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Journal ArticleDOI

A minimum cost heterogeneous sensor network with a lifetime constraint

01 Jan 2005-IEEE Transactions on Mobile Computing (IEEE)-Vol. 4, Iss: 1, pp 4-15
TL;DR: A heterogeneous sensor network in which nodes are to be deployed over a unit area for the purpose of surveillance is considered, finding optimum node intensities and node energies that guarantee a lifetime of at least T units, while ensuring connectivity and coverage of the surveillance area with a high probability.
Abstract: We consider a heterogeneous sensor network in which nodes are to be deployed over a unit area for the purpose of surveillance. An aircraft visits the area periodically and gathers data about the activity in the area from the sensor nodes. There are two types of nodes that are distributed over the area using two-dimensional homogeneous Poisson point processes; type 0 nodes with intensity (average number per unit area) /spl lambda//sub 0/ and battery energy E/sub 0/; and type 1 nodes with intensity /spl lambda//sub 1/ and battery energy E/sub 1/. Type 0 nodes do the sensing while type 1 nodes act as the cluster heads besides doing the sensing. Nodes use multihopping to communicate with their closest cluster heads. We determine them optimum node intensities (/spl lambda//sub 0/, /spl lambda//sub 1/) and node energies (E/sub 0/, E/sub 1/) that guarantee a lifetime of at least T units, while ensuring connectivity and coverage of the surveillance area with a high probability. We minimize the overall cost of the network under these constraints. Lifetime is defined as the number of successful data gathering trips (or cycles) that are possible until connectivity and/or coverage are lost. Conditions for a sharp cutoff are also taken into account, i.e., we ensure that almost all the nodes run out of energy at about the same time so that there is very little energy waste due to residual energy. We compare the results for random deployment with those of a grid deployment in which nodes are placed deterministically along grid points. We observe that in both cases /spl lambda//sub 1/ scales approximately as /spl radic/(/spl lambda//sub 0/). Our results can be directly extended to take into account unreliable nodes.

Summary (3 min read)

1 INTRODUCTION

  • SENSOR networks are dense networks of low cost,wireless nodes that sense certain phenomena in the area of interest and report their observations to a central base station for further analysis.
  • Type 0 nodes do the basic sensing as well as the relaying of packets since multihop communication is used within each cluster.
  • Each visit of the aircraft triggers a sensing and data gathering cycle on the ground during which every node sends a packet to its cluster head.
  • {mhatre, cath, mazum, shroff}@ecn.purdue.edu., also known as E-mail.

3 PROBLEM FORMULATION AND MODELS USED

  • There are two types of nodes; nodes of energy level E0 deployed Authorized licensed use limited to: University of Waterloo.
  • Thus, the aircraft serves as the remote base station.
  • For the sake of simplicity, in the rest of this paper, the authors assume that all the clusters send information to the base station during every data gathering cycle.
  • In the second scenario, nodes are deterministically placed along grid points.

3.1 Cost Model

  • Let C0 and C1 be the cost per node for each type of node.
  • Then, a simple model for a cost function is,.

Ci ¼ i þ Ei;

  • Where i and are some constants that depend on the manufacturing process.
  • The constant i is the cost of the hardware of a type i node (excluding the battery cost), while is the proportionality constant for the battery cost.

3.2 Connectivity and Coverage Model

  • For the sensor network to provide sensing coverage of the region and for the nodes to successfully use multihop communication, it is necessary to ensure that the conditions for node connectivity and area coverage be met.
  • Using a similar approach, the authors can study the case in which nodes are deployed over a unit area with a two-dimensional homogeneous Poisson point process.
  • The authors restrict ourselves to omni- directional sensing in order to keep the model generic.
  • For simplicity, the authors assume that r is also the critical distance between any two nodes for successful transmission.

3.3 Lifetime Constraint Model

  • This is because the critical nodes are the last hop nodes for all the paths (see Fig. 1).
  • Hence, among all the type 0 nodes in a cluster, the critical nodes have the highest burden of relaying data.
  • In order to have a sharp cutoff effect, the authors also require that almost all the nodes in the network expire at about the same time.
  • This ensures that very little residual energy is left behind when the system becomes unusable, i.e., when coverage and/or connectivity are lost.
  • Let P0 be the average energy spent by a typical critical node during each cycle, and let P1 be the average energy spent by a typical cluster head during each cycle.

