fourth semester probability and queuing theory two marks with answers regulation 2013

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DEPARTMENT OF MATHEMATICS

QUESTION BANK

PROBABILITY AND QUEUEING THEORY

Subject Code: MA6453 (II CSE & II I T) Semester : IV

UNIT - I RANDOM VARIABLES

PART A 1. Define random variable. 2. In a company, 5% defective components are produced. What is the probability

that at least 5 components are to be examined in order to get 3 defectives? 3. If X and Y are independent random variables with variances 2 and 3

respectively. Find the variance of 3X + 4Y. 4. Obtain the probability function or probability distribution from the following

distribution Function:

x : 0 1 2 3 F(x) : 0.1 0.4 0.9 1.0

5. Let X be a discrete random variable whose cumulative distribution is

0 , x < -3

1

, -3 < x < 6 6

F (x) 1

, 6 < x < 10

3

1 , x > 10

(i) Find P(X < 4), P(-5 < X < 4) (ii) Find the probability distribution of X.

6. If var(X) = 4, find Var (3X+8), where X is a random variable. 7. The first four moments of a distribution about A = 4 are 1 , 4 , 10 and 45

respectively. Show that the mean is 5, variance is 3, 3 0 and 4 26 . 8. If a Random variable X takes the values 1, 2, 3, 4 such that

2 P(X=1) = 3 P(X=2) = P(X=3) = 5 P(X =4). Find the probability distribution of X .

9. Find the cumulative distribution function F(x) corresponding to the p.d.f of

f ( x ) 1 1

, - < x <

1 + x 2

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10. Define continuous random variable. Give an example.

11. A random variable X has p.d.f f(x) given by f (x) C x e- x

, x > 0.

Find the value of C and C. D. F. of X. 12. The cumulative distribution function of a random variable X is

- x

, x > 0. Find the density function of X.

13. Define r th

central moment of a random variable. 14. If a Random variable X has the moment generating function

MX (t) 2 , determine the Variance of X.

2 - t

15. Define binomial B(n,p) distribution. Obtain its m.g.f , mean and variance. 16. The mean of the binomial distribution is 20 and standard deviation is 4.

Find the Parameters of the distribution. 17. Obtain m.g.f of a Poisson distribution. 18. The number of monthly breakdowns of a computer is a r.v having a Poisson

distribution with mean equal to 1.8. Find the probability that this computer will function for a month with only one breakdown.

19. If X is a Poisson variate such that P[X=2] = 9 P[X=4] + 90 P[X=6],

find the Variance. 20. Define geometric distribution and give its m.g.f, mean and variance.

21. Show that for the uniform distribution f ( x) = 1

, - a < x < a the m.g.f about

2 a

s i n h a t the origin is .

a

t

22. If X is uniformly distributed in ( - 1 , 1 ), find the p.d.f of Y = Sin X

.

2

23 . If X is a uniform random variable in [ - 2 , 2 ], find the p.d.f of X and Var[X].

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

with parameter = 1

. What is the probability that the repair time exceeds 3 3

hours?.

25. The life time of a component measure in hours is Weibull distribution with

parameters = 0.2 , = 0.5 . Find the mean and variance of the component.


F (x) = 1 - ( 1 + x ) e


PART B

1. A random variable X has the following distribution.

x : - 2 - 2 0 1 2 3

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

Find (i) the value of k. (ii) P ( X < 2 ) and P ( -2 < X < 2 ). (iii) the cumulative distribution (iv) the mean of X

2. The monthly demand for Allwyn watches is known to have the

following probability distribution: Demand : 1 2 3 4 5 6 7 8

Probability: 0.08 0.12 0.19 0.24 0.16 0.10 0.07 0.04 Determine the expected demand for watches. Also compute the variance.

3. If X is a continuous Random variable with potential distribution function given by f(x) = Kx , 0 < x < 2

= 2K , 2 < x < 4 = 6K - Kx , 4 < x < 6 = 0 , elsewhere.

Find the value of K and C.D.F of X.

4. If the density functions of a continuous random variable X is given by

f(x) = ax , 0 < x < 1 = a , 1 < x < 2 = 3a - x , 2 < x < 3 = 0 , elsewhere.

Find i) the value of a ii) the cumulative distribution function of X. iii) if x1, x2, & x3 are 3 independent observations of X,

what is the probability that exactly of these 3 is greater than1?

5. If X is a continuous random variable, the probability density is given by f(x) = k x (2-x) , 0


2 ( 1 - x ) , 0 < x


1 -- x

e

f (x) =

2

2

13. Let X be a random variable with p.d.f

0

Find 1. m.g.f of X.

, x > 0 , otherwise

2. P[X > 3] 3. E[X] 4. Var[X]

14. Ten coins are simultaneously tossed. Find the probability of at least 7 heads.

15. If X is a binomial distribution with E[X] = 2 and Var[X] = 4/ 3, find P[X = 5]

16. For a binomial distribution mean is 6 and standard deviation is 6 . Find the first two terms of the distribution.

17. Let the random variable X follows binomial distribution with the parameters n, p.

Find 1) the p.m.f of X 2) m.g.f of X. 3) mean and variance.

18. The monthly breakdowns of a computer is a random variable having Poisson distribution with mean equal to 1.8. Find the probability that this computer will function for a month

1) without breakdown 2) with only one breakdown

19. Prove that Poisson distribution is the limiting case of Binomial distribution. 20. Find the m.g.f of a Poisson variable and hence deduce that the sum of two

independent Poisson variables is a Poison variate, while the difference is not a Poisson variate.

21. If Poisson variate X such that P[X = 1 ] = 2 P[ X = 2 ], find P[X = 0] and Var[X].

22. Find the m.g.f, mean and variance of Poisson distribution. 23. Define Geometric distribution. Find the m.g.f of Geometric distribution and hence find

its mean and variance. www.cseitquestions.in


24. Suppose that the trainee soldiers shots independently. If the probability that the target shot an any one shot is 0.7

1) What is the probability that the target would be hit on 10th

attempt? 2) What is the probability that it takes him less than 4 shots? 3) What is a probability that take him on even number of shots? 4) What is a average number of shots needed to hit the target?

25. State and prove memory less property on Geometric distribution.

X 1

1

26. If X is uniformly distributed in [ - 2 , 2 ], find P[X


34. If the life time (in hrs) X of a certain type of car has a weibull distribution with the

parameter = 2 . Find the parameter , given that the probability that the life time of the car exceeds 5 years is e

- 0.25 . For both parameters find mean and variance of X.

35. Each of the six tubes of a radio set has a life length(in years) follows a WEIBULL

distribution with parameters = 25 , = 2 . If these tubes function independently of one another, what is the probability that no tube will have to be replaced during the first two months of operation?

36. If X is uniformly distributed in

2

37. If X is a continuous r.v having

p.d.f the p.d.f of Y.

, , find the p.d.f of Y = X

2 .

2 2 x , 0 x 1

f (x) =

, otherwise

and Y = e - X

, find

0

******************** END

********************



Unit-1

RANDOM VARIABLES

1. Define a Random Variable

Solution:

A random variable is a function that assigns a real number X(s) to every element s S Where S is a sample space corresponding to a random experiment E.

