Fourth Semester Probability And Queuing Theory Two Marks With Answers Regulation 2013 4l3b4z

Find 1) m.g.f th 2) r moment 3) mean 4) variance 9. The diameter of an electric cable X is a continuous random variable with p.d.f f(x) = k x (1 - x) , 0 < x < 1 . Find 1) the value of k. 2) the c.d.f of X. 3)

Px



1 2

/

1 3

<x<

2 3

10. Find the moment generating function of the random variable with the probability law

P X  x = q p , x = 1 , 2 , 3 ..... . Also find the mean and variance. x-1

x 11. For the triangular distribution

f ( x) = 2-x 0

, 0<x<1 , 1<x<2 , otherwise

find the mean , variance and moment generating function. 12. In each of the following cases Mx (t), the moment generating function is given. Determine the distribution function of X and its mean. n 1 t 3 4  1) MX (t ) = e  2) M X (t ) =

4

1

4

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

f (x) =

2

e 0

Find

-- x 2

,

x>0

,

otherwise

1. m.g.f of X. 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 of the distribution.

17.

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

6 . Find the first two

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.



24.

Suppose that the trainee soldiers shots independently. If the probability that the target shot an any one shot is 0.7 1) 2) 3) 4)

25.

th

What is the probability that the target would be hit on 10 attempt? What is the probability that it takes him less than 4 shots? What is a probability that take him on even number of shots? What is a average number of shots needed to hit the target?

State and prove memory less property on Geometric distribution.

26. If X is uniformly distributed in [ - 2 , 2 ], find P[X<0] and

P

X1 

1

.

2

27.

Starting at 5.00 A.M every half an hour there is a flight from San Francisco Airport to Los Angels International Airport. Suppose that none of these planes is completely sold out and then they always have room for engers. A person who wants to fly to Los Angels arrives at the airport at a random time between 8.45 A.M and 9.45 A.M. Find the probability that she waits i) at most 10 mins ii) at least 15 mins.

28.

If X is uniformly distributed with E[X] = 1 and Var[X] = 4/3 , find P[X<0].

29. The number of personal computers (PC) sold at a computer world is uniformly distributed with minimum of 2000 PC and a maximum of 5000 PC. Find

30.

1) the probability that daily sales will fall between 2,500 and 3000 PC. 2) what is the probability that the computer world will sell at least 4000 Pc? 3) what is the probability that the compute world will sell at most 2500 PC?

Find the m.g.f of exponential distribution and hence find its mean and variance.

31. The daily consumption of milk in excess of 20,000 gallons is approximately e exponentially distributed with  = 3000 . The city has a daily stock of 35,000 gallons. What is the probability that of two days selected at random, the stock is insufficient for both days. 32. The life time in (in hrs) X of a component is followed Weibull distribution with shape parameter  = 2 . Starting with a large number of components, it is observed that 15 % of the components that have lasted 90 hrs fail before 100 hrs. Determine the scale parameter. 33.

Define Weibull distribution; find its mean and variance.



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 - 0.25 the car exceeds 5 years is e . 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



37.

2

 ,

2

, find the p.d.f of Y =

If X is a continuous r.v having f (x) =

2x

0

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

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


X2

.

, 0 x  , otherwise

1

and Y

= e - X , find


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−λ λ )/x! , x=0, 1, 2,….. 3 P(X=3) = e−λ λ )/3! By Poisson distribution λ = n p = 5(5/100) = 0.25 x

∴ P(X=3) = e−

0.25

(0.25)3

2!

 0.002

3. If X and Y are independent r. v’s with variance 2 and 3. Find the Variance of 3X+4Y Solution: WKT 2 2 Var ( a X+ bY) = a Var X + b Var Y 2 2 ∴ Var ( 3 X+ 4Y) = 3 Var X + 4 Var Y = 9 χ 2 +16 χ 3 = 18+48 = 66

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4. Obtain the probability function or probability distribution from the following Distribution function X f(x)

0 0.1

1 0.4

2 0.9

3 1

0 0.1

1 0.3

2 0.5

3 0.1

Solution: X p(x)

5. Let x be a discrete random variable whose cumulative distribution is 0,x  −3 1 ,− 3  x  6 F(x)=

6

1 ,6  x  10 3 1,x  10

i) find P(x>4) , p(-5<x<4) ii) find the probability distribution of X Solution: P(X<4) = F(4) = 0+

