respectiveservice rates of 2 per hour and 3 per hour. If the average rate of arrivals is 1 per hour, find (1)


II Year / IV Sem 76

the average number of machines in the system (2) the average waiting time in the system (3)probability that both service stations are idle (April/ May 2015)(May/ June 2016) (AP)

27. Patients arrive at a clinic in a poisson fashion at the rate of 3 per hour. Each arriving patientshas to pass through two sections. The assistant in the first section take 15 minutes per patientand the doctor in the second section takes nearly 6 minutes per patient. IF the service time intwo sections is approximately exponential find the probability that there are 3 patients in thefirst sections and 2 patients in the second section. Find the average number of patients in theclinic and the average waiting time of a patient? (AP)(April/ May 2015)

28. The police department has 5 petrol cars. A petrol car breaks down and repairs service one every30 days. The police department has two repair workers, each of whom takes an average of 3says to repair a car. Breakdown times and repair time are exponential. Determine the averagenumber of patrol cars in good condition; also find the average down time for a patrol car thatneeds repairs. (AP)(May/ June 2016)

A Laundromat has 5 washing machines. A typical machine breaks down once every 5 days. A repairtakes an average of 2.5 days to repair a machine. Currently, there are three repair workers on duty.The owner has the option of replacing them with a super worker, who can repair a machine in anaverage of (5/6) day. The salary of the super worker equals the pay of the three repair workers.Breakdown time and repair time are exponential. Should the Laundromat replace the three repairerswith a super worker? (Nov./Dec.2015) (AP)

COURSE OUTCOME: Able to analyze simple queuing networks, Model communication networksand I/O computer systems

COURSE OUTCOMES

Course Name : PROBABILITY AND QUEUEING THEORY (MA6453)

Year/Semester : II / IV

Year of Study : 2016 –2017 (R – 2013)

On Completion of this course student will be able to

C210.1Understand the characteristics of probability distributions by identifying thediscrete and continuous random variables.

C210.2Identify and inculcate adequate knowledge in multi random variables tounderstand covariance, correlation and transformation of random variables.

C210.3Analyze the relation between random input and output signals using the basics ofrandom process and its characteristics to solve problems and model situationsusing techniques of Markov process

C210.4Analyze basic properties of Markov chains and their applications in modelingqueuing systems and acquire skills in analyzing Markovian queueing models.

C210.5Analyze probability and stochastic models which evolve with respect to time in aprobabilistic manner and ability to analyze simple queuing networks , Modelcommunication networks and I/O computer systems


II Year / IV Sem 77

CO – PO MATRIX:

CO PO1 PO2 PO3 PO4 PO5 PO6 PO7 PO8 PO9 PO10 PO11 PO12

CO1 3 3 - - - - - - - - - 2

CO2 3 3 - - - - - - - - - 2

CO3 3 3 - - - - - - - - - 2

CO4 3 3 - - - - - - - - - 2

CO5 3 3 - - - - - - - - - 2

AVG 3 3 - - - - - - - - - 2

COURSE OUTCOMES AND PROGRAM SPECIFIC OUTCOMES MAPPING

CO – PSO MATRIX:

CO PSO1 PSO2 PSO3

CO1 - - --CO2 - - -CO3 - - -CO4 - - -CO5 - - -AVG - - --

Documents

MA6453 PROBABILITY & QUEUEING THEORY UNIT I · 2018. 7. 23. · 10. A continuous random variable X has the probability function f (x) k(1 x), 2 x 5. Find P(X< 4). Solution: 27 2 1