2. In a Company 5% defective components are produced. What is the probability that At least 5 components are to be examined in order to get 3 defectives?

Solution:

p=5/100 q = 1-p = 1-5/100 =95/100 n = 5

P(X=x) = (e x )/x! , x=0, 1, 2,..

P(X=3) = e 3 )/3!

By Poisson distribution = n p = 5(5/100) = 0.25

P(X=3) = e0.25

(0.25)3 0.002 2!

3. If X and Y are independent r. vs with variance 2 and 3. Find the Variance

of 3X+4Y

Solution: W K T

Var ( a X+ bY) = a2 Var X + b

2 Var Y

Var ( 3 X+ 4Y) = 32 Var X + 4

2 Var Y

= 9 2 +16 3 = 18+48 = 66

1



4. Obtain the probability function or probability distribution from the following

Distribution function

X 0 1 2 3

f(x) 0.1 0.4 0.9 1

Solution:

X 0 1 2 3

p(x) 0.1 0.3 0.5 0.1

5. Let x be a discrete random variable whose cumulative distribution is

0,x 3

1

, 3 x 6

6

F(x)= 1

,6 x 10

3

1,x 10

i) find P(x>4) , p(-5


6. If Var (x) = 4 Find var (3X+8), where X is a random variable.

Solution: W K T

Var (aX+b) = a2 Var X

(3X+8) = 32 var X

= 9x4=36

7. The first four moments of a distribution about A=4 are 1,4,10 and 45

Respectively. Show that the mean is 5 variance is 3, 3 =0 and 4 =26.

Solution:

Given 1' 1, 2

' 4, 3

' 100& 4

' 25 the point

A=5 Mean = A + 1' =4+1=5

Variance = 2 2' 1

'2 4 1 3

3 3' 3 2

' 1

' 21

'3

10 3(4)(1) 2(1)

3 0

4 4' 4 3

' 1

' 6 2

' 1

' 31

'4

45 4(10)(1) 6(4)(1)

2 3(1)

4

45 40 24 3 8. If a random variable X takes the value 2P(X=1)=3P(X=2)=P(X=3)=5P(X=4)

Find the probability distribution of X

Solution: 2P(X=1)=3P(X=2)=P(X=3)=5P(X=4)=K

P(X=1)=k/2 P(X=2)=k/3 P(X=3)=k P(X=2)=k/5

k

k

k k

=1 2

3 5

61k =1 30

K= 30

61

3



P(x=1)=k/2 =15/61 P(X=2)=k/3 =10/61 P(X=3)=k = 30/61 P(X=4)=k/5 = 6/61

X 1 2 3 4

p(x) 15/61 10/61 30/61 6/61

9. Find the Cumulative distribution function F(X) corresponding to the p.d.f of

1 1

f(x)=

( 1 x2

),

x

Solution: W K T

f(X)=P(X x)

= f(x)dx

1 1

=

(

)dx

1 x2

= 1

[ tan1 x ]

x

1 [tan

1 x tan

1( )]

F(x) =

1

[tan1 x 2

10. Define continuous random variable. Given an example

Solution:

A Random variable X is said to be continuous if it is uncountable and lies in Specified interval.

f(x)= x , a


11. A random variable X has p d f f(X)=Cxex , x>0. fine the value of C and C d f

of X Solution:

f(x)dx 1 0

Cxexdx 1

0

C[x( e x ) (1)(e

x )] 1

C[(0 0) (0 1)] 1 C 1 F(x) P(X x)

x

Cxexdx

0 x

xe x dx Cxe

xdx

0

0 [ x( e x ) 1(e

[( xe x e

x ) (0

xe x e

x 1

F(x) 1 ex ( x 1)

x )]0x

1)]

12. The C.d.f of a random variable X is F(X) =1-(1+x) ex , x>0. Find the

density Function of X

Solution: WKT

f(x) d(F(x)) dx

d [1 (1 x)e

x ]

dx

0 d

((1 x)e x )

dx

[(1 x)(e x ) e

x (1)]

[ e x xe

x e

x ]

xex,x 0

5



13. Define r th

central moment of a random variable

Solution:

e(X x)r

(x x)r p(x),ifxisdiscrete

r

(x x) f(x)dx,ifxiscontinuous

2

14. If a random variable X has the m.g.f Mx (t) 2 t , determine the variance of X Solution:

' d [ M ( t )]

1 dx x t0

[ (2 t ).0 2( 1) ]

(2 t)2 t 0

2 1 (2 0)

2

2

' d2 [ M ( t )]

dt2

2 x t 0

d 2 dt

[ (2 t)2

]t0

[ (2 t )2 .0 2(2 t ).2)( 1) ]

(2 t)4 t 0

4 1

8 2

var 2 2

' ( 1

' )

2

1 (

1)2

2 2

1 1 1 2 4 4

15. Define Binomial distribution. Write Mean and variance and its m g f.

Solution:

B ( n, p ) nCr p x q

n

x , x 0,1,2...n

mgf ( q pe

t )

n

mean np

var iance npq

6



16. The Mean of the binomial distribution is 20 and S D is 4, Find the

parameters of the Distribution.

Solution: n p =20

npq 4

npq 1

20q 16

q 16

4

20 5

p 1 q 1 4

1

5 5

n. 1

20 5

n 100

P ( X x ) nC x p x q

n

x , x 0,1,2,....n

100C x (

1 )

x (

4)100x , x

0,1,.......100 5 5

17. Obtain the mgf of poisson distribution

Solution:

M x (t) etx p(x) x0

etx e x

x0

x!

e (et

)x x0

x!

e [1 e ( e )2 .....]

e ( et 1) 18. The No. of monthly breakdowns of a computer is a r. v having a poission

Distribution with mean equal to 1.8 . Find the probability that this computer Will function for a month with only one break down.

Solution:

e1.8

(1.8)

Here =1.8 P(X=x) =

1!

0.297

7


www.cseitquestions.in 19. If X is a poisson variable such that P(X=2)=9P(X=4)+90P(X=6). Find

the variance.

Solution: Given P(X=2)=9P(X=4)+90P(X=6).

e 2 9 e

4 90 e

6

2! 4!0 6!

1

2 90

6

2

24 720

1 3 2

4

2 8 8

1 3 2

4

4 3

2

4 4

4 3

2 4 0

2i,isnotpossible 1 var iance 1

20. Define geometric distribution and give its mgf , mean and variance.

Solution: A distribution r.v X is said to follow geometric distribution if its p m f is

P(X=x)=

qx1p,x 1,2,...

mgf

p 1

qet

var iance q

p

2

21. Show that the mgf for the uniform distribution is sinhat wheref(x)

1 , a x a at

2a

Solution:

8



Mx (t) E[ex ]

a

etx .f (x)dx

a

a 1

etx dx

a 2a

1

a

etxdx 2a

M.g.f a

1

[ etx

]aa

2a

t

1

[eat

eat

]

2at

1 eat eat sinhat

[

]

at 2 at

22. If X is uniformly distributed (-1,1) find the p d f of Y=sin(

x

) 2

Solution:

P d f of X is f(X) = 1 , -1


23. If X is a uniform random variable is (-2,2). Find the p d f of X and var X

Solution: Here a=-2 , b=2

f(x) 1 1 , 2 X 2

b a 4

var(X) (b a)2

12

(2 2)2 14 4

12 12 3

24. The time (in hour) required to repair a machine is exponentially distributed

with Parameter 1

. With is the probability that the repair time exceeds 3 3

hours

Solution:

P d f of exponential distribution is

f(X) = e x , 0

Here =1/3 1 1 x

f(X)=

e 3

3

1 1 x

P(X>3) f (x)dx

e 3 dx ,

0 0 3 1 1 x 1

=

) 3 (0 e

1 ) e

1

( e 3

3 e

10



25. The life time of a component measure in hours is weibull distribution with

Parameter 0.2, 0.5 . Fine the mean and variance of the component. Solution:

Here 0.2, 0.5

Mean 1 1

1

1

1

( 1

1)

1 0.5

(0.2)0.5

1 2 0.50

0.04

Variance 1 [ ( 2 1) ( ( 2 1))2 ]

2

1 [ ( 2 1) ( ( 2 1))2 ]

2 0.5 0.5 (0.2)

0.4

1

(24 4)

(0.2) 4

20

(0.2)4

11



UNIT II TWO DIMENSIONAL RANDOM VARIABLES

PART A

1. The joint p.d.f of the two dimensional random variable (X,Y) is given by

f (x , y) = K x y e -- ( x

2 + y

2 ) , x > 0 , y > 0. Find the value of K

2. If f X Y (x , y ) 1

x ( x - y ) , 0


PART B

1) Find the correlation co efficient for the following data:

X : 10 14 18 22 26 30

Y : 18 12 24 06 30 36

2) The joint p.m.f of ( X,Y ) is given by

p ( x , y ) = K ( 2 x + 3 y ) , x = 0 , 1 , 2. and y = 1 , 2 , 3 .

Find all marginal and conditional densities.

3) Two independent random variables are defined by

4 a x , 0 x 1 4 a y , 0 y 1

f (x) = , otherwise

, f ( y) = , otherwise

.

0

0

Show that U = X + Y and V = X Y are uncorrelated.

4) The joint p.d.f of ( X, Y ) is given by f ( x , y ) = e - ( x + y )

, x > 0 , y > 0 .

Find the probability density function of U = X + Y .

2

5) Given is the joint p.m.f of ( X , Y ) .

X

0 1 2

0 0.02 0.08 0.10

Y 1 0.05 0.20 0.25

2 0.03 0.12 0.15

Find i) marginal distributions ii) the conditional distribution of X given Y = 0

6) If the joint density of X and Y is given by

6 e - ( 3 x + 2 y ) , x > 0 , y > 0

f ( x , y ) = 0 , otherwise

,

find the probability density function of U = X + Y.

2



7) Suppose the joint probability density function is given by 6

x + y

2

, 0 < x < 1 , 0 < y < 1

f ( x , y ) = 5 ,

0

, otherwise

find the correlation co efficient of ( X , Y ).

8) The joint probability mass function of ( X , Y ) is given by

X Y -1 1

0 1/8 3/8

1 2/8 2/8 Find the correlation co efficient of ( X , Y ).

9) The probability mass function of ( X , Y ) is given by

X

0 1 2

0 0.10 0.04 0.02

Y 1 0.08 0.20 0.06

2 0.06 0.14 0.30

1) Compute the marginal densities of X and Y 2) Find P[ X 1 , Y 1 ] . 3) Are X and Y independent?

10) Two random variables X and Y have the joint p.d.f f ( x , y ) = 2 - x - y , 0 < x < 1 , 0 < y < 1 .

Show that C ov ( X , Y ) = - 1

.

11

11) Two dimensional random variable ( X , Y ) has the joint p.d.f

f ( x , y ) = 8 x y , 0 < x < y < 1. Find 1) marginal densities of X and Y

2) conditional densities of X and Y. 3) X and Y are independent?

3



12) Two random variables X and Y defined as Y = 4 X + 9. Find the co efficient of correlation between X and Y

13) If the j.p.d.f of a two dimensional random variable is x y

x3 + , 0 < x < 1 , 0 < y < 2

3

f (x , y) =

, otherwise 0

Find 1) P[ Y < X ] 2) P [ X > 1/2 ] 3) P [ Y < 1/2 / X < 1/2)] 4) P [ X < / Y < ]

14) Two dimensional random variable ( X , Y ) has the j.p.d.f

2 , 0 < y

www.cseitquestions.in 17) If are the two regression lines, find the mean

values of X and Y. Find the Correlation co efficient between them. Find an estimate of x when y = 1.

18) The j.p.d.f of a two dimensional random variable is given by

f ( x , y) x y2 + x

2 , 0 < x < 2 , 0 < y < 1

8

Find 1) P[X > 1]

1

2) P Y<

2

1

3) P X>1 / Y<

2

4) 1 P Y>1 / X< 2

5) P [ X Y ] 6) P [ X Y 1]

19) If the joint probability distribution of X and Y is given by 1 e - x 1 e - y , x > 0 , y > 0

F (x , y) =

0

, otherwise

1) Find the marginal densities of X and Y 2) Are X and Y independent? 3) P[1 < X < 3, 1 < Y < 2]

20) The life time of a certain brand of an electric bulb may be considered a r.v with mean 1200 hrs and s.d 250 hrs. Find the probability using Central Limit

Theorem, that the average life time of 60 bulbs exceeds 1250.

21) State and prove Central Limit Theorem.

22) A random sample of size 100 is taken from a population whose mean is 60 and variance is 400. Using Central Limit Theorem, with what probability can we

assert that the mean of the sample will not differ from = 60 by more than 4?.

5


y = 2 x - 3 and y = 5 x + 7


23) Let ( X,Y ) be a two dimensional random variable having the j.p.d.f - ( x

2 + y 2 ) , x > 0 , y > 0

f ( x , y ) 4 x y e

, otherwise

0

Find the density function of U X 2 Y

2 .

24) Find the co efficient of correlation and also obtain the lines of regression from the

data given below.

X : 62 64 65 69 70 71 72 74

Y : 126 125 139 145 165 152 180 208

25) The joint p.d.f of the two dimensional random variable (X,Y) is given by

f (x , y) = K x y e - ( x

2 + y

2 ) , x > 0 , y > 0. Find the value of K.

Prove also that X and Y are independent.

26) Determine the correlation co efficient between the random variables X and Y whose

x + y , 0 < x < 1, 0 < y < 1

j.p.d.f is f ( x , y ) = 0 , otherwise

.

27) If X and Y are independent variates uniformly distributed in ( 0 , 1)

find the densities of XY and X

. Y

28) If the j.p.d.f of (X,Y) is given by

find the p.d.f of U = XY.

f ( x , y) x + y , 0 < x , y < 1 ,

29) A distribution with unknown mean has variance equal to 1.5 . Use CLT to find how large a sample should be taken from the distribution in order that the probability will be at least 0.95 that the sample will be with in 0.5 of the population mean.

30) The j.p.d.f of a bivariate random variable (X,Y) is

f (x , y) K(x + y) , 0 < x


UNIT-2

TWO DIMENSIONAL RANDOM VARIABLES 1. The joint p.d.f of the two dimensional random variable ( X , Y ) is given by

f ( x , y ) kxye ( x 2 y2)

, x 0, y 0. Find the value of k.

