pqt formula vidhyatri

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SUBJECT NAME : Probability & Queueing Theory

SUBJECT CODE : MA 2262MATERIAL NAME : Formula Material

MATERIAL CODE : JM08AM1007

Name of the Student: Branch:

UNIT-I (RANDOM VARIABLES)

1) Discrete random variable:A random variable whose set of possible values is either finite or countably

infinite is called discrete random variable.

Eg: (i) Let X represent the sum of the numbers on the 2 dice, when two

dice are thrown. In this case the random variable X takes the values 2, 3, 4, 5, 6,7, 8, 9, 10, 11 and 12. So X is a discrete random variable.

(ii) Number of transmitted bits received in error.

2) Continuous random variable:A random variable X is said to be continuous if it takes all possible values

between certain limits.

Eg: The length of time during which a vacuum tube installed in a circuit

functions is a continuous random variable, number of scratches on a surface,

proportion of defective parts among 1000 tested, number of transmitted in

error.

3)

Sl.No. Discrete random variable Continuous random variable1

( ) 1i

i

p x

( ) 1f x dx

2 ( )F x P X x ( ) ( )xF x P X x f x dx

3 Mean ( )i i

i

E X x p x Mean ( )E X xf x dx

4 2 2 ( )

i i

i

E X x p x 2 2( )E X x f x dx

5 22Var X E X E X 22Var X E X E X 6 Moment = r r

i i

i

E X x p Moment = ( )r rE X x f x dx

7 M.G.F M.G.F
http://www.vidyarthiplus.com/vp/


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( )tX tx Xx

M t E e e p x ( )tX tx X

M t E e e f x dx

4) E aX b aE X b 5) 2Var VaraX b a X 6) 2 2Var VaraX bY a X b Var Y 7) Standard Deviation Var X 8) ( ) ( )f x F x 9) ( ) 1 ( )p X a p X a 10) / p A Bp A B

p B , 0p B

11)If A and B are independent, then p A B p A p B .12)1stMoment about origin = E X =

0X

t

M t

(Mean)2ndMoment about origin = 2E X =

0X

t

M t

The co-efficient of

!

rt

r= rE X (rth Moment about the origin)

13)Limitation of M.G.F:i) A random variable X may have no moments although its m.g.f exists.ii) A random variable X can have its m.g.f and some or all moments, yet the

m.g.f does not generate the moments.

iii) A random variable X can have all or some moments, but m.g.f does notexist except perhaps at one point.

14)Properties of M.G.F:i) If Y = aX + b, then bt

Y XM t e M at .

ii)cX X

M t M ct , where c is constant.

iii) If X and Y are two independent random variables thenX Y X Y

M t M t M t .15)P.D.F, M.G.F, Mean and Variance of all the distributions:

Sl.

No.Distributio

nP.D.F ( ( )P X x ) M.G.F Mean Variance

1 Binomial x n xx

n c p q ntq pe

np npq

2 Poisson

!

xe

x

1t

e

e

3 Geometric 1xq p(or)

xq p

1

t

t

pe

qe

1

p

2

q

p


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4 Negative

Binomial 1( 1) k x

kx k C p p

1

k

t

p

qe

kq

p

2

kq

p

5 Uniform1

,( )

0, otherwise

a x bf x b a

( )

bt at e e

b a t

2

a b

2( )

12

b a

6 Exponential, 0, 0

( )0, otherwise

xe x

f x

t

1 21 7 Gamma 1

( ) , 0 , 0( )

xe x

f x x

1

(1 )t

8 Weibull 1( ) , 0, , 0

xf x x e x

16)Memoryless property of exponential distribution/P X S t X S P X t .

UNIT-II (RANDOM VARIABLES)

1) 1ij

i j

p (Discrete random variable)( , ) 1f x y dxdy

(Continuous random variable)

2) Conditional probability function X given Y, ,/( )

i i

P x yP X x Y y

P y .

Conditional probability function Y given X , ,/( )

i i

P x yP Y y X x

P x .

,/( )

P X a Y b P X a Y b

P Y b

3) Conditional density function of X given Y, ( , )( / )( )

f x yf x yf y

.

Conditional density function of Y given X,( , )

( / )( )

f x yf y x

f x .


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4) If X and Y are independent random variables then( , ) ( ). ( )f x y f x f y (for continuous random variable)

, .P X x Y y P X x P Y y (for discrete random variable)5) Joint probability density function , ( , )d b

c a

P a X b c Y d f x y dxdy .