3.4 Energy Model

  • This consists of energy spent on relaying packets of other nodes that are in the same cell (Pr0 ), and transmitting one’s own data (Et0 per packet).
  • The authors also assume that during each data gathering cycle a type 0 node sends one packet of its own to its cluster head.
  • This consists of energy spent on receiving data from other nodes in the cell (Er0 per packet), processing and compressing the received data (Ef per packet), and transmitting the compressed data to the aircraft (Et1 per packet).
  • The authors also assume that the cluster heads coordinate MAC and routing in their respective clusters so that no energy is wasted on packet collisions or idle listening (ideal MAC assumption).

4.1 Random Deployment

  • For that, the authors need some results from stochastic geometry [14].
  • Since type 0 nodes as well as type 1 nodes are deployed using a homogeneous Poisson point process, the authors can shift the origin to one of the type 1 points and use Campbell’s theorem and Slivnyak’s theorem [16] to compute the expected number of type 0 nodes in a typical Voronoi cell.
  • The authors have introduced the constants c0, c1, c2, and c3 for ease of notation.
  • The authors assume a propagation loss model of 1x 2 for communication between a node and its cluster head, and 2x 4 for communication between the cluster heads and the base station.

4.2 Grid Deployment

  • Now, the authors consider a simple grid of nodes in which nodes are placed along grid points with distance r between them.
  • The authors would like to compare the results that they get for the random node deployment scenario with those for the grid deployment scenario.
  • The proof is along similar lines as before and is therefore skipped.
  • As the energy spent in countering the propagation loss to communicate with the aircraft ( Hk) is much larger than the other energy terms ( rk, l, Ef ), c2 dominates over other ci (see (14) and (15)).
  • If the nodes are unreliable and the authors use the random deployment model, all their results are still valid with minor modifications because even in this case, the coverage-connectivity constraint retains the same log = form .

5 NUMERICAL RESULTS

  • The authors provide justifications for the approximations that they made in obtaining (23) by using some typical transceiver radio parameters.
  • Hence, for the above settings, all the distances should be divided by 10km.
  • These are typical values for practical surveillance networks.
  • From Fig. 2, it is clear that the approximation works quite well for the settings of practical interest since the curves corresponding to the exact and the approximate solutions overlap.
  • As H decreases, the communication between the cluster heads and the aircraft becomes less expensive.

6 CONCLUSIONS

  • The authors consider two types of hierarchical sensor networks: one that uses random uniform deployment and the other that uses grid deployment.
  • Type 0 nodes have energy level E0 and are deployed with intensity 0, while type 1 nodes have energy level E1 and are deployed with intensity 1.
  • The authors provide results that guarantee a minimum lifetime (i.e., at least T successful data gathering trips) of the sensor network.
  • For this, the authors note that the cluster heads as well as the nodes within one hop of the cluster heads, i.e., the critical nodes have the maximum relaying burden and, therefore, these nodes are likely to run out of battery before other nodes.
  • The authors compare the results for the random deployment with those of the grid deployment.

