Maciej Stasiak, Mariusz GłąbowskiArkadiusz Wiśniewski, Piotr Zwierzykowski

Models of Links Carrying Multi-Service Traffic

Chapter 7

Modeling and Dimensioning of Mobile Networks: from GSM to

LTE

2

Multi-rate systems

Integrated services systemIntegrated services system

L,1

MM t,

11, t

22 , tC,1

L,2 LM ,

C,2

CM ,

3

Multi-rate systems - parameters

arrival rate for class-i call stream,

arrival rate for carried call stream of class-i,

arrival rate for lost call stream of class-i,

service rate for class-i call stream,

number of demanded units of service resources for class-i call,

number of offered call streams in the system,

offered traffic of class-i: iiia /

i

Ci ,

Li ,

i

ia

M

it

4

Full-Availability GroupFAG

5

FAG with multi-rate traffic

A mixture of different multi-rate trafficstreams

},,,{ 21 Mxxx ),,,( 21 Mxxxp

Microstate:

Microstate probability:

12

V

1l

2l

Ml

il

PJP

6

Multidimensional Markov process

},,,{ 21 Mxxx ),,,( 21 Mxxxp

Microstate:

Microstate probability:

Mi xxx ,...1,...,1 Mi xxx ,...,...,1 Mi xxx ,...1,...,1

Mi xxx ,...,...,11 Mi xxx ,...,...,11

1,...,...,1 Mi xxx 1,...,...,1 Mi xxx

...),1( 111 x ,...)( 111 x

,...)1(..., iii x

)(..., MMM x

11x 11 )1( x

iix iix )1(

MMx MMx )1(

,...)(..., iii x

)1(..., MMM x

1

i

M

i

1

M

7

Reversibility of multi-dimensional Markov process

• Necessary and sufficient condition for reversibility (Kolmogorov criteria):o The circulation flow (product of streams parameters) among

any four neighboring states in a square equals zero.o Flow clockwise = flow counter clockwise

• State equations:o Reversibility property leads to local balance equations

between any two neighboring microstates of the process.

8

Reversibility of multi-dimensional Markov process

Mi xxx ,...1,...,1 Mi xxx ,...,...,1 Mi xxx ,...1,...,1

Mi xxx ,...,...,11 Mi xxx ,...,...,11

1,...,...,1 Mi xxx 1,...1,...,1 Mi xxx

M

1 1

i i

MM

11x 11 1 )( x

iix iix )( 1

MMx MMx )( 1i

iix )( 11,...,...,1 Mi xxx

MMx )( 1

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

9

Product form solution of multi-dimensional distribution (multi-rate)

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

M

1jjMi1 xpxxxp )(),,,,(

M

1j j

xj

Mi1 x

A

G

1xxxp

j

!),,,,(

,!0 0 0 11

V

z

l

z

l

z

M

i i

zi

j

j

M

M

i

z

AG

1

1

j

kkkj tzVl

10

Product form solution of multi-dimensional distribution (single-rte)

M

1jjMi1 xpxxxp )(),,,,(

M

1j j

xj

Mi1 x

A

G

1xxxp

j

!),,,,(

,!0 0 0 11

V

z

l

z

l

z

M

i i

zi

j

j

M

M

i

z

AG

1

1

j

kkkj tzVl

11

Macro-states

• Macro-state: {n}, where n is the integer number of BBUs in the group.

• Macro-state probability:

• where: is the set of such subsets , that the following equation is fulfilled:

i

M

iitxn

1

),,,( 21)(

Mn

Vn xxxpP

},,,,{ 1 Mi xxx )(n

12

Macro-states and micro-states

Example: V=10, t1=1, t2=2, t3=4

{5,0,0} {3,1,0} {1,2,0} {1,0,1}

Micro-states associated with macro-states {5}

13

Markov process in FAG – micro-state level

),.,1,.,(),.,,.,( 11 MiiMiii xxxpxxxpx

Mi xxx ,...1,...,1 Mi xxx ,...,...,1 Mi xxx ,...1,...,1

Mi xxx ,...,...,11 Mi xxx ,...,...,11

1,...,...,1 Mi xxx 1,...,...,1 Mi xxx

...),1( 111 x ,...)( 111 x

,...)1(..., iii x

)(..., MMM x

11x 11 )1( x

iix iix )1(

MMx MMx )1(

,...)(..., iii x

)1(..., MMM x

1

i

M

i

1

M

14

Markov process in FAG – macro-state level

• Solution:

),.,,.,(),.,,.,( Mi1iMi1ii x1xxpxxxpx

M

iVtniiVn i

PtAPn1

M

iMiii

M

iMiii xxxptAxxxptx

11

11 ),,1,,(),,,,(

15

Kaufman-Roberts recursion

• One-dimensional Markov chain - graphic interpretation (t1=1, t2=2):

M

iVtniiVn i

PtAPn1

1n n11ta

11 tny )(1n 2n

11ta 11ta

22 ta 22 ta

11 1 tny )( 11 2 tny )(

22 1 tny )( 22 2 tny )(

16

Blocking probability in FAG

• The Kaufman-Roberts model for multi-rate systems is a generalization of the Erlang model for one-rate systems.

