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Probability Model Based Energy Efficient and
Reliable Topology Control Algorithm
Ning Li *, Jose-Fernan Martinez-Ortega, Lourdes Lopez Santidrian and
Juan Manuel Meneses Chaus
Research Center on Software Technologies and Multimedia Systems for Sustainability (CITSEM),
Universidad Politécnica de Madrid (UPM), Madrid 28031, Spain; [email protected] (J.-F.M.-O.);
[email protected] (L.L.S.); [email protected] (J.M.M.C.)
* Correspondence: [email protected]; Tel.: +34-688-500-639
Academic Editor: Robert Lundmark
Received: 20 June 2016; Accepted: 12 October 2016; Published: 20 October 2016
Abstract: Topology control is an effective method for improving the performance of wireless
sensor networks (WSNs). Many topology control algorithms can achieve high energy efficiency
by dynamically changing the transmission range of nodes. However, these algorithms prefer to
choose short multihop communication links rather than the long directly communication links which
also energy efficient probabilistic. Note that these fact, in this paper, we propose a mathematic model
to explore the probability that the long directly communication links are more energy efficient than
the short links. We investigate the properties of this probability and find out the optimal transmission
range which has highest probability of energy efficient. Based on this conclusion, we propose the
energy efficient and reliable topology control algorithm (ERTC) to maintain the r-range for the nodes
instead of the k-connection; moreover, ERTC can achieve energy efficient and network connection at
the same time.
Keywords: wireless sensor network (WSN); topology control; energy efficient; reliable
1. Introduction
Wireless sensor networks (WSNs) have become more and more widely used and important
in recent years. The properties of WSNs—which contain hundreds or thousands of sensors—are
limited by the energy, the bandwidth, the capability of computing, etc. Moreover, in most applications,
the WSNs are arranged in the remote area where changing the sensor nodes always impossible or
inconvenient, so how to save the node energy and prolong the network lifetime is important for WSN.
There are many algorithms have been proposed to improve the network reliable and energy efficient for
WSN. One remarkable approach is topology control. Topology control has been proposed to address
many problems in WSNs by adding or deleting nodes/links according to certain algorithms. The aim
of topology control is to reduce energy consumption and preserve other fundamental properties for
the network at the same time [1], such as network connectivity, reliability, fault-tolerant, coverage, etc.
In WSNs, topology control can be implemented in three approaches [2]: (1) Power Adjustment
Approach: minimizing the transmission power by adjusting the transmission range of node; in this
approach, the long distance communication links will be eliminated while the short links will be
chosen; (2) Power Model Management: controlling the feature of the operating mode to reduce
energy consumption; there are four operating modes: sleep mode, idle mode, transmission mode,
and receiving mode; since the energy consumption during the transmission mode and receiving
mode is generally higher than that in the sleep mode [3], so switching the redundant nodes into
sleep mode can save energy obviously [4]; (3) Clustering Approach [5]: selecting a set of nodes
in the network to construct an efficiently hierarchical topology; the clusterheads are restricted to
Energies 2016, 9, 841; doi:10.3390/en9100841
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Energies 2016, 9, 841
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certain tasks like collecting data, processing packets, or forwarding packets to non-clusterheads; the
non-clusterheads nodes collect data and transmit the data packets to the clusterheads. In this paper,
we mainly concentrate on the first one, i.e., the power adjustment approach.
To the power adjustment approach, on one hand, for reducing interference and energy
consumption, each node transmits packets with relative low power [6]. The algorithms are generally
localized, i.e., each node uses only the information that is one or two hops away. The problem of
minimizing the total energy consumption for the whole network is NP-hard in both two and three
dimensional space [7,8]. In addition, if the WSN consists thousands of nodes, it is difficult to calculate
the optimal transmission ranges for transmitting the packets to the concerned nodes [3]. On the other
hand, even reducing the transmission range of nodes is the most common and effective approach
to control the network topology and reduce the energy consumption in WSN, but in this paper we
will show that the long directly communication links can probabilistically spend less energy than the
short indirectly communication links, i.e., it is probabilistic when reducing the energy consumption of
network by reducing transmission range.
Motivated by these, in this paper, we explore the probability of reducing energy consumption
by reducing the transmission range, and investigate the properties of this probability under different
scenarios. Based on the conclusions, we propose an energy efficient and reliable topology control
algorithm (ERTC) which meets the requirements of network connection and energy efficient at the
same time. The contributions of this paper are as follows:
•
•
•
we propose a mathematic probability model for energy consumption analysis when applying
the transmission power adjustment approach. To the best of our knowledge, this is the first
probability analysis model for this kind of issue;
we analyze the probability model in detail and explore the features of this model under different
network parameters;
we propose an ERTC based on these conclusions, which maintain the r-range of the node instead
of the k-connection and can adapt the network dynamic.
The rest parts of the paper are organization as follows: in Section 2, we will introduce the related
works of the and topology control; Section 3 will provide the network model and state the problems
which will be investigated in this paper; we will introduce the probability model and analysis the
properties of this model in Section 4; in Section 5, we will introduce the ERTC method in detail;
Section 6 explores the performance of ERTC based on simulation; in Section 7, we conclude this paper.
2. Related Works
The latest surveys of network topology control algorithms can be found in [2,9–11]. The primary
goals of topology control are to guarantee the network connection and reduce the energy consumption
as far as possible. Many heuristic algorithms have been proposed, such as, Local Minimum Spanning
Tree (LMST) [12], Local Tree-based Reliable Topology (LTRT) [6], A1 [13], Poly [14], Centralized Robust
Topology Control Algorithm (CRTCA) [15], Cooperative topology control scheme with Opportunistic
Interference Cancelation (COIC) [16], Local Mean Neighbor (LMN) [17], Local Mean Algorithm
(LMA) [17], Smart Boundary Yao Gabriel Graph (SBYaoGG) [18], BRASP [19], etc. Almost all of
these protocols regard topology control as a technique in which nodes dynamically change their
transmission ranges to gain energy efficient and network connection. In [12], each node builds its
own LMST independently and only on-tree nodes that one-hop away are kept in the final topology.
Considering the fact that the LMST always constructs one-connected network in the final topology,
in [6], the authors propose LTRT algorithm, which combines the idea of LMST and Tree-based Reliable
Topology (TRT) together to guarantee k-edge connectivity in the resulting topology. LTRT can maintain
the network connection at low computational cost and energy consumption. In [19], due to the lossy
links in the real environment (which can provide only probabilistic connection), the authors propose
a novel probabilistic network model, in which the network connectivity is metered by the network
Energies 2016, 9, 841
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transmissionThe
power
for each
nodethe
when
the network
reachability
above
given
threshold.
Based
reachability.
authors
explore
minimal
transmission
powerisfor
eachanode
when
the network
on
the
conclusion,
the
authors
propose
BRASP
algorithm
to
improve
the
energy
efficiency
and
reachability is above a given threshold. Based on the conclusion, the authors propose BRASP algorithm
reduce
the average
node
degree. and
A1 assumes
theaverage
networknode
topology
as aA1
connected
and
to
improve
the energy
efficiency
reduce the
degree.
assumes network
the network
finds
a
set
of
active
nodes
to
form
connected
dominating
set
(CDS)
[13].
This
algorithm
can
form
topology as a connected network and finds a set of active nodes to form connected dominating seta
reduced
topology
while keeping
network
connection
coverage
at the connection
same time.and
In
(CDS)
[13].
This algorithm
can formthe
a reduced
topology
while and
keeping
the network
addition,
A1
forms
the
CDS
which
comprising
high
energy
nodes
in
a
single
phase
construction
coverage at the same time. In addition, A1 forms the CDS which comprising high energy nodes in
and a construction
set of active process
nodes for
andfor
better
sensing
coverage,
respectively.
aprocess
single phase
andenergy
a set ofefficiency
active nodes
energy
efficiency
and better
sensing
Similarly
with
A1,
Poly
[14]
is
also
the
algorithm
based
on
CDS.
In
Poly,
the
network
is
modeled
as
coverage, respectively. Similarly with A1, Poly [14] is also the algorithm based on CDS. In Poly,
a
connected
graph.
The
protocol
can
turn
off
the
unnecessary
node
and
keep
the
network
connection
the network is modeled as a connected graph. The protocol can turn off the unnecessary node and
and coverage
at the
same time.
LMN
and LMA
thetime.
two typical
power
topology
keep
the network
connection
and
coverage
at theare
same
LMN and
LMAadjustment
are the two
typical
control
algorithms
[17].
In
LMA,
all
the
nodes
can
get
their
node
degree.
The
algorithm
sets
the
power adjustment topology control algorithms [17]. In LMA, all the nodes can get their node degree.
minimum
threshold
and
maximum
threshold
for
this
number;
if
the
node
degree
is
less
than
the
The algorithm sets the minimum threshold and maximum threshold for this number; if the node degree
minimum
the transmission
will be increased;
transmission
will be
is
less thanthreshold,
the minimum
threshold, therange
transmission
range willotherwise,
be increased;
otherwise, range
transmission
reduced.
The
principle
of
LMN
is
similar
with
LMA,
but
LMN
does
not
set
the
maximum
and
range will be reduced. The principle of LMN is similar with LMA, but LMN does not set the maximum
minimum
thresholds
for
the
node
degree.
In
LMN,
the
nodes
use
the
mean
neighbors’
node
degree
and minimum thresholds for the node degree. In LMN, the nodes use the mean neighbors’ node
as the threshold
to adjust
transmission
ranges.ranges.
degree
as the threshold
to the
adjust
the transmission
3. Network
NetworkModel
Modeland
andProblem
ProblemStatement
Statement
models can express
express the
the connection
connection mode between nodes, which are
In general, there are three models
shown
in
Figure
1
[20].
Figure
1a
illustrates
the
k
nearest
neighbor
model;
each
node
in this
model
shown in Figure 1 [20]. Figure 1a illustrates the k nearest neighbor
model;
each
node
in this
model
has
has constant
node degree
and maintains
node
bycommunication
changing communication
range
constant
node degree
and maintains
the nodethe
degree
by degree
changing
range dynamically.
dynamically.
Figure the
1b illustrates
the
model, therange
transmission
range
modeled
Figure
1b illustrates
disc model;
indisc
this model;
model, in
thethis
transmission
is modeled
as aisdisk
with
as
a
disk
with
radius
r;
the
nodes
connect
with
other
nodes
that
fall
into
its
communication
range.
radius r; the nodes connect with other nodes that fall into its communication range. Figure 1c illustrates
Figure
1c illustrates
the graph
Erdos-Renyi
randomany
graph
connects
any two
nodes by
the issame
the
Erdos-Renyi
random
that connects
two that
nodes
by the same
probability
which
not
probability
which
is
not
appropriate
in
WSNs.
Disc
model
is
more
plausible
in
WSN
since
obtaining
appropriate in WSNs. Disc model is more plausible in WSN since obtaining k neighbors is not always
k neighbors
feasible due
to the
communication
range limitation
[20].we
Therefore,
in this
feasible
due is
tonot
thealways
communication
range
limitation
[20]. Therefore,
in this paper,
only interested
paper,
we only
interested
in the
Discmodel
model.are
The
nodes distributed
in this model
uniform
distributed
and
in
the Disc
model.
The nodes
in this
uniform
andare
fixed;
moreover,
the nodes
fixed;
moreover,
the
nodes
have
different
initial
transmission
ranges
and
can
change
their
have different initial transmission ranges and can change their transmission ranges from zero to
transmission
the
maximum.ranges from zero to the maximum.
1
1
1
p
4
3
(a)
4
3
(b)
p
2
2
2
4
p
p
p
p
3
(c)
Figure 1. Different network models: (a) k nearest neighbor model; (b) disc model; and (c)
Figure 1. Different network models: (a) k nearest neighbor model; (b) disc model; and (c) Erdos-Renyi
Erdos-Renyi random graph.
random graph.
n
rn
r

