Chapter
3
Capacity of GSM Systems
3.1 List of Mathematical Symbols
a
i
represents path loss and shadow fading effects for the ith
MS
B TDMA carrier frequency
dA area occupied by an MS
da area occupied by an MS, or an area of ring centred around
a BS at a distance r
j
and having thickness dr
j
d
j
distance between an MS in jth cell and the zeroth BS
dφ infinitesimal change in φ
D distance between a cell site and the nearest co-channel cell
site
E

ϒ
j
]
average interference experienced at the zeroth BS due to all
MSs in the jth cell in the presence of frequency hopping
E
()]
expectation of

CHAPTER 3. CAPACITY OF GSM SYSTEMS
M cluster size
N
f
number of carriers allocated to each BS
N
ts
time slots per carrier
P
j
(
r
j
)
transmitted power from the jth BS
P
T
MS transmit power
Q
(
k
)
Q-function
r distance between an MS and a BS in the same cell
R cell radius
S power received from an MS at a BS that is just sufficient to
maintain good communications
SIR signal-to-interference ratio
SIR
min

p
2 σ
e
λ shadow fading random variable for path r
λ
0
shadow fading random variable for path d
j
µ voice activity factor (VAF)
(=
E

v
j
])
φ angle between a line from an MS to its BS in the jth cell,
and a line between the zeroth and jthBSs(seeFigure3.3)
σ standard deviation of λ and λ
0
σ
ε
standard deviation of δ
ϒ
j
average interference power at the zeroth BS during slot-k,
frequency f
i
, from MSs over the jth cell in the presence of
frequency hopping
ϒ

will calculate the signal-to-interference ratios (SIRs) for different cluster sizes and identify
the cluster size (M) that will support the minimum acceptable SIR, namely SIR
min
. Knowing
M we will determine the number of channels per MHz per cell site, allowing for signalling
channels. The traffic carried by the network can then be computed for a given blocking
probability.
The SIR needs to be calculated for both up-link (reverse) and down-link (forward), and
for omnicells and sectorised cells. In computing the SIR we will make the following as-
sumptions: that the power control is applied; that frequency hopping (FH) is used where
the carrier in each frame hops beyond the coherence bandwidth; and that discontinuous
transmission (DTX) enables transmissions to be suspended on a link when the user is not
speaking [9–11]. All the traffic channels are considered to be occupied, and initially we will
ignore the signalling channels. The GSM radio link is assumed to be able to combat the ef-
fect of fast fading by means of its channel coding, bit interleaving, channel equalisation
and signal processing sub-systems. The radio channel is subjected to log-normal shadow
fading, and path losses that increase with distance raised to the power α.
Section 3.3 provides SIR calculations for transmissions in macrocellular GSM networks,
dealing with omnidirectional and sectorised cells, and examining the effect of power control
errors. The analysis is repeated in Section 3.4 for down-link transmissions. Armed with the
SIR calculations and the knowledge of cluster size M required to ensure SIR
min
, the capacity
of the hexagonal macrocellular network is determined in Section 3.5, along with the effect
of sectorisation on the teletraffic performance.
Section 3.6 is concerned with a street microcellular GSM network. The models used are
cross-shaped microcells formed by placing the base stations (BSs) at street intersections,
and rectangular-shaped microcells where the BSs are mid-way along the sides of the city
blocks. The cluster sizes for these two models are two and four, respectively. The other
154

10
S
=
10log
10
P
T

α10 log
10
r
+
λ

(3.1)
where the path loss is α 10log
10
r ,andα is called the path loss exponent. Note that λ is in
dBs. Measurements of λ show that it is normally distributed between

4σ and 2σ [2]. Since
λ is in dBs, it is said to be log-normally distributed between

4α and 2σ.Inordertomake
S a constant, P
T
is varied using a closed-loop power control system. From Equation (3.1)
the power transmitted by the MS is
P
T

j
MS
R
Figure 3.3: Up-link: an MS in the jth cell interfering with the zeroth BS.
156
CHAPTER 3. CAPACITY OF GSM SYSTEMS
distance d
j
between the MS and the zeroth BS, namely
d
j
=
q
D
2
j
+
r
2

