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Fast Algorithms for Resource Allocation inRadio-Over-Fiber Access Networks
Pedro Henrique Gomes and Nelson L. S. da FonsecaInstitute of Computing
State University of Campinas
Campinas, Brazil
[email protected], [email protected]
Omar C. BranquinhoPontifical Catholic University of Campinas
Campinas, Brazil
Abstract—This article presents algorithms for the optimizationof radio resources and proposes an optimization model that canbe implemented in dynamic mobile networks based on RoF.The algorithms are based on linear relaxation technique forinteger linear programming (ILP) problems. The formulationmodels a multi-tier structure of antennas with increasing ra-dius. Considering this antenna structure the optimizer performsdynamic cell merging and cell splitting in the coverage area forsaving resources and yet improving network availability. Both theinteger and the relaxation formulations showed similar results forall the experiments, but the time of processing required by therelaxation algorithms was much shorter than that required bythe integer algorithm for large instances of the problem, whichhighlights the advantages of relaxation techniques under timeconstraints.
Index Terms—Radio-over-fiber, Radio Resource Management,Optimization, IP Relaxation Algorithms, Mobile Networks.
I. INTRODUCTION
An important step in wireless network design is the Base
Station positioning problem [1], which tries to determine the
best position for radio equipments to minimize the number of
cells and to keep network quality above certain requirements.
In traditional cellular networks, radio resource allocation
is static, since each cell correspond to one Base Station.
Employing static resource allocation is reasonable for fixed
and small networks, but it can waste resources in mobile
and dynamic networks, since bandwidth requirements are not
always spatially distributed in a uniform way and they can
change dynamically according to users’ mobility.
The Radio-over-fiber (RoF) technology can improve net-
work cost and provide a flexible infrastructure for radio
resource allocation in mobile and bandwidth-consuming net-
works. It integrates radio transmissions and optic fibers, taking
advantage of the best of each technology for the design of
efficient access networks. RoF networks usually interconnect
a great number of simple antennas, called Remote Antenna
Units (RAUs), through a fiber link backhaul, to a few central
sites, called Base Station Controllers (BSCs), where the radio
equipment (Base Stations - BSs) are located [2]. All radio
resources are centralized at the BSCs and they can be dynam-
ically distributed along the network. This centralized network
architecture calls for a global optimization algorithm that
allows the search for the best solution of resources distribution.
This paper introduces a solution based on integer program-
ming formulation for the resources allocation problem. The
solution of the problem considers two distinct objectives: i)
minimization of the number of used BSs, which represents
the main network cost, and ii) maximization of the revenue
obtained by the served users, which represents the main
network profit. Since the proposed architecture is focused on
mobile networks the optimization algorithm is subject to very
strict time constraints, which is difficult to deal with whsen
considering integer programming problems. To circumvent
this issue two different relaxation algorithms based on linear
relaxation techniques are proposed, which make it feasible to
solve the optimization problem in a short time.
The paper is organized as follows. Section II presents related
work. Section III introduces the proposed architecture and
the integer programming formulation. Section IV presents the
corresponding relaxed algorithms. Section V shows simulation
results. Finally, Section V concludes the paper.
II. RELATED WORK
Radio Resource Management (RRM) techniques applied to
Radio-over-Fiber architecture have been under investigation
recently. These algorithms have been extensively used in
cellular networks planning, but most of them were proposed
for static decision making. A central problem for wireless
networks optimization is the positioning of Base Stations for
optimal use of radio resources. Solutions for this traditional
problem are usually based on static methods and can lead to
resource waste in dynamic networks such as mobile environ-
ments. Novel RRM techniques have been proposed [3], so that
the dynamic clustering of users can be accounted for. Next,
recent work on RRM in wireless networks is briefly surveyed.
In [4], solutions for dynamic transmission power control,
dynamic channel allocation and load balancing for Wi-Fi
networks based on centralized agents are proposed. In [5]
and [6], the Base Station positioning problem is surveyed.