1 1

6

= 6

P(-5<x<4) = F(4) –F(-5) =

1

-0=

1

66 WKT f(x) =

d[F(X )]

dx

=0



6. If Var (x) = 4 Find var (3X+8), where X is a random variable. Solution: WKT 2 Var (aX+b) = a Var X 2 (3X+8) = 3 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 '

'2

Variance =  2   2 − 1  4 − 1 3 '

'

'

 3   3 − 3 2 1  21

'3

3

 10 − 3(4)(1)  2(1)  0 '

'

'

'

'

 4   4 − 4 3 1  6 2 1 − 31 2

'4

4

 45 − 4(10)(1)  6(4)(1) − 3(1)   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 35

k

=1 2

61k =1 30 30 K=

61



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 p(x)

1 15/61

2 10/61

3 30/61

4 6/61

9. Find the Cumulative distribution function F(X) corresponding to the p.d.f of 1

1

f(x)= ( ),  x  Π 1 x2 − Solution:

WKT f(X)=P(X ≤ x) ∝

= ∫ f(x)dx −∝

1 1 2 = ∫ Π ( 1 x )dx ∝

−∝

=

1 1 x [ tan− x ] − ∝ Π 1

[tan− x − tan− (− ∝)] Π F(x) = Π 1

1

1

Π [tan−

1

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<X



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

Solution: ∞

∫f(x)dx  1 0 ∞

∫Cxe− dx  1 x

0

C[x(− e − ) − (1)(e− )]  1 C[(0 − 0) − (0 − 1)]  1 ∴C1 F(x)  P(X ≤ x) x

x

x

 ∫ Cxe− dx x

−∞

0x



∫ xe−



x

dx  ∫Cxe− dx  x

0

 0  [ x(− e 

−x

x −x ) −  1(e  )]0

 [(− xe− − e− ) − (0 − 1)]   x

x

 − xe− − e−  1  x

x

F(x)  1 e− (− x −1) x

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

Solution: WKT

f(x)  d(F(x))   

 

dx

d

x

[1− (1 x)e− ] dx 

0 −

d

((1 x)e− dx 

x

)

 −[(1 x)(e− )  e− (1)]  x

x

 −[ −e − − xe−  e− ]  x

x

x

 xe− ,x  0  x



13. Define r

th

central moment of a random variable

Solution: r

  e(X − x) 



 ∑(x −x) p(x),ifxisdiscrete  r

 

∞

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 )] dx x t0  [ (2 − t ).0 − 2( − 1) ] 2 (2 − t) t 0 2 1  2 (2 − 0) 2 1

'

  d2 [ M ( t )] 2 2



dt

d

2

x

t 0

[ ]  dt (2 − t)2 t0 2

 [ (2 − t ) .0 − 2(2 − t ).2)( − 1) ] 4 (2 − t) t 0  4  8 2

1

var  2   2 − ( 1 )  '

' 2

1

12

−( ) 22

1−11 2 4 4

 

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

B ( n, p )  nCr p q

n−x

, x  0,1,2...n

t n

mgf  ( q  pe ) mean  np

var iance  npq



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

16 4

q

 20 5

p  1 − q  1− 5 n.

1

4 1  5

 20

5

n  100 x

P ( X  x )  nC x p q

1

x

n−x

4 100

, x  0,1,2,....n

−x

 100C x ( ) ( ) ,x 0,1,.......100 5 5

17. Obtain the mgf of poisson distribution Solution: ∞

Mx (t)  ∑e p(x) tx

x0

 ∑∞ etx e−λ λ x!

x

x0

 e −λ



∑∞

( e t )x λ

x!

x0

 e− λ





[1 λ e − λ  ( λe−λ )  .....]  2

 e− λ( e −1)  t



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: Hereλ =1.8

e−

P(X=x) =

1.8

(1.8) 1!  0.297



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− λ λ  9 e− λ λ 2! 4!0 2 6 λ 1 λ   90 2 24 720 2

4

 90 e−λ λ

6

6!

1  3λ  λ 2 8 8 2

1 3λ

2

4

λ

4

4 4 3λ  λ  4 2

λ  3λ − 4  0 λ  2i,isnotpossible λ 1 var ianceλ  1 4

2

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)= q

x−1 p,x

mgf 

p

 1,2,...

