Opportunistic Sub-channel and Transmit Power Radio... · Opportunistic Sub-channel and Transmit Power ... interference management ... macrocell and femtocell network are using OFDMA

Opportunistic Sub-channel and Transmit PowerAllocation in an OFDMA Based Cognitive FemtocellNetwork

Sandip Karar1,2 • Abhirup Das Barman1

Published online: 19 May 2015� Springer Science+Business Media New York 2015

Abstract The paper proposes a scheme to minimize the co-tier and cross-tier interference

by properly allocating the sub-channels and power among the femtocell users (FUs) in an

OFDMA based cognitive femtocell network. An efficient graph coloring based sub-channel

allocation scheme is adapted in which the sub-channels in the uplink band of the macrocell

are shared by the femtocells dynamically in an opportunistic way through spectrum

sensing. A price-based power allocation scheme using game theory is proposed to assign

the transmission power among the FUs, whereby the macrocell base-station (MBS) con-

trols the transmission power of the FUs by pricing their resulted interference power levels

at the MBS with the constraint that the total interference created at the MBS in each sub-

channel is kept below a tolerable threshold. The game theoretic approach ensures that no

FUs can improve its utility by changing its own transmission power selfishly. The allo-

cation of sub-channels among the FUs is centrally controlled whereas the power allocation

is handled in a distributed way which makes the process efficient. Finally, numerical

examples are presented to analyze the proposed scheme. The results show that substantial

number of FUs with good quality of services can be accommodated in the macrocell

reusing the uplink frequency sub-channels and thereby enhancing the throughput and

coverage of the network.

Keywords Femtocell networks � Graph coloring � OFDMA � Power control �Stackelberg game � Sub-channel allocation

& Sandip [email protected]

Abhirup Das [email protected]

1 Institute of Radio Physics and Electronics, University of Calcutta, 92, Acharya Prafulla ChandraRoad, Kolkata, West Bengal 700009, India

2 ITRA project ‘‘Mobile Broadband Service Support over Cognitive Radio Networks’’, IRPE-CU,Kolkata, India

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Wireless Pers Commun (2015) 84:1303–1323DOI 10.1007/s11277-015-2689-3

http://crossmark.crossref.org/dialog/?doi=10.1007/s11277-015-2689-3&domain=pdf

http://crossmark.crossref.org/dialog/?doi=10.1007/s11277-015-2689-3&domain=pdf

1 Introduction

With the current cellular technology and services it would be very difficult to cope up

with the growing demand for increased users, high data rate and better quality of services

(QoS). A significant fraction of the users are indoor users. Recent studies have suggested

that more than 50 % of all voice calls and 70 % of data traffic originates indoors [1]. The

macrocell coverage becomes challenging to serve those massive indoor users with high

service demands due to wall penetration loss inside buildings. As an efficient solution the

application of so called femtocell access points (FAPs) or femtocell base stations (FBSs)

in indoors have been considered [1, 2]. Femtocells are small-coverage, low cost and low

power wireless network system deployed by the users inside a building to provide good

coverage and improved data rate. Femtocells are usually connected to the operator’s

broadband network through optical fiber or digital subscriber line (DSL) or separate RF

backhaul link while on the air interface femtocells use standard cellular technology (e.g.

GSM, UMTS, WiMAX, LTE etc.) [3]. Along with good coverage, low transmit power

and improved data rate the femtocell technology also brings many technical challenges

that have not been sufficiently addressed such as interference mitigation, synchronization

issues, access control, mobility management and so on. This work will focus only on the

problem of interference mitigation in femtocell networks. The deployment of femtocells

in the cellular network can lead to severe interference to the macrocell users (MUs)

which degrade the QoS of the MUs and at the same time the deployed femtocell users

(FUs) can also suffer from significant interference from the transmission of existing

MUs. This type of interference is known as the cross-tier interference. On the other hand,

since the femtocells are often deployed inside the residential apartments, many of those

femtocells are likely to be overlapped or deployed very close to each other so as to cause

mutual interference which is termed as inter femtocell interference or co-tier

interference.

Different techniques have been proposed in literature in order to manage the inter-

ference issues in femtocell networks. In [4], the authors studied the downlink cross-tier

interference problem in macro-femto two-tier networks with shared spectrum by power

control. Li et al. in [5] discusses about the resource allocation problem in orthogonal

frequency division multiple access (OFDMA) femtocell network and a sub-carrier and

power allocation approach to manage the cross-tier interference in underlay femtocell

network is presented in [6]. In [7], uplink capacity analysis and interference avoidance

strategy for a shared spectrum two-tier DS-CDMA network has been shown. In [8], the

authors propose a novel radio resource management scheme of joint power and sub-

carrier allocation to maximize the system capacity in indoor dense environment where

the users are exposed to intercell interference. In [9], the authors showed a downlink

spectrum sharing technique in an overlay mode of cognitive femtocell networks which

can achieve more average capacity than fixed power control scheme. Li et al. in [5] and

Xiang et al. in [9] tackle the power control problem in a distributed manner due to self-

organizing feature of the femtocell networks and therefore they all involves cognitive

radio technology whereas [10] assigns dedicated spectrum to femtocells for eliminating

cross-tier interferences. In [11] an efficient frequency assignment scheme has been

proposed for femtocells considering the practical issues such as hand off and coverage

along with interference. Several other research works have been carried out on various

interference management schemes for femtocell networks such as using power control

1304 S. Karar, A. Barman

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[12, 13], spectrum allocation [14], multiple antennas [15–17], adaptive resource allo-

cation [18], cognitive radio [19, 20] and many others.