2D
j
r cos φ

(3.3)
where D
j
and φ are shown in Figure 3.3. The interference also depends on the shadow
fading affecting the MS’s transmissions over the path between it and the zeroth BS. The
interference due to the MS in the jth cell at the zeroth cell site is

j
=
S

r
d
j

α
10
(
λ
0

λ
)=
10
(3.5)
and
ζ
=
λ
0

λ (3.6)
is a random variable having a normal distribution with standard deviation
p
2σ, with

4

time slots, resulting in N
ts
N
f
traffic channels per cell. The carriers are hopped on a frame-
by-frame basis. Each user is assigned a specific slot in a time frame and it stays in this slot
as frequency hopping occurs and the call progresses. Consider the kth timeslot supported
by carrier frequency, f
i
.Inthejth interfering cell, although all the users change their carrier
frequency during each TDMA frame, there is always a user who occupies the kth timeslot of
the carrier f
i
. Owing to frequency hopping at each TDMA frame the interference associated
with the channel specified by the kth timeslot and f
i
carrier can come from users in different
locations within the jth cell, although the interference is only from one user during any
frame. Eventually frequency f
i
will be used by a subset of N
f
users who occupied the kth
timeslot in the jth cell. If there is a sufficiently large number of carriers, and the MSs are
3.3. MACROCELLULAR GSM NETWORK: UP-LINK TRANSMISSIONS
157
uniformly distributed over the jth cell, then the probability that an MS will be using the
kth slot on carrier f
i
is f

cell
Z
area
I
j
f
(
a
)
da

(3.9)
where I
j
is the interference from location da. Substituting I
j
from Equation (3.5) into Equa-
tion (3.9), and using Equation (3.6) yields
ϒ
j
=
S
πR
2
Z
R
0
Z
2π
0

j
]=
S
πR
2
Z
R
0
Z
2π
0

r
d
j

α
E

10
ζ
=
10
]
r dr dφ

(3.11)
where E
()]
means the expectation of

x
d
j
=
R

α
E
h
10
ζ
=
10
i
x dx dφ
:
(3.12)
We now need to determine E
h
10
ζ
=
10
i
.
3.3.1.1 Expectation of E
h
10
ζ
=

<
2
p
2σ .
Because ζ is a normal random variable,
E

10
ζ
=
10
]=
Z
∞

∞
10
ζ
=
10

exp


ζ
2
4σ
2

(

E
h
10
ζ
=
10
i
=
Z
∞

∞
exp

ζ
10
ln
(
10
) 
1
2
ζ
2
2σ
2

(
4πσ
2

p
2σ
ln
(
10
)
10

2

(
4πσ
2
)
1
=
2

dζ
=
exp

σ ln
(
10
)
10

2
Z


dζ
and the integral is unity because it represents a normal distribution of mean
p
2σln
(
10
)=
10.
The expectation is, therefore,
E

10
ζ
=
10
]=
exp

σ ln
(
10
)
10

2

(3.14)
If we truncate the normal distribution of ζ at


10
ζ
=
10
:
exp


ζ
2
4σ
2

(
4πσ
2
)
1
=
2

dζ

(3.15)
where
Q
(
k
) =
1


2

Z
2
p
2σ

4
p
2σ
exp


1
2
n
ζ
p
2σ

p
2σ
ln
(
10
)
10
o
2

dx
=
dζ
p
2σ
; (3.18)
E

10
ζ
=
10
] =
1
:
023 exp

σ ln
(
10
)
10

2

p
2σ
(
4πσ
2

dx
=
1
:
023 exp

σ ln
(
10
)
10

2
"
Q
(

4

p
2σ ln
(
10
)
10
)

Q
(
2

(3.20)
and µ is called the voice activity factor (VAF). The mean of v
j
is
E

v
j
]=
µ
:
(3.21)
When we include DTX, the total interference from the MSs in the jth cell is decreased to
E

v
j
]
E

ϒ
j
]
. Extending the situation to include J co-channel cells, we have a total interfer-
ence of
I
T
=
J
∑

and normally J is set to six.
160
CHAPTER 3. CAPACITY OF GSM SYSTEMS
3.3.1.3 Computing the SIR
We need to compute the graph of SIR as a function of cluster size M (or reuse factor), and
then note the minimum value of M that provides an SIR above SIR
min
. By using this value
of M the radio links will operate with an acceptable bit error rate (BER). Figures 3.4 and
3.5 show examples of tessellated cells in three and seven cell clusters. The first step is to
note that the distance between co-channel cell sites, D,isgivenby
D
=
R
p
3M

