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

psilva@lrc.ic.unicamp.br, nfonseca@ic.unicamp.br

Omar C. BranquinhoPontifical Catholic University of Campinas

Campinas, Brazil

branquinho@puc-campinas.edu.br

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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