Wireless Sensor Network, 2012, 4, 76-83
http://dx.doi.org/10.4236/wsn.2012.43011 Published Online March 2012 (http://www.SciRP.org/journal/wsn)
Framework for Random Power Allocation of Wire less
Sensor Networks in Fading Channels
Mohammed Elmusrati1, Naser Tarhuni2, Riku Jantti3
1Communication and Systems Engineering Group, University of Vaasa, Vaasa, Finland
2Department of Electrical and Computer Engineering, Sultan Qaboos University, Muscat, Oman
3Department of Communications and Networking, Aalto University, Helsinki, Finland
Received December 13, 2011; revised January 23, 2012; accepted February 11, 2012
ABSTRACT
In naturally deaf wireless sensor networks or generally when there is no feedback channel, the fixed-level transmit
power of all nodes is the conventional and practical power allocation method. Using random power allocation for the
broadcasting nodes has been recently proposed to overcome the limitations and problems of the fixed power allocation.
However, the previous work discussed only the performance analysis when uniform power allocation is used for
quasi-static channels. This paper gives a general framework to evaluate the performance (in terms of outage and aver-
age transmit power) of an y truncated probab ility den sity fun ction of th e random allocat ed power. Furthermore, d ynamic
Rayleigh fading channel is considered during the performance analysis which gives more realistic results that the
AWGN channels assumed in the previous work. The main objective of this paper is to evaluate the communication per-
formance when general random power allocation is used. Furthermore, the truncated inverse exponential probability
distribution of the random power allocation is proposed and compared with the fixed and the uniform power allocations.
The performance analysis for the proposed schemes are given mathematically and evaluated via intensive simulations.
Keywords: Power Allocation; Deaf Networks; Rayleigh Channels; Inverse Exponential Distribu tion
1. Introduction
There are many MAC protocols suggested for self-orga-
nized wireless sensor networks during the last two dec-
ades. All protocols are designed to achieve certain targets
such as minimizing the MAC delay, maximizing the throu-
ghputs, minimizing the energy consumption, maximizing
the network life-time, and many other objectives [1,2].
However, all these protocols assume that the sensor
nodes have full listening (reception) capabilities. There-
fore, the nodes can receive feedbacks and/or sense the
environment (e.g., in CSMA/CA). The transmit power
value is one of the radio resources which can be adjusted
during the transmission. Efficient Power control sch emes
requires feedback channel between the receiver and the
transmitter, where the receivers inform the transmitter
about the channel quality in terms of different indicators
such a s the r eceiv ed s igna l-to-in terference and no ise ratio
(SINR). This is known as closed-loop power control whi-
ch is used mainly to mitigate distance and shadowing
losses, fast fading, and most importantly to overcome the
near-far effect in co-channel multiuser wireless commu-
nication systems (see e.g. [3-8]). However, when we dis-
cuss about naturally deaf sensor nodes, it becomes totally
tricky how to optimize the transmission parameters (po-
wer, modulation-level, etc.). In such networks, the sen-
sors’ nodes are just transmitters (broadcasting) and do
not have any receiving capabilities. Naturally deaf net-
work is different than the temporary deafness of some
networks because of channel fading or directive antennas.
Generally, the transmission of deaf sensors can be event-
based or periodic. An example for both cases can be
found the health structure monitor of the bridges. The
even-based sensor will send the measured vibration of
the structure if the vibration value exceeds a certain pre-
defined value otherwise the sensors will stay silent. In
periodic sensors, the sensors send the measured vibration
of the structure periodically, for example every 10 sec-
onds [9]. The transmit-only (TX) sensor nodes are much
simpler and cheaper than the transceiver (TRX) sensor
nodes. Moreover, it is observed that some TRX sensors
consumes during the reception period more than 60% of
the consumed energy during the transmission [1]. Hence,
TX sensor nodes consume much less energy than TRX
nodes. However, there are very few algorithms in the li-
terature to improve the performance of deaf sensor nodes.
Looking only to the power allocation for the TX sensor
nodes, it is usually assumed that the nodes transmit with
C
opyright © 2012 SciRes. WSN

