f-pqt

Engineering Mathematics Material 2010

Prepared by C.Ganesan, M.Sc., M.Phil., (Ph: 9841168917) Page 1

SUBJECT NAME : Probability & Queueing Theory

SUBJECT CODE : MA 2262

MATERIAL 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 variable 1

( ) 1ii

p x∞∞∞∞

====∑∑∑∑ ( ) 1f x dx∞∞∞∞

−∞−∞−∞−∞

====∫∫∫∫

2 [[[[ ]]]]( )F x P X x= ≤= ≤= ≤= ≤ [[[[ ]]]]( ) ( )x

F x P X x f x dx−∞−∞−∞−∞

= ≤ == ≤ == ≤ == ≤ = ∫∫∫∫

[[[[ ]]]]Mean ( )i ii

E X x p x= == == == =∑∑∑∑ [[[[ ]]]]Mean ( )E X xf x dx∞∞∞∞

−∞−∞−∞−∞

= == == == = ∫∫∫∫

4 2 2 ( )i ii

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 ii

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

∞∞∞∞

−∞−∞−∞−∞

==== ∫∫∫∫

7 M.G.F M.G.F

3

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(((( )))) ( )tX txX

x

M t E e e p x = == == == = ∑∑∑∑ (((( )))) ( )tX txXM 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) 1st Moment about origin = [[[[ ]]]]E X = (((( ))))0

Xt

M t====

′′′′

(Mean)

2nd Moment about origin =

2E X = (((( ))))0

Xt

M t====

′′′′′′′′

The co-efficient of !

rtr

= rE X (r

th 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 not

exist except perhaps at one point. 14) Properties of M.G.F:

i) If Y = aX + b, then (((( )))) (((( ))))btY XM t e M at==== .

ii) (((( )))) (((( ))))cX XM t M ct==== , where c is constant.

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

(((( )))) (((( )))) (((( ))))X Y X YM t M t M t++++ = ⋅= ⋅= ⋅= ⋅ .

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

No. Distributio

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

1 Binomial x n xxnc p q −−−− (((( ))))ntq pe++++

np npq

2 Poisson

!

xex

λλλλ λλλλ−−−−

(((( ))))1te

eλλλλ −−−−

λλλλ λλλλ

3 Geometric 1xq p−−−− (or) xq p

1

t

t

peqe−−−−

1p 2

qp

via

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4 Negative

Binomial 1( 1) k xkx k C p p−−−−+ −+ −+ −+ −

1

k

t

pqe

−−−−

kqp 2

kqp

5 Uniform 1,

( )0, otherwise

a x bf x b a

< << << << <==== −−−−

( )

bt ate eb a t

−−−−−−−−

2

a b++++

2( )12

b a−−−−

6 Exponential , 0, 0( )

0, otherwise

xe xf x

λλλλλ λλ λλ λλ λ−−−− > >> >> >> >====

t

λλλλλλλλ −−−−

1λλλλ 2

1λλλλ

7 Gamma 1

( ) , 0 , 0( )

xe xf x x

λλλλ

λλλλλλλλ

− −− −− −− −

= < < ∞ >= < < ∞ >= < < ∞ >= < < ∞ >ΓΓΓΓ

1

(1 )t λλλλ−−−−

λλλλ λλλλ

8 Weibull 1( ) , 0, , 0xf x x e xβββββ αβ αβ αβ ααβ α βαβ α βαβ α βαβ α β− −− −− −− −= > >= > >= > >= > >

16) Memoryless property of exponential distribution

(((( )))) (((( ))))/P X S t X S P X t> + > = >> + > = >> + > = >> + > = > .

UNIT-II (RANDOM VARIABLES)

1) 1iji 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 bP X a Y b

P Y b

< << << << << < =< < =< < =< < =

<<<<

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

( / )( )

f x yf x y

f y==== .

