DTIC ADA465201: Deployment Strategies for Differentiated Detection in Wireless Sensor Networks

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by  Defense Technical Information Center

Deployment Strategies for Differentiated Detection
in Wireless Sensor Networks
Jingbin Zhang, Ting Yan, and Sang H. Son
Department of Computer Science
University of Virginia, Charlottesville 22903
{jz7q, ty4k, son}@cs.virginia.edu
Abstract—In this paper, we address the deployment problem of the paper is to develop a deployment strategy to satisfy the
for differentiated detection requirements, in which the required different detection probability thresholds at different locations
detection probability thresholds at different locations are dif-
using minimum number of nodes.
ferent. We focus on differentiated deployment algorithms that
Minimizing the number of nodes deployed may not be
are applied to the probabilistic detection model, since it is
more realistic than the binary detection model. We show that critical when the cost of node is negligible and we have
the relationship between the node deployment strategy and the excessive number of nodes. However, as long as the prices of
logarithmiccollectivemissprobabilitydistributionisLinearShift sensornodesarenotnegligible,toreducethenumberofnodes
Invariant (LSI). Using this property, we formulate the differen-
deployed is always necessary. For example, while MicaZ
tiated deployment problem as an integer linear programming
motes, which are commonly used in terrestrial environments,
problem,whichisawellknownNP-hardproblem.Weproposea
differentiatednodedeploymentalgorithmcalledDIFF DEPLOY, cost over 120 dollars each [7], some sensors used in undersea
whichachievesmuchbetterperformancethanthestate-of-the-art surveillanceapplicationsaretensoforevenhundredsoftimes
node deployment algorithm for both uniform and differentiated more expensive. While it is expected that the cost of sensors
detection requirements.
will decrease as the technologies advance, it may not happen
quickly, since the advances of the technologies also result in
I. INTRODUCTION
more powerful features being incorporated into a single node.
The emergence of wireless sensor networks gives rise to This is especially true for undersea sensor nodes. Another
many applications, such as military surveillance [1] and habi- issue is stealthiness. By minimizing the number of sensors
tat monitoring [2]. In these applications, it is an important deployed, theriskofthesystembeing detected by adversaries
requirement to provide adequate sensing coverage to achieve isalsoreduced.Evenifthecostandthestealthinessarenotan
acceptable quality of service. issue, knowing the minimal number and the deployment loca-
Inpreviouspapersonsensingcoverage[3][4][5][6],abinary tions of the nodes provides us the guidance on how to deploy
detection model is assumed. In a binary detection model, the redundant sensors to improve reliability. For example, the
sensor node can detect a target with a 100% probability redundant sensors can be deployed proportionally, based on
providedthatthetargetiswithinitssensingrange,andcannot the density of the sensors after a minimum number of sensors
detect a target beyond the range. In this paper, we use a are deployed.
probabilistic detection model, because in reality the detection We assume that we have good control of the node deploy-
ofatargetisnotdeterministicduetotheuncertaintyassociated ment, i.e., we can place the sensors into the exact targeted
with sensor detections. When a target is within a sensor’s locations, either manually [8] or air-dropped [9]. However,
sensing range, whether the target is detected by the sensor when spatial error is inevitable in the node deployment, we
is probabilistic. can model the uncertainty of nodes’ actual locations using
With a probabilistic detection model, we can hardly have Gaussian distribution [10]. We show that to consider uncer-
a 100% detection probability for all the geographic points in tainty in the node deployment, we only need to modify the
the target area. In many surveillance applications, the system node detection model. Therefore, our algorithm is applicable
requiresdifferentdegreesofsurveillanceatdifferentlocations. to both precise node deployment and node deployment with
Thesystemmayrequireextremelyhighdetectionprobabilities uncertainty.
at certain sensitive areas. However, for some not so sensitive In this paper, we show that the relationship between the
areas, relatively low detection probabilities are required to node deployment strategy and the logarithmic collective miss
reduce the number of sensors deployed so as to decrease the probability distribution of the sensor field is Linear Shift
cost. In this case, different areas require different densities of Invariant (LSI). By taking advantage of this property, we
deployed nodes. When the detection probability thresholds at formulate the node deployment problem as an integer lin-
different subareas are specified, the minimum number and the ear programming problem, which is a well known NP-hard
deployment locations of the sensors need to be decided based problem. Further, based on the linear relationship, we devise
on the specification and the probabilistic detection model. We a differentiated deployment algorithm called DIFF DEPLOY,
nameitasthe“DifferentiatedDeployment”problem.Thegoal which outperforms the state-of-the-art probability based node
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This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE SECON 2006 proceedings.
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deployment algorithm MIN MISS for both uniform and dif- in both cases.
ferentiated detection requirements.
The remainder of the paper is organized as follows. We III. SENSORANDTERRAINMODELS
review related work in Section II. We describe the sensor In this paper, we use a two-dimensional grid to cover the
and terrain models in Section III. Section IV presents how sensor field. We only consider the detection probability at the
to relate the detection probability to system performance. We grid points. The granularity of the grid is adjusted based on
formulate the node deployment problem in Section V. In Sec- the precision we want and the time and space we can afford
tion VI, we present our differentiated deployment algorithm forcomputation.SupposethedimensionofthegridisU ×V.
