Analysis of Cell Load Coupling for LTE
Network Planning and Optimization
Iana Siomina and Di Yuan
Linköping University Post Print
N.B.: When citing this work, cite the original article.
©2012 IEEE. Personal use of this material is permitted. However, permission to
reprint/republish this material for advertising or promotional purposes or for creating new
collective works for resale or redistribution to servers or lists, or to reuse any copyrighted
component of this work in other works must be obtained from the IEEE.
Iana Siomina and Di Yuan, Analysis of Cell Load Coupling for LTE Network Planning and
Optimization, 2012, IEEE Transactions on Wireless Communications, (11), 6, 2287-2297.
http://dx.doi.org/10.1109/TWC.2012.051512.111532
Postprint available at: Linköping University Electronic Press
http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-79705
PAPER-TW-AUG-11-1532.R1 1
Analysis of Cell Load Coupling for LTE Network
Planning and Optimization
Iana Siomina and Di Yuan, Member, IEEE
Abstract—System-centric modeling and analysis are of key
significance in planning and optimizing cellular networks. In
this paper, we provide a mathematical analysis of performance
modeling for LTE networks. The system model characterizes
the coupling relation between the cell load factors, taking
into account non-uniform traffic demand and interference
between the cells with arbitrary network topology. S olving
the model enables a network-wide performance evaluation in
resource consumption. We develop and prove both sufficien t
and necessary conditions for the feasibility of the load-coupling
system, and provide results related to computational aspects for
numerically approaching the solution. The theoretical findings
are accompanied with experimental results to instructively illus-
trate the application in optimizing LTE network configuration.
Index Terms—3.5G and 4G technologies, cell load coupling,
network planning, optimization, system modeling
I. INTR ODUCTION
Planning and optimization of LTE network deployment,
such as base station (BS) lo c ation and antenna parame-
ter configuration, necessitate mode ling and algorithmic ap-
proach e s for network-level performance evaluation. Finding
the optimal network d esign and configuration a mounts to
solving an optimization problem of combinatorial nature.
Toward this end, system modeling admitting r apid perfor-
mance assessment in order to facilitate the selection among
candidate configuration solutions, of which the nu mber is
typically huge, is essential. In this paper, we provide a
rigoro us analysis of an LTE system performanc e model that
works for general network to pology and explicitly accounts
for non-uniform traffic demand. The performanc e model
that we study is referred to as the load-coup ling system, to
emphasize the fact that the model characterizes the coupling
relation between the cells in their load factors. For each
cell, the load factor is defined as the amount of resource
consumption in relation to that is available in the cell. The
load value grows with the cell’s traffic demand a nd the
amount of inter-cell interferenc e. Intuitively, low load means
that the network has more tha n enough capacity to meet
the dem and, whilst high load indicates po or perfo rmance
in terms of congestio n and potential servic e outage. In the
latter case, the network design and configuration solution
in question should be revised, by reconfiguration or adding
BS infra structure. Thus, simple m eans for evaluating the
I. Siomina is with Ericsson Research, Ericsson AB, Sweden. (e-mail:
iana.siomina@ericsson.com).
D. Yuan is with Department of Science and Technology, Link¨oping
University, SE-601 74, Sw eden, and Ericsson Research, Ercisson AB,
Sweden. (e-mail: diyua@itn.liu.se).
cell load for a given candidate design solution is of high
importance, particularly because the evaluation may have to
be conducted for a large number of user demand and network
configuration scenarios.
The load-coupling model for LTE networks takes the form
of a non- trivial system of non-linear equations. Calculating
the solution to the model, or determining solution existence,
is not straightforward. In th is paper, we present con tribu-
tions to characterizing and solving th e load -couplin g system
model. First, we present a rigo rous mathematical ana lysis
of fundamental properties of the system and its solution.
Second, we develop an d prove a sufficient and necessary
condition for solution existence. Third, we provide theoret-
ical results that are impo rtant for numerically approaching
the solu tion or delivering a bounding interval. Fourth, we
instructively illustrate the application of the system model
for optimizing LTE network configuration.