V

tVnVni

i

PE1

M

iVtniiVn i

PtaPn1

17

Blocking probability – graphic interpretation

Example: V=10, t1=1, t2=2, t3=4

B1=P(10)

B3=P(10)+P(9)+P(8)+P(7)

B2=P(10)+P(9)

{10} {9} {8} {7}

18

Blocking probability – results

OFFERED TRAFIC

FULL AVAILABILITY GROUP

V=30

Stream 1Stream 2Stream 3

Simulations

Calculations

OFFERED TRAFIC

BL

OC

KIN

G

PR

OB

AB

ILIT

Y

19

Service streams

n11ta

)(nyt 11

1tn

iita

11ta

22ta 22ta

1tn 2tn 2tn )( 111 tnyt

iita

MM ta MM ta

)(nyt 22

)(nyt ii

)(nyt MM

)( 222 tnyt

)( iii tnyt

)( MMM tnyt

State equations for state {n}:

)(11

nyttaP i

M

iii

M

iiVn

Vtnii

M

iiVtni

M

ii ii

PtnytPta

)(11

20

Service streams

M

iVtniii

M

iVnii i

PtnytPtA11

)( )(1

nytn i

M

ii

Vtni

M

iiVn i

PtAPn

1

VtniiiVnii i

PtnytPtA )(

Vtn

VtnPPAtny

i

iVtnVniii

i

dla0

dla)(

/

Vtnii

M

iiVtni

M

iii

M

iii

M

iiVn ii

PtnytPtAnyttAP

)()(

1111

• Balance equation for state n:

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

21

Calculation algorithm for Kaufman-Roberts distribution

Vtni

M

iiVn i

PtAPn

1

VVn

V

iV

M

iiii

GnqP

iqG

tnqtAn

nq

q

/)(

)(

)(1

)(

1)0(

0

1

22

Calculation algorithm for Kaufman-Roberts distribution

• Let us assume that a full-availability group with capacity V services two traffic classes: t1=1, t2=2.

2) q(2) value calculations:

22

21 )0(

2

]2)([)2( xq

AAq

)0(2)0()2(2

)0(2)1()2(2

22

1

21

qAqAq

qAqAq

)q(nA)q(nAq(n)n

q

221

1)0(

21

11 )0()1( xqAq

1) q(2) value calculations:

23

Calculation algorithm for Kaufman-Roberts distribution

4) Using normalization procedure we calculate the value q(0):

Note that the results of calculation are expressed

as coefficients xi multiplied by constant q(0)=1.

1)0(0

V

nnxq

V

nnxq

0

/1)0(

GV

V

nniVi xxP

0

/

5) Calculation of the real values of probabilities : ViP

3) q(i) values calculations :

ixiq )(

24

Calculation algorithms for multi-service distributions

Analytical models

Recurrence algorithms Convolution algorithms

Poisson traffic model Any kind traffic model

State – independent systems

State – dependent systems

State – independent systems

State – dependent systems ?

25

Convolution operation for two distributions

V

yVyVyV

n

yVyVyn

yVyVyVV

VVVn

pppppppp

ppP

0

)2()1(

0

)2()1(1

0

)2()1(1

)2(0

)1(0

)2()1(

2

,,,,,

*

Convolution of two distributions: )2()1( , VV pp

V

iVi

Vn

V

iViVnVn

Pk

PkPPP

02

20

22

/1

/

Normalization of the state space 2V V

26

Convolution algorithm

• 3 steps of algorithm:

o Calculation of the occupancy distribution for each traffic class

o Calculation of the aggregated occupancy distribution [P]V

o Calculation of the blocking probability Ei for the class i traffic stream

27

Convolution algorithm – step 1

[p0]14 [p1]1

4 [p2]14 [p3]1

4 [p4]14

[p0]24 [p2]2

4 [p4]24

state

28

Convolution algorithm – step 2

[p3]14[p1]1

4[p0]14 [p2]1

4 [p4]14

*[p0]2

4 [p2]24 [p4]2

4

[p0]124 [p3]12

4 [p4]124

=

[p2]128 = [p0]1

4 [p2]24 + [p2]1

4 [p0]24

(0+2=2) (2+0=2)