Solution:

We know that f ( x , y ) dxdy 1

kxye ( x 2 y2 )dxdy 1 0 0

2 2

k xe x

1

dx ye y dy

0 0

Put t x2 u y

2

dt = 2xdx du=2ydy

x dx = dt / 2 y dy=du/2

dt

du

k e t e

u 1

2

0 0 2

k e t eu 4

0 0

k[(0 1)(0 1)] 4

K=4.

2. The joint p.d.f of a two dimensional random variable is given by

f ( x , y ) 1 x ( x y ),0 x 2, x y x. Find f ( y / x ) XY Y

8

X

Solution:

f ( y / x ) f ( x

, y

)

f ( x )



x

f ( x ) f ( x , y ) dy x

x

1 ( x

2 xy ) dy

8 x

xy

2

x

1 x

2 y

8 2

x

x3

4

1

x ( x y)

x y

f ( x , y )

f ( y / x ) 8 ,0 x 2, x y x

2x2

f ( x ) x3

4

3. If X and Y are independent random variables having the jointp.d.f

f ( x , y ) 6 x y ,0 x 2,2 y 4. XY

X Y 3 .

8

Find P

Solution:

Given f ( x , y ) 6 x y ,0 x 2,2 y 4. XY

8

( 0 , 3 )

( 1, 2 ) y = 2

x + y =3

X varies from x=0 to x=3 y

Y varies from y=2 to y=3



3 3 y

P X Y 3 1

(6 x y ) dxdy

2 0

8

x2 3 y

1 3 5

6x

xy dy

8 2

24

2

0

4. Find the acute angle between two regression lines

Solution:

The angle between the two lines of regression is given by

tan 1 r

2

Y

X

, where r is the correlation coefficient between X and 2

Y

2

r

X

Y and X andY are standard deviations of X and Y respectively.

5. State the equations of two regression lines.

Solution:

Regression equation of X on Y is X

X

b

XY

Y

Y

x

y

x y

Where b

x

2

XY x

Regression equation of Y on X is

Y

bYX

X

Y X

x

y

Where b x y

y

2

YX

y

6. State Central limit theorem in Lindberg-Levys form.

Statement:

If X 1 , X 2 , ...............Xn be a sequence of independent identically

distributed random variables with E Xi and Var X i 2 , i 1,2,......and



if S n X 1 X 2 ............ Xn then under certain general conditions , Sn follows

Normal distribution with mean n and variance n 2 as n tends to infinity.

7. If X and Y are un correlated, what is the angle between the regression lines.

Solution: Since X and Y are uncorrelated r=0

tan 1 r

2

X

Y

r

2 2

X Y

tan

2

8. Let X and Y are integer valued random variables with

P ( X m, Y n ) q 2 p

m n

2 , m 1,2,....., n 1,2,........ and p + q=1. Are

X and Y independent?

Solution: The two random variables X and Y are independent, if.

P X x , Y y P X x

i P Y y

i j j P

ij Pi * P* j

P X 1, Y 2 q 2 p 1 2 2 q 2 p

P X 2, Y 1 q 2 p 2 1 2 q 2 p

Pij

Pi *

P* j

Therefore X and y are independent.

9. Distinguish between Correlation and Regression

Solution:

Correlation Regression 1.Correlation means relation 1. Regression means between two variables measure of average

relationship between 2.Correlation coefficient is the two variables symmetric 2. Regression coefficient r

xy

ryx is not symmetric



bxy

byx

10. Can the joint distribution of two random variables X and Y be got of their

marginal distributions are known?

Solution:

We can find joint distribution of X and Y provided X and Yare

independent. 11. State central limit theorem in Liapounoffs form

Solution:

If X 1 , X 2 , ...............Xn is a sequence of independent random variables with

E Xi i andVar X i i2 , i 1,2,...... and if S n X 1 X 2 ............ Xn then

under certain general conditions, Sn follows a normal distribution with mean

n n

iand variance2

i2 as n tends to infinity.

i1 i1

12. If two random variables X and Y have joint p.d.f f

( x , y ) k (2 x y ),0 x 2,0 y 3 . Evaluate k.

Solution:

2 3

k (2 x y ) dydx 1 0 0

2 y2 3

k 2 xy dx 1

2

0 0

2 9

k 6 x dx 1

0 2

k[12 9] 1

k 1

21

13.Define joint distribution of two random variables X and Y and state its properties.

Solution:

If(X,Y) is a two dimensional random variables the function F ( x , y ) P ( X x , Y y ) P ( X x , Y y) is called the joint c.d.f of (X,Y)



(i) For discrete, F ( x , y ) Pij j i

x y

(ii) For continuous, F ( x , y ) f ( x , y ) dxdy

Properties:

i) At points of continuity, f ( x , y ) 2F

( x

, y

)

xy (ii) F ( , y ) 0 F ( x , ) and F( , ) 1

14. The two equations of the variables X and Y are X= 19.13 0.87y and

Y=11.64 0.5x.Find the correlation coefficient between X and Y.

Solution: Given X 19.13 0.87 y , Y 11.64 0.5x

bxy 0.87, b yx 0.5

r bxy byx

r ( 0.87)( 0.5) 0.66 15. Prove that the correlation coefficient lies between -1 and 1.

Solution:

We know that r ( X , Y ) cov( X , Y )

X Y

1 ( xi

)( yi

x y)

n

1 x

i

2 yi

2

x y

n

ab

x x i i ,a

a 2 b2 i i

i i

bi yi y

Squaring on both sides,

r 2 ( X , Y ) ab

i i

ai 2 bi

2

By Schwartz inequality,

abii 2 ai2 bi2 ab

2 i i 1

ai2 bi2



r 2 X , Y 1

| r | 1 1 r 1

16. The regression equation of X on Y and Y on X respectively are 5x y = 22 and 64x 45 y = 24. Find the means of X and Y.

Solution: Since the regression lines passes through the means we have

5 x y 22 64 x 45 y 24

Solving the above two equations, we have

x 6, y 8

17. Show that Cov 2 ( X , Y ) Var ( X )Var (Y )

Solution:

Let X and Y be any two random variables For any real number a,

2

X X

E a must be always non-negative YY

2 2

E

a 2 X X 2 a X X Y Y Y Y 0

2 2

X X

a E 2 aE XXYY

E Y Y 2 0

a 2Var ( X ) 2 aCov ( X , Y ) Var (Y ) 0

This is a quadratic equation in a and is always non-negative, so

the discreminent must be non-positive Cov ( X , Y )

2 Var ( X )Var (Y )

18. If X and Y are linearly related, find the angle between the two regression lines. Solution:

Let Y a b yx X and X d bxyY be the two regression lines.

L et be the angle between the above equations.