(3.24)
where R is the cell radius and M is the cluster size. The path loss exponent α is set to
4 (although others might prefer 3.5). The standard deviation σ of the shadow fading is 8
dB, and E

10
ζ
=
10
]
is calculated using Equation (3.19). Three values of the VAF, i.e. µ,are
used: 1.0, when all users are speaking, which is a worse case scenario, and 1/2 and 3/8.
In normal conversation a speaker on average may speak for only 40% or so of the time.

shaded.
Figure 3.5: Tessellated clusters with seven cells per cluster, omnidirectional sites.
162
CHAPTER 3. CAPACITY OF GSM SYSTEMS
Figure 3.6: Up-link SIR versus number of cells per cluster.
The SIR will be calculated for different cluster sizes where each cell has three sectors. This
will eventually lead to us understand how sectorisation affects the traffic carried by the
network.
3.3.2.1 Sectorisation for three- and four-cell clusters
Figure 3.7 shows a three-cell cluster with three sectors per cell, while Figure 3.8 displays
a four-cell cluster with three sectors per cell. Note that sectorisation produces sectors that
are essentially smaller cells, and that the spectrum at each site has been partitioned into
three parts. The most apparent effect of sectorisation is to decrease the number of first tier
co-channel cells from six to just two sectors. For the sectorised situation, Equation (3.12)
becomes
E

ϒ
j
]
S
=
3
π
Z
1
0
Z
2π
=

significant interfering sectors, and allowing for DTX, the SIR for the three cell clusters with
three sectors per cell is
SIR
=
S
2µE

ϒ
j
]
:
(3.26)
Employing Equation (3.19), the SIR for the four cell clusters is 17.8, 20.8 and 22 dB for
µ
=
1

1
=
2and3
=
8, respectively. The corresponding SIRs for the three-cell clusters are
3.3. MACROCELLULAR GSM NETWORK: UP-LINK TRANSMISSIONS
163
14.9, 17.9 and 19.1 dB, respectively. The improvement in SIR due to sectorisation is 9.5 dB
as a result of the lower E

ϒ
j
]

using Equation (3.25). The interference is computed for mobiles (which use omnidirectional
antennas) from each sector, and the SIR is computed using Equation (3.23), where J is either
4or5.
The computed SIR values for sectors 1 and 2 are 7.7, 10.7 and 11.9 dB for VAFs, i.e.
values of 1, 1/2, and 3/8, respectively. The SIR of sector 3 is higher because although there
are five co-channel sectors, they are all spaced by at least two cells. The corresponding
SIR values are 13.9, 16.9 and 17.9 dB. However, note that the interfering sectors 3 are in a
contiguous line, and mobiles near the corner of one sector are likely to interfere with mobiles
in an adjacent sector. Consequently, the SIR values for sectors 3 are somewhat optimistic.
Indeed, this contiguous group of five sectors 3 may invalidate the two-cell cluster option.
3.3.2.3 Sectorisation for the single-cell cluster
Since the spectral efficiency in fixed channel allocation networks is inversely proportional to
the cluster size, it follows that it would be highly desirable if the entire bandwidth allocation
could be deployed at every cell site. Let us determine if sectorisation will facilitate single-
cell frequency reuse by calculating the SIR. Figure 3.10 shows the cellular arrangement.
The reuse distances D
j
for sectors E and F are
p
3R . The next closest sectors are D, B and
GatD
j
=
3R, while the remaining sectors A and C are at 2
p
3R. The procedure outlined
in Section 3.3.2.2 is employed again, and we display our SIR values, along with the others
evaluated in Section 3.3.2, in Table 3.1. We also include the SIR values for a three-cell
cluster without sectorisation as a bench marker. Observe that as with the two-cell cluster,
the single-cell cluster has contiguous co-channel sectors. The interference between adjacent

3.3. MACROCELLULAR GSM NETWORK: UP-LINK TRANSMISSIONS
165
sector 1:
sector 3:
sector 2:
MS:
B
BS
MS
BS
BS
BS
C
A
Figure 3.9: Up-link: two-cell clusters with three sectors per cell.
d
j
D
j
MS:
BS
0-th cell
j-th cell
A
B
C
E
F
G
D

tributed random variable having a standard deviation of σ
e
[13]. Because of power control
errors the mobile transmitted power in Equation (3.2) becomes
P
T
=
S
0
r
α
10

