EUROPEAN TRANSACTIONS ON TELECOMMUNICATIONS
Eur. Trans. Telecomms. 2007; 18:651–660
Published online 24 May 2007 in Wiley InterScience
(www.interscience.wiley.com) DOI: 10.1002/ett.1230
Interference and outage probability evaluation in
UMTS network capacity planning†
Tuo Liu∗ and David Everitt
School of Information Technologies, University of Sydney, NSW, Australia
SUMMARY
In this paper, we develop efficient analytic techniques for characterisation of other-cell interference in the
uplink of Wideband Code Division Multiple Access (WCDMA) cellular networks. Heterogeneous service
classes are supported in such systems with different data rate, bit error rate and activity factors. Additionally,
the considered propagation model includes both distance loss and log-normal shadowing effects. First we
suggest analytical methods to derive the whole distribution function (d.f.) of other-cell interference, followed
by corresponding log-normal approximation introduced and validated to simplify the computation. An
iterative method for solving fixed-point equations is employed to determine both d.f. and the mean and
variance for log-normal approximation of the other-cell interference. Finally, one important application of
the derived distribution of other-cell interference, the calculation of outage probability, is demonstrated.
Copyright © 2007 John Wiley & Sons, Ltd.
1. INTRODUCTION
The Universal Mobile Telecommunication System (UMTS)
is the European standard of third generation mobile
communication systems. It is distinguished from the
previous second generation systems (e.g. GSM, IS-95)
by supporting various service classes with different data
rates, quality of service (QoS) requirements, etc. Wideband
Code Division Multiple Access (WCDMA) which shares
most operational features with CDMA, is employed as the
air interface of UMTS. During the deployment process
of such systems, the capacity issue is one of the most
important parameters for the network providers, since larger
capacity translates to more revenue. It has been shown
in Reference [1] that CDMA systems can achieve much
higher capacity than FDMA/TDMA systems, while the
capacity evaluation techniques differ much from those for
FDMA/TDMA systems due to distinct operational features
[2]. The capacity of WCDMA network depends on the total
interference from all the users in system (i.e. interferencelimited system), in which the interference generated by
users located in inter-cells (known as other-cell interference) is the key component [3] for capacity evaluation. Once
the interference is well analytically modelled, the operators
can benefit much from it since the computation task during
the capacity planning process is significantly reduced when
a new WCDMA network is to be deployed.
With the assumption that an equal and fixed number
of users are evenly spread over each cell and perfect
power control to a fixed signal power, Gilhousen et al.
[1] and Kim [4] study the first and second moment of
other-cell interference under different propagation models,
respectively. Then Viterbi et al. [5] not only extend the
work in Reference [1] in terms of the cell membership
issue, but also introduce the concept of ‘relative othercell interference factor’, which can be understood as the
‘virtual’ effective interference contributed by each user in
the cell concerned. This concept is popularly used in many
subsequent works such as References [2, 6].
The paper by Evans et al. [7] gives the entire
approximated distribution function (d.f.) of interference
generated by an individual other-cell user, still with received
* Correspondence to: Tuo Liu, Advanced Network Research Group, School of Information Technologies, University of Sydney, Darlington, NSW 2006,
Australia. E-mail: tliu@it.usyd.edu.au
† A previous edition of the paper has been presented in the 12th European Wireless Conference (EW 2006), Athens, Greece.
Copyright © 2007 John Wiley & Sons, Ltd.
Accepted 1 December 2006
652
T. LIU AND D. EVERITT
power presumed to be controlled to a constant. This allows
us to model the other-cell interference with varying traffic
loads in the system, rather than the static case as before.
Another paper by Karmani et al. [8] suggests a numerical
integral over two-dimensional hexagonal regions to study
the moments of the individual interference random variables
(r.v.s), but in essence, these two computations share the
same principle. The latter one is more efficient compared
with obtaining the whole d.f.
An analytical approximation for the other-cell interference under the fixed signal-to-interference-ratio (SIR)
power control mechanism is presented in Reference [9],
in which an iterative method is proposed to solve the
fixed-point equations in order to determine the distribution.
Furthermore, the other-cell interference is proved to
follow a log-normal distribution approximately in this
scenario, and thus the complexity of computation reduces
dramatically since only the mean and variance are sufficient
to characterise this r.v. However, in doing so, it requires
some empirical results, which can be regarded as semianalytical approximation.