1 b yx bxy whereb and b are the regression coefficients.

yx

b yx

bxy

xy



19. The joint probability mass function of (X,Y) is given by p(x,y)=k(2x+3y),x=0,1,2;

y=1,2,3 . Find the marginal probability distribution of X:i , pi* Solution:

Y

X 1 2 3 0 3k 6k 9k 1 5k 8k 11k 2 7k 10k 13k

Let P(x,y) be the p.m.f ,we have

3 2

p ( xi , y j ) 1 j 1 i 0

3k+6k+9k+5k+8k+11k+7k+10k+13k =1

K= 1

72

The marginal probability distribution of X:i , pi*

Y

X 1 2 3 Pi * P ( X xi )

0 3 6 9 P(X=0)= 18

72 72 72 72

1 5 8 11 P(X=1)= 24

72 72 72 72

2 7 10 13 P(X=2)= 30

72 72 72 72

Hence the marginal probability distribution of X are given by

p ( X 0) 18

72

p ( X 1) 24

72

p ( X 2) 30

72

20. Show that the correlation coefficient r bxy b yx where bxy and byx are regression coefficients.



Solution: Regression coefficient of y on x is

r y b ......................... (1)

yx x

Regression co efficient of x on y is

r x b .............................(2)

y xy

Multiplying (1) and (2)

r 2 x y b b

y yx xy

x

r 2 bxy byx

r bxy byx



UNIT III

MARKOV PROCESSES AND MAROKOV CHAINS

PART A

1) Define a Markov process and a Markov chain. Give an example. 2) Examine whether the Poisson process {X (t)} given by the law.

P[X(t) = n ] = e

- t (

t)

n , n = 0,1,2... is covariance stationary or not.

n !

3) Let X be the random variable which gives the inter arrival time (time between successive arrivals), where the arrival process is a Poisson process. What will be the distribution of X? How?

4) Define strict sense stationary process and give an example. 5) Define wide sense stationary. 6) Let X (t) be a Poisson process with rate . Find the correlation co efficient of X (t). 0 1

7) If the transition probability matrix of a Markov chain is 1 1 ,

2

2

find the limiting distribution of the chain..

8) Define random process and its classification. 9) Show that the sum of two independent Poisson processes is a Poisson process.

1



10) The transition probability matrix of a Markov chain {X n}, n=1,2,3 with three

3 1 0

4 4

1 1 1

1

1

1

states 1, 2, 3. is P

=

with initial distribution P ( 1 ) =

.

4 2 4

3

3

3

3 1

0

4 4

Find P[ X 3 1, X 2 2, X1 1] .

1 0

11) Draw the transition diagram for the Markov chain whose TPM is

P =

1

1

.

2 2

12) What is a Markov chain? When can you say that a Markov chain is Homogeneous?

13) Consider the random process X (t) = cos ( 0 t + ) where uniformly distributed in the interval - to . Check whether X(t) is stationary or not?

14) What is continuous random sequence? Give an example. 15) Define irreducible Markov chain? And state Chapman-Kolmogorov Theorem. 16) What is meant by steady state distribution of Markov chain? 17) State any four properties of Poisson process. 18) If {X(s, t)} is a random process, what is the nature of X(s, t) when s is fixed and

t is fixed? 19) What is a stochastic matrix? When it is said to be regular?

20) A man tosses a fair coin until three heads occur in a row. Let {X n} denotes the

longest string of heads ending at the nth

trial. i.e. X n = k, if at the nth trial the last

tail occurred at the (n-k)th

trial. Find the TPM.

2



PART B

1) Show that the random process X (t) = A cos ( t + ) is a wide sense stationary, if A and are constants and is uniformly distributed random variable in ( 0 , 2 ) .

2) The process {X (t)} whose probability distribution is given by a t

n - 1

, n = 1 , 2 ....

n + 1

1 + a t

P[X (t) = n] = Show that it is not stationary. a t

, n = 0

1 + a t

3) A raining process is considered as a two state Markov chain. If it rains, it is considered to be in state 0 and if it does not rain that the chain is in state 1. The

0.6 0.4 . Find the

transition probability of the Markov chain is defined as P =

0.2 0.8

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

4) A machine goes out of order, whenever a component fails. The failure of this part

follows a Poisson process with a mean rate of 1 per week. Find the probability that

2 weeks have elapsed since last failure. If there are 5 spare parts of this component

in an inventory and that the next supply is not due in 10 weeks, find the probability

that the machine will not be out of order in the next weeks.

5) The TPM of a Markov chain { X n }, n = 1 , 2 , 3 having 3 states 1 , 2 , 3 is 0 .1 0 .5 0 .4

P =

0 .6 0 .2 0 .2

and the initial distribution is P ( 0 )

= 0 .7 0 .2 0 .1 .

0 .3 0 .4 0 .3

Find P[ X 2 3, X 1 3, X 0 2] .

3



6) 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 P.M each 0.6 0.2 0.2

day, we get the transition probability matrix as P =

0.1 0.8 0.1 rd

. Find out the 3

0.6 0 0.4

step transition probability matrix. Determine the limiting probabilities. 7) A person owning a scooter has the option to switch over to scooter; bike or a car

next time with the probability of (0.3 0.5 0.2). If the transition probability

0 .4 0 .3 0 .3

matrix is P =

0 .2 0 .5 0 .3

, what are the probabilities vehicles related to

0.25 0 .25 0 .5

his fourth purchase?

8) A stochastic process is described by X (t) = A sint + B cost where A and B are independent random variables with zero means and equal standard deviations show that the process is stationary of the second order.

9) The One step TPM of a Markov chain { X n , n = 0 , 1 , 2 } having state space 0.1 0.5 0.4

S = { 1 , 2 , 3} is P =

0.6 0.2 0.2

and initial distribution is P ( 0 ) = 0.7 0.2 0.1 .

0.3 0.4 0.3

Find (i) P[X 2 = 3 / X 0 = 1] (ii) P [ X 2 = 3] (iii) P[X 3 = 2,X 2 = 3 ,X 1 = 3,X 0 =1]

10) If the process {N(t); t >0} is a Poisson process with parameter , obtain

P[N(t) = n] and E[N(t)]

11) Derive the probability law of the Poisson process and find its mean and variance 12) A man either drives a car of catches a train to go to office each day. He never

goes 2 days in a row by train but if he drives one day, then the next day he is just as likely to drive again he is to travel by train. Now suppose that the first day of the week the man tossed a fair die and drove to work if and only if a 6 appeared Find 1) the probability that he takes a train on the third day.

2) the probability that he drives to work in long run.

4



13) Consider the random process X (t) = cos ( t + )where a random variable

with density function is f ( ) = 1 , -

x

, Check whether the process is

2 2

stationary or not.

14) Given a random variable with density function f () and another random variable uniformly distributed in ( - , ) and independent of and X (t) = a cos ( t + ) , prove that { X(t) } is a WSS process.

15) Show that when events occur as a Poisson process, the time interval between successive events follow exponential distribution.

16) For a random process X(t) = Y sin t , Y is an uniform random variable in

the interval 1 to 1. Check whether the process is WSS or not.

17) A stochastic process is described by X(t) = Y cos t + Z sin t , where Y and Z

are two independent random variables with zero means and equal standard

deviations. Show that the process is stationary of the second order.

18) Two random processes X (t) and Y (t) are defined by X(t) = A cos t + B sin t

and Y(t) = B cos t - A sin t . Show that X (t) and Y (t) are jointly WSS if A and B are uncorrelated random variables with zero means and the

same variances and is constant. 19) Write short notes on each of the following:

1) Binomial process 2) Sine wave process 3) Ergodic process

20) Consider a Markov chain { X n ; n 1 } with state space S = { 1 , 2 } and one step

0.9 0.1

TPM P =

0.2 .