λ
10
=
Sr
α
10

λ
10
10
δ
10

(3.28)
The interfering power at the zeroth cell BS from an MS in the jth cell is obtained from
Equation (3.5) as
I

(3.29)
where δ
0
is the random variable of the interference power due to imperfect power control.
The average of the received power from mobiles located anywhere within the jth cell using
the kth timeslot can be found by inserting 10
δ
0
10
into Equation (3.10), namely
ϒ
0
j
=
S
πR
2
Z
R
0
Z
2π
0

r
d
j

α
10


α
10
ζ
10
10
ε
10
xdxdφ

(3.31)
3.4. MACROCELLULAR GSM NETWORK: DOWN-LINK TRANSMISSIONS
167
where ε is a random variable
ε
=
δ
0

δ (3.32)
having a normal distribution with a standard deviation of
p
2σ
e
.Sinceζ and ε are two
mutually independent random variables, the mean value of Equation (3.31) is
E

ϒ
0

R

α
xdxdφ
:
(3.33)
Compared with the interference-to-signal ratio for perfect power control given by Equa-
tion (3.12), there is an increase of E
h
10
ε
10
i
in the interference-to-signal ratio due to imper-
fect power control. The corresponding decrease in the SIR is0.2dB,0.9dB,and2.1dBfor
δ having a standard deviation of 1 dB, 2 dB and 3 dB, respectively, irrespective of the voice
activity factor and cluster size.
3.4Macrocellular GSM Network: Down-link Transmissions
3.4.1 The SIR for omnidirectional macrocells
The BSs in the tessellated hexagonal structures adjust the transmitted power to their mo-
biles such that each mobile receives a power S. This statement implies perfect power con-
trol. Again frequency hopping beyond the coherence bandwidth is assumed, and DTX is
applied. Mobiles receive interference from nearby BSs, and not from other mobiles. Thus,
as the MSs move around the cell the interference they experience varies, but the sources
of interference are static. The SIR on the down-link is dependent upon the location of an
MS, and this leads us to consider the average SIR for the mobiles in a cell, or the SIRsfor
particular mobiles.
Figure 3.11 shows an MS at a distance r from its BS in the zeroth cell. The MS occupies
an area dA. In the figure the source of interference is from the jth BS in the jth cell.
The distance between the two base sites is D

dr
j

(3.34)
All MSs in this ring receive a power S, and this means that the jth BS transmitted power is
P
j
(
r
j
)=
Sr
α
j
10

λ
j
=
10

(3.35)
168
CHAPTER 3. CAPACITY OF GSM SYSTEMS
Figure 3.11: Down-link: two-cell cluster with three sectors per cell.
where α is the path loss exponent and λ
j
is the shadow fading random variable. For MSs
uniformly distributed over the jth cell, the average transmitted power per channel from the
jth BS is

(
r
j
) =
Z
R
0
Sr
α
j
10

λ
j
=
10

1
πR
2

2πr
j
dr
j
=
2
α
+
2


α
j

(3.38)
where λ
0
is the shadowing random variable for path d
j
. Substituting P
j
(
r
j
)
from Equa-
tion (3.37) into Equation (3.38) yields
I
j
=
2
α
+
2
S 10
ζ
=
10

R

carriers in the zeroth cell, associated with N
f
users assigned to the kth slot, are
uniformly distributed over the area of the cell. It is important to realise that for a uniform
distribution to have validity, N
f
must be sufficiently large. This in turn means that the op-
erator has a wide bandwidth at his disposal. We will now consider the average interference
experienced by mobiles in the zeroth cell from the jth BS. The area dA shown in Figure 3.11
is rdφdr,andthePDF f
(
A
)
is again 1
=
πR
2
, and hence the interference average over the ze-
roth cell is, with the aid of Equation (3.39),
ϒ
j
=
Z
2π
0
Z
R
0

2S

(
α
+
2
)
π
Z
2π
0
Z
1
0
10
ζ
=
10

R
d
j

α
xdφdx
:
(3.42)
Note that

d
j
R

2

(3.43)
giving

d
j
R

2
=
3M
+
x
2

2
p
3Mxcos φ
:
(3.44)
Owing to frequency hopping, the expectation of
I
j
is made with respect to the random
variable ζ, because ζ is different for each hop. This expectation is
E