DIFF DEPLOY. Then we briefly describe MIN MISS and Thedeploymentstrategyofthesensorfieldcanberepresented
the way uncertainty is handled in Section VII. Section VIII byaU ×V matrixD,inwhichD(x,y)denotesthenumberof
contains a complete evaluation of different deployment al- nodes deployed at grid point (x,y). If D(x,y)=1, it means
gorithms. We present the conclusions and future work in that one sensor is deployed at grid point (x,y). If D(x,y)=
Section IX. 0, it means no sensor is deployed at that location. Typically,
D(x,y) is either 0 or 1. However, in very rare cases when
II. RELATEDWORK the detection requirement is so high that more than one node
In sensor network based surveillance applications, it is needtobedeployedinasmallneighborhood,itispossiblethat
critical to determine where to deploy the sensors to satisfy D(x,y) takes a value greater than 1. We use d((m,n),(i,j))
the sensing coverage requirement. A large number of pub- todenotetheEuclideandistancebetweenlocations(m,n)and
lications exist in the literature [10][3][11][4][5][12][6][13]. (i,j).
However, most of them are based on binary detection models In reality, it is challenging to obtain an exact and precise
[3][4][5][6]. node detection model, because the detection of a target de-
When the binary detection model is used, if we only pends on many factors. We classify the factors into three
considersensingcoverage,todeterminetheminimumnumber categories. Factors in the first category are those related to
of nodes to cover the whole area is actually a circle covering the sensors, such as the type and the quality of the sensor,
problem [5][14]. A lot of work also considers communication and the sensing algorithm used to detect the target. Factors
connectivity requirements while minimizing the number of in the second category are those related to the target. They
nodes deployed [5][6]. include: the type, size and shape of the target; the duration
Althoughthebinarydetectionmodelsimplifiestheanalysis, of a target staying at a certain location, which is related to
itisnotrealisticinmanycases.Inreality,asensordetectionis the speed of the target; the distance between the target and
a probabilistic event, in which a target is not always detected the sensor. Factors in the third category are those related to
byasensorduetoitslimitedsensingcapabilityandprocessing the environment where sensors are deployed. This category
power. In [12], Meguerdichian et. al. assume that the sensor includes the following factors: the surrounding obstacles; the
coverage decreases as the distance from the sensor increases weather, such as the temperature, humidity and whether it is
and propose an algorithm based on Voronoi diagram and rainyorwindy;backgroundnoise,suchasthemagneticfieldof
Delaunay triangulation to find the maximal breach path and theearth.Almostallthesefactorsareexperiencedinrealfield
maximalsupportpath.Althoughthatalgorithmcanbeusedas experiments [15]. Many of these factors are hard to model.
a guide to deploy extra sensors to improve breach coverage, Therefore,itisverydifficulttogetatotallyaccuratedetection
it can not be used for differentiated deployment problem model.
unless both different node detection models and differentiated However, it is possible to obtain a conservative detection
detection requirements are considered. In [11], Dhillon et. model,aswhathasbeendonetoobtainaconservativesensing
al. propose to use probabilistic detection models to better range [16]. This kind of model can be obtained by using
address this problem. In these models, the probability of conservativevaluesintheoreticalanalysisorthroughextensive
target detection by a sensor decreases exponentially when the experiments. For example, we can run the experiments many
distance between them increases. times under different weather conditions, and use the lowest
Several deployment algorithms [10][11][13] are proposed detection probability if the results under different weather
when the probabilistic detection models are used. The algo- conditions are different. In this way, we rule out several
rithm described in [13] assumes the mobility of the sensors, hard to model factors and the obtained model provides the
whileweconsiderstaticsensorsthatdonotmoveoncetheyare worst case detection probability. Although it may be time
deployed. The deployment algorithms described in [10][11] consuming,itispossibletogetaconservativedetectionmodel
are similar, between which MIN MISS proposed in [10] is with current technologies. Obtaining the detection model is
more sophisticated. MIN MISS is mainly used for uniform not only useful for the node deployment problem, but helps
detection requirements, in which the specified detection prob- usbetterunderstandtheperformanceofthesystem.Providing
ability threshold for the entire surveillance area is the same. a totally accurate detection model is out of the scope of this
Different from MIN MISS, DIFF DEPLOY is designed for paper.
both uniform and differentiated detection requirements. Fur- In [10][13], a probabilistic node detection model consider-
thermore, DIFF DEPLOY achieves much better performance ingthefactorofthedistancebetweenthetargetandthesensor
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE SECON 2006 proceedings.
is used, which is shown in Equation (1):
(cid:2)
M(x,y) = pmiss((x,y),(i,j))D(i,j)
p((m,n),(i,j))= (i,j)∈Grid
(cid:1) (cid:2)
e0−ad((m,n),(i,j)) iiffdd((((mm,,nn)),,((ii,,jj))))><=rsrs (1) = (i,j)∈Grid(1−p((x,y),(i,j)))D(i,j) (4)
Grid in Equation (4) denotes the whole sensor field where
In the above equation, p((m,n),(i,j)) is the probability of
we want to deploy the sensors.
atargetatgridpoint(m,n)beingdetectedbythenodeatgrid
point(i,j),r isthesensingrange,andaisaparameterrelated
s IV. DETECTIONPROBABILITYvsSYSTEMPERFORMANCE
tothephysicalcharacteristicsofthesensingdevice,whichcan
Inthedifferentiateddeploymentproblem,realizingtherela-
be obtained from field experiments. This model assumes that
tionshipbetweenthedetectionprobabilityandthesystemper-
thedetectionprobabilityvariesexponentiallywiththedistance
formance helps the designer to specify the detection probabil-
between the target and the sensor. Although this model is
itythresholdsbasedonthedifferentperformancerequirements
not accurate, it reasonably characterizes the behavior of range
at different locations. In this section, we briefly discuss how
sensing devices, such as infrared and ultrasound sensors [17].
toinferthesystemperformancefromthedetectionprobability
This equation does not take into consideration of the time
and vice versa.
duration a target stays at a certain location. In reality, if a
In surveillance applications, primary performance require-
targetstaysatacertainlocationforalongertime,itresultsina
ments include detection performance and tracking perfor-
higherprobabilityofbeingdetected.Therefore,weincorporate
mance. The detection probability is closely related to both of
the factor of the duration into the node detection model:
them.Thedetectionperformanceisaboutwhetheratargetcan
be detected when it appears in the network, and the detection
p((m,n),(i,j),t)=
(cid:1) delay if the target is detected. The tracking performance
1−(1−p((m,n),(i,j),1))(cid:1)t/T(cid:2) ift>T
p((m,n),(i,j)) if0<t(cid:1)T (2) represents how well the target’s trajectory is obtained, given
that the target is detected.