M. ELMUSRATI ET AL. 77
fixed power. This solution has many problems and limi-
tations such as the near-far problem and the unnecessary-
ily power consumptions for good channel sensors.
New randomized power allocation strategy was sug-
gested in [10], which proposed the use of uniformly dis-
tributed transmitter power levels to mitigate the near-far
effect in congested systems without any channel feed-
back. That work was based on so called snapshot analy-
sis approach and thus neglected the effects of the channel
fading. The performance analysis of the uniform random
power allocation in Rayleigh fading channel is evaluated
in [11].
In this paper, a framework of the performance analysis
for a general distributed random power allocation is in-
troduced. The paper is organized as follows. In the next
section, a description of the system model is given. For
the logical information flow and for comparison purpose
we introduce the system performance of the fixed power
transmission in Section 3. In Section 4, a general treat-
ment of the performance analysis of random power allo-
cation algorithms is given. New empirical random power
distribution is suggested in Section 5. Simulation results
are shown in Section 6. Finally the paper conclusion is
presented in Section 7.
2. System Model
In this paper we assume multiuser environment with broad-
casting devices (sensor nodes) randomly distributed in
certain region. We refer to the transmitters as terminals
and sometimes as sensors. All terminals send their sig-
nals to one or more access points with CDMA multiple
access method. Because of the lack of the feedback
channels it is not possible to use CSMA/CA or any other
protocols that require receiving capabilities in the sensor
nodes. Multi-hop scenarios are not possible as well be-
cause of natural deafness. Every transmitter has different
spreading code, however we do not assume that they are
perfectly orthogonal at the access point.
We consider dynamic scenario, where the terminals or
the access points may have mobility or the environment
is highly dynamics. The transmitted signals arrive from
sensors to the access point in multi-path manner without
dominant path, in other words we assume Rayleigh chan-
nel. The time slot length is small enough to assume that
second order effects such as shadow fading and distance
based attenuation remain constant during the time dura-
tion of the time slot. Although the mean of the received
signal is constant but the instant value of the received
signal magnitude is random variable with Rayleigh pro-
bability density function . In case of Rayleigh fading whi-
ch is considered here, the link gain, i.e. the fraction be-
tween received power and transmitted power becomes
Exponential distributed random variable.
Time is assumed to be slotted such that slot duration is
approximately the same as the coherence time of the
channel. For instance in some sensor network applica-
tions, the duty cycle of the transmitters is low and thus
the channel state in consecutive time slots allocated to
single transmitter node become independent of each other.
Let G denote the link gain between transmitter and re-
ceiver. In case of frequency-non-selective Rayleigh fad-
ing, it can be shown to follow the Exponential distribu-
tion with parameter 1
g


where
g
denotes the ex-
pected channel gain which depends on the distance based
attenuation and shadow fading. Let


1
g
G
F
ge




de-
notes the cdf of the link gain G and
g
G
f
ge



de-
notes its pdf.
Let I and 2
n

denote the received interference and
noise powers, respectively. Let γ denotes the minimum
required signal to interference and noise ratio (SINR) at
the receiver. When the received SINR is less than γ, we
assume that the receiver cannot decode the transmitted
packet correctly. The requ ired SINR depends on the util-
ized modulation and coding method and is out of the
scope of the paper. The outage probability of sen sor i, i.e .
the probability that a packet error occurs, is given by
2
Pr ii
in
GP
I













(1)
where Gi is the channel gain of sensor i, Pi is the trans-
mitted power from sensor i, and the interference term is
given by
N
i
ji
jj
I
GP

 (2)
Note that in this model interference is treated as noise.
The use of multi-user detection could be taken into ac-
count by scaling down the interference power I by some
factor 01