Conditional density function of Y given X, ( , )

( / )( )

f x yf y x

f x==== .

via

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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, ( ) ( ) ( , )Yf y f y f x y dx∞∞∞∞

−∞−∞−∞−∞

= == == == = ∫∫∫∫

7) ( 1) 1 ( 1)P X Y P X Y+ ≥ = − + <+ ≥ = − + <+ ≥ = − + <+ ≥ = − + <

8) Correlation co – efficient (Discrete): ( , )

( , )X Y

Cov X Yx yρρρρ

σ σσ σσ σσ σ====

1( , )Cov X Y XY XY

n= −= −= −= −∑∑∑∑ ,

2 21X X X

nσσσσ = −= −= −= −∑∑∑∑ ,

2 21Y Y Y

nσσσσ = −= −= −= −∑∑∑∑

9) Correlation co – efficient (Continuous): ( , )

( , )X Y

Cov X Yx yρρρρ


(((( )))) (((( )))) (((( ))))( , ) ,Cov X Y E X Y E X E Y= −= −= −= − , ( )X Var Xσσσσ ==== , ( )Y Var Yσσσσ ====

10) If X and Y are uncorrelated random variables, then ( , ) 0Cov 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 yb

y y

− −− −− −− −====

−−−−∑∑∑∑∑∑∑∑

Regression line Y on X is (((( ))))yxy y b x x− = −− = −− = −− = − , (((( )))) (((( ))))

(((( ))))2yx

x x y yb

x x

− −− −− −− −====

−−−−∑∑∑∑∑∑∑∑

Correlation through the regression, .XY YXb bρρρρ = ±= ±= ±= ± Note: ( , ) ( , )x y r x yρρρρ ====

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13) Regression for Continuous random variable:

Regression line X on Y is (((( ))))( ) ( )xyx E x b y E y− = −− = −− = −− = − , xxy

y

b rσσσσσσσσ

====

Regression line Y on X is (((( ))))( ) ( )yxy E y b x E x− = −− = −− = −− = − , y

yxx

b rσσσσσσσσ

====

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 X

dxf y f x

dy==== (One dimensional random variable)

( , ) ( , )UV XY

u ux y

f u v f x yv vx y

∂ ∂∂ ∂∂ ∂∂ ∂∂ ∂∂ ∂∂ ∂∂ ∂

====∂ ∂∂ ∂∂ ∂∂ ∂∂ ∂∂ ∂∂ ∂∂ ∂

(Two dimensional random variable)

15) Central limit theorem (Liapounoff’s form)

If X1, X2, …Xn be a sequence of independent R.Vs with E[Xi] = µi and Var(Xi) = σi2, i

= 1,2,…n and if Sn = X1 + X2 + … + Xn then under certain general conditions, Sn

follows a normal distribution with mean 1

n

ii

µ µµ µµ µµ µ====

====∑∑∑∑ and variance 2 2

1

n

ii


====∑∑∑∑ as

n → ∞→ ∞→ ∞→ ∞ .

16) Central limit theorem (Lindberg – Levy’s form)

If X1, X2, …Xn be a sequence of independent identically distributed R.Vs with E[Xi]

= µi and Var(Xi) = σi2, i = 1,2,…n and if Sn = X1 + X2 + … + Xn then under certain

general conditions, Sn follows a normal distribution with mean nµµµµ and variance

2nσσσσ as n → ∞→ ∞→ ∞→ ∞ .

Note: nS n

zn

µµµµσσσσ

−−−−==== ( for n variables), X

z

n

µµµµσσσσ−−−−==== ( for single variables)

via

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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 Є S and t Є 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 -∞< t <∞ and -∞ < x < ∞.

Continuous random process

Continuous random sequence

Discrete random process

Discrete random sequence

Continuous random process:

If X and t are continuous, then we call X(t) , a Continuous Random Process.

Example: If X(t) represents the maximum temperature at a place in the

interval (0,t), {X(t)} is a Continuous Random Process.

Continuous Random Sequence:

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

called a Continuous Random Sequence.

Example: If Xn represents the temperature at the end of the nth hour of a day, then

{Xn, 1≤n≤24} is a Continuous Random Sequence.

