Maciej Stasiak, Mariusz GłąbowskiArkadiusz Wiśniewski, Piotr Zwierzykowski

Models of Links Carrying Single-Service Traffic

Chapter 7

Modeling and Dimensioning of Mobile Networks: from GSM to

LTE

2

Two-dimensional Erlang distributionAssumptions

• V channels in the full-availability groupo each of them is available if it is not busy

• Arrivals create two Poisson streams with intensities 1 and 2

• Service times has exponential distribution with parameters 1 and 2

• Rejected call is lost

3

Two-dimensional Erlang distribution

• Offered traffic:

111 / A

222 / A

Full-availability group with two call streams

4

Microstate definition

• Operation of the system is determined by the so-called two-dimensional Markov chain with continuous time: {x(t), y(t)},where x(t) and y(t) are the numbers of channels occupied at the moment t by calls of class 1 and 2, respectively

• Steady-microstate probabilities

Vtytx )()(0

ytyxtxppt

Vyx

)(;)(lim,

5

State transition diagram for two-dimensional Markov chain

• state {0,0} – all channels are free,

• state {x,y} – x channels are servicing calls of class 1, y channels are servicing calls of class 2,

6

Statistical equilibrium equations

1

)(for)(

)(for)1()1(

)(

0for)(

0

1,2,11,21

1,2,11

1,2,11

,2121

1,020,110,021

N

iNi

VyxVyxVyx

VyxVyx

VyxVyx

Vyx

VVV

p

Vyxpppyx

Vyxpypx

pp

pyx

yxppp

7

Two-dimensional Erlang distribution

• Distribution:

• Blocking probability:

v

i

iv

j

ji

yx

Vyx

j

A

i

A

y

A

x

A

p

0 0

21

21

,

!!

!!

v

i

iv

j

ji

VQ

yx

j

A

i

A

y

A

x

A

EE

0 0

21

)(

21

21

!!

!!}:,{)( VyxyxVQ where

8

Reversibility of the two-dimensional Markov process

Necessary and sufficient condition for reversibility (Kolmogorov criteria):

The circulation flow (product of streams parameters) among any four neighboring states in a square equals zero.

Flow clockwise = Flow counterclockwise

2121

1212

)1()1(

)1()1(

yx

xy

11x

11x

2

11x

11x

1

y1 2 2)1( y 2

1,1 yx

x y, x y1,

x y, 1

1

9

Reversibility of the two-dimensional Markov process

Reversibility property leads to local balance equations between any two neighboring microstates of the process.

If there exists a possibility to achieve the microstate {x2 y2} outgoing from the microstate {x1 y1}, then there exists the possibility to achieve the microstate {x1 y1}, outgoing from the microstate {x2 y2}.

Reversibility property leads to local balance equations between any two neighboring microstates of the process.

If there exists a possibility to achieve the microstate {x2 y2} outgoing from the microstate {x1 y1}, then there exists the possibility to achieve the microstate {x1 y1}, outgoing from the microstate {x2 y2}.

10

Reversibility of two-dimensional Markov process

• Noteo Between any two neighboring microstates of the process we

can (as in the case of one-dimensional birth and death process) write local balance equations

VyxVyx pxp ,11,1 )1(

VyxVyx pyp 1,2,2 )1(

11x

11x

2

11x

11x

1

y1 2 2)1( y 2

1,1 yx

x y, x y1,

x y, 1

1

11

Product form solution of two-dimensional distribution

• Independently on the chosen path between microstates {x, y} and {0, 0}, we always obtain:

• Where GV is the normalization constant :

V

yx

V

yx

Vyx px

A

y

Ap

xyp 0,0

210,0

2

2

1

1

, !!!!

V

yx

VyxVyx Gx

A

y

AGppp

!!21

,

V

i

iV

j

jiV

jA

iA

G

0 0

21

!!

1

12

Example of the two-dimensional Erlang distribution

1

2

21

1

21

0 2

2

1

1

0

2

2

1

1

21

!!

!!),( xV

x

xxV

x

xx

xA

xA

xA

xA

xxp

1.0)2,0()0,2(

2,0)1,1()1,0()0,1()0,0(

pp

pppp

4.0)2,0()1,1()0,2(

),( 21

21

ppp

xxpEVxx

1,2 2121 V

1

1

1

1

1 1

1

1

1 1

1 1

0,0 1,0 2,0

0,1

0,2

1,1

13

Macrostate probability

)(

,n

VyxVn pP

where:

!!21

)(,

)( y

A

x

AGpP

yx

nVVyx

nVn

}:),{()( nyxyxn

V

i

i

n

Vn

i

AAn

AA

P

0

21

21

!

!

)(

2121 !!

!n

yxn

y

A

x

AnAA

14

Blocking probability – macrostate level

V

i

i

n

VV

iAA

nAA

PE

0

21

21

!

!

Example: 4.0)2()( 221 EAAEE V

15

Multi-dimensional Erlang distribution

• Assumptions:o V channels in the full availability trunk group; each of

them is available if it is not busy;

o Arrivals create M Poisson streams with intensities 1, 2, ..., M

o Service times have exponential distribution with parameters 1, 2, ..., M

o Rejected call is lost

16

Multi-dimensional Erlang distribution

g Call streams

1

BBU

1

V

2

2

M

iiiA /

Full-availability group with M call streams

offered traffic:

17

Microstates in multi-dimensional Erlang distribution

• state {x1,..., xi ,..., xM }

o x1 channels are servicing calls of class 1,

o . . .

o xi channels are servicing calls of class i,

o . . .

o xM channels are servicing calls of class M.