Puvγ
Puv
ρ
The notations and network definitions used in this paper are as follows:
The notations and network definitions used in this paper are as follows:
The node number of the whole network;
The Euclidean
node number
of thebetween
whole network;
The
distance
two nodes u and v, in this paper, we also use r to represent the
The Euclidean
distance
between two nodes u and v, in this paper, we also use r to represent the
initial
transmission
range;
initial
transmission range;
The
distance-power
gradient;
The energy
distance-power
The
required gradient;
to transmit data from node u to node v;
The
energy
required
to transmit
data from
nodethe
u to
nodeadjustment
v;
The probability of energy
efficient when
applying
power
topology control algorithm.
The probability of energy efficient when applying the power adjustment topology control algorithm.
Energies 2016, 9, 841
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Definition 1.
1. The communication range of node uu is
is defined
defined as
as the area where other nodes can receive
Definition
receive u’s
correctly.
packet correctly.
thispaper,
paper,thethe
communication
ranges
of nodes
are circles,
not
be the
same.
In this
communication
ranges
of nodes
are circles,
but maybut
notmay
be the
same.
Moreover,
Moreover,
the
transmission
range
can
be
changed
from
zero
to
maximum.
the transmission range can be changed from zero to maximum.

Definition
toto
transmit
data
packet
from
node
u to
v isv defined
as rasγ , rwhere
r is
Puv required
Definition 2.
2. The
The energy
energy Puv
required
transmit
data
packet
from
node
u node
to node
is defined
, where
the
distance
between
nodenode
u andunode
v, andv,γand
is a γconstant
called called
the distance-power
gradient
whose
r is Euclidean
the Euclidean
distance
between
and node
is a constant
the distance-power
gradient
typical
value
is
between
2
and
4
[21–23].
whose typical value is between 2 and 4 [21–23].
Definition
areare
defined
as as
thethe
nodes
which
cancan
receive
the the
packet
fromfrom
nodenode
u and
can
Definition 3.
3. The
Theneighbors
neighborsofofnode
nodeu u
defined
nodes
which
receive
packet
u and
reply
message
to
node
u.
can reply message to node u.
Consequently,
Consequently, as
as shown
shown in
in Figure
Figure 2,
2, according
according to
to the
the definition
definition of
of the
the neighbors,
neighbors, if
if node
node vv is
is
the
of node
node u,
u, then
the neighbor
neighbor of
then the
the node
node uu is
is also
also the
the neighbor
neighbor of
ofnode
nodev.v.In
In Figure
Figure 2,
2, even
even the
the node
node m
m
locates
range of
of node
node u,
u, but
but it
locates in
in the
the communication
communication range
it can
can not
not send
send packet
packet to
to node
node u
u due
due to
to the
the small
small
transmission
range,
so
node
m
is
not
the
neighbor
of
node
u.
Node
v
is
the
neighbor
node
of
node u,
u,
transmission range, so node m is not the neighbor of node u. Node v is the neighbor node of node
since
since node
node uu also
also locates
locates in
in the
the transmission
transmission range
range of
of node
node v.
v.
m
u
v
Figure 2. Definition of communication range and neighbor.
Figure 2. Definition of communication range and neighbor.
Definition 4. r-range of the node is defined as the optimal transmission range which has high probability of
energy efficient.
Definition
4. r-range of the node is defined as the optimal transmission range which has high probability of
energy efficient.
Like the definition of k-connection for the network reliability and considering the probability
Like the definition of k-connection for the network reliability and considering the probability that
that reduce the energy consumption by reducing the transmission range, if the nodes can maintain
reduce the energy consumption by reducing the transmission range, if the nodes can maintain the
the optimal transmission range, i.e., r-range, the probability of energy efficient will be high.
optimal transmission range, i.e., r-range, the probability of energy efficient will be high.
As discussed in Section 2, many energy efficient topology control algorithms change the
As discussed in Section 2, many energy efficient topology control algorithms change the
transmission range dynamically to gain energy efficient, but they fail to give the strict proof of
transmission range dynamically to gain energy efficient, but they fail to give the strict proof of
whether this approach always effective or not; if not, what is the probability of this issue, and how
whether this approach always effective or not; if not, what is the probability of this issue, and how to
to improve this probability? In this paper, we will analyze these issues in detail.
improve this probability? In this paper, we will analyze these issues in detail.
4. Probability Analysis
4. Probability Analysis
Theorem 1. After the transmission range adjustment, the probability that the energy consumption is less
1
Theorem 1. After the transmission
range adjustment,
the probability that the energy consumption is less than
R 2 r  1 
than
the
previous
one
is
2 r 2γ (r γ γx ) dx  1 .
(r − x ) dx − 1.
the previous one is ρ =
r2 0
r
0
Proof.
v is
thethe
neighbor
of node
u. when
nodenode
u transmits
packet
to node
Proof. In
In WSN,
WSN,supposing
supposingthat
thatnode
node
v is
neighbor
of node
u. when
u transmits
packet
to
v,
the
energy
consumption
is
related
to
the
distance
between
two
nodes,
which
can
be
expressed
as:
node v, the energy consumption is related to the distance between two nodes, which can be
expressed as:
Puv1 ∝ rγ ,
Puv1  r  ,
(1)
(1)
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where r is the Euclidean distance between node u and node v, γ is the distance-power gradient that
where
r 2016,
isonthe
Euclidean
distanceofbetween
node u and node
v,  (2
is ≤
theγdistance-power
gradient
Energies
9,
841
5 ofoutdoor
17
depending
the
characteristics
the communication
medium
≤ 4, γ ≥ 2 for
that
depending
on
the
characteristics
of
the
communication
medium
(
,
for
outdoor
2



