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Published in IET Communications
Received on 27th May 2012
Revised on 15th June 2012
doi: 10.1049/iet-com.2012.0297
ISSN 1751-8628
Game-theoretic approach for interference
management in heterogeneous multimedia
wireless personal area networks
A. Mehbodniya S. Aı
¨ssa
INRS-EMT, University of Quebec, Montreal, QC, Canada
E-mail: mehbod@emt.inrs.ca
Abstract: Emergence of new wireless technologies has facilitated the way to higher data rates and more robust communication
links. Ultra wideband (UWB) is one of these promising technologies that, despite its several outstanding benefits, can cause
interference to other networks operating in the same frequency range because of its large bandwidth. In this study, the authors
introduce a solution for interference management in heterogeneous UWB networks. The analysis can be generalised for other
coexistence scenarios where we have wireless networks with different specifications. Game theory is used to study the joint
power and rate control problem under mutual interference between two UWB standards, namely, multiband orthogonal
frequency division multiplexing UWB and direct-sequence UWB. A non-cooperative joint rate and power control game with
pricing (NRPGP) in which each node seeks to choose its possible transmit power and rate in order to maximise its own utility
while satisfying its target signal-to-interference-and-noise ratio as quality-of-service requirement is introduced. Simulation
results are provided to evaluate the performance of the proposed game, NRPGP.
1 Introduction
One of the newly emerged coexistence scenarios is between
the two ultra wideband (UWB) standards. These standards,
which have been proposed to the IEEE 802.15.3a task
group, are multiband orthogonal frequency division
multiplexing (MB-OFDM) UWB [1] and direct-sequence
(DS) UWB [2]. The IEEE 802.15.3a task group could not
finalise selection between these two standards. As a result,
both standards will have to coexist together. Interference
mitigation when these two technologies operate in the
vicinity of each other is very challenging and appropriate
solutions are yet to be developed. Efficient resource
allocation such as power and rate adaptation can be useful
to increase performance and reduce interference. In contrast,
game-theoretic approaches have recently found their way to
solve different resource allocation problems in wireless
communications. Algorithms obtained using game theory
are usually distributed (decentralised). In this context, we
use a game-theoretic approach to study resource allocation
in heterogeneous multimedia UWB wireless personal area
networks (WPANs) [3 – 5]. As a result of their high
transmission rates, UWB standards are the main candidates
to be used in future multimedia WPANs.
Previous literature related to this work can be classified in
two categories. First, research addressing the two UWB
technologies’ coexistence, for which works done so far are
very limited. In [6], an approach based on multi-carrier
waveform design is recommended to reduce the effect of
MB-OFDM interference on a pulse-based UWB receiver;
this work does not take into consideration the reverse case
of interference on MB-OFDM. In contrast, in the work
reported in [7], the authors simulated the MB-OFDM UWB
and DS-UWB standards to study the effect of the mutual
interference between both standards, and proposed to
reduce the interference by means of power control. As will
be discussed, power control by itself is not sufficient to
improve the performance. In [8], an accurate bit error rate
(BER) analysis for MB-OFDM UWB transmission in the
presence of DS-UWB and time hopping UWB interference
is proposed, considering the Nakagami-m fading channel
model for both the interferers and the MB-OFDM
transmitter. The latter work uses a waveforming technique
for the purpose of interference mitigation between the two
standards. The work in [9] treats the reverse case of the
analysis considered in [8] and evaluates the performance
of DS-UWB transmission when exposed to MB-OFDM
interference, using a pulse collision model.
The second category concerns research related to parameter
control for interference management. Several research works
have used power control for interference mitigation [10– 13].
In addition to power control, in order to support a variety of
applications with diverse transmission rates, rate adaptation
is also critical for wireless data networks. Thus, joint power
and rate control are necessary. Relevant works in this
regard, such as [14] and [15], mostly use mathematical
optimisation tools to simultaneously optimise the transmit
powers and rates. These global optimisation approaches
imply that a centralised algorithm should be implemented.
However, owing to several problems such as huge network
2278 IET Commun., 2012, Vol. 6, Iss. 15, pp. 2278–2286
&The Institution of Engineering and Technology 2012 doi: 10.1049/iet-com.2012.0297
www.ietdl.org
signalling, centralised solutions are usually not practical.