In [5], an algorithm for such problem based on Nelder-Mead
978-1-4244-7173-7/10/$26.00 c©2010 IEEE
method is investigated, and in [6], a solution via simulated
annealing is presented.
In [7] and [8], the RAU positioning problem in hybrid
Wireless-Optical networks is addressed. A greedy algorithm
for solving this problem is proposed in [8], and it is compared
to random and deterministic approaches. Based on a set of
pre-established positions the algorithm tries to minimize the
euclidean distance between RAUs and users. In [7], a solution
based on simulated annealing is proposed; results show sig-
nificant cost reduction. All these solutions, however, provide
last-mile access for fixed users and thus are not appropriate to
mobile users since the clustering of such users is dynamic.
The solution proposed in this paper deals with mobility and
yet optimizes the network radio resources. This work proposes
an optimization model based on integer programming and
approximations for the obtainment of solution rapidly.
III. OPTIMIZATION MODEL
The architecture considered in this paper employs a single
wireless technology and a hierarchical structure of cells with
different radius disposed in a multi-tier fashion (Figure 1). The
centralized network management can perform cell splitting
and cell merging in order to optimize the arrangement and
size of the cells. Cell splitting consists of diving large cells
into smaller ones and, consequently, increasing the network
capacity as well as its cost. Cell merging consists of joining
small and contiguous cells into a larger one, minimizing net-
work cost and decreasing capacity in the location. Deploying
statically cells with fixed sizes can lead to resource waste,
since users density varies along the network and congested
areas can “migrate“ from one region to another. One solution
to cope with such mobility is to implement cells with different
sizes, adopting small cells only when necessary, and a dynamic
process of cell rearrangement.
Tier 24 Micro-cells
Tier 316 Pico-cells
Tier 11 Macro-cell
Fig. 1: Multi-tier structure of cells used by splitting process.
The solution of the optimization problem needs to find the
cells that optimize the network cost and the operator revenue.
The splitting process can form a cluster of n smaller cells
at tier N from cells at tier N − 1. The merging process do
the opposite. If a RAU at a higher tier is active, all RAUs
located in its coverage at a lower level should be deactivated,
which avoids channel overlapping, facilitating frequency reuse.
We assume that there is a sufficiently large number of RAUs
installed and that these RAUs can be dynamically “turned on
and off”, splitting or merging the cells, according to the need.
The solution of the optimization problem determines the
BSCs that should operate in the network, the BSs that should
be activated in the operating BSCs, the RAUs associated to
the active BSs and the MSs served by the associated RAUs.
The problem is formulated as an integer programming (IP)
model. The following notation is used:
C = {C1, C2, ..., Cp} : set of BSCs;
B = {B1, B2, ..., Bo} : set of BSs;
R = {R1, R2, ..., Rm} : set of RAUs;
M = {M1,M2, ...,Mn} : set of MSs;
v : minimum percentage of served MSs;
t : number of tiers of RAUs;
Ui : sets of RAUs in the tier i, i ≤ t;q : number of RAUs per cluster;
ai,j : 1 if BS Bi is located at the BSC Cj ; 0 otherwise;
ci : capacity of BS Bi;
bi,j : 1 if RAU Ri is connected to BSC Cj ; 0 otherwise;
ri : coverage radius of RAU Ri;
PRi = (XRi , YRi) : geographical location of RAU Ri;
di : demand of MS Mi;
PMi= (XMi
, YMi) : geographical locations of MS Mi;
wi : class type of MS Mi;
disti,j =√(XMi −XRj )
2 + (YMi − YRj )2 : distance
between MS Mi and RAU Rj .
The decision variables are the following.
xi,j,k : 1 if the RAU Ri is associated to the BS Bj , located
at the BSC Ck; 0 otherwise;
yi,j : 1 if the MS Mi is served by the RAU Rj ; 0 otherwise.