1− t qe

var iance 

q 2

p

21. Show that the mgf for the uniform distribution is

sinhat

wheref(x) 

1

,− a  x  a at

2a

Solution:



x

Mx (t)  E[e ] a

 ∫ e .f (x)dx tx

−a a

1 dx 2a

 ∫ etx −a

 1 2a M.g.f

a tx ∫ e dx

−a

1 etx a ] −a [ 2a t 1 at at [e − e− ]  2at 1 eat − e−at sinhat  at [ ]  2 at 

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

P d f of X is f(X) =1 , 2 Y

= sin π

x

)2

-1<X<1

x

2 π x = sin−1 Y ⇒ X  2 sin−1 Y π 2 dx  2 . 1 2 1−Y dy π 1 1 ∴ f ( y )  f ( x )| dx |  1 . 2 , -1 <1 1 2 2 π 1−Y dy 2 π 1−Y To find range of Y: 1 −π  sin−1 Y  π ⇒ − 1  2 sin− Y  1 ⇒ -1<X<1 π 2 2 ⇒ sin(−

π )  Y  sin(π )

2 ⇒ −1  Y 1

2



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 1  1 ,− 2  X  2 b−a 4 2 var(X)  (b − a) 12 2  (2  2)  14  4 12 12 3

f(x) 

24. The time (in hour) required to repair a machine is exponentially distributed 1 with Parameter λ  . With is the probability that the repair time exceeds 3 3 hours Solution: P d f of exponential distribution is

f(X) = λ e −λx , λ  0 Here λ =1/3 1

f(X)=

3

∞

∞

1

P(X>3) ∫ f (x)dx  ∫ 0

0

1

−1 x

e3 −1x

e 3 dx ,

3

1 1 1 ∞ − − )  − (0 − e )  e  =3 ( e 3 e −1x

3



25. The life time of a component measure in hours is weibull distribution with Parameter α  0.2,β  0.5 . Fine the mean and variance of the component. Solution: Here α  0.2,β  0.5 Mean  11 1  1

β

αβ 

1 1

0.5

( 1  1) 0.5

(0.2) 1  2  0.50  0.04

2 Variance  1 [ ( 2  1) − ( ( 2  1)) ] 2

α 

1 2

0.4

β

β

β

2 [ ( 2  1) − ( ( 2  1)) ] 0.5 0.5

(0.2) 1 (24 − 4)  (0.2) 4 

20 4 (0.2)



UNIT – II TWO DIMENSIONAL RANDOM VARIABLES PART – A 1.

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

f (x , y) = K x y e 2. If f X 3.

2

f



6-x-y

13. 14.

15. 16. 17. 18. 19. 20.

)

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

8

, 0 < x < 2 , 2 < y <4 . Find P[X + Y < 3] .

Find the acute angle between two regression lines. State the equations of two regression lines. What is the angle between them? State Central Limit Theorem in Lindberg – Levy’s form. If X and Y are uncorrelated what is the angle between the regression lines? Let X and Y are integer valued random variables with 2

11. 12.

2

1 (x , y )  x ( x - y ) , 0 <x < 2 , - x < y < x , find f Y \ X ( y \ x) . 8

P[ X = m , Y = n ] = q p 9. 10.

+y

If X and Y are independent random variables having the t p.d.f X Y

4. 5. 6. 7. 8.

Y

-- ( x

m+n-2

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

Are X and Y independent? Distinguish between correlation and regression. Can the t distribution of two random variables X and Y be got of their marginal distributions are known? State Central Limit Theorem in Liapounoff’s form. If two random variables X and Y have t p.d.f f ( x , y) = K ( 2 x + y ) , 0 < x < 2 , 0 < y < 3 , evaluate K. Define t distribution of two random variables X and Y and state its properties. The two equations of the variables X and Y are x = 19.13 - 0.87 y and y = 11.64 - 0.50 x . Find the correlation co efficient between X and Y. Prove that the correlation co efficient lies between – 1 and 1 The regression equation of X on Y and Y on X respectively 5 x - y = 22 and 64 x - 45 y = 24 . Find the means of X and Y. 2 Show that Cov (X,Y) ≤ Var(X) . Var(Y). If X and Y are linearly related, find the angle between the two regression lines. The t 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 distribution function of X. Show that correlation co efficient r =  bx y x by x where bx y and by x are regression co efficient. 1



PART – B 1) Find the correlation co efficient for the following data: X Y

: :

10 18

14 12

18 24

22 06

26 30

30 36

2) The t 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 4ay , 0y1 4ax , 0x1 f (x)0 = , f ( y) =0 , otherwise , otherwise Show that U = X + Y and V = X – Y are uncorrelated. 4) The t p.d.f of ( X, Y ) is given by

f(x,y) = e

Find the probability density function of U = X + Y

-(x+y)

.

