Transcript
8/12/2019 Pqt Formula Vidhyatri
1/13
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/8/12/2019 Pqt Formula Vidhyatri
2/13
( )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
http://www.vidyarthiplus.com/vp/8/12/2019 Pqt Formula Vidhyatri
3/13
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 .
http://www.vidyarthiplus.com/vp/8/12/2019 Pqt Formula Vidhyatri
4/13
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
http://www.vidyarthiplus.com/vp/8/12/2019 Pqt Formula Vidhyatri
5/13
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)
http://www.vidyarthiplus.com/vp/8/12/2019 Pqt Formula Vidhyatri
6/13
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 -
8/12/2019 Pqt Formula Vidhyatri
7/13
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
http://www.vidyarthiplus.com/vp/8/12/2019 Pqt Formula Vidhyatri
8/13
(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
http://www.vidyarthiplus.com/vp/8/12/2019 Pqt Formula Vidhyatri
9/13
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)
http://www.vidyarthiplus.com/vp/8/12/2019 Pqt Formula Vidhyatri
10/13
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
http://www.vidyarthiplus.com/vp/8/12/2019 Pqt Formula Vidhyatri
11/13
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
http://www.vidyarthiplus.com/vp/8/12/2019 Pqt Formula Vidhyatri
12/13
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
http://www.vidyarthiplus.com/vp/8/12/2019 Pqt Formula Vidhyatri
13/13
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
---- ll the Best ----
http://www.vidyarthiplus.com/vp/
top related