[p1]124 [p2]12

4

29

Convolution algorithm – step 3

[p0]124 [p1]12

4 [p2]124 [p3]12

4 [p4]124

state

E2

E1

30

Convolution algorithm for M class of traffic

Convolution algorithm – step 1

Convolution algorithm – step 2

Convolution algorithm – step 3

31

Convolution algorithm for different distributions

Convolution algorithm – step 1

Convolution algorithm – step 2

32

Convolution algorithm for different distributions

Blocking / loss probability

Convolution algorithm – step 3

33

Example of link dimensioning

• Offered traffic parameters:

• To find the number of channels for blocking probabilities B(i) <0.005

class 1 class 2 class 3

t 1 2 6

ai [Erl.] 21 10.5 3.5

ai ti [Erl.] 21 21 21

34

Example of link dimensioning

Variant class 1 class 2 class 3

2 x 30 0.034 0.1 0.44

3 x 30 0.001 0.006 0.064

4 x 30 B<0.0001 B<0.0001 0.001

Variant 2: 3 x 30, a=63/90=0.7 Variant 3: 4 x 30, a=63/120=0.525

Variant 1: 2 x 30, a=63/60=1.05

35

FAG – multi-service Erlang-Engset model

• PROBLEMo Calculation of blocking probabilities Ei and loss probabilities

Bi for M1 traffic streams of PCT1type and M2 traffic streams of PCT2 type :

12

V

1111 ,, , t

1111 MM t ,, ,

BBUtraffic streams

121212 ,,, ,, tN

222222 MMM tN ,,, ,,

PCT1

PCT2

36

FAG – multi-service Erlang-Engset model

• Assumptions

PCT1 stream intensity of class i: 1,i,

PCT2 stream intensity of class j: jjjj yN,2

)( ,2,2,2

PCT1 traffic of class i offered to the group : iiiA ,1,1,1 /

PCT2 traffic offered to the group by one free source of class j:

jjj ,2,2,2 /

PCT2 stream intensity offered by one free source of class j: 2,i

37

Multi-service Erlang-Engset model – recurrence algorithm

• Idea of the algorithmo It was assumed in the algorithm that the number of occupied

BBU’s y2,j(n) by PCT2 stream of class j in each macro-state  {n} is the same as the number of occupied BBU’s by equivalent PCT1 stream with traffic intensity A2,j =N 2,j 2,j .

• Approximation rule: the number of serviced calls in the given state of the group is the same for both Erlang and Engset models.

Vtnjj

M

jjVtni

M

iiVn ji

PtNPtAPn,2

2

,1

1

,2,21

,2,11

,1

38

Recurrence algorithm – step 1

• Determination of occupancy distribution under the assumption that all offered streams are PCT1 type (Erlang streams):

VnP

Vtnjj

M

jjVtni

M

iiVn ji

PtNPtAPn,2

2

,1

1

,2,21

,2,11

,1

1n n11ta

11 tny )(1n 2n

11ta 11ta

22 ta 22 ta

11 1 tny )( 11 2 tny )(

22 1 tny )( 22 2 tny )(

ii tA ,1,1 ii tA ,1,1ii tA ,1,1

jjj tN ,2,2,2 jjj tN ,2,2,2

ii tny ,1,1 )2( ii tny ,1,1 )1( ii tny ,1,1 )(

jj tny ,2,2 )1( jj tny ,2,2 )2(

Erlang

39

Recurrence algorithm – step 2

• Determination of busy BBU’s y2,i(n), occupied by PCT2 calls in each macro-state {n}

Vn

VnPPNny VnVtnjj

jj

dla0

dla)( ,2,2,2

,2

/

40

Recurrence algorithm – step 3

• Determination of occupancy distribution , under the assumption that offered streams are PCT1 and PCT2 type :

VnP

Vtnjj

M

jjjjVtni

M

iiVn ji

PttnyNPtAPn,2

2

,1

1

,2,21

,2,2,2,11

,1 )(

1n n11ta

11 tny )(1n 2n

11ta 11ta

22 ta 22 ta

11 1 tny )( 11 2 tny )(

22 1 tny )( 22 2 tny )(

ii tA ,1,1 ii tA ,1,1ii tA ,1,1

jjjj tnyN ,2,2,2,2 )1(

ii tny ,1,1 )2( ii tny ,1,1 )1( ii tny ,1,1 )(

jj tny ,2,2 )1( jj tny ,2,2 )2(

jjjj tnyN ,2,2,2,2 )( Engset

41

Recurrence algorithm – step 4

• Calculation of the blocking probability E, and loss probability B, for PCT1 and PCT2 streams

• PCT1 stream:

• PCT2 stream:

Vn

V

tVnj PE

j

1

,2

,2

jjjVn

V

n

jjjVn

V

tVn

j

nyNP

nyNP

B j

,2,2,20

,2,2,21

,2

)(

)(,2

Vn

V

tVnii PBE

i

1

,1,1

,1

42

Full availability group with Engset traffic

S=400 S - infinity