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A Minimum Cost Heterogeneous Sensor
Network with a Lifetime Constraint
Vivek P. Mhatre, Student Member, IEEE, Catherine Rosenberg, Senior Member, IEEE,
Daniel Kofman, Member, IEEE, Ravi Mazumdar, Senior Member, IEEE, and
Ness Shroff, Senior Member, IEEE
Abstract—We consider a heterogeneous sensor network in which nodes are to be deployed over a unit area for the purpose of
surveillance. An aircraft visits the area periodically and gathers data about the activity in the area from the sensor nodes. There are two
types of nodes that are distributed over the area using two-dimensional homogeneous Poisson point processes; type 0 nodes with
intensity (average number per unit area)
0
and battery energy E
0
; and type 1 nodes with intensity
1
and battery energy E
1
. Type 0
nodes do the sensing while type 1 nodes act as the cluster heads besides doing the sensing. Nodes use multihopping to communicate
with their closest cluster heads. We determine the optimum node intensities (
0
,
1
) and node energies (E
0
, E
1
) that guarantee a
lifetime of at least T units, while ensuring connectivity and coverage of the surveillance area with a high probability. We minimize the
overall cost of the network under these constraints. Lifetime is defined as the number of successful data gathering trips (or cycles) that
are possible until connectivity and/or coverage are lost. Conditions for a sharp cutoff are also taken into account, i.e., we ensure that
almost all the nodes run out of energy at about the same time so that there is very little energy waste due to residual energy. We
compare the results for random deployment with those of a grid deployment in which nodes are placed deterministically along grid
points. We observe that in both cases
1
scales approximately as
ffiffiffiffi
0
p
. Our results can be directly extended to take into account
unreliable nodes.
Index Terms—Sensor networks, energy, lifetime, stochastic geometry, Voronoi cells.
æ
1INTRODUCTION
S
ENSOR networks are dense networks of low cost,
wireless nodes that sense certain phenomena in the
area of interest and report their observations to a central
base station for further analysis. Sensor networks have
been predicted to have a wide range of applications in
both civilian as well as military domains [2]. An important
application of sensor networks is surveillance of a battle-
field or sensitive borders of countries. A simple way to
monitor such areas is to deploy sensors. Deployment
could either be deterministic, i.e., placing the nodes along
grid points (may not be practical in general), or the nodes
could be deployed randomly (e.g., from an aircraft). In
this paper, we study a scenario in which an aircraft
(possibly unmanned) or a LEO satellite passes over these
areas periodically and collects updates from the deployed
nodes. Thus, in the above scenario, the aircraft acts as the
(mobile) base station. We assume that the nodes are
organized as clusters and the cluster heads perform data
aggregation. Data aggregation is used because it reduces
the amount of data that is sent from the sensor nodes to
the base station and, thereby, improves energy efficiency
of the network [3], [4].
We consider a heterogeneous network with two types of
nodes, type 0 nodes deployed with intensity
0
and battery
energy E
0
, and type 1 nodes deployed with intensity
1
and
battery energy E
1
. Type 0 nodes do the basic sensing as well
as the relaying of packets since multihop communication is
used within each cluster. Type 1 nodes are the cluster
heads. They do data fusion within each cluster and directly
transmit the aggregated data to the aircraft. Type 1 nodes
also participate in sensing. Since type 0 nodes communicate
over short range, their hardware requirements are simple.
On the other hand, type 1 nodes perform long range
transmissions to the aircraft, perform data aggregation, and
coordinate MAC and routing within the clusters. Hence, the
type 1 nodes have more complex hardware than the type 0
nodes. We first assume that all the nodes are reliable. We
show later that our analysis can be easily extended to the
case of unreliable nodes.
Each visit of the aircraft triggers a sensing and data
gathering cycle on the ground during which every node
sends a packet to its cluster head. This model of periodic
monitoring is different from an event detection sensor
network where nodes send data only when an event occurs.
The total battery energy of each node is finite and, so, there
is a limit on the number of successful data gathering trips
that the aircraft can make. Our objective is to determine the
optimum node deployment parameters that will ensure a
certain minimum number of successful data gathering
cycles before the sensor system becomes unusable, i.e.,
connectivity and/or coverage can no longer be ensured.
Each type of node has a cost function associated with it that
takes into account its hardware and battery cost. We
4 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 4, NO. 1, JANUARY/FEBRUARY 2005
. V.P. Mhatre, C. Rosenberg, R. Mazumdar and N. Shroff are with the
School of Electrical and Computer Engineering, Purdue University, West
Lafayette, IN 47907.
E-mail: {mhatre, cath, mazum, shroff}@ecn.purdue.edu.
. D. Kofman is with ENST, 46, rue Barrault 75634 Paris Cedex 13 France.
E-mail: daniel.kofman@enst.fr.
Manuscript received 4 Mar. 2003; revised 25 Sept. 2003; accepted 10 Jan.
2004; published online 1 Dec. 2004.
For information on obtaining reprints of this article, please send e-mail to:
tmc@computer.org, and reference IEEECS Log Number TMC-0032-0303.
1536-1233/05/$20.00 ß 2005 IEEE Published by the IEEE CS, CASS, ComSoc, IES, & SPS
Authorized licensed use limited to: University of Waterloo. Downloaded on February 25, 2009 at 10:05 from IEEE Xplore. Restrictions apply.