For different target types, we use different requirements.
In this model, we use discrete time. The time unit T is the
We divide the targets into two categories: static targets and
time period during which the sensing algorithm is executed
movingtargets.Forthefirstcategory,wearemainlyinterested
and a decision is made on whether a target is detected. t
in the detection performance. The detection probability at a
denotes the duration the target stays at location (i,j). This
certain location indicates whether a target can be detected if
model assumes that the target detection in different time units
the target stays there for a certain period of time. We use
is independent. We also assume that p((m,n),(i,j),1) =
p (x,y) to denote the detection probability at grid point
p((m,n),(i,j)). detect
(x,y) when the target stays at (x,y) for one time unit. If
Note that the detetion probability at a certain location
p (x,y)=w, we know that if a target stays at (x,y) for
for one time unit can be easily converted to the detection detect
l time units, the probability of being detected is 1−(1−w)l.
probabilityformultipletimeunits.Ifthedetection probability
The detection probability also indicates the detection delay.
for one time unit is w, the detection probability for l time
If p (x,y) = w, with c% confidence level, the detection
units is 1 − (1 − w)l. For simplicity, we assume that the detect
delayiswithinln(1−c%)/ln(1−w)timeunits.Forexample,
detection probability thresholds specified by the users are the
if p (x,y) = 0.5, the detection delay is within 5 time
probabilitiesforonetimeunit.Tobeconsistentwiththedetec- detect
units with 95% confidence level. If the system requirement is
tion probability thresholds, all the probabilities mentioned in
specified by the desired detection performance, the designer
the following sections are the probabilities for one time unit.
can compute the detection probability thresholds at different
Since p((m,n),(i,j)) = p((m,n),(i,j),1), we directly use
locations based on the relationship between the detection
p((m,n),(i,j)) instead of p((m,n),(i,j),1) for brevity.
probability and the detection performance.
Similar to p((m,n),(i,j)), p ((m,n),(i,j)) is used to
miss For moving targets, we are interested in both the detection
denote the probability of a target at grid point (m,n) being
performance and the tracking performance. Given the node
missed by the node at grid point (i,j), which is shown in
detection model for moving targets, the relationship between
Equation (3):
the detection performance and the detection probability for
pmiss((m,n),(i,j))=1−p((m,n),(i,j)) (3) moving targets is the same as that for static targets. The
tracking performance mainly depends on the frequency of the
detectionreportsfromthenetwork.Afinergrainedtrackingis
We use a U × V matrix M to denote the collective miss achievedifthetimeintervalbetweentheconsecutivedetection
probability distribution of the whole field. The collective miss reports is smaller. The frequency of the detection report is
probability M(x,y) means the probability of a target at grid indicatedbythedetectionprobability.Ifp (x,y)=w,the
detect
point (x,y) being missed by all the sensors deployed in the averagetimeintervalbetweentheconsecutivedetectionreports
field, which is given by: at(x,y)is1/w.Forexample,ifthedetectionprobabilityinan
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE SECON 2006 proceedings.
areais0.5,thenetworkonaveragegeneratesdetectionreports Theorem 1: The system described in Equation (5) that
everytwotimeunitswhenatargetappearsinthatarea.Inthis transforms the node deployment strategy D to I is a Linear
case,weareinformedthelocationsofthetargetforhalfofthe Shift Invariant (LSI) System.
time, but we do not exactly know where the target is for the Proof: We prove that the system described in Equation
other half of the time. However, if the detection probability (5) has the three properties of an LSI system as follows:
in the area is 0.99, we get the detection reports almost every 1. Scaling. Considering two inputs D1 and D2, and D2 =
time unit. In this case, we obtain a finer grained trajectory of λD1, in which λ is an arbitrary nonnegative integer constant.
the target, together with the duration of the target associated Suppose I1 is the output of D1 and I2 is the output of D2.
with each location of the trajectory. If the desired tracking Then for any (x,y)∈Grid, we have:
performance at different locations is provided, the designer
(cid:3)
can obtain the detection probability threshold at each location I2(x,y)= [D2(i,j)×lnpmiss((x,y),(i,j))]
based on the relationship between the detection probability (i,j)∈(cid:3)Grid
and the tracking performance. =λ [D1(i,j)×lnpmiss((x,y),(i,j))]
(i,j)∈Grid
V. PROBLEMFORMULATION =λI1(x,y)
In this section, we formulate the node deployment problem
as an integer linear programming problem, which has been Therefore,I2 =λI1 andthesystemconformstothescaling
proventobeNP-hard.Notethatboththeproblemformulation property.