4


2
propagation modes [23]); Puv1 is the power needed for link between node u and node v.
where r is the Euclidean distance between node u and node v,  is the distance-power gradient
propagation modes [23]); Puv1 is the power needed for link between node u and node v.
If the
transmission
of node
node v are medium
reduced( 2based
control
that
depending
on the ranges
characteristics
of u
theand
communication
, the
for outdoor
   4on
 2 topology
If thethen
transmission
ranges
of node
u andcommunicate
node v are reduced
based
on the
topology
control
algorithm,
node
u
and
node
v
cannot
directly,
which
is
shown
in
Figure
3.
propagation modes [23]); Puv1 is the power needed for link between node u and node v.
then n
node
ube
and
node vascannot
communicate
directly,
which
is
shown
in Figure
3. Asnode
a
Asalgorithm,
a result,
node
will
chosen
the
relay
node,
where
node
n
is
the
neighbor
of
both
If the transmission ranges of node u and node v are reduced based on the topology control
result, node n will be chosen as the relay node, where node n is the neighbor of both node u and
u andalgorithm,
node v. Thus,
the energy
needed
to transmit
packets
from node
node in
v will
be:3. As a
then node
u and node
v cannot
communicate
directly,
whichuistoshown
Figure
node v. Thus, the energy needed to transmit packets from node u to node v will be:
result, node n will be chosen as the relay node, where node n is the neighbor of both node u and
Puv2
∝ r1 γ from
+ r2 γnode
,
node v. Thus, the energy needed to transmit
u to node v will be:
Ppackets
(2) (2)
uv 2  r1  r2 ,

Puv 2 
 ru2  node
(2)
,
r2 Euclidean
where
is the
Euclidean
distance
between
node
and
node
n, the
is the Euclidean
distance
where
r1 isr1 the
Euclidean
distance
between
node
ur1 and
n, r2 is
distance
between
node
n
and
node
v;
P
is
the
power
needed
for
communicating
between
node
u
and
node
v
by
using
uv2 node v;distance
Puv 2 is between
between
n and
the power
needed
communicating
between node
u and
r1 is the
where node
Euclidean
node
u andfor
node
n, r2 is the Euclidean
distance
relay
node
node
v byn.using
between
node relay
n andnode
noden.v; P is the power needed for communicating between node u and
uv 2
node v by using relay node n.
Figure 3. Transmission range adjustment.
Figure3.3.Transmission
Transmission range
Figure
rangeadjustment.
adjustment.
Therefore,
the issues we need to solve are when P is smaller than P and what is the
Therefore, the issues we need to solve are when P uv2 2 is smaller than Puv1 uv1and what is the
Therefore, the issues we need to solve are when Puv2 is uvsmaller
than Puv1 and what is the probability
PuvP2 smaller
probability
that
than
For exploring
exploringthese
theseissues,
issues,we
wedefine
define
Energy
efficient
Puvuv11 ..these
probability
that
than P
For
thethe
Energy
efficient
uvP
2 smaller
that P
smaller
than
.
For
exploring
issues,
we
define
the
Energy
efficient
Dominating
Sets
uv2
uv1
Dominating Sets
(EDS) as follows:
(EDS)Dominating
as follows:Sets (EDS) as follows:


Puv1 ≥
 PPuvPuv2uv2
P
uv1
2

P

 r uv+1 r ≥
r
2
r

r

r
1
r11  r22  r
(3)
,, ,

(3) (3)


0
<
r
≤
r
1


 00  rr11 rr
0 < r2 ≤ r
00  rr22 rr
where the first constraint means the energy consumption after power adjustment is smaller than the
where
the
firstconstraint
constraintmeans
means the
the energy
energy consumption
is smaller
than
thethe
where
the
first
consumptionafter
afterpower
poweradjustment
adjustment
is smaller
than
previous one; the second constraint make sure node u still can communication with node v by using
previous
one;
the
second
constraint
make
sure
node
u
still
can
communication
with
node
v
by
using
previous one; the second constraint make sure node u still can communication with node v by using
a relay
node;
the third
andand
the the
fourth
constraints
guarantee
the transmission
rangesofofeach
eachnodes
nodes are
a relay
node;
thethird
third
fourth
constraints
guarantee
ranges
a relay
node;
the
and the
fourth
constraints
guaranteethe
thetransmission
transmission
ranges of each
nodes
smaller
than
thethan
previous.
The EDS
is shown
in Figure
4.
smaller
theprevious.
previous.
The EDS
EDS
is shown
areare
smaller
than the
The
is
shown in
inFigure
Figure4.4.
r
r22
r
r
C
A
C
A
r1  r2  r 
r1  r2  r 
r1  r2  r
r1  r2  r
B
B r r1
r1
O
r
Figure 4. The Energy efficient Dominating Sets (EDS) shown in Equation (3).
Figure 4. The Energy efficient Dominating Sets (EDS) shown in Equation (3).
Figure 4. The Energy efficient Dominating Sets (EDS) shown in Equation (3).
O
Energies 2016, 9, 841
6 of 17
According the definition of EDS and the principle of linear programming, the EDS is the shadow
area in Figure 4. The area ABC means the whole values which satisfy the second, third, and fourth
>
constraints in Equation (3); the area AB is the set that the energy consumption is smaller than the
previous. Therefore, the probability that the energy consumption is less than the previous one is the
>
proportion of area AB in area ABC, which can be calculated as:
ρ=
2
r2
Z r
0
1
(rγ − xγ ) γ dx − 1,
(4)
where x is the transmission range of nodes and 0 < x < r.
Lemma 1. With the increasing of γ (2 ≤ γ ≤ 4), the probability ρ will increase from 0.5708 to 0.8541.
Proof. Considering the first derivative function of ρ on γ:
0
ργ
=
d[ r22
Rr
0 (r
γ
1
− xγ ) γ dx − 1]
,
dγ
(5)
In order to simplify the denotation, we define:
1
f (γ) = (r γ − x γ ) γ ,
(6)
Therefore, Equation (5) can be rewritten as:
0
ργ
=
d[ r22
Rr
0
f (γ)dx − 1]
dγ
,
(7)
Since 0 < x < r, so if we can prove f (γ) is an increasing function, then we can conclude that ρ(γ)
is also the increasing function with γ. The first derivative function of f (γ) on γ is:
1
γ − xγ )
1
γ − xγ )
f 0 (γ) = e γ (r
> e γ (r
·
·
γ· rγ −1 xγ ·(rγ ln(r )− xγ lnx )−ln(rγ − xγ )
γ2
ln(rγ )−ln(rγ − xγ )
>0
γ2
,
(8)
As f 0 (γ) > 0, so f (γ) is an increasing function. Thus, the probability ρ will increase with the
increasing of γ. In addition, the maximum and minimum values of ρ are as follows:
ρmin = ρ2 = 0.5708,
(9)
ρmax = ρ4 = 0.8541,
(10)
This can also be found in Figure 5.
As shown in Figure 5, with the increasing of γ, the probability ρ increases. The maximum value of
ρ is 0.8741 and the minimum value is only 0.5708. This demonstrates that the algorithm which reduces
the energy consumption by adjusting the transmission range is probabilistic, i.e., short transmission
range does not mean small energy consumption. The reason why the probability increases with the
increasing of γ is that with the increasing of γ, the energy consumption is more and more seriously
effected by the distance between two nodes, which can be concluded from Equation (1); so if the
transmission range changes, the energy consumption will be changed obviously.
Energies 2016, 9, 841
Energies 2016, 9, 841
7 of 17
7 of 17
0.95
0.9
Energies 2016, 9, 841
7 of 17
Probability (ρ)
Probability (ρ)
0.85
0.95
0.8
0.9
0.75
0.85
0.7
0.8
0.65
0.75
0.6
0.7
0.55
2
2.2
2.4
0.65
2.6
2.8
3
3.2
3.4
distance-power gradient (γ)
3.6
3.8
4
Figure
and ρ.ρ .
Figure5.5.The
Therelationship
relationshipbetween
between γγ and
0.6
0.55
2.4
2.6 ρ2.8 will
3 keep
3.2
3.4
3.6
3.8
Lemma 2. With constant γ , 2the 2.2
probability
constant
with 4 the variation of the initial
distance-power
gradient (γ)with the variation of the initial transmission
Lemma 2. With constant γ, the probability ρ will
keep constant
transmission
range r.
range
r.
Figure 5. The relationship between γ and ρ .
Proof. We
can prove
prove this
this conclusion
conclusion by
by simulation.
simulation. The
The result
result can
can be
be found
found in
in Figure
Figure 6.
6.
Proof.
We can
Lemma 2. With constant γ , the probability ρ will keep constant with the variation of the initial
0.95with the increasing of r, the probability ρ will keep constant. In addition,
Figure 6 illustrates that
γ=2
transmission range r.
γ=3
when γ increases, the probability
increases,
too; and the bigger the γ, the small increasing rate is, which
0.9
γ=4
is consistent
with
thethis
conclusion
of Lemma
1 (shown
in Figure
5). be found in Figure 6.
Proof.
We can
prove
conclusion
by simulation.
The
result can
Probability (ρ)
Probability (ρ)
0.85
0.95
0.8
γ=2
γ=3
γ=4
0.9
0.75
0.85
0.7
0.8
0.65
0.75
0.6
0.7
0.55
0
0.1
0.65
0.6 Figure
0.2
0.3
0.4
0.5
0.6
Transmission range (r)
0.7
0.8
0.9
1
6. Relationship between r and ρ for γ .
0.55
0.1 the
0.2 increasing
0.3
0.4
0.5
0.8
0.9
ρ 1 will keep constant. In
Figure 6 illustrates that0 with
of r,0.6the0.7probability
Transmission range (r)
addition, when γ increases, the probability increases, too; and the bigger the γ , the small
Figure 6. Relationship between r and ρ for γ .
Figure 6.with
Relationship
betweenof
r and
ρ for γ.
increasing rate is, which is consistent
the conclusion
Lemma
1 (shown in Figure 5).
The Lemma 2 indicates that the probability is nothing to do with the initial transmission range,
Figure 6 illustrates that with the increasing of r, the probability ρ will keep constant. In
The
Lemma
2 indicates
the probability
is nothing
to be
dothe
with
the initial transmission range,
γ , the probability
i.e., no
matter
what
r is, withthat
constant
will
same.
γ rincreases,
addition,
when
the probability
increases,
and
the bigger the γ , the small
i.e., no matter
what
is, with constant
γ, the probability
willtoo;
be the
same.
(1/ 2)1/γ
γ , when with
Lemma 3. With
fixed
value
the transmission
range
, the probability
increasing
rate is,
which
is of
consistent
the conclusion
ofisLemma
1 r(shown
in Figureρ5).
e can get the
Lemma
3.
With
fixed
value
of
γ,
when
the
transmission
range
is
(
1/2
)1/γthe
r, the
probability
ρe can range,
get the
The
Lemma
2
indicates
that
the
probability
is
nothing
to
do
with
initial
transmission
maximum value.
maximum
value.
i.e.,
no matter
what r is, with constant γ , the probability will be the same.
Proof. When applying the power adjustment topology control algorithm, the EDS of r1 and r2 are
Proof. When applying the power adjustment topology control algorithm,
the EDS of r1 and r2 are
1/γ
(1/
2)
r
γ
Lemma
3.
With
fixed
value
of
,
when
the
transmission
range
is
,
the
probability
the
ρ e ofcan
r1 and
shown
in
Equation
(3)
and
Figure
4.
Thus,
similar
with
the
definition
of
EDS,
theUn-EDS
Un-EDS
ofr get
shown in Equation (3) and Figure 4. Thus, similar with the definition of EDS, the
r2
1 and
maximum
r2 can
bevalue.
shown
as follows:
can
be shown
as follows:


Puv1 ≤ Puv2


Proof. When applying the power adjustment
 rtopology
P+1 r2 P≥uv 2rcontrol algorithm, the EDS of r1 and r2 are
1 uv
,
(11)

shown in Equation (3) and Figure 4. Thus,similar
with
 rr the definition of EDS, the Un-EDS of r1 and
0r1<rr12 ≤


,
 
(11)
r2 can be shown as follows:
00< rr2 ≤rr
1

P
r2 P
r
 0uv1 
uv 2

r

r

1 2 r
,

0  r1  r
0  r2  r
(11)
Energies 2016, 9, 841
8 of 17
The meanings of each constraint are similar with Equation (3). The values which satisfy
Un-EDS
mean
the
Energies 2016,
9, 841 that the energy consumption of WSN does not decrease after reducing8 of
17
transmission range. Therefore, how to reduce the size of Un-EDS is an effective method to increase
the probability of energy efficient. A possible way is to eliminate some values of r1 and r2 from
The meanings of each constraint are similar with Equation (3). The values which satisfy Un-EDS
Un-EDS. Therefore, the new probability of energy efficient will be:
mean that the energy consumption of WSN does not decrease after reducing the transmission range.
1
Therefore, how to reduce the size of Un-EDS ris an effective
method to increase the probability of energy
γ
γ γ
(
r

x
)
dx
r r 2and
/ 2 r from Un-EDS. Therefore, the(12)

efficient. A possible way is to eliminate some
values
of
new
1
, 2
ρe  02
probability of energy efficient will be:
r / 2  ( r  r1 )( r  r2 )
1
r γ
where s  (r  r1 )(r  r2 ) is the eliminatedR Un-EDS.
2
γ γ
0 (r − x ) dx − r /2
ρ
=
,
(12)
e
According the principle of linear
programming,
r2 /2 − (r − r1 )(when
r − r2 ) r1 and r2 are in the boundary of
Un-EDS, the eliminated Un-EDS s can get the maximum value, i.e., the probability ρ e can get the
where s = (r − r1 )(r − r2 ) is the eliminated Un-EDS.
maximum value, which can be found in Equation (12). This means that r1 and r2 should satisfy
According the principle of linear programming, when r1 and r2 are in the boundary of Un-EDS,
the
follows:s can get the maximum value, i.e., the probability ρ can get the maximum
the constraint
eliminatedasUn-EDS
e
 means
 1/γ that r1 and r2 should satisfy the constraint
value, which can be found in Equation (12).
This
r2  ( r  r1 ) ,
(13)
as follows:
γ 1/γ
r1 rcan
The first derivative function of s andr2 r=
on
rγ −
(13)
1 ) be ,expressed as:
2 (
The first derivative function of s and
rds
as:
2 on r1 can be expressed
sr'1 
 r2  r  (r1  r )r2' r1 ,
(14)
dr1
ds
0
sr0 1 =
= r2 − r + (r1 − 1r)γr2r
,
(14)
1
dr1dr
'
 1 