formulate an optimization problem with the above men-
tioned constraints and find a solution that minimizes the
overall cost of the network. We dimension the network (in
terms of cluster head intensity and battery energies of the
nodes) so that both types of nodes expire at about the same
time in order to reduce waste of residual energy. We study
two deployment scenarios; namely, grid and random
deployment and obtain results for
0
, E
0
,
1
, and E
1
for
both scenarios. We observe that, in both scenarios,
1
, i.e.,
the cluster head intensity scales approximately as
ffiffiffiffi
0
p
.
This paper is organized as follows: In Section 2, we
discuss some of the related work. In Section 3, we first
discuss t he models that we use for communication,
connectivity and coverage, lifetime, etc., and then formulate
the design problem. In Section 4, we provide an exact
solution to the problem and then obtain an approximate
solution for some typical radio parameter settings. Section 5
contains some numerical results that validate the approx-
imations that we make in our analysis and give insights on
the nature of solutions that we get. We conclude in Section 6.
The proofs of some of the results that we use in Sections 3
and 4 are provided in Appendices A and B.
2RELATED WORK
In [12], Heinzelman et al. consider a homogeneous clustered
network in which each cluster head collects data from its
one hop neighbors, aggregates the gathered data, and
transmits it directly to the remote base station. The cluster
heads are periodically rotated for efficient load balancing.
All the nodes in the network are identical, and there is no
multihop communication. In this scenario, the authors
provide results supporting their idea that, even though
the cluster heads have the highest energy drainage rate,
periodic rotation ensures good load balancing and, hence, a
higher lifetime. It has also been shown in [12] that, in order
to minimize the total energy spent in the network, the
required number of cluster heads has to scale as the square
root of the total number of sensor nodes; a result similar to
what we obtain.
The scenario that we study consists of a multihop
network with heterogeneous nodes. In [12], cluster head
rotation requires that all the nodes be capable of performing
data aggregation as well as long range transmissions to the
remote base station. This results in extra hardware complex-
ity in all the nodes. In contrast, in our approach, the above
complexity is embedded in only a few nodes (type 1 nodes).
As a result, the vast majority of the nodes (type 0 nodes)
have less complex hardware. Also, in our system, since the
cluster heads are predetermined and fixed, there is no need
for a cluster head election protocol.
In [11], Bandyopadhyay and Coyle consider a homo-
geneous sensor network in which nodes are uniformly
deployed using a two-dimensional Poisson point process
over a unit area. Cluster heads are chosen from among these
nodes randomly with a probability p. The nodes send their
data to their closest cluster head node. This leads to the
formation of Voronoi cells wherein the cluster heads are the
nuclei of the cells. The nodes use multihopping to
communicate with the cluster heads, and the cluster heads
use multihopping to communicate with the base station.
The authors find an expression for the expected value of the
total energy that is spent in the network during each data
gathering cycle. The authors minimize this total energy to
determine an optimum value of p.
In [7], Bhardwaj et al. provide loose bounds on the
lifetime of a sensor network. In [20], the authors provide
upper bounds on the lifetime of a sensor network by taking
into account all the possible collaborative data gathering
strategies over all the possible network routes. However,
Bhardwaj and Chandrakasan [20] do not provide any
practical ways to achieve these bounds. In [10], Kalpakis
et al. provide bounds on the lifetime of a sensor network by
assuming a communication model in which any node can
communicate with any other node, as well as with the base
station; clearly, a limited model. In [8], Chiasserini et al.
determine an optimum strategy to allocate sensor nodes to
different clusters so as to improve the system lifetime. In
[22], Meguerdichian et al. study the coverage problem in a
wireless sensor network by using Voronoi diagram tools.
The authors propose an algorithm to determine a maximal
breach path for a given sensor network topology. In [19],
Zussman and Segall study energy-efficient routing in
emergency sensor networks by using the principles of
network flows. In [21], Cerpa and Estrin propose using
redundant nodes to improve the connectivity of the net-
work. Nodes use a distributed protocol to determine if they
need to stay awake to improve the network connectivity. In
[18], Gupta and Younis study a heterogeneous sensor
network (with two types of nodes) and study the impact
on clustering of node failure at the higher level of the node
hierarchy.
In our approach, we observe that, in a sensor network,
energy drainage is not uniform over the entire network. The
cluster heads have the highest energy burden due to the
long-range transmissions to the base station. Also, the nodes
close to the cluster heads have a high energy burden due to
relaying of packets. Hence, these nodes are likely to expire
before other nodes. To determine a bound on the lifetime of
the network, these observations have to be taken into
account. Besides, it is also important to ensure a sharp cutoff
effect, i.e., to ensure that almost all the nodes expire at about
the same time when the network becomes unusable. This
guarantees that there is very little residual energy left
behind. The above related papers mainly focus on homo-
geneous networks and do not take all the above factors into
account. We also note that in a heterogeneous network the
right objective function to minimize is not the overall battery
energy, but the overall network cost. This cost takes into
account the cost of the battery as well as the node hardware.
We also take into account the conditions for connectivity
and coverage of the area by extending the results in [13]. In
our approach, we model all the above factors as constraints
of an optimization problem. Our results can be directly
extended to the case of unreliable nodes.
3PROBLEM FORMULATION AND MODELS USED
A circular disk of unit area is to be covered with sensor
nodes. If the region to be covered has an area A, all the
distances are normalized by dividing them by
ffiffiffi
A
p
. There
are two types of nodes; nodes of energy level E
0
deployed
MHATRE ET AL.: A MINIMUM COST HETEROGENEOUS SENSOR NETWORK WITH A LIFETIME CONSTRAINT 5
Authorized licensed use limited to: University of Waterloo. Downloaded on February 25, 2009 at 10:05 from IEEE Xplore. Restrictions apply.