The remainder of th e paper is o rganized as follows. In
Section II we review some related works. The system model
is p resented in Section III, and its fundamental pro perties
are discussed in Section IV. In Section V, we present linear
equation systems fo r the purpo se of determining solution
existence. In Section VI, we provide the relation betwe en
solving the load-coupling system and convex optimization,
and discuss approximate solu tions. The application of the
system model and our theoretical results to LTE network
optimization is illustrated in Section VII, and conclusions
are given in Section VIII.
II. RELATED WORKS
Planning and perf ormance optimization in cellular net-
works form a very active line of research in wireless commu-
nications. Ther e are many works on UMTS network planning
and optimization. The research topics ra nge from BS location
and coverage planning [3]–[5], [25], [43], antenna parameter
configuration [15], [16], [33], to cell load balancing [18],
[34]. For UMTS, the power control mechanism that links
together the ce lls in resource consumption is an importa nt
aspect in pe rformance modeling [2], [3], [19], [41], [42]. By
power control, the transmit power of each link is adjusted to
meet a given signal-to-inter ference-a nd-noise ratio (SI N R)
threshold. By the SINR re quirement, the power exp enditure
of one cell is a linear function in those of the other cells.
As a result, the power control mech anism is represented by
a system of linear equation s, which sometimes is referred
to as UMTS interference coupling [15], [16]. Interference
coupling can be modeled for both downlink and uplink. For
network p la nning, th e interference coupling system needs
PAPER-TW-AUG-11-1532.R1 2
to be solved many times for performance evaluation of
different can didate network configur ations and multiple or
aggregate user demand snapshots. In [26], it is shown th at,
for both downlink and uplink radio network planning, the
dimension of the power-control-ba sed system of equations
can be reduced from the number of users in the system
to the number of cells. The observation stems from system
characteristics that also form the f oundation of distributed
power control mechanisms, see, e.g., [19], [42]. In [12], the
authors provide theoretical proper ties of the power-control-
based system, and feasibility conditions in te rms of target
data rates and Qo S require ments. Motivated by the fact
that full-scale dynamic simulation is not compu ta tionally
affordable for large networks, the authors of [44] extend
the UMTS power-control system by a ra ndomization-based
proced ure of service and rate adaptatio n for HSUPA network
planning.
In c ellular network planning, the power-control equation
system is considered under given SINR th reshold. Thus
the system solution and its existence are induced by the
(candida te ) network configuration in question. In a more
general context of wireless co mmunications, power control
is often a m e ans for pe rformance optimization, that is, the
powers are optimization variables in minimizing or ma xi-
mizing objectives representing error probability, utility, QoS,
etc., that are all functions of SINR. There is a vast amount
of theoretical analy sis and algorithmic approaches for power
optimization under various (typically non-linear) objective
functions, where a gain matrix defines interferenc e coupling
[37]–[40]. In [37], the authors identify o bjective fun ctions
admitting a convex formulatio n of power optimization, and
develop a distributed gr adient-pr ojection-based algorithm.
Further developments inc lude algorithmic design utilizing
Kuhn-Tucker condition [39 ], conditional Newton iteration
yielding quadratic convergence [40], and mo del extension
to include explicit SINR-threshold constrain ts [38].
Another line of research of power control is the character-
ization of the achievable performance region under various
utility and interference functions. The authors of [11] show
the strict convexity of the region for logarithmic functions
of SINR. In [7], the auth ors cha racterize utility functions
and function transform a tion of power, for which the resulting
power optimization problem is convex. The investigation in
[9] provides conditions under which the boundar y points of
the region are Pareto-optimal. In [8], the authors pre sent
graph representations of power and interference, and study
the relation between grap h structure, irreducibility of the
interference coupling matrix, and the convexity of the utility
region.
In contrast to the power-control model, the service re-
quirement of rate-control scheme in cellular networks is
not a pre-de fined SINR target, but the amount of data
to be served over a given time period. Among other ad-
vantages, this approach makes it possible to capture the
effect of scheduling without th e need of explicitly modeling
full details of scheduling algorith ms. The rate-control-based
approa c h is primarily targeting, altho ugh not limited to, non-
power-controlled systems or systems with a target rate traffic
demand . The approach has been less studied, but is of a
high interest for OFDMA-based n etworks. In gene ral, the
rate-control scheme exhibits non-linear relations between the
cell-coupling elements (in our c ase, cell loads). Th e resulting
model is therefore more complex than the power-control
model for UMTS. For power control, fundamental solution
characterizations are well-established for linear as well as
more general interference functions. For the latter, see, for
example, [10]. For rate-control-based coupling systems (see
[23] and Section III), a structural difference from power c on-
trol is that, in the former, one element cannot be expressed
as a sum of terms, each being a function de noting the im pact
of ano ther element, and the coupling is not scale-invariant.
