opportunistic sub-channel and transmit power radio... · opportunistic sub-channel and transmit...
Post on 12-Apr-2018
222 Views
Preview:
TRANSCRIPT
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 Kararsk7632@gmail.com
Abhirup Das Barmanabhirup_rp@yahoo.com
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
123
Wireless Pers Commun (2015) 84:1303–1323DOI 10.1007/s11277-015-2689-3
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
123
[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
123
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
1306 S. Karar, A. Barman
123
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’
Opportunistic Sub-channel and Transmit Power Allocation in… 1307
123
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
1308 S. Karar, A. Barman
123
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.
Opportunistic Sub-channel and Transmit Power Allocation in… 1309
123
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
1310 S. Karar, A. Barman
123
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
Opportunistic Sub-channel and Transmit Power Allocation in… 1311
123
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
1312 S. Karar, A. Barman
123
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:
Opportunistic Sub-channel and Transmit Power Allocation in… 1313
123
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
1314 S. Karar, A. Barman
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
Opportunistic Sub-channel and Transmit Power Allocation in… 1315
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
1316 S. Karar, A. Barman
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
Opportunistic Sub-channel and Transmit Power Allocation in… 1317
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
1318 S. Karar, A. Barman
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
Opportunistic Sub-channel and Transmit Power Allocation in… 1319
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
1320 S. Karar, A. Barman
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
Opportunistic Sub-channel and Transmit Power Allocation in… 1321
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’’.
References
1. Chandrasekhar, V., Andrews, J. G., & Gatherer, A. (2008). Femtocell networks: A survey. IEEECommunication Magazine, 46(9), 59–67.
2. Claussen, H., Ho, L. T., & Samuel, L. G. (2008). An overview of the femtocell concept. Bell LabsTechnical Journal, 13(1), 221–245.
3. Lopez-Perez, D., Valcarce, A., De La Roche, G., & Zhang, J. (2009). OFDMA femtocells: A roadmapon interference avoidance. IEEE Communications Magazine, 47(9), 41–48.
4. Chandrasekhar, V., Andrews, J. G., Muharemovict, T., Shen, Z., & Gatherer, A. (2009). Power controlin two-tier femtocell networks. IEEE Transactions on Wireless Communications, 8(8), 4316–4328.
5. Li, L., Xu, C., & Tao, M. (2012). Resource allocation in open access OFDMA femtocell networks. IEEEWireless Communications Letters., 1(6), 625–628.
6. Gupta, N. K., & Banerjee, A. (2011). Power and subcarrier allocation for OFDMA femto-cell basedunderlay cognitive radio in a two-tier network. In Proceedings internet multimedia systems architectureand application (IMSAA), (pp. 1–6).
7. Chandrasekhar, V., & Andrews, J. G. (2009). Uplink capacity and interference avoidance for two-tierfemtocell networks. IEEE Transactions on Wireless Communications, 8(7), 3498–3509.
8. Kim, J., & Cho, D. H. (2010). A joint power and subchannel allocation scheme maximizing systemcapacity in indoor dense mobile communication systems. IEEE Transactions on Vehicular Technology,59(9), 4340–4353.
9. Xiang, J., Zhang, Y., Skeie, T., & Xie, L. (2010). Downlink spectrum sharing for cognitive radiofemtocell networks. IEEE Systems Journal, 4(4), 524–534.
10. Sun, Y., Jover, R. P., & Wang, X. (2012). Uplink interference mitigation for OFDMA femtocellnetworks. IEEE Transactions on Wireless Communications, 11(2), 614–625.
11. Guvenc, I., Jeong, M. R., Watanabe, F., & Inamura, H. (2008). A hybrid frequency assignment forfemtocells and coverage area analysis for co-channel operation. IEEE Communications Letters, 12(12),880–882.
12. Ngo, D. T., Le, L. B., & Le-Ngoc, T. (2012). Distributed Pareto-optimal power control for utilitymaximization in femtocell networks. IEEE Transactions on Wireless Communications, 11(10),3434–3446.
13. Tan, C. W., Friedland, S., & Low, S. H. (2011). Spectrum management in multiuser cognitive wirelessnetworks: Optimality and algorithm. IEEE Journal on Selected Areas in Communications, 29(2),421–430.
14. Chandrasekhar, V., & Andrews, J. G. (2009). Spectrum allocation in tiered cellular networks. IEEETransactions on Communications, 57(10), 3059–3068.
15. Jeong, Y., Quek, T. Q., & Shin, H. (2011). Beamforming optimization for multiuser two-tier networks.Journal of Communications and Networks, 13(4), 327–338.
16. Jeong, Y., Shin, H., & Win, M. Z. (2011). Interference rejection combining in two-tier femtocellnetworks. In Proceormation theory and applicationo communications (PIMRC), (pp. 137–141).
1322 S. Karar, A. Barman
123
17. Guler, B., & Yener, A. (2014). Selective interference alignment for MIMO cognitive femtocell net-works. IEEE Journal on Selected Areas in Communication, 32(3), 439–450.
18. Ko, C. H., & Wei, H. Y. (2011). On-demand resource-sharing mechanism design in two-tier OFDMAfemtocell networks. IEEE Transactions on Vehicular Technology, 60(3), 1059–1071.
19. Cheng, S. M., Lien, S. Y., Chu, F. S., & Chen, K. C. (2011). On exploiting cognitive radio to mitigateinterference in macro/femto heterogeneous networks. IEEE Wireless Communications, 18(3), 40–47.
20. Adhikary, A., Ntranos, V., & Caire, G. (2011). Cognitive femtocells: Breaking the spatial reuse barrierof cellular systems. In Information theory and applications workshop (ITA), (pp. 1–10).
21. Kang, X., Zhang, R., & Motani, M. (2012). Price-based resource allocation for spectrum-sharingfemtocell networks: A stackelberg game approach. IEEE Journal on Selected Areas in Communications,30(3), 538–549.
22. Lee, H., Han, K., Hwang, Y., & Choi, S. (2009). Opportunistic band sharing for point-to-point linkconnection of cognitive radios. In Proceedings cognitive radio oriented wireless networks and com-munications (CROWNCOM), (pp. 1–6).
23. Nash, J. (1951). Non-cooperative games. Annals of Mathematics, 54(2), 286–295.24. Saraydar, C. U., Mandayam, N. B., & Goodman, D. (2002). Efficient power control via pricing in
wireless data networks. IEEE Transactions on Communications, 50(2), 291–303.25. Yates, R. D. (1995). A framework for uplink power control in cellular radio systems. IEEE Journal on
Selected Areas in Communications, 13(7), 1341–1347.
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.
Opportunistic Sub-channel and Transmit Power Allocation in… 1323
123
top related