. However, this is not considered in this
paper. We assume that the sensor in outage whenever its
power at the access point is less than some threshold.
3. Performance Analysis of Fixed Power
Allocation
In this section we analyze the system performance of fi-
xed power transmission strategy. The results of this sec-
tion is well known in the literature [6,11], however it is
given here for the subject integrity and for comparison
purposes. Moreover some intermediate results have been
used in next sections. First we assume fixed average in-
terference power (or single sensor scenario). In order to
simplify our notation, let us define n

2
ii
I


  Con-
ditioned on Pi = P the outage probability becomes

Pr, i
iii iG
i
GPPF P







(3)
1
i
i
P
e

 
 (4)

M. ELMUSRATI ET AL.
78
For a given outage level pout we can find the corres-
ponding fi x ed transmission po wer Pi = P* to be

*
out
ln 1
Pp

  (5)
In order to achieve low outage probability P* must be
large. Consider now the case, where we have multiple
sensors. In this case, I cannot be treated as constant any-
more, but rather as a random variable. Assume that all
the N transmitters are communicating with the same ac-
cess point. In such case, the interference power at the
receiver Ii is given by (2). In this case we should average
(4) over i

.
From (2) we can deduce that the pdf of the interference
power is the ( N – 1) fold convolution of exponential distri-
bution pdf. This distribution is called Hypo-exponential dis-
tribution. Hence, the outage probability can be written as:


2
0
Pr 1d
in i
ii
i
g
PP
ii iI
i
GPPeefg g
 

 

 (6)
We note that the integral in the above formula is in
fact the moment generating function of the interference


i
i
tI
I
M
tEe where i
i
tP


 . Let Zi = GiPi de-
notes the received power. In case of fixed transmission
power, this still follows the exponential distribution with
the following pdf function

i
i
i
z
P
i
Z
i
f
ze
P



 (7)
The moment generating function of Zi can be easily
found to be



i
i
tZ
Z
M
tEe (8)

0d
i
tz
Z
ef zz

 (9)
1
1i
i
P
t
e



(10)
Now the moment generating function for the inter-
ference power $I_i$ can be written as:


j
ji
i
tZ
I
Mt Ee




(11)


j
tZ
ji
Ee

 (12)

i
Z
ji
M
t

 (13)
Let us revisit the outage probability (6). With the help
of (10), (13), and (6) the outage probability of fixed po-
wer transmission is:

2
Pr1 in
i
i
Pi
iii iI
i
GPPe MP





 


21
1
1
ni
i
P
ij ji
j
i
eP
P
 






 (15)
The outage probability shown in (15) is valid for any
deterministic (fixed) or slowly changing power (power
update rate is slower than the frame duration) transmis-
sion [12]. For randomly selected transmission power we
need to average (6) over the probability density function
of the transmitted power fP(p) as will be shown in the
next section.
4. General Distribution Random Power
Allocation
In this section we generalize the results of the previous
section to the case of random transmitted power and ana-
lyzing the resultant performance in terms of outage pro-
bability and power consumption. Assume that sensor i
uses random transmitted power Pi which has probability
density function fP(p). First i

is assumed to be con-
stant. This assumption could be justified for a single sen-
sor-node scenario but not for multi sensor scenario. The
outage probability in Rayleigh fading channel is given by


0
Pr1 d
i
p
i
ii iP
i
GPefp p





 (16)
where


i
P
f
pis the pdf of the random pow er allocation.
Define a variable 1
x
p

it follows that 2 ddpxx ,
so (16) can be rewritten as

2
0
1d
Pr1 i
i
x
iii iP
x
GPef x
x



 

 (17)
Note that

2
1d
XP
x
fx fx
x



is called the inverted distribution of fP(p) [13]. Theoreti-
cally any pdf (should be truncated to be between some
positive minimum and maximum values) could be used
as


fp
i
P. The probability density function that mini-
mizes the outage probability (17) under mean power con-
straint can be shown to be the fixed power solution P =
Pmax, i.e.,




max
fp pP  
i
P, where is the dirac
delta function. This property follows directly from The-
orem 4.5.2 in [14]. However, in multi-user case finding
the optimal distribution is not trivial. Randomization of
the transmitted power implies randomization of the inter-
ference power which in turn can help to solve the nearfar
problem.

p 
Considering multi sensor scenario, where all sensors
use the same power distribution

P
f
p. In this case Ii
should be considered as a random variable. We assume
that


P
f
p is selected so that the moment generating
function for the received power Zi = GiPi is well-defined.


