1
Pricing and Power allocation in Femtocells using
Stackelberg Game Theory
V. Udaya Sankar and Vinod Sharma
Department of Electrical Communication Engineering,
Indian Institute of Science, Bangalore 560012, India
Email: {uday,vinod}@ece.iisc.ernet.in
Abstract—We consider a system with multiple Femtocells
operating in a Macrocell. The transmissions in one Femtocell
interfere with its neighboring Femtocells as well as with the
Macrocell Base Station. We model Femtocells as selfish nodes and
the Macrocell Base Station protects itself by pricing subchannels
for each usage. We use Stackelberg game model to study this
scenario and obtain equilibrium policies that satisfy certain
quality of service.
Keywords- Femtocell, cellular communication, power control, QoS, Game theory, Decentralized algorithms, Stackelberg
Game.
I. I NTRODUCTION
Carbon footprint of Information and communication Technology (ICT) industry has been a cause of concern in recent
years [15]. The energy consumed in cellular networks is a
significant part of it. Small cells and Femtocells (FCs) ([5],
[6]) are an important part of the overall strategy to reduce the
RF transmission in a cellular system. These cells have small,
low power Access Points (APs) deployed indoors. These are
connected to DSL lines or cable TV and are used to provide
service to indoor users. Since data generated indoors is about
60% of the total data carried in a cellular network, these APs
can divert a substantial part of the network traffic from Base
Station (BS) and improve the coverage and capacity of the
indoor users. Also, then the BS has to transmit at a much
lower power.
As envisaged, there will be many FCs/small cells in a
(macro)cell, each servicing a building. These small cells will
share the spectrum with each other and also with the macro cell
(MC). Thus the users in different FCs and MC will interfere
with each other. Therefore, to be able to utilize the channels
efficiently the FCs and the Macro Base Station (MBS) will
need to coordinate with each other such that their users get
their Quality of Service (QoS). We address this problem in
this paper.
In the following we provide related literature survey. Game
theoretic approaches for multiple access are surveyed in [1].
The studies on Interference management in FCs are summarized in ([9], [10], [18]). In [17], authors studied distributed
multichannel power allocation for spectrum sharing cognitive
radio networks with QoS guarantee. Down link power allocation in FC based cellular network was considered in [7] using
Stackelberg game theory. Spectrum sharing between MBS and
FBSs is studied in [2]. Cross tier interference mitigation using
power allocation in two tier FC networks is studied in [8].
In [3], authors studied the problem of pricing uplink (UL)
power in wideband cognitive radio networks under the objective of revenue maximization for the service provider while
ensuring incentive compatibility for the users. The distributed
power allocation strategies for spectrum sharing FCs deployed
in a MC using Stackelberg game was investigated in [11]. In
[19], authors studied the joint pricing and power allocation for
dynamic spectrum access using a Stackelberg game.
In this paper we consider the problem of UL/Downlink
(DL) subchannel power allocation to MSs in a multi FC multi
channel environment by considering interference at the MBS
using the Stackelberg Game model. The MBS computes the
price per subchannel usage under the constraints of QoS to
MC users and maximum interference limit at each subchannel.
At each FC, joint power allocation is done under the QoS
constraints to its users. In the previous studies, authors considered only single channel case UL subchannel allocation with
no QoS guarantee. Furthermore, in our case the followers’s
game becomes a potential game and we use regularizaton
to obtain a unique equilibrium (NE) for the follower’s game
resulting in algorithms providing a Stackelberg equilibrium
under much weaker conditions than in literature [11], [19].
We also guaranteed a minimum QoS (rate) to the FCs and
MBS and use minimum power at the MBS and FCs.
This paper is organized as follows. Section II describes the
system model and formulates the problem. Section III provides
algorithms to obtain a Stackelberg equilibrium. Section IV
shows the efficacy of the algorithms via simulations. Section
V concludes the paper.
II. S YSTEM M ODEL AND P ROBLEM FORMULATION
We consider a two tier cellular system in which within
a MC there are K FCs (Fig 1). The cellular system has N
subchannels, perhaps using OFDMA (e.g., LTE or WiMAX)
and the subchannels are shared by the FCs and the outdoor
users in the MC. The transmission between a Femto Access
Point (FAP) and its Mobile Stations (MSs) may be in the
uplink or downlink or in both directions. Each of the FCs
would like to ensure a minimum rate for its users. The
transmission considered in the MBS is only in the uplink (this
is because we will consider a constraint on the interference
at the MBS). Also, the MBS wants to ensure that its users
get their rate requirements. To ensure that the different users
in the FCs and in the MC use channels in such a way that
2
on its channels, i = 1, 2, ..., N . The interference limit Qi is
fixed such that (1) is satisfied with a power P̂i ≤ P̂i,max when
Ii = Qi .