2

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

Y

X 1 0.08 0.20 0.12

0 0.02 0.05 0.03

0 1 2

2 0.10 0.25 0.15

Find i) marginal distributions ii) the conditional distribution of X given Y = 0 6) If the t density of X and Y is given by

f(x,y) =

6 e-(3x+2y)

0

, x>0,y>0 , otherwise

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

2


,

.

, x>0,y>0.


7) Suppose the t probability density function is given by 2 6 x+y , 0<x<1,0< y<1  f(x,y) = 5  , 0 , otherwise find the correlation co efficient of ( X , Y ). 8) The t probability mass function of ( X , Y ) is given by X Y -1 1 1/8 3/8 0 2/8 2/8 1 Find the correlation co efficient of ( X , Y ). 9)

The probability mass function of ( X , Y ) is given by X 0 1 0 0.10 0.04 Y 1 0.08 0.20 2 0.06 0.14 1) Compute the marginal densities of X and Y 2) Find P[ X  1 , Y  1 ] . 3) Are X and Y independent?

2 0.02 0.06 0.30

10)

Two random variables X and Y have the t p.d.f f(x,y) = 2-x-y, 0<x<1,0 <1. -1 Show that C ov ( X , Y ) = . 11

11)

Two dimensional random variable ( X , Y ) has the t 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) 13)

Two random variables X and Y defined as Y = 4 X + 9. Find the co efficient of correlation between X and Y If the j.p.d.f of a two dimensional random variable is

f (x , y) =

x

3

+

xy

, 0<x<1,0 <2

3

0

14)

, otherwise

Find 1) P[ Y < X ] 2) P [ X > 1/2 ] 3) P [ Y < 1/2 / X < 1/2)] 4) P [ X < ½ / Y < ¼] Two dimensional random variable ( X , Y ) has the j.p.d.f

f (x , y) = Find

15)

, 0 < y <x < 1

0

, otherwise

.

1) marginal densities of X and Y 2) conditional densities of X and Y. 3) t distribution F( x , y ) 4) Check whether X and Y are independent or not.

From the following data find 1) two regression equations 2) the co efficient of correlation between the mathematics marks and statistics marks. 3) the most likely marks in statistics when mark in mathematics is 30. Mathematics Statistics

16)

2

25 43

28 46

35 49

32 41

31 36

36 32

29 31

Given f X Y ( x , y ) = C x ( x - y ) , 0 < x < 2 , - x < y < x . 1) Find C. 2) Find f X ( x) 3) Find f Y \ X ( y \ x) 4) Find f Y ( y) . 5) Find f X \Y ( x \ y) 4


38 30

34 33

32 39


y = 2 x - 3 and y = 5 x + 7 17) If two regression lines, find the mean

are the

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 2

2 f ( x , y)  x y + x , 0 < x < 2 , 0 < y < 1 8 Find 1) P[X > 1] 1

2) P Y<

2 1

3) P X>1 / Y<

1

4) P Y>1 / X<

2

2

5) P [ X  Y ] 6) P [ X  Y  1] 19)

If the t probability distribution of X and Y is given by

F (x , y) =

1e 

-x

1 e

-y





0

, x>0,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



23)

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

f(x,y)

-(x

4xye

2

+y

2

25)

,x>0,y>0 , otherwise

0

Find the density function of U  24)

)

X

2

Y

2

.

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 The t 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 j.p.d.f is

27)

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

If X and Y are independent variates uniformly distributed in ( 0 , 1) X find the densities of XY and . Y

28) If the j.p.d.f of (X,Y) is given by find the p.d.f of U = XY. 29)

30)

.

f ( x , y)  x + y , 0 < x , y < 1 ,

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

f (x , y)  1) 2) 3) 4)

K(x + y)

0

, 0 < x <2 , 0 <2

, otherwise

Find K. find the marginal densities of X and Y. Are X and Y independent? find f X \Y ( x \ y) and f Y \ X ( y \ x) 6


.


UNIT-2

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

( x 2  y2)

f ( x , y )  kxye −

, x  0, y  0. Find the value of k.