A practical approach to mitigate the co-tier and cross-tier interference is proper allo-

cation strategy of frequency channels and power among the users which we will be our

focusing area in this work. Throughout the paper we will be focusing our attention to a

particular macrocell C which contains several femtocells. We consider that both the

macrocell and femtocell network are using OFDMA technology. The OFDMA based

femtocells are preferred over CDMA based femtocell network because of its intracell

interference avoidance properties and robustness to multipath [3]. In an OFDMA system,

the total available bandwidth is divided into orthogonal subcarriers which are then grouped

into sub-channels or sub-bands. We assume that in a macrocell only one such sub-channel

can be allocated to a particular MU and similarly in a femtocell also one such sub-channel

can be allocated to a FU. The transmission strategy and channel selection tasks for MUs

are functioning independently of the femtocell network as governed by the macrocell base

station (MBS). In this work, we focus on proper spectrum and power allocation among the

FUs without hampering the transmission of the MUs with the objective of minimizing the

co-tier and cross-tier interference in the network. One easy way to assign frequencies to the

FUs is by allocating the set of sub-channels that has not been assigned to the macrocell

C. This will definitely increase the network capacity, but in addition to this, to achieve the

full spectrum utilization we propose to use the same set of sub-channels that has been

allotted to the uplink band for macrocell C which will be shared among the femtocells

within C dynamically in an opportunistic way. The reason why the FUs will be utilizing the

uplink band of the macrocell C rather than the downlink band is discussed in Sect. 2. We

study on the sub-channels as well as power allocation issues for the FUs and want to find

out the maximum number of FUs with decent QoS can be accommodated within the

macrocell C through such opportunistic sharing.

The sub-channel allocation problem is solved by considering the femtocell network as

an interference graph with the FUs acting as vertices or nodes of the graph and edges

between two nodes implies that the corresponding FUs cannot be assigned the same sub-

channel. Each FU which is equipped with sensing equipments can proactively sense and

acquire information about the vacant sub-channels available to it. The sub-channel allo-

cation problem is then modeled as a list coloring problem where each node can be colored

from a list of available colors and an efficient algorithm is proposed to color the maximum

number of nodes possible. The determination of suitable transmission power for the as-

signed FUs is also a significant issue because each of the FUs can act selfishly to increase

its own SINR by increasing its transmission power indefinitely. This may hamper the

transmission of other users. To prevent this situation a non-cooperative power control

game based approach is presented to assign the transmission power to the FUs keeping in

mind that the total cumulative interference created at the MBS at each sub-channel is kept

below a tolerable threshold. The MBS controls the transmit power of the FUs by pricing

their resulted interference power levels at the MBS. So a FU cannot increase its trans-

mission power indefinitely because it has to pay the price for it. A Stackelberg game has

been formulated to jointly maximize the revenue of the macrocell as well as individual

utilities of different FUs for the proposed price-based power allocation. The power control

work is mainly inspired from [21], but in that work the aggregate co-tier interference is

avoided for sparse femtocell deployment and has been kept limited within a constant bound

for dense deployment scenario, which simplifies the situation and making it quite

straightforward whereas in our work the interference from each user has been taken care of

Opportunistic Sub-channel and Transmit Power Allocation in… 1305

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individually. The whole process of sub-channel assignment and power allocation among

the FUs must be repeated dynamically depending on the spectrum usage of the MUs.

The main contributions of this paper are summarized as follows. A general framework

for sub-channel allocation among the FUs within a particular macrocell using the same

frequency subset that has also been used by the associated macrocell using a list coloring

algorithm has been presented. Secondly a price-based power allocation scheme is proposed

for the FUs, whereby the MBS controls the transmit power of the FUs by pricing their

resulted interference power levels at the MBS receiver subject to a maximum tolerable

interference margin.

The rest of the paper is organized as follows. First the system model of the macrocell

and femtocell based two-tier network is introduced in Sect. 2. The frequency sub-channel

allocation algorithm among the FUs is given in Sect. 3 and in Sect. 4 a price-based power

control scheme is proposed for the FUs. The implementation protocol for the dynamic

allocation of sub-channel and power among the FUs is discussed in Sect. 5. Numerical

results are given in Sect. 6. Finally, Sect. 7 concludes the paper.

2 System Model

The system consists of a dense single macrocell C providing cellular coverage in a par-

ticular region along with a few femtocells located within as shown in Fig. 1. For uplink

transmission of the MUs in the macrocell, different MUs will transmit at different fre-

quencies and the spatial location of the MUs within the macrocell would result in a spatial

distribution of frequency holes within the macrocell which can properly be utilized to

operate the femtocell network.

But this is not the case for downlink transmission, as in downlink transmission the MBS

transmits at all frequencies over the whole macrocell area thereby creating no spectrum

vacancy at any place when all the sub-channels are occupied by the MUs. The primary

uplink band sharing by secondary users is not new in cognitive radio. Lee et al. in [22] has

demonstrated that such an uplink band sharing can enhance the throughput performance of

point-to-point link where the primary users (MUs) are assumed to form a cellular network.

In this work, we therefore consider the uplink sub-channel allocation to the FUs from the

uplink band of the MUs. So here the MBS as well as the FBSs will suffer the interference.

Fig. 1 System model of a macrocell and femtocell based two-tier network


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The analysis for downlink transmission for the FUs can be done in the same way. Let S be

the set of frequency sub-channels allotted to the macrocell C for uplink transmission. These

set of frequency sub-channels are shared between the MUs and FUs in a cognitive way.