Game-theoretic-based rate and power control solutions in
wireless networks have also been proposed, for example see
[16– 19]. In particular, the work in [19] presents a general
game-theoretic approach for distributed asynchronous power
and rate control for wireless ad hoc networks. This work
allows nodes to freely choose their rate and power
adaptation function without any special technological
assumption.
Works in the two aforementioned categories are mostly
developed for scenarios such as CDMA networks or
cellular environments, which are different from distributed
networks such as the distributed WPANs considered in this
paper. Moreover, only few of the above-mentioned research
works consider fading in their analysis.
In this study, we present a non-cooperative game solution
for the coexistence of WPANs using two different UWB
technologies, binary phase-shift keying direct sequence
UWB and MB-OFDM. The game-theoretic approach is
used as a resource management technique to jointly control
the transmit power and rate for the UWB transceivers,
while indirectly contributing to the mutual interference
reduction. The problem is formulated by introducing a
utility function for the proposed game, where each node
seeks to determine its transmission power and rate so as to
maximise its own utility while satisfying its target signal-to-
interference-and-noise ratio (SINR) as quality-of-service
requirement. A Rayleigh approximation of the IEEE
802.15.3 UWB standard channel model [20] is considered
to derive the utility function based on a procedure similar to
[21], for both cases of fast and quasi-static slow fading. A
pricing of transmit power and rate is applied into the utility
function to achieve Pareto improvement compared with the
game with no pricing and enhance the overall system
performance. Unlike conventional pricing schemes as in
[10], which consider only a linear function of the transmit
power, our scheme considers joint pricing for the transmit
power and rate. The existence, uniqueness, best-response
strategies and Pareto efficiency of Nash equilibrium (NE)
for the proposed game are proved. To evaluate the
performance, simulations are carried out within the
framework of multimedia WPAN based on (i) a coexistence
scenario with two transceiver pairs, a DS-UWB pair and a
MB-OFDM one and (ii) a general network topology with
arbitrary numbers of transceiver pairs. Our work also
investigates the necessary compensations for the channel
fading to be taken into consideration in the design. Game
theoretic approaches for design of wireless systems are
usually easier for practical system implementations because
they are usually distributed in nature. In this work, we do
not address directly the networking issues regarding
practical system implementation for our algorithm, as this is
out of the scope of this work. However, the huge growth of
wireless networks’ computational capacity as well as system
designs based on game theoretic approaches in recent years
[19, 22, 23], is a proof to feasibility of our algorithm for
practical wireless scenarios.
The remainder of this paper is organised as follows. In
Section 2, the system model is presented. Section 3 derives
the utility function for Rayleigh quasi-static slow and fast
flat-fading channels. In Section 4, the non-cooperative rate
and power control game with pricing (NRPGP) is
introduced and the NE and Pareto optimality conditions are
proven. Section 5 addresses the numerical results and
performance in terms of transmission rate and power.
Finally, concluding remarks are drawn in Section 6.
2 System model and preliminaries
In this section we present the basic definitions for different
aspects of the system, which comprise the network
topology, utility function, channel model and BER.
2.1 Network topology
In the heterogeneous coexistence environment, a number of
nodes with different UWB technologies share the UWB
spectrum. We consider the WPAN as network topology.
Fig. 1 shows a typical heterogeneous WPAN topology
consisting of two piconets. In the WPAN standard
terminology, a piconet is a collection of nodes which form
a network with the same technology. We call the
combination of neighbouring networks, whether of the
same technology or different technology, a ‘system’. Each
‘node’ is technically called a device (DEV). The piconet
controller (PNC) is chosen by other DEVs and is
responsible for coordinating and synchronising the DEVs
within the piconet. For this purpose, PNC broadcasts
beacons to the DEVs at regular superframe intervals. After
coordination, the communication is performed directly
between the transmitter and receiver, forming a ‘transceiver
pair’. The first piconet in Fig. 1 consists of N
DS
number of
DS-UWB transceiver pairs and the other piconet consists of
N
MB
number of MB-OFDM UWB transceiver pairs. Each
receiver in a pair is subject to interference from other
simultaneously transmitting nodes, whether located in the
same piconet or in other piconets. The distance between the
centre points of each transceiver pair with other interfering
pairs is called ‘vertical distance’ and will be used in the
simulation study as the measure of distance between
different transceiver pairs. Each node desires to achieve a
high quality of reception, that is, high SINR, while using
the minimum possible amount of power in order to extend
its battery life. The conflicting goal of each node to have a
high SINR, which results in increasing the transmission
power and, as a result, increases interference onto other
nodes, is another reason to use game theory to find a
tradeoff for these conflicting optimisation goals.