ϒ
j

where E
h
10
ζ
10
i
is given by Equation (3.19). The SIR can be found from Equations (3.23)
and (3.45), namely
S
I
=
S
∑
J
j
=
1
E

v
j
]
E

ϒ
j

=
1
2µ

(3.46)
170
CHAPTER 3. CAPACITY OF GSM SYSTEMS
For a hexagonal pattern of cells there are six close interferers, as shown in Figure 3.12. We
compute E
h
10
ζ
=
10
i
in Equation (3.19) for a path loss exponent of α
=
4, and for shadow
fading with a standard deviation of σ
=
8. This expectation value is inserted into Equation
(3.46) and the SIR found. When the number of cells per cluster M is changed, d
j
changes
(as evident in Equation (3.44)), which in turn changes the SIR. Graphs of the SIR as a
function of the number of cells per cluster are displayed in Figure 3.13 for three different
voice activity factors (VAFs) of 1, 1
=
2, and 3
=
8. For three cells per cluster, the average SIRs
forVAFsof1,1
=
2, 3

j
=
1
E

v
j
]
E
h
ϒ
j
S
i
=
1
2µ
α
+
2
∑
J
j
=
1
E
h
10
ζ
10

=
2, and 3
=
8, respectively. The worst
case SIR is about 1 dB less than the average SIR, while the SIR difference between the best
and worst case is about 2 dB.
3.4.2 The SIR for sectorised macrocells
By dividing each cell into three sectors, as we did for the up-link in Section 3.3, we have
the arrangement shown in Figure 3.15. By averaging over the sector area, the average SIR
can be found from Equation (3.46), namely
S
I
=
S
∑
J
j
=
1
E

v
j
]
E

ϒ
j

=

α
xdφdx

(3.48)
From Equation (3.47), the SIR for mobiles at different locations is
S
I
=
S
∑
J
j
=
1
E

v
j
]
E

ϒ
j

=
1
2µ
α
+
2

2, and 3
=
8are14
:
1dB,17
:
1dB,and18
:
4 dB, respec-
tively, for the three-cell cluster. The SIRs for mobiles located along the line AB in Figure
3.15 are shown in Figure 3.16. The SIRs for mobiles at location B (where r
=
R
=
0
:
87) are
15
:
7dB,18
:
7dB,and19
:
9 dB for VAF values of 1, 1
=
2, and 3
=
8, respectively. The worst
SIR conditions occur when the mobiles are located near the BS in Figure 3.15, where the
SIRsare11

:
7dB,10
:
7dB,and11
:
9dBfora
VA F o f 1 , 1
=
2, and 3
=
8, respectively, while for mobiles in sector 3 the average SIR values
3.4. MACROCELLULAR GSM NETWORK: DOWN-LINK TRANSMISSIONS
173
Figure 3.15: Down-link: three cells per cluster with three sectors per cell. Primary interfering sec-
tors are shown shaded.
Figure 3.16: Down-link SIR for mobiles as a function of normalised distance r
=
R from the BS to
the boundary along the line AB in Figure 3.15 for different VAFs; three sectors per cell
and three cells per cluster.
174
CHAPTER 3. CAPACITY OF GSM SYSTEMS
sector 1:
sector 3:
sector 2:
MS:
B
BS
MS
BS

=
8are4
:
2dB,7
:
2dB,and
8
:
4 dB, respectively. From Figure 3.18, the difference between the two extreme values of
the SIR due to different mobile locations is about 3
:
5dB.
3.4.2.2 The special case of a one-cell cluster
The down-link SIR of the one-cell cluster TDMA system with three sectors per cell is cal-
culated using the same assumptions as for the up-link case. The average SIR is calculated
from Equation (3.48), but now with different co-channel separation D
j
. From Figure 3.19,
there are seven significant co-channel interfering BSs. Sectors E and F have a reuse distance
of
p
3R , sectors B, D and G have a reuse distance of 3R, and sectors A and C have a reuse
distance of 2
p
3R . We find that the average SIR is 4
:
4dB,7
:
4dB,and8
:

i
into the denominator of Equations (3.46) and (3.47) for
the average SIR and the SIR for mobiles at different locations in a cell, while for sectorised
cells it can be calculated by inserting it into the denominator of Equations (3.48) and (3.49).
The decrease in SIR is 0
:
2dB,0
:
9dB,and2
:
1 dB for a standard deviation of power error of
1 dB, 2 dB, and 3 dB, respectively, irrespective of VAF, cluster size and sectorisation.