γ
2
r2 r1 
 r1 ( r  r1 ) ,
(15)
1−γ
drdr
2 1
γ−1 γ
γ γ
0
r2r1 =
(15)
= −r1 (r − r1 ) ,
dr
Substitute Equation (15) into Equation1 (14), when sr'1  0 , the eliminated Un-EDS s can get the
Substitute Equation (15) into Equation
(14), when sr0 1 = 0, the eliminated
Un-EDS s can get
1/γ
0  r  (1/ 2)1/γ r , s '  0 ; when
extremum value when r1  (1/ 2) r1/γ
. Furthermore, when
the extremum value when r1 = (1/2) r. Furthermore, when 0 < 1r1 < (1/2)1/γ r, rs1 r0 1 > 0; when
1/γ
2))1/γ
r r <
r1 r1r <
2)1/γ)r1/γ
) is
sr'1 s0 0<; so
, r,
thethe
maximum
((1/
1/2
0; sos((1/
s((1/2
r ) is
maximumvalue
valueofofs.s.This
Thisconclusion
conclusionalso
also can
can be
be
r1
found in
in Figure
Figure 7.
7. In
InFigure
Figure7,7,we
weset
setthe
theinitial
initialtransmission
transmissionrange
ranger rtoto1,1,i.e.,
i.e.,rr =1 1in
inthis
thissimulation.
simulation.
found
0.95
γ=2
γ=3
γ=4
0.9
Probability (ρ)
0.85
0.8
0.75
0.7
0.65
0.6
0.55
0
0.1
0.2
0.3
0.4
0.5
0.6
Transmission range (r)
0.7
0.8
0.9
1
Figure 7. Relationship between r and ρ e for γ .
Figure 7. Relationship between r and ρe for γ.
When the probability ρ e get the maximum value, the values of r1 are 0.7r, 0.8r, and 0.85r
When the probability ρe get the maximum value, the values of r1 are 0.7r, 0.8r, and 0.85r where
ρ e in7 Figure
where
and
The maximum
value
7 is consistent
γ  3γ, =
γ  4 , respectively.
γ = 2, γγ 
=23,, and
4, respectively.
The maximum
value of
ρe inofFigure
is consistent
with the
with
the conclusion
inwhich
Table 1,
are got
from Equations
(4) and (12).
conclusion
in Table 1,
arewhich
got from
Equations
(4) and (12).
Energies 2016, 9, 841
9 of 17
Table 1. Probabilities before and after optimizing.
Value of γ
γ=2
γ=3
γ=4
Probabilities
r1
ρe
ρ
0.7071
0.7937
0.8409
0.689
0.8379
0.8996
0.5708
0.7666
0.8541
From Table 1, we can find that after eliminating some values from the Un-EDS, the probabilities of
energy efficient increase obviously: 11% when γ = 2, 7% when γ = 3, and 5% when γ = 4. Thus, in the
power adjustment based topology control algorithm, we can use (1/2)1/γ r as the optimal transmission
range of nodes.
5. Energy Efficient and Reliable Topology Control Protocol
In Section 4, we proved that the optimal transmission range for getting high probability of energy
efficient is (1/2)1/γ r. In this section, we propose an energy efficient and reliable topology control
protocol based on this conclusion.
In Section 4, we have explored the probability of energy efficient by reducing the transmission
range in power adjustment based topology control algorithm. For guaranteeing the network
connection, in this paper, we introduce the conclusions in [24] into our algorithm as the constraints of
network reliable. In [24], the authors prove that when every node connects to its nearest 5.1774logn
neighbors, the network is asymptotic connectivity (the asymptotic connectivity means that when the
number of neighbor nodes is larger than m, then the probability that the network is connected is
asymptotic to 1); when each node connects to less than 0.074logn nearest neighbors, the network is
asymptotic disconnectivity (the asymptotic disconnectivity means that when the number of neighbor
nodes is smaller than k, then the probability that the network is disconnected is asymptotic to 1).
The simulation result also shows that if the number of neighbors larger than 1.5logn, the probability of
connectedness increases rapidly to 1 for a modest number of nodes (e.g., n ≈ 30). Therefore, in ERTC,
1.5logn will be used as the lower limitation of the neighbors number, i.e., the node degree.
There are two stages in the ERTC: (1) neighbor information collection; (2) transmission
range adjustment.
5.1. Neighbor Information Collection
In this section, node i broadcast HELLO message mi using initial transmission range ri to calculate
the node degree and the distances to the neighbor nodes. As shown in Section 3, the transmission range
is a circle, but may not same for each node. The HELLO message mi includes the transmission power
Pi , the source node ID Ii , and the version number vsi which is used to decide whether the received
HELLO message is a new one or not. When the neighbor nodes receive this HELLO message, firstly,
comparing the node ID Ii in the HELLO message mi with the node IDs that in the neighbors-list; if the
node ID already exist, then check the version number vsi to find out whether this HELLO message is
a new one or not; if not, the HELLO message will be dropped immediately; otherwise, updating the
neighbor-list; in case the node ID Ii does not exist in the neighbors-list, then adding the node ID to
the neighbors-list. The distances dij between two nodes are calculated when the node i receives the
HELLO message m j from the neighbor nodes by using received signal strength indicator (RSSI) [25,26].
When the node i receive the HELLO message m j from other nodes, they will update the neighbors-list
based on the same principle which described above and calculate the node degree N1i .
As shown in Lemma 4, the optimal transmission range for node i is (1/2)1/γ ri , when the source
node i receive the HELLO message m j from the neighbor nodes, it will compare the distance dij with
(1/2)1/γ ri ; the number of neighbor nodes whose distances to the source node i are smaller than
(1/2)1/γ ri will be the node degree of node i with transmission range (1/2)1/γ ri , which is N2i .
Energies 2016, 9, 841
10 of 17
5.2. Transmission Range Adjustment
In this stage, the node i adjusts their transmission range according the node degree N1i and
N2i . As discussed in Section 3, the optimal transmission range for energy efficient is (1/2)1/γ ri and
for guaranteeing the network connection, the lower limitation of the neighbors number is 1.5logn;
therefore, for meeting the requirements of both the energy efficient and the network reliability, the
node degree N1i and N2i should be compared with 1.5logn for deciding the transmission range of node
i. There are three relationships between the node degree and the lower limitation of neighbor numbers
in ERTC: (1) N2i ≥ 1.5logn; (2) N2i ≤ 1.5logn ≤ N1i ; and (3) N1i ≤ 1.5logn; different relationships will
have different transmission range adjustment strategies:
(i) when N2i ≥ 1.5logn, it means that when the transmission range of node i is (1/2)1/γ ri , it has the
highest probability to satisfy the requirements of both the energy efficient and network connection.
Therefore, the transmission range of node i is reduced to (1/2)1/γ ri , which is reasonable.
(ii) when N2i ≤ 1.5logn ≤ N1i , this means that when the transmission range of node i is ri , the
network connection can be satisfied; however, when the transmission range is (1/2)1/γ ri , it can
not meet the requirement of network connection. As shown in Figure 7, when the transmission
range is close to (1/2)1/γ ri , the probability is close to the highest probability, too. In addition,
considering the node in ERTC is uniform distributed, so the node degree ni is proportional
with the coverage area πri2 ; therefore, the transmission range in this situation can be set to
((1.5logn)/N2i )1/2 · (1/2)1/γ ri .
(iii) when N1i ≤ 1.5logn, this means the initial transmission range of node i ri can not meet the
requirement of network connection. Therefore, the transmission range should be increased.
Similar with the reason in (ii), the transmission range closer to (1/2)1/γ ri has higher probability
of energy efficient than that far from (1/2)1/γ ri and considering the node distribution in ERTC is
uniform, so the transmission range in this range can be set to ((1.5logn)/N1i )1/2 · ri .
The process of the ERTC is:
Energy Efficient and Reliable Topology Control Algorithm (ERTC)
1. ERTC:
Input:
2. The length of the configuration area, Border_length;
3. The number of the nodes in the network, n;
4. The value of distance-power gradient, γ;
Ensure:
5. Broadcast the HELLO message mi with initial transmission range ri ;
6. Receive the HELLO message m j ;
7. Update the neighbors-list;
8. Calculate the node degree N1i ;
9. Compare the distance between node i and the neighbor nodes with (1/2)1/γ ri ;
10. Calculate the node degree N2i ;
11.
if N2i ≥ 1.5logn then
TR = (1/2)1/γ ri ;
12.
else if N2i ≤ 1.5logn ≤ N1i then
TR = ((1.5logn)/N2i )1/2 · (1/2)1/γ ri ;
13.
else
TR = ((1.5logn)/N1i )1/2 · ri ;
14.
end if
15. ri = TR;
Energies 2016, 9, 841
11 of 17
As shown in the table above, the runtime complexity for ERTC is O(n), which is the same as the
runtime complexity of LMA and LMN [17]. Therefore, the ERTC can improve the network performance
without increasing the algorithm complexity seriously.
6. Simulation and Discussion
In this section, we will evaluate the performance of ERTC and discuss the properties in detail.
ERTC is power adjustment based topology control algorithm; we compare the performance of