2.2 Utility function
Utility functions are a measure of the satisfaction level that a
player attains by choosing an action from its strategy profile,
given that other players’ actions are known. A utility function
maps the players’ preferences into real numbers. In this paper,
we consider a generally used form for the utility function of
the transmitter node in the ith transceiver pair given by [10]
ui(ri,pi)=Lri
Mpi
f(
g
i)∀i=1, 2, ...,N(1)
Fig. 1 Typical heterogeneous WPAN topology
IET Commun., 2012, Vol. 6, Iss. 15, pp. 2278–2286 2279
doi: 10.1049/iet-com.2012.0297 &The Institution of Engineering and Technology 2012
www.ietdl.org
where Mis the transmitted packet length (bits) for transmitter
node i,Lis the number of information bits in a packet, r
i
is the
transmission rate (bits per second) for transmitter node i,p
i
represents the average transmit power level for transmitter
node i,f(
g
i
) is a function of SINR which approximates the
average probability of correct reception of a packet (defined
as P
c
) at the receiver node i, and N¼N
DS
+N
MB
is the
total number of transceiver pairs in the system. Hence, u
i
provides the number of information bits that are
successfully received, per joule of energy.
Assuming no error correction, the random packet’s correct
reception rate is given by M
l=1(1 −˜
Pe(l)), where ˜
Pe(l) is the
random BER of the lth bit at the given SINR.
2.3 UWB node’s BER
In general, the SINR of the ith (DS-UWB or MB-OFDM)
receiver node,
g
i
, in the ith transceiver pair can be written as
g
i=Bi
ri
pici
a
2
i
N
k=ipkck
a
2
k+
s
2=
g
′
i
a
2
i(2)
where
a
i
is the path-fading coefficient between the transmitter
node and its corresponding receiver node, which is constant
for each bit in a fast flat-fading channel, and constant for
each packet in a quasi-static slow flat-fading channel. p
i
is
the transmit power of the ith transmitter node, c
i
is the
path attenuation between the ith transmitter and its
corresponding receiver. B
i
is the two-sided bandwidth of the
ith receiver node, p
k
is the transmitted power of the kth
interfering transmitter node, c
k
is the path gain between the
kth interfering transmitter node and the ith receiver node,
and
s
2
is the variance of the additive white Gaussian noise
(AWGN) which models the background thermal noise at the
receiver. For simplicity, interference from all other
transmitter nodes onto the ith receiver node is denoted by
Ii=N
k=ipkck
a
2
kwhich comprises all the interferers, being
nodes of the same or different technology.
We consider a conditional upper bound for the BER of the
UWB received signal as [7]
˜
P(MB,DS)(e)=1
2exp −
u
(MB,DS)
g
k
i
2
(3)
where
k
≃2 and
u
(MB,DS)
is another constant defined based
on the kind of UWB technology used (MB or DS).
2.4 Channel model
For the channels, we consider the UWB IEEE 802.15.3
standard channel model [20], which is common for both
DS-UWB and MB-OFDM UWB. This model is based on a
modified version of the Saleh – Valenzuela multipath model
and is generally described by
h(t)=C
Lc
l=0
K
k=0
a
k,l
d
(t−Tl−
t
k,l) (4)
where
a
k,l
¼
q
k,l
c
k,l
is the gain coefficient of the kth ray in the
lth cluster (channel measurements showed that signal rays
arrive in clusters),
q
k,l
takes the values +1or21 with
equal probability (to consider signal inversion because of
reflections),
c
k,l
is a lognormal random variable,
t
k,l
is the
arrival time of the kth ray with respect to the lth cluster
arrival time (T
l
), L
c
denotes the number of clusters, and K
indicates the number of rays within each cluster. The term
Cin (4) models the path attenuation
C=GTGR
h
2
(4
p
)2dnf2(5)
where G
T
is the UWB transmit antenna gain, G
R
is the UWB
receive antenna gain, dis the distance between the receiver
and the considered transmitter node, fis the centre
frequency for the UWB signal,
h
is the speed of light and n
is the path loss exponent.