The constraints of the problem are the following.
xi,j,k ∈ {0, 1} ∀i ∈ R, ∀j ∈ B, ∀k ∈ C (C1)yi,j ∈ {0, 1} ∀i ∈M, ∀j ∈ R (C2)xi,j,k ≤ bi,k ∀i ∈ R, ∀j ∈ B, ∀k ∈ C (C3)xi,j,k ≤ aj,k ∀i ∈ R, ∀j ∈ B, ∀k ∈ C (C4)∑
j∈B
∑
k∈Cxi,j,k ≤ 1 ∀i ∈ R (C5)
∑
i∈R
∑
k∈Cxi,j,k ≤ 1 ∀j ∈ B (C6)
∑
j∈Ryi,j ≤ 1 ∀i ∈M (C7)
∑
i∈M
∑
j∈Ryi,j ≥ n.v (C8)
∑
l∈B
∑
k∈Cxj,l,k ≥ yi,j ∀i ∈M, ∀j ∈ R (C9)
yi,j .disti,j ≤ rj ∀i ∈M, ∀j ∈ R (C10)
∑
i∈Myi,j .di ≤
∑
l∈B
∑
k∈Cxj,l,k.cl
∀j ∈ R (C11)∑
f∈R
∑
g∈C(xk,f,g + xU�k/qj−i�,f,g) ≤ 1
∀i, j ∈ N+|i < t, i+ 1 < j ≤ t, ∀k ∈ Ui (C12)
Constraints C1 and C2 ensure that decision variables are
binary. Constraint C3 states that a RAU can be associated to
a BS in a certain BSC only if there is a fiber link between
the RAU and the BSC. Constraint C4 establishes that a RAU
can be associated to a BS in a certain BSC only if the
BS is located at that BSC. Constraints C5 and C6 provide
a one-to-one association between the RAUs and the BSs.
Constraint C7 ensures that each MS will be served by only one
RAU. Constraint C8 guarantees that a minimum percentage
of MSs will be served. Constraint C9 establishes that only
RAUs associated to a BS can serve users. Constraint C10enforces that RAUs can only serve users in their coverage
area. Constraint C11 limits the aggregated demand to be less
or equal the BS’s capacity. Finally, constraint C12 prevents
that RAUs from different tiers to be used in a certain cluster.
IV. ALGORITHMS FOR FAST OBTAINTION OF RESULTS
The problem of radio resource allocation in RoF is an
extension of the classical problem of Base Station positioning.
At each round of the algorithm, the BSs are associated to the
best available RAUs. The problem of resource allocation in
RoF is an NP-hard [1]. Optimal solutions in real time are only
possible for small instances of the problem. Large instances
require either heuristics or approximations.
The nature of resource allocation problems leads to an
integer optimization problem. Although integer linear pro-
gramming (ILP) results in optimal solution the required time
can be infeasible for mobile and dynamic networks. To cir-
cumvent this timing issue two algorithms are proposed that
employ linear relaxation technique for finding optimal (or
quasi-optimal) solutions in a short time. The linear relaxation
consists of obtaining partial fractional solutions and, then,
converting the real solutions into integer ones. The relaxed
(real) solutions can be considered as probabilities and, by
using iterative randomized rounding techniques, it is possible
to transform them into integer values that satisfy the original
constraints and, fortunately, are close to the optimal ones. The
relaxation algorithms replace constraints C1 and C2 by C1′
and C2′.xi,j,k ∈ [0, 1] ∀i ∈ R, ∀j ∈ B, ∀k ∈ C (C1′)yi,j ∈ [0, 1] ∀i ∈M, ∀j ∈ R (C2′)
The algorithm receives as input an initial linear program-
ming solution. During the approximation process other linear
programming problems can be executed; each execution in-
cluding previous approximation decisions. This way, the relax-
ation algorithm refines the real solutions through the solution
of sucessives linear programming problems that increasingly
approximate the real solution to integer ones. The relaxation
algorithm is beneficial due to the fact that the time required for
solving linear problems is much shorter than the time required
for solving integer problems for large instances. Algorithm 1minimizes the network cost and Algorithm 2 maximizes the
operator revenue.