and our differentiated deployment algorithm are not restricted 2. Superposition. Assume that the outputs of two arbitrary
to the detection model we use. While we use the detection inputs D1 and D2 are I1 and I2, correspondingly. Suppose I(cid:1)
model described by Equation (1), any detection model can is the output D1+D2. Then for any (x,y)∈Grid, we have:
be applied. We first assume that the node detection model
(cid:3)
remains the same no matter where the node is deployed. A I(cid:4)(x,y)= [(D1(i,j)+D2(i,j))×lnpmiss((x,y),(i,j))]
solutionisprovidedwhendifferentlocationsresultindifferent (i,j)∈Grid
(cid:3)
node detection models, which is mainly caused by terrain = [D1(i,j)×lnpmiss((x,y),(i,j))]
complexities, such as obstacles in the environment. (i,j)(cid:3)∈Grid
In this paper, we study the system that transforms the + [D2(i,j)×lnpmiss((x,y),(i,j))]
(i,j)∈Grid
node deployment strategy D (the input of the system) to
=I1(x,y)+I2(x,y)
the logarithmic collective miss probability distribution lnM
(the output of the system). The input-output relationship of
the system can be characterized by the node detection model Therefore, I(cid:1) = I1 +I2 and the system conforms to the
p((m,n),(i,j)). We will prove later in this section that this superposition property.
system is a Linear Shift Invariant (LSI) system. From the 3.ShiftInvariant.AssumetheoutputofanarbitraryinputD
property of the LSI system, the logarithmic collective miss is I. Shift the input D by m in the x-axis and n in the y-axis
probability distribution is the two-dimensional convolution of toformanewinputD(cid:1),inwhichD(cid:1)(x,y)=D(x−m,y−n)
thenodedeploymentstrategyandtheimpulseresponseofthis and m and n are arbitrary integers. Denote the corresponding
system, which is characterized by the node detection model. output of D(cid:1) by I(cid:1). Then for any (x,y) ∈ Grid(cid:1), in which
Inordertosimplifytherelationshipamongthenodedeploy- Grid(cid:1) is the sensor field after we shift the original sensor
ment strategy D, the node detection model p((m,n),(i,j)) field Grid by m in the x-axis and n in the y-axis, we have:
and the collective miss probability distribution M, we first (cid:3)
replace p((m,n),(i,j)) with p ((m,n),(i,j)). As shown I(cid:4)(x,y) = [D(cid:4)(i(cid:4),j(cid:4))×lnpmiss((x,y),(i(cid:4),j(cid:4)))]
miss
in Equation (4), the collective miss probability M(x,y) is the (i(cid:4),j(cid:4)(cid:3))∈Grid(cid:4)
product of all the sensors’ probabilities to miss the target at = [D(i(cid:4)−m,j(cid:4)−n)×lnpmiss((x,y),(i(cid:4),j(cid:4)))]
(i(cid:4),j(cid:4))∈Grid(cid:4)
location (x,y). We use logarithm to convert the multiplicative (cid:3)
= [D(i(cid:4)−m,j(cid:4)−n)
relationship shown in Equation (4) to an additive relationship
(i(cid:4),j(cid:4))∈Grid(cid:4)
shown in Equation (5). ×ln(cid:3)pmiss((x−m,y−n),(i(cid:4)−m,j(cid:4)−n))]
(cid:3) = [D(i,j)×lnpmiss((x−m,y−n),(i,j))]
lnM(x,y) = [D(i,j)×lnpmiss((x,y),(i,j))] (5) (i,j)∈Grid
(i,j)∈Grid = I(x−m,y−n)
Forsimplicity,weuseI todenotethelogarithmiccollective
miss probability distribution lnM. It is easy to see that Therefore, the system conforms to the shift invariant prop-
M(x,y) = eI(x,y) for any (x,y) ∈ Grid. Through the erty.
conversions, the system described in Equation (5) is indeed From the above three properties, we have shown that the
a Linear Shift Invariant System (LSI), which takes D as the system is an LSI system.
input and I as the output. This is shown by the following Since it is an LSI system, we use the impulse response to
theorem. characterize how an input D is transformed into the output I.
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE SECON 2006 proceedings.
We use g to denote the impulse response in this LSI sytem. stillholdsandbothourproblemformulationanddifferentiated
When we use the node detection model shown in Equation node deployment algorithm are not affected.
(1), g(x,y) is expressed in the following equation: Based on Equation (8), we are able to formulate the node
deployment problem as an integer linear programming prob-
g(x,y) == (cid:4)lnp0lmn(is1s−((xe−,ya)√,(x02,+0y)2)) iiff(cid:5)(cid:5)xx22++yy22<>=rsrs (6) ldpeirmsotbr.iabWbuitleiiotyunstoehfrmethstehhowltdhooaldteenlfiooectlaedt,itoihnnew(mxhi,icsysh).mprOtohub(raxbo,iyblij)teyicsttihtvhreeesimhsoiltsdos
find the minimal number of nodes to satisfy that, after these
Further, as an LSI system, the output I can be represented nodesaredeployed,forany(x,y)∈Grid,thecollectivemiss
by the convolution result of the input D and the impulse probability M(x,y) is smaller than or equal to mth(x,y). We
response g, which is shown in Equation (7). setI tolnmth.Thenweconstructthecorrespondingmatrices
for D, g and I. The problem can be formulated as follows:
I(x,y) = D(x,y)∗g(x,y)
(cid:3)(cid:3)
= D(i,j)×g(x−i,y−j) (7)
i j Minimize: sum(D )
p
Subject to: G ·D <=I
p p p
Two dimensional convolution is a well studied area in
The elements in D are nonnegative integers
p
image processing. We apply the same techniques used in
image processing to transform the convolution to the ma-
Fig.1. Integerlinearprogrammingmodel.
trix multiplication by constructing the corresponding matrices
[18][19]. We do not discuss the details of how to construct This problem is a typical integer linear programming prob-
thecorrespondingmatricesinthispaperduetothepagelimit. lem, which has been proven to be NP-hard. An NP-hard
Readers are referred to [18][19] for details. By using the problem indicates that it is almost impossible to find the
corresponding matrices, Equation (7) is represented by the optimal solution when the input size is large, since we can
following matrix Equation: not afford the computation time. Therefore, we devise an
optimization algorithm called DIFF DEPLOY to obtain the
Ip=Gp·Dp (8)
suboptimal result.
