modeling and dimensioning of mobile networks: from gsm...

Maciej Stasiak, Mariusz Głąbowski

Arkadiusz Wiśniewski, Piotr Zwierzykowski

Groups models

Modeling and Dimensioning of Mobile Networks: from GSM to LTE

Erlang Model

Modeling and Dimensioning of Mobile Networks: from GSM to LTE 2

Full-availability group (FAG)

• Assumptions:

o V channels in the full-availability trunk group. Each of them is

available if it is not busy;

o Arrival process is the Poisson process;

o Service time has exponential distribution with parameter 1/μ;

o Rejected call is lost


1

2

V

State transition diagram

• state „0” - all channels are free,

• state „1” - one channel is busy, others are free,

• . . .,

• state „i” - i channels are busy and (V-i) are free,

• . . .,

• state „V” - all channels are busy.


0 1

i

i-1

(i+1)

i i+1

V

V-1 V

Statistical equilibrium equations


0 1

i

i-1

(i+1)

i i+1

V

V-1 V

V

i

ik

Vkik

p0 !!

10

1

1

10

V

iVi

VVVV

ViVi

VV

p

pVp

pip

pp

Interpretation λ/μ


h

1

h

/ determines the average number of arrivals within average service time

Erlang’s distribution


Distribution of busy channels in the FAG, capacity V=5, offered traffic: A=1 Erl. (a); A=3 Erl. (b); A=8 Erl. (c).

/where!!!

)/(

!

)/(

00

Ai

A

k

A

ikp

V

i

ikV

i

ik

Vk

0

0,1

0,2

0,3

0,4

0,5

0,6

0 1 2 3 4 5

0

0,1

0,2

0,3

0,4

0,5

0,6

0 1 2 3 4 5

0

0,1

0,2

0,3

0,4

0,5

0,6

0 1 2 3 4 5

a) b) c)

Erlang formula


V

i

i

V

VVN

i

A

V

A

pAE

0

,1

!

!)(

Blocking probability = f ( offered traffic, capacity)

Recurrence property of Erlang formula


)(

)()(

1,1

1,1

,1AAEN

AAEAE

N

N

N

1!0!0

)(00

0,1 AA

AE

Characteristics of carried traffic

• Mean value of carried traffic (average number of

simultaneously busy channels)

• Variance of carried traffic


][ (A)EApkY ,V

V

kVkV 1

0

1

)()()( ,1

2

0

22

VVVV

V

kVk YVAAEYYpk

V




.

!

])!2()!1(

[

!

)!1(]1)1[(

!

!)(

2

0

2

2

1

1

2

0

1

02

0

0

2

2

0

22

VV

k

k

N

k

kV

k

k

VV

k

k

kV

kVV

k

k

V

k

k

V

V

kVkV

Y

k

A

k

AA

k

AA

Y

k

A

k

AkA

Y

k

A

k

Ak

Ypk



o Taking into account:

o we obtain:


),(1

!

)!1(,1

0

1

1

AE

k

A

k

A

VV

k

k

V

k

k

)()(1

!

)!2(,1,1

0

2

2

AEA

NAE

k

A

k

A

VVV

k

k

V

k

k

)()(,1

2

VVV YVAAEYV


• NOTE !

o Variance of offered traffic

o is equal to

o mean value of offered traffic


Erlang tables

• Two kinds of Erlang tables in engineering practice:

o

o N A1 A2 A3 N N B1 B2 B3 N

o 1 B11 B21 B31 1 1 A11 A21 A31 1

o 2 B12 B22 B32 2 2 A12 A22 A32 2

o 3 B13 B23 B33 3 3 A13 A23 A33 3


Erlang table


Capacity V

Blocking probability B

E=0.02

E=0.01

E=0.005

E=0.001

Offered traffic intensity A

1

0.02

0.01

0.005

0.001

5

1.70

1.40

1.10

0.80

10

5.10

4.50

4.00

3.10

15

9.00

8.10

7.38

6.08

20

13.20

12.00

11.10

9.41

25

17.50

16.10

15.00

13.00

30

21.90

20.30

19.00

16.70

35

26.40

24.60

23.20

20.50

40

31.00

29.00

27.40

24.40

45

35.60

33.40

31.70

28.40

50

40.30

37.90

36.00

32.50

60

49.60

46.90

44.80

40.80

70

59.1

56.1

53.70

49.20

80

68.70

65.40

62.70

57.80

90

78.30

74.70

71.80

66.50

100

88.00

84.10

80.90

75.20

Group principle


A1

V1

A2

V2

A1 + A2

V1 + V2

Two groups joint group

)(,)(max)( 2121 2121AEAEAAE VVVV

Group principle - example


group 1 group 2

joint group

0,001 ) (

0,01 ) ( 0,02 ) (

20 , Erl. 12,0 10 , Erl. 5,1

2 1 ) (

2 1

1 2 1 1

2 1

2 1

A A E

A E A E

V A V A

V V

V V

Poisson distribution

• Border case of Erlang distribution

• The number of channels is infinite, so there is no

blocking in the system


V

A

kV

i

ik

VVk e

k

A

i

A

k

Ap

!!

/!

lim0

Poisson distribution

• Approximation of blocking probability

o If the number of servers is equal to V, the blocking probability

can be approximated by the Poisson model:


!1

!!

