-
A Study of k-Coverage and Measures ofConnectivity in 3D Wireless
Sensor Networks
Habib M. Ammari, Member, IEEE, and Sajal K. Das, Senior Member,
IEEE
AbstractIn a wireless sensor network (WSN), connectivity enables
the sensors to communicate with each other, while sensing
coverage reflects the quality of surveillance. Although the
majority of studies on coverage and connectivity in WSNs
consider
2D space, 3D settings represent more accurately the network
design for real-world applications. As an example, underwater
sensor
networks require design in 3D rather than 2D space. In this
paper, we focus on the connectivity and k-coverage issues in 3D
WSNs,
where each point is covered by at least k sensors (the maximum
value of k is called the coverage degree). Precisely, we propose
the
Reuleaux tetrahedron model to characterize k-coverage of a 3D
field and investigate the corresponding minimum sensor spatial
density. We prove that a 3D field is guaranteed to be k-covered
if any Reuleaux tetrahedron region of the field contains at
least
k sensors. We also compute the connectivity of 3D k-coveredWSNs.
Based on the concepts of conditional connectivity and forbidden
faulty sensor set, which cannot include all the neighbors of a
sensor, we prove that 3D k-coveredWSNs can sustain a large number
of
sensor failures. Precisely, we prove that 3D k-covered WSNs have
connectivity higher than their coverage degree k. Then, we
relax
some widely used assumptions in coverage and connectivity in
WSNs, such as sensor homogeneity and unit sensing and
communication model, so as to promote the practicality of our
results in real-world scenarios. Also, we propose a placement
strategy of
sensors to achieve full k-coverage of a 3D field. This strategy
can be used in the design of energy-efficient scheduling protocols
for
3D k-covered WSNs to extend the network lifetime.
Index Terms3D k-covered wireless sensor networks, Reuleaux
tetrahedron, coverage, connectivity.
1 INTRODUCTION
A wireless sensor network (WSN) consists of a largenumber of
resource-limited (such as CPU, storage,battery power, and
communication bandwidth), tiny de-vices, which are called sensors.
These sensor nodes can sensetask-specific environmental phenomenon,
perform in-net-work processing on the sensed, and communicate
wirelesslyto other sensor nodes or to a sink (also, called data
gatheringnode), usually via multihop communications. WSNs can
beused for a variety of applications dealing with
monitoring,control, and surveillance.
In the literature, most of the works on WSNs dealt with2D
settings, where sensors are deployed on a planar field.However,
there exist applications that cannot be effectivelymodeled in the
2D space. For instance, WSNs deployed onthe trees of different
heights in a forest, or in a buildingwith multiple floors, or
underwater applications [2], [3]require the design in the 3D space.
Oceanographic datacollection, pollution monitoring, offshore
exploration, dis-aster prevention, and assisted navigation are
typicalapplications of underwater sensor networks [2]. In
[27],different deployment strategies have been proposed for 2D
and 3D communication architectures in underwater acous-tic
sensor networks, where sensors are anchored at thebottom of the
ocean for the 2D design and float at differentdepths of the ocean
to cover the entire 3D region. As shownin [31], both coverage of
space and energy-efficient datarouting for a telepresence
application require 3D design.
In this paper, we study the coverage and connectivityissues in
3D k-covered WSNs, where sensors are deployed ina 3D field, such
that every location is covered by at leastk sensors (a property
known as k-coverage, where themaximum value of k is called the
coverage degree). Thelimited battery power of sensors and the
difficulty ofreplacing and/or recharging batteries in hostile
environ-ments require that the sensors be deployed with highdensity
(up to 20 sensors per cube meter [30]) in order toextend the
network lifetime. Moreover, to cope with theproblem of faulty
sensors due to low battery power andachieve high data accuracy,
redundant coverage (in parti-cular, k-coverage) of the same region
is necessary. Asmentioned, the essential function of a WSN is to
monitor afield of interest and report the sensed data to the sink
forfurther processing. However, sensing coverage would
bemeaningless if the sensed data cannot reach the sink due tothe
absence of communication paths between it and thesource sensors (or
data generators). In other words, networkconnectivity must be
guaranteed. A WSN is said to be faulttolerant if it remains
functionally connected in spite of somesensor failures. That is why
each source sensor has to beconnected to the sink by multiple
communication paths.This is why we focus on connectivity. Hence, a
WSN will beable to function properly if both coverage and
connectivityare maintained simultaneously. This implies that all
sourcesensors and the sink should belong to the same
connectedcomponent.
IEEE TRANSACTIONS ON COMPUTERS, VOL. 59, NO. 2, FEBRUARY 2010
243
. H.M. Ammari is with the Wireless Sensor and Mobile Ad-hoc
Networks(WiSeMAN) Research Lab, Department of Computer Science,
HofstraUniversity, 205 Adams Hall, Hempstead, NY 11549.E-mail:
[email protected].
. S.K. Das is with the Center for Research in Wireless Mobility
andNetworking (CReWMaN), Department of Computer Science and
En-gineering, University of Texas at Arlington, Room 249B,
Nedderman Hall,Arlington, TX 76019-0015. E-mail:
[email protected].
Manuscript received 9 June 2008; revised 31 Jan. 2009; accepted
9 June 2009;published online 28 Oct. 2009.Recommended for
acceptance by R. Marculescu.For information on obtaining reprints
of this article, please send e-mail to:[email protected], and
reference IEEECS Log Number TC-2008-06-0282.Digital Object
Identifier no. 10.1109/TC.2009.166.
0018-9340/10/$26.00 2010 IEEE Published by the IEEE Computer
Society
Authorized licensed use limited to: IEEE Xplore. Downloaded on
May 13,2010 at 11:46:58 UTC from IEEE Xplore. Restrictions
apply.
-
Traditional (or unconditional) connectivity assumes thatany
subset of nodes can potentially fail at the same time,including all
the neighbors of a given node. While suchconnectivity can measure
the fault tolerance of small-scalenetworks, it is not effective for
large-scale dense networks,such as highly dense, 3D k-covered WSNs.
In such networkswith thousands of sensors, it is highly unlikely
that all theneighbors of a given sensor node fail simultaneously.
This isdue to high-density deployment and heterogeneity of
thesensors. Assuming a uniform sensor distribution, the ratioof the
size of the neighbor set of a given sensor to the totalnumber of
sensors in a WSN covering a field of volume V is4R3=3V , where R V
1=3 is the radius of the communica-tion range of the sensors. The
probability of failure of all theneighbors of a given sensor could
be identified with thisratio, and hence, is very low. Furthermore,
in real-worldsensing applications [19], WSNs have
heterogeneoussensors with unequal energy levels and different
sensing,processing, and communication capabilities, thus
increasingthe network reliability and lifetime [36]. This also
impliesthat the probability that the entire neighbor set of a
sensorfails at the same time is very low.
In this study, we use a more general (and realistic)concept of
connectivity, called conditional connectivity [15]with respect to
some property P, defined as follows: Theconditional connectivity of
a connected graph G is thesmallest number of nodes of G whose
removal disconnectsG into components each of which has property P.
Anothergeneralization of connectivity, called restricted
connectivity[12], is based on the concept of forbidden faulty set,
where allneighbors of a node cannot simultaneously fail.
1.1 Contributions
In this paper, we address the following questions concern-ing
coverage and connectivity in 3D k-covered WSNs:
1. Given a 3D field, what is the minimum sensor spatialdensity
to guarantee full k-coverage of the field?
2. What is the connectivity and conditional connectiv-ity of 3D
k-covered WSNs?
To this end, our main contributions are summarized
asfollows:
1. We characterize k-coverage of 3D WSNs by usingthe fundamental
result of the intersection of convexsets stated in Hellys Theorem.
Based on thischaracterization and the concept of Reuleaux
tetra-hedron, we compute the minimum sensor spatialdensity required
to guarantee full k-coverage of a3D field. We find that this
density is k=0:422r30 forhomogeneous 3D k-covered WSNs, where r0
r=1:066and r is the sensing radius of sensors. For hetero-geneous
3D k-covered WSNs, r is replaced with rmin,the minimum sensing
radius of all sensors. Oursimulation results match well with the
analyticalresults.
2. Based on the minimum sensor spatial density, weprove that the
connectivity of 3D k-covered WSNs ismuch higher than the degree k
of sensing coverageprovided by the network. More precisely,
thisconnectivity is 9:926R=r03k for homogeneousk-covered WSNs,
where r0 r=1:066 and R is the
radius of the communication range of sensors. Forheterogeneous
3D k-coveredWSNs, r is replacedwithrmin and R with Rmin, where Rmin
is the minimumradius of the communication range of all sensors.
3. We show that the traditional connectivity metricused to
capture network fault-tolerance underesti-mates the resilience of
3D k-covered WSNs. Inparticular, we find that the conditional
connectivityof 3D k-covered WSNs is equal to r0 2R3 r30=r30k for
homogeneous k-covered WSNs, wherer0 r=1:066. For heterogeneous 3D
k-coveredWSNs,r andR are replaced with rmin andRmin,
respectively.
The remainder of this paper is organized as follows:Section 2
reviews related work on coverage and connectiv-ity. Section 3
introduces the basic assumptions anddefinitions. Section 4
discusses coverage and connectivityof 3D k-covered WSNs, while
Section 5 focuses on theirconditional connectivity. Section 6 shows
how to relax someof the assumptions used in our analysis. Section 7
concludesthe paper.
2 RELATED WORK
In this section, we review a sample of approaches forcoverage
and connectivity in 2D and 3D WSNs. For morecomprehensive review on
coverage and connectivity inWSNs, the interested reader is referred
to [8], [13].
2.1 Coverage and Connectivity of 2D WSNs
In [1], a directional sensors-based approach is proposed forWSN
coverage, where the coverage region of a directionalsensor depends
on the location and the orientation ofsensors. A differentiated
coverage algorithm is presented in[9] for heterogeneous WSNs, where
different network areasdo not have the same coverage degree. In
[14], the authorsproposed centralized and distributed algorithms
for con-nected sensor cover, so the network can self-organize
itstopology in response to a query and activate the
necessarysensors to process the query. The relationship
betweensensing coverage and communication connectivity of WSNsis
studied and distributed protocols to guarantee that bothcoverage
and connectivity are proposed in [18]. Based onthe Voronoi diagram
and graph search algorithms, anoptimal polynomial time (worst and
average case) algo-rithm for coverage calculation is proposed in
[26].