If FC k uses power Pik on channel i and has channel gain
k
hi then its interference to MBS on channel i is Pik hki and it
pays βi Pik hki to the MBS as interference price. Thus the FC
k would like to use powers Pk = (P1k , P2k , ..., PNK ) on the N
channels such that it minimizes
N
X
βi hki Pik ,
(4)
i=1
while ensuring that it can have at least total rate R̄k in the
FC, i.e.,
N
X
Cik ≥ R̄k ,
(5)
Fig. 1.
i=1
Block diagram of the system
P k Gk
from the
where
= log2 1 + σ2 +αk P̂ +iP i Gk,l P l
l6=k
i i
i
i
k
Shannon capacity formula and Gi is the channel gain at FC
k for channel i, αik is the channel gain from the MBS user to
FC k for channel i and Gk,l
i is the channel gain from FC l to
Cik
every one’s QoS is satisfied without a centralized control, the
MBS fixes a price for the usage of each channel. Then each
FC decides its channel usage. We use Stackelberg game frame
work to use this approach for the FCs and the MBS to utilize
the channels.
Let P̂i be the power used by a user in the MC on the ith
channel and let ĝi be its channel gain. Let Pik be the transmit
power used in FC k on channel i. Also, let hki be the channel
gain of the ith channel from the k th FC to the MBS. All the
channel gains will be assumed constant during a slot for which
decisions need to be made. Also we assume that all channels
are Additive White Gaussian Noise (AWGN) channels with
receiver noise σ 2 . The receiver noise power can be different for
different receivers and it will not make any difference in our
power management algorithms. The MBS charges βi ≤ β̄i <
∞ as the interference price per unit
PKof interference from the
FCs. Thus, the MBS charges βi k=1 hki Pik as interference
price from all the FCs together for channel i. A user in the
MBS using channel i requires a minimum rate R̂i and uses
power P̄i in the absence
of any interference from FC users,
P̄i ĝi
i.e. , R̂i = log2 1 + σ2
where σ 2 is the receiver noise
FC k on channel i. Also, in FC k there is a maximum power
constraint,
Pik ≤ P̄ik , ∀i = 1, 2, ..., N.
(6)
We will take the utility of FC k as negative of (4) and denote
by Uk .
To obtain an equilibrium solution for the system at which
all the users can agree to operate in a distributed manner
while satisfying all the constraints, we formulate it as a
Stackelberg game where the MBS is a leader while the FCs
act as followers. The MBS fixes a price for the channel and the
FCs (and the MBS) adjust their power allocations accordingly
so as to satisfy their rate requirements. By pricing, the MBS
influences the equilibrium point obtained so that it satisfies the
constraints (3). For this we divide the optimization problem
at the MBS into two parts, namely power allocation problem
and interference price computation problem:
Interference Price Computation Problem: For each P̂ and P ,
power and uses P̂i ≤ P̂i,max in the presence of interference
which satisfies,
log2 (1 +
σ2
+
P̂i ĝi
PK
k k
k=1 hi Pi
) ≥ R̂i ,
∀i = 1, 2, ..., N. (1)
For this the MBS has to pay price A(P̂i − P̄i ) where A is a
constant representing a marginal power cost. Thus MBS has
a utility
UM BS (β, P̂ , P ) =
N X
K
X
βi hki Pik − A
i=1 k=1
N
X
(P̂i − P̄i )
(2)
i=1
where P̂ = (P̂1 , P̂2 , ..., P̂N ), P = (P1 , P2 , ..., PN ) the powers
at the FCs and β = (β1 , β2 , ..., βN ). The MBS would like to
choose the price β to maximize its UM BS while satisfying
QoS constraints (1) and interference limit
Ii =
K
X
k=1
hki Pik ≤ Qi ,
(3)
N X
K
X
max
0≤βi ≤β̂i i=1
k=1
βi hki Pik
∀i = 1, 2, ..., N,
(7)
such that
Ii =
K
X
hki Pik ≤ Qi ,
∀i = 1, 2, ..., N.