It is known that if L
c
and Kin (4) are large enough, it is
reasonable to approximate the standard channel model
fading amplitude as Rayleigh fading [24]. Hence, the
probability density function (PDF) of |
a
2
i|can be
approximated by the exponential distribution as
Pr{|
a
2
i|}≃1
E{|
a
2
i|}e−((|
a
2
i|)/E{|
a
2
i|}) (6)
where E{|
a
2
i|}=1.269 for the four common channel
realisations in the IEEE 802.15.3 standard (CM1, CM2,
CM3 and CM4, defined in [24]) and E{.}denotes the
expectation operator. With respect to (6) and (2), the PDF
of
g
i
for a given total interference, I
i
, is expressed by [24]
f
g
i|Ii(
v
)=0.78
g
′
i
e−0.78
v
/
g
′
i(7)
3 Utility function under channel fading
In this section we derive the utility function for Rayleigh
fast and quasi-static slow flat-fading channels. In impulse ratio
(IR)-UWB channel models the fading is usually frequency
selective. However, to simplify the analysis, we assume a flat-
fading channel (in analytical approach we replace flat fading
with a special case of frequency-selective multipath fading
where the number of incoming paths equals one). This
assumption is true for comparative purposes, as it has been
used in several other research works on UWB [25–27].Even
in practical environments this assumption is not far from
reality regarding that the two UWB standard use smaller
portions of the UWB bandwidth instead of the whole
bandwidth like in MB-OFDM where the entire available
bandwidth is divided into 14 smaller sub-bands. In addition,
in some special applications such as 4-G WLAN [28] or
MIMO channels [29], the flat-faded assumption may be
considered reasonably met. In our flat-faded Rayleigh channel
model, we also investigate two specific cases of quasi-static
slow fading and fast fading. In quasi-static slow fading, the
fading is assumed to be static during the duration of one
packet and independent between two packets. Quasi-static
assumption is usually used in simulation to provide an
averaged performance over many channel realisations. We
have provided our analysis and the results for fast fading as
well as the quasi-static slow fading. In general fast fading
may not be a significant issue in UWB communications.
However, this may be a matter of concern in some UWB
applications which operate at large distances with low rates
like radars. In highly dynamic environments, fast fading may
be also considered for UWB propagation as investigated in [30].
2280 IET Commun., 2012, Vol. 6, Iss. 15, pp. 2278–2286
&The Institution of Engineering and Technology 2012 doi: 10.1049/iet-com.2012.0297
www.ietdl.org
3.1 Rayleigh fast flat-fading channel
For the lth bit in a packet, the SINR and the interference term
can be written as
g
i=Bi
ri
pici
a
2
i(l)
Ii(l)+
s
2(8)
Ii(l)=
N
k=i
pkck
a
2
k(l) (9)
Assuming that {
a
i(l)}M
l=1and {Ii(l)}M
l=1are independent and
identically distributed (i.i.d.) random variables, the averaged
probability of correct reception for all bits of a packet is
given by (1 2P
e
)
M
, where P
e
is the average BER and
equals Pe=E{˜
Pe}. Hereafter, we omit the index (MB, DS)
for
u
and ˜
P(e) in (3), for notation simplicity. The derived
P
e
is general for either of the two kinds of UWB
techniques. However, to compute P
e
for each UWB
technology,
u
should be replaced by its corresponding
value. To find P
e
, we start by finding ˜
P(e|Ii) as follows
˜
P(e|Ii)=E{˜
P(e|
g
i,Ii)} =1
0
˜
P(e|
v
,Ii)f
g
i|Ii(
v
)d
v
=0.78
2
g
′
i1
0
e−((1.56+
g
′
i
uv
)/2
g
′
i)
v
d
v
=0.78
2
p
√e0.3042/
g
′2
i
u
4
u
√
g
′
i
1−err 39
2
√
100
g
′
i
u
√
(10)
where err(x)=2/
p
√
x
0e−t2dtis the error function. For
large
g
′
i, we can approximate ˜
P(e|Ii)by0.195
2
p
√/
u
√
g
′
i
.