Studies on k-coverage to maintain connectivity in 2DWSNs have
been the focus of several works. The first tostudy coverage and
connectivity in WSNs in an integratedfashion is due to [34], where
the authors proved that if thecommunication radius of sensors is
double their sensingradius, a network is connected given the
coverage isprovided. They also proposed a coverage
configurationprotocol based on the degree of coverage of the
application.In [17], polynomial-time algorithms are presented for
thecoverage problem to check whether every point in a field isat
least k-covered, depending on whether sensors have thesame or
different sensing ranges. Efficient distributedalgorithms are
proposed in [24] to optimally solve the bestcoverage problem with
the least energy consumption. In[37], Zhang and Hou proposed a
distributed algorithm inorder to keep a small number of active
sensors in a WSNregardless of the relationship between sensing and
com-munication ranges. The results in [34] and [37] are
244 IEEE TRANSACTIONS ON COMPUTERS, VOL. 59, NO. 2, FEBRUARY
2010
Authorized licensed use limited to: IEEE Xplore. Downloaded on
May 13,2010 at 11:46:58 UTC from IEEE Xplore. Restrictions
apply.
-
improved in [32] by proving that if the original network
isconnected and the identified active nodes can cover thesame
region as all the original nodes, then the networkformed by the
active nodes is connected when thecommunication range is at least
twice the sensing range.An optimal deployment strategy is described
in [5] toachieve both full coverage and 2-connectivity regardless
ofthe relationship between communication and sensing radiiof
sensors. A distributed and localized algorithm using theconcept of
the kth-order Voronoi diagram is proposed in[40] to provide fault
tolerance and extend the networklifetime, while maintaining a
required degree of coverage.
Necessary and sufficient conditions for 1-covered,1-connected
wireless sensor grid networks are presented in[29]. A variety of
algorithms have been proposed tomaintainconnectivity and coverage
in large WSNs. In [20], Kumaret al. proposed k-barrier coverage for
intrusion detection.The authors established an optimal deployment
pattern forachieving k-barrier coverage and developed efficient
globalalgorithms for checking such coverage. As shown in [21],
theminimum number of sensors needed to achieve k-coveragewith high
probability is approximately the same regardlessof whether sensors
are deployed deterministically orrandomly, if sensors fail or sleep
independently with equalprobability. The changes of the probability
of k-coveragewith the radius of the sensing range of sensors or the
numberof sensors are studied in [33]. The k-coverage set and
thek-connected coverage set problems are formalized in termsof
linear programming with two nonglobal solutions in [35].The
coverage problem in heterogeneous planar WSNs as aset intersection
problem is formulated in [22] and [23] whichalso derived analytical
expressions to which quantify thestochastic coverage. It is proved
in [38] that the requireddensity to k-cover a square region depends
on both the sidelength of the square field and the coverage degree
k.2.2 Coverage and Connectivity of 3D WSNs
Coverage and connectivity in 3D WSNs have gainedrelatively less
attention in the literature. A placementstrategy based on Voronoi
tessellation of 3D space isproposed in [4], which creates truncated
octahedral cells.Several fundamental characteristics of randomly
deployed3D WSNs for connectivity and coverage are investigated
in[28], which compute the required sensing range to
guaranteecertain degree of coverage of a region, the minimum
andmaximum network degrees for a given communicationrange as well
as the hop-diameter of the network. Related toour work is the novel
result discussed in [43], which provedthat the breadth of the
Reuleaux tetrahedron is not constant.This shows that the properties
of 2D space cannot be directlyextended to 3D space. Indeed, the
Reuleaux triangle [44](counterpart of Reuleaux tetrahedron in 2D
space) has aconstant width. Note that the Reuleuax tetrahedron is
thesymmetric intersection of four congruent spheres such thateach
sphere passes through the centers of the other threespheres.
However, the Reuleaux triangle [44] corresponds tothe symmetric
intersection of three congruent disks such thateach disk passes
through the centers of the other two disks.Thus, its constant width
is equal to the radius of these disks.
Our work can be viewed as an extension of [4] byconsidering
k-coverage in 3D WSNs. Moreover, existingworks on coverage and
connectivity in WSNs assumed thenotion of traditional connectivity,
while our work considersamore realisticmeasure, namely, conditional
connectivity [18],
which is based on the concept of forbidden faulty set [15].Also,
our work exploits the result given in [43] with regardto the
breadth of the Reuleaux tetrahedron discussed earlier.This helps us
provide correct measures of connectivity andfault tolerance of 3D
WSNs based on an accurate character-ization of k-coverage of 3D
fields.
3 ASSUMPTIONS AND DEFINITIONS
In this section, we present the assumptions made in ouranalysis
and define some key concepts. Relaxation of someof the assumptions
will be discussed in Section 6 andAppendix B.
We consider staticWSNs,which represent the sensing
andcommunication ranges of sensors as spheres. From now on,the
sensing and communication ranges of a sensor si arecalled sensing
and communication spheres of radii ri and Ri,respectively, centered
at iwhich represents the location of si.In a homogeneous
(heterogeneous)WSNs, the sensors have (donot have) the same sensing
ranges and the same communica-tion ranges. The communication links
between sensors aresupposed to be perfectly reliable, while sensor
nodes can failor die independently due to low battery power. We
considera cubic field of volume V in 3D euclidean space.
Moreover,we assume that the volumes of the sensing and
communica-tion spheres of sensors are negligible compared to V
suchthat ri, Ri V 1=3.
Let us define the following terms. A location of afield is said
to be 1-covered (or sensed) if it belongs to thesensing sphere of
at least one sensor. A 3D convexregion C is said to be k-covered if
each location in C iscovered by at least k sensors.
The neighbor set of a sensor si is defined as Nsi fsj : ji jj
Rig, where j is the location of the neighbor sjof si and ji jj is
the euclidean distance between i and j.
The breadth of 3D convex region C is the maximumdistance between
tangential planes on opposing faces oredges of C.
A communication graph of a WSN is a graph G S; L,where S is a
set of sensors and L is a set of communicationlinks between them
such that for all si, sj 2 S, si; sj 2 L ifji 0jj Ri.
The connectivity of a communication graph G of a WSNis equal to
k if G can be disconnected by removing at leastk sensors.
4 UNCONDITIONAL CONNECTIVITY OF 3Dk-COVERED WSNS
In this section, we show how to guarantee k-coverage of a3D
field, derive the corresponding minimum sensor spatialdensity, and
compute measures of unconditional (or tradi-tional) connectivity
for homogeneous 3D k-covered WSNs,where any subset of sensors can
fail. The results forheterogeneous WSNs are summarized in Appendix
A.
4.1 Minimum Sensor Density for k-Coverage
Lemma 1 characterizes the breadth of a 3D k-coveredconvex
region.
Lemma 1. A 3D convex region C is guaranteed to be k-coveredwith
exactly k sensors if its breadth is less than or equal to r,the
radius of the sensing spheres of the sensors.
AMMARI AND DAS: A STUDY OF k-COVERAGE AND MEASURES OF
CONNECTIVITY IN 3D WIRELESS SENSOR NETWORKS 245
Authorized licensed use limited to: IEEE Xplore. Downloaded on
May 13,2010 at 11:46:58 UTC from IEEE Xplore. Restrictions
apply.
-
Proof (By contradiction). Assume that the breadth of a 3Dconvex
region C does not exceed r and C is not k-coveredwhen exactly k
sensors are deployed in it. Notice thateach of these k sensors is
located on the boundary orinside of C. Thus, there must be at least
one location 2 C that is not k-covered. In other words, there is
atleast one sensor si located at
0 such that j 0j > r,which contradicts our hypothesis that
the breadth of Cdoes not exceed r. tuIt is true that the deployment
of k sensors in a 3D convex
region, say C, whose breadth is larger than r cannotguarantee
its k-coverage, where r is the radius of the sensingrange of the
sensors. Let pi and pj be two points inC such thatone sensor si is
located at pi and pi; pj b > r, where b isthe breadth of C and
pi; pj is the euclidean distancebetween pi and pj. Given that b
> r, it is impossible for si tosense any event that occurs at
pj. Thus, there is at least onesensor (i.e., si) among those k
sensors, which cannot cover pj,and hence, C cannot be
k-covered.
Next, we compute the minimum sensor spatial densityrequired for
guaranteeing k-coverage of a 3D field. To thisend, we compute the
maximum volume of a 3D convexregion C that is guaranteed to be
k-covered when exactlyk sensors are deployed in it. First, we state
Hellys Theorem[6], a fundamental result that characterizes the
intersectionof m convex sets in n-dimensional space, where m n
1.Hellys Theorem (Intersection of convex sets) [6]. Let be a
family of convex sets in IRn such that for m n 1, anym members
of have a nonempty intersection. Then, theintersection of all
members of is nonempty.
From Hellys Theorem [6], we infer that givenk 4 sensors, a 3D
convex region C is k-covered by thosek sensors if and only if C is
4-covered by any four of those ksensors. Given that the breadth of
C is r, the networkinduced by sensors located in C is guaranteed to
beconnected if R r, where r and R are, respectively, theradii of
the sensing and communication spheres of sensors.Now, let us
address the first question: What is the minimumsensor spatial
density necessary to guarantee full k-coverage of a3D field?
Theorem 1 computes the minimum sensor density.
Theorem 1. Let r be the radius of the sensing spheres of
sensorsand k 4. The minimum sensor spatial density required tofully
k-cover a 3D field is computed as
r; k k0:422r30
; 1
where r0 r=1:066.Proof. Let Ck be the intersection of k sensing
spheres and
assume that their centers do not coincide in a 3D field.