(8)
k=1
Power Allocation Problem:For each interference price β
min
A
0≤P̂i ≤P̂i,max
N
X
(P̂i − P̄i ) ∀i = 1, 2, ..., N,
(9)
i=1
such that
log2 (1 +
σ2
+
P̂i ĝi
PK
k=1
hki Pik
) ≥ R̂i ,
∀i = 1, 2, ..., N. (10)
The strategy space for MBS ( allocated power per subchannel) for the power allocation problem is
ΩM BS−C (Pk ) = {P̂ : P̂i ≤ P̂i,max ,
3
log2 (1 +
σ2
+
P̂i ĝi
PK
k k
k=1 hi Pi
) ≥ R̂i , ∀i}
(11)
which is convex and compact.
For each price β = (β1 , β2 , ..., βN ) announced by the
MBS, the FCs and the MBS adjust their powers to arrive
at a Nash point which satisfies (if possible) each player’s
rate requirements. Those powers are announced to the MBS.
Then the MBS again modifies its prices (according to an
algorithm in Algorithm 1 below). The game may eventually
converge to an equilibrium, called Stackelberg equilibrium.
At this equilibrium point we obtain the highest utility for
MBS at which all the rate requirements at FCs and MBS are
satisfied, the interference constraints at the MBS are met. In
the following we discuss conditions which may guarantee the
existence of an equilibrium point and then provide distributed
algorithms to converge to an equilibrium point.
III. S OLUTION CONCEPTS AND D ISTRIBUTED A LGORITHM
Let Ωl and Πl be the strategy space and utility function
of the leader and Ωk and Πk of the k th follower for k =
1, 2, ..., K. Let xk ∈ Ωk be a strategy of follower player k, x−k
be the strategy of other followers and xl ∈ Ωl be a strategy of
player l. Then xk is called the best response of player k for
(x−k , xl ) if Πk (xk , x−k , xl ) = supx Πk (x, x−k , xl ) and then
we write xk ∈ BRk (x−k , xl ). Similarly we define the best
responses of the leader and the other followers.
Definition 1. The strategy (xl , xk , k = 1, 2, .., K) is a Stackelberg equilibrium if
(1) xl ∈ BRl (xk , k = 1, 2, .., K) and
(2) xk ∈ BRk (x−k , xl ), ∀k = 1, 2, .., K
From [4] we obtain conditions for existence of a Stackelberg
equilibrium.
Proposition 1. There exists at least one Stackelberg NE if:
(a) Ωk is nonempty, compact and convex, ∀k = 1, 2, ..., K,
(b) Ωl is nonempty, compact and convex,
(c) Πk is once continuously differentiable and pseudo concave
with respect to xk given any (xl , x−k ), ∀k = 1, 2, ..., K,
(d) Πl is continous.
We assume that Rate requirements at the leader and the
follower’s are feasible within the given constraints. Then, it is
easy to see that our problem in section II satisfies the above
assumptions and hence we have existence of a Stackelberg
NE. Now we develop distributed algorithms using which the
system can converge to a Stackelberg equilibrium.
A. Convergence condition
To develop our distributed algorithm further, we will use
the concept of a Potential non-cooperative game.
Definition 2. A strategic game GQ
= {I , X , (Φk )k∈I } where
I is the set of players, X = k Xk is the strategy space
and Φk is the utility of player k, is called an exact Potential
game if there exists a function f : X → R such that ∀k ∈ I ,
xk , yk ∈ Xk and x−k ∈ X−k ,
Φk (xk , x−k ) − Φk (yk , x−k ) = f (xk , x−k ) − f (yk , x−k ). (12)
Theorem 1. ([14]) Let G = {I , X , (Φk )k∈I } be a potential
game, with a potential function f , and let Pmax denote the
set of maxima of f (assumed non empty). Then each point
x ∈ Pmax is a NE of G . If f is strictly concave then the
game has a unique NE.
It is easy to verify that the power allocation game played
by FAPs and MBS is a potential game with potential function
f (x) = −
K X
N
X
k=1 i=1
βi hki Pik − A
N
X
(P̂i − P̄i ).
(13)
i=1
which they try to maximize. The strategy spaces of FCs and
the power game of the MBS are compact and convex sets and
the Potential function f is continuous. Hence there exists at
least one NE for this game for each price β. Also, the best
response iteration converges to a Nash Point ([12]).