The main focus of this paper is to obtain a purely
analytical characterisation of other-cell interference, with
the practical fixed SIR power control mechanism employed
for the use of UMTS network capacity evaluation. The
employed practical power control mechanisms aim to
maintain the SIR, rather than received power, for each
mobile at the same level so as to maximise the system
capacity. The system behaviour with multiple service
classes and log-normal shadowing effects are studied.
Furthermore, in the above literature, the number of users in
each cell is generally modelled as an independent M/G/∞
queue [10], which implies there is no call admission control
(CAC) mechanism applied in the system. In contrast, in
this work, one simple CAC scheme is employed. After
obtaining the fully analytical characterisation of other-cell
interference, our computation of the interference is directly
applied to the analysis of outage probability, which is a
significant parameter during network capacity planning.
This paper is organised as follows. In Section 2, the
system assumptions as well as propagation and traffic
models used throughout the paper are introduced. Section 3
describes the techniques for other-cell interference
characterisation, first with fixed user locations then these
techniques are extended to stochastic user locations.
In Section 4, the calculation of outage probability for
capacity planning is demonstrated. Numerical results are
illustrated to validate the proposed analytical approximation
in Section 5. The paper is concluded in Section 6, and the
scope for future work is highlighted.
Copyright © 2007 John Wiley & Sons, Ltd.
2. SYSTEM MODEL DESCRIPTION
2.1. System assumptions
The standard uniform hexagonal cell layout with a NodeB at
the centre of each cell is assumed. The uplink and downlink
are assumed to utilise disjoint frequency bands, and all
mentions in the sequel (e.g. path loss, received power, etc.)
refer to the uplink only as it has been generally accepted to
be the limiting factor in capacity evaluation.
Each user equipment (UE) chooses the NodeB with least
attenuation as its home NodeB. The transmit power of all
the UEs in one cell are adjusted by the employed power
control mechanism so as to keep the received SIR at their
home NodeBs at the same level.
Furthermore, it is assumed that there are always sufficient
available codes so that the system is interference limited
only. Finally, without loss of generality, all the distance
values are normalised by the distance between any two
adjacent NodeBs.
2.2. Propagation model
The simplest propagation model for a communication
channel in the mobile radio environment is the log-distance
path loss model, where the attenuation incurred is inversely
proportional to the distance d between the transmitter and
the receiver raised to the path loss exponent (PLE) γ.
However, in order to employ such a simple model, many
restrictions, such as minimum and maximum distances,
terrain profile variation, should apply. In practice, due to
variations in terrain contour and shadowing from buildings
along the propagation path, measurements have shown
that the path loss at a particular location is random and
distributed log-normally about the above mean distancedependent value [11]. Incorporation of this phenomena,
which is generally referred to as log-normal shadowing,
leads to the following equation:
PR (d) = PT C0 d −γ 10ζ/10
(1)
where PT , d and γ denote the transmit power, T −
R separation distance and PLE, respectively. C0 is a
function of carrier frequency, antenna gains, etc., which is
independent of distance and thus assumed to be a constant
in this model. ζ is a zero mean Gaussian r.v. with standard
deviation σ typically in the range 6–12. Thus the received
Eur. Trans. Telecomms. 2007; 18:651–660
DOI: 10.1002/ett
653
INTERFERENCE AND OUTAGE EVALUATION
power PR (d) has the density function given in Reference [7]
fP (z) =
σ
1
√
′
2 /2σ ′ 2
2π
e−(ln z−µ)
(2)
3.1. Analytical model with deterministic user patterns
In the uplink of a WCDMA system, the received bit-energyto-interference-density ratio of a UE k with class t at its
NodeB x, with system bandwidth W and bit rate Rt , can be
represented by the following equation:
′
where µ = ln PT C0 − γ ln d and σ = σ ln 10/10.
2.3. Traffic model
We assume there are T available service classes in the total
system, each of which has its particular data rate, target SIR,
etc. The stochastic user population and distribution of each
class on a two-dimensional surface are generated according
to a spatial homogeneous Poisson process [12]. Therefore,
the number of active calls on a surface with area A for any
arbitrary observation instant is distributed according to the
product form in Reference [13] as:
P(N) =
P0 T
t=1
0,
(λt Aυt )Nt
Nt !