FromLemma 1, it follows that Ck is guaranteed to be k-coveredby
exactly k sensors if its breadth does not exceed theradius r of the
sensing spheres of sensors. Thus, themaximum volume of Ck is
obtained when its breadth isequal to r. From Hellys Theorem [6], it
follows that theintersection of k sensing spheres is not empty if
theintersection of any four of these k spheres is not empty. Onthe
other hand, the intersection set operator requires thatthe maximum
intersection volume of these k sensingspheres be equal to that of
four spheres provided that themaximum distance between any pair of
these k sensors
does not exceed r. Let us focus on the analysis of foursensing
spheres. The maximum overlap volume of foursensing spheres such
that every point in this overlapvolume is 4-covered corresponds to
the configuration: thecenter of each sensing sphere is at distance
r from thecenters of all other three sensing spheres. Precisely,
thesensing sphere of each of the four sensors passes throughthe
centers of the other three sensing spheres, as shown inFig. 1. The
edges between the centers of these four spheresform a regular
tetrahedron and the shape of the intersectionvolume of these four
spheres is known as the Reuleauxtetrahedron [43] and denoted by RT
r. Unfortunately, itwas proved that the Reuleaux tetrahedron does
not have aconstant breadth [43]. Indeed, while the distance
betweensome pairs of points on the boundary of the
Reuleauxtetrahedron RT r is equal to r, the maximum distancebetween
other pairs of points on the boundary ofRT r isequal to 1:066r
[43], i.e., slightly larger than r. This impliesthat the Reuleaux
tetrahedron RT r cannot be k-coveredwith exactly k sensors given
that the distance betweensome pairs of points on the boundary of RT
r is largerthan r, where r is the radius of the sensing spheres of
thesensors. Therefore, the Reuleaux tetrahedron that isguaranteed
to be k-covered with exactly k sensors shouldhave a side length
equal to r0 r=1:066. The volume of theReuleaux tetrahedron RT r0 is
given by [43]
volr0 3ffiffiffi2
p 49 162 tan1
ffiffiffi2
pr30=12 0:422r30:
Thus, RT r0 is the maximum volume that can bek-covered by
exactly k sensors, where k 4. Weconclude that the maximum volume of
Ck, denoted by
volmaxCk, is equal to volmaxCk 0:422r30. Given thatvolmaxCk has
to contain k sensors to k-cover Ck, weconclude that the minimum
sensor spatial density per
unit volume required for full k-coverage of a 3D field is
computed as
r; k k=volmaxCk k=0:422r30:ut
We should mention that the notion of the arc lengthdiscussed in
[43] corresponds to the (maximum) breadth ofthe Reuleaux
tetrahedron. Indeed, it is possible to find twoparallel plans that
bound the Reuleaux tetrahedron suchthat the maximum distance
between these two plans isequal to r only.
246 IEEE TRANSACTIONS ON COMPUTERS, VOL. 59, NO. 2, FEBRUARY
2010
Fig. 1. (a) Intersection of four symmetric spheres and (b)
theircorresponding Reuleaux tetrahedron.
Authorized licensed use limited to: IEEE Xplore. Downloaded on
May 13,2010 at 11:46:58 UTC from IEEE Xplore. Restrictions
apply.
-
It is worth emphasizing that the value of r; k is tight inthe
sense that it is minimum given that k sensors should belocated
within volmaxCk, such that Ck is guaranteed to bek-covered with
exactly these k sensors. Also, notice thatr; k depends only on the
coverage degree k dictated by asensing application and the radius r
of the sensing range ofsensors. Moreover, r; k increases with k.
Indeed, highcoverage degree k requires more sensors to be deployed,
andhence, a denserWSN is necessary. Also, r; k decreases as
rincreases. When the sensing range gets larger, a fewernumber of
sensors is needed to fully k-cover a 3D field. Bothof these
observations reflect the real behavior of the sensors.Thus, r; k
does not depend on the size of the field asopposed to the result of
the stochastic approach in [38].
Lemma 2 uses Theorem 1 and states a sufficient conditionfor
k-coverage of a 3D field.
Lemma 2. A 3D field is guaranteed to be k-covered if anyReuleaux
tetrahedron region of side r0 in the field contains atleast k
sensors, where r0 r=1:066 and k 4.
It is worth noting that there may be some differencesbetween
analytical and simulation/implementation resultsdue to the boundary
effects. In fact, it is impossible todecompose a 3D field into
complete Reuleaux tetrahedra suchthat the Reuleaux tetrahedra close
to the border of a 3D fieldlie entirely inside it. Hence, more than
necessary number ofsensors is used to k-cover the border of the 3D
field.
Next, based on the minimum sensor spatial density(Theorem 1) and
another criterion to be specified in thefollowing section, we
compute the network connectivity of3D k-covered WSNs.
4.2 Computing Network Connectivity
Data accuracy depends on the size of the connectedcomponent that
contains the sink. It reaches the highestvalue when the sink
belongs to the largest connectedcomponent of the network. Thus,
high-quality coveragerequires all source sensors be connected to
the sink. That iswhy we focus on the sink to compute the
connectivity of3D k-covered WSNs. In other words, connectivity of
WSNsshould be so defined as to take into consideration theinherent
structure of this type of network. Indeed, sensorshave neither the
same role nor the same impact on thenetwork performance. Thus,
measuring the connectivity ofWSNs should account for their specific
morphology, wherethe sink is the most critical node in the network.
Hence, wecompute the connectivity of 3D k-covered WSNs based onthe
size of the connected component that includes the sink.Theorem 2
deals with homogeneous WSNs.
Theorem 2. Let G be a communication graph of a homogeneous3D
k-covered WSN deployed in a cubic field of volume V .
Theconnectivity of G, which is denoted by G, is given by
1G G 3G; 2where
1G 12:0243k;
3G RV2=3k
0:422r30;
in which r0 r=1:066, R=r, and k 4.
Proof. The optimum position of the sink in terms of
energy-efficient data gathering is the center of the cubic
field[25]. Let 0 be the location of the sink s0. We consider
thefollowing three cases depending on the size of theconnected
component that includes the sink. Also, giventhat sensor failure is
due to low battery power, weassume that the sink has infinite
source of energy, thusexcluding the possibility of a faulty
sink.
Case 1: Single-node connected component. In thiscase, there are
at least two components, one of thembeing the single-node component
containing the sink.Finding the minimum number of nodes to
disconnect thenetwork requires that the disconnected network has
onlytwo components. In order to isolate the sink, all itsneighbors
must fail. Hence, at least the communicationsphere of the sink
should contain no sensor but the sink.
Let N be a random variable that counts the number ofsensor
failures to isolate the sink s0. Given that sensorsare randomly and
uniformly deployed in a volume Vwith density r; k per unit volume,
where R V 1=3,the expected number of neighbors of the sink is given
by
EN r; kkB0; Rk; 2awhere jB0; Rj 4R3=3 is themeasure of the
volume ofthe communication sphere B0; R of the sink s0 locatedat 0.
Thus, the expected minimum number of sensor
failures to isolate s0 is equal to EN. Substituting (1) in(2a),
we find that the network connectivity is given by
1G EN 12:0243k; 2bwhere R=r. Figs. 2 and 3 plot the function 1G
in(2b). Clearly, 1G increases with the ratio and thedegree of
coverage k. More importantly, 1G is muchhigher than k.
Case 2: Nontrivial connected components. Twoconfigurations of
the disconnected network are ofparticular interest where the two
connected componentsof the disconnected network are separated by a
vacantregion (or gap). Furthermore, any pair of sensors, onefrom
each component, are separated by a distance atleast equal to R in
order to prohibit any communicationbetween the two components. In
the first configuration(Fig. 4a), the component including the sink
is reduced to
AMMARI AND DAS: A STUDY OF k-COVERAGE AND MEASURES OF
CONNECTIVITY IN 3D WIRELESS SENSOR NETWORKS 247
Fig. 2. Plot of the function 1G (fix k and vary ).
Authorized licensed use limited to: IEEE Xplore. Downloaded on
May 13,2010 at 11:46:58 UTC from IEEE Xplore. Restrictions
apply.
-
its communication sphere. Thus, the volume of thevacant region,
denoted by gap0; R, and surroundingthe component of the sink, is
given by
kgap0; Rk 42R3=3 4R3=3 9:333R3: 2cThus, the expected minimum
number of sensor failures
to isolate the component of the sink is given by
EN r; kkgap0; Rk: 2dSubstituting (1) in (2d), we find that the
networkconnectivity is equal to
2G EN 84:1643k; 2e
where R=r.In the second configuration, the original network
is
split into two components such that the vacant regionforms a
cuboid, denoted by cubR, and whose sides areR; V 1=3 and V 1=3, as
shown in Fig. 4b. Now, thisconfiguration corresponds to the
smallest connectedcomponent containing the sink if the field has to
bedivided into two regions such that none of themsurrounds the
other. Thus, the expected minimumnumber of sensor failures to
isolate the connectedcomponent containing the sink is given by
EN kcubRk; 2fwhere
kcubRk RV 2=3: 2gSubstituting (1) and (2g) in (2f), it follows
that network
connectivity is equal to
3G EN RV2=3k
0:422r30: 2h
It is easy to check that 3G > 2G since R V 1=3.Case 3:
Largest connected component. This config-
uration is totally opposite to the one given in Case 1 andhas
only one isolated sensor. Since we are interested ink-coverage of
the entire field, such a network isconsidered as disconnected. The
network connectivityis the same as in Case 1. Thus,
1G G 3G:It is easy to check that G > k. tuAll these bounds
and those to be derived in next sections
are based on our fundamental result stated in Theorem 1.Since
these bounds are based on the minimum sensorspatial density for
full k-coverage of a 3D field, they arelower bounds.
4.3 Boundary Effects
Network connectivity is defined as the minimum number ofsensors
whose failure (or removal) disconnects the network.Given the cubic
geometry of a 3D field we considered,sensors located close to the
border of the field are affectedby the boundary effects. Indeed,
the communication rangesof these sensors cover areas outside of the
deployment area,and hence, have fewer number of communication
neighborscompared to all other sensors (especially the ones
whosecommunication ranges lay entirely inside the field). How-ever,
the proposed network connectivity measures considerthe sink as the
most critical node in the network and whoseisolation would
definitely kill the entire network. Becausethe optimum position of
the sink in terms of energy-efficientdata gathering is the center
of the cubic field [25], theboundary effects do not exist at all,
and thus, our derivedbounds on the connectivity are correct. Even
when our
248 IEEE TRANSACTIONS ON COMPUTERS, VOL. 59, NO. 2, FEBRUARY
2010
Fig. 3. Plot of the function 1G (fix and vary k).