Now we can find the averaged BER as
Pe=E{˜
P(e|Ii)} =0.195
2
p
√
u
√E{Ii}+
s
2
(Bi/ri)pici=0.195
2
p
√
u
√
g
i
(11)
where
g
iis given by
g
i=Bi
ri
pici
1.269 N
k=ipkck+
s
2(12)
With respect to (1) and (11), the utility function of the ith
transmitter node is given by
ui(ri,pi)=Lri
Mpi
1−0.195
2
p
√
u
√
g
i
M
(13)
3.2 Rayleigh quasi-static slow flat-fading channel
In a quasi-static slow-fading channel, the fading coefficient,
a
i
, changes independently for each packet so that
a
i
(1) ¼
a
i
(2) ¼... ¼
a
i
(b), where
a
i
(b) is the fading
affecting bit bin the packet. It is also assumed that
a
i
, the
fading coefficient of the main signal, and
a
k
, the fading
coefficient of the kth interfering signal, are independent. In
this case, the conditional utility function is written as
u(i|
g
i,Ii)(ri,pi)=Lri
Mpi
(1 −e−
ug
2
i/2)M(14)
Similar to (10), we derive u(i|Ii)(ri,pi) by averaging over the
conditional utility function in (14)
u(i|Ii)(ri,pi)=1
0
u(i|
v
,Ii)(ri,pi)f
g
i|Ii(
v
)d
v
=1
0
0.78Lri
Mpi
(1 −e−
uv
2/2)M1
g
′
i
e−0.78
v
/
g
′
id
v
=0.78Lri
Mpi
g
′
i
M
k=0
M
k
(−1)k
×1
0
e−((k
uv
/2)+(0.78/
g
′
i))
v
d
v
=0.78Lri
Mpi
g
′
i
g
′
i+
M
k=1
M
k
2
p
√(−1)k
2
k
u
√
×e0.304/k
ug
′2
i1−err 0.7
2k
u
√
g
′
i
(15)
Considering
g
′
i≫1 we can approximate (15) as
u(i|Ii)(ri,pi)=0.78Lri
Mpi
1+1
g
′
i
M
k=1
M
k
2
p
√(−1)k
2
k
u
√
(16)
and finally uican be obtained as
ui(ri,pi)=E{u(i|Ii)}
≃0.78Lri
Mpi
1+E[Ii]+
s
2
(Biri)pici
M
k=1
M
k
2
p
√(−1)k
2
k
u
√
=0.78Lri
Mpi
1+1
g
i
M
k=1
M
k
2
p
√(−1)k
2
k
u
√
=0.78Lri
Mpi
1−
b
g
i
(17)
where
b
=−M
k=1
M
k
2
p
√(−1)k
2
k
u
√.0
4 Non-cooperative rate and power control
game with pricing
In this section, the existence, uniqueness and Pareto
efficiency, of NE for the proposed game are discussed.
4.1 Transmission rate nash equilibrium for NRPGP
In non-cooperative rate and power control, each transmitter
node maximises its own utility by adjusting its rate and
power, but ignores the harm it may cause to other receiver
IET Commun., 2012, Vol. 6, Iss. 15, pp. 2278–2286 2281
doi: 10.1049/iet-com.2012.0297 &The Institution of Engineering and Technology 2012
www.ietdl.org
nodes by the interference it produces. Pricing is an effective
tool in this condition. An efficient pricing mechanism
makes the decentralised decisions compatible with the
overall network efficiency by encouraging efficient sharing
of the resources as opposed to the aggressive competition in
pure non-cooperative games. In this section we define a
non-cooperative game with pricing in which the price is
proportional to the rate of the node. Let
Gp=[N,{r,p}, {uc
i(r,p)}] denote the NRPGP, where
N={1, 2, ...,N} is the index set for the active
transceiver pairs, r={ri}N
i=1is the rate strategy set for all
nodes, p={pi}N
i=1is the power strategy set for all nodes
and uc
i(r,p)(.) is the complementary utility function of the
ith transmitter node in the ith transceiver pair. The utility
for the NRPGP is
uc
i(r,p)=ui(r,p)−ci(r,p) (18)
where c
i
:<×<<
+
is the pricing function of i[Nand
ui(r,p)=ui(ri,pi|R−i,P−i) (19)
where p
i
and r
i
are the transmit power and rate of the ith
transmitter node and P
2i
and R
2i
are the transmit power
and rate vector of all other other nodes. We propose a
pricing scheme in which the price increases monotonically
with the rate and power of the transmitter node i
ci(r,p)=
l
iripi(20)
The pricing factor
l
i
is set based on the type of the node (DS-
UWB or MB-OFDM), such that each node’s self-interest
leads to overall improvement of the system performance. In
our distributed environment, these pricing factors are
broadcasted to DEVs through PNCs. The NRPGP with
linear pricing is defined by
NRPGP max
ri[r,pi[puc
i(r,p)∀i[N(21)
Theorem 1: (a) Existence: For the Rayleigh fast flat-fading
case, NE in the transmission rates exists in the NRPGP
game (called G
p1
hereafter) if, for all i¼1, 2, ...,N, first, r
is a nonempty, convex and compact subset of the Euclidean
space <
N
, and second, uc
i(r,p) is continuous in rand quasi-
concave. (b) Uniqueness: The NE point of G
p1
is unique.