ERTC with LMA and LMN in this paper. The reasons why the LMA and LMN are chosen as the
contrasts are: (1) the ERTC is most similar to LMA and LMN and can be regarded as the extensional of
LMN and LMA based on the theory analysis in Section 4; (2) LMN and LMA are the two typical and
basic power adjustment based topology control algorithms. As a contrast, we use NONE (in which
there is no topology control algorithm used) as the control group.
The topology control algorithms that will be simulated in this section are as follows.
•
•
•
•
NONE: without using topology control algorithm, i.e., forming the network topology randomly
and do not control the network topology artificially.
LMA: in LMA, there are two node degree thresholds: the minimum threshold and maximum
threshold. If the node degree is smaller than the minimum threshold, the node will increase the
transmission range by certain factor Ainc ; otherwise, reducing the transmission ranges by Adec .
The nodes in which the node degrees are between the minimum threshold and the maximum
threshold will not change their transmission ranges.
LMN: in LMN, each node collects the neighbor information from their neighbors, and calculates
the average neighbors’ node degree. The value will be set as the node degree threshold. If the
node degree is large than this threshold, the transmission range will be reduced; otherwise, it will
be increased.
ERTC: the algorithm proposed in this paper.
6.1. The Properties of Energy Efficient and Reliable Topology Control Algorithm
In this section, the performance and properties of ERTC will be discussed in detail. We built
the simulation platform by MATLAB. The simulation parameters are presented as follows: (1) the
node number: 50–200; (2) distribution range: 1 km × 1 km; (3) initial transmission range: 0–200 m;
(4) distance-power gradient: γ = 3; (5) simulation time: 3000 s; (6) initial energy supply: 100 J;
(7) transmit power: 0–1 mW; (8) receive power: 0.5 mW; (9) transmission rate: 10 kbit/s.
In Figure 8, the network is formed randomly (Figure 8a) and by the ERTC (Figure 8b), respectively.
From Figure 8, we can clearly find that the ERTC reduces the number of communication links of
the original network and guarantees the network connection at the same time. The communication
links in Figure 8b are less than that in Figure 8a, which means that after using the ERTC, the energy
consumption will be reduced. The node degree in Figure 8b is obviously smaller than that in Figure 8a;
the conclusion can be found in Figure 9, too.
Figure 9 shows the node degree of the original network and the network which uses the ERTC.
From Figure 9, we can conclude that with the increasing of the node number, the overall trend of
node degree is increasing. However, the node degree does not always increase with the rising of the
node number, e.g., as shown in Figure 8, when the node number is 100, 105, 110, and 120, the node
degree does not keep increasing when the node number rises. The reason is that the network is created
randomly, so the node degree oscillates near the average node degree; however, the overall trend is
increasing. The increasing trend in ERTC is similar with the original one, but node degrees are smaller
than that. In addition, when the node number is large than 100, the increasing rate of the original
network is faster than that in ERTC. Furthermore, as shown by the blue points in Figure 9, with the
increasing of the node number, the increasing trends are different between the original network and
ERTC. The reason of this issue will be explained in the next section. Moreover, in Figure 9, the node
Energies 2016, 9, 841
12 of 17
degree
in ERTC is larger than the minimum node degree threshold, so the network connection
can be
Energies 2016, 9, 841
12 of 17
guaranteed.
This
is
consistent
with
the
conclusion
in
Figure
8b.
Energies 2016, 9, 841
12 of 17
Km Km
343
51
3
62 50
63
63
62 50
2
24
52
7
2
57 71
16
32
7 24 46
12 52
16
32 65
46
12 45
36
475965
45
36
4759
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
79
55
0
79
0.1 550.2
0
0.1
0.2
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48
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69
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73
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0.4
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20
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42
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75
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73
410.5
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0.5
Km
29
49
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15
53
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28
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0.9
28
67
0.9
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0.8
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60
6
27
9
6 31 27
78
9
31
78
74
56
13 18
Km Km
1
28
1
0.9
28
67
0.9
0.8
67
0.8
0.7
0.2
0.1
0.919
1
0.1
0
0.9
1
0
70
0.6
0.7
70
0.8
54
0.6
0.7
0.8
11
51
63
10
0.7
0.6
62 50
2
57 71
10
24
0.6
52
7
0.5
2
57 71
16
32
52 24 46
0.5
12 7
0.4
16
32 65
46
12 45
0.4
36
0.3
475965
45
36
0.3
4759
0.2
56
68
74
13 18 19
54
33
34
69
17
48
33 58
34 39
343
69
17
48
51
25
58 39 66
3
25
63
66
62 50
43
79
0
55
79
0.1 550.2
0
0.1
0.2
35
29
37
15
11
35
53
37
15
1
30
5
30
73
4
64
22
64 14
22
14 21
4
76
26
76
0.3
0.4
26
0.3
0.4
21
41
410.5
Km
0.5
Km
75
75
8
618072
6180
38
60
38
60
40 23
42
20
42
20
72
73
49
77
49
77
40 23
53
1
5
29
6
27
9
6 31 27
78
9
31
44
78
68
74 56
13 18
68
74 56
13 18 19
70
54
44
8
0.6
0.7
70
540.8
0.6
0.7
0.8
0.919
1
0.9
1
(a)
(b)
(a)
(b)
Figure 8. The simulation result: (a) NONE; and (b) energy efficient and reliable topology control
Figure 8. The simulation result: (a) NONE; and (b) energy efficient and reliable topology control
Figure
8. The
simulation
result:and
(a)Y-axis
NONE;
and the
(b) node
energy
efficient and
reliable topology control
algorithm
(ERTC).
* The X-axis
means
distribution
area.
algorithm
(ERTC). * The X-axis and Y-axis means the node distribution area.
algorithm (ERTC). * The X-axis and Y-axis means the node distribution area.
22
22
20
20
18
Node degree of orignial network
Node degree
degree of
of orignial
EERTC network
Node
Minimum
threshold
of node degree
Node
degree
of EERTC
Minimum threshold of node degree
NodeNode
Degree
Degree
18
16
16
14
14
12
12
10
10
8
8
6
6
4
4
2
50
2
50
100
100
Node Number
150
200
150
200
Node Number
Figure 9. The relationship between node number and node degree.
Figure 9. The relationship between node number and node degree.
Figure 9. The relationship between node number and node degree.
Figure 9 shows the node degree of the original network and the network which uses the ERTC.
shows
the conclude
node degree
the the
original
network
the network
which
uses thetrend
ERTC.
FromFigure
Figure9 9,
we can
thatofwith
increasing
of and
the node
number,
the overall
of
Figure
10
shows
the
numbers
of
nodes
that
use
different
transmission
range
adjustment
From
Figure
9,
we
can
conclude
that
with
the
increasing
of
the
node
number,
the
overall
trend
of
node degree is increasing. However, the node degree does not always increase with the rising strategies
of the
(introduced
in Section
5)shown
to adjust
the the
transmission
Since
the
number
of nodes
which
use
node
is e.g.,
increasing.
However,
degree
doesnumber
not
always
increase
with
the120,
rising
the the
node degree
number,
as
in Figure
8,node
when
the range.
node
is 100,
105, 110,
and
theof
node
node
number,
e.g.,
as
shown
in
Figure
8,
when
the
node
number
is
100,
105,
110,
and
120,
the
node
ruledegree
(i) is pretty
so in
Figure 10,
we show
the logarithm
value
this number.
In Figure
10,iswith
does huge,
not keep
increasing
when
the node
number rises.
Theofreason
is that the
network
degree
increasing
when
the node
number
rises.
The
thattransmission
the network
is
the increasing
ofnot
thekeep
node
number,
the
nodes
which
use
rule (i)
toreason
adjust
range
created does
randomly,
so the
node degree
oscillates
near
thethe
average
node
degree;istheir
however,
the overall
so
theincreasing
node degree
oscillates
near
the degree
average
node
degree;
however,
thedegrees
overall in
trend
is randomly,
increasing.
The
trend
in
ERTC
is
similar
with shown
the
original
one,
but
has created
the similar
increasing
trend
with
the
average
node
in Figure
9. node
Additionally,
trend
isthe
increasing.
The
increasing
trend
in ERTC
is similar
the
original
one,
but node
degrees
are 10,
smaller
than that.
addition,
number
iswith
large
than
100,
the
increasing
of(ii)
the and
Figure
number
ofInnodes
usewhen
rule the
(i)
isnode
huge,
and the
number
of nodes
that
userate
rule
are
smaller
than that.
In addition,
when
the node
number isaslarge
than
100,
the
rate
of the
1/γincreasing
original
network
is
faster
than
that
in
ERTC.
Furthermore,
shown
by
the
blue
points
in
Figure
9,