1

0 i

Ae

i

Aee

i

AE

iV

i

A

i

Vi

AA

i

Vi

Channel load – random hunting

• Traffic carried by V channels:

• Traffic carried by any channel:

• For V=10, A=10 Erl.:


)](1[ AEAY VV

V(A)EAVY VV /]1[/

Erl.79.010/]101[10/ 10 )(EVYV

Channel load – random hunting


0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

group: A=10 Erl., V=10

Load

Channel number

Channel load – successive hunting

• Traffic carried by i channels:

• Traffic carried by i-1 channels:

• Traffic carried by channel i:


)](1[ AEAY ii

)](1[ 11 AEAY ii

(A)E(A)EA

(A)EA(A)EAYY

ii

iiiii

1

11 ]1[]1[

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

group: A=10 Erl., V=10 Load

Channel number

Channel load – successive hunting


Palm – Jacobaeus formula

• Formula defines occupancy probability of x exactly

determined servers


occupancy probability of

any i channels

Conditional occupancy probability of

x exactly determined servers under

condition that i servers are busy:

)!(

!

!

)!()|(

xi

i

V

xV

i

V

xi

xVixP

V

k

ki

Vik

A

i

Ap

0 !!

)(

)()()(

AE

AEpixPxH

xV

VV

xiVi

Engset Model


Full availability group – Engset model

• Assumptions:

o V channels in the full availability trunk group. Each of them is

available if it is not busy;

o Arrivals create a stream generated by a finite number of N

(N>V) traffic sources. Each free source generates arrivals with

intensity γ;

o Service time has exponential distribution with parameter 1/μ;

o Rejected call is lost


State transition diagram

• state „0” - all channels are free, N sources are free

• state „1” - one channel is busy, (N-1) sources are free,

• . . .,

• state „i” - i channels are busy and (N-i) sources are free,

• . . .,

• state „V” - all channels are busy (V-N) sources are free


N

0 1

i

( N -i+1)

i-1

( i+1)

( N -i)

i i+1

V

( N -V+1)

V-1 V

Statistical equilibrium equations


traffic offered by one free source

1

)1(

)1(

0

1

1

10

V

iVi

VVVV

ViVi

VV

p

pVpVN

pipiN

ppN

N

0 1

i

( N -i+1)

i-1

( i+1)

( N -i)

i i+1

V

( N -V+1)

V-1 V

V

j

ji

Vij

N

i

Np

0

/

Blocking / loss probability

• Blocking probability

• Loss probability:

o The loss probability in the group with traffic generated by N

sources is equal to the blocking probability in the group with traffic generated by N-1 sources


V

j

jV

VVj

N

V

NNVEEp

0

/),,(

V

j

jV

V

j

Vj

VV

j

N

V

N

pjN

pVNNVB

0

0

1/

1

)(

)(),,(

)1,,(),,( NVENVB

),1,()1(

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

NVEVNV

NVEVNNVE

1),0,( NE

Recurrence property of Engset formula


Engset formula – another form of notation

• Blocking probability:

• Parameter a expresses the ratio of the average time of source activity (occupancy) to the sum of the average time of source activity and the average time between the moment of terminating the activity and the moment of activity related to the generation of the next call. Therefore, the parameter a can be interpreted as the mean traffic offered by one source.

• Note that parameter is the mean traffic offered by one free source.


V

j

jV

a

a

j

N

a

a

V

NNVaB

0 1

1/

1

1),,(

/1/1

/1

1

a /1 /1

Engset model – carried traffic

• Mean value of carried traffic is equal to the average

number of simultaneously busy channels:

o where y is traffic carried by one source

• It can be proved:


yNNVB

NVBNPiY

V

iVi

1 )],,(1[1

)],,(1[

),,()(1

NVEVNNY

)],,(1[1

)],,(1[

NVB

NVBy

NaNPiYAV

iViN

1 1

Engset model – offered traffic

• Mean value of offered traffic is equal to the average

number of busy channels in the group with capacity of

N channels (system without losses):


Engset model – variance

• Variance of Engset distribution:

• Peakedness factor:


ANN

AYpk

V

kVk

2

0

22

N

A

YZ 1

2

Engset model – lost traffic

• Lost traffic intensity:

• Traffic loss probability (traffic congestion ) – relation of

lost traffic to offered traffic:


),,()(1

NVEVNYAR

),,(),,( NVEN

VN

A

RNVC

Engset model – paradox of call stream

• Stream parameter averaging all over the states

(expresses mean number of calls per mean service

time, i.e. mean call intensity)

• Mean call intensity resulting from evaluation of offered

traffic:


V

iVi YNPiN

1

)()(

1NAA A

1(

1YNA

CANVEN

VNN

),,(

1

Engset model – paradox of call stream

• Product AC determines the lost traffic intensity, i.e. the average number of sources which should be free as a result of blocking. The g parameter is the traffic intensity per one free source. If we assume that each blocked source within mean service time 1/m is not active, then Δ=0 and LA=L.

• The parameter L determines the mean call intensity under the assumption that each lost call (as a results of blocking) immediately causes the source to be free within hypothetical service time.

• The parameter LA determines the mean call intensity under the assumption that each lost call (as a results of blocking) immediately causes the source to be blocked within hypothetical service time


Palm – Jacobaeus formula for Engset

• Formula defines occupancy probability of x exactly

determined servers


),,(

),,()()(

NxVE

NVEpixPxH

V

xiVi

occupancy probability of

any i channels

Conditional occupancy probability

of x exactly determined servers

under condition that i servers are

busy:

V

j

ji

Vij

N

i

Np

0

/ )!(

!

!

)!(/)|(

xi

i

V

xV

i

V

xi

xVixP

Erlang and Engset Models


Erlang and Engset model

• Engset formula is a generalization of the Erlang formula

when the number of traffic sources N tends to infinity,

and parameter γ is decreased in such a way that the

product N γ remains constant.


NNlim

ii

N

i

N ii

iNNN

i

N

!

1

!

)1()1(limlim

Engset distribution Erlang distribution

N

V

j

ji

Viji

p0 !

/!

V

j

ji

Vij

N

i

Np

0

/

modeling and dimensioning of mobile networks: from gsm...

Documents