modeling and dimensioning of mobile networks: from gsm...
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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
Modeling and Dimensioning of Mobile Networks: from GSM to LTE 3
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.
Modeling and Dimensioning of Mobile Networks: from GSM to LTE 4
0 1
i
i-1
(i+1)
i i+1
V
V-1 V
Statistical equilibrium equations
Modeling and Dimensioning of Mobile Networks: from GSM to LTE 5
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
Modeling and Dimensioning of Mobile Networks: from GSM to LTE 6
h
1
h
/ determines the average number of arrivals within average service time
Erlang’s distribution
Modeling and Dimensioning of Mobile Networks: from GSM to LTE 7
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
Modeling and Dimensioning of Mobile Networks: from GSM to LTE 8
V
i
i
V
VVN
i
A
V
A
pAE
0
,1
!
!)(
Blocking probability = f ( offered traffic, capacity)
Recurrence property of Erlang formula
Modeling and Dimensioning of Mobile Networks: from GSM to LTE 9
)(
)()(
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
Modeling and Dimensioning of Mobile Networks: from GSM to LTE 10
][ (A)EApkY ,V
V
kVkV 1
0
1
)()()( ,1
2
0
22
VVVV
V
kVk YVAAEYYpk
V
Characteristics of carried traffic
• Variance of carried traffic
Modeling and Dimensioning of Mobile Networks: from GSM to LTE 11
.
!
])!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
Characteristics of carried traffic
• Variance of carried traffic
o Taking into account:
o we obtain:
Modeling and Dimensioning of Mobile Networks: from GSM to LTE 12
),(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
Characteristics of carried traffic
• NOTE !
o Variance of offered traffic
o is equal to
o mean value of offered traffic
Modeling and Dimensioning of Mobile Networks: from GSM to LTE 13
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
Modeling and Dimensioning of Mobile Networks: from GSM to LTE 14
Erlang table
Modeling and Dimensioning of Mobile Networks: from GSM to LTE 15
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
Modeling and Dimensioning of Mobile Networks: from GSM to LTE 16
A1
V1
A2
V2
A1 + A2
V1 + V2
Two groups joint group
)(,)(max)( 2121 2121AEAEAAE VVVV
Group principle - example
Modeling and Dimensioning of Mobile Networks: from GSM to LTE 17
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
Modeling and Dimensioning of Mobile Networks: from GSM to LTE 18
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:
Modeling and Dimensioning of Mobile Networks: from GSM to LTE 19
!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.:
Modeling and Dimensioning of Mobile Networks: from GSM to LTE 20
)](1[ AEAY VV
V(A)EAVY VV /]1[/
Erl.79.010/]101[10/ 10 )(EVYV
Channel load – random hunting
Modeling and Dimensioning of Mobile Networks: from GSM to LTE 21
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:
Modeling and Dimensioning of Mobile Networks: from GSM to LTE 22
)](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
Modeling and Dimensioning of Mobile Networks: from GSM to LTE 23
Palm – Jacobaeus formula
• Formula defines occupancy probability of x exactly
determined servers
Modeling and Dimensioning of Mobile Networks: from GSM to LTE 24
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
Modeling and Dimensioning of Mobile Networks: from GSM to LTE 25
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
Modeling and Dimensioning of Mobile Networks: from GSM to LTE 26
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
Modeling and Dimensioning of Mobile Networks: from GSM to LTE 27
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
Modeling and Dimensioning of Mobile Networks: from GSM to LTE 28
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
Modeling and Dimensioning of Mobile Networks: from GSM to LTE 29
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
Modeling and Dimensioning of Mobile Networks: from GSM to LTE 30
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.
Modeling and Dimensioning of Mobile Networks: from GSM to LTE 31
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:
Modeling and Dimensioning of Mobile Networks: from GSM to LTE 32
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):
Modeling and Dimensioning of Mobile Networks: from GSM to LTE 33
Engset model – variance
• Variance of Engset distribution:
• Peakedness factor:
Modeling and Dimensioning of Mobile Networks: from GSM to LTE 34
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:
Modeling and Dimensioning of Mobile Networks: from GSM to LTE 35
),,()(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:
Modeling and Dimensioning of Mobile Networks: from GSM to LTE 36
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
Modeling and Dimensioning of Mobile Networks: from GSM to LTE 37
Palm – Jacobaeus formula for Engset
• Formula defines occupancy probability of x exactly
determined servers
Modeling and Dimensioning of Mobile Networks: from GSM to LTE 38
),,(
),,()()(
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
Modeling and Dimensioning of Mobile Networks: from GSM to LTE 39
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.
Modeling and Dimensioning of Mobile Networks: from GSM to LTE 40
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
/
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