Solution: ∞ ∞

We know that ∫

∫ f ( x , y ) dxdy  1 −∞ −∞

∞∞

∫∫kxye − ( x 2  y2 )dxdy 1 00 ∞

∞

k ∫ ∫ xe − 0

x

2

dx ye − y

2

dy  1

0 2

uy du=2ydy y dy=du/2

Put t  x dt = 2xdx x dx = dt / 2 2

∞

k∫e

−t

0

dt 2

∞

k −e −

t

∞

∫ e− 0

u

du 1 2 ∞

−e−u  0

4 0

k[(0 − 1)(0 − 1)]  4

K=4.

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

f XY ( x , y )  1 x ( x − y ),0  x  2,− x  y  x. Find f Y ( y / x ) X 8 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

2

1 8

2

xy

x

x y−

2

−x

3

x 4

1 f ( y / x )  f ( x , y )  8 x ( 3x − y)  x − 2y ,0  x  2,− x  y  x 2x f(x) x 4 3. If X and Y are independent random variables having the tp.d.f f XY ( x , y )  6 − x − y ,0  x  2,2  y  4. 8 Find P

XY3.





Solution: Given f

XY

( x , y )  6 − x − y ,0  x  2,2  y  4. 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  ∫

∫ 8

1

(6 − x − y ) dxdy

20

13

 ∫ 82

6x

x

−

3− y

2

5

−xy dy  24

2

0

4. Find the acute angle between two regression lines Solution: The angle θ between the two lines of regression is given by

tanθ 

2

1− r

σ σX Y

r

σX

2

2

, where r is the correlation coefficient between X and

σY

Y andσ X andσY are standard deviations of X and Y respectively.

5. State the equations of two regression lines.



Solution:

X X  b XY Y − Y Regression equation of X on Y is − y Where



b XY

Σ x − x  − y  ∑ x −x 2









Regression equation of Y on X is Y − Y 



Σ x − x

Where b YX



b

YX

y

X



 − y 

− X 

∑ y − y 2

6. State Central limit theorem in Lindberg-Levy’s form. Statement: If X 1 , X 2 , ...............Xn be a sequence of independent identically

distributed random variables with E  Xi    and Var  X i   σ , i  1,2,......and 2



if S n  X 1  X 2  ............ Xn then under certain general conditions , Sn follows Normal distribution with mean n and variance nσ 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

σσ

2

tanθ 

1− r

Y

X

2

σ X σ

r

∴ tanθ  ∞

2

Y

θπ 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

P ij

i

j

 Pi * Χ P * j



Xx i



PYy

P  X  1, Y  2  q p

12−2

q p

P  X  2, Y  1  q p

21−2

q p

2 2

j

2 2

∴ Pij  Pi * Χ P* j Therefore X and y are independent. 9. Distinguish between Correlation and Regression Solution: Correlation 1.Correlation means relation between two variables 2.Correlation coefficient is symmetric

r r xy

yx

Regression 1. Regression means measure of average relationship between the two variables 2. Regression coefficient is not symmetric



b ≠b xy yx

10. Can the t distribution of two random variables X and Y be got of their marginal distributions are known? Solution: We can find t distribution of X and Y provided X and Yare independent. 11. State central limit theorem in Liapounoff’s form Solution: If X 1 , X 2 , ...............Xn is a sequence of independent random variables with E  Xi   i andVar  X i   σi , i 1,2,...... and if S n  X 1  X 2  ............ Xn then 2

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

∑iand varianceσ  ∑σi2 n

2

i1

as n tends to infinity.

i1

12. If two random variables X and Y have t p.d.f f ( x , y )  k (2 x  y ),0  x  2,0  y  3 . Evaluate k.

Solution: 23

∫∫k (2 x  y ) dydx  1 0 0

2

k

∫ 2 xy 

y2 3

dx  1

2

0

2

k

∫6 x 

9

0

dx  1

2 0 k[12  9]  1 k

1 21

13.Define t 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 t 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)

(ii)

At points of continuity, f ( x , y )  ∂

2

F(x,y)

