maciej stasiak, mariusz głąbowski arkadiusz wiśniewski, piotr zwierzykowski models of links...
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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
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2
Multi-rate systems
Integrated services systemIntegrated services system
L,1
MM t,
11, t
22 , tC,1
L,2 LM ,
C,2
CM ,
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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
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Full-Availability GroupFAG
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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
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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
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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.
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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(
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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
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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
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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
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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}
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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
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14
Markov process in FAG – macro-state level
• Solution:
),.,,.,(),.,,.,( Mi1iMi1ii x1xxpxxxpx
M
iVtniiVn i
PtAPn1
M
iMiii
M
iMiii xxxptAxxxptx
11
11 ),,1,,(),,,,(
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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 )(
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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
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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}
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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
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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
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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 :
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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
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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:
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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 )(
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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 ?
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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
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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
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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
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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
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29
Convolution algorithm – step 3
[p0]124 [p1]12
4 [p2]124 [p3]12
4 [p4]124
state
E2
E1
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30
Convolution algorithm for M class of traffic
Convolution algorithm – step 1
Convolution algorithm – step 2
Convolution algorithm – step 3
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Convolution algorithm for different distributions
Convolution algorithm – step 1
Convolution algorithm – step 2
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32
Convolution algorithm for different distributions
Blocking / loss probability
Convolution algorithm – step 3
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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
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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
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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
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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
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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
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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
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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
/
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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
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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
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42
Full availability group with Engset traffic
S=400 S - infinity