The MUs, being the primary users, have the higher priority to access the frequency sub-

channels whereas the FUs are the secondary users which can access only those frequency

sub-channels that have not been used by any nearby MU. We assume that both the MUs

and the femtocells are distributed uniformly within the macrocell C. The transmission zone

of a MU can be defined as a circular area of radius RMT so that the received power outside

the area is negligibly small. The value of RMT should be carefully chosen depending on the

transmission power of the MU. If a MU is using the frequency sub-channel j then a FU can

only use the sub-channel j if the FU and its corresponding FBS are located outside the

transmission zone of the MU. Each of the FUs as well as the FBSs prepares a list of vacant

sub-channels available to it by proactively sensing the radio environment and sends the

information to a central femtocell controller (CFC). Then the CFC prepares a modified list

of available sub-bands for uplink for each of the FUs by taking the set of common sub-

bands available to the FU and its associated FBS. The CFC now assigns sub-channels to a

maximum number of FUs with the objective of minimizing the co-tier interference among

the femtocells using an allocation algorithm as described in the next section. The sub-

channel assignment process must be repeated dynamically depending on the sub-channel

usage by the MUs.

3 Frequency Sub-channel Allocation Among FUs

Our objective is to find the optimum allocation of the sub-bands among the FUs from their

modified lists of available sub-bands so as to maximize the system throughput by assigning

the sub-bands to a maximum number of FUs possible. The frequency assignment scheme

must take care of the fact that within each femtocell more than one FU cannot be assigned

the same frequency sub-channel. Also it may sometime happen that two FUs under the

coverage of different femtocells may be very closely located so as to interfere each other if

they share the same frequency. So care must be taken to resolve this issue during sub-band

allocation.

For simplicity, we model the interference situation as a binary condition that means any

two users can either interfere each other or not at all. The femtocell network can be

represented by an interference graph G V ;Eð Þ and the frequency allocation problem can be

translated into a sort of vertex coloring problem in graph theory. All the FUs can be

represented by a set of vertices V in the graph G V;Eð Þ where E is the set of all the edges.

Two vertices u and v are connected by an undirected edge (u, v) [ E if and only if their

associated FUs are either under the same femtocell coverage or under different femtocell

coverage but the separation between one FU and the FBS corresponding to the other FU is

less than a critical distance RFT . The critical distance RFT should be carefully chosen

depending on the maximum transmission power of a FU.

The sub-band allocation problem is equivalent to the coloring of the vertices of the

graph G V;Eð Þ where each color corresponds to a sub-band and two adjacent vertices must

be colored with different colors with each color chosen from the available list of colors

corresponding to the modified list of available sub-bands at the FU. Throughout the paper

we will use the terms ‘FU’ and ‘vertex’/‘node’ interchangeably and also the terms ‘color’


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and ‘sub-channel’/‘sub-band’. If there are sufficiently large numbers of FUs in the

macrocell then a complete solution may not be obtained. Incomplete solution means some

of the FUs remain unassigned. Our objective is to color as many vertices as possible i.e. to

minimize the number of unassigned FUs to achieve the maximum throughput from the

system.

We represent all the FUs in the macrocell C by the vertex set V = {v1, v2, …, vK} and

the set of sub-bands by S ¼ 1; 2; . . .;Wf g. The set of available sub-bands at FU vk is

denoted by Yk ¼ yk1; yk2; . . .ykrkf g, Yk ( S for all k. The function L:V ? S denotes the

frequency assignment function or the vertex coloring function which assigns a sub-band to

each of the FUs in the macrocell such that L vkð Þ 2 Yk. We define a sub-band allocation

matrix D, the elements of which are given by:

Djk ¼1 if sub-band jis allocated to FUvk0 Otherwise

�ð1Þ

Note that Djk ¼ 0 for j 62 Yk and if Djk ¼ 1 then Djl ¼ 0 for all neighbors vl of vk in G.

Therefore for the graph G V;Eð Þ our objective is:

maxDjk

Xvk

Xj2Yk

Djk ð2aÞ

s:t: DjkDjl ¼ 0 if vl; vkð Þ 2 E ð2bÞ

and

Djk ¼ 0 for j 62 Yk ð2cÞ

To achieve an optimal solution of this problem is very difficult. Hence a suboptimal

heuristic algorithm is adopted to solve the problem. Detailed description of the algorithm is

given below:

1. Group the vertices of the graph G(V, E) into sets w1, w2, …, wW such that wi contains

the set of all vertices that has sub-band i in there modified list of available sub-bands.

The sets w1;w2; . . .;wW are in general overlapping sets i.e. not mutually exclusive. If

there is no any vertex vk such that Yk ¼ ; thenSW

i¼1 wi ¼ V .

2. Select a set from w1, w2, …, wW with minimum cardinality. Let the corresponding

sub-band of the minimum cardinality set be q, then wq ¼ argmin

jw1j; w2j j; . . .; wWj jf g. If there are two or more sets having the same minimum

cardinality then any one of those sets can be selected at random.

3. Find a maximal independent set from the set of vertices of wq and assign the sub-band

q to those nodes. To find the maximal independent set, first select a vertex u from wq

with minimum degree in the sub-graph consisting of the vertices in wq. If there is more

than one vertex that has the same minimum degree, then any one of those vertices can

be chosen randomly. Starting with the selection of a vertex of minimum degree

improves the chance of getting an independent set with maximum number of vertices.