, N∈S
otherwise
(3)
where P0 is a normalisation constant and S represents the
admissible region. λt denotes the spatial traffic intensity (i.e.
mean number of users per unit area size) and υt the activity
factor of UEs with class t.
3. OTHER-CELL INTERFERENCE
CHARACTERISATION
Eb
I0
k
=
Sk
W
·
Rt Ixown + Ixother + N0 − Sk υt
(4)
where Sk denotes the received power of UE k at NodeB x,
and N0 the background thermal noise power. Ixown refers
to the total power received from all the UEs that belong to
the same NodeB as k, and Ixother corresponds to the sum of
power received at NodeB x from the UEs which connect to
all the NodeBs other than x.
Then following the assumption that the power control
mechanism aims to maintain (Eb /I0 )k for any UE k with
service t to be at the same constant level, ε∗t is used to
represent this target (Eb /I0 )k of class t and it is identical
for each k in one class. Furthermore, since all the UEs of one
NodeB experience the same level interference, the received
powers Sk are in turn controlled equal to each other for
every UE with same class connecting to NodeB x (i.e. Stx =
Sk:[k∈t]&[NodeB(k)=x] , ∀k). Thus, solving Equation(4) for Stx
yields
Stx =
ε∗t Rt
Ixown + Ixother + N0
∗
W + εt Rt υt
(5)
The own-cell interference is given by
T
In most literature regarding other-cell interference analysis,
power control is assumed to achieve a fixed received
power globally. However, in a real system, power control
algorithms manage to maintain the received SIR rather than
power at a constant level. This ensures that the mutual
interference is minimised so that significant capacity gain
can be achieved. However, the analysis in this scenario is
not an easy task, since higher other-cell interference in
one cell results in higher transmission power of all the
mobiles in this cell for fixed SIR, which in turn causes
higher other-cell interference for all neighbour cells and
then higher interference from outer cells again. This is
defined as ‘feedback behaviour’ in Reference [9]. In this
section, we investigate the system behaviour under such
scenario, by starting with the model under deterministic
user patterns, and then extend to that under stochastic user
patterns.
Copyright © 2007 John Wiley & Sons, Ltd.
Ixown =
k:NodeB(k)=x
Sk υk =
Sm x nm x υm
(6)
m=1
where nxm is the number of UEs with class m of NodeB
x. Combining Equation (5) with Equation (6) cancels the
component of Ixown and thus yields the intra-cell received
power
ε∗t Rt
W + ε∗t Rt υt 1 −
Stx =
T
m=1
·
Ixother
+ N0
nxm υm ε∗m Rm
W+ε∗m Rm υm
(7)
To model the Ixother , we first investigate the inter-cell
inter , which is the power caused by one UE
received power Sk,x
with class t that does not belong to the target NodeB x (i.e.
Eur. Trans. Telecomms. 2007; 18:651–660
DOI: 10.1002/ett
654
T. LIU AND D. EVERITT
NodeB(k) = x), and depends on the propagation loss from
the UE k to the target NodeB as well as to the UE’s home
NodeB (assumed to be y). If we initially assume any UE
connects to its closest NodeB, by the suggested propagation
inter can be represented as:
model, Sk,x
y
inter
Sk,x
= Sk
y
= St
dk,y
dk,x
dk,y
dk,x
γ
γ
10(ζk,x −ζk,y )/10
10ζk /10
(8)
y
inter
Sk,x
= St · min 1,
dk,y
dk,x
γ
10ζk /10
(9)
And then the other-cell interference Ixother is the sum of
inter from the UEs connecting to
inter-cell received power Sk,x
all the NodeBs in the system other than x:
Ixother =
Fy,x =
1−
T
where dk,x and dk,y refer to the distance from UE k to NodeB
x and y, respectively. ζk is the difference of two independent
Gaussian r.v. with zero mean and standard deviation σ, and
thus
√ is a zero mean Gaussian r.v. with standard deviation
2σ.