Proof 1a: We assume that each transmitter node has a strategy
space that is defined by the maximum and minimum rates,
and all rates in between. So the first condition on the
strategy space, r, is satisfied. To check whether the function is
quasi-concave, we first derive the second derivative (see (22))
Then (see (23))
To show that the utility function is a quasi-concave function
of riwe should have ((∂2uc
i(r,p))/∂r2
i),0∀i[N. This
condition is fulfilled if
∂2uc
i(r,p)
∂r2
i
,0∀
g
i.39
2
√
p
(M+1)
400
u
√=
g
i−min
(24)
where
g
i−min is defined as the minimum required SINR, in order
for the NE to exist.
Proof 1b: The NE point is unique if the best response vector
of all transmitter nodes is a standard vector function. The best
response vector of a transmitter node iis the best strategy that
the node ican take to attain the maximum utility. The best
response vector of all transmitter nodes is defined as
r
(r)¼(
r
1
(r),
r
2
(r), ...,
r
N
(r)) and the best response vector
of transmitter node iis given by
r
i(r)=min(rmax
i,ri−max) (25)
where r
i2max
is the maximum allowed transmission rate of the
transmitter node iin the strategy space rand rmax
iis the the
maximising transmission rate of transmitter node ibased on
the utility function concavity constraint and can be derived
by setting (22) to zero and solving it numerically. In the
case of the game with no pricing we have
rmax
i=200
u
√Bi
39
2
√
p
(M+1)
pici
N
k=ipkck+
s
2(26)
A function is said to be standard if it satisfies the following
properties [10]: (i) positive:
r
(r).0; (ii) monotone: if
r≥r′then
r
(r).
r
(r′); and (iii) scalable: for all
m
.1,
mr
(r).
r
(
m
r). These properties can be easily verified for
fixed point rmax
i=
r
i(ri). Therefore the NE is unique.
Theorem 2: Existence: For the Rayleigh quasi-static slow flat-
fading channel, NE in the transmission rates exists in the
NRPGP game (called G
p2
hereafter) and is unique.
Proof 2: To show the concavity of the utility function we find
the first and second order derivatives based on (17) as
∂uc
i(r,p)
∂ri=0.78L
Mpi
1−2
b
g
i
−
l
ipi(27)
and
∂2uc
i(r,p)
∂r2
i=−1.56L
b
Mripi
g
i
(28)
Considering
b
.0 we get ((∂2uc
i(r,p))/∂r2
i),0 and the
utility function is concave. Therefore NE exists for the
gi=∂uc
i(r,p)
∂ri=2−5M/2L100
2
u
√−(39
p
(M+1)/
g
i)
4
2
√−(39
p
)/25
u
√
g
i
M
Mpi100
2
u
√−(39
p
/
g
i)
−
l
ipi(22)
∂2uc
i(r,p)
∂r2
i=39.2−5M/2
p
L39
p
(M+1) −200
2
u
√
g
i
4
2
√−(39
p
/25
u
√
g
i)
M
piri100
2
u
√
g
i−39
p
2(23)
2282 IET Commun., 2012, Vol. 6, Iss. 15, pp. 2278–2286
&The Institution of Engineering and Technology 2012 doi: 10.1049/iet-com.2012.0297
www.ietdl.org
game G
p2
. The uniqueness can be easily verified similarly to
proof 1.