Assign the sub-channel q to the node u and then modify wq by deleting u and

neighbors of u from wq. Now from the modified set of wq select another node u0 with

minimum degree and assign frequency sub-channel q to it using the same procedure as

mentioned above. Repeat this process till there are no vertices left in wq. Finally we get


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an independent set of vertices from wq and those vertices have been assigned the

frequency q.

4. Eliminate the nodes from V that have already been assigned a frequency sub-band in

the previous steps. Also the sub-band q is removed from the list of available sub-bands

for each remaining FU if it is within the list. Hence we get a modified graph G0 V 0;E0ð Þ.5. The first round of the algorithm stops here. In the next round repeat the steps 1, 2, 3

and 4 again for the modified graph G0. The algorithm will finally stop at the last round

when there is no frequency sub-channel left to be assigned or the vertex set for the

final modified graph is completely empty.

3.1 Algorithm

To illustrate the above algorithm using an example we consider a simple interference graph

G as shown in Fig. 2. This graph G in Fig. 2 is just a prototype for illustration purpose to

show the functioning of our proposed algorithm, which has nothing to do with the actual

femtocell network interference graph, topology-wise which may be immensely dissimilar.

The graph G consists of a vertex set V of eleven vertices or nodes labeled as {1, 2, 3 …11}

each corresponds to a FU. There are a total number of four frequency sub-bands or colors

labeled as 1, 2, 3, and 4. Not all the four colors are available to all the vertices. The lists of

numbers within the braces shown in the graph of Fig. 2 are the list of available colors to the

corresponding nodes. Each node can select a color only from the corresponding available

color list.

In the first step of the algorithm, the nodes are grouped into overlapping sets w1, w2, w3

and w4 such that wj (for j ¼ 1; 2; 3; 4Þ consists of those vertices that has the color j in its listof available colors e.g. w1 ¼ 1; 2; 3; 8; 9f g. Similarly for the other three sets

w2 ¼ 1; 2; 3; 4; 5; 9; 10f g, w3 ¼ 2; 3; 4; 5; 6; 7; 8; 9f g and w4 ¼ 4; 5; 6; 7; 8; 10; 11f g.


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In second step, one of the sets from w1, w2, w3 and w4 needs to be selected with the

minimum cardinality. Obviously the set w1 has the minimum cardinality since,

w1j j ¼ 5; w2j j ¼ 7; w3j j ¼ 8; and w4j j ¼ 7.

Next find the maximum independent set of vertices from w1 ¼ 1; 2; 3; 8; 9f g in graph G

and those independent set of vertices are assigned the color 1. For that we have to start with

minimum degree vertex from the sub-graph containing the vertex set w1. The degree of

nodes 8 and 9 is one whereas all the other nodes in w1 have the degree two in the sub-

graph. So there is a tie between the nodes 8 and 9 which have the minimum degree. From

these nodes, node 9 is arbitrarily selected at random and the color 1 is assigned to node 9.

Now the vertices 8 and 9 are deleted from the set w1 as vertex 8 is the neighbor of vertex 9.

Hence the set w1will now contain the nodes 1, 2 and 3 all of which have the degree two in

the sub-graph containing the vertex set w1. The color 1 is also randomly assigned to node 1.

Now the vertices 1, 2 and 3 are eliminated from the set w1 which leaves w1 completely

empty. This completes the third step and the color 1 has been assigned to the nodes 1 and 9.

In the final step of the first round the graph G is modified by eliminating the nodes 1 and

9 from the vertex set V. Also the color 1 is deleted from the lists of available colors for

those vertices which have the color 1 in their lists i.e. for the vertices 2, 3 and 8. The

modified graph is shown in Fig. 3 from which the second round starts. All the steps as

stated previously are performed again to find out that the nodes 10, 2 and 5 can be assigned

the color 2. Eliminating those vertices and the color 2 we obtain the modified graph for

third round shown in Fig. 4. The process has been repeated all over again to assign the

color 3 to the nodes 3 and 6. Finally in the last round, we are left with the graph in Fig. 5 in

which four nodes are left and they have to be properly colored only using color 4. The

nodes 7, 8 and 11 cannot be assigned the same color simultaneously. Applying the algo-

rithmic steps the sub-band 4 can be assigned to at most three nodes 4, 7 and 11. So the node

8 remains to be unassigned. The final result of the algorithm is shown in Table 1.

4 Power Allocation Among FUs

After the sub-channel assignment step, each of the FUs must determine a suitable trans-

mission power to optimize its own utility as well as ensures that the total cumulative

interference power at the MBS in each sub-channel is kept below a predefined threshold.

Fig. 2 Initial interference graphalong with the list of availablefrequency sub-channels at eachvertex


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We look at the power control problem for each particular sub-channel individually. A

price-based power allocation scheme is proposed for the FUs in each sub-channel, whereby

the MBS controls the transmit power of the FUs by pricing their resulted interference

power levels at the MBS receiver subject to a maximum tolerable interference margin.

A Stackelberg game can be formulated to jointly maximize the revenue of the macrocell as

well as the individual utilities of different FUs for the proposed price-based power allo-

cation. In this work, we assume uniform pricing i.e. same price is charged to all the FUs in

a particular sub-channel. The MBS tries to increase the total revenue whereas the FUs try

to increase the throughput as well as minimize the price spent for creating interference at

the MBS. Here in the Stakelberg game the MBS acts as a leader and the FUs act as

followers. The leader (MBS) always moves first and the followers (FUs) move subse-

quently. The MBS imposes a price c on per unit amount of received interference power

from each FU for a particular sub-channel. Then, the FUs figure out their power allocation

strategies to maximize their individual utilities based on the assigned interference price.