∂x∂y 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 ) σ XσY 

1 ∑( xi − x )( yi − y) n 1∑x − i x n

 2 ∑  yi − y 2

∑ ab



i

i

∑ a 2 ∑ b2 i

x−x

,a i

i

i

bi  yi − y Squaring on both sides, ∑ ab r2 ( X , Y )   ∑ ai  2  ∑ bi 2 ii

By Schwartz inequality,

 ∑ abii  2 ≤  ∑ ai2 ∑bi2  ∑ab2 ii

≤1

 ∑ ai  ∑ bi2  2



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 es through the means we have

 x  −  y  22 64  x  − 45  y  24 5

Solving the above two equations, we have x  6, y  8 2

17. Show that Cov ( X , Y ) ≤ Var ( X )Var (Y ) Solution: Let X and Y be any two random variables’ For any real number ‘a’, E a

∴E a



2

2

X−X− 2

X − X 

aE X−X



Y− Y

2



must be always non-negative

−2a X−X

2

− 2 aE

Y − Y    Y − Y   −  −   X

X

Y

E

2

≥0

 Y − Y 2 ≥ 0

Y

2

a Var ( 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 etθ be the angle between the above equations.

∴θ  1− b yx bxy whereb

b b yx

and b are the regression coefficients. y x

xy

xy



19. The t 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:

X 0 1 2

1 3k 5k 7k

Y 2 6k 8k 10k

3 9k 11k 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*

X 0 1 2

Y 1 3 72 5 72 7 72

2 6 72 8 72 10 72

Pi *  P ( X  xi )

3 9 72 11 72 13 72

P(X=0)= 18 72 P(X=1)= 24 72 P(X=2)= 30 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 σ y  b yx......................... (1)

σx

Regression co efficient of x on y is

rσx b

σy

.............................(2)

xy

Multiplying (1) and (2) 2 r σ x σ y  b b yx σy σx xy

r  bxy byx 2

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!

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

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 λ. Find the correlation co efficient of X (t). 0

7)

1

If the transition probability matrix of a Markov chain is 2

1 1

,

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

states 1, 2, 3. is

P=

3

1

4 1

4 1

4

2

4 with initial distribution

3

1

0

0 1 P(1)

=

1

1

1

3

3

3

.

4

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

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

P =

0 1

2

1 . 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 ( ω 0 t + θ ) where θ uniformly distributed in the interval -π to π. 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 th longest string of heads ending at the n trial. i.e. X n = k, if at the nth trial the last th tail occurred at the (n-k) 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

 at  1+at



P[X (t) = n] =



3)

a t 1+at

n-1

, n = 1 , 2 ....

n+1

Show that it is not stationary. ,n=0



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 transition probability of the Markov chain is defined as P =

0.6

0.4

0.2

0.8

. Find the

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

0 .3

0 .4 0 .3

and the initial distribution is P

Find P[ X 2  3, X 1  3, X 0  2] .

3


(0)

= 0 .7 0 .2 0 .1  .


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

day, we get the transition probability matrix as P =

7)

0.2

0.2

0.1 0.8

0.1

0.6

0.4

0

rd

. Find out the 3

step transition probability matrix. Determine the limiting probabilities. 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 matrix is P =

0 .4

0 .3

0 .3

0 .2

0 .5

0 .3

0.25

0 .25

0 .5

, what are the probabilities vehicles related to

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  P[N(t) = n] and E[N(t)]

, obtain

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  stationary or not.

,-



x 

2



, Check whether the process is 2

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 tly 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 TPM

P=

0.9

.

0.2 1) 2) 3)

0.1 0.8

Is chain irreducible? Find the mean recurrence time of states ‘1’ and ‘2’ 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





t t

= P X (t n )  an / X (t n −1 )  an−1 for all 1  Markov Chain: If for all n, P



2 3



n

 X (t n )  an / X (t n −1 )  an −1 , X (t n − 2 )  an − 2 ... X (t 1 )  a1 =P

then the process

t .... t

X

 X (t n )  an / X (t n −1 )  an−1 

 is called a Markov Chain.

n

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.

e − λ t  λtn

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

P  X (t )  N  

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

n! Solution:

 

Var X t  λt

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





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

1

λ .

parameter λ has an exponential distribution with mean Let the two consecutive occurrences of the event be E

and E . Let i1

i

t

. Let T be the inter arrival between the occurrences of

time instant

i

E

take place at

i

E

and

i

E

. Then

i1

T is a continuous random variable.

P Tt PE





does not occur in i 1

t, t

i

i1



 PNo event occuer in an interval of length t  P X (t)  0

e−λt λt 0  P0 (t)     e−λt 0!