4.2 Transmission rate Pareto optimality for NRPGP
The NE is usually the solution to the power and rate control
problems where no transmitter node can increase its utility
any further unilaterally. Therefore such a distributed
scenario is supposed to be less efficient than a possible
power and rate allocation obtained through cooperation
between transceiver pairs as a result of centralised
optimisation. Indeed, it is well known that in general NEs
are inefficient. A resource (power or rate) allocation is said
to be more efficient (or Pareto dominant) if it is possible to
increase the utility of some of the transmitter nodes without
hurting any other receiver nodes. Mathematically speaking,
a transmission rate vector r0=(r0
1,r0
2,...,r0
N) in game G
p
,
is said to be Pareto optimal if there exists no other vector ˆ
r
in the strategy space such that ui(r0,p).ui(ˆ
r,p), ∀i[N.
Theorem 3: A NE point for the game G
p
is Pareto rate optimal
for the fast flat-Rayleigh fading case.
Proof 3: Let us consider r0
i=
w
iˆ
ri,i[Nwhere
w
i.1, ∀i[N. Then based on (13) and (21) we have
uc
i(r0,p)
=Lr0
i
Mpi
1−0.195
2
p
√
u
√Bi/r0
i
pici/N
k=ipkck+
s
2
M
−
l
iripi(29)
To analyse the behaviour of uc
i(r0,p) with
w
i
, we find the first
derivative as follows (see (30))
It is easy to see that gi(ˆ
ri/
w
i),gi(ˆ
ri)=0, ∀i[N. Hence
((∂uc
i(r0,p))/∂
w
i),0, ∀i[Nwhich means that uc
i(r0,p)is
a decreasing function of
w
i
.1. Thus the NE point of game
G
p1
is Pareto rate optimal.
Theorem 4: A NE point of the game G
p2
is Pareto rate optimal
for the quasi-static slow flat-Rayleigh fading case.
Proof 4: Can be easily proven similar to Theorem 3.
Similarly, the same theorems can be proven for transmit
power equilibrium point.
5 Numerical results
An asynchronous rate and power control algorithm similar to
[10] is considered, which converges to the unique NE point.
In this algorithm, the nodes update their transmission rate
and power in the same step. In the beginning, an initial
power and rate vector is assigned to the system, then at
each step the NE is calculated using (13) for both, the
rates and powers, and the new values are chosen as the
minimum between the value for NE and the maximum
defined values.
To get the most out of the system performance, the pricing
factor is considered adaptive and for each iteration an
optimum value,
l
opt
iis calculated [10]. Once the NE with
no pricing is obtained, the NPRGP is played again after
incrementing the pricing factor,
l
i
, by a positive value, D
l
i
.
If the utilities improve with this new equilibrium, that is,
uc
i(
l
i),uc
i(
l
i+D
l
i)∀i[N, the pricing factor is
incremented and the procedure is repeated. We let this
process continue until an increase of the pricing factor
results in utility levels worse than the previous equilibrium
values for at least one node. This last value is taken as the
optimum pricing factor. For the DS-UWB standard (channel
4) [2] the maximum data rate is considered as 220 Mbps,
the two-sided bandwidth, B
i
, is considered to be 1352 MHz,
the centre frequency is 4056 MHz and the coefficient
u
is
0.74. For the MB-OFDM (band group 1) standard [1],
theses values are taken as 200 Mbps, 1584 MHz,
3960 MHz and 1.05, respectively. The noise power is
2174 dBm/Hz and the maximum allowed transmit power
of both networks in the system is 29.9 dBm. Two
scenarios are simulated for the purpose of performance
evaluation.
In the first scenario, we consider two transceiver pairs in
two heterogeneous piconets. One DS-UWB transceiver pair
with a transmit–receive distance of 1.2 m in the DS-UWB
piconet and another transceiver pair of MB-OFDM with a
distance of transmit–receive 4 m in the MB-OFDM
piconet. These two transceiver pairs are positioned
parallel to each other, whereas their parallel distance
varies. Fig. 2 shows the transmission rate performance of
each transceiver pair. As observed, in the case of NPRGP
parameter control, both pairs achieve their maximum
transmission rate even at much nearer interfering
distances. The plots are drawn starting from 7-m distance
because below this value the system is supposed to be
shut off because of enormous performance degradation.
The other noticeable result in the plot is the effect of
fading. We can observe that the quasi-static slow flat-
fading case has a slightly better performance. Fig. 3
shows the power consumption of each transceiver pair.
We observe that the power consumption is reduced more
than half in the case of NPRGP. This performance can be
interesting in scenarios like sensor networks where power
consumption is a critical issue.