With the inclusion of shadowing effects, not only does
the received power become more varied, but even the cell
membership becomes more complicated than before, since
now the NodeB with least distance is not always to be
the one with the least attenuation. Suppose now the UE
connects to the NodeB offering least attenuation instead. To
be practically feasible, any UE can only select from a limited
number of closest candidate NodeBs. This scenario has been
studied in Reference [5], which leads to much complicated
computations and thus only mean values are obtained. To
reduce complexity, we follow the same approach as in
References [1, 7], where the choice is made merely between
the closest NodeB and target NodeB, thus the inter-cell
received power would be either take the same value as in
Equation (8) if controlled by the target NodeB, or simply
the own-cell received power if controlled by the closest one.
In order to include such effects into our model, the received
power at each cell are assumed to be equal to each other as
in References [1, 7] at this stage so that the min operator
can be applied as:
the total number of NodeBs in the system, regarding to the
variables I other can be constructed. For easy representation,
out and F , are introduced, where I out
two new variables, Iy,x
y,x
y,x
denotes the part of inter-cell interference experienced at
NodeB x caused by all the UEs associated with NodeB y,
and Fy,x is defined as a coefficient as follows:
inter
υk Sk,x
(10)
y=x k:NodeB(k)=y
Hence, if we substitute Equation (7) into Equation (10)
y
for the expression of St , a set of N equations, where N is
Copyright © 2007 John Wiley & Sons, Ltd.
·
t=1
1
T
m=1
y
nm υm ε∗m Rm
W+ε∗m Rm υm
y
υt ε∗t Rt
W + ε∗t Rt υt
nt
min 1,
k=1
dk,y
dk,x
γ
10ζk /10
(11)
Then Equation (10) can be rewritten as:
Ixother =
out
Iy,x
(12)
y=x
where
out
Iy,x
= Iyother + N0 · Fy,x
(13)
If the user pattern is deterministic, which means the
y
variables in the expression of Fy,x such as nt , dk,x , dk,y
and ζk are known, the other-cell interference in each cell
can be easily obtained by solving this set of equations
numerically. Accordingly, a Monte-Carlo simulation is built
based on these formulae and the simulation results are used
for verification purposes.
3.2. Iterative model with stochastic user patterns
With stochastic user patterns, the user population and
distribution are now r.v.s with d.f. specified in the previous
section, and all these random factors are included in the
variable Fy,x in Equation (11). The other-cell interference
I other also becomes a r.v., and its distribution can be
determined from the fixed-point Equations (12) and (13)
based on a modified version of the iterative method
suggested in Reference [9], which will be elaborated in the
following.
To characterise the r.v. Fy,x , which is the stochastic
representation of the variable Fy,x , we can first focus on
the conditional distribution by fixing the number of users
in each class, and then uncondition it as a sum over all
the possible combinations of number of active UEs with
Eur. Trans. Telecomms. 2007; 18:651–660
DOI: 10.1002/ett
655
INTERFERENCE AND OUTAGE EVALUATION
each class controlled by y according to the total probability
theorem
P Fy,x z =
P(n̄y )·P Fy,x z|n̄y
origin, and the target NodeB is assumed to be at (a, π) (i.e.
the distance between these two NodeBs is a), the desired
r.v. y,x becomes
(14)
n̄y ∈S
(r, φ, a) =
y
y
n̄1 , . . . , n̄T
where P n̄y is the probability that
active
UEs connecting to NodeB y. According to the spatial
homogeneous Poisson process assumption, it is given as
a product form in Equation (3)
P n̄y = P0
y
T
Mt υt
y
n̄t
(15)
y
t=1
n̄t !
and
1
P0 =
n̄′y ∈S
y
where Mt
y
Mt υt
y′
t=1
nt !
T
y′
nt
(16)
t=1
r 2 + a2 + 2ar cos φ
(18)
(19)
where
g1 (z) =
(17)
For the conditional distribution P Fy,x z|n̄y , since
the number of users in each class are fixed in
Fy,x , the only part that remains undetermined is
γ
Dy,x = min 1, dk,y /dk,x 10ζk /10 , which depends on the
individual user location within one cell and shadowing
factor. One approach based on integration of the conditional
d.f. to derive the whole d.f. of a similar r.v. is outlined in
Reference [7], thus we can follow this idea to obtain the d.f.
of Dy,x , and then get the first and second moments for later
approximation use. For the sake of clarity, we first focus
γ
on the d.f. of y,x = dk,y /dk,x , then extend it to that of
Dy,x .
