On the Probability Bound for Uncovered Areas
in Sensor Networks
Wei Yen
Department of Computer Science and Engineering
Tatung University
Taipei, R.O.C.
Abstract—In a sensor network, a large number of sensors are
deployed in the sensing field. Although there are guidelines
describing the recommended number of sensors, not many
analyses have been done to capture the characteristics of the
uncovered areas. Given the system parameters, this paper
provides a probability bound that estimates the likeliness of the
uncovered areas. We find that the probability bound is equal to
the probability that no sensor is found in a corresponding circle.
This bound has a concise form and provides quantitative
measurement to deployment efficiency. In addition, it impacts
other related research topics such as finding minimum full
coverage sets for lengthening sensor expectancy. Numerical
results are included to show how the bound changes along with
various system parameters.
sensor
sensing field
backbone
sink
user A
user B
user C
sensor region
Figure 1. Architecture of the Sensor Network
Keywords—sensor networks, coverage, probability distribution,
I. INTRODUCTION
S
ensor networks have attracted a lot of research interest in
recent years. In general, it includes two major portions
(Figure 1) [1]. The first part is accounted for the name of the
network. That is, a large volume of sensors are deployed in an
area, referred to as the sensing field. Each sensor has limited
power supply and generally transmits information on demand.
To save cost, both its memory size and computing capability are
constricted. The range of a sensor’s detection function is called
the sensor region. It may or may not be the same as its
communication range. The second portion is called the sink.
This is a critical device focal to the communications in the
sensor network. Retrieval commands and collected data need to
go through the sink to be exchanged between the sensor network
and the outside world. The sink is usually connected to the
Internet or satellites. It is common to assume sink has greater
resources.
Sensor networks have a broad spectrum of applications
including but not limited to inventory and environmental
control, military and civil surveillance, monitoring hazardous
environment, distributed recognition and computing, and
environmental monitoring [2]. For example, every year
hundreds of thousands trees and wild lives are lost to fires. With
the help of the sensor network, effective monitoring can prevent
such disasters from taking place. In another application, the
sensor network makes intelligent building possible.
Although the sensor network sounds similar to the ad hoc
network on surface, there are some important distinctions. The
former, thought not completely static, is confined in a small
neighborhood, possesses much limited power and computation
capability, and rarely initiates peer-to-peer communications. In
fact, most communications in the sensor network are between
the sink and the sensors. One of the more interesting problems
in the sensor network is the deployment strategy [3, 4]. If the
sensors are placed in the field simultaneously, it is possible that
many of them will fail during a short period of time and cause
functional disruption. Other noticeable impacts are in the
domains of routing, power reservation, deployment
environment, timing synchronization, and security requirements
[1, 5, 6, 7, 8, 9, 10, 11, 12].
In many applications, minimizing the number of sensors is
not a priority because of two main reasons. First, it is foreseen
that the cost of sensors will be in the range of less than one U.S.
dollar. Secondly, excessive sensors can provide good fault
tolerance [13, 14, 15, 16]. A good benchmark of this
excessiveness is the sensor density, measuring the number of
sensors in a unit area. Although abundant sensors seem
desirable, all sensors transmitting collected information in
response to retrieval requests is usually not. Contrarily, people
strive to reduce the number of messages transmitted in the
network. Since the sensor density is high, many messages
contain overlapped or redundant information. By eliminating
unnecessary messages, we can reduce power consumption and
lengthen the expectancy of the network [17, 18, 19].
This paper aims to answer the probability bound of  percent
of the sensing field is not covered by any sensors given the
following parameters, the total number of sensors (N), the radius
of sensor range (r), the area of the sensing field (A), and uniform
distribution of the sensors. As mentioned earlier, minimizing
the number of sensors is not of critical importance. However,
knowing the probability of uncovered areas is an essential
design objection for the sensor network. Furthermore, the
probability bound provides a new perspective to the problem of
finding minimum full coverage sensor sets. With sufficient
sensors, the selection process can simply be distributed tosses of
dice. As long as the characteristics of the sensor distribution are
maintained, we can use the bound to compute the performance
of the minimum sensor sets.