Fig. 3 Interference graph after1st round along with the list ofavailable frequency sub-channelsat each vertex

Fig. 4 Interference graph after2nd round along with the list ofavailable frequency sub-channelsat each vertex

Fig. 5 Interference graph in thefinal round along with the list ofavailable frequency sub-channelsat each vertex


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Consider a particular sub-channel s is allocated to N number of FUs labeled by

1; 2; . . .;N by the proposed sub-carrier allocation algorithm discussed in the previous

section. Given the price c for the sub-channel s as determined by the MBS, N number of

FUs will participate in a non-cooperative power control game GP ¼ N ; Pif g; Ui :ð Þf g½ �whereN ¼ 1; 2; . . .;Nf g refers to the set of players i.e. the FUs, Pi is the strategy set of the

transmission power for user i and Ui :ð Þ is the utility for FU i. We can define a concrete

expression of the utility function for the FU i as [21]

Ui pi; p�i; cð Þ ¼ li log 1þ hiipiPj 6¼i hijpj þ r2n

!� chBipi ð3Þ

where pi is the power transmission strategy for FU i and p-i = [p1, p2, …, pi-1,

pi?1, …pN] is the strategy vector for all the N - 1 FUs except user i, li is the utility gain

per unit transmission rate for user i, hij is the channel power gain from FU j to the FBS

associated with the FU i and hBi is the channel power gain from FU i to the MBS. Each

player i 2 N selects a proper power transmission strategy within the strategy space Pi to

maximize its utility function Ui pð Þ where p, pi; p�i½ � ¼ p1; p2; . . .; pN½ � is the power vectorof all the N FUs. Formally, for user i 2 N , this power control game can be expressed as

maxpi

Ui pi; p�ið Þ 8i 2 N ð4aÞ

s:t: 0� pi� pmax ð4bÞ

We are interested of a Nash equilibrium solution [23] of (4) of the power control game

GP. A set of strategies is in Nash equilibrium if no player can improve its utility by

unilaterally changing its own strategy. In the non-cooperative power control game

GP ¼ N ; Pif g; Ui :ð Þf g½ �, the power vectorp� ¼ ½p�1; p�2; . . .; p�N � is a Nash equilibrium so-

lution if for every i 2 N ,

Ui p�i ; p��i

� ��Ui pi; p

��i

� �8pi 2 Pi; pi 6¼ p�i ð5Þ

At Nash equilibrium, no user can unilaterally improve its individual utility. The de-

velopment of a Nash equilibrium solution for a game requires the investigation of its

existence and uniqueness. We can state the following proposition regarding the existence

of Nash equilibrium solution of (4) which is given in [24].

Proposition 1 A Nash equilibrium exists in a non-cooperative power control game

GP ¼ N ; Pif g; Ui :ð Þf g½ �; if for all i 2 N ,

1. Pi is a non-empty, convex and compact subset of some Euclidean space RN .

2. Ui pð Þ is continuous in p and quasi-concave in pi.

Table 1 Final outcome of the sub-channel allocation algorithm

Rounds Colors to be assigned Nodes to which thecolor would be assigned

1st Round (1) 1, 9

2nd Round (2) 10, 2, 5

3rd Round (3) 3, 6

4th Round (4) 4, 7, 11


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Following the Proposition 1 we can state the following theorem:

Theorem 1 A Nash equilibrium exists in the non-cooperative power control game

GP ¼ N ; Pif g; Ui :ð Þf g½ �.

Proof Firstly, the power set Pi which is defined to be Pi ¼ pi : 0� pi� pmaxf g, is a non-empty, convex and compact subset of some Euclidean space R

N . The utility Ui pð Þ for ithFU is given in (3) which is obviously continuous in p. Now we compute the second order

partial derivative of Ui pð Þ w.r.t pi to prove its concavity.

oUi pð Þopi

¼ hiilihiipi þ

Pj 6¼i hijpj þ r2n

� hBic ð6Þ

o2Ui pð Þop2i

¼ � h2iili

hiipi þP

j 6¼i hijpj þ r2n

� �2 ð7Þ

The second order derivative of Ui pð Þ w.r.t pi is always negative 8pi� 0, therefore Ui pð Þ isa concave in pi. According to Proposition 1, a Nash equilibrium exists in the non-coop-

erative power control game GP ¼ N ; Pif g; Ui :ð Þf g½ � h

Theorem 2 The non-cooperative power control game GP ¼ N ; Pif g; Ui :ð Þf g½ � has a

unique Nash equilibrium.