With the assumption that users are uniformly distributed
within one cell, which is equivalent to the spatial
homogeneous Poisson process, this problem can be
simplified by approximating the hexagonal cells by circles
with radius b and consequently the coordinate system is
converted from Cartesian to polar, such that the problems
due to diverse orientations of a hexagon as well as to the
dependence of coordinates are eradicated. Then if the user
at location (r, φ) is connecting to its home NodeB at the
Copyright © 2007 John Wiley & Sons, Ltd.
γ
By firstly fixing φ and calculating the conditional d.f.
Fa,b (z|φ) and then unconditioning it with a definite integral
on φ from 0 to π, the d.f. of r.v. y,x is given in Reference [7]
as:
0,
z<0
−γ
g1 (z) , 0 z < ba + 1
−γ
−γ
a
z < ba − 1
, z = 1
Fa,b (z) = g2 (z) ,
b +1
−γ
g3 (z) , 1 < ba − 1
, z=1
−γ
1,
a
z
−1
y
n̄t ε∗t Rt
<1
W + ε∗t Rt υt
r
b
is the mean number of UEs with service t having
the least attenuation to NodeB y. The admissible region S
is defined by the pole capacity as n̄y ∈ S if
T
g2 (z) =
g3 (z) =
a2 z2/γ
b2 z2/γ − 1
2
1
1
arccos (h1 (z)) + g1 (z) [π
π
π
a
− 2z2/γ h1 (z) h2 (z) − z2/γ h2 (z)
b
− arccos (h1 (z)) − arcsin z1/γ h2 (z)
a
1
a 2
4b − a2
+
arccos −
π
2b
4πb2
and
−a2 − b2 + b2 z−2/γ
2ab
h2 (z) = 1 − h21 (z)
h1 (z) =
A few possible values for the approximating circle radius
b have been compared with the Monte-Carlo simulation
results in Reference [7], with the conclusion that when
b is chosen such that the area of hexagonal
√ √ cell and
approximating circle are equal (i.e. b = 4 3/ 2π with the
Eur. Trans. Telecomms. 2007; 18:651–660
DOI: 10.1002/ett
656
T. LIU AND D. EVERITT
distance between two adjacent NodeBs normalised to 1),
the approximation matches the simulation best.
Next in order to compute the whole d.f. of Dy,x , we begin
with deriving the d.f. of D̂y,x = y,x 10ζk /10 , followed by
combining the d.f. of both r.v.s in the min operator to get the
final form for the d.f. of Dy,x . Since the randomness due to
diverse locations has been defined by the above result, we
can first give the d.f. of D̂y,x conditioned on known y,x as:
FD̂y,x z|y,x = Q
ln z − ln y,x
√
2βσ
(20)
where
1
Q (x) = √
2π
x
Figure 1. Normal probability plot of the logarithm of F.
t2
e− 2 dt
(21)
−∞
√
β = ln 10/10 and 2σ is the standard deviation of ζk as
before.
Then Equation (20) can be unconditioned with known
d.f. of in Equation (19)
FD̂y,x (z) =
( ba +1)−γ
Q
0
+
( ba −1)−γ
( ba +1)−γ
ln z − ln y,x
√
2βσ
g1′ () d
ln z − ln y,x
Q
√
2βσ
g2′ () d
(22)
And eventually because of the min operator the whole d.f.
of Dy,x can be expressed as:
0,
z<0
(z)
F
,
0
z<1
FD (z) =
D̂
1,
1z
(23)
The d.f. of Fy,x can be then derived in theory since the
d.f. for all the r.v.s in Equation (14) have been obtained,
however, it is still quite a hard task since the computation
would involve many convolutions which is numerically
intractable, hence in the next step some approximation
techniques have been employed to reduce the computational
complexity.
Copyright © 2007 John Wiley & Sons, Ltd.
3.3. Log-normal approximation of the
analytical model
Through some trial experiments, the normal probability
plot in Figure 1 demonstrates comparison between the
simulation results of the logarithm of Fy,x and the
corresponding normal distribution. In this figure, the closer
the dashed line to the solid line, the more similar the
distribution between the simulation results and the normal
r.v. From the figure, it is shown that the logarithm of majority
samples of Fy,x match the normal distribution very well,
thus Fy,x can be well approximated by the log-normal
distribution, which reduces the problem to determining only
the mean and variance of Fy,x . To calculate these two
moments of Fy,x , we can fix the number of users in each
y
class as n̄t and develop the moments of conditional Fy,x (n̄)
first.