Algorithm 1 can be divided into two parts. In the first part
(lines 1 to 19), the algorithm analyses all RAUs and, based
on the real solutions, it decides to associate or not the RAUs
to the BS with the highest probability. From the top to the
lowest tier (line 1), the RAUs are sort randomly. For each
Algorithm 1 Linear relaxation for minimizing network cost
Input: Relaxation linear programming solution.Probthr : threshold probability for choosing the MSs.
Output: Integer linear programming solution1: for each tier l in decreasing order of radius do2: for each RAU r of tier l in random order do3: Find the highest probability xr,j,k , ∀j ∈ B and ∀k ∈ C4: Draw a uniform random variable between [0, 1]5: if highest probability (xr,j,k) ≥ drawn value then6: Add constraint xr,j,k = 1 and run again linear programming formula-
tion7: if new problem is infeasible then8: Remove constraint xr,j,k = 19: Add constraint
∑∀p∈C
∑∀o∈B xr,o,p = 0
10: end if11: else12: Add constraint
∑∀p∈C
∑∀o∈B xr,o,p = 0 and run again linear
programming formulation13: if new problem is infeasible then14: Remove constraint
∑∀p∈C
∑∀o∈B xr,o,p = 0
15: Add constraint xr,j,k = 116: end if17: end if18: end for19: end for20: sumy ← Amount of MSs already served by the linear programming formulation21: for i ← 0; (sumy < v.n) and (i < n); i + + do22: if MS i has not yet been served then23: Find an RAU j randomly such as yi,j ≥ Probthr
24: if exists such RAU j then25: Calculate the aggregated demand of all MSs already served by RAU j26: if capacity of the BS associated to RAU j supports the aggregated
demand plus di then27: Add constraint yi,j = 128: sumy ← sumy + 129: else30: Add constraint
∑∀l∈R yi,l = 0
31: end if32: else33: Add constraint
∑∀l∈R yi,l = 0
34: end if35: end if36: end for37: while i < n do38: Add constraint
∑∀l∈R yi,l = 0
39: i ← i + 140: end while
41: Run again linear programming formulation
RAU it is drawn an uniform random variable U [0, 1] that is
used as a threshold for the decision on associating the RAU
to the BS with highest real solution (line 4). If the highest
real solution found by the optimizer for the chosen RAU (line
3) is greater or equal to the drawn value, then the chosen
RAU is associated; otherwise it is not. If the chosen RAU is
associated, a new constraint is added and another execution
is performed (line 6). Different constraints are added if the
chosen RAU is not associated. In this case, the linear problem
is also executed again (line 12). As an attempt to circumvent
possible misleading decisions, the algorithm checks whether
the new linear problem becomes infeasible after the addition of
the constraint. In such case of making the problem infeasible,
it removes the last added constraint and adds a new constraint
ensuring that the chosen RAU will not be associated to any
BS (lines 7 to 10 and 13 to 16). Since all RAUs are analyzed
and new constraints defining their association to the BSs are
added, all the network infrastructure is defined, which means
that all variables x are integer.
Part two of Algorithm 1, which correspond to lines 20
to 41 finishes the rounding of the variables y. For all MSs
that have not been covered (line 22), the algorithm finds a
random RAU to which the MS has a large probability of being
served (line 23). If the algorithm finds such RAU it tries to
associate the MS to it, considering the required demand and
the RAU capacity. If the association attempt is successful, a
new constraint is added (line 27); otherwise the MS is not
served by any RAU (lines 30 and 33). The association attempts
are performed until all MSs are verified or the minimum
percentage of served users is reached (line 21). Since new
constraints were added in part two, the linear problem is
executed again in order to find the final solution.
Algorithm 2 Linear relaxation for maximizing revenue
Input: Relaxation linear programming solution.Probthr : threshold probability for choosing the MSs.