Fig. 4. 2D projection of nontrivial-connected components of
thedisconnected network.
Authorized licensed use limited to: IEEE Xplore. Downloaded on
May 13,2010 at 11:46:58 UTC from IEEE Xplore. Restrictions
apply.
-
approach for computing network connectivity measuresconsiders
all sensors as critical and peer-to-peer (seeAppendix C for
detailed discussion), the boundary effectsdo not have any impact on
the derived bounds on networkconnectivity. Indeed, we are
interested in the minimumnumber of sensor failures to disconnect
the network.
As far as sensor deployment to achieve 3D k-coverage
isconcerned, we should mention that the boundary effectshave an
impact on the performance of the network. ByLemma 2, a 3D field is
guaranteed to be fully k-covered if andonly if each Reuleaux
tetrahedron region in the field containsat least k sensors.
However, it is impossible to randomlydecompose a 3D field into an
integer number of Reuleauxtetrahedron regions because of the
boundary of the field.Indeed,most of the Reuleaux tetrahedron
regions close to theborder of a 3D field do not entirely lay inside
the deploymentarea. Therefore, more than necessary sensors would
beneeded to achieve k-coverage of these Reuleaux tetrahedronregions
on the border of the field. Simulation results reportedin Section
4.1.2 show that the sensor spatial density necessaryto fully
k-cover a cubic field is a bit higher than the boundgiven in
Theorem 1, mainly due to the boundary effects.
Next, we introduce new measures of connectivity of3D k-covered
WSNs by placing a specific constraint on asubset of sensors that
would fail.
5 CONDITIONAL CONNECTIVITY OF 3D k-COVEREDWSNS
In this section, we use the concepts of conditional
connectivity[15] and forbidden faulty set [12], as a remedy to the
aboveshortcomings of the traditional (unconditional)
connectivitymetric.Ourapproach isbasedon forbidden faulty sensor
sets.
Let G S;E be a communication graph representing a3D k-covered
WSN. Define a forbidden faulty sensor set of Gas a set of faulty
sensors that includes the entire neighborset of a given sensor.
Consider the property P: A faultysensor set cannot include the
entire neighbor set of a givensensor. A faulty sensor set
satisfying property P is denotedby FP, where FP S and defined
by
FP fU Sj8si 2 S : Nsi 6 Ug:The conditional connectivity of G
with respect to P, denotedby G : P, is the minimum size of FP such
that thegraph Gd S FP; ESFP is disconnected, where ESFPis a set of
remaining communication edges between thenonfaulty sensors.
As we will see, our results prove that 3D k-coveredWSNscan
sustain a larger number of sensor failures under therestriction
imposed on the faulty sensor set. According toTheorem 1, the
minimum number of sensors necessary tok-cover a cubic field of
volume V is given by jSminj V k=0:422r3, where r is the radius of
the sensing spheres ofsensors. The probability of the failure of
the entire neighborset of a given sensor can be identified with the
ratio of thesize of the neighbor set of a given sensor to the total
numberof sensors jSminj. This ratio is equal to 4R3=3V , which is
verylow given that R V 1=3. This shows that the
traditionalconnectivity, which does not impose any restriction on
thefaulty sensor set, is not a useful metric for 3D k-coveredWSNs,
which are highly dense networks.
5.1 Homogeneous 3D k-Covered WSNs
Theorem 3 computes the conditional connectivity of
homogeneous 3D k-covered WSNs.
Theorem 3. The conditional connectivity of a homogeneous 3D
k-covered WSN k 4 is given by
G : P r0 2R3 r30k
r30; 3
where r0 r=1:066.Proof. We consider the following two cases
based on the
type of connected component that contains the sink.Case 1:
Smallest connected component. According to
our conditional connectivity model, no sensor can beisolated,
and hence, no trivial component can be part ofthe disconnected
network. Under the assumption offorbidden faulty sensor set, the
smallest connectedcomponent that is disconnected from the rest of
thenetwork and contains the sink can be determined asfollows: In
order to achieve k-coverage of the cubic field,every location must
be k-covered, including the location0 of the sink s0. Otherwise,
the k-coverage property willnot be satisfied. Therefore, the
smallest connectedcomponent that includes the sink consists of k
sensorsdeployed in the Reuleaux tetrahedron of side r0 andcentered
at 0. In order to disconnect the sink under theforbidden faulty
sensor set constraint, the Reuleauxtetrahedron should be surrounded
by an empty annulusof width equal to R (sensors located in the
annulus havefailed), as can be seen in Figs. 5 and 6. The
Reuleauxtetrahedron together with this annulus forms a
largerReuleaux tetrahedron of side r0 2R. The volume of theannulus,
denoted by A0; R, is equal to
kA0; Rk 0:422r0 2R3 0:422r30:Therefore, the expected conditional
minimum number of
sensor failures to disconnect the smallest component
including the sink can be computed as
EN : P r; kkA0; Rk: 3a
AMMARI AND DAS: A STUDY OF k-COVERAGE AND MEASURES OF
CONNECTIVITY IN 3D WIRELESS SENSOR NETWORKS 249
Fig. 5. Two nested concentric Reuleaux tetrahedra.
Authorized licensed use limited to: IEEE Xplore. Downloaded on
May 13,2010 at 11:46:58 UTC from IEEE Xplore. Restrictions
apply.
-
Substituting (1) in (3a), we find that the
conditionalconnectivity is given by
1G : P EN : P r0 2R3 r30k
r30: 3b
It is easy to check that the forbidden faulty setconstraint is
satisfied for both the faulty sensors (locatedinside the annulus
and which have failed) and nonfaultysensors (located outside the
annulus). Indeed, any sensorin the inner Reuleaux tetrahedron still
has nonfaultyneighbors in the inner Reuleaux tetrahedron. Besides,
anysensor outside the outer Reuleaux tetrahedron still hasnonfaulty
neighbors outside the outer Reuleaux tetrahe-dron. Also, any faulty
sensor within the annulusAR hasnonfaulty neighbors located in the
inner Reuleauxtetrahedron and outside the outer Reuleaux
tetrahedron.
Case 2: Largest connected component. This case issimilar to the
previous one. However, the sink belongs tothe largest connected
component. Hence, the discon-nected network consists of two
components: the oneincluding the sink and a second component
associatedwith sensors located in a Reuleaux tetrahedron of side
r.Using the same analysis as in Case 1, we obtain the
sameconditional network connectivity:
2G : P 1G : P : 3cFrom both cases 1 and 2, the conditional
connectivity of
homogeneous 3D k-covered WSNs is computed as
G : P 1G : P :ut
5.2 Heterogeneous 3D k-Covered WSNs
We observe that computing the conditional connectivity
ofheterogeneous 3D k-covered WSNs is not a
straightforwardgeneralization of the approach used previously for
homo-geneous 3D k-covered WSNs. If we use the same process
aspreviously, we either violate the forbidden faulty sensor
setconstraint or maintain network connectivity. Precisely, ifthe
width of the annulus containing the faulty sensors isRmax, then
sensors whose communication radii are lessthan or equal to Rmax=2
may be located in the annulus.
Thus, the entire neighbor set of this type of sensors wouldfail
at the same time, and hence, the property P would beviolated (Fig.
7a). Now, if the width of the annuluscontaining the faulty sensors
is less than Rmax, then thenonfaulty sensors of one connected
component will be ableto communicate with the nonfaulty sensors of
the otherconnected component of the disconnected network. Hence,the
network is still connected (Fig. 7b). In this case, it isimpossible
to find an exact value of conditional connectiv-ity for
heterogeneous 3D k-covered WSNs. Next, wecompute their lower and
upper bounds based on the typesof sensors in and around the
annulus.
Lemma 3. The conditional connectivity of the heterogeneous
3D
k-covered WSNs is given by
1G : FP G : P 2G : FP ; 4where
1G : P r0min 2Rmin3 r0min3k
r0min3
;
2G : P rmax 2Rmax3 r3maxk
r0min3
;
250 IEEE TRANSACTIONS ON COMPUTERS, VOL. 59, NO. 2, FEBRUARY
2010
Fig. 6. 2D projection of an annulus of width R surrounding a
Reuleauxtetrahedron of side r.
Fig. 7. 2D projection: (a) forbidden fault sensor set constraint
violated
and (b) connectivity maintained.
Authorized licensed use limited to: IEEE Xplore. Downloaded on
May 13,2010 at 11:46:58 UTC from IEEE Xplore. Restrictions
apply.
-
in which
k 4; r0min minfrj=1:066 : sj 2 Sg;rmax maxfrj=1:066 : sj 2
Sg;Rmin minfRj : sj 2 Sg; andRmax maxfRj : sj 2 Sg:
Proof. As above, we consider the following two casesdepending on
the size of the connected component thatincludes the sink:
Case 1: Smallest connected component. In order tocompute a lower
bound on the conditional connectivityof heterogeneous 3D k-covered
WSNs, we consider theReuleaux tetrahedron centered at location 0 of
the sinks0, which will be disconnected from the network. First,we
assume that the annulus containing the faulty sensorsas well as the
volume surrounding it contain only leastpowerful sensors, and
hence, the width of this annulus isequal to Rmin. Also, to
guarantee that the sink will not beisolated, which would violate
the forbidden faulty sensorset constraint, the Reuleaux tetrahedron
centered at 0should have a side equal to rmin. These two
conditionshelp disconnect the network while satisfying the
for-bidden faulty sensor set constraint. The volume of theannulus
A0; Rmin is given by
kA0; Rmink 0:422rmin 2Rmin3 0:422r3min: 4aHence, the expected
conditional minimum number ofsensor failures to disconnect the
connected componentincluding the sink (or the inner Reuleaux
tetrahedron)from the rest of the network is computed as
EN : P rmin; kkA0; Rmink; 4bwhere
rmin; k k0:422r3min
and
rmin minfrj=1:066 : sj 2 Sg:Thus, the conditional network
connectivity is given by
1G : P EN : P rmin 2Rmin3 r3mink
r3min; 4c
where k 4 and Rmin minfRj : sj 2 Sg.In order to compute an upper
bound on the condi-
tional connectivity of heterogeneous 3D k-coveredWSNs, we assume
that sensors inside the annulus arethe most powerful ones. Thus,
the side of the innerReuleaux tetrahedron should be equal to rmax,
while thewidth of the annulus surrounding it should be equal
toRmax. It is easy to check that this setup will disconnectthe
network, while satisfying the forbidden faulty setconstraint. The
upper bound on the conditional con-nectivity is given by
2G : P EN : P rmin; kkA0; Rmaxk
rmax 2Rmax3 r3maxk
r3min:
4d
where
rmin; k k0:422r3min
; k 4;
rmin minfrj=1:066 : sj 2 Sg;rmax maxfrj=1:066 : sj 2 Sg; andRmax
maxfRj : sj 2 Sg:
Case 2: Largest connected component. In this case,the sink
belongs to the largest connected component ofthe disconnected
network. Hence, the previous analysisapplies to any sensor in the
network instead of the sink.Thus, the conditional connectivity of
heterogeneous 3Dk-covered WSNs satisfies
1G;P G;P 2G;P :ut
Next, we relax the assumptions used in our previousanalysis to
enhance the practicality of our results.