(14)

M. ELMUSRATI ET AL. 79
That is, we assume that the Pi has finite moments. In
practice Pi must be bounded in the interval [Pmin, Pmax]
from which this condition automatically follows. For a
given i
Z

M
twe can find the corresponding


i
I
M
t
using (13) in Section 3. Let’s first condition on Pi, then
the outage probability can be found using (14) as,

2
Pr GP 1in
i
i
Pi
ii iI
ii
Pe MP





 




(18)
Thus the outage can be found using single integral,
 
2
0
Pr 1dp
in
i
pi
I P
i
eMf p
P
 



 



ii i
GP 

(19)
Now taking into account that Pi is bounded, hence, we
can rewrite the above equation as

12
min
1
max
Pr 1in
i
Pxt
I ip
P2
1d
ii i
x
GPeMx fx
x





 




t
(20)
where p
f
p is the truncated version of the utilized
probability density function. Without any loss of general-
ity we will assume that Pmin = 0 in remaining of this pa-
per. Using (20), it is possible to evaluate the outage pro-
bability for arbitrary any truncated pdf. The average tran-
smitted power is found by


max
0d
i
Pt
p
EPpfp p (21)
It is clear that randomizing the transmitted power re-
sults in an average power which is always less than Pmax.
This is one advantage of using random power over fixed
power. By assuming random uniform distribution of the
power between 0 and Pmax the outage in this case is given
by [11]

2
1
max
11 1
max
max
11
Pr 1
ln1d
in
x
ii
GP

t
p
iNNNN
P
i
i
j
ji j
e
Px
Pxx






 













(23)
Moreover, it is clear that in this case the average
power will be Pmax/2.
Now the most interesting point is to find the optimum
distribution function which can achieve the following
target:
Find

f
p such that

12
min
1
max 2
1d
max
i
in
i
Pxt
Ii p
P
x
eM xf
x
x








 (23)


max
0
S.T. d
i
Pt
p
pfp pP
 (24)
It is possible also to change the order, i.e., minimize
the average power subject to certain outage. So far, we
are not sure if the above optimization problem is solvable
or not. However, we leave it as an open problem for fur-
ther research. Nevertheless, a random power distribution
is proposed based on empirical assumptions as shown in
the next section.
5. Inverted Exponential Distribution
Since the received power has an exponential distribution
when the transmitted power is fixed, we propose random
transmitted power with inverted exponential distribution
to mitigate the fading channel. This selection was not
based on any optimization criteria. However, this em-
pirical selection shows few features and advantages over
the uniform distribution of the power allocation as will
be discussed in the next section. The inverted exponential
distribution is given by

2,
p
P
fpe p
p


0

 (5)
where

. The cumulative probability density function is
given by

p
P
F
pe


 (26)
The outage probability for single sensor scenario utili-
zing random power allocation with inverted exponential
distribution can be computed using (16) and (25) such as

Pr 1
ii
iii i
ii ii
GP
 






 

(27)
From (27) the relation between pdf parameter

and
the outage is given by
out
out
1
ii
p
p




 (28)
Thus the smaller the larger
out
p

, for zero outage

.
The probability that the power allocation exceeds
so me maximu m value Pmax is given by

out
out max
1
max max
Pr 11
ii
p
pP
P
PPFP e







  (29)
In order to avoid using excessive power, the power
distribution should be truncated to be between 0 and Pmax.
The truncated inverse exponential distribution is given
by:

max
20
0otherwis
p
P
ep
fp p


 



e
P
(30)
In this case, we have

max
Pr 1
ii
P
iii i
ii
GP e



 