with intensity
0
and nodes of energy level E
1
deployed
with intensity
1
. A surveillance aircraft flying at an altitude
of H sweeps the area periodically and triggers a data
sensing cycle during which all the type 0 nodes in each
cluster send their data to the closest type 1 node using
multihop communication. The type 1 nodes aggregate the
received data and then send it to the aircraft using a direct
transmission. Thus, the aircraft serves as the remote base
station. In this paper, we assume that the base station
receives updates from every cluster. However, if the base
station is interested in receiving updates from only a few
clusters (extra sensitive regions), then our analysis can be
modified as follows: Deploy more nodes over the regions of
frequent updates and take these nodes into account in the
overall network cost. The redundant nodes stay inactive
while the battery energy of other nodes lasts and join the
cluster when other nodes start to expire. A similar problem
in which redundant nodes are used to improve network
connectivity is studied in [21]. Thus, it is easy to modify our
approach when frequent updates are desired from certain
clusters. For the sake of simplicity, in the rest of this paper,
we assume that all the clusters send information to the base
station during every data gathering cycle.
There are two ways to deploy the nodes. In the first
scenario (which is more realistic), nodes are thrown from an
aircraft and this can be modeled using a two-dimensional
homogeneous Poisson point process for each type of nodes.
In the second scenario, nodes are deterministically placed
along grid points. In the case of random deployment,
0
and
1
are the intensities of the corresponding (independent)
Poisson point processes. In the case of random deployment,
clustering leads to the formation of Voronoi cells with type
1 nodes being the nuclei of these cells. In the case of a grid,
0
and
1
are simply the number of type 0 and type 1 nodes
that are used. The topology will consist of
1
equispaced
type 1 nodes and
0
equispaced type 0 nodes placed along
the grid points.
We now describe the various models that we use in our
analysis, and we then formulate the design problem.
3.1 Cost Model
Let C
0
and C
1
be the cost per node for each type of node.
Then, a simple model for a cost function is,
C
i
¼
i
þ E
i
;
where
i
and are some constants that depend on the
manufacturing process. The constant
i
is the cost of the
hardware of a type i node (excluding the battery cost),
while is the proportionality constant for the battery cost.
The overall cost of the network as a function of the vector
½
;
EE¼½
0
;
1
;E
0
;E
1
is
Cð
;
EEÞ¼
0
0
þ
1
1
þ ð
0
E
0
þ
1
E
1
Þ: ð1Þ
3.2 Connectivity and Coverage Model
For the sensor network to provide sensing coverage of the
region and for the nodes to successfully use multihop
communication, it is necessary to ensure that the conditions
for node connectivity and area coverage be met. Results for
connectivity and coverage of an area when unreliable nodes
are placed along grid points have been obtained in [13].
Using a similar approach, we can study the case in which
nodes are deployed over a unit area with a two-dimensional
homogeneous Poisson point process. We assume that the
sensing radius of each node is r. In reality, the model for
node sensing coverage is highly application dependent. For
example, there are scenarios in which the sensor nodes can
only sense in a certain direction as in the case of certain
acoustic sensors. However, we restrict ourselves to omni-
directional sensing in order to keep the model generic. For
simplicity, we assume that r is also the critical distance
between any two nodes for successful transmission. This
distance r depends on the allowable signal to noise ratio for
successful packet reception, modulation scheme, propaga-
tion loss exponent, etc. If the communication and sensing
radii are different, we can use a similar approach to
determine conditions for connectivity and coverage with
some modifications (see [13]).
Let be the intensity of the Poisson process and p be the
reliability probability of each node. The probability of
connectedness of nodes and coverage of area is given as
follows (see Appendix A).
P ðnetwork is connected and region is coveredÞ
1
1
r

2
e

2
pr
2
8; > 0 such that þ 2 ¼ 1:
ð2Þ
The result in (2) is similar in form to the result that the
authors obtained in [13]. In [13], node placement is
deterministic but randomness is induced due to node
failure, while, in our scenario, randomness is due to the
deployment process. Note that the result stated above is
valid for r<<1, i.e., >>1. So, all the results are
applicable only when the dimensions of the area are much
larger than r. In practical applications where an area is to be
covered by sensors, r<<1 indeed holds. This is because we
are interested in deploying a large number of nodes, each
having a coverage area that is much smaller than the total
area of the region.
For simplicity, we assume that all the nodes are reliable,
i.e., p ¼ 1. The case of unreliable nodes is not fundamentally
different since the functional form of the above relation is
not affected. With unreliable nodes, we simply have to
replace by p (the resultant thinned process is also a
Poisson process). In our scenario, there are two types of
nodes with intensity
0
and
1
and both deployment
processes are independent homogeneous Poisson point
processes. Assuming that both types of nodes do the
sensing and have the same sensing radius r ,
P ðnetwork is connected and region is coveredÞ
1
1
r