6 DISCUSSIONS
The analysis of the minimum sensor spatial densitynecessary for
k-coverage of a 3D field and networkconnectivity of 3D k-covered
WSNs is based on the unitsphere model, where the sensing and
communication rangesof sensors are modeled by spheres. In other
words, sensorsare supposed to be typically isotropic. Although
thisassumption is the basis for most of the protocols forcoverage
and connectivity in WSNs, it may not holduniversally, and thus, may
not be valid in practice. InAppendix B, we show how to relax this
assumption in orderto promote the applicability of our results to
real-world3D WSN scenarios, and summarize our results for theconvex
model, where the sensing and communicationsranges of sensors are
convex but not necessarily spherical.Moreover, we assumed that our
results for networkconnectivity hold for a degree of sensing
coverage k, wherek 4. In this section, we show how to relax
severalassumptions to account for more realistic scenarios.
6.1 Relaxing the Assumption of k 4The analysis of k-coverage and
connectivity for 3Dk-covered WSNs are valid for all k 4. Since the
breadthof the Reuleaux tetrahedron is equal to r, our results
canalso be used for k 3. That is, a 3D field can be k-coveredby
deploying k sensors in the Reuleaux tetrahedron, wherek 3. However,
the network would be denser thannecessary (especially for k 1) and
the coverage degreewould be higher than that dictated by the
application.
6.2 Sensor Placement Strategy
Notice that under the assumption of spherical model, it
isimpossible to achieve a degree of coverage exactly equal to kin
all the locations of the cube. Thus, a sensor placementstrategy to
achieve k-coverage should be devised in such away that every
location in the cube is k0-covered, where k0 isvery close to k.
This placement strategy should benefit fromthe geometry of the
Reuleaux tetrahedron. The sensorplacement problem can be
transformed into a problem ofcovering a cube with overlapping sets
of congruent
AMMARI AND DAS: A STUDY OF k-COVERAGE AND MEASURES OF
CONNECTIVITY IN 3D WIRELESS SENSOR NETWORKS 251
Authorized licensed use limited to: IEEE Xplore. Downloaded on
May 13,2010 at 11:46:58 UTC from IEEE Xplore. Restrictions
apply.
-
Reuleaux tetrahedra. An optimal covering consists in usinga
minimum number of Reuleaux tetrahedra by minimizingthe overlap
volume between them. As described inSection 4.1.2, two adjacent
Reuleaux tetrahedra overlapsuch that the faces of their
corresponding regular tetrahedraare entirely coinciding with each
other. Thus, the curvededges of the Reuleaux tetrahedra should
overlap, so thesame subset of sensors deployed on these curved
edgescould participate in covering the space associated with
bothReuleaux tetrahedra, thus minimizing the number ofsensors
required to k-cover the cube. Notice that sensorslocated in a 3D
lens, which corresponds to the overlapvolume of two adjacent
Reuleaux tetrahedra of side length r,are at distance r from any
point in their volumes, and hence,participate in k-covering both
tetrahedra. The design ofduty-cycling protocols to k-cover a 3D
field with a minimumnumber of sensors should select sensors based
on thisobservation.
6.3 Sink-Independent Connectivity Measures
Although in centralized algorithms, the concept of sink iswell
defined, it is likely that distributed algorithms, such asthe
consensus-based algorithm, will be implemented forWSNs. In this
case, the concept of (fixed) sink would losevalue. It would be
interesting to revise the definition ofconnectivity to take this
concept into account. We suggestthat all the nodes be considered as
peer-to-peer. Thus, wedefine connectivity with respect to all
sensors in thenetwork. Given the geometry of the deployment field
thatwe consider (cube), the boundary sensors, i.e., sensors
locatedat the boundary of the cube, have small neighbor sets.
Inparticular, the sensors located at the eight corners of thecube
sb1, . . . , sb8 as shown in Fig. 8 are highly likely to havethe
smallest neighbor set. Our measures of connectivity willbe based on
one of these boundary sensors to find theminimum number of sensors
of its neighbor set that shouldfail in order to disconnect the
network. It is easy to checkthat compared to the sink, the size of
the neighbor set of aboundary sensor is equal to a quarter of that
of the neighborset of the sink. Thus, the previous connectivity
measurescomputed in Sections 4 and 5 with respect to the sinkremain
the same for a boundary sensor but are weighted bya coefficient
equal to 1
4. For more details, the interested
reader is referred to Appendix C.
6.4 Stochastic Sensing and Communication Models
In the deterministic sensing model, which has been consideredso
far, a point in a field is covered by a sensor si based onthe
euclidean distance ; si between and si. In thispaper, we use
coverage of a point and detection of an
event interchangeably. Formally, the coverage Cov; si ofa point
by a sensor si is equal to 1, if ; si r, and 0,else. As can be
seen, this sensing model considers thesensing range of a sensor as
a sphere, and hence, all sensorreadings are precise and have no
uncertainty. However,given the signal attenuation and the presence
of noiseassociated with sensor readings, it is necessary to
consider amore realistic sensing model by defining Cov; si
usingsome probability function. That is, the sensing capability ofa
sensor must be modeled as the probability of successfuldetection of
an event, and hence, should depend on thedistance between it and
the event as well as the type ofpropagation model being used
(free-space versus multi-path). Indeed, it has been showed that the
probability thatan event in a distributed detection application can
bedetected by an acoustic sensor depends on the distancebetween the
event and the sensor [10]. A realistic sensingmodel for passive
infrared (PIR) sensors that reflect theirnonisotropic range was
presented in [7]. This sensingirregularity of PIR sensors was
verified by simulations [7].Thus, in our stochastic sensing model,
the coverage Cov; siis defined as the probability of detection p;
si of an event atpoint by sensor si as follows:
p; si e;si ; if ; si r;0; otherwise;
5
where represents the physical characteristic of the
sensorssensing units and 2 4 is the path-loss exponent.Precisely,
we have 2 for the free-space model and 2 < 4 for the multipath
model. Our stochastic sensing modelis motivated by Elfes one [11],
where the sensing capabilityof a sonar sensor is modeled by a
Gaussian probabilitydensity function. A probabilistic sensing model
for coverageand target localization in WSNs was proposed in [42].
Thissensing model considers ; si r re instead of ; si,where r is
the detection range of the sensors and re < r is ameasure of
detection uncertainty.
Under the stochastic sensing model, a point in a field issaid to
be probabilistically k-covered if the detection prob-ability of an
event occurring at by at least k sensors is atleast equal to some
threshold probability 0 < pth < 1.
In this section, we exploit the results of Section 4.1
tocharacterize probabilistic k-coverage in 3D WSNs based onour
stochastic sensing model. Theorem 4 computes theminimum k-coverage
probability pk;min such that every point ina field is
probabilistically k-covered.
Theorem 4. Let r be the radius of the nominal sensing range
ofthe sensors, r0 r=1:066, and k 4. The minimumk-coverage
probability so that each point in a 3D field isprobabilistically
k-covered by at least k sensors under ourstochastic sensing model
defined in (5) is computed as
pk;min 1 1 er0 k
: 6
Proof. First, we identify the least k-covered point in a
3Dfield, so we can compute pk;min. By Lemma 2, k sensorsshould be
deployed in a Reuleaux tetrahedron region ofside r0 r=1:066 in the
field to achieve k-coverage of a3D field with a minimum number of
sensors. It is easy tocheck that the least k-covered point lc is
the one thatcorresponds to the configuration where all deployed
252 IEEE TRANSACTIONS ON COMPUTERS, VOL. 59, NO. 2, FEBRUARY
2010
Fig. 8. Eight boundary sensors located on the corners of a
cube.
Authorized licensed use limited to: IEEE Xplore. Downloaded on
May 13,2010 at 11:46:58 UTC from IEEE Xplore. Restrictions
apply.
-
k sensors are located at a distance r0 r=1:066 from lc.In other
words, lc is the farthest point from all thosek sensors. Hence, the
distance between lc and each ofthese k sensors is equal to r0
r=1:066. Thus, theminimum k-coverage probability for the least
k-coveredpoint lc by k sensors under the stochastic sensing modelin
(5) is given by
pk;min 1Yki11 p; si 1 1 er
k:
utThe stochastic k-coverage problem is to select a mini-
mum subset Smin S of sensors such that each point in a3D field
is k-covered by at least k sensors and the minimumk-coverage
probability of each point is at least equal to somegiven threshold
probability pth, where 0 < pth < 1. Thishelps us compute the
stochastic sensing range rs, whichprovides probabilistic k-coverage
of a field with a prob-ability no less than pth. Lemma 4 computes
the value of rs.
Lemma 4. Let k 3 and 2 4. The stochastic sensingrange rs of the
sensors that is necessary to probabilisticallyk-cover a 3D field
with a minimum number of sensors andwith a probability no lower
than 0 < pth < 1 is given by
rs 1log1 1 pth1=k1=
; 7
where represents the physical characteristic of the
sensorssensing units.