0 0

, ( , )

b a

P X a Y b f x y dxdy 6) Marginal density function of X, ( ) ( ) ( , )

Xf x f x f x y dy

Marginal density function of Y,( ) ( ) ( , )Y

f y f y f x y dx

7) ( 1) 1 ( 1)P X Y P X Y 8) Correlation coefficient (Discrete): ( , )( , )

X Y

C ov X Yx y

1( , )Cov X Y XY XY

n , 2 21

X X Xn

, 2 21Y

Y Yn

9) Correlation coefficient (Continuous): ( , )( , )X Y

C ov X Yx y

( , ) ,Cov X Y E X Y E X E Y , ( )X

Var X , ( )Y

Var Y 10)If X and Y are uncorrelated random variables, then ( , ) 0C o v X Y .11) ( )E X xf x dx

, ( )E Y yf y dy

, , ( , )E X Y xyf x y dxdy

.

12)Regression for Discrete random variable:Regression line X on Y is xyx x b y y , 2xy x x y y b

y y

Regression line Y on X is

yxy y b x x ,

2yx

x x y y b

x x

Correlation through the regression, .X Y YX

b b Note: ( , ) ( , )x y r x y


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13)Regression for Continuous random variable:Regression line X on Y is ( ) ( )

xyx E x b y E y , x

xy

y

b r

Regression line Y on X is ( ) ( )yx

y E y b x E x , yyxx

b r

Regression curve X on Y is / /x E x y x f x y dx

Regression curve Y on X is / /y E y x y f y x dy

14)Transformation Random Variables:( ) ( )

Y Xdxf y f x dy

(One dimensional random variable)

( , ) ( , )UV XY

x x

u vf u v f x y

y y

u v

(Two dimensional random variable)

15)Central limit theorem (Liapounoffs form)If X1, X2, Xnbe a sequence of independent R.Vs with E[Xi] = iand Var(Xi) = i2, i

= 1,2,n and if Sn= X1 + X2+ + Xnthen under certain general conditions, Sn

follows a normal distribution with mean1

n

i

i

and variance 2 21

n

i

i

asn .

16)Central limit theorem (LindbergLevys form)If X1, X2, Xnbe a sequence of independent identically distributed R.Vs with E[Xi]

= iand Var(Xi) = i2, i = 1,2,n and if Sn= X1 + X2+ + Xnthen under certain

general conditions, Snfollows a normal distribution with mean n and variance2

n as n .Note:

nS n

zn

( for n variables), Xz

n

( for single variables)


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UNIT-III (MARKOV PROCESSES AND MARKOV CHAINS)

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

functions of a real variable, namely time t where s S and t T.

2) Classification of Random Processes:We can classify the random process according to the characteristics of time t

and the random variable X. We shall consider only four cases based on t and X

having values in the ranges -


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3) Condition for Stationary Process: ( ) ConstantE X t , ( ) constantVar X t .If the process is not stationary then it is called evolutionary.

4) Wide Sense Stationary (or) Weak Sense Stationary (or) Covariance Stationary:A random process is said to be WSS or Covariance Stationary if it satisfies thefollowing conditions.

i) The mean of the process is constant (i.e) ( ) constantE X t .ii) Auto correlation function depends only on (i.e) ( ) ( ). ( )XXR E X t X t

5) Property of autocorrelation:(i) 2

( ) lim

XXE X t R

(ii) 2( ) 0XX

E X t R

6) Markov process:A random process in which the future value depends only on the present value

and not on the past values, is called a markov process. It is symbolically

represented by 1 1 1 1 0 0( ) / ( ) , ( ) ... ( )n n n n n n P X t x X t x X t x X t x 1 1

( ) / ( )n n n n

P X t x X t x Where

0 1 2 1...

n nt t t t t

7) Markov Chain:If for all n,

1 1 2 2 0 0/ , , ...

n n n n n n P X a X a X a X a

1 1/

n n n n P X a X a then the process nX , 0,1,2,...n is called the

markov chain. Where0 1 2, , , ... , ...

na a a a are called the states of the markov chain.

8) Transition Probability Matrix (tpm):When the Markov Chain is homogenous, the one step transition probability is

denoted by Pij. The matrix P = {Pij} is called transition probability matrix.

9) ChapmanKolmogorov theorem:If P is the tpm of a homogeneous Markov chain, then the n step tpm P(n)is

equal to Pn. (i.e) ( ) n

n

i j i j P P .

10) Markov Chain property:If 1 2 3, , , then P and1 2 3

1 .11) Poisson process:

If ( )X t represents the number of occurrences of a certain event in (0, )t ,then

the discrete random process ( )X t is called the Poisson process, provided thefollowing postulates are satisfied.