2
e

2
pr
2
ð
0
þ
1
Þ
:
ð3Þ
8; > 0 such that þ 2 ¼ 1. In a network dimensioning
problem, the designers provide a parameter such that the
probability of connectivity and coverage be at least 1 .
Therefore, we require
6 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 4, NO. 1, JANUARY/FEBRUARY 2005
Authorized licensed use limited to: University of Waterloo. Downloaded on February 25, 2009 at 10:05 from IEEE Xplore. Restrictions apply.

0
þ
1
1

2
pr
2

log
1
ðrÞ
2
!
: ð4Þ
8; > 0 such that þ 2 ¼ 1 . We assume that r
2
< 1.
Otherwise, (4) holds trivially for any positive value of
0
þ
1
. This scenario corresponds to the case when is
large which is not of much interest to us. When r
2
< 1, the
right-hand side of (4) can be minimized as a function of
under the constraint of þ2 ¼ 1. This is because the right-
hand side can be rewritten as a function of only (by
eliminating )
uðÞ¼
1

2
pr
2

log
1
ðð1 2ÞrÞ
2
!
: ð5Þ
Note that uðÞ approaches 1 as approaches 0 as well as
1=2. Hence, there is a point in between where uðÞ is
minimized since uð:Þ is continuous and lower bounded by 0.
In fact, we can show that there is a unique point where the
function takes this minimum value. The proof is technical
and, therefore, has been omitted. If
0
is the point where this
minimum is attained (and correspondingly
0
þ 2
0
¼ 1),
then ensuring
0
þ
1
uð
0
Þ is sufficient for (4) to be true.
Note that this
0
depends only on , p, and r which are
constants.
Hence, the constraint in (4) reduces to a constraint of the
form
0
þ
1
uð
0
Þ¼a; ð6Þ
where a is completely determined by , p, and r.
Note that, in the grid case, the required number of nodes
is exactly
0
þ
1
(all the nodes are assumed to be reliable).
Hence, the connectivity-coverage requirement for a unit
area takes the simple form:
0
þ
1
¼ 1=r
2
: ð7Þ
3.3 Lifetime Constraint Model
We call those type 0 nodes which are within a distance r
from a cluster head as the critical nodes. Since the
transmission radius of each type 0 node is r, we observe
that every transmission of a type 0 node to its cluster head
has to go through one of these critical nodes. This is because
the critical nodes are the last hop nodes for all the paths (see
Fig. 1). Hence, among all the type 0 nodes in a cluster, the
critical nodes have the highest burden of relaying data. As a
result, these nodes are likely to exhaust their battery energy
before other type 0 nodes. When critical nodes in a cluster
expire, connectivity is lost. We also observe that a cluster
head has a high energy burden due to the long range
transmissions to communicate with the aircraft, as well as
due to the computations that it has to perform during data
aggregation. Hence, the energy drainage rates of the critical
nodes and the cluster heads determine the lifetime of the
system. In light of these facts, it is natural to define the
lifetime of the system to be the number of cycles until which
all the cluster heads as well as all the critical nodes are
active. In order to have a sharp cutoff effect, we also require
that almost all the nodes in the network expire at about the
same time. This ensures that very little residual energy
(waste) is left behind when the system becomes unusable,
i.e., when coverage and/or connectivity are lost. We cannot
ensure a sharp cutoff for all the nodes in the network due to
the inherent nonuniform nature of energy drainage in a
cluster. For example, the type 0 nodes that are near the
periphery of a cluster have very little relaying to do and, so,
whenever the critical nodes or the cluster heads expire, the
residual energy in the peripheral nodes is wasted. How-
ever, this is inevitable. The best we can do is to ensure that
the two types of nodes which determine the lifetime of the
system, i.e., the cluster heads and the critical nodes, expire
at about the same time.
Let P
0
be the average energy spent by a typical critical
node during each cycle, and let P
1
be the average energy
spent by a typical cluster head during each cycle. Then,
E
0
=P
0
(respectively, E
1
=P
1
) is the average number of cycles
that the critical nodes (respectively, the cluster heads) can
sustain. For a sharp cutoff effect, we require:
E
1
P
1
¼
E
0
P
0
:
To ensure a lifetime of at least T cycles, we require
E
1
P
1
¼
E
0
P
0
T: ð8Þ
3.4 Energy Model
Let P
0
be the amount of energy spent by a critical node
during one cycle. This consists of energy spent on relaying
packets of other nodes that are in the same cell (P
r
0
), and
transmitting one’s own data (E
t
0
per packet). We assume
that type 0 nodes do not perform data fusion. We also
assume that during each data gathering cycle a type 0 node
sends one packet of its own to its cluster head. Let P
1
denote
the amount of energy spent by a type 1 node during a single
data gathering cycle. This consists of energy spent on
receiving data from other nodes in the cell (E
r
0
per packet),
processing and compressing the received data (E
f
per
packet), and transmitting the compressed data to the
aircraft (E
t
1
per packet). We assume the radio model used
in [12] wherein the energy required to transmit a packet
over distance x is l þ x
k
. l is the constant per packet energy
spent in the transmitter electronics circuitry while x
k
is the
energy spent in the RF amplifier to counter the propagation
loss. The energy required to receive a packet is just l. Hence,
E
t
1
¼ l þ H
k
;E
t
0
¼ l þ r
k
;E
r
0
¼ l: ð9Þ
MHATRE ET AL.: A MINIMUM COST HETEROGENEOUS SENSOR NETWORK WITH A LIFETIME CONSTRAINT 7
Fig. 1. A typical Voronoi cell.
Authorized licensed use limited to: University of Waterloo. Downloaded on February 25, 2009 at 10:05 from IEEE Xplore. Restrictions apply.