0.8

1) Is chain irreducible? 2) Find the mean recurrence time of states 1 and 2 3) Find the invariant probabilities.

5



UNIT - III

Markov Processes and Markov Chains 1. Define Markov Process and Markov Chain. Give an example.

Solution: Markov Processes:

A random process X(t) is called a Markov Process if

P X (t n ) an / X (t n 1 ) an 1 , X (t n 2 ) an 2 ... X (t 1 ) a1

= P X (t n ) an / X (t n 1 ) an1 for all t1

t 2

t 3

....

tn

Markov Chain:

If for all n, P X (t n ) an / X (t n 1 ) an 1 , X (t n 2 ) an 2 ... X (t 1 ) a1 = P X (t n ) an / X (t n 1 ) an1

then the process X

n is called a Markov Chain.

Example:

Let X ( t ) N of birth up to time t so that the sequence X(t) forms a

pure birth process since the future is independent of the past given current state.

2. Examine whether the Poisson process X(t) given by the law

P X (t ) N e

t

tn

, n 0,1,2.....is covariance stationary or not.

n!

Solution:

Poisson Process {X (t)} is not covariance stationary since the which is not a constant.

Var X t t


www.cseitquestions.in 3. Let X be the random variable which gives the inter arrival time , where the arrival process

is a Poisson process .What will be the distribution of X ? How?

Solution:

The interval between two successive occurrences of a Poisson process with

parameter has an exponential distribution with mean

1

.

Let the two consecutive occurrences of the event be E and E . Let E take place at

i i1 i

time instant t

i . Let T be the inter arrival between the occurrences of E and E . Then

i i1

T is a continuous random variable.

P T t

P E

i 1 does not occur in t , t

i i1

PNo event occuer in an interval of length t P X (t) 0

P0 (t) et

t0 et

0! The cumulative distribution of T is given by F (t ) P T t P T t 1 e t et

1

Which is an exponential distribution with mean .

4. Define strict sense stationary process and give an example.

Solution:

A random process is called strict sense stationary if all its finite dimensional distribution are invariant under translation of time parameter.

Example:

X (t ) A cos0 t where A and 0 are constants and is

uniformly distributed in , www.cseitquestions.in


5. Define wide sense stationary process.

Solution:

A random process X(t) is said to be wide sense stationary if its mean is constant and its autocorrelation depends only on time difference.

i.e ) E X (t ) cons tant and R XX t , t RXX

6. Let X(t) be a Poisson process with rate . Find the correlation co efficient of X(t).

Solution:

The correlation co efficient is given by

XX t1 , t2 CXX t1 , t2

VarX (t1 )

VarX (t2 )

t1 if t2 t1

t1t2

t

1 t1t 2

t

1 t2

0 1

7. If the transition probability matrix of a Markov chain is

1 1

, find the limiting

2 2

distribution of the chain

Solution:

Limiting distribution of the chain is p

0 1

1 2 1 1 1 2

2 2



2 1

2

1 2

2 2

2 1 2 1 2

2

1 2

2 2

Since 1 2 1, 1 21 1

1 1

, 2 2

3 3

1 2

Limiting distribution of the chain is 1 2 .

3 3

8. Define Random Process and its classification.

Solution:

A random process is a collection of random variables that {X(s,t)} that are

functions of a real variable where s S , S is the sample space and t T , T is an

index set.

Random process is classified into four types

1. Continuous random process 2. Discrete random process 3. Continuous random sequence 4. Discrete random sequence

9. Show that the sum two independent Poisson processes is a Poisson process.

Solution:

Let X t X 1 t X 2 t

n

P X ( t ) n P X t r P X 2 t n r 1

r0 www.cseitquestions.in


n e

1

t 1 t

r e 2

t 2t

n

r

r !

n r ! r 0

n

r n r

1t 2t

e 1

2 t

r0 r ! n r!

1

n

e 1 2 t nc

r 1t r 2t

n

r

n! r0

e 1

2 t 1 t n

n! 1 2

Hence

X

t

X 1

t

X 2

t is a Poisson process.

3 1 0

4 4

1 1 1

10. The t.p.m of a Markov chain {Xn} is P

4 2 4 with three states 1,2,3, is with initial

3 1

0

4 4

P1 1 1 1 P X 1, X 2, X 1 distribution is Find 3 2 1 . 3 3 3

Solution:

P X 3 1, X 2 2, X 1 1 P X 3 1/ X 2 2, P X 2 2/ X 1 1 P X1 1

P 1 1 1

1 1 1 1

P . . = 3 4

4 48

21 12 4



11. What is Markov Chain ? When can you say that a Markov chain is homogeneous?

Solution:

If for all n,

P X (t n ) an / X (t n 1 ) an 1 , X (t n 2 ) an 2 ... X (t 1 ) a1 = P X (t n ) an / X (t n 1 ) an1 then

the process X

n is called a Markov Chain. P

X

(t n

)

an

/

X

(t n 1

)

an1 is called the one step transition

probability . If the one step probability does not depend on the step

P ( n 1, n ) P ( m 1, m ) i.e) ij ij the Markov chain is called homogeneous.

12. Consider the random process X ( t ) cos( 0 t ) where is uniformly

distributed in the interval , . Check whether X ( t ) is stationary or not.

Solution:

Since is uniformly distributed in the interval , , f ( ) 1

2

E X

t E Cos

t

0

1

Cos 0 t

d

2

1 sin t

sin

t

2 0 0

2sin0t

2 which is a function in t.HenceX(t) is not stationary



13. What is continuous random sequence, give an example.

Solution:

A random process for which X is continuous but time takes only discrete values is called continuous random sequence

Example:

If

Xn represents the temperature at the end of the nth hour of a day then

is a continuous random sequence, since the temperature can take any

value in the interval and hence it is continuous. 14. Define irreducible Markov chain and state the chapman Kolmogorov theorem.

Solution:

A Markov chain is said to be irreducible if every state can be reached from every

other states, where P

n

0

for some n and for all i and j. ij

Chapman Kolmogorov theorem:

If P is the t.p.m of a homogeneous Markov chain then the nth step Pn

is equal to pn .

P n P n

i.e)

ij ij

15. What is steady state distribution of a Markov chain.

Solution:

A Markov chain whose tpm P is said to be a steady state distribution if p where

1 2 1 and 1 2

16. State any four properties of Poisson process.

Solution:

1. The Poisson process is a Markov process 2. Sum of the two independent Poisson processes is a Poisson process 3. Difference of two independent is not a Poisson process

4. The inter arrival time of a Poisson process is exponentially distributed with mean

1


X n ,1 n 24


17. If { X( s, t ) } is a random process , what is the nature of X( s , t ) when s is fixed and t is fixed?

Solution:

X(s, t) becomes a number.

18. What is a stochastic matrix? When it is said to be regular?

Solution:

n

A matrix A is stochastic if a

i 1

and all entries are positive. It is said to be regular if i1

all entries are positive and sum of the entries in all the rows of A2 is 1.