Proof By Theorem 1, we know that there exists a Nash equilibrium in G. Let p� be the

Nash equilibrium solution, which must satisfy the best response function r(p) so that

p* = r(p*), where r pð Þ ¼ r1 p�1ð Þ; r2 p�2ð Þ; . . .rN p�Nð Þð Þ: ri p�ið Þ is the best response

function of player i given the strategy of other players, which is given by

ri p�ið Þ ¼ argmax0� pi � pmax

Ui pi; p�ið Þ ð8Þ

The fixed point p* = r(p*), which is the Nash equilibrium point is unique for a standard

function [25]. Therefore we only need to prove that the best response function r(p) is a

standard function. The function r(p) is a said to be standard if for feasible values of p all the

following properties are satisfied:

1. Positivity: r(p)[ 0;

2. Monotonicity: If p� p0 then r pð Þ� r p0ð Þ i.e. r pð Þ is monotonically decreasing;

3. Scalability: For alla[ 1, ar pð Þ[ r apð Þ.Since we’ve already proved that Ui pi; p�ið Þ is concave w.r.t. pi, therefore the best responseis achieved by solving the maximization problem (4) using KKT condition. The solution is

given by:

ri p�ið Þ ¼

0 iflichBi

\Ri

hiilichBi� Ri

hiiif 0� li

chBi� Ri

hii� pmax

pmax iflichBi

[Ri

hii

8>>>>><>>>>>:

ð9Þ

where Ri ¼P

j 6¼i hijpj þ r2n:


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Given the condition 0� lichBi� Ri

hii� pmax; the best response function ri p�ið Þ ¼ li

chBi�

Ri

hii[ 0 is always positive. Therefore r pð Þ[ 0:

To prove the monotonicity, we know that ri(p-i) is a linear function ofP

j 6¼i hijpj and is

decreasing monotonically whenP

j=ihijpj is increasing. Therefore r pð Þ is a monotonically

decreasing function of p:

For scalability we have, ari p�ið Þ � ri ap�ið Þ ¼ a� 1ð Þ lichBi� r2n

hii

� �[ 0 for a[ 1 given

the condition 0� lichBi� Ri

hii� pmax: Hence ari p�ið Þ[ ri ap�ið Þ; this satisfies the scalability

property.

In conclusion, r pð Þ is a standard function as it is positive, monotonic and scalable.

Therefore there exists a unique Nash equilibrium solution for the non-cooperative power

control game GP ¼ N ; Pif g; Ui :ð Þf g½ �. h

Theorem 3 The unique equilibrium of the non-cooperative game GP ¼N ; Pif g; Ui :ð Þf g½ � is given by

p�i ¼0

ðH�1bÞipmax

if ðH�1bÞi� 0

if 0�ðH�1bÞi� pmax

if ðH�1bÞi� pmax

8<: ð10Þ

where

H ¼

h11 h12h21 h22

� � � h1Nh2N

..

. . .. ..

.

hN1 hN2 � � � hNN

26664

37775; b ¼ b1; b2; . . .; bN½ �T

where bi ¼ hiilihBic� r2n for all i ¼ 1; 2; . . .N and xð Þi means the ith element of vector x.

Proof Equating the first order derivative of Ui pð Þ to zero for all i 2 N we get from (6)

hiilihiip

�i þ

Pj 6¼i hijp

�j þ r2n

� hBic ¼ 0 8i 2 1; 2; . . .;Nf g

or,

hiip�i þ

Xj 6¼i

hijp�j ¼

hiilihBic� r2n 8i 2 1; 2; . . .;Nf g ð11Þ

In matrix form we can write (11) as Hp� ¼ b where p� ¼ p�1; p�2; . . .p

�N

� �T; H and b are

defined above. From the above matrix equation we can calculate the unique equilibrium

power vector as p� ¼ H�1b. Since the transmit power is bounded by the constraint

0� p�i � pmax, so we have the result given in (10). h

Based on the analytical result of the non-cooperative power control game among the

FUs, the leader of the Stackelberg game, the MBS, can choose the price ‘c’ for unit amount

of interference created at the MBS in the sub-channel s in order to maximize its revenue


123

keeping an eye to the constraint that the total cumulative interference at the MBS does not

cross a pre-defined threshold ITh. Mathematically we can write the optimization problem as

maxc� 0

XNi¼1

chBip�i ð12aÞ

s:t:XNi¼1

hBip�i � ITh ð12bÞ

From (10) we can write

p�i ¼XNj¼1

~hijbj ¼XNj¼1

~hijhjjljhBjc� r2n

!þð13Þ

where ~hij is the element of the ith row and jth column of H�1 and xþ means max x; 0ð Þ.Here we neglect the upper-bound of pi

* because the values of pmaxand ITh would be so

chosen that very few values of pi* would cross pmax. Now using the value of pi

* we can write

the problem (12) as

maxc� 0

XNi¼1

ai � bicð Þþ ð14aÞ

s:t:XNi¼1

ai=c� bi� �

� ITh ð14bÞ

where

ai ¼XNj¼1

~hijhjjhBiljhBj

!

and bi ¼PN

j¼1~hijhBir2n� �

. For each FU i 2 N we can define an indicator function fi such

that

fi ¼1 if c\

aibi

0 Otherwise

(ð15Þ

Now the problem (14) can be written as

maxc� 0

XNi¼1

fi ai � bicð Þ ð16aÞ

s:t:XNi¼1

fi ai=c� bi� �

� ITh ð16bÞ

The above problem is a non-convex optimization problem due to fi and so is difficult to

solve. The problem formulated above is similar to the one in [21]. To simplify the above

problem we first consider that the values of c and ITh are such that all users are admitted. In


123

this case c\ai=bi for all i 2 N . The maximum value of c to accommodate all the FUs is

given by cmax ¼ minj

aj=bj� �

. The MBS cannot charge a price more than cmax when all the

FUs need to be admitted and at the same time the MBS must set a tolerable interference

threshold value

ITh [XN1¼1

aicmax � bi

¼XN1¼1

aiiminj aj=bj

� �� b

!:

So under the above conditions on the values of c and ITh, when all the FUs will be

admitted, the problem (16) can be written as

minc� 0

XNi¼1

bic ð17aÞ

s:t:XN1¼1

aic� bi

� ITh ð17bÞ

This is a convex optimization problem the solution of which is given by

c� ¼PN

i¼1 ai.ðPN

i¼1 bi þ IThÞ ð18Þ

With the result abovewe can now solve the problem (16). The FUs are first sorted in the order

a1b1

[a2b2

[ � � � [ aNbN

:

If we define

Qk,

Xki¼1

aiak=bkð Þ � bi

8k ¼ 1; 2; 3; . . .N

then all the FUs will be admitted ifITh [QN . If QN�1\ITh\QN , then Nth FU in the

ordered list will be discarded and the optimum price in this case is given by

c� ¼PN�1

i¼1 aiPN�1i¼1 bi þ ITh

� � :

In general if Qk\ITh\Qkþ1 then the price c� is given by

c� ¼Pk

i¼1 aiPki¼1 bi þ ITh

� � ð19Þ


123

5 Implementation Protocol

In this section, we will propose a protocol to dynamically allocate the sub-channels and

power among the FUs. The sub-channel allocation part is centrally controlled by the CFC

whereas the power allocation among the FUs is handled in a distributed way. The total time

is divided into slots of duration T. It has been assumed that the primary activity do not

change much during each time slot. In the beginning of each time slot all the FUs within

the macrocell C can sense the radio spectrum and make lists of the available sub-channels

and inform the lists to their associated FBS through the backhaul connection. Each of the

FBSs also creates such lists itself by sensing and sends all of the lists to the CFC. The CFC

then prepares a modified list of available sub-bands for uplink for each of the FUs by

taking the set of common sub-bands of the FU and its associated FBS. The CFC now

assigns the sub-channels to the FUs using the graph coloring algorithm described above.

This channel assignment information is then disseminated among the FBSs as well as the

MBS through backhaul link. The FBSs then inform the information to their associated FUs.

The MBS uses this channel allocation information to determine the price for unit amount of

received interference for each of the sub-channels. It has been assumed that the channel

state information from each FU to the MBS and from each FU to each FBS are known at

the MBS in advance and also each FU knows the channel state information from each FBS

to itself. We assume that the channel conditions do not change with time. The pricing

information is then disseminated among the FUs through CFC and FBSs. Each FU now

knows the sub-channel allocated to itself and also the information about other FUs which

have been allocated the same sub-channel. Based on this information and the price for unit

amount of interference created at the MBS, each FU now individually determines the

equilibrium transmission power and transmits till the next time slot arrives. At the be-

ginning of the next time slot the whole process is repeated all over again. The imple-

mentation protocol of the whole process is shown in Fig. 6.

6 Simulation Results

The simulation model comprises of one circular macrocell of 500 m radius with an MBS at

the centre and multiple femtocells located within. Each femtocell is assumed to be of

circular shape with a radius of 20 m consisting of one FBS at the centre and four numbers

of active FUs. Two or more femtocells may overlap in some places. The FBSs, FUs, and

MUs are deployed randomly within the macrocell area. The system performance is

evaluated for different femtocell deployment densities. The performance of the proposed

algorithm has been evaluated for different snapshots of the two-tier network. We generate

100 different snapshots of the network for each of the different femtocell deployment

densities and the performance is averaged over all the snapshots. We assume that there are

20 sub-channels that have been allotted to the macrocell for uplink transmission and all the

sub-channels are occupied by the MUs. We assume that all the MUs transmit with equal

power (45 dBm). Beyond a transmission zone of 500 m radius from any MU the received

signal strength is negligibly small (less than -100 dB) so that any femtocell using the

same frequency beyond the zone would not be interfered by the MU. The maximum

transmission power pmax of a FU is assumed to be 30 dBm. The FU critical separation

distance RFT is selected as 60 m because beyond that distance the path loss is so high

(greater than 102 dB) that the received interference becomes negligibly small. The average


123

noise power level for each sub-channel is assumed to be -120 dBm. The value of li i.e.the utility gain per unit transmission rate for user i is taken to be 1 for all FUs. All the

simulation parameters are summarized in Table 2.

The channel propagation model is represented as a combination of path loss and wall

penetration loss. The effect of log-normal shadowing is left out for simplicity of calcu-

lation. The wall penetration loss (WL) is assumed to be 10 dB. The following five types of

radio link situations may arise in a femtocell based two-tiered network:

1. MBS $ MU

2. MBS $ FU

3. FBS $ FU in the same femtocell

4. FBS $ FU in different femtocell

5. FBS $ MU

Fig. 6 Implementation protocol


123

The path loss models for these links are given in Table 2.

For each of the 100 different snapshots of two-tier femtocell network the proposed

algorithm for the sub-channel allocation among the FUs has been applied and averaged

considering different femtocell densities with the number of femtocells varying from 30

to 80. It has been observed that for all these cases there remain some FUs that cannot be

assigned any sub-channel. Figure 7 shows the average number of such unassigned FUs

after the sub-channel allocation step. It is obvious that the number of unassigned FUs

will increase with the number of femtocell. But if we see in terms of percentage, it is

found that the percentage of unassigned FUs is nearly same (approx. 8 %) for all the

different femtocell densities. After the sub-channel allocation step, the equilibrium