rule (iii) are quite small. Since most transmission ranges will be set to (1/2) r (which is the optimal
original
network
is
faster
than
that
in
ERTC.
Furthermore,
as
shown
by
the
blue
points
in
Figure
9,
with the increasing
the
node number,
thehigh
increasing
trendstoare
different
betweenconsumption.
the original
transmission
range), soof
the
network
will have
probability
reduce
the energy
with
the and
increasing
thereason
node number,
the increasing
trends are
different
betweenMoreover,
the original
network
ERTC.ofThe
of this issue
will be explained
in the
next section.
in
Due to the randomly formation of the network and the small number of nodes which use rule
(ii)
network
The reason
of this
issue will
explained
in node
the next
section.
Moreover,
in
Figure 9,and
the ERTC.
node degree
in ERTC
is larger
thanbethe
minimum
degree
threshold,
so the
andFigure
rule (iii),
in
Figure
10,
the
statistic
characteristics
of
these
values
are
not
regularly,
so
we
can
not
the node degree
in ERTC is This
larger
than the minimum
node degree
threshold,
network9,connection
can be guaranteed.
is consistent
with the conclusion
in Figure
8b. so the
get anetwork
clear trend
from these
data.
connection
can
be
guaranteed.
This
is
consistent
with
the
conclusion
in
Figure
8b.
Figure 10 shows the numbers of nodes that use different transmission range adjustment
TheFigure
inconformity
shown
in Figure
(by the
blue
points)
can
be
explained
by theadjustment
conclusion
10 shows
theSection
numbers
of9adjust
nodes
that
use
different
transmission
strategies
(introduced
in
5) to
the
transmission
range.
Since the range
number
of nodes in
Figure
10.
In
Figure
10,
the
number
of
communication
links
that
use
rule
(ii)
when
node
numbers
strategies
Section
5) to
the 10,
transmission
range.
Since the
number
nodes are
which use(introduced
the rule (i) isinpretty
huge,
soadjust
in Figure
we show the
logarithm
value
of this of
number.
use10,
thewith
rule0,
(i)5,increasing
isand
pretty
huge,
in Figure
we(iii),
show
logarithm
170, which
175,
and
180
are
0, respectively;
to
the10,
rule
thisthe
numbers
arevalue
6, 0,(i)of
and
0, number.
respectively.
In
Figure
the
of thesonode
number,
the
nodes
which
use the
rule
tothis
adjust
their
Figure
10, with
the
increasing
of175,
the node
thenodes
nodes
whichnode
use
the
rule (i)
to adjust
theirthan
NoteIn
that
when
the
node
number
is
therenumber,
are
more
decrease
the
transmission
transmission
range
has
the similar
increasing
trend
with
the average
degree
shown
inranges
Figure
range
has the10,
similar
increasing
trend
with
the
average
node
degree
shown
inmore
Figure
9.
Additionally,
innumbers
Figure
the170
number
of nodes
use
rule
(i)node
is huge,
and the
number
of is
nodes
that
thattransmission
when
the node
are
and 180;
and
when
the
number
is 170,
there
nodes
9.
Additionally,
in
Figure
10,
the
number
of
nodes
use
rule
(i)
is
huge,
and
the
number
of
nodes
that
increase
the transmission
ranges
than that
when
node
numbers are
175 and
soto
in Figure
(1/ 2)1/γ r9, the
use rule
(ii) and rule (iii)
are quite
small.
Sincethe
most
transmission
ranges
will 180,
be set
1/γ
use
rule (ii)
and ruletrends
(iii) areofquite
small.
Sinceare
most
transmission
ranges
willblack
be set
to (1/ 2) r
node
degree
increasing
the blue
points
different
with that
in the
points.
Energies 2016, 9, 841
13 of 17
Energies 2016,
9, 841
(which
is the optimal transmission range), so the network will have high probability to reduce the 13 of 17
energy consumption.
10
Logarithmic Number of nodes use rule (i)
Number of nodes use rule (ii)
Number of nodes use rule (iii)
9
Different kind fo Node Number
8
7
6
5
4
3
2
1
0
50
100
150
200
Node Number
Figure 10. The number of different kind of communication links in different scenario.
Figure 10. The number of different kind of communication links in different scenario.
Due to the randomly formation of the network and the small number of nodes which use rule
6.2. Compare
Performance
of 10,
Energy
Efficient
and Reliabile
Topology
Control
with
(ii) andthe
rule
(iii), in Figure
the statistic
characteristics
of these
values
are notAlgorithm
regularly, so
weOther
can
Topologynot
Control
Protocols
get a clear
trend from these data.
The inconformity shown in Figure 9 (by the blue points) can be explained by the conclusion in
In this section, the performance of ERTC will be compared with two typical transmission power
Figure 10. In Figure 10, the number of communication links that use rule (ii) when node numbers
adjustment
based
control
LMA and
LMN.
ofare
LMA
have
are 170,
175,topology
and 180 are
0, 5, algorithms:
and 0, respectively;
to the
rule The
(iii), principles
this numbers
6, 0,and
andLMN
0,
been introduced
at the
Section
respectively.
Notebeginning
that whenofthe
node 6.
number is 175, there are more nodes decrease the
transmission
rangesthat
than the
thatnumber
when the of
node
numbers are 170links
and 180;
and when
the smallest,
node number
Figure
11 indicates
communication
in ERTC
is the
which can
is 170,
there is11b.
moreIn
nodes
increase
thedifferent
transmission
ranges
when
the node numbers
arekinds of
be found
in Figure
Figure
11c,d,
color
linesthan
are that
used
to represent
different
175 and 180, so in Figure 9, the node degree increasing trends of the blue points are different with
communication links. In Figure 11c, the black links represent the communication links that have been
that in the black points.
increased, while the blue lines mean the communication links which have been reduced. Similarly, in
Figure 11d,
the black
lines indicate
that the
communication
rangeControl
are notAlgorithm
changed,
the
red lines mean
6.2. Compare
the Performance
of Energy
Efficient
and Reliabile Topology
with
Other
Topology
Control
Protocols
the communication
links
are
increased,
and
the
blue
lines
show
the
communication
links
are
reduced.
Energies 2016, 9, 841
14 of
17
Km
Km
In this section, the performance of ERTC will be compared with two typical transmission
1
1
30
51
power adjustment based topology 30control algorithms:
LMA and LMN. The principles
of LMA
and
51
0.9
77
0.9
77
68
68
39
39
69
69
72
LMN have been
introduced
at the beginning
of Section 6.
72
73
73
3
3
56
56
80
80
0.8
0.8
61
7
7
Figure 11 25
indicates
that the
number
of19communication links25
in ERTC
is the61smallest,
which
can
47
19
47
146
146
6
60
6
49
60
49
52
52
0.7
0.7
27
43
43 different kinds of
be found in Figure2811b. In27 Figure
11c,d,
different
color
lines
are
used
to
represent
31
31
33
33
28
54
54
8 74
8 74
40
40
0.6
0.6
2911 links represent the communication links that
2911have
communication links. In Figure 11c, the 16black
16
24
24
35
13
35
13
9
9
44
44
0.5
63
63
been0.5increased, while the blue
lines mean
the
communication
links
which
have
been
reduced.
20
20
34
34
0.4
0.4
70
70
Similarly,
in36Figure 11d, the black
range 66are1415not changed,
the
36
15 lines indicate that the communication
66 14
59
59
64
64
0.3 76
0.3 76
red lines
mean the communication
links are increased,
and
the blue lines show
65
10 75 the17 communication
65
10 75
17
50
57
50
42
57
42
0.2
53
53
62
71
62
links0.2are reduced.
58
71
5
58
32
5
32
23
21
26
23
21
26
0.1
2
0
37
0
0.1
79
22
18
55
67
0.3
0.2
45
0.4
0.5
Km
0.6
0.7
0.1
484
38
0.8
0.9
2
0
37
0
0.1
67
0.3
0.2
0.4
0.7
Km
35
0.5
52
61
49
43
3
80
7
25
146
16
24
44
9
20
34
36
57
62
58
42
0.2
71
21
0.1
2
55
0
18
0.1
0.2
10
75
50
59
26
23
79
45
0.4
0.5
Km
(c)
0.6
0.7
0.8
0.9
3
80
7
19
6
31
16
24
44
2911
9
20
34
0.3 76
71
2
55
0
18
0.1
0.2
70
15
14
66
10 75
50
57
62
58
42
21
0
61
49
43
63
64
0.1
1241
78
1
52
36
32
484
38
51
33
28
40
13
0.2
53
5
37
22
67
0.3
65
17
35
0.5
0.4
70
15
14
66
64
27
54
74
8
0.6
2911
73
47
60
0.7
31
63
0.3 76
1241
78
1
0.9
77
39
68
69
72
56
0.8
19
6
33
28
40
13
0.4
0
73
27
54
74
8
0.6
0.8
30
0.9
77
39
68
69
47
60
0.7
51
Km
25
146
0.6
(b)
0.9
56
0.5
Km
1
30
0.8
38
45
(a)
1
72
484
79
22
18
55
1241
78
1
59
26
53
5
37
23
79
22
67
0.3
65
17
45
0.4
0.5
Km
0.6
0.7
32
484
38
0.8
0.9
1241
78
1
(d)
Figure 11. The simulation result: (a) NONE; (b) ERTC; (c) Local Mean Neighbor (LMN); and
Figure 11. The simulation result: (a) NONE; (b) ERTC; (c) Local Mean Neighbor (LMN); and (d) Local
(d) Local Mean Algorithm (LMA). * The X-axis and Y-axis means the node distribution area.
Mean Algorithm (LMA). * The X-axis and Y-axis means the node distribution area.
Both the LMA and LMN do not take the energy consumption into consideration. Moreover,
since the maximum and minimum thresholds of node degrees in LMA are set by users without
strict definition, the network topology will change greatly under different thresholds. In ERTC, the
node degree thresholds are variation with different network conditions (such as the node numbers,
the different topology, etc.), which aims to maintain r-range instead of k-connection for the nodes.
Energies 2016, 9, 841
14 of 17
Both the LMA and LMN do not take the energy consumption into consideration. Moreover,