 (31)
The outage in this case depends on the value of Pmax as

M. ELMUSRATI ET AL.
80
well as the value of

. We can now find the required
peak power for given outage level
out
p

max
out
ln1ln 1
ii
ii
P
p



 



(32)
for
out
out
1
ii
p
p



 (33)
We note that as

, given in (5).
Thus in single sensor scenario, the random power alloca-
tion requires higher peak transmit power than the fixed
allocation to achieve same average outage. This result
can be generalized for any truncated distribution function.
Figure 1 shows the truncated inverted exponential distri-
bution for different values of
*
max
PP

. It shows that as

increases the random power becomes more close to .
From (30) when max
P

the distribution tends to be
fixed power at , i.e.,
max
P


fp p  

max
P
i
P
The average transmitted power of using truncated in-
verted exponential distribution can be computed using
(21) and (30) as
.

max
max
0d
P
Pp
EP eep
p



 (34)
max
max
1d
Px
P
ee
x




x (35)
max 1max
P
eE
P







(36)
where

1
1dd
nxt n nt
nx
Extet xtet




Figure 1. Truncated inverted exponential pdf.
is callthat ed the exponential integral of order n. Note


0
n
Ex for 0x. The asymptotic expansion of


n
Ex can be was ritten
 
2
1
1
x
n
nn
en
Ex xx
x

  


 (37)
Figure 2 shows the relation between the average tran-
smitted power and

when max 1P.
The probability of outage in multi-user environment
where all transmitters utilize trun cated inverted exponen-
tial distribution is discussed next. Without loss of gener-
ality we assume that all transmitters use same distribu-
tion parameter

. This assumption is practical since no
information is ailable for transmitters about neither
their channel quality nor their locations. We will discuss
the influence of the selection of
av

on the system per-
formance in the next section. Thcdf of the received
power Zi = GiPi, denoted by

i
Z
e
F
z, is given by (31) by
replacing i

by z. Let us nove the moment gen-
erating funion w deri ct


i
Z
M
t using

i
Z
F
z. We first note
that


00
i
Z
F

and


1z. Let
s
i
Z
F as z
0t

 g integraarts w. Usintion by pe get,
 
0
dd
d
ii
sz
ZZ
sFzez
z


 (38)

M
 
0 0d
ii
sz sz
ZZ
s
FzesFze z


  (39)
max
0
1d
i
i
i
sz
P
s
ez
z








 
 
 (40)
Now from [15], we can find that,

1
0
1d,
st as
eteEasa
at
0



 (41)
Hence,
Figure 2. Average transmitted power with respect to α.

M. ELMUSRATI ET AL. 81

max 1max
1
it
P
i
i
i
Z
ii
M
tte Et
P 

 





(42)
Once again we can solve the moment generating func-
tio
 

 






n for the interference power

i
I
M
t using (13). From
the general outage probability formu and (30) we
can find the mathematical formula for the outage prob-
ability in case on inverted exponential power allocation
such as,
la (20)


2
max 1
max
max 1max
Pr 1
1d
in
i
j
x
P
ii iP
x
P
i
ji j
GPee
x
i
j
x
eE
P

 




 












 
 

  







x





 