For the best response iteration, each user in turn responds
with a strategy that maximizes its utility, given the strategies of
the other players. This can be done by solving the optimization
problem using Karush-Kuhn-Tucker (KKT) conditions. Since
our utility function is linear we will get a global optimal
solution. Thus for the k th FC, the best response power
allocation is
1
λk
− k ]+
(14)
Pik = [
βi hki + µik
di
where λk , µik are Lagrange multipliers such that eq(5)-(6) are
Gk
satisfied, (x)+ = max(0, x) and dki = σ2 +αk P̂ +Pi Gk,l P l .
k
i
i
l6=k
i
i
At this point if feasible the total rate achieved by FC k is R̄k .
Similarly, for the MBS,
PK
σ 2 + k=1 hki Pik +
λi
] , ∀i = 1, 2, ..., N,
(15)
P̂i = [ −
A
ĝi
where λi ’s are the Lagrange multipliers.
The FCs do not explicitly need to exchange their best responses. Rather, each FC at its turn, measures the interference
and nose power at each channel and then computes its best
response powers (14), (15) and transmits in each channel at
that power.
The overall distributed algorithm to compute the Stackelberg
NE works as follows. The MBS starts with an initial price
vector β0 = β̂ = (β̂1 , ..., β̂N ) and announces it to all FCs. The
FCs and the MBS iteratively update their power strategies via
best response strategies (14) and (15) to eventually converge
to a NE. Existence and convergence to a NE corresponding
to any price β is obtained because we have a potential game
corresponding to each β. This NE is sent to the MBS which
updates its price according to an algorithm in Algorithm
1 below and announces it to the FCs. Basically, we start
with the maximum possible prices βi = β̂i , ∀i (to maximize
revenue). If from the corresponding Nash Point Interference
Ii in channel i exceeds Qi , lower the price of channel j with
Ij < Qj − by a small amount ∆. This will make the FCs
increase their powers in these channels and reduce in others.
The FCs and the MBS now again play the non cooperative
game to converge to another NE and send to the MBS. This
is repeated till we obtain convergence of the overall game to a
Stackelberg NE. At equilibrium if feasible the rate constraints
4
will be satisfied with equality everywhere if possible. This
distributed algorithm is summarized in Algorithm 1. We now
discuss the issue of uniqueness of a NE for a given β.
In general a potential game may have multiple NE. Therefore, corresponding to each price β, our algorithm may pick
one of the NE. It is desirable to have a unique NE. It is shown
in [16] that if the set of channel gains satisfies
k Gi k2 < 1,
∀i = 1, 2, ..., N
where k . k2 is the L2 norm and
( k,l
Gi
,
i
Gk
[G ]l,k =
i
0,
(16)
if k 6= l
otherwise,
IV. N UMERICAL R ESULTS
then there exists a unique NE for the follower’s game.
Conditions
(16) are satisfied if ∀i = 1, 2, ..., N ,
P
a.
b.
k
Gk,l
i +αi
< 1, ∀k ∈ K,
k
Gi
PK
k
k=1 hi
< 1, ∀i = 1, 2, ..., N .
ĝi
l6=k
Algorithm 1 Distributed Algorithm
a. Initialize β 0 = (β̂1 , β̂2 , ..., β̂N ) and P̂ = (P̂1 , P̂2 , ..., P̂N )
and broadcast to all FAP’s, n = 1.
b. Power Iteration ( Do until convergence of power)
1) solve (4)-(6) using KKT for all FC’s for getting Pk =
(P1k , P2k , ..., PNk ), ∀k = 1, 2, ..., K using eq (14).
2) By using PK , solve for P̂ = (P̂1 , P̂2 , ..., P̂N ) at MBS
using eq (15) iteratively until convergence.( can use
fmincon function to compute power)
PK
c. From eq (3) measure Iin = k=1 hki Pik , ∀i = 1, 2, ..., N at
MBS and update the interference price based on the following
algorithm.
• Take a small positive constant. This controls the algorithm accuracy. Let NS = {i : Iin > Qi + } and
S = {i : Iin < Qi − }.
n
• if (Ii ≤ Qi + , ∀i) then
stop and return β n
• else if (| S |6= ∅ and | NS |6= ∅) then
βin+1 = βin − ∆, ∀i ∈ S
• end if
• n=n+1
d. Repeat (b),(c) until | Ii − Qi |≤ , ∀i = 1, 2, ..., N .
The above conditions are quite restrictive. Thus we also
use another method [13] by adding regularization
terms to
PN
the utilities of the players as Φk (x) = − i=1 βi hki Pik −
PN
PN
c i=1 (Pik )2 and ΦM BS (x) = −A i=1 (P̂i − P̄i ) −
PN
c0 i=1 (P̂i )2 , where c and c0 are small positive constants.