19. A man tosses a fair coin until 3 heads occur in a row. Let Xn denotes the longest string

of heads ending at the nth trial X

n k : if at the nth trial the last tail occurred at the

(n-k)th

trial, find the t.p.m.

Solution:

The state space = { 0,1,2,3 } since the coin is tossed until 3 heads occur in a row.

1 1 0 0

2 2

1

1

0 0

P 2 2

1

1

0 0

2 2

0 0 0 1

1 0

20. Draw the transition diagram for the Markov chain whose t.p.m is P

1 1

2 2

Solution:

0

1/2

1

0 1


www.cseitquestions.in www.cseitquestions.in


1

UNIT IV

QUEUEING THEORY

1. What are the basic characteristics of a queueing system? 2. What are the basic characteristics of a queueing process? 3. State littles formula. 4. State kendalls notation. 5. What do you meant by transient state and steady state queueing

system? 6. In the usual notation of an M / M /1 queueing system, if 12 per hour

and 24 per hour, find the average number of customers in the

system.

7. Derive the average number of customers in the system for

(M / M /1) : ( /FIFO) .

8. What is the probability that an arrival of an infinite capacity with

2 p 0 1

c 3 and

enters the service without waiting?

9

9. Obtain the steady state probabilities of an (M / M /1) : ( N / FIFO)

queueing model.

10. In a given M/M/1 queue, the arrival rate 7 customers per hour

and service rate 10 customers per hour. Find P[ X 5] , where X

is the number of customers in the system.



2

11. In a duplicating machine maintained for office use is operated by an

office assistant. If the jobs arrive at rate of 5 per hour and the time to

complete each job varies according to an exponential distribution with

mean 6 minutes , find the percentage of idle time of the machine in a

day. Assume that the jobs arrive at according to a Poisson process.

12. In a given (M / M /1) : /FCFS queue, 0.6 , what is the probability

that the queue contains 5 or more customers.

13. For (M / M /C) : ( N / FIFO) model, write down the formula for

average number of customers in the queue.

14. For (M / M / C ) : N/FIFO model, write down the formula for average

waiting time in the system. 15. What is the probability that a customer has to wait more than 15

minutes to get his service completed in (M / M /1) : /FIFO

queueing system if 6 and 10 per hour.



3

PART B

1. Arrivals ate telephone booth are considered to be Poisson with an

average time of 12 min between one arrival and the next. The length of

a phone call is distributed exponentially with mean 4 minutes.

1 . What is the average number of customers in the system?

2 . What fraction of the day the phone will be in use?

3 . What is the probability that arriving customers have to wait? 2. Customers arrive at a process at 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. 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 customers 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 customers spend in the queue.

5. What is the probability that there will be 6 or more

customers waiting for service? 3. Derive the formula for average number of customers in the queue and

the probability that an arrival has to wait for M/M/c with infinite

capacity. Also derive the same model the average waiting time of a

customer in the queue as well as in the system.



4

4. Define Kendalls notation. What are the assumptions that are made for

simplest queueing model?

5. Arrival rate of telephone calls at telephone booth are according to

Poisson distribution with an average time of 12 minutes between two

consecutive calls arrival. The length of telephone call is assumed to be

exponentially distributed with mean 4 minutes.

1. Determine the probability that person arriving at the booth will

have to wait.

2. Find the average queue length that is formed from time to time.

3. The telephone company will install second both when convinced

that an arrival would expect to have to wait at least 5 minutes for

the phone. Find the increase in flows of arrivals which will

justify a second booth.

4. What is the probability that an arrival will have to wait for mare

than 15 minutes before the Phone is free.

6. A petrol pump station has 2 pumps. The service times follow the

exponential distribution with mean of 4 minutes and cars arrive for

service is a Poisson process at the rate of 10 cars per hour. Find the

probability that a customer has to wait for service. What is the

probability that the pump s remain idle?

7. Obtain the steady state probabilities for M/M/1/N/FCFS queuing model. www.cseitquestions.in


5

8. In a given M/M/1 queueing system, the average arrivals in 4 customers

per minute 0.7 .

What are

1. Mean number of customers L s in the system.

2. Mean number of customers L q in the queue.

3. Probability that the server is idle.

4. Mean waiting time W s in the system. 9. A two person barbershop has 4 chairs to accommodate waiting

customers. Potential customer who arrive when all 5 chairs are full,

leaving without entering barber shop. Customers arrive at the average

rate of 4 per hour and spend an average of 12 minutes in the barbers

chair. Compare P 0 , P 7 and average number of customers in the

queue. 10. Derive the formula for

1. Average number L q of customers in the queue.

2. Average waiting time of customer in the queue for

M / M /1 : ( /FIFO) model. www.cseitquestions.in


UNIT-4

QUEUEING THEORY

1. what are the basic characteristics of a queueing system.

Solution: The basic the basic characteristics of a queueing system are

1) arrival pattern of customers. 2) Service pattern of servers 3) Queue discipline. 4) System capacity.

2. what are basic characteristics of a queueing process

Solution: The basic queueing process describes how customers arrive at and

proceed through the queueing system. This means that the basic queueing process describes the operation of a queueing system a) the calling process b) the arrival process c) the queue configuration d) the queue discipline e) the service mechanism.

3. state littles formula.

Solution: Littles formula

Ls = Ws Lq = Wq Ls = Lq+ /

Where - arrival rate - service rate.

4. state kendalls notation.

Solution:

kendalls notation is given by (a/b/c) : (d/e/f) Where a= Arrival distribution

b= Service distribution c= Number of parallel servers (n=1,2,3,..) d= Queue discipline e= Maximum number allowed in system f= size of calling source (finite or infinite)



5. what do you mean by transient state and steady state queueing system. Solution:

If the system has been in operation for a sufficiently long time then it is called steady state behavior of the queueing situation and the early operation of the system is said to be transient behavior of the queueing situation.

6. In the usual notation of an M/M/1 queueing system, if - 12 per hour and - 24 Per hour, find the average number of customers in the system.

Solution:

Ls = / - = 12/24-12 = 1.

7. Derive the average number of customers in the system for (M/M/1) : ( /F/FO). Soultion:

Let Ls denotes number of customers in the system and N denotes numbers of customers in the queueing system.

W.K.T P0 =1- /

Pn = ( / )n (1- / )

Ls = E(N) N

=nPn n0

N

=n ( / )n (1- / ) n0

= ( / )(1- / ) n ( / )n1 n1

= ( / )(1- / ) (1 / )2 = ( / ) /(1- / ) = ( / )

8. What is the probability that an arrival of an infinite capacity 3 server poisson

queue with /c =2/3 and P0 = 1/9 enters the service without waiting. Solution:

P(without waiting) = 1-P(w>0)

= 1-( / )c P0 /c!(1- /c)



/ , C=3, P0 = 1/9 Therefore probability = 1- (2)

3 (1/9) / 3! (1-

2/3) = 1- (8/9) / 6(1/3) = 1-8/(9)(2) =1-4/9

= 5/9 9. Obtain the steady state probability of an (M/M/1) : (N/F/FO) queueing model Solution:

steady probabilities

1

,if

(

N 1

P0 = 1

)

1 ,if

N 1

1

n

Pn = /

if .

if

N 1

1 ( ) N 1

10. In a given M/M/1 queue, the arrival rate =7 customers/hour and service rate = 10 customers/hour. Find P(X>=5), where x is the number of

customers in the system. Solution:

=7, =10

P(X k) =( / )k

P(X 5)= (7/10)5

11. A duplicating machine mainfained for office use is operated by an office assistant. If the jobs arrive at a rate of 5 per varies according to an exponential distribution with mean 6 minutes, find the percentage of idle time of machine in a day. Assume that jobs arrive according to a poisson process

Solution: =

5 .