(43)
It can be shown that when

the above equ
wase w
6. Numerical Results
aluated in this section. In the
ation
ill be reduced to (15), the chen the transmitted
power from sensors are fixed and identical.
Different aspects will be ev
first part we validate Equation (43) by condu cting exten-
sive Monte-Carlo simulations. And in th e second part we
test the performance of the random truncated exponential
allocation algorithm for different scenarios and compare-
ing it with fixed and random uniform power allocation
schemes. All simulations are carried out for Rayleigh
channels with an average of d4, where d is the distance
between the sen sor and the access point. Th e background
white noise power is fixed at –70 dBm. The multiple
access method for all nodes is CDMA and the processing
gain is 27 dB. The target bo EN is 7 dB. It means that
the minimum required SNout 0.01 for all sensors.
Figure 3 shows the outage with respect with number of
sensors using exact formula (43) and Monte-Carlo simu-
R is ab
lation. The simulation has been carried out by generating
30 sensors with randomly distributed distance from the
access-point from 20 up to 150 meters. The outage is
calculated by counting the total number of packets where
the SINR is less than 0.01, and then dividing this number
by the total number of sensors at that stage. This proce-
dure repeated 1000 times to obtain a reliable measure for
the outage. The sensors are increased one by one in as-
cending manner, i.e., the first sensor is the nearest to the
access point and the second is the next further one and so
on. The derived formula gives directly the probability of
outage by substituting the average channel losses of sen-
sors in (43).
To compare the performance of inverted exponential
random power (IERP) allocation with fixed and uniform
power, we repeat the previous simulation including other
power allocation methods. Figure 4 shows the outage for
all cases at 0.5


, maximum power of 1 Watt and the
uniform distribution is truncated between 0 and 1. In
terms of outage, it is clear that IERP allocation outper-
forms the uniform power when the number of sensors is
less than about 22 sensors. For larger network size the
IERP allocation outperform the fixed power allocation
and becomes very close to the uniform. In terms of po-
wer consumption the IERP consumes less average power
than both other methods. The average required power is
1, 0.5, and 0.46 Watt for fixed, uniform and IERP meth-
ods, respectively. This result shows one benefit of using
IERP allocation over fixed or uniform allocation. It gives
less average outage for large network size than the fixed
power allocation at lower average power consumption.
Figure 5 shows the average outage of worst sensor with
respect to the minimum required SINR for different val-
ues of α, i.e., with different average power values.
In this scenario we mean by worst sensor is the one
which is located on the cell boarder, i.e., at 150 meters.
However, because of the fading behavior it is not neces-
Figure 3. Average outage using the exact formula and Monte
Carlo simulations. Figure 4. The average outage using different power alloca-
tion methods.

M. ELMUSRATI ET AL.
82
Figure 5. The average outage of worst sensor for different
values of α.
sor has all time the worst channel. The
gure shows that as the α becomes smaller, the outage of
sary that this sen
fi
the worst sensor reduces, and it gets a better chance to
access the network. The reason is clearly because of re-
ducing the interference coming from closer sensors by
reducing their average transmitted power. However, re-
ducing α has a negative impact on the average outage of
all sensors as shown in Figure 6. Figure 6 shows also
the average outage when using uniform and fixed power
allocations. This last simulation has been carried out for
15 sensors; other simulation parameters are as before.
The average power used for IERP increases with α. At
0.61

 the average power is 0.5 Watt which is the
same average power of uniform allocation. However, the
hives the same outage as uniform allocation at
0.165
IERP arc

 which means average power of 0.26 Watt.
This result is rather interesting where smaller average
be achieved at less average power. The fixed
power scenario all the time has the highest average
power consumption (1 Watt).
7. Conclusion
outage can
his paper is to introduce a general fra-
random power allocation methods in
The main aim of t
mework analysis for
Rayleigh fading channels. Simple mathematical proce-
dure has been given to analysis the system performance
when using any arbitrary truncated random power distri-
bution. We extend our work by proposing the truncated
inverted exponential probability density function (IERP)
for the random power. Mathematical representation of
the outage as well as the average transmitted power is
given. IERP shows many advantages over fixed as well
as over random uniform power allocations. At small net-
work size IERP gives less average outage than the uni-
form power allocation. For large network size IERP
gives less outage than th e fixed power allocation method.
At the same outage the IERP needs less average power
Figure 6. The average outage sensors versus α.
than thends
n the network size and nodes spatial distribution. An-
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o
other good advantage of the IERP that it has only one
parameter to tune, this reduces the difficulties of optimi-
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intensive mathematical and simulation analysis in this
work, it was obvious that using random power allocation
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such as find the optimum power distribution (if any), di-
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simplify the problem by using discrete power values with
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