(i) 1 occurrence in ( , )P t t t t O t


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(ii) 0 occurrence in ( , ) 1P t t t t O t (iii) 2 or more occurrences in ( , )P t t t O t (iv) ( )X t is independent of the number of occurrences of the event in any

interval.

12) Probability law of Poisson process: ( ) , 0,1,2, ...!

nte t

P X t n n n

Mean ( )E X t t , 2 2 2( )E X t t t , ( )Var X t t .

UNIT-IV (QUEUEING THEORY)

nNumber of customers in the system.

Mean arrival rate. Mean service rate.

nPSteadyState probability of exactly n customers in the system.

qL Averagenumber of customers in the queue.

sL Average number of customers in the system.

qW Average waiting time per customer in the queue.

sW Average waiting time per customer in the system.

ModelI (M / M / 1): ( / FIFO)

1) Server Utilization 2) 1n

nP (P0no customers in the system)

3)1

sL

4) 21

qL

5) 11

sW


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6)1

qW

7) Probability that the waiting time of a customer in the system exceeds t is( )

( ) t

sP w t e

.

8) Probability that the quue size exceeds t is 1nP N n where 1n t.

ModelII (M / M / C): ( / FIFO)

1)s

2)1

1

0

0 ! ! 1

n ss

n

s sPn s

3)

1

02

1

. ! 1

s

q

sL P

s s

4)

s qL L s

5) qq

LW

6) ss

LW

7) The probability that an arrival has to wait:0

! 1

s

sP N s P

s

8) The probability that an arrival enters the service without waiting = 1P(anarrival hat to wait) = 1 P N s

9)( 1 )

0

( ) 1

1 !(1 )( 1 )

s t s s

ts e

P w t e P s s s

ModelIII (M / M / 1): (K / FIFO)

1)


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2)0 1

1

1 k

P

(No customer)3)

01 P (effective arrival rate)

4) 11

1

1 1

k

s k

kL

5)q s

L L

6) ss

LW

7) qq

LW

8) 0a customer turned away kkP P P ModelIV (M / M / C): (K / FIFO)

1)s

2)1

1

0

0 ! !

n ss k

n s

n n s

s sP

n s

3) 00

,!

,!

n

n n

n s

sP n s

nP

sP s n k

s s

4) Effective arrival rate: 10

s

n

n

s s n P

5) 1

02

1

! 11

s k s k s

q

s k s

L Ps

6)

s qL L

7) qq

LW


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8) ss

LW

UNIT-V (NONMARKOVIAN & QUEUEING NETWORK)

1) PollaczekKhintchine formula:

22

( ) ( )( )

2 1 ( )S

Var t E t L E t

E t

(or)

2 2 2

2 1S

L

2) Littles formulas:

2 2 2

2 1S

L

q SL L

S

S

LW

q

q

LW

3) Series queue (or) Tandem queue:The balance equation

00 2 01P P 1 10 00 2 11

P P P 01 2 01 1 10 2 1b

P P P P 1 11 2 11 01

P P P 2 1 1 11bP P

Condition00 10 01 11 1

1b

P P P P P 4) Open Jackson networks:

i) Jacksons flow balance equation1

k

j j i ij

i

r P


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Where knumber of nodes, rjcustomers from outside

ii) Joint steady state probabilities1 2

1 2 1 1 2 2, , ... 1 1 ... 1k

nn n

k k kP n n n

iii) Average number of customers in the system1 2

1 2

...1 1 1

k

S

k

L

iv) Average waiting time of a customers in the system

S

S

LW where 1 2 ... kr r r

5) Closed Jackson networks:In the closed network, there are no customers from outside, therefore 0jr then

i) The Jacksons flow balance equation1

k

j i ij

i

P

0jr (or)

11 12 1

221 22

1 2 1 2

1 2

...

... ... ...

...

k

k

k k

k k kk

P P P

PP P

P P P

ii) If each nodes single server1 2

1 2 1 2, , ... ... k

nn n

k N kP n n n C Where 1 2

1 2

1

1 2

...

... k

k

nn n

N k

n n n N

C

iii) If each nodes has multiple servers

1 2

1 21 2

1 2

, , ... ...

knn n

kk N

k

P n n n C a a a

Where1 2

1 2

1 1 2

... 1 2

...k

k

nn n

k

N

n n n N k

Ca a a


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Prepared by C.Ganesan, M.Sc., M.Phil., (Ph: 9841168917) Page 13

! ,

! ,i ii i i

i n s

i i i i

n n sa

s s n s

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