T
E Fy,x (n̄) =
y
t=1
n̄t ·
υt ε∗t Rt
∗R υ
W+ε
(
t t t)
T
1−
· E Dy,x
y
m=1
,
(24)
n̄m υm ε∗m Rm
W+ε∗m Rm υm
2
E Fy,x (n̄)2 = VAR Fy,x (n̄) + E Fy,x (n̄)
(25)
where the conditional variance is given by
T
VAR Fy,x (n̄) =
y
n̄t
t=1
·
2
υt ε∗t Rt
∗
(W+εt Rt υt )
1−
T
m=1
· VAR Dy,x
2
y
n̄m υm ε∗m Rm
W+ε∗m Rm υm
(26)
Eur. Trans. Telecomms. 2007; 18:651–660
DOI: 10.1002/ett
657
INTERFERENCE AND OUTAGE EVALUATION
Then again applying the theorem of total probabilities to
characterise the stochastic number of users, the first and
second moments of Fy,x are given as:
E Fy,x =
E F2y,x =
P(n̄y )E Fy,x (n̄)
(27)
P(n̄y )E Fy,x (n̄)2
(28)
n̄y ∈S
n̄y ∈S
where P(n̄y ) and S are the corresponding user distribution
and admissible region in Equations (15) and (17),
respectively. And finally the variance of Fy,x can be simply
derived from these two moments as:
2
VAR Fy,x = E F2y,x − E Fy,x
(29)
One thing to be noticed is that during the calculation
of moments of D̂y,x ∈ [0, 1], it is hard to compute the
derivative of Equation (21) with respect to z directly,
therefore we use the following alternative equations instead.
E D̂y,x =
=
1
0
−
1
FD̂ (z) dz
(30)
0
E
2
D̂y,x
=
1
z
2
2
VAR Iout
y,x = N0 ·VAR Fy,x
=
E Iother
x
VAR Iother
=
x
FD̂′ (z) dz
0
= z2 FD̂ (z) |10 − 2
E Iout
y,x = N0 ·E Fy,x
(32)
(33)
Then following the same assumption in Reference [9]
that Iout
y,x are independent of each other, the other-cell
, which comprises the sum of a series of
interference Iother
x
log-normally distributed r.v.s, can also be approximated by
a log-normal r.v. with both parameters as:
zFD̂′ (z) dz
zFD̂ (z) |10
initial zero for all NodeBs. In the next iteration, the new
are used to compute d.f. of Iout
d.f. of Iother
y,x again and then
x
other
other
Ix in turn. The d.f. of Ix would finally converge if the
relative changes fall below some thresholds.
The analysis is exceedingly tedious since again numerous
convolutions are required, however, the excellent lognormal approximation of Fy,x suggests that Iother
also
x
follows log-normal distribution [14], which implies that the
mean and variance are sufficient to characterise this r.v. The
iterative approach mentioned in Reference [9] is applied to
as follows.
calculate the moments of Iother
x
are set to zero in the first step, Iout
Since Iother
y,x is simply
x
the product of N0 and Fy,x , which in turn follows lognormal distribution as well. The mean and variance can
thus be easily determined as:
1
zFD̂ (z) dz
(31)
0
Once Fy,x has been characterised, we can analyse the d.f.
of other-cell interference by iteratively solving the fixedand Iout
point Equations (12) and (13), where Iother
y,x are
x
used to be the stochastic representation of corresponding
set
variables within. Starting with the desired r.v. Iother
x
are
determined
to zero for all NodeBs, the d.f. of Iout
y,x
for all possible pairs in a two-tier cell rings system by
Equation (13). Then, the sum of Iout
y,x for all NodeBs
according to
y = x yields the other-cell interference Iother
x
Equation (12) and thus the d.f. of Iother
are updated from
x
Copyright © 2007 John Wiley & Sons, Ltd.