Proof. From (6), we deduce that pk;min pth ) rs 1 log1 1
pth1=k1=. Since we are interested inthe minimum number of sensors
to probabilisticallyk-cover a 3D field, we should consider the
maximumvalue of rs, i.e., the maximum stochastic sensing range
ofthe sensors. This will allow the sensors to probabilisti-cally
k-cover as much space of the 3D deployment field aspossible. Thus,
rs 1 log1 1 pth1=k1=. tuThe upper bound on the stochastic sensing
range rs of the
sensors computed in (7) will be used to compute ourmeasures of
connectivity and fault tolerance of 3D k-coveredWSNs under the
assumption of more realistic, stochasticsensing, and communication
models. Fig. 9 shows rs fordifferent values of pth and kwhile
considering the free-spacemodel ( 2) (Fig. 9a) and the multipath
model ( 4)(Fig. 9b). Note that rs decreases as a function of pth,
k, and .This is due to the fact that the minimum probability pk;min
ofk-coverage of the same location bymultiple sensors decreasesas
pth, k, and increase.
Lemma 5 states a sufficient condition for
probabilistick-coverage of a 3D field based on our stochastic
sensingmodel in (5), the threshold probability pth, and the degree
kof coverage.
Lemma 5. Let k 4. A 3D field is probabilistically k-coveredwith
a probability no lower than 0 < pth < 1 if any
Reuleauxtetrahedron of maximum breadth rs=1:066 in the field
containsat least k sensors.
Lemma 6 states a sufficient condition for connectivitybetween
sensors under our stochastic sensing model.
Lemma 6. Let k 4. The sensors that are selected to k-cover a3D
field with a probability no less than 0 pth 1 under thestochastic
sensing model defined in (5) are connected if the
radius of their stochastic communication range Rs is at
leastequal to their stochastic sensing range rs., Rs rs.Given that
network connectivity is defined as theminimum
number of sensors of a neighbor set which need to fail inorder
to disconnect the network, we should consider theminimum stochastic
communication range of the sensors,i.e., Rs rs, to minimize the
size of the neighbor set of eachsensor. Therefore, to find our
unconditional and conditionalmeasures of network connectivity and
fault tolerance for3D k-coveredWSNs usingmore realistic scenarios,
includingstochastic models for sensing and communications, we
needto replace r by rs and R by rs in Sections 4 and 5,
andAppendices A, B, and C.
6.4.1 Simulation Results
In this section, we present the simulation results using
ahigh-level simulator written in the C language. We considera cubic
field of side length 1,000 m. All simulations arerepeated 200 times
and the results are averaged.
Fig. 10 plots the sensor spatial density as a function ofthe
degree of coverage k for different values of thethreshold
probability pth and for a path-loss exponent 2. As expected, the
density increases with pth. Indeed,as we increase pth, more sensors
would be needed toachieve the same degree of coverage k. Recall
that thebreadth of the Reuleaux tetrahedron that is guaranteed tobe
covered with exactly k sensors decreases as pth and increase.
Precisely, this breadth is equal to rs=1:066.However, for pth 0:8,
the sensor density tends to decreasewhen k goes from 4 to 5 and
increases afterward. Thisbehavior is clearly noticeable for pth
0:9. This is mainly
AMMARI AND DAS: A STUDY OF k-COVERAGE AND MEASURES OF
CONNECTIVITY IN 3D WIRELESS SENSOR NETWORKS 253
Fig. 9. Upper bound of rs versus k.
Authorized licensed use limited to: IEEE Xplore. Downloaded on
May 13,2010 at 11:46:58 UTC from IEEE Xplore. Restrictions
apply.
-
due to the stochastic nature of the sensing range of thesensors,
which depends on the logarithm of pth, thethreshold probability, k,
and .
Fig. 11 plots the achieved degree of coverage k versus thetotal
number of deployed sensors. Moreover, we vary bothpth and fix 2.
Definitely, higher number of deployedsensors would yield higher
coverage degree. Here, also, anyincrease in pth would require a
larger number of deployedsensors to provide the same degree of
coverage. Notice thatthe same observation holds for pth 0:8 and pth
0:8 as inthe previous experiment.
6.5 3D Sensing Applications
In the case of WSNs deployed on the trees of differentheights in
a forest, the sensors could be seen almosteverywhere in the space.
Pompili et al. [27] proposeddifferent deployment strategies for 2D
and 3D communica-tion architectures for underwater acoustic sensor
networks,where the sensors are anchored at the bottom of the
oceanfor the 2D design and float at different depths of the oceanto
cover the entire 3D region. Indeed, oceanographic datacollection,
pollution monitoring, offshore exploration, dis-aster prevention,
and assisted navigation are all typicalapplications of underwater
sensor networks [2], [3]. ForWSNs deployed in buildings with
multiple floors, sensorsare placed on the ground and/or the wall,
but the networksseldom contain sensors floating in the middle of
the room.The first examples show that our proposed 3D model isvalid
and can be applied to choose the sensor density inpractical
problems. The last example, however, shows thelimited validity of
our model due to the restriction imposedon the placement of sensors
inside buildings or rooms.
7 CONCLUSION
In this paper, we investigated coverage and connectivity in3D
k-covered WSNs. Indeed, emerging applications, such asunderwater
acoustic sensor networks, require 3D design.We proposed the
Reuleaux tetrahedron model to guaranteek-coverage of a 3D field.
Based on the geometric propertiesof Reuleaux tetrahedron, we
derived the minimum sensorspatial density to ensure k-coverage of a
3D space. Also, wecomputed the connectivity of homogeneous and
hetero-geneous 3D k-covered WSNs. Our approach takes intoaccount an
inherent characteristic of WSNs in that the sinkhas a critical role
in terms of data processing and decision
making, compared to the rest of the network. Thus, wecomputed
the connectivity of 3D k-covered WSNs based onthe size of the
connected component that includes the sink.We conclude that the
connectivity of 3D k-covered WSNs ismuch higher than the degree of
sensing coverage kprovided by the network. The traditional
connectivitymetric, however, is defined in an abstract way.
Moreprecisely, it does not consider the inherent properties ofWSNs
because it assumes that any subset of nodes can failsimultaneously.
This assumption is not valid for hetero-geneous 3D k-covered WSNs.
To compensate for theseshortcomings, we proposed more realistic
measures ofconnectivity based on the concept of forbidden faulty
set.Using this concept, we found that 3D k-covered WSNs cansustain
a large number of sensor failures.
We believe that our results have practical significance
forsensor network designers to develop 3D applications
withprescribed degrees of coverage and connectivity.
Theseconnectivity measures can be exploited in the design
offault-tolerant topology control protocols for 3D k-coveredWSNs.
Our future work will focus on the design of efficientsensor
deployment strategies for 3D k-covered WSNs. Also,we are interested
in the design of data routing protocols onduty-cycled 3D k-covered
WSNs. Indeed, joint k-coverageand routing poses major challenges
due to the time-varyingconnectivity of the network. This is mainly
due to the factthat the sensors are turned on or off to save energy
andextend the network lifetime.
APPENDIX A
HETEROGENEOUS 3D k-COVERED WSNS
A.1 Minimum Sensor Density for k-Coverage
Achieving k-coverage of a 3D field by heterogeneoussensors would
depend on the least powerful ones in termsof their sensing
capabilities. Lemmas 7 and 8 correspond toLemma 1 and Theorem 1,
respectively.
Lemma 7. If the breadth of a 3D convex region C is at most
equalto the minimum radius rmin of the sensing spheres of
sensors,then C is guaranteed to be k-covered if k (k 4) sensors
aredeployed in it, where rmin minfrj=1:066 : sj 2 Sg.From Lemma 7,
it follows that connectivity between
sensors located in the Reuleaux tetrahedron of side rminrequires
that Rmin be at least equal to rmin, i.e., Rmin rmin.
254 IEEE TRANSACTIONS ON COMPUTERS, VOL. 59, NO. 2, FEBRUARY
2010
Fig. 10. Sensor spatial density versus k. Fig. 11. k versus
nd.
Authorized licensed use limited to: IEEE Xplore. Downloaded on
May 13,2010 at 11:46:58 UTC from IEEE Xplore. Restrictions
apply.
-
Lemma 8. Let rmin be the minimum radius of the sensing spheresof
sensors and k 4. The minimum sensor spatial densityneeded for
k-coverage of a 3D field by heterogeneous WSNs isgiven by
rmin; k k0:422r3min
; 8
where rmin minfrj=1:066 : sj 2 Sg.A.2 Network Connectivity
Lemma 9 computes the connectivity measures for hetero-geneous 3D
k-covered WSNs.
Lemma 9. Let G be a communication graph of a heterogeneous3D
k-covered WSN with Rmin rmin and k 4. Theconnectivity of the graph
G is given by
4G G 3G; 9where
3G RmaxV2=3k
0:422r3min; 4G 12:02432k;
2 Rmin=rmin; k 4; rmin minfrj=1:066 : sj 2 Sg;Rmin minfRj : sj 2
Sg; and Rmax maxfRj : sj 2 Sg:
APPENDIX B
CONVEX SENSING AND COMMUNICATION MODEL
The assumption of spherical sensing and communicationranges of
sensors given in Section 3 may not hold in real-world WSN
platforms. It has been observed in [39] that thecommunication range
of MICA motes [16] is asymmetricand depends on the environments. It
is also found in [41]that the communication range of radios is
highly probabil-istic and irregular.
In this Appendix, for problem tractability, we consider aconvex
model, where the sensing and communication rangesof sensors are
convex but not necessarily spherical.
First, we define the notion of largest enclosed sphere of a3D
convex region C as a sphere that lies entirely inside Cand whose
diameter is equal to the minimum distancebetween any pair of points
on the boundary of the region C.
B.1 Homogeneous 3D k-Covered WSNs
We consider homogeneous sensors that have the sameconvex sensing
ranges and communication ranges. Lem-mas 10 and 11 correspond to
Lemma 1 and Theorem 1,respectively. Their proof is similar to that
in Section 4 byusing the largest enclosed sphere instead of the
sensingsphere.
Lemma 10. If k 4 homogeneous sensors are deployed in a3D convex
region C, then the region C is k-covered; if itsbreadth does not
exceed rled, the radius of the largest enclosedsphere of the
sensing range.
Lemma 11. The minimum sensor spatial density required
toguarantee k-coverage of a 3D field is given by
rled; k k0:422rled0
3; 10a
where rled stands for the radius of the largest enclosed sphere
ofthe sensing range, r0led rled=1:066, and k 4.