InEquation(8),matrixG isconstructedbasedong,matrix VI. DIFFERENTIATEDDEPLOYMENTALGORITHM
p
D is constructed based on D and matrix I is constructed In this section, we present the differentiated deployment
p p
based on I. By using an LSI system to represent the rela- algorithm DIFF DEPLOY. DIFF DEPLOY makes use of the
tionship among the node deployment strategy D, the node Linear Shift Invariant (LSI) property, which justifies the rep-
detectionmodelp((m,n),(i,j))andthecollectivemissproba- resentation of the relationship among the node deployment
bilitydistributionM,wefinallyreachasimpleequationwhich strategy, the node detection model and the collective miss
characterizes the relationship among all three elements. Note probability distribution by matrix multiplication as shown in
thatifN =max(U,V),boththedimensionsofI andD are Equation (8). Based on matrix algebra, it is clear that if we
p p
N2×1, and the dimension of G is N2×N2. Also note that know the miss probability threshold distribution and the node
p
iftheindicesinG startfromzero,theelementsinG canbe detection model, we can directly compute the deployment
p p
interpreted in the following way: the value of G (x,y) equals strategy D based on Equation (8) as follows:
p p
to the value of lnp (((cid:2)x/N(cid:3),x%N),((cid:2)y/N(cid:3),y%N)).
By now, we assmuimsse that all the nodes share the same Dp=G−p1·Ip (9)
detection model. However, this assumption does not hold in
certain situations. For example, the sensing capability of a The result D computed from Equation (9) indicates the
p
nodemaybeunderminedbytheobstaclesintheenvironment. number of sensors we should place at each location to satisfy
This problem is solved by modifying the matrix G . For that M = m , based on the node detection model. If we
p th
example, if a node at grid point (m,n) has a detection could place the sensors according to the computed result D ,
p
model different from others, we change the values of the thetotalnumberofnodesdeployedwouldbeminimal,because
elements in the (m×N +n)th column of G . If the value G ·D would equal to I . However, the computed result D
p p p p p
of p ((x,y),(m,n)) is changed to be v , G ((x×N+ does notreflecttherealworldsituation.Inreality,thenumber
miss new p
y),(m×N +n)) is set to lnv . In this way, Equation (8) of nodes we can deploy at a place must be non-negative
new
allows different detection models at different locations and integers, while the results computed from Equation (9) can
the impact of obstacles is incorporated. By allowing different beanyrealnumbers,includingnegative values.Therefore,we
detectionmodelsatdifferentlocations,thesystemisnolonger can not directly use the result D as the deployment strategy.
p
an LSI system. However, it is still a linear system which However, the result D is a good heuristics to decide where
p
can be proven in the same way as the proof of the scaling to deploy sensors. In the result D , a larger numeric value
p
and superposition properties of Theorem 1. So Equation (8) requires a higher number of nodes to be deployed to satisfy
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE SECON 2006 proceedings.
therequirement.Notethatthenumberofnodesrequiredcanbe than MIN MISS, especially for differentiated detection re-
any real number. We perceive the location which corresponds quirements. For brevity, we use RANDOM to denote the
to the maximum value in D as the location that contributes random node deployment algorithm.
p
the most in satisfying the detection requirement if a sensor In MIN MISS, for each iteration, the algorithm decides
is deployed at the location. When we decide the location to the location for the next node to be deployed. We call each
deploy the next sensor, the most contributing location is our location at which no sensor node is deployed a “candidate
first choice. location”. If we deploy a new node at candidate location
(i,j), the collective miss probability at location (x,y) is then
p ((x,y),(i,j))×M(x,y). The overall miss probability
miss
Procedure DIFF DEPLOY(Gp,Ip,N){ Moverall(i,j) is defined as the sum of the collective miss
//N =max(U,V) probabilities at all possible locations when the new node is
invG =inv(G ); //Get the inverse matrix of G deployed at location (i,j), or
p p p
remainI =I ;
p p (cid:3)
Dp =zeros(N ×N,1); /*Dp is a (N ×N)×1 Moverall(i,j)= pmiss((x,y),(i,j))×M(x,y). (10)
(x,y)∈Grid
matrix, initialized with all zeros.
while sum(remainIp)<0 /*there are still some In MIN MISS, candidate location (i,j), which has the
locations whose miss probability minimum M (i,j) among all the candidate locations,
overall
thresholds are not satisfied*/
is selected for the next sensor to deploy. The algorithm
nextD =invG ×remainI ;
p p p terminates when the miss probability of each grid point is
Find the maximum value nextD (k) in
p smaller than or equal to its miss probability threshold.
nextD , which satisfies the following
p The paper [10] states that MIN MISS is an “uncertainty-
constraints: remainI (k)<0&&D (k)==0;
p p awaresensornodeplacementalgorithm”.Theuncertaintyhere
D (k)=D (k)+1; /*Place the next sensor
p p meansthatwhenwedeploysensors,sensorsmaynotbeplaced
at location k*/
at the exact intended positions because of uncontrollable
remainI =I −G ×D ; //Update remainI
p p p p p conditions,suchaslocalizationerror,anddriftingofsensorsin
Set any positive value in remainI to zero;
p underseasensornetworksduetowaterflows.Byincorporating
end
the uncertainty in the node deployment, the node detection
}
modelismodified.However,itdoesnotaffectthedeployment
algorithm. We show it based on the equations from [10].