We also assume that the cluster heads coordinate MAC
and routing in their respective clusters so that no energy
is wasted on packet collisions or idle listening (ideal
MAC assumption). A cluster head performs fusion of the
data packets that it receives from all the sensors in its
cluster, and transmits a single packet to the aircraft
during each cycle.
P
0
¼ E
t
0
þ P
r
0
; ð10Þ
P
1
¼ E½N
v
ðE
r
0
þ E
f
ÞþE
t
1
; ð11Þ
where E
f
is the processing energy spent on data fusion. We
denote by E½N
v
the expected number of type 0 nodes in a
typical cluster. Note that the type 0 nodes use multihopping
to communicate with the cluster head, but the cluster head
still has to receive as many as E½N
v
packets during each
cycle since no aggregation is performed at the type 0 nodes.
3.5 Problem Statement
We would like to determine
xx ¼½
0
;
1
;E
0
;E
1
, the para-
meters of the minimum cost network so that a lifetime of at
least T cycles is guaranteed while ensuring connectivity and
coverage with a probability of at least 1 . Hence, we have
the following optimization problem for the random
deployment scenario:
minimize
0
0
þ
1
1
þ ð
0
E
0
þ
1
E
1
Þ
subject to
0
þ
1
a
E
1
P
1
¼
E
0
P
0
T:
For a grid deployment, the problem formulation is along
similar lines with (7) as the connectivity-coverage con-
straint. Note that P
0
and P
1
are functions of
0
and
1
and
are yet to be determined.
4SOLUTIONS FOR THE TWO DEPLOYMENT
SCENARIOS
4.1 Random Deployment
We begin the solution by first determining an expression for
P
r
0
, i.e., the energy spent by a critical type 0 node to relay
packets. For that, we need some results from stochastic
geometry [14].
When the deployment is random, each cluster is a
Voronoi cell because sensor nodes choose the closest type 1
node as their cluster head. We first find the expected
number of critical nodes in a typical Voronoi cell. This
number is simply the expected number of type 0 nodes in a
circle of radius r around a type 1 node. We then find the
expected number of type 0 nodes in that Voronoi cell that lie
outside this circle of radius r. From this, we can find the
average relaying load on a critical node.
We use the approach used in [14] to determine the
expected number of type 0 nodes in a cluster, as well as the
expected number of critical type 0 nodes. Let ð
1
Þ denote
the sigma algebra generated by the point process corre-
sponding to the type 1 nodes. Since type 0 nodes as well as
type 1 nodes are deployed using a homogeneous Poisson
point process, we can shift the origin to one of the type 1
points and use Campbell’s theorem and Slivnyak’s theorem
[16] to compute the expected number of type 0 nodes in a
typical Voronoi cell. Let C
0
denote the Voronoi cell of a
typical type 1 node located at the origin, and fx
i
2
0
g
denote the set of all the type 0 points. Then, 1
fx
i
2C
0
g
is the
indicator function which is one when a type 0 node i lies in
cell C
0
. Let E½N
v
be the expected number of type 0 nodes in
cell C
0
.
E½N
v
¼E½
X
x
i
2
0
1
fx
i
2C
0
g
¼ E½E½
X
x
i
2
0
1
fx
i
2C
0
g
jð
1
Þ
¼
Z
2
0
Z
1
0
e
1
x
2
0
xdxd:
The event that a type 0 point located at ðx; Þ belongs to
the Voronoi cell C
0
is equivalent to the event that there is a
point of type 0 in a small area xdxd located at ðx; Þ, and
there is no other point of type 1 in a circle of radius x
around that type 0 point. The latter condition ensures that
this type 0 node belongs to the Voronoi cell C
0
. From this,
we get
E½N
v
¼
0
1
: ð12Þ
This is a well-known result. Using a similar argument, we
can find the expected number of type 0 nodes located
within a distance of r from a type 1 node as follows:
E½N
v
ðrÞ ¼ E½
X
x
i
2
0
1
fx
i
2C
0
;jx
i
j<rg
¼ E½E½
X
x
i
2
0
1
fx
i
2C
0
;jx
i
j<rg
jð
1
Þ
¼
Z
2
0
Z
r
0
e
1
x
2
0
xdxd;
which gives
E½N
v
ðrÞ ¼
0
1
ð1 e
1
r
2
Þ: ð13Þ
These E½N
v
ðrÞ number of critical nodes relay the data of the
E½N
v
E½N
v
ðrÞ type 0 nodes that are located outside the
circle of radius r in the same Voronoi cell (see Fig. 1). Hence,
the average relaying load on a typical critical node (P
r
0
)is
P
r
0
¼ðE
r
0
þ E
t
0
Þ
E½N
v
E½N
v
ðrÞ
E½N
v
ðrÞ