The resulting Potential function is given by
g(x) = −
K X
N
N
X
X
(
βi hki Pik + c
(Pik )2 )
k=1 i=1
−(A
N
X
i=1
(P̂i − P̄i ) + c0
i=1
N
X
i=1
(P̂i )2 ).
It is observed that g(x) is strictly concave. Hence, the corresponding game has a unique NE. We use this NE in our
algorithm. This NE will be close to a NE of the original
problem which has low total power requirement.
In the next section we will run the algorithm and show
convergence of our algorithm to a Stackelberg NE (SNE)
on an example. We have seen the convergence in several
other examples also. The proof of convergence of the overall
algorithm to a SNE is still under investigation although at an
intuitive level we can see why it converges.
(17)
We consider two FC’s with 4 subchannels and one
scheduled user deployed in MC environment. We consider
the following parameters for simulation. We are simulating
for uplink only.
Bandwidth of each channel is : B = 1KHz,
Background Noise σ 2 = 0.02, = 1.5 × 10−2 .
MBS:
channel
gain from MMS
to MBS is : ĝ
=
0.9, 0.7, 0.8, 0.9 .
Channel
gain from FAP-1 user
to MBS is :
h1 = 0.04, 0.03, 0.05, 0.01 .
Channel
gain from FAP-2 user
to MBS is :
h2 = 0.02, 0.01, 0.04, 0.05 .
Maximum
constraint is (in watts) :
subchannel power
P̂max = 10, 4, 7, 8 .
Rate requirement per subchannel
in the MBS is (in Kbps):
R̂ = 3, 4, 2, 3 .
Maximum
tolerable interference
limit is : Q
=
0.02, 0.005, 0.033, 0.05 .
Interference price initial value is : βinit (= β̂) =
2, 3, 4, 5 .
Price increment is : ∆ = 0.07.
FAP-1:
0.8, 0.5, 0.6 . Subchannel Gain is :G1 = 0.7,
Cross channel gain is : G1,2 = 0.2, 0.4, 0.1, 0.3 .
=
Channel
gain from MMS
to FAP-1 is : α1
0.05, 0.1, 0.2, 0.03 .
Maximum rate requirement (in Kbps) is : R̄1 = 5,
Power per subchannel (in watts) is : P̄1 = 5, 5, 5, 5 .
FAP-2:
0.6, 0.4, 0.7 . Subchannel Gain is :G2 = 0.9,
Cross channel gain is : G2,1 = 0.1, 0.3, 0.2, 0.4 .
2
Channel
gain from MMS
=
to FAP-1 is : α
0.02, 0.2, 0.03, 0.01 .
Maximum rate requirement (in Kbps) is : R̄2 = 7,
Power per subchannel (in watts) is : P̄2 = 5, 5, 5, 5 .
For regularization we considered c = c0 = 0.05.
With regularization case:
Total Powers allocated at FAP1 is 1.2964, FAP2 is 2.0642
and MMS is 1.6703 and the total utility for MBS is 0.1675
with exact rates achieved at both Femtocells as 5 and 7.
Similarly the Rate requirements at the MBS also satisfied
with equality for each user.
Without regularization case: Total Powers allocated at
FAP1 is 1.2348, FAP2 is 2.1937, MMS is 1.6887 and total
5
Fig. 2.
Variation of interference price, without regularization
Fig. 3.
Variation of interference price using reqularization
utility for MBS is 0.2079 with exact rates achieved at both
Femtocells as 5 and 7. Similarly Rate requirements at the
MBS also satisfied with equality for each user.
Fig 2 shows the variation in Interference price when we
do not use regularization (condition (16) is satisfied) in our
example. Fig 3 shows the price evolution for the regularization
case. In Figs 4 and 5 we show the power allocation in the four
channels at the FC 1 and FC 2 respectively for regularization
case. We see that as the price of a channel in Fig 3 reduces,
the power in that channel at FC 1 and FC 2 is increases. Also,
if a channel has lower Qi , this channel uses less power at FCs.
V. S UMMARY
We have considered the problem of interference management in a Macrocell with multiple Femtocells and multiple
channels. Each Femtocell and the users in the Macrocell need
guaranteed minimum rate and the interference at the MBS
needs to be below a threshold. Using Stackelberg Game theory
we have developed a distributed algorithm which achieves the
above aims.
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