For network plannin g, one known approach is to consider an
approximate linear function, obtained from system-specific
adaptive modulation and coding (AMC) parameters, to rep-
resent the r e la tion between da te r ate and SINR [24], and
thereby arrive at a equation system being similar to that of
UMTS.
From an engineering standpoint, LTE n etwork optimiza-
tion is becoming inc reasingly important. In [13], the authors
provide the fundame ntal principles of LTE network operation
and rad io resource a llocation. Among the optimization is-
sues, the research theme o f scheduling strategies and radio re-
source management (RRM) algorithms has bee n extensively
investigated. See, for example, [6], [20]–[22], [29]–[31] and
the references ther ein. Two major aspe cts considered in
the references are the b alance between resource efficiency
and fairness, and quality of service awareness. In [17], the
author gives a survey of tools enabling service and subscriber
differentiation. For cell planning, propagation modeling, link
budget consideration, and perf ormance parameters have been
investigated in [36].
High-level and accurate performance modeling is of high
value in pla nning cellular networks, as full-scale dynamic
simulations are not affordable for large p la nning scenarios
(e.g., [44]). The LTE system mod el that we analyze has
been introduced by Siomina et al. [32] for studying OFDM
network capacity region with QoS consideration. The work
in [3 2] does not, however, provide a general analysis of the
model, a nd the major part of the study relies on a simplifica-
tion assuming uniform traffic distribution. In the fo rthcoming
sections, we present both analytical and nu merical results
overcoming these limitations.
Recently, the authors of [23] have presented a non-linear
LTE performance mo del being very similar to the one studied
in the current paper. That our performance model has been
indepen dently proposed by others supports the modeling
approa c h. Th e work in [23] provides further an approxi-
mation of load coupling via another non-line ar but simpler
equation system, along with incorporating continuous user
distribution. Our study differs fro m [23], as the f ocus of the
current paper is a detailed investigation of key properties and
solution characterization of the load-coupling system.
III. THE SYSTEM MODEL
Denote by N = {1, . . . , n} the set of ce lls in a given
network design solution. Without loss of generality, we
PAPER-TW-AUG-11-1532.R1 3
assume that each cell has one a ntenna to simplify notation.
The service area is represented by a grid of pixels or
small areas, each being characterized by uniform signal
propagation conditions. The set of pixels is denoted by J .
The total power g ain between antenna i and pixel j is denoted
by g
ij
. We use J
i
⊂ J to denote the serving area of cell i.
In a network planning context, both the gain m atrix as well
as the cells’ serving areas are determined by BS location and
antenna config uration.
For realistic network planning scenario s, the traffic de-
mand is irregularly distributed. Let the user de mand in pixel
j be denoted by d
j
. The demand re presents the amount of
data to be delivered to the users located in pixel j within
the time interval under consideration. By defining a service-
specific index, the demand paramete r and the system model
can be extended to multiple types of services (see [32]). We
will, however, consider one service type merely for the sake
of compactness.
We use ρ
i
to denote the level of resource consumption in
cell i. The entity is also ref erred to as cell load. In LTE sys-
tems, the cell load can be interpreted as the expected fraction
of the time-frequency resources that are scheduled to deliver
data. The network-wide load vector, ρ = (ρ
1
, ρ
2
, . . . , ρ
n
)
T
,
plays a key role in performance mode ling. In p articular,
a well-designed ne twork shall be able to meet the target
demand scenarios without overloading the cells. Hence the
load vector for ms a natural perf ormance metric in network
configuration (cf. power consumption in UMTS networks).