Fig.2. DIFFDEPLOYnodedeploymentalgorithm. In [10], a Gaussian probability distribution is used to
model the deviation between the intended sensor locations
DIFF DEPLOYmakesuseoftheheuristicsprovidedbythe and the sensors’ actual locations. In this Gaussian probability
resultD .Figure2showsthepseudocodeofDIFF DEPLOY. distribution model, the intended location (x,y) is the mean
p
The algorithm runs iteratively and places one sensor in the value and the standard deviations are σ in the x dimension
x
sensor field at a time. During each iteration, nextD is andσ intheydimension.Withthismodel,theprobabilityof
p y
computed based on the remaining requirement remainI , a node intended to be deployed at location (x,y) but actually
p
whichisthedifferencebetweenI specifiedintherequirement goes to location (x(cid:1),y(cid:1)) is:
p
apmnroodsvtitdcheoedntlrboiybguatrthiintehgmnlooidcceasctiootlhnleabcttahisvaeevdemoanilsrtsehaepdrryeombbaeabeininliintdygeprdeloiqsyutreiirdbe.umtTieohnnet pdeploy((x,y),(x(cid:4),y(cid:4)))= e−(x(cid:4)2−σ2x2xπ)σ2x−σ(yy(cid:4)2−σy2y)2 (11)
remainI is used as the location to place the next node,
p
which corresponds to the maximum value in nextD . Then Let Area denote all the possible locations a sensor can be
p
the remaining requirement remainI is updated based on the deployedduetouncertainty.Thenthepossibilityofatargetat
p
newly deployed node. The algorithm terminates when all the gridpoint(m,n)beingdetectedbyasensorwhichisintended
elements in remainI are zero, which means that the miss to be deployed at location (x,y) can be denoted in Equation
p
probability thresholds at all the locations are satisfied. (12):
VII. MIN MISSALGORITHMANDUNCERTAINTY p(cid:6)uncertainty((m,n),(x,y))=
(x(cid:4),y(cid:4))∈(cid:6)Areap((m,n),(x(cid:4),y(cid:4)))pdeploy((x,y),(x(cid:4),y(cid:4))) (12)
In this section, we briefly describe the MIN MISS de- (x(cid:4),y(cid:4))∈Areapdeploy((x,y),(x(cid:4),y(cid:4)))
ployment algorithm presented in [10], which to the best
of our knowledge is the state-of-the-art node deployment As shown in Equation (12), by introducing uncertainty,
algorithm based on probabilistic detection model. Although the node detection model changes from p((m,n),(i,j)) to
MIN MISSachievesbetterperformancethantherandomnode p ((m,n),(i,j)). We call the new model that con-
uncertain
deployment algorithm, we show that in Section VIII our siders uncertainty p ((m,n),(i,j)), the uncertainty
uncertain
algorithm DIFF DEPLOY achieves much better performance aware node detection model. Figure 3 shows the uncertainty
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE SECON 2006 proceedings.
aware node detection model. We assume that σ equals σ ,
wittAshohset0ircsiu.nhh5ntoceiwtasnotndedcedeiodnnmaolnFotpdeciudgatwuetaireosetnhss3eeta,dtnceatdovhrelrtiahenrmsegptaehoaxrecnimstdfiutiduagnelmugvrleodds.ceiiagsTtettnhaicoinefitniccoGeantoaneutpxrl3sryros.oirbdaWanbebecemirtlewsioaeteisdteeeyesansl. Number of nodes deployed 1111357913570000000000000000 DMRIIAFNNF_D_MODIMESPSLOY
thedetectionprobabilityneartheintendedsensorlocation,but 100
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
slightly increases the detection probability far away from the The miss probability threshold
intended sensor location.
Fig. 4. Performance of DIFFDEPLOY, MINMISS and RANDOM with
variousmissprobabilitythresholdsforuniformdetectionrequirements.
1
Detection Probability 00000000........23456789 Osssrtttidddgeeeivvvn===a012l. 5Model AmBosistshshpMorwoINnbaibMnilIFiStiySgutarhenred4s,hDotIhlFdeFninDucmErePbaeLsrOesoYfinnuosaedlelmstuhdcreehcerfeeaawlsgeeosrriantshomdtheses.
0.1 thanRANDOMforuniformdetectionrequirements.However,
0
-7 -5 -3 -1 1 3 5 7 DIFF DEPLOY achieves better performance than MIN MISS
Distance to the intended Sensor location
even for uniform detection requirements. In DIFF DEPLOY
when we deploy a new sensor, both the remaining miss
Fig.3. Sensordetectionmodelwhenuncertaintyisconsidered.
probabilitythresholddistributionandthenodedetectionmodel
are directly integrated into the computation as shown in
Since MIN MISS is a general node deployment algorithm
Equation (9). The performance advantage of DIFF DEPLOY
as DIFF DEPLOY, which is not designed just for one spe-
overMIN MISSismoresignificantwhenthemissprobability
cific node detection model, it makes more sense to decouple
threshold is larger. Overall, DIFF DEPLOY uses 14%-42%
this uncertainty from the node deployment algorithm. When
fewer sensors than MIN MISS.