• Total number of busy channels:

VxM

ii

1

18

State transition diagram for multi-dimensional Markov chain

State interpretation:

state  (x1,..., xi,..., xM) - group services x1 calls of class 1, ..., xi calls of class i, ..., xM calls of class M.

x x M1 ,...,

x x M1 1+ ,...,

x x M1 1,..., +

x x M1 1- , ...,

x x M1 1,..., -

m1 1x

l 1

mM Mx

l M

m1 1 1x +

l 1

mM Mx +1

l M

19

Statistical equilibrium equations

g

x1, ..., xi, ..., xM +1

x1 +1, ..., xi, ..., xM

M

x1, ..., xi, ..., xM -1

x1 -1, ..., xi, ..., xM

x1, ..., xi -1, ..., xM i

1

i

M xM M

x1 1

(x1 +1)1

(xi +1)i

xi i

(xM +1)M

1

x1, ..., xi +1, ..., xM x1, ..., xi ..., xM

1),..,.,..,.(

...

),..,.1,..,.(

),..,.1,..,.()1(),..,.,..,.()(

1

11

11

11

VMi

VMi

M

ii

VMi

M

iiiVMiii

M

ii

xxxp

xxxp

xxxpxxxxpx

20

Reversibility of multi-dimensional Markov process

g x1 +1, ..., xi, ..., xM

M

x1, ..., xi, ..., xM -1

x1 -1, ..., xi, ..., xM

i

(xi +1)i

i

1

i

M xM M

x1 1

(x1 +1)1

(xi +1)i

xi i M

1

(xM +1)M (xM +1)M

x1, ..., xi -1, ..., xM x1, ..., xi, ..., xM x1, ..., xi +1, ..., xM

x1, ..., xi, ..., xM +1 x1, .., xi +1, ..., xM +1

iiMMiMMMiiMi xxxx )1()1()1()1(

21

Multi-dimensional Erlang distribution

• All offered streams are considered to be mutually independent and the service process in the group is reversible, so we can rewrite each microstate in product form:

M

jVjVVMi xpGxxxp

11 )(),,,,(

M

j j

xj

VVMi x

AGxxxp

j

11 !

),,,,(

V

z

l

z

l

z

M

i i

zi

Vj

j

M

M

i

z

AG

0 0 0 11!

1

1j

1kkj zVlwhere:

22

Macro-state probability

VMn

Vn xxxpP ),,,( 21)(

M

i i

xi

n

nM

ii x

AnA

i

1)(1 !!

V

k

kM

jj

nM

jj

Vn k

A

n

A

P0 !

/!

)(n },,,,{ Mi1 xxx

M

iixn

1

where: is the set of such subsets in which the following

equation is fulfilled:

23

Interpretation of macrostates distribution

call stream with intensity:

Service time – hyper-exponential distribution (weighed sum of exponential distributions) with average value:

Blocking probability:

Multi-dimensional distribution is the Erlang distribution for traffic:

M

iiAA

1

This distribution can be treated as a model of the full-availability group with parameters:

M

ii

1

M

1ii

M

1i i

i

i

M

1i

i A1111

M

iiV AEBE

1

24

Recurrent form of multidimensional Erlang distribution

Vn

M

iiVn PAPn 1

1

Calculation algorithm:

)(

)(/1

)1(1

)(

1)0(

0

1

nqGP

iqG

nqAn

nq

q

VVn

V

iV

M

ii

25

Birth and death process calibration (calibration constant)

• A section of a state transition diagram for the birth and death process in the full availability group:

• A section of the calibrated state transition diagram for the birth and death process in the full availability group (calibration constant 1/ ):

n

n-1 n

n+1

(n+1)

n

A

n-1 n

n+1

A

(n+1)

26

Interpretation of recurrent notation form of multidimensional Erlang distribution

Vn

M

iiVn

M

ii PAPny 1

11

)(

Vn

M

iiVn PAPn 1

1

y1(n+1)

AM

A1 A1

AM

y1(n)

yM(n) yM(n+1)

n+1

n

n-1

A fragment of a state transition diagram which interprets the recurrent form of multidimensional Erlang distribution

Calibration constant:

Each component process is calibrated by „own” calibration constant 1/i

27

Service streams

y1(n+1)

AM

A1 A1

AM

y1(n)

yM(n) yM(n+1)

n+1

n

n-1

Vn

VnPPAny VnVni

i 1dla0

1dla)1( 1/

M

iVni

M

iVni PnyPA

11

1

)1(

Balance equation for state n:

)(1

nyn i

M

i

Vn

M

iiVn PAPn 1

1

Vni

M

iVn

M

iii

M

i

M

iiVn PnyPAnyAP 1

11

111

)1()(

VniVni PnyPA 1)1(

This equation is fulfilled when the local balance equations are fulfilled for each stream i :

)(1

nyn i

M

i

Vn

M

iiVn PAPn 1

1

VniVni PnyPA 1)1(