The cumulative distribution of T is given by F (t )  P T ≤ t   P  T  t   1 − e − λ  λe−λ t

1 Which is an exponential distribution with mean λ .

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 cosω0 t θ  where A and ω0 are constants and θ is uniformly distributed in

−π , π 


t


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  τ   RXX τ 

6. Let X(t) be a Poisson process with rate λ . Find the correlation co efficient of X(t). Solution: The correlation co efficient is given by

ρXX  t1 , t2  



  

CXX t1 , t2  VarX (t1 ) VarX (t2 )

λt1 if t2 ≥ t1 λ t1λt2

λt1



λ t 1t 2  t 1



t2 0 7. If the transition probability matrix of a Markov chain is

1 2

distribution of the chain Solution: Limiting distribution of the chain is π p  π

π1 π 2

0 1 2

1 1 π1π

2



2


1 1 , find the limiting 2


π

π1

2

π2

2

π1 π2 

2

π2  π 1 ⇒ 2π 1  π2 2 π 1  π2  π2 2

Since π 1  π 2  1, π 1  2π1  1

π1 

1

,π2 3 3

Limiting distribution of the chain is π 1

8.

π2

2



1

2

3

3

.

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 ε S , S is the sample space and t ∈ T , T is an index set. Random process is classified into four types 1. 2. 3. 4.

9.

Continuous random process Discrete random process Continuous random sequence Discrete random sequence

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

1

2

t 

r0


n−r


−λ  λ 1 t  e − λ  λ 2t  − n e ∑  n − r ! r! r0 t

r

t

1

λ

 λ1t  λ2t r !  n − r! r0

∑

t

2

n−r

r

n

− λ e  1

n r

2

e−λ1 λ2 t

1

n

∑

nc

 λ1t  r  λ2tn − r

r

n!

r0

 e −  λ1

λ t 2

1 λλ n!  1

2



t

n

X t X t X t Hence    1    2   is a Poisson process.

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

distribution is

P1  1 3

1

P

1 4 1

4

2

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

0

3

1

4

4

0 1

P  X  1, X  2, X

1 Find

3

3 4 1

3

2

1

1 .

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 21

1 1

1

P 12

3



11 1 .

.

1 =

4 4 4 48



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 )  an−1  then the process



X

 is called a Markov Chain.

P X (t )  a / X (t )  a  n n n −1 n−1  is called the one step transition n

probability . If the one step probability does not depend on the step P ( n − 1, n )  P ( m − 1, m ) ij

i.e)

ij

the Markov chain is called homogeneous.

X ( t )  cos(ω 0 t  θ ) where θ is uniformly

12. Consider the random process

,

distributed in the interval

−π π  . Check whether X ( t ) is stationary or not.

Solution: Since θ is uniformly distributed in the interval

t  E Cos

EX





π



ω t θ 0

f (θ ) 

1

2π



 ∫ Cos  ω 0 t  θ  −π

− π , π  ,

1 2π dθ

 −1 sin π  ω t  sin π − ω t    0  0 2π



−2sinω0t 2π

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:

X If

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

 X n ,1  n  24 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

Pn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 .

Pn i.e) 15.

P

n

ij

ij

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 π p  π where

π 1  π2  1 and π  π 1 π2  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

λ


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:

A matrix A is stochastic if

n

a 1

∑

i

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

i1

all entries are positive and sum of the entries in all the rows of 19.

2

A

is 1.

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 th

(n-k)



X

n

k

 : if at the nth trial the last tail occurred at the

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 2 1 P 2

1 2

0

0

0

1 2

0

1

0

0

1

2

2

0

0

0

1 1

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

0 1

2

1 2

Solution: 0 1/2 1 0

1




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 little’s formula.

4.

State kendall’s 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.

9.

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

 2 p 0 1 c  3 and 9

enters the service without waiting? 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.



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.



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.

Define Kendall’s 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.



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 barber’s 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.



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 little’s formula. Solution: Little’s formula

Ls =λ Ws Lq = λ W q L s = L q + λ /



Where λ - arrival rate  - service rate. 