Discrete Random Process:

If X assumes only discrete values and t is continuous, then we call such random

process {X(t)} as Discrete Random Process.

Example: If X(t) represents the number of telephone calls received in the interval

(0,t) the {X(t)} is a discrete random process since S = {0,1,2,3, . . . }

Discrete Random Sequence:

A random process in which both the random variable and time are discrete is

called Discrete Random Sequence.

Example: If Xn represents the outcome of the nth toss of a fair die, the {Xn : n≥1} is a

discrete random sequence. Since T = {1,2,3, . . . } and S = {1,2,3,4,5,6}

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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 the

following conditions.

i) The mean of the process is constant (i.e) (((( ))))( ) constantE X t ==== .

ii) Auto correlation function depends only on ττττ (i.e)

[[[[ ]]]]( ) ( ). ( )XXR E X t X tτ ττ ττ ττ τ= += += += +

5) Property of autocorrelation:

(i) (((( )))) (((( ))))2

( ) lim XXE X t R

ττττττττ

→ ∞→ ∞→ ∞→ ∞ ====

(ii) (((( )))) (((( ))))2( ) 0XXE 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 nP X t x X t x X t x X t x+ + − −+ + − −+ + − −+ + − −≤ = = =≤ = = =≤ = = =≤ = = =

1 1( ) / ( )n n n nP 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 nP X a X a X a X a− − − −− − − −− − − −− − − −= = = == = = == = = == = = = 1 1/n n n nP X a X a− −− −− −− −= = == = == = == = =

then the process {{{{ }}}}nX , 0,1,2, ...n ==== is called the markov chain. Where

0 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) Chapman – Kolmogorov theorem:

If ‘P’ is the tpm of a homogeneous Markov chain, then the n – step tpm P(n) is

equal to Pn. (i.e)

( ) nnij ijP P ==== .

10) Markov Chain property: If (((( ))))1 2 3, ,Π = Π Π ΠΠ = Π Π ΠΠ = Π Π ΠΠ = Π Π Π , then PΠ = ΠΠ = ΠΠ = ΠΠ = Π and

1 2 3 1Π + Π + Π =Π + Π + Π =Π + Π + Π =Π + Π + Π = .

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 the

following postulates are satisfied.

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(i) [[[[ ]]]] (((( ))))1 occurrence in ( , )P t t t t O tλλλλ+ ∆ = ∆ + ∆+ ∆ = ∆ + ∆+ ∆ = ∆ + ∆+ ∆ = ∆ + ∆

(ii) [[[[ ]]]] (((( ))))0 occurrence in ( , ) 1P t t t t O tλλλλ+ ∆ = − ∆ + ∆+ ∆ = − ∆ + ∆+ ∆ = − ∆ + ∆+ ∆ = − ∆ + ∆

(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 tP X t n n

n

λλλλ λλλλ−−−−

= = = ∞= = = ∞= = = ∞= = = ∞

Mean [[[[ ]]]]( )E X t tλλλλ==== , 2 2 2( )E X t t tλ λλ λλ λλ λ = += += += + , [[[[ ]]]]( )Var X t tλλλλ==== .

UNIT-IV (QUEUEING THEORY)

n – Number of customers in the system.

λλλλ – Mean arrival rate.

µµµµ – Mean service rate.

nP – Steady State probability of exactly n customers in the system.

qL – Average number 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.

Model – I (M / M / 1): (∞ / FIFO)

1) Server Utilization λλλλρρρρµµµµ

====

2) (((( ))))1nnP ρ ρρ ρρ ρρ ρ= −= −= −= − (P0 no customers in the system)

3) 1sL

ρρρρρρρρ

====−−−−

4) 2

1qLρρρρ

ρρρρ====

−−−−

5) (((( ))))1

1sWµ ρµ ρµ ρµ ρ

====−−−−

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6) (((( ))))1qWρρρρ

µ ρµ ρµ ρµ ρ====

−−−−

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

( )( ) tsP w t e µ λµ λµ λµ λ− −− −− −− −> => => => = .