The load o f a cell is a result o f th e user demands in the pixels
in the cell serving area , the channel c onditions, as well as
the amount of interferenc e. The last aspe c t interconnects the
elements in the load vector, a s the load of a cell is determin ed
by th e SINRs an d the resulting bit rates over the cell’s serv ing
area, and these values are in turn depende nt on th e lo ad
values of the other cells. To derive the performanc e model,
we consider the SINR in pixel j ∈ J
i
defined as follows,
γ
j
(ρ) =
P
i
g
ij
P
k∈N \{i}
P
k
g
kj
ρ
k
+ σ
2
. (1)
In (1 ), P
i
is the power spectral density per minimum
resource unit in scheduling (in LTE, this correspo nds to a
pair of time- consecutive resource blocks), a nd σ
2
is the noise
power. By (1 ), the inter-cell in terference grows by the load
factor. In effect, ρ
k
can be interpreted as the probability of
receiving interference originating from cell k on all the sub-
carriers of the resource unit. Let B log
2
(1 + γ
j
(ρ)) be the
function describing the effective bitrate per resource unit.
This formula is shown to be very accurate for LTE downlink
[27]. Thus to serve demand d
j
in j,
d
j
B l og
2
(1+γ
j
(ρ))
resource
units are required .
Let K denote the total number of resource units in the
frequency-time domain in qu e stio n, and denote by ρ
ij
the
proportion of resou rce consumption of cell i due to serving
the users in j ∈ J
i
. By these definitions, we obtain the
following equation,
Kρ
ij
=
d
j
B log
2
(1 + γ
j
(ρ))
. (2)
From (2), it is clear that the load of a cell is a function of
the load levels of o ther cells. Observing that ρ
i
=
P
j∈J
i
ρ
ij
and putting th e previous equations toge ther lead to the
following equation,
ρ
i
=
X
j∈J
i
ρ
ij
=
X
j∈J
i
d
j
KB log
2
(1 + γ
j
(ρ))
=
X
j∈J
i
d
j
KB log
2
1 +
P
i
g
ij
P
k∈N \{i}
P
k
g
kj
ρ
k
+σ
2
.
(3)
The equation above represents the coupling relation
between cells in their resource consump tion. In vector
form, we have ρ = f(ρ, g, d, K, B), where f =
(f
1
, . . . , f
i
, . . . , f
n
)
T
, and f
i
, i = 1, . . . , n, represents the
R
n−1
+
→ R
+
function as d efined by (3); here, R
+
and R
n−1
+
are used to denote the single- and (n − 1)-dim e nsion space
of all real non-negative numbers, re spectively. Since in the
subsequen t discussions there will be no ambiguity in the
input parameters, we use the following compact notation to
denote the non-line a r equation system,
ρ = f(ρ). (4)
From (3), three immediate observations follow. First, for
all i = 1, . . . , n , the load function f
i
is strictly increasing in
the load of other ce lls. Second, for non -zero σ
2
, this function
is strictly positive when the load values of other cells (and
thus interference) are all zeros, i.e., f (0) > 0. Third, the
function is continuous, and at least twice differentiable for
ρ ≥ 0.
From the network performance standpoint, the capacity is
sufficient to support the traffic demand, if equation system
(4) admits a load vector ρ with 0 ≤ ρ
i
≤ 1, i ∈ N . In our
analysis, however, we do not restrict ρ to be at most one,
in order to avoid any loss of generality. In addition, even
if the solution contains elements being g reater than one, the
values are of significance in network planning, because they
carry information ab out the amount of shortage of resource
in relation to the dem and.
Solving (4) deals with finding a fixed point (aka invariant
point) of function f in R
n
+
, or determin ing that such a point
does not exist. In the remainder of the paper, w e use S a s
a general notation for the space of non-negative solutions to
systems of equation s or inequalities. The system in qu estion
is identified using subscr ipt. Thus, S
ρ=f (ρ)
denotes th e
solution space of (4). Note that, for (4) as well as the linear
equation systems to be introduced later, only non-negative
solutions are of interest. Henc e , throughout the article, a
(linear or non-linear) system is said to be fea sib le , if there
exists a solution for which non-negativity holds, otherwise
the system is said to be infeasible (even if a solution of
negative values exists). Th e case that (4) is infeasible is
denoted by S
ρ=f (ρ)
= ∅.
A useful optimization formulation in our analysis is the
minimization of the total cell load, subject to the inequality
PAPER-TW-AUG-11-1532.R1 4
0 1 2 3
0
0.5
1
1.5
2
2.5
3
ρ
1
ρ
2
(a) Feasible solution within network capacity:
0 ≤ ρ ≤ 1.