Now, we discuss how those results can be derived usinga convex
model, where the sensing and communicationranges of the sensors may
not necessarily be spherical.
B.2 Heterogeneous 3D k-Covered WSNs
Lemmas 12 and 13 correspond to Lemma 1 and Theorem
1,respectively.Theirproof is also the sameas that inSection4byusing
the largest enclosed sphere instead of sensing sphere.
Lemma 12. A 3D convex region C is guaranteed to be k-coveredwhen
exactly k heterogeneous sensors are deployed in it, ifthe breadth
of C does not exceed rminled , where r
minled
minfrled=1:066 : sj 2 Sg and k 4.Lemma 13. The minimum sensor
spatial density required to
k-cover a 3D field is given by
rminled ; k
k0:422rminled 2
; 10b
where rminled minfrled=1:066 : sj 2 Sg and k 4.
The measures of network connectivity can be derivedusing the
same approach as in Section 4.1.2 and Lemma 13.Thus, the assumption
of unit sphere model for sensing andcommunication ranges of sensors
can be relaxed with theaid of the largest enclosed sphere of their
sensing range.
APPENDIX C
CASE OF UNDERWATER WSNS
The results in Sections 4 and 5 and in Appendices A and Bare
only applicable to the connectivity of sink node.Although the
connectivity of sink is critical, in somescenarios, such as
underwater WSNs [2], [3], any sensormay be critical due to the high
cost, for instance.
In the following, we extend our network connectivitymeasures for
3D k-covered WSNs to the case where anysensor in the network is
critical. Specifically, we consider aboundary sensor, i.e., a
sensor located at one corner of thecubic field. Such a boundary
sensor has the minimumnumber of communication neighbors given that
all sensorsare located within the deployment regionthe cube,
andhence, the actual communication range of a boundarysensor is
only a quarter of its communication sphere. In [34],a boundary
sensor is considered to compute the connectiv-ity of 2D k-covered
WSNs.
Theorem 4 summarizes the connectivity measures withrespect to a
boundary sensor for homogeneous 3D k-coveredWSNs. The case of
heterogeneous WSNs and the case ofsensors with convex sensing and
communication ranges canbe treated similar to Sections 4 and 5
aswell as Appendices Aand B. We omit the proof of Theorem 4 as we
use the sameanalysis in Sections 4 and 5, and Appendices A and
B.
Theorem 4. Let G be a communication graph of a homogeneous3D
k-covered WSN deployed in a cubic field, where the radii ofthe
sensing and communication spheres of sensors are r and
R,respectively. The connectivity of G is computed as
G 3:023k; 11
AMMARI AND DAS: A STUDY OF k-COVERAGE AND MEASURES OF
CONNECTIVITY IN 3D WIRELESS SENSOR NETWORKS 255
Authorized licensed use limited to: IEEE Xplore. Downloaded on
May 13,2010 at 11:46:58 UTC from IEEE Xplore. Restrictions
apply.
-
whereas the conditional connectivity of G is given by
G : P 3:02r0 R3 r30k
r30; 12
where r0 r=1:066, R=r, and k 4.
ACKNOWLEDGMENTS
The authors gratefully acknowledge the insightful com-ments of
the anonymous reviewers which helped improvethe quality and
presentation of the paper significantly. Thework of H.M. Ammari is
partially supported by theUS National Science Foundation (NSF)
grant 0917089 and aNew Faculty Start-Up Research Grant from Hofstra
Collegeof Liberal Arts and Sciences Deans Office. The work of
S.K.Das is partially supported by the AFOSR grant A9550-08-1-0260
and NSF grants IIS-0326505 and CNS-0721951. Hiswork is also
supported by (while serving at) the NSF. Anyopinion, findings, and
conclusions or recommendationsexpressed in this material are those
of the authors and do notnecessarily reflect the views of the
NSF.
REFERENCES[1] J. Ai and A. Abouzeid, Coverage by Directional
Sensors in
Randomly Deployed Wireless Sensor Networks, J.
CombinatorialOptimization, vol. 11, no. 1, pp. 21-41, Feb.
2006.
[2] I.F. Akyildiz, D. Pompili, and T. Melodia, Underwater
AcousticSensor Networks: Research Challenges, Ad Hoc Networks, vol.
3,pp. 257-279, Mar. 2005.
[3] I. Akyildiz, D. Pompili, and T. Melodia, Challenges for
EfficientCommunications in Underwater Acoustic Sensor Networks,ACM
SIGBED Rev., vol. 1, no. 2, pp. 3-8, July 2004.
[4] S. Alam and Z. Haas, Coverage and Connectivity in
Three-Dimensional Networks, Proc. ACM MobiCom, pp. 346-357,
2006.
[5] X. Bai, S. Kumar, D. Xuan, Z. Yun, and T.H. Lai,
DeployingWireless Sensors to Achieve Both Coverage and
Connectivity,Proc. ACM MobiHoc, pp. 131-142, 2006.
[6] B. Bollobas, The Art of Mathematics: Coffee Time in
Memphis.Cambridge Univ. Press, 2006.
[7] Q. Cao, T. Yan, J. Stankovic, and T. Abdelzaher, Analysis
ofTarget Detection Performance for Wireless Sensor Network,
Proc.Intl Conf. Distributed Computing in Sensor Systems (DCOSS),pp.
276-292, 2005.
[8] M. Cardei and J. Wu, Energy-Efficient Coverage Problems
inWireless Ad-Hoc Sensor Networks, Computer Comm., vol. 29,no. 4,
pp. 413-420, Feb. 2006.
[9] X. Du and F. Lin, Maintaining Differentiated Coverage
inHeterogeneous Sensor Networks, EURASIP J. WCN, vol. 5,no. 4, pp.
565-572, 2005.
[10] M. Duarte and Y. Hu, Distance Based Decision Fusion in
aDistributed Wireless Sensor Network, Proc. Intl
WorkshopInformation Processing in Sensor Networks (IPSN), pp.
392-404, 2003.
[11] A. Elfes, Using Occupancy Grids for Mobile Robot
Perceptionand Navigation, Computer, vol. 22, no. 6, pp. 46-57, June
1989.
[12] A. Esfahanian, Generalized Measures of Fault Tolerance
withApplication to N-Cube Networks, IEEE Trans. Computers, vol.
38,no. 11, pp. 1586-1591, Nov. 1989.
[13] A. Ghosh and S.K. Das, Coverage and Connectivity Issues
inWireless Sensor Networks, Mobile, Wireless and Sensor Net-works:
Technology, Applications and Future Directions. John Wiley&
Sons, Inc. Mar. 2006.
[14] H. Gupta, Z. Zhou, S.R. Das, and Q. Gu, Connected
SensorCover: Self-Organization of Sensor Networks for Efficient
QueryExecution, IEEE/ACM Trans. Networking, vol. 14, no. 1, pp.
55-67,Feb. 2006.
[15] F. Harary, Conditional Connectivity, Networks, vol. 13, pp.
347-357, 1983.
[16] M. Horton, D. Culler, K. Pister, J. Hill, R. Szewczyk, and
A. Woo,MICA: The Commercialization of Microsensor Motes,
SensorsMagazine, pp. 40-48, Apr. 2002.
[17] C.-F. Huang and Y.-C. Tseng, The Coverage Problem in
aWireless Sensor Network, Proc. ACM Intl Workshop WirelessSensor
Networks and Applications (WSNA), pp. 115-121, 2003.
[18] C.-F. Huang, Y.-C. Tseng, and H.-L. Wu, Distributed
Protocolsfor Ensuring Both Coverage and Connectivity of a Wireless
SensorNetwork, ACM Trans. Sensor Networks, vol. 3, no. 1, pp. 1-24,
Mar.2007.
[19] R. Kumar, V. Tsiatsis, and M.B. Srivastava,
ComputationHierarchy for In-Network Processing, Proc. ACM Intl
WorkshopWireless Sensor Networks and Applications (WSNA), pp.
68-77, 2003.
[20] S. Kumar, T.H. Lai, and A. Arora, Barrier Coverage with
WirelessSensors, Proc. ACM MobiCom, pp. 284-298, 2005.
[21] S. Kumar, T.H. Lai, and J. Balogh, On k-Coverage in a
MostlySleeping Sensor Network, Proc. ACM MobiCom, pp.
144-158,2004.
[22] L. Lazos and R. Poovendran, Coverage in Heterogeneous
SensorNetworks, Proc. Intl Symp. Modeling and Optimization in
Mobile,Ad Hoc and Wireless Networks (WiOpt), pp. 1-10, 2006.
[23] L. Lazos and R. Poovendran, Stochastic Coverage in
Hetero-geneous Sensor Networks, ACM Trans. Sensor Networks, vol.
2,no. 3, pp. 325-358, Aug. 2006.
[24] X.-Y. Li, P.-J. Wan, and O. Frieder, Coverage in Wireless
Ad-HocSensor Networks, IEEE Trans. Computers, vol. 52, no. 6, pp.
753-763, June 2003.
[25] J. Luo and J.-P. Hubaux, Joint Mobility and Routing for
LifetimeElongation in Wireless Sensor Networks, Proc. IEEE
INFOCOM,pp. 1735-1746, 2005.
[26] S. Megerian, F. Koushanfar, M. Potkonjak, and M.
Srivastava,Worst and Best-Case Coverage in Sensor Networks, IEEE
Trans.Mobile Computing, vol. 4, no. 1, pp. 84-92, Jan./Feb.
2005.
[27] D. Pompili, T. Melodia, and I.F. Akyildiz, Deployment
Analysisin Underwater Acoustic Wireless Sensor Networks, Proc.
IntlWorkshop Underwater Networks (WUWNet), pp. 48-55, 2006.
[28] V. Ravelomanana, Extremal Properties of
Three-DimensionalSensor Networks with Applications, IEEE Trans.
Mobile Comput-ing, vol. 3, no. 3, pp. 246-247, July-Sept. 2004.
[29] S. Shakkottai, R. Srikant, and N. Shroff, Unreliable Sensor
Grids:Coverage, Connectivity and Diameter, Ad Hoc Networks, vol.
3,no. 6, pp. 702-716, Nov. 2005.