8) Probability that the quue size exceeds “t” is (((( )))) 1nP N n ρρρρ ++++> => => => = where

1n t= += += += + .

Model – II (M / M / C): (∞ / FIFO)

1) s

λλλλρρρρµµµµ

====

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

(((( ))))

11

00 ! ! 1

n ss

n

s sP

n s

ρ ρρ ρρ ρρ ρρρρρ

−−−−−−−−

====

= += += += +

−−−− ∑∑∑∑

3) (((( ))))(((( ))))

1

02

1. ! 1

s

q

sL P

s s

ρρρρρρρρ

++++

====−−−−

4) s qL L sρρρρ= += += += +

5) qq

LW

λλλλ====

6) ss

LW

λλλλ====

7) The probability that an arrival has to wait: (((( )))) (((( ))))(((( )))) 0! 1

ss

P N s Ps

ρρρρρρρρ

≥ =≥ =≥ =≥ =−−−−

8) The probability that an arrival enters the service without waiting = 1 – P(an

arrival hat to wait) = (((( ))))1 P N s− ≥− ≥− ≥− ≥

9) (((( ))))( 1 )

0

( ) 11

!(1 )( 1 )

s t s s

ts e

P w t e Ps s s

µ ρµ ρµ ρµ ρµµµµ

ρρρρρ ρρ ρρ ρρ ρ

− − −− − −− − −− − −−−−−

−−−− > = +> = +> = +> = + − − −− − −− − −− − −

Model – III (M / M / 1): (K / FIFO)

1) λλλλρρρρµµµµ

====

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2) 0 1

11 kP

ρρρρρρρρ ++++

−−−−====−−−−

(No customer)

3) (((( ))))01 Pλ µλ µλ µλ µ′′′′ = −= −= −= − (effective arrival rate)

4) (((( )))) 1

1

1

1 1

k

s k

kL

ρρρρρρρρρ ρρ ρρ ρρ ρ

++++

++++

++++= −= −= −= −

− −− −− −− −

5) q sL Lλλλλµµµµ

′′′′= −= −= −= −

6) ss

LW

λλλλ====

′′′′

7) qq

LW

λλλλ====

′′′′

8) [[[[ ]]]] 0a customer turned away kkP P Pρρρρ= == == == =

Model – IV (M / M / C): (K / FIFO)

1) s

λλλλρρρρµµµµ

====

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

11

00 ! !

n ss kn s

n n s

s sP

n s

ρ ρρ ρρ ρρ ρρρρρ

−−−−−−−−

−−−−

= == == == =

= += += += + ∑ ∑∑ ∑∑ ∑∑ ∑

3)

0

, !

, !

n

n n

n s

sP n snP

sP s n k

s s

ρρρρ

−−−−

≤≤≤≤====

4) Effective arrival rate: (((( ))))1

0

s

nn

s s n Pλ µλ µλ µλ µ−−−−

====

′′′′ = − −= − −= − −= − − ∑∑∑∑

5) (((( )))) (((( ))))

(((( ))))(((( )))) 1

02

1

! 11

s k s k s

q

s k sL P

s

ρ ρρ ρρ ρρ ρρ ρρ ρρ ρρ ρρρρρρρρρ

−−−− − +− +− +− + −−−− −−−− = −= −= −= −

−−−− −−−−

6) s qL Lλλλλµµµµ′′′′

= += += += +

7) qq

LW

λλλλ====

′′′′

ρρρρ

0

≤≤≤≤ ≤≤≤≤

( )

( )

via

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8) ss

LW

λλλλ====

′′′′

UNIT-V (NON – MARKOVIAN & QUEUEING NETWORK)

1) Pollaczek – Khintchine formula:

(((( ))))[[[[ ]]]]

22 ( ) ( )( )

2 1 ( )S

Var t E tL E t

E t

λλλλλλλλ

λλλλ

++++ = += += += +

−−−−

(or)