uncertainty is introduced into the node deployment, we only
need to feed the uncertainty aware node detection model,
2000
idStnoeespcstltehoiaooydwnmoVetfhnIeIttIh-VarCeolgIbIowouIrrs.hiitgtenhPinenmEsauRs.lnFWocnOfeoeRrdotMageuiirAnvdtepNeyrtCsoeioEnpcmtoEidsoeeVenpdAelxmoLapyUloegmAdroiTeermlnIi,tOtehniNimnstat.oclorntehsseiudlentsroedidne, Number of nodes deployed 11111246802468000000000000000000 DMRIIAFNNF_D_MODIMESPSLOY
0
0.2 0.3 0.4 0.5 0.6 0.7 0.8
WesimulateDIFF DEPLOY,MIN MISS,andRANDOMin
Value of a
Matlabandcomparetheirperformance.Inoursimulations,we
setthegriddimensionto50×50.AllthepointsinRANDOM
Fig. 5. Performance of DIFFDEPLOY, MINMISS and RANDOM with
arethemeanvaluescomputedfrom20trials.Unlessotherwise variousavaluesforuniformdetectionrequirements.
specified, the sensing range is set to 7; the value of a is set
to 0.5. Figure 5 shows the performance of different deployment
For this performance evaluation, four sets of experiments algorithms with various a values, when the miss probability
are designed. In the first set, we compare the performance threshold is set to 0.1. As shown in Figure 5, the number
of different deployment algorithms for uniform detection re- of nodes deployed increases as the value of a increases,
quirements.Inthesecondset,theperformanceofdifferentde- because the larger the value of a is, the faster the detection
ployment algorithms for differentiated detection requirements probability drops when the distance between the target and
is evaluated. In the third set, we investigate the performance the node increases. Both DIFF DEPLOY and MIN MISS
of different deployment algorithms when uncertainty in the greatly outperforms RANDOM with all a values. Overall,
node deployment is introduced. In the forth set, we compare DIFF DEPLOY achieves 20%-34% better performance than
the performance of DIFF DEPLOY to the optimum for small MIN MISS.
scale problems. Figure 6 shows the performance of different deployment
algorithms when we change the sensing range. The miss
A. Evaluation Set 1: Uniform Detection Requirements
probability threshold is set to 0.1. As shown in the figure,
We first study the performance of DIFF DEPLOY, the number of nodes deployed first decreases fast in all three
MIN MISS and RANDOM when the miss probability thresh- algorithms as the sensing range increases. Then the number
olds at all locations are the same. of nodes starts to stabilize when the sensing range is above
Figure 4 shows the performance of different deployment 7, because the detection probability becomes small when the
algorithms when we change the miss probability threshold. distance between the target and the sensor is large. Overall,
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE SECON 2006 proceedings.
2500 the issue of different miss probability thresholds at different
Number of Nodes Deployed 112050500000000 DMRIIAFNNF_D_MODIMESPSLOY lrt5Mohe7qcrI%eaNustiihrfoMeoenmwlIsdS.eesnSrTtahnsti.eosddBpiemefysfreoftorarheedranmantpsaitMlngioncncIgeNiafitaticdMooavnnIatstShn,SetfaD.ogdrIeFifFdofeiffrDfDeenErIteFPnFLmtOiiDasYtsEedPpuLrsdoeOebsYtae4bco7itliv%ioetnyr-
0
2 3 4 5 6 7 8 9
Sensing Range
2500
Fvaigri.o6u.ssePnseirnfogrmraanngceesfoofrDunIFifForDmEdPeLteOcYti,onMrIeNquMireISmSenatnsd. RANDOM with Number of Nodes Deployed 112050500000000 DMRIIAFNNF_D_MODIMESPSLOY
0
2 3 4 5 6 7 8 9
Sensing Range
Fig. 9. Performance of DIFFDEPLOY, MINMISS and RANDOM with
varioussensingrangesfordifferentiateddetectionrequirements.
Figure 9 shows the performance of DIFF DEPLOY,
MIN MISS and RANDOM with various sensing ranges.
(a) WithoutObstacle (b) WithObstacle
With different sensing ranges, the performance advantage of
Fig.7. Missprobabilitythresholddistribution. DIFF DEPLOYoverMIN MISSisalsomuchmoresignificant
fordifferentiateddetectionrequirements.Thenumberofnodes
used in DIFF DEPLOY is 50%-54% fewer than that used in
DIFF DEPLOY uses 9%-34% fewer sensors than MIN MISS MIN MISS.
with various sensing ranges.
B. Evaluation Set 2: Differentiated Detection Requirements 1800
1600 DIFF_DEPLOY
aa7dnt(isadWt)drRiiesbfhAfuseottNriwueoDdnsnytOtohtMflheotechswaepptheiefioecrenniffilosdtcrh.maaeFtraiieosgnpnucedercoeififofif8eftehradDeennmdImtFiFFisisinsgsDpuptErrhroePoibs9LbaaObssbhieYliocil,tiwtytiMyottnhhIt.hNreerreFsMehsishgouIoulSldlrtSdess Number of Nodes Deployed 111024246800000000000000 MRIANN_DMOIMSS
0
of performance evaluation when there are no obstacles in the 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Value of a
environment.Thenweshowtheirperformancewhenthereare
obstacles in the environment in Figure 10.
Fig. 10. Performance of DIFFDEPLOY, MINMISS and RANDOM with
variousavaluesfordifferentiateddetectionrequirements.
Number of nodes deployed 111112468024680000000000000000000 DMRIIAFNNF_D_MODIMESPSLOY tioiinsbavtsDseethsadItFocidlwgFeeasntDteeicEnitnthPitoeLhFneOpigreYeeurnqfrvoseuihrirmroo7ewn(ambmns)ec.epnenrtWsoot.mfweTDihshaIeiesnFnsgFdutimhpsDeeterrEriefbPofuaoLrtrmriOeoasYnnniomcowepofhlbftieohscntreiattydcohlibeeftfrssheet.aarWetacnrleeae-
0.2 0.3 0.4 0.5 0.6 0.7 0.8 sensor can not detect a target if there is an obstacle lying
Value of a
between the sensor and the target. Thus, the nodes deployed
near the obstacle have different detection models than others.