In the second scenario, we study the impact of increasing
the number of active transceiver pairs on the average
throughput and power performance, without considering
any specific WPAN piconet structure. An initial topology
is considered similar to the first scenario but with a fixed
vertical distance of 10 m. Later different transceiver pairs,
∂uc
i(r0,p)
∂
w
i
=ˆ
ri2−5M/2L100
2
u
√−39
p
(M+1)/(Bi/
w
iˆ
ri)pici/N
k=ipkck+
s
2
42
√−39
p
/25
u
√Bi/
w
iˆ
ri
.pici/N
k=ipkck+
s
2
M
Mpi100
2
u
√−(39
p
/Bi/
w
iˆ
ri)pici/N
k=ipkck+
s
2
−
l
ipi
=ˆ
rigi
ˆ
ri
w
i
,∀i[N(30)
IET Commun., 2012, Vol. 6, Iss. 15, pp. 2278–2286 2283
doi: 10.1049/iet-com.2012.0297 &The Institution of Engineering and Technology 2012
www.ietdl.org
that is, either DS-UWB or MB-OFDM, are placed around
the initial DS-UWB transceiver pair, whereas the same
10-m vertical distance is kept constant between the newly
placed transceiver pair and the initial transceiver pair. At
each iteration, we have N
MB
¼N
DS
, that is 50% DS-
UWB and 50% MB-OFDM pairs. Fig. 4 plots the average
network throughput for different number of active
transceiver pairs around the initial DS-UWB transceiver
pair. It is observed that our proposed game, NPRGP, has
the best performance. The results are also compared with
the algorithm in [10] based on power control only. It is
seen that the joint power and rate control algorithm
outperforms the algorithm based on power control only.
Fig. 4 also demonstrates that the average system
throughput decreases as the number of active pairs
increases. Fig. 5 shows the average system power
consumption against the number of active transceiver
pairs. As expected the performance is better than the case
without any control. However, the algorithm based on
power control only, has a slightly better performance.
Owing to not having the transmission rate constraint, this
can be interpreted as having more freedom in choosing
transmission power values in the power strategy space.
Fig. 6 shows the throughput of one DS-UWB transceiver
pair while the number of active interfering MB-OFDM
transceiver pairs varies. Fig. 7 show similar results as in
Fig. 6 but for a MB-OFDM transceiver pair while the number
of active DS-UWB interfering transceiver pairs varies. It is
observed that although the maximum achievable throughput
for the MB-OFDM pair is a bit lower than for a DS-UWB
pair, the rate of performance deterioration, as the number of
interfering pairs increases, is better than DS-UWB, which
implies better robustness of MB-OFDM to external
interference. In Figs. 4–7, we have fixed a minimum rate of
30 Mbps for all the nodes to obtain the results based on only
power control algorithm. Therefore the curves are
comparable with the best case scenario of the algorithm only
based on power control.
Fig. 4 Average system throughput against number of active
transceiver pairs (N
MB
¼N
DS
at each iteration)
Fig. 5 Average system power consumption against number of
active transceiver pairs (N
MB
¼N
DS
at each iteration)
Fig. 2 Throughput against vertical distance between two
transceiver pairs (N
MB
¼N
DS
¼1)
Fig. 3 Power consumption against vertical distance between two
transceiver pairs (N
MB
¼N
DS
¼1)
2284 IET Commun., 2012, Vol. 6, Iss. 15, pp. 2278–2286
&The Institution of Engineering and Technology 2012 doi: 10.1049/iet-com.2012.0297
www.ietdl.org
6 Conclusion
In this paper, the coexistence issue between MB-OFDM
and UWB standards has been addressed, considering a
WPAN topology. Specifically, a game-theoretic approach has
been proposed for interference mitigation between WPANs
using the two UWB technologies. A non-cooperative game
model was used for joint rate and power control in these
heterogeneous networks, which can lead to overall network
interference reduction. To derive the expression for the utility
function, Rayleigh fading model was considered. A pricing
mechanism has been integrated into the defined game to
compensate for the effect of interference. Later, the
existence, uniqueness and Pareto efficiency of the NE for the
proposed game were discussed. It has been observed that in
the case of NPRGP parameter control, both kinds of
networks achieve their maximum transmission rate even at
much closer distances and the power consumption is
reduced. The latter specification is notable, specifically for
applications like sensor networks where power consumption
is a critical issue. The effect of the number of transceiver
pairs on performance was also investigated.
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