(((( ))))2 2 2

2 1SLλ σ ρλ σ ρλ σ ρλ σ ρρρρρ

ρρρρ++++= += += += +

−−−−

2) Little’s formulas:

(((( ))))2 2 2

2 1SLλ σ ρλ σ ρλ σ ρλ σ ρρρρρ

ρρρρ++++= += += += +

−−−−

q SL L ρρρρ= −= −= −= −

SS

LW

λλλλ====

qq

LW

λλλλ====

3) Series queue (or) Tandem queue:

The balance equation

00 2 01P Pλ µλ µλ µλ µ====

1 10 00 2 11P P Pµ λ µµ λ µµ λ µµ λ µ= += += += +

01 2 01 1 10 2 1bP P P Pλ µ µ µλ µ µ µλ µ µ µλ µ µ µ+ = ++ = ++ = ++ = +

1 11 2 11 01P P Pµ µ λµ µ λµ µ λµ µ λ+ =+ =+ =+ =

2 1 1 11bP Pµ µµ µµ µµ µ====

Condition 00 10 01 11 1 1bP P P P P+ + + + =+ + + + =+ + + + =+ + + + =

4) Open Jackson networks:

i) Jackson’s flow balance equation 1

k

j j i iji

r Pλ λλ λλ λλ λ====

= += += += +∑∑∑∑

via

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Where k – number of nodes, rj – customers from outside

ii) Joint steady state probabilities

(((( )))) (((( )))) (((( )))) (((( ))))1 21 2 1 1 2 2, , ... 1 1 ... 1knn n

k k kP n n n ρ ρ ρ ρ ρ ρρ ρ ρ ρ ρ ρρ ρ ρ ρ ρ ρρ ρ ρ ρ ρ ρ= − − −= − − −= − − −= − − −

iii) Average number of customers in the system

1 2

1 2

...1 1 1

kS

k

Lρρρρρ ρρ ρρ ρρ ρ

ρ ρ ρρ ρ ρρ ρ ρρ ρ ρ= + + += + + += + + += + + +

− − −− − −− − −− − −

iv) Average waiting time of a customers in the system

SS

LW

λλλλ==== where 1 2 ... kr r rλλλλ = + + += + + += + + += + + +

5) Closed Jackson networks:

In the closed network, there are no customers from outside, therefore 0jr ====

then

i) The Jackson’s flow balance equation 1

k

j i iji

Pλ λλ λλ λλ λ====

====∑∑∑∑ 0jr ====∵∵∵∵

(or)

(((( )))) (((( ))))11 12 1

221 221 2 1 2

1 2

...

... ... ...

...

k

kk k

k k kk

P P P

PP P

P P P

λ λ λ λ λ λλ λ λ λ λ λλ λ λ λ λ λλ λ λ λ λ λ

====

� ��

ii) If each nodes single server

(((( )))) 1 2

1 2 1 2, , ... ... knn nk N kP n n n C ρ ρ ρρ ρ ρρ ρ ρρ ρ ρ====

Where 1 2

1 2

11 2

...

... k

k

nn nN k

n n n N

C ρ ρ ρρ ρ ρρ ρ ρρ ρ ρ−−−−

+ + + =+ + + =+ + + =+ + + ===== ∑∑∑∑

iii) If each nodes has multiple servers

(((( ))))1 2

1 21 2

1 2

, , ... ...knn n

kk N

k

P n n n Ca a a

ρρρρρ ρρ ρρ ρρ ρ====

Where 1 2

1 2

1 1 2

... 1 2

...k

k

nn nk

Nn n n N k

Ca a a

ρρρρρ ρρ ρρ ρρ ρ−−−−

+ + + =+ + + =+ + + =+ + + =

==== ∑∑∑∑

via

http://csetube.tk/






! ,

! ,i i

i i i

i n si i i i

n n sa

s s n s−−−−

<<<<==== ≥≥≥≥

---- All the BestAll the BestAll the BestAll the Best ----

via

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