60

10 .

hr 6 hr www.cseitquestions.in


P(machine is idle) =P(N=0) = (1- / ) = 1/2.

Percentage of idle time =50% .

12. In a given M/M/1/ /FCFS queue, =0.6, what is the probability that the

queue contains 5 or more customers. Solution:

P(N k) = ()k P(N 5)= (0.6)

5

= 0.0467

13. For (MM//C): (N/F/FO) model, write down the formula for average number

of customers in the queue. Solution:

L = 2

( )

q

Where - arrival rate, - service rate.

14. For (M/M//C): (N/F/FO) model, write down the formula for average waiting time in the system.

Solution:

Ws=

+

1

( )

= ( )

=

( ) = 1

( ) 15.What is the probability that a customer has to wait more than 15 minutes

to get his service completed in (M/M//1): ( /F/FO) queue system if =6 per hour and =10 per hour

Solution:

P(w>t) = e (

)(

t )

P(w>15)= e (10

6)(15)

= e4(15)

= e60



1

UNIT - 5

Non-Markovian queues and queue networks

PART - A

1. Write Pollaczek-Khintchine formula and explain the notations. 2. What is the effective arrival rate for M/M/1/N queueing system

3. What is the effective arrival rate for M/M/1/4/FCFS queueing model when = 2

and = 5.

4. In M/G/1 queueing model write the formula for the average number

of customers in the queue.

5. In M/G/1 queueing model write the formula for the average number

of customers in the system.

6. Write the formula for average waiting time of a customer in the queue of

M/G/1 model?

7. What is the average waiting time in the system in the M/G/1 model? 8. Maruti cars arrive according to Poisson distribution with mean of 4 cars per

hour and may wait in the facilitys parking lot. If the bug is busy if the

service time for all the cars is K and equal to 10 minutes .Find Ls and Lq. 9. In a hospital patients arrive according to Poisson distribution with mean of 6

patients per hour and may wait if the clinic is busy. If the consulting time for

all the patients is K and equal to 20 minutes. Find Ws and Wq.

10. What is the probability that an arrival to an infinite capacity 3 server

Poisson distribution with _ / = 2 and P0 =1/9 enters the service without

Waiting? www.cseitquestions.in


2

PART - B

1. In a heavy machine job the overhead crane is 75% utilized. Time study

observations gave the average service time as 10 - 5 minutes with a standard

deviation of 8 8 minutes. What is the average calling rate for the services of the

crane and what is the average delay in getting service? If the average service time

is cut to 8 minutes with standard deviation of 6 minutes. How much reduction will

occur an average in the delay of getting served?

2. Derive the Pollaczek-Khintchine formula for M/G/1 queueing model

3. Automatic car Wash facility operates with only one bay. Cars arrive according

to a poisson distribution with a mean of 4 cars per hour and may wit in the facilitys parking lot if the bay is busy.If the service time for all cars is constant and equal to

10 minutes determine Ls,Lq,Ws,Wq.

4. Derive the formula for the average waiting time in the system from

Pollaczek-Khintchine formula.

5. Maruti cars arrive according to Poisson distribution with mean of 4 cars per hour

and may wait in the facilitys parking lot if the bug is busy. If the service time for

all cars is K and equal to 10 minutes. Find Ls, Lq, Ws, Wq.

6. Discuss about open and closed networks.

7. Explain in detail about series queues.

8. Customers arrive at a watch repair shop according to a Poisson process at a

rate of one per every 10 minutes and the service time is an exponential random

variable with mean 8 minutes. Find the average number of customers Ls, the

average waiting time of a customer spends in the shop Ws and the average time

a customer spends in the waiting for service Wq www.cseitquestions.in


UNIT-V

NON-MORKOVIAN QUEUES AND QUEUE NETWORKS

1. Write Pollaczek- Khintchine formula and explain the notations. Soln:

V ( s ) (1 )(1 s ) B * ( s)

B * ( s ) s

Where

s Number of servers

B * ( s ) e st d B ( t )

0

B

(t

) = p.d.f of service time .

2. What is the effective arrival rate for M/M/1/N Queuing system Soln:

Effective arrival rate

1

' (1 P )

P0

Where N1

0 1

3. What is the effective arrival rate for M/M/1/4/FCFS Queuing model when 2 and 5

Soln: Effective arrival rate

' (1 P0 )

Here 2 , 5 , N 4

1

P0

1 N1

1 2

5

2 41

1

5



3 3

5 5

3125 25

1 25

55 55

3x54 1875 0.6062

3125 32 3093

' 5(1 0.6062) 1.96

4. In M / G / 1queuing model write the formula for the average number

of customers in the queue.

Soln: The average number of customers in the queue

L 2

2

2 ,

2

Variance of service

q 2(1 )

5. In M / G / 1queuing model write the formula for the average number

of customers in the system. Soln:

The average number of customers in the system

L 2

2

2

s 2(1 )

6. Write the formula for average waiting time of a customer in the queue of M / G / 1 model.

Soln; Average waiting time of a customer in the queue

w 2

2

2

q

2 (1 )

7. What is the average waiting time in the system in the M / G / 1 model? Soln:

Average waiting time of customer in the system

w 2

2

2 1

s 2 (1 )



8. Maruti cars arrive according Poisson distribution with mean of 4 cars per

hour and may wait in the facilitys parking lot. If the bug is busy, if the

service time for all the cars is K and equal to 10 minutes. Find

Ls and

Lq .

Soln:

4 Cars / hours

1

10

Mins, 6 Cars /

hours Var=0. 0

4

2

6 3

2

2

2

Lq

2(1 )

4 2 (0) 2 2

= 3

0.667 cars.

2

1 2

3

L 2

2

2

s 2(1 )

2 2

0

2

3

= 1.333 1

2 3

2 1

3

9. In a hospital, patients arrive according to Poisson distribution with mean of 2

patients per hour and may wait if the clinic is busy. If the consulting time

for all the patients is K and equal to 20 mins. Find

ws and

wq .

Soln:

2 1

20

mts. 3 patients / hour. www.cseitquestions.in


Var=0,

2

3

w 2

2

2 1

s 2 (1 )

2 2

0

1

4

= 3

2 3

9

2x6 1

3

w 2

2

2

q 2 (1 )

2 2

0

1

3

=

2x6

1 2 9

3

10. What is the probability that an arrival to an infinite capacity 3 server

Poisson distribution with 2 and P 1 enters the service without

0 9

waiting.

Soln:

P(without waiting time) P(N 3)

P0 P1 P2

Pn

1 n when n c 3

. P0

n!

P(N 3) 1 2 1 x22 x 1 5

9 9 2 9 9


fourth semester probability and queuing theory two marks with answers regulation 2013

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