since the maximum and minimum thresholds of node degrees in LMA are set by users without
strict definition, the network topology will change greatly under different thresholds. In ERTC, the
node degree thresholds are variation with different network conditions (such as the node numbers,
the different topology, etc.), which aims to maintain r-range instead of k-connection for the nodes.
Furthermore, as shown in Figures 11b and 13, the protocol can meet the requirements of network
connection and energy efficient at the same time in ERTC.
Figure 12 shows the node degrees of different algorithms under different scenarios. The node
degree of ERTC increases with the increasing of node number, and the trend is similar with that of
the original network. For LMA, the node degree will keep oscillating between 8 and 11. The reason is
that the minimum and maximum node degree thresholds have been set to 8 and 11 in this simulation.
However, the performance of LMA is affected by the value of thresholds greatly. The node degree
trend is2016,
totally
is
Energies
9, 841different in LMN, it looks randomly with the increasing of the node number. This
15 of 17
because the LMN decides the node degree only based on their neighbors’ average node degree, which
is easy
to fall
into
the locally
optimalperformance
solutions. However,
overallaffected
node degree
LMN
has
stable
node
degree,
the network
of LMA isthe
seriously
by thetrend
nodeindegree
is increasing.
thresholds.
Moreover, how to set the node degree threshold has not been discussed strictly in LMA.
22
20
18
Node degree of orignial network
Node degree of EERTC
Node degree of LMN
Node degree of LMA
Node Degree
16
14
12
10
8
6
4
2
50
100
150
200
Node Number
Figure
Figure 12.
12. The
The relationship
relationship between
between node
node number
number and
and node
node degree
degree in
in four
four protocols.
protocols.
Energy Consumption
Figure 13 displays the energy consumption in different topology control algorithms. From
In Figure 12, when the node number smaller than 140, the node degree of ERTC is smaller than
Figure 13, we can conclude that the energy consumption of ERTC is the smallest in these
that of LMA; otherwise, when the node number is larger than 145, the node degree of ERTC is large
algorithms, and ERTC saves approximately 67.4% energy compare with none topology control
than LMA. The reason is that ERTC does not maintain constant node degree which is popular in the
network. The energy consumption in LMA and LMN are larger than ERTC; moreover, due to LMN
topology control algorithm to guarantee network connection. Maintaining constant neighbors can not
can maintain the node degree in a stable level, so the energy consumption is smaller than LMA.
reflect the dynamic of WSN, so it can not guarantee that the node works at the optimal transmission
However, the node degree in LMN cannot adapt the topology changing.
range which is energy efficient with high probability. As a result, in ERTC, the algorithm maintains the
r-range for the node rather than
the k-connection. The r-range is the optimal transmission range with
900
high probability of energy efficient.
In addition, although LMA has stable node degree, the network
800
performance of LMA is seriously affected by the node degree thresholds. Moreover, how to set the
700
node degree threshold has not been discussed strictly in LMA.
600
Figure 13 displays the energy consumption in different topology control algorithms.
500
From Figure 13, we can conclude
that the energy consumption of ERTC is the smallest in these
algorithms, and ERTC saves 400
approximately 67.4% energy compare with none topology control network.
The energy consumption in LMA
and LMN are larger than ERTC; moreover, due to LMN can maintain
300
the node degree in a stable level,
so
the energy consumption is smaller than LMA. However, the node
200
degree in LMN cannot adapt the topology changing.
100
0
ERTC
LMN
LMA
Topology control algorithms
NONE
Figure 13. The energy consumption of different topology control protocols.
1/γ
In Section 4, we proved that when the transmission range of node i is (1/ 2) ri , then the
probability that the communication link is energy efficient is highest than others. Correspondingly,
Figure 13 displays the energy consumption in different topology control algorithms. From
Figure 13, we can conclude that the energy consumption of ERTC is the smallest in these
algorithms, and ERTC saves approximately 67.4% energy compare with none topology control
network. The energy consumption in LMA and LMN are larger than ERTC; moreover, due to LMN
can maintain
the node degree in a stable level, so the energy consumption is smaller than 15
LMA.
Energies
2016, 9, 841
of 17
However, the node degree in LMN cannot adapt the topology changing.
900
800
Energy Consumption
700
600
500
400
300
200
100
0
ERTC
LMN
LMA
Topology control algorithms
NONE
Figure
topology control
control protocols.
protocols.
Figure 13.
13. The
The energy
energy consumption
consumption of
of different
different topology
1/γ
In Section 4, we proved that when the transmission range of node i is (1/ 2) r , then the
In Section 4, we proved that when the transmission range of node i is (1/2)1/γ rii , then the
probability that
that the
the communication
communication link
is highest
highest than
than others.
others. Correspondingly,
probability
link is
is energy
energy efficient
efficient is
Correspondingly,
1/γ
1/γr consumes less energy than
(1/
2)
in
Figure
13,
the
ERTC
whose
transmission
range
is
related
to
in Figure 13, the ERTC whose transmission range is related to (1/2) ri i consumes less energy than
other topology control algorithms. This demonstrates
demonstrates that
that the conclusion
conclusion that we get in Section 4 is
effective in
inreducing
reducingthe
theenergy
energy
consumption
WSN.
Additionally,
as shown
in when
[24], when
the
effective
consumption
forfor
WSN.
Additionally,
as shown
in [24],
the node
node degree
is than
larger1.5logn,
than 1.5log
is connected
high probability
1). 8b
In
degree
is larger
the network
is connected
with highwith
probability
(nearly 1). (nearly
In Figures
n , the network
and
9,
when
the
node
degree
in
ERTC
is
larger
than
1.5logn,
then
the
network
is
connected
as
shown
Figures 8b and 9, when the node degree in ERTC is larger than 1.5log n , then the network in
is
Figure
8b.
Therefore,
the
simulation
results
show
that
the
theoretical
analyses
are
correct
and
effective.
connected as shown in Figure 8b. Therefore, the simulation results show that the theoretical
analyses are correct and effective.
7. Conclusions
In this paper, for diminishing the energy consumption of WSN as far as possible, first, we
propose a probability model for energy efficient in WSN; second, we analyze the properties of the
probability model and find out the optimal transmission range; finally, we propose an ERTC based on
the conclusions in Section 4, which maintains r-range for the nodes instead of k-connection. To the
best of our knowledge, this paper is the first one that provides the strict mathematic analysis model
for these kinds of issues. In addition, in this paper, we also investigate the performance of ERTC
and compare the performance with LMN and LMA, which are the typical power adjustment based
topology control algorithms.
We have proved that when the transmission ranges are changed, the probability of energy efficient
Rr
1
is ρ = r22 0 (rγ − xγ ) γ dx − 1, which varies with the distance-power gradient γ and keeps constant
with different r, i.e., the probability is nothing to do with the initial transmission range. The probability
varies from 0.5708 to 0.8541. We have also proved that when the transmission range is (1/2)1/γ r, the
probability ρ will get the maximum value. In this paper, we also propose an ERTC algotithm, which
can guarantee both the energy efficient and network connection at the same time.
Acknowledgments: The research leading to the presented results has been undertaken within the SWARMs
European Project (Smart and Networking Underwater Robots in Cooperation Meshes), under Grant Agreement
No. 662107-SWARMs-ECSEL-2014-1. The China Scholarship Council (CSC), which has partially supported the
first author’s research described in this paper, is also acknowledging as a fund contributor.
Author Contributions: Ning Li and Jose-Fernan Martinez conceived and designed the experiments; Ning Li,
Jose-Fernan Martinez, Lourdes Lopez Santidrian and Juan Manuel Meneses Chaus performed the experiments;
Ning Li, Jose-Fernan Martinez, Lourdes Lopez Santidrian and Juan Manuel Meneses Chaus analyzed the data;
Ning Li and Jose-Fernan Martinez contributed analysis tools; Ning Li and Jose Fernan Martinez wrote the paper.
Conflicts of Interest: The authors declare no conflict of interest.
Energies 2016, 9, 841
16 of 17
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