The remainder of this paper is structured in the following
fashion. In Section II, we introduce some related works. We
derive the probability bound and prove its validity in Section III.
Numerical results are, then, demonstrated in Section IV. Finally,
we conclude our paper and point out future research directions
in Section V.
II. EXISTING WORKS
Doherty and Pister performed numerous simulations regarding
node self-selection in [18]. The mechanisms examined in this
paper are, in one way or another, variations of random selection.
The paper assumed no data fusing capability in the sensors.
Instead, each node through some self-selection mechanisms
decides if it should send its data or not. They showed the energy
cost of reconstructing data to a demanded resolution for the
schemes of interest.
A data fusion method for node selection is proposed in [19].
In this work, a node analyzes the similarity of its data with that
of other nodes. Then, a self-scheduling mechanism is
formulated to let nodes with similar data (or equivalently with
large overlapped sensing regions) to transmit in turn. This
method is capable of balancing workload of the sensors.
In [20, 21], algorithms are designed to compute the best and
worst coverage-distance with respect to two end points in the
sensor network. It combines the techniques of Voronoi diagram
and Delaunay triangulation. GPS support is demanded in [20]
while a distributed approach using only local information is
proposed in [21].
The research works focus on the
observability. It does not address the full coverage problem
proposed in this research.
for global information of sensor location. It shows the how
parallel configuration can be achieved for sharing the load.
Before we start Section III, we wish to briefly comment on
the positioning techniques used in the sensor networks. There
are numerous proposals on locating the sensor nodes using a
GPS system, a small number of beacons, and signal strength
[24, 25, 26]. In [27], an algorithm is proposed to discover
sensor locations with connectivity information. However, it
should be pointed out, this method assumes a unique
identification of each sensor. It is shown that the complexity of
the algorithm is of O(n3) and a small number of beacons are still
necessary.
III. DERIVATION AND VERIFICATION OF THE PROBABILITY
BOUND
In this section, we will derive the upper bound for the
probability that  percent of the sensing field is not covered by
any sensors. The system parameters used are listed below. In
our analysis, we assume static and uniformly distributed sensors.
We also assume that the information associated within a sensor
region can be collected by the corresponding sensor. Unless
explicitly stated, the boundary conditions are excluded in the
following discussions. Although we believe these boundary
conditions will not affect the result, the complete proofs are not
included here.
N: the number of sensors in the network.
A: the area of the sensing field of the network.
: the sensor density.  = N/A in the uniform distribution.
r: the radius of the sensor region.
: the percentage of uncover area in the sensing field.
Theorem 1. In a uniformly distributed sensor network, the
probability that a point, say a, is not in any of the sensor regions
is denoted by pa. And pa = (1-r2/A)N .
PROOF. Since a is not in any of the sensor regions, there is no
sensor in the disk centered at a with radius r (Figure 2). The
probability that a sensor is in the sensing field A but outside the
disk is 1 - r2/A. The distributions of sensor locations are
identical and independent. Hence,
pa 
 1  r
N
i 1
2
  1  r A
A
2
N
.
□
Next we would like to show that given two points a and b,
Shakkottai et al. discussed coverage problem with unreliable
sensor grids in [22]. It suggested the necessary and sufficient
conditions for the random grid network to cover the unit square
as well as remain connected. Due to the limited applications of
deterministic sensor placement, this research may not be
extended easily to more realistic environment.
A
r
Coordinating grid (Co-Grid) is proposed in [23] to maintain a
coverage area with a minimum number of active sensors. This
work coped with two design objectives. One is the minimum
sensor set and another is the scheduling for load balancing.
Specifically, the paper assumes a centralized positioning system
Figure 2. An Example of Uncovered Point
v1
A
v2
D
r
a
b
v3
v5
v4
Figure 3. Independence of Uncovered Points
D
PROOF. Since D does not intersect with any of the sensor
regions, there is no sensor in the disk D’ of radius r, co-centered
with D (Figure 4). The probability that a sensor is in the sensing
field A but outside the disk D’ is 1 - r+d