Output: Integer linear programming solution1: Define M ′ as a list of all MSs in decreasing order of revenue2: Defines R′ as a empty set {This set will store the RAUs already associated}3: for each MS i in list M ′ do4: Find a probability yi,j , ∀j ∈ R randomly such that yi,j ≥ Probthr
5: if probability yi,j was found then6: if RAU j is already in set R′ then7: Calculate the aggregated demand of all MSs already served by RAU j8: if capacity of the BS associated to RAU j supports the aggregated
demand plus di then9: Add constraint yi,j = 1
10: else11: Add constraint
∑j∈R yi,j = 0
12: end if13: else14: Find the highest probability xj,k,l, ∀k ∈ B and ∀l ∈ C15: if capacity ck supports di then16: Add constraint xj,k,l = 117: Add constraint yi,j = 118: Run again linear programming formulation19: if new problem is infeasible then20: Remove constraint xj,k,l = 121: Remove constraint yi,j = 122: Add constraint
∑j∈R yi,j = 0
23: else24: Add RAU j to set R′
25: end if26: end if27: end if28: else29: Add constraint
∑j∈R yi,j = 0
30: end if31: end for32: for all RAU i not included in set R′ do33: Add constraint
∑∀p∈C
∑∀o∈B xi,o,p = 0
34: end for
35: Run again linear programming formulation
Algorithm 2 can be divided into three parts. The first part
(lines 1 and 2) corresponds to the initialization of auxiliary
data structures M ′ and R′. List M ′ has all MSs ordered by
decreasing value of revenue. Set R′ is initially empty and will
store the used RAUs in the final network structure. Part two
involves lines 3 to 31. The algorithm tries to serve each MS in
list M ′, seeking for any probability greater or equal Probthr(line 4). If such probability exists and the RAU is already
in set R′, the algorithm calculates the aggregated demand of
all MSs already served by this RAU (line 7) and, if possible,
associates the MS to it. If RAU is not in set R′ the algorithm
verifies if the MS’s demand is supported by the BS with
highest probability of association with the RAU (lines 14 and
15); if the MS is supported by the new RAU, constraints are
added and another execution is performed (lines 16 to 18). It is
important to notice that a new execution of the linear problem
is only performed when new RAUs are added to set R′. In
order to avoid reaching infeasible solutions, the algorithm can
modify some association constraints (lines 19 to 23) when new
problem becomes infeasible. After the second part, all MSs are
analyzed and all variables y are integer. In the third part, the
variables x of all the RAUs not associated by the algorithm
receive values 0 (line 33). At the end, a new linear problem is
executed, which ensures that all variables x are also integer.
V. NUMERICAL RESULTS
The RoF network infrastructure considered in all the ex-
periments is shown in Figure 2. It consists of one BSC and
several RAUs distributed uniformly in an area of 2Km x 2Km.
RAUs are organized in a multi-tier fashion with 4 tiers and
clusters with 4 cells. The highest tier (tier one) consists of a
single RAU with radius of 1420m. The second tier consists of
4 RAUs with radius of 710m, disposed in a grid 2 x 2. The
third tier has 16 RAUs with radius of 360m. The lowest tier
(fourth one) involves 64 RAUs with radius of 180m. RAUs
of two different tiers cannot operate simultaneously to cover a
certain area. In the worst case, the network will operate with
64 BSs, which happens if all RAUs from fourth tier are active.
. . .
. . .
...
2,000m X 2,000m
1 BSC
Tier 11 RAU
Tier 24 RAUs
Tier 316 RAUs
Tier 464 RAUs
BS1
BS3
BS4
BSo
BS2
...
MS1MS2
MS3
MS4
MS5
MS6
MS7
MSn
Fig. 2: Radio-over-Fiber infrastructure used in the evaluation
The optimization model was implemented using the C
programming language and the optimization library FICO
Xpress 7.0 [9]. All experiments were executed in a work-
station with Intel Core 2 Quad core processor at 2.6 GHz,
3 GB of RAM and Debian GNU/Linux kernel 2.6.23.1 op-
erating system. To evaluate mobile network we employed
the Random Trip Model [10] with a scenario that in-
cludes the one in Houston, Texas/USA, near West University
(http://www.cs.rice.edu/∼amsaha/Research/MobilityModel/ ).
