Tutorial 12

Cover: *Conditional Distribution and Expectation*Assignment 7 8

Conica, Cui Yuanyuan

Erlang

r-stage Erlang

1

1

0

1

0

( )1 !

( ) 1!

( )!

r r t

k tr

k

krt

k

t ef t

r

t eF t

k

tR t e

k

2

2

( )

( ) 1 (1 )

( ) (1 )

t

t

t

r

f t te

F t e t

R t e t

Cold Standby

Hypoexponential

r-stage Hypo

1 2

1 2

1 2

1 2

2 1

2 1

2 1 2 1

2 1

2 1 2 1

2

( ) ( )

( ) 1

( )

t t

t t

t t

r

f t e e

F t e e

R t e e

Warm Standby

( )

( )

( )( ) ( )

t t

t t

R t e e

e e

Laplace Transform

0

( ) ( ) [ ]sx sXf s e f x dt E e

Continuous non-negative r.v.

22

2

0 0

[ ] ( )

( ) [ ]

( ) ( ) [ ] [ ]

sX

sX

s s

E e f s

df sE Xe

ds

df s d f sE X E X

ds ds

Z Transform

0

( ) ( ) [ ]i XX

i

g z p i z E Z

Discrete non-negative r.v.

1

22

2

1 1

[ ] ( )

( ) [ ]

( ) ( ) [ ] [ ] [ ]

X

X

z z

E z g z

dg zE Xz

dz

dg z d g zE X E X E X

dz dz

Z Transform

Conditional Distribution

Discrete X and Y

( ) ( )

( ) ( )

( ) ( ) ( )

Y X

Y X

Y XY Xx

p y x P Y y X x

F y x P Y y X x

p y p y x p x

Example

( )!

xep x

x

Continuous X and Y

( , )( )

( )

( ) ( ) ( )

( , )( )

( )

Y XX

y

Y X Y X

Y xX

f x yf y x

f x

F y x P Y y X x f u x du

f x yf y dx

f x

Example

Discrete X and Continuous Y

( , )( )

( )

( ) ( ) ( )

( ) ( , ) ( ) ( )

Y XX

y

Y X Y X

Y XY Xi i

f i yf y x i

p i

F y x P Y y X i f u i du

f y f i y f y i p i

Example Consider a file server whose work load may be divided into r distinct

classes. For class i, the CPU service time is exponential distributed with i. Y : the service time of a job X : the job class

1 1

( )

( )

( ) ( ) ( )

i

i

yiY X

X i

r ry

Y X i iY Xi i

f y i e

p i

f y f y i p i e

Continuous X and Discrete Y

( ) ( )

( ) ( )

( ) ( ) ( )

Y X

Y X

Y XY Xx

p y x P Y y X x

F y x P Y y X x

p y p y x f x dx

Example

X : the service time of a request to a web server X~EXP() Y : the number of requests arriving Y~Poisson(t)

0 0

( ) ( )!

( , ) ( ) ( )!

( ) ( , )!

y

xY X

y

xX Y X

y

x yY

xp y x P Y y X x e

y

xf x y f x p y x e

y

p y f x y dx e x dxy

Conditional Expectation

[ ] ( )

[ ] ( )y

E Y x yf y x dy

E Y X x yP Y y X x

Example: r-stage Hyperexponential

1

22

1

1 1

( ) [ ]

21[ ] [ ]

( ) ( ) ( )

i

i

ry i

iY Xi i

ri

ii i

r ri i

X i Y i Yi i i

f y i e E Y

E Y X i E Y

p i L s L ss

Example

Example

Imperfect fault coverage and R(t)

X=lifetime of a system

Y=fault class

( )( , ) ( ) ( ) ( )

( )

X YX X X

Y

f t yf t y f t F t R t

f y

( )( )

( )

X YX

Y

R t yR t

f y

Example

Random Sum

1 2

22

2 2 2

: [ ] [ ] [ ] [ ] [ ]

: [ ] [ ] [ ]

[ ] [ ] [ ] [ ] [ ]

Var[ ] [ ] [ ] [

L NT X X X

Expectation E T N n nE X E T E X E N

Variance E T N n nVar X nE X

E T Var X E N E N E X

T Var X E N Var N

2] [ ]

: ( ) ( ) : G ( ) ( )

( ) ( ) ( ) G ( ) ( ) ( )

n n

X XT N T N

n n

T X N T X Nn n

E X

LST L s n L s PGF z n G z

L s L s p n z G z p n

Random Sum

1

1

Especially, when N is geometric distribution with p

:

( ) ( ) (1 )

( ) =

1 (1 ) ( )

:

( ) ( ) (1 )

=

n nT X

n

X

X

n nT X

n

X

LST

L s L s p p

pL s

p L s

PGF

G z G z p p

pG

( )

1 (1 ) ( )X

z

p G z

Example

Example

Thanks for coming!Questions?