(34)
y=x
E Iout
y,x
(35)
y=x
VAR Iout
y,x
, Iout
Based on these updated moments of Iother
x
y,x becomes
other
the product of two log-normal r.v., (N0 + Iy ) and Fy,x ,
which again can be regarded as a log-normal r.v. in all
the following iterations. The multiplication is performed
by summing up the corresponding parameters of these lognormal r.v.s
= µ
µIout
y,x
σI2out = σ2
y,x
N0 +Iother
y
N0 +Iother
y
+ µFy,x
(36)
+ σF2 y,x
(37)
2 are the median and variance of the r.v.
where µX and σX
X’s logarithm, which can be calculated from the mean and
Eur. Trans. Telecomms. 2007; 18:651–660
DOI: 10.1002/ett
658
T. LIU AND D. EVERITT
variance of X as:
2
σX
representation as:
VAR[X]
+1
= log
E[X]2
ωt =
(38)
ε∗t Rt
W + ε∗t Rt υt
(41)
then
σ2
µX = log (E[X]) − X
2
(39)
Stx =
Iother
x
can be determined
Again the mean and variance of
by Equations (34) and (35), with the moments of Iout
y,x
derived from the inversion of Equations (38) and (39). In
the following, Equations (34)–(37) constitute one whole
iteration, which would be repeated until the relative change
of interested parameters fall below certain thresholds. The
mean and variance of other-cell interference can be obtained
after convergence.
ωt
T
1−
m=1
nxm υm ωm
Ixother + N0
(42)
It can be seen that within a given cell, the received power
Stx of each class only differ from each other in ωt . Hence,
if the received power of the user class with maximum ωt is
less than the threshold, the other classes must also satisfy the
requirements. Then the outage probability can be rewritten
as:
x
∗
Poutage = 1 − Pr max St < S
t∈T
4. OUTAGE PROBABILITY ANALYSIS
In Reference [7], a QoS indicator called outage probability
is introduced for capacity analysis in radio network
planning. It is assumed in this work that received power
of UEs at all their home NodeBs are controlled at the
same level, thus the outage probability is defined as the
probability that a UE receives an insufficient SIR, which can
be easily translated into a constraint on the total interference
at one NodeB. Then the calculation of outage probability
reduces to the evaluation of the probability that a compound
Poisson r.v. exceeds a certain threshold.
However, in our current model, since it is presumed
that power control aims to maintain SIR for each UE to
be constant, a UE can always satisfy the requirement for
acceptable QoS unless the requested transmission power,
which is proportional to the received power at NodeB,
goes beyond its capability. In UMTS networks where
multiple classes are supported, the outage probability is
approximately defined as:
Poutage = 1 − Pr S1x < S ∗ , S2x < S ∗ , . . . , STx < S ∗ (40)
where S ∗ represents a certain received power threshold and
Stx is given in Equation (7).
With certain traffic class, the parameters such as ε∗t
and Rt are fixed values, thus Stx can be characterised as
a function of user number nxt and other-cell interference
Ixother . If we introduce a class-dependent term ωt for simple
Copyright © 2007 John Wiley & Sons, Ltd.
= 1 − Pr
St̃x
∗
< S , t̃ : max |ωt |
t∈T
(43)
Based on the Poisson d.f. for r.v. nxt and acquired log-normal
d.f. for Ixother from the previous section, the following
analysis is quite straightforward. We have
Pr St̃x < S ∗
T
x
1−
nm υ m ω m
m=1
other
∗
<S
= Pr Ix
− N0
ωt̃
=
P (n̄x )
n̄x ∈S
T
xυ ω
1
−
n̄
m m m
m=1
other
∗
− N0
· Pr Ix
<S
ωt̃
(44)
where the notation n̄x , S and P(n̄x ) can be referred to
Equations (15)–(17).
Under some user allocations, the right hand of the above
inequality might be less than zero, in which cases the
probability of this summand component simply takes zero
since the other-cell interference is always positive. When the
Eur. Trans. Telecomms. 2007; 18:651–660
DOI: 10.1002/ett
659
INTERFERENCE AND OUTAGE EVALUATION
T
x
1−
n̄m υm ωm
∗
m=1
other
Pr ln Ix
< ln S
− N0
ωt̃
ln
S∗
= Q
T
1−
m=1
n̄xm υm ωm
ωt̃
σI
− N0
− µI
8
7
6
5
4
3
6
8
10
12
14
16
Mean number of UEs per cell
(45)
where Q(x) is the function given in Equation (21).
Hence, together with the parameters µI and σI of othercell interference derived in the previous section, the outage
probability in a certain cell can be finally characterised.