[30] E. Shih, S. Cho, N. Ickes, R. Min, A. Sinha, A. Wang, and
A.Chandrakasan, Physical Layer Driven Protocol and AlgorithmDesign
for Energy-Efficient Wireless Sensor Networks, Proc.ACM MobiCom,
pp. 272-287, 2001.
[31] S. Soro and W.B. Heinzelman, On the Coverage Problem
inVideo-Based Wireless Sensor Networks, Proc. Intl Conf. Broad-band
Networks (BaseNets), pp. 932-939, 2005.
[32] D. Tian and N. Georganas, Connectivity Maintenance
andCoverage Preservation in Wireless Sensor Networks, Ad
HocNetworks, vol. 3, no. 6, pp. 744-761, Nov. 2005.
[33] P.-J. Wan and C.-W. Yi, Coverage by Randomly
DeployedWireless Sensor Networks, IEEE Trans. Information
Theory,vol. 52, no. 6, pp. 2658- 2669, June 2006.
[34] G. Xing, X. Wang, Y. Zhang, C. Lu, R. Pless, and C.
Gill,Integrated Coverage and Connectivity Configuration for
EnergyConservation in Sensor Networks, ACM Trans. Sensor
Networks,vol. 1, no. 1, pp. 36-72, Aug. 2005.
[35] S. Yang, F. Dai, M. Cardei, and J. Wu, On Connected
MultiplePoint Coverage in Wireless Sensor Networks, Intl J.
WirelessInformation Networks, vol. 13, no. 4, pp. 289-301, Oct.
2006.
[36] M. Yarvis, N. Kushalnagar, H. Singh, A. Rangarajan, Y. Liu,
and S.Singh, Exploiting Heterogeneity in Sensor Networks, Proc.
IEEEINFOCOM, pp. 878-890, 2005.
[37] H. Zhang and J. Hou, Maintaining Sensing Coverage
andConnectivity in Large Sensor Networks, Ad Hoc and SensorWireless
Networks, vol. 1, nos. 1/2, pp. 89-124, Mar. 2005.
[38] H. Zhang and J. Hou, On the Upper Bound of -Lifetime
forLarge Sensor Networks, ACM Trans. Sensor Networks, vol. 1, no.
2,pp. 272-300, Nov. 2005.
[39] J. Zhao and R. Govindan, Understanding Packet
DeliveryPerformance in Dense Wireless Sensor Networks, Proc.
ACMConf. Embedded Networked Sensor Systems (SenSys), pp. 1-13,
2003.
[40] Z. Zhou, S. Das, and H. Gupta, Fault Tolerant Connected
SensorCover with Variable Sensing and Transmission Ranges,
Proc.IEEE Conf. Sensor and Ad Hoc Comm. and Networks (SECON),pp.
594-604, 2005.
256 IEEE TRANSACTIONS ON COMPUTERS, VOL. 59, NO. 2, FEBRUARY
2010
Authorized licensed use limited to: IEEE Xplore. Downloaded on
May 13,2010 at 11:46:58 UTC from IEEE Xplore. Restrictions
apply.
-
[41] G. Zhou, T. He, S. Krishnamurthy, and J. Stankovic, Impact
ofRadio Irregularity on Wireless Sensor Networks, Proc. MobiSys,pp.
125-138, 2004.
[42] Y. Zou and K. Chakrabarty, Sensor Deployment and
TargetLocalization in Distributed Sensor Networks, ACM Trans.
SensorNetworks, vol. 3, no. 2, pp. 61-91, 2004.
[43]
http://mathworld.wolfram.com/ReuleauxTetrahedron.html,2009.
[44] http://mathworld.wolfram.com/ReuleauxTriangle.html,
2009.
Habib M. Ammari received the Diploma ofEngineering and Doctorat
de Specialite degreesin computer science from the Faculty
ofSciences of Tunis, Tunisia, in 1992 and 1996,respectively, the MS
degree in computer sciencefrom Southern Methodist University in
Decem-ber 2004, and the PhD degree in computerscience and
engineering from the University ofTexas at Arlington (UTA) in May
2008. He is anassistant professor of computer science in the
Department of Computer Science at Hofstra University, where he
is thefounding director of the Wireless Sensor and Mobile Ad-Hoc
Networks(WiSeMAN) Research Laboratory. He was on the faculty of the
SuperiorSchool of Communications of Tunis (SupCom Tunis), from 1992
to2005 (an engineer of computer science, 1992-1993; a lecturer
ofcomputer science, 1993-1997; an assistant professor of
computerscience, 1997-2005; received tenure in 1999). His main
researchinterests lie in the areas of wireless sensor and mobile ad
hocnetworking, and multihop mobile wireless Internet architectures
andprotocols. In particular, he is interested in coverage,
connectivity,energy-efficient data routing and information
dissemination, faulttolerance, security in wireless sensor
networks, and the interconnectionbetween wireless sensor networks,
mobile ad hoc networks, and theglobal IP Internet. He received the
US National Science Foundation(NSF) Research Grant Award and the
Faculty Research and Develop-ment Grant Award from Hofstra College
of Liberal Arts and Sciences,both in 2009. He published his first
book Challenges and Opportunitiesof Connected k-Covered Wireless
Sensor Networks: From SensorDeployment to Data Gathering (Springer,
2009). He received the JohnSteven Schuchman Award for 2006-2007
Outstanding Research by aPhD Student and the Nortel Outstanding CSE
Doctoral DissertationAward, both from UTA in 2008 and 2009,
respectively. He was arecipient of the TPC Best Paper Award from
EWSN 08 and the BestContribution Paper Award from IEEE PerCom
08-Google PhD forum.He also was an ACM Student Research Competition
(ACM SRC)nominee at ACM MobiCom 05. He was selected for inclusion
in the2006 edition of Whos Who in America and the 2008-2009
HonorsEdition of Madison Whos Who Among Executives and
Professionals.He serves as an associate editor of the International
Journal ofCommunication Systems and the International Journal of
NetworkProtocols and Algorithms. He is on the Editorial Board of
theInternational Journal of Mobile Communications and the
InternationalJournal on Advances in Networks and Services. He also
is on theEditorial Review Board of the International Journal of
DistributedSystems and Technologies. He is the cofounder and a
coeditor, withDr. Sylvia Silberger, Associate Professor and Chair
of the Department ofMathematics at Hofstra University, of the
Sciences Undergraduate AndGraduate REsearch Experiences (Sciences
U-AGREE) Journal, whichis published at Hofstra University. He
served as the program cochair/workshop cochair of WiMAN 10, IWCMC
10, IQ2S 09, and WiMAN 09.He has served as a reviewer for several
international journals, includingthe IEEE Transactions on Mobile
Computing, the IEEE Transactions onParallel and Distributed
Systems, the ACM Transactions on SensorNetworks, the IEEE
Transactions on Vehicular Technology, the IEEETransactions on
Wireless Communications, the Mobile and NetworkApplications, the
Wireless Networks, the Ad Hoc Networks, theComputer Networks, the
Ad Hoc & Sensor Wireless Networks, theInternational Journal of
Sensor Networks, the Information ProcessingLetters, the Computer
Communications, the Journal of Parallel andDistributed Computing,
the Information Sciences, the InternationalJournal of Computer and
Applications, and the Data and KnowledgeEngineering Journal, and as
a technical program committee member ofnumerous IEEE and ACM
conferences and symposia, including theIEEE Infocom, IEEE ICDCS,
IEEE PerCom, SSS, IEEE MASS, IEEEMSN, IEEE LCN, and EWSN. He is a
member of the IEEE.
Sajal K. Das is a university distinguishedscholar professor of
computer science andengineering and the founding director of
theCenter for Research in Wireless Mobility andNetworking (CReWMaN)
at the University ofTexas at Arlington (UTA). He is currently
aprogram director at the US National ScienceFoundation (NSF) in the
Division of ComputerNetworks and Systems. He is also an
E.T.S.Walton professor of Science Foundation of
Ireland; a visiting professor at the Indian Institute of
Technology (IIT)at Kanpur and IIT Guwahati; an honorary professor
of Fudan Universityin Shanghai and international advisory professor
of Beijing JiaotongUniversity, China; and a visiting scientist at
the Institute of InfocommResearch (I2R), Singapore. His current
research interests includewireless and sensor networks, mobile and
pervasive computing, smartenvironments and smart heath care,
pervasive security, resource andmobility management in wireless
networks, mobile grid computing,biological networking, applied
graph theory, and game theory. He haspublished more than 400 papers
and more than 35 invited book chaptersin these areas. He holds five
US patents in wireless networks and mobileInternet, and coauthored
the books Smart Environments: Technology,Protocols, and
Applications (Wiley, 2005) and Mobile Agents inDistributed
Computing and Networking (Wiley, 2009). He is a recipientof the
IEEE Computer Society Technical Achievement Award (2009),the IEEE
Engineer of the Year Award (2007), and several Best PaperAwards in
various conferences such as EWSN 08, IEEE PerCom 06,and ACM MobiCom
99. At UTA, he is a recipient of the Lockheed MartinTeaching
Excellence Award (2009), UTA Academy of DistinguishedScholars Award
(2006), University Award for Distinguished Record ofResearch
(2005), College of Engineering Research Excellence Award(2003), and
Outstanding Faculty Research Award in Computer Science(2001 and
2003). He is frequently invited as keynote speaker atinternational
conferences and symposia. He serves as the foundingeditor-in-chief
of Elseviers Pervasive and Mobile Computing (PMC)Journal, and also
as an associate editor of the IEEE Transactions onMobile Computing,
the ACM/Springer Wireless Networks, the IEEETransactions on
Parallel and Distributed Systems, and the Journal ofPeer-to-Peer
Networking. He is the founder of the IEEE WoWMoMSymposium and the
cofounder of the IEEE PerCom Conference. He hasserved as the
general and technical program chair as well as TPCmember of
numerous IEEE and ACM conferences. He is a seniormember of the
IEEE.
. For more information on this or any other computing
topic,please visit our Digital Library at
www.computer.org/publications/dlib.
AMMARI AND DAS: A STUDY OF k-COVERAGE AND MEASURES OF
CONNECTIVITY IN 3D WIRELESS SENSOR NETWORKS 257
Authorized licensed use limited to: IEEE Xplore. Downloaded on
May 13,2010 at 11:46:58 UTC from IEEE Xplore. Restrictions
apply.