The experiments were divided into two groups. The first
group aimed at minimizing network cost, considering inte-
ger programming solution with objective function O1 and
Algorithm 1. The second group aimed at maximizing the
operator revenue, considering integer programming solution
with objective function O2 and Algorithm 2. In all the
experiments, four different types of network infrastructures
were considered: infrastructure 1 involved only the lowest tier
of RAUs (64 RAUs); infrastructure 2 consisted of the lowest
2 tiers (64 + 16 RAUs); infrastructure 3 was composed by
3 tiers (64 + 16 + 4 RAUs); and infrastructure 4 involved
all 4 tiers (64 + 16 + 4 + 1 RAUs). For each experiment
with different number of MSs or different number of BSs 20
samples were taken to compute the desired statistics. Intervals
with 95% confidence were derived.
A. Minimizing network cost
In these experiments, the network consisted of 1 BSC
with 64 BSs and the four different infrastructures of RAU
previously described. The number of MSs varied from 1 to
1000 and the algorithm was set to serve 100% of users.
No bound to the execution duration was set to the integer
programming solver.
Figure 3 shows the network cost considering all four pro-
posed infrastructures and both integer and relaxation algo-
rithms. For all the infrastructures both algorithms give very
close results. Infrastructure 1 presents the highest network
cost, demanding a much higher number of active BS than the
others. For a number of MS lower than 100, while infrastruc-ture 1 with a single tier of RAUs demands from 30 to 35 active
BS, all the other types of infrastructure demand less than 15
BSs. For a number of MSs larger than 400 infrastructure 2 and
infrastructure 3 give better results and the relaxation algorithm
(Algorithm 1) shows good performance, since its results are
very close to the results of integer algorithm.
0
5
10
15
20
25
30
35
40
45
50
55
60
65
0 100 200 300 400 500 600 700 800 900 1000
Num
ber
of
BS
s
Number of MSs
Inf. 1 - Integer algorithmInf. 1 - Relaxation algorithm
Inf. 2 - Integer algorithmInf. 2 - Relaxation algorithm
Inf. 3 - Integer algorithmInf. 3 - Relaxation algorithm
Inf. 4 - Integer algorithmInf. 4 - Relaxation algorithm
Fig. 3: Number of used BSs as a function of the number of
MSs
The duration of the execution of experiments was also
analyzed. Considering infrastructure 1, the required execution
time was very close to both the integer and the relaxation
algorithms. The infrastructures 2, 3 and 4 presented better
performance when employing the relaxation algorithm. For
networks with more than 400 MSs, the time required by the
integer programming algorithm grows drastically and over-
passes 150s for the infrastructure 2 and more than 250s for
the infrastructure 3 and infrastructure 4. Figure 4 shows the
execution duration for the infrastructures 2 (Figure 4a) and
the infrastructure 3 (Figure 4b), which are the infrastructure
types that most reduce the cost. To solve the problem for
Infrastructure 2 it is required up to three times more time when
using the integer programming solver for networks with more
than 700 MSs, than it is required when using the relaxation
algorithm, and fours times more time for infrastructure 3 for
networks with more than 600 MSs.
0
20
40
60
80
100
120
140
160
180
0 100 200 300 400 500 600 700 800 900 1000
Tim
e (s
)
Number of MSs
Inf. 2 - Integer algorithmInf. 2 - Relaxed algorithm
(a) Infrastructure 2
0
50
100
150
200
250
300
350
400
450
0 100 200 300 400 500 600 700 800 900 1000
Tim
e (s
)
Number of MSs
Inf. 3 - Integer algorithmInf. 3 - Relaxed algorithm
(b) Infrastructure 3
Fig. 4: Optimization duration of minimization algorithm -
Integer and Relaxed solutions
The large reduction in the execution time when using Algo-rithm 1 highlights the benefits of linear relaxation techniques
for solving linear programming problems. Infrastructure 2shows the best trade-off between network cost reduction and
computational complexity, specially when using relaxation
algorithm.