¼ðE
r
0
þ E
t
0
Þ
e
1
r
2
1 e
1
r
2
!
:
Substituting the above in (10) and the value of E½N
v
from
(12) in (11), we have the following:
P
0
¼ E
t
0
þðE
r
0
þ E
t
0
Þ
e
1
r
2
1 e
1
r
2
!
¼ c
0
þ c
1
e
1
r
2
1 e
1
r
2
!
:
ð14Þ
8 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 4, NO. 1, JANUARY/FEBRUARY 2005
Authorized licensed use limited to: University of Waterloo. Downloaded on February 25, 2009 at 10:05 from IEEE Xplore. Restrictions apply.

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References
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TL;DR: This work develops and analyzes low-energy adaptive clustering hierarchy (LEACH), a protocol architecture for microsensor networks that combines the ideas of energy-efficient cluster-based routing and media access together with application-specific data aggregation to achieve good performance in terms of system lifetime, latency, and application-perceived quality.
Abstract: Networking together hundreds or thousands of cheap microsensor nodes allows users to accurately monitor a remote environment by intelligently combining the data from the individual nodes. These networks require robust wireless communication protocols that are energy efficient and provide low latency. We develop and analyze low-energy adaptive clustering hierarchy (LEACH), a protocol architecture for microsensor networks that combines the ideas of energy-efficient cluster-based routing and media access together with application-specific data aggregation to achieve good performance in terms of system lifetime, latency, and application-perceived quality. LEACH includes a new, distributed cluster formation technique that enables self-organization of large numbers of nodes, algorithms for adapting clusters and rotating cluster head positions to evenly distribute the energy load among all the nodes, and techniques to enable distributed signal processing to save communication resources. Our results show that LEACH can improve system lifetime by an order of magnitude compared with general-purpose multihop approaches.

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Abstract: Preface. MATHEMATICAL REVIEW. Methods of Proof and Some Notation. Vector Spaces and Matrices. Transformations. Concepts from Geometry. Elements of Calculus. UNCONSTRAINED OPTIMIZATION. Basics of Set--Constrained and Unconstrained Optimization. One--Dimensional Search Methods. Gradient Methods. Newton's Method. Conjugate Direction Methods. Quasi--Newton Methods. Solving Ax = b. Unconstrained Optimization and Neural Networks. Genetic Algorithms. LINEAR PROGRAMMING. Introduction to Linear Programming. Simplex Method. Duality. Non--Simplex Methods. NONLINEAR CONSTRAINED OPTIMIZATION. Problems with Equality Constraints. Problems with Inequality Constraints. Convex Optimization Problems. Algorithms for Constrained Optimization. References. Index.

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Frequently Asked Questions (1)
Q1. What are the contributions mentioned in the paper "A minimum cost heterogeneous sensor network with a lifetime constraint" ?

The authors consider a heterogeneous sensor network in which nodes are to be deployed over a unit area for the purpose of surveillance.