Table 2 Simulation parameters

Parameters Value

Macrocell coverage radius (Rm) 500 m

Femtocell coverage radius (Rf) 20 m

MU transmission power (Pmu) 45 dBm

Maximum transmission power of a FU (pmax) 30 dBm

Number of FUs in a femtocell 4

Number of sub-channels in uplink band 20

MU transmission zone radius (RMT) 500 m

FU critical separation distance (RFT) 60 m

Wall penetration Loss (WL) 10 dB

Noise power per sub-channel -120 dBm

MBS $ MU path loss 15.3 ? 37.6log10 (d)

MBS $ FU path loss 15.3 ? 37.6log10 (d) ? WL

FBS $ FU (same femtocell) path loss 38.46 ? 20log10 (d) ? 0.7d

FBS $ FU (different femtocell) path loss 15.3 ? 37.6log10 (d) ? 2WL

FBS ! MU path loss 15.3 ? 37.6log10 (d) ? WL

Fig. 7 Number of unassigned FUs after sub-channel assignment


123

transmission power for each of the assigned FUs is calculated using the rule described

before. Figure 8 shows cumulative probability distribution (CDF) of the received SINR

at the corresponding FBSs for all the FUs considering the transmission of MUs as

interference when the tolerable interference margin Ith at the MBS set at -120 dB in

each sub-channel with the number of femtocells as parameter. Figure 9 shows the CDF

of received SINR of the FUs for different tolerable interference margin at the MBS when

the number of femtocells is fixed at 50. We consider a received SINR of 10 dB as a

minimum requirement for a FU which is sufficient for a quality video transmission with

data rate of 384 kbps. A FU that cannot achieve this minimum requirement is said to be

in outage. Figure 10 depicts the percentage of the FUs in outage for different femtocell

densities which can easily be derived from the CDF of the received SINR plots. From

the figure it is evident that the fraction of the total FUs in outage increases when the

number of femtocells is increased for a fixed interference margin Ith at the MBS and also

when the Ith value is decreased for a fixed femtocell density.

Fig. 8 CDF of the SINR of theFUs after the sub-channel andpower allocation for differentfemtocell densities when thetolerable interference threshold atthe MBS (Ith) is -120 dB

Fig. 9 CDF of SINR of the FUs after the sub-channel and power allocation for different tolerableinterference threshold at the MBS (Ith) when the number of femtocells is 50


123

Total revenue earned by the MBS for different femtocell densities with different values

of tolerable interference threshold at the MBS is shown in Fig. 11. It is obvious from the

plot that for a fixed interference threshold Ith at the MBS, the macrocell would earn more

revenue when femtocell density increases, but for a fixed femtocell density the interference

threshold has practically no effect on the earned revenue of the macrocell.

7 Conclusion

In this paper, we studied the sub-channel and power allocation in an OFDMA based

cognitive femtocell network and proposed a scheme to minimize the co-tier and cross-tier

interference. Firstly, the sub-channels are allocated among the FUs from the uplink band of

Fig. 10 Percentage of the FUs in outage for different tolerable interference thresholds at the MBS fordifferent femtocell densities

Fig. 11 Total revenue earned by the MBS for different femtocell densities with the tolerable interferencethreshold at the MBS (Ith) kept at -115 dB, -120 dB and -125 dB


123

the macrocell opportunistically using an efficient graph coloring algorithm. The objective

was to assign sub-channels to the maximum number of FUs possible. The determination of

suitable transmission power for the assigned FUs is also a significant issue because each of

the FUs can act selfishly to increase its own SINR by increasing its transmission power

indefinitely. A price-based power allocation scheme is proposed to assign the transmission

power among the FUs, whereby the MBS controls the transmission power of the FUs by

pricing their resulted interference power levels at the MBS with the constraint that the total

interference created at the MBS at each sub-channel is kept below a tolerable threshold.

The results show that substantial number of FUs with good QoS can be accommodated in

the macrocell using the same frequency sub-channels and thereby increasing the total

network throughput and coverage. Also the macrocell can earn some revenue by allowing

some interference from the FUs.

Acknowledgments The work is undertaken as part of Media Lab Asia Project entitled ‘‘Mobile BroadbandService Support over Cognitive Radio Networks’’.

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Sandip Karar received the bachelor’s degree (B.Tech) and masterdegree (M.Tech.) in Radio Physics and Electronics from University ofCalcutta, India, in 2010 and 2012, respectively. Presently he is workingtoward Ph.D. degree in the Institute of Radio Physics and Electronics,University of Calcutta, India. His research interests include wirelesscommunication and cognitive radio.

Abhirup Das Barman received the bachelor’s degree (B.Tech) inElectronics & Communication Engineering (ECE) from the Institute ofRadio Physics & Electronics (IRPE), Calcutta University and masterdegree (M.Tech.) in Electrical Engineering (EE) from IIT Kanpur andPh.D. from IRPE, Calcutta Univ. He worked few years in IndianBroadcasting Engineering Service (IBES), Government of India in theareas of Satellite Communications, Digital Terrestrial Transmission(DTT), etc. Then he joined as a faculty in the Dept. of Radio Physics &Electronics, Univ. of Calcutta. Presently he is an Associate Professorin this Department. In 2007 and 2009 he worked as a visiting re-searcher at the CNIT Photonic Networks National Laboratory, Pisa,Italy. In 2010-11 he worked as a researcher in Aalborg University,Denmark in the Dept. of Electronic Systems. His current researchinterests include signal processing in communication systems, broad-band cognitive radio and next-generation optical access networktechnologies.


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Opportunistic Sub-channel and Transmit Power Radio... · Opportunistic Sub-channel and Transmit Power ... interference management ... macrocell and femtocell network are using OFDMA