0 1 2 3
0
0.5
1
1.5
2
2.5
3
ρ
1
ρ
2
(b) Feasible solution beyond network capacity:
∃(i ∈ N )|ρ
i
> 1.
0 1 2 3
0
0.5
1
1.5
2
2.5
3
ρ
1
ρ
2
(c) Infeasible system: S
ρ=f(ρ)
= ∅.
Fig. 1: An illustration of a load-coupling system of two cells.
form of (4). The formu la tion is given below.
min
X
i∈N
ρ
i
(5a)
ρ ≥ f(ρ) (5b)
ρ ∈ R
n
+
(5c)
For (5), its solution space S
ρ≥f (ρ)
is also referred to as the
feasible load region. Recall that f (ρ) is strictly increa sing,
hence if S
ρ≥f(ρ)
6= ∅, then for any optima l solution to (5),
(5b) holds with equality, as otherwise (5a) can be improved,
contradicting that th e solution is optimal. In c onclusion, any
optimum of ( 5) is a solution to (4).
We end the section by an illustration of the load-coupling
system for two cells in Figure 1. The two cells have symmet-
ric parameters. In the figure, the two non-linear functions are
given by the blue solid lines. In the first two cases, system
(4) has solu tions in R
2
+
, thoug h one o f them represents a
solution beyond the network capacity. In the last case, the
system is infeasible, a s the two curves will never intersect
in the first quadrant. The red straight lines with markers in
the figure represent linear equations related to (4). Details of
these linear equations are deferred to Section V.
IV. FUNDAMENTAL PROPERTIES
In this section, we presen t and prove some fund amental
properties of the load-coupling system (4). These theoreti-
cal results are of key importance in the study of solution
existence and computation. For compactness, we introduc e
additional notation to simplify (3) while keeping the essence
of the equation. De fine a
j
=
KB
d
j
, b
ikj
=
P
k
g
kj
P
i
g
ij
, and
c
ij
=
σ
2
P
i
g
ij
. These parameters contain, respectively, the
relation between the de mand in pixel j and the resource in
cell i, the inter-cell coupling in gain between ce lls k and i in
pixel j, and the channel quality of cell i in relation to noise
in pixel j. The load equation (3) can then be written in the
following form,
ρ
i
= f
i
(ρ) =
X
j∈J
i
1
a
j
log
2
(1 +
1
P
k∈N \{i}
b
ikj
ρ
k
+c
ij
)
. (6)
The first fundamental property of (4) is how fast the load
of a cell asymptotically grows in the load of a nother cell.
We formulate and prove the fact that, in the limit, the first-
order partial derivative of the load function converges to a
constant. For any two cells i, k (i 6= k),
∂ρ
i
∂ρ
k
is equal to
X
j∈J
i
ln(2)
b
ikj
a
j
1
ln
2
(1 +
1
P
h∈N \{i}
b
ihj
ρ
h
+c
ij
)
×
×
1
(
P
h∈N \{i}
b
ihj
ρ
h
+ c
ij
)
2
(1 +
1
P
h∈N \{i}
b
ihj
ρ
h
+c
ij
)
.(7)
Theorem 1: lim
ρ
k
→∞
∂f
i
∂ρ
k
=
X
j∈J
i
ln(2)
b
ikj
a
j
Proof: Consider the compon e nt for pixel j in the sum
in (7), and ignore the c onstant multiplier ln(2)
b
ikj
a
j
. Letting
u =
P
h∈N \{i}
b
ihj
ρ
h
+ c
ij
, Equation (7) can be written as
the following expression,
1
u
2
(1 +
1
u
) ln(1 +
1
u
) ln(1 +
1
u
)
=
1
ln(1 +
1
u
)
u
ln(1 +
1
u
)
u
+ ln(1 +
1
u
)
u
ln(1 +
1
u
)
.
The theo rem follows then from the facts that u is linear
in ρ
k
and lim
u→∞
(1 +
1
u
)
u
= e.
By Theorem 1, the load of a cell increases linearly in th e
load of another cell in the limit, i.e., the function co nverges
to a line in the high-load region. Moreover, the slo pe of the
line is strictly positive.