5. NUMERICAL RESULTS
We consider an area with two-tier hexagonal cell rings
surrounding one cell, thus 19 NodeBs in total. Three service
classes are assumed in the UMTS system, which are voice
users with R1 = 12.2 kbps, ε∗1 = 5.5 dB, low-rate data users
with R2 = 28.8 kbps, ε∗2 = 4.0 dB and high-rate data users
with R3 = 64 kbps, ε∗3 = 3.5 dB, respectively. The ratio for
the number of users of these three classes is 75% : 20% :
5%. The system chip rate W = 3.84 MHz, and background
thermal noise N0 = −108 dBm. The activity factor υt are
assumed to be 1 for all classes and the PLE γ takes the value
of 4.
The validation of the analytical model can be performed
by Monte-Carlo simulation described above, where the
user patterns are generated randomly according to a spatial
homogeneous Poisson process, followed by solving a set
of linear equations to get the other-cell interference for
each NodeB. However, to avoid the border effect due to
less neighbours for the cells located at the border, only the
sample values at the central cell are counted.
Figures 2 and 3 illustrate the comparison of mean
and standard deviation between numerical results from
Copyright © 2007 John Wiley & Sons, Ltd.
Analytical
Simulation
9
2
18
20
Figure 2. Comparison of mean other-cell interference.
analytical model and simulation results with different
average number of users per cell. From the figures,
we can see the mean values from both models match
quite well, while the standard deviations show slightly
greater discrepancy. The reason for this is explained in
Reference [9], which is due to the mutual independence
out made during the iterative calculation.
assumption of Iy,x
It impacts on the variance computation and thus
underestimates the standard deviation of I other with high
loads.
The analytical and simulation results of logarithm outage
probability Poutage with assumed maximum received power
−12
2.6
Std. dev. of other−cell interference [mW]
−12
x 10
10
Mean of other−cell interference [mW]
right hand expression is larger than zero, natural logarithm
can be applied on both sides. Because ln Ixother follows
standard Gaussian distribution with known parameters µI
and σI , the probability inside the summand component is
actually the tail of Gaussian distribution
x 10
Analytical
Simulation
2.4
2.2
2
1.8
1.6
1.4
1.2
1
0.8
6
8
10
12
14
16
Mean number of UEs per cell
18
20
Figure 3. Comparison of standard deviation of other-cell
interference.
Eur. Trans. Telecomms. 2007; 18:651–660
DOI: 10.1002/ett
660
T. LIU AND D. EVERITT
mechanisms, each of them results in different othercell interference. Thus, our future work is to provide
analytical and approximation models with the incoming
traffic regulated by various CAC algorithms. Moreover, the
system behaviour in the downlink, or inclusion of packet
access data type, are also some interesting topics which
remain to be studied.
0
−0.5
Analytical
Simulation
Log outage probability
−1
−1.5
−2
−2.5
ACKNOWLEDGMENTS
−3
The authors would like to thank the Smart Internet CRC, Australia
for supporting this work as well as to Dr Dirk Staehle and Mr
Andreas Mäder for constant fruitful discussions.
−3.5
−4
−4.5
10
12
14
16
Mean number of UEs per cell
18
20
Figure 4. Analytical and simulation results for logarithm outage
probability.
S ∗ = −119 dBm and same user distribution are illustrated
in Figure 4. Again, it can be seen that the suggested
analytical approximation model can estimate the outage
probability excellently.
6. CONCLUSION
In this paper, we have presented a purely analytical
model for the characterisation of other-cell interference
in UMTS networks with log-normal shadowing effects,
which is crucial for capacity evaluation during network
planning. In theory, the distribution function of othercell interference can be computed based on solving fixedpoint equations iteratively where many convolutions may
be involved. A log-normal approximation model is then
suggested and verified so that the calculation can be
simplified significantly. Finally, one important metric for
network capacity planning, outage probability, is introduced
and its derivation is demonstrated with fully analytically
characterised other-cell interference.
In this work, the system behaviour is evaluated under
only the simplest CAC scheme. However, in practice, there
are many different CAC mechanisms, such as various
power-based and number-based CAC. Since the number
of active users in the system is controlled by these CAC
Copyright © 2007 John Wiley & Sons, Ltd.
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