4. state kendall’s notation. Solution: kendall’s 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 λ - 12 per hour and  - 24 Per hour, find the average number of customers in the system.  Solution:

Ls = λ / -λ = 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-λ / Pn = ( λ /  )n (1- λ / ) Ls = E(N) N

=∑nPn n0 N

=∑n (λ / )n (1- λ / ) n0

∞

= (λ / )(1- λ / ) ∑n (λ / )n−1 n1

= (λ / )(1- λ / ) (1− λ / )2 = (λ / ) /(1- λ / ) = (λ /  − λ) 8. What is the probability that an arrival of an infinite capacity 3 server poisson queue with λ /c  =2/3 and P0 = 1/9 enters the service without waiting. Solution: P(without waiting) = 1-P(w>0)

= 1-(λ / )c P0 /c!(1- λ /c)



λ/,

C=3, P0 = 1/9

Therefore probability = 1- (2)3 (1/9) / 3! (12/3) = 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− λ 

λ N 1 −( ) 

P0 = 1

1

,if λ ≠ 

,if λ  

N 1

λ

Pn =  λ /  

n

1−  1 − ( λ ) N 1 

λ if

N 1

if λ   .

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

P(X≥ k) =( λ /  )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: λ

=

5

.

60



10

.

hr 6 hr



P(machine is idle) =P(N=0) = (1- λ / ) = 1/2. Percentage of idle time =50% . 12. In a given M/M/1/ ∞ /FCFS queue, ρ =0.6, what is the probability that the queue contains 5 or more customers. Solution:

P(N≥ k) = (ρ)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= q

λ2 (  − λ)

Where λ - 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:

λ 1 Ws=  (  − λ) + λ

= =

λ−λ  (  − λ) 

 (  − λ) 1 = ( − λ) 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 λ =6 per hour and  =10 per hour Solution: P(w>t) = e−

(  −λ )( t )

(10 − 6)(15) P(w>15)= e− −4(15)

=e

= e−60



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 λ= 2 and µ = 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 facility’s 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 _ / µ = 2 and P0 =1/9 enters the service without Waiting?



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 facility’s 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 facility’s 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



UNIT-V NON-MORKOVIAN QUEUES AND QUEUE NETWORKS 1. Write Pollaczek- Khintchine formula and explain the notations. Soln:

V ( s )  (1 − *ρ )(1 − s ) B Where

ρ

*

B ( λ − λs ) − s

λ

( λ − λs)



s  Number of servers ∞

B ( s )  ∫e − d  B ( t ) *

st

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

λ  (1− P ) '

0 Where

1− λ



P0 

N1

1−

λ

 3. What is the effective arrival rate for M/M/1/4/FCFS Queuing model when λ  2 and   5 Soln: Effective arrival rate λ  (1− P0 ) Here λ  2 ,   5 , N  4 '

1− λ



P0  1− λ N1  1− 2 5  2 41 1−

5



3 3 5  5  5 5 3125 − 2 1− 2

55 55 4  3x5  1875  0.6062 3125 − 32 3093 ' λ  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

2 L λ σ ρ ,σ − Variance of service q 2(1− ρ) 2

2

2

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  λ σ  ρ ρ s 2(1 − ρ) 2

2

2

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

λ σ ρ 2λ (1 − ρ) 2

wq

2

2

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

ws  λ σ  ρ  1 2λ (1 − ρ )  2

2

2



8. Maruti cars arrive according Poisson distribution with mean of 4 cars per hour and may wait in the facility’s parking lot. If the bug is busy, if the

L

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

s and

L

q.

Soln:

λ  4 Cars / hours



1 10 

Mins,   6 Cars /

hours Var=0. σ  0

ρλ

4 2 



6 3 2 2 λ σ ρ

Lq 

=

2

2(1− ρ) 2

4 2 (0)  2 3 2 1−

 0.667 cars.

2 3

L  λ σ  ρ ρ s 2(1 − ρ) 2 2 2 0 2

2

2

3

=

2

 

3

1.333 ≃1

21−

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

w

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

λ2 1



 20

mts.   3 patients / hour.


w

s and

q .


ρ λ

Var=0,





2 3

ws  λ σ  ρ 1 2λ (1− ρ )  2

0

=

2

2

0

=

1 4

 

2

3

3 2

9

λ σ ρ 2λ (1− ρ) 2



2

3

2x6 1 −

wq

2

2 3

2x6 1−

2

2

1 2 9 3

10. What is the probability that an arrival to an infinite capacity 3 server Poisson distribution with λ  2 and P  1 enters the service without  9 0 waiting. Soln: P(without waiting time)  P(N  3)  P0  P1  P 2

Pn 

1

λn

. P0 when n ≤ c  3

n!  2 1 P(N  3)   2  1 x2 x 1  5 9 9 2 9 9


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