B. Maximizing operator revenue
In these experiments, the optimizer tries to serve as many
users as possible, prioritizing them according to their type
of service. We considered four classes of service (w), with
values 1, 2, 3 and 4, which is proportional to the revenue
the operator will receive by serving the users in these classes.
Users of class 4 generate four times more revenue for the
operator than do users of class 1. The proportion of users was
set to 40%, 30%, 20% and 10%, respectively for classes 1, 2,
3 and 4. In the experiments, the network consisted of 1000
MSs and a number of BSs varying from 1 to 64.
Figure 5 shows the operator revenue when varying the num-
ber of BSs. In this figure it is shown results for infrastructures1 and infrastructure 2, since the results for infrastructure 3 and
infrastructure 4 were quite similar to those of infrastructure2. It is possible to see that the solution of both integer
and relaxation algorithms are very close for all experiments,
showing the good performance of the approximation algorithm
(Algorithm 2). The total revenue increases as more BSs are
available since more MSs can be served. The difference of
the total revenue for infrastructure 1 and infrastructure 2reaches its maximum value which is a little higher than 200
(revenue units) for networks with 15 BSs. As the number of
BSs increases the revenue of both infrastructures reach the
maximum value of 2000, which happens when all MSs are
served.
0
200
400
600
800
1000
1200
1400
1600
1800
2000
0 10 20 30 40 50 60
Oper
ator
Rev
enue
Number of BSs
Inf. 1 - Integer algorithmInf. 1 - Relaxation algorithm
Inf. 2 - Integer algorithmInf. 2 - Relaxation algorithm
Fig. 5: Operator revenue as a function of the number of BSs
Figure 6 shows the required time for solving the integer
and relaxed algorithms considering infrastructures 1 and 2.
Infrastructure 1 requires little time for all experiments when
using integer algorithm, much less than the relaxed one, which
shows that in this infrastructure relaxation techniques are not
worth it. To solve the integer program for infrastructures 2,3 and 4 required very long time, which led to set the bound
for the execution time to 300s. When reaching the maximum
time, the integer algorithm returns the best integer solution
found, even if it is not an optimal solution. As it can be seen
in Figure 6, from 20 to 50 BSs the time to produce results for
infrastructure 2 when using the integer linear programming is
up to three times longer than those when using the relaxation
algorithm. The same behavior could be noticed when solving
for infrastructure 3 and infrastructure 4.Results shown in Figure 5 and the time reduction for
infrastructures 2 (Figure 6) envince the effectiveness of Al-gorithm 2. Again, infrastructure 2 can be considered to
furnish the best trade-off between quality of results and
computational requirement, mainly when linear relaxation is
used (Algorithm 2).
VI. CONCLUSION
This paper presents a centralized resource optimization
model for mobile users, which consists of a dynamic cell
0
50
100
150
200
250
300
350
400
0 10 20 30 40 50 60
Tim
e (s
)
Number of BSs
Inf. 1 - Integer algorithmInf. 1 - Relaxation algorithm
Inf. 2 - Integer algorithmInf. 2 - Relaxation algorithm
Fig. 6: Optimization duration of maximization algorithm -
Integer and Relaxed solutions
splitting/merging process executed in a multi-tier infrastructure
of RAUs. Two objectives can be achieved in such architecture:
network cost reduction and operator revenue improvement. For
both cases algorithms based on linear relaxation technique
were presented. For all the experiments, solutions of the
relaxation problems were similar to the optimal ones. Time
execution requirements were drastically reduced, mainly for
high-congested networks and for infrastructures with more
than one tier. Infrastructure 2 has been shown to be the best
option considering the quality of results and computational
cost, specially when using the relaxation algorithms.
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