Fig. 8. Performance of DIFFDEPLOY, MINMISS and RANDOM with
variousavaluesfordifferentiateddetectionrequirements. We show the performance of the three deployment algorithms
in Figure 10 when there are obstacles. Compared to the
Figure 8 shows the performance of the three deployment performance shown in Figure 8 when there are no obstacles,
algorithms with various a values. When the miss probability each deployment algorithm uses a little larger number of
thresholds at different locations are different, the performance nodes.Theexistenceofobstaclerestrictsthesensingabilityof
advantage of MIN MISS over RANDOM is smaller than that the nodes deployed close to it, which results in more nodes.
when the miss probability thresholds at all locations are the Overall, DIFF DEPLOY requires 47%-54% less nodes than
same, because MIN MISS does not take into consideration MIN MISS with various a values when there are obstacles.
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE SECON 2006 proceedings.
C. Evaluation Set 3: Uncertainty to the optimum for small scale problems, because it is
computationally infeasible to obtain the optimum when the
In this section, we show the performance of the three de-
input size is large. For this set of experiments, the miss
ploymentalgorithmswhenuncertaintyinthenodedeployment
probability thresholds at different locations are the same; the
is considered. In this set of experiments, we use the same
grid dimension is set to 5×5; the sensing range is set to 5;
missprobabilitythresholddistributionasthatshowninFigure
the value of a is set to 0.5. We use OPTIMUM to represent
7. Note that by considering uncertainty, it only changes the
the optimal node deployment algorithm, which always uses
node detection model. It means that in DIFF DEPLOY, only
the minimum number of nodes to satisfy the requirements.
Y needstobereplacedwiththeonethatisconstructedbased
p
on the uncertainty aware node detection model.
Number of node deployed 1234567890000000000000000000 DMRIIAFNNF_D_MODIMESPSLOY Number of nodes deployed 111112024680246800 0.05 0T.h1e mi0s.s1 5Prob0a.b2ilit0y. 2t5hres0hODMR.oPIIA3lTFNNdIF_DM_MOADIM0LES.PS3L5OY 0.4
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Value of stdev
Fig. 13. Performance of OPTIMUM, DIFFDEPLOY, MINMISS and
RANDOM.
Fig.11. PerformanceofDIFFDEPLOY,MINMISSandRANDOMwhen
uncertaintyisconsidered.
Figure 13 shows the number of nodes needed for OP-
TIMUM, DIFF DEPLOY, MIN MISS and RANDOM when
Figure 11 shows the performance of the three deployment
the miss probability threshold varies. Overall, for small scale
algorithmswhenwechangethestdevvalues.Inthissetofex-
problems,DIFF DEPLOYisthealgorithmwhichachievesthe
periments, the maximum distance error between the intended
closest performance to the OPTIMUM, except that when the
location and the actual location is set to 3. With uncertainty
miss probability threshold is 0.01, DIFF DEPLOY actually
considered, DIFF DEPLOY still maintains great performance
uses one more sensor than MIN MISS. It is not surprising
improvement over MIN MISS. Overall, DIFF DEPLOY uses
to see that in some special cases, DIFF DEPLOY uses a
42%-54% fewer sensors than MIN MISS with various values
few more sensors than MIN MISS. Both DIFF DEPLOY and
of stdev, when uncertainty is considered.
MIN MISS are heuristic based algorithms and there is no
guarantee that one algorithm always performs better than the
800
Number of Nodes Deployed 1234567000000000000000 DMRIIAFNNF_D_MODIMESPSLOY woDthIhIFiencFrhthDooinEsnePpIaXLauvpnOe.erdYarCe,grsOewhaNeoulClwsfeLosssUcequt1SutsiIinmOtoegNonscrS.ledoAFissfNoeefernDprseetoFhrnrfUitositTarhmtUsaemnRdanaEOdlclWeePpsTtOlocoIaRMyOlKemPUeTpMnrItoM.bpUlreoMmb-,,
1 2 3 4 5 6 7
lem, in which the required detection probability thresholds at
Maximum Distance Error
different locations are different. We apply the probabilistic
detection model, which is more realistic than the binary
Fig.12. PerformanceofDIFFDEPLOY,MINMISSandRANDOMwhen
uncertaintyisconsidered. detection model. We show that the relationship between the
node deployment strategy and the logarithmic collective miss
Figure 12 shows the performance of the three deployment probability distribution is Linear Shift Invariant (LSI). We
algorithms when we change the maximum distance error be- formulate the differentiated deployment problem as an in-
tween the intended location and the actual location. The stdev teger linear programming problem, which is a well known
issetto2inthissetofexperiments.Asshowninthefigure,by NP-hard problem. The main contribution of this paper is
varying the maximum distance error, the performance of each DIFF DEPLOY, a differentiated node deployment algorithm
algorithm does not change much. Overall, DIFF DEPLOY that achieves much better performance than the state-of-the-
uses 45%-51% fewer sensors than MIN MISS with different art algorithm MIN MISS, especially for differentiated detec-
maximum distance errors. tion requirements. DIFF DEPLOY provides the flexibility to
enforce the differentiated detection requirements and at the
D. Evaluation Set 4: Towards Optimum
same time minimize the total number of nodes deployed.
We have shown the performance superiority of Inthefuture,wewillinvestigateotherapplicationsofusing
DIFF DEPLOY over MIN MISS in various settings. In the LSI relationship between the node deployment strategy
this section, we compare the performance of DIFF DEPLOY and the logarithmic collective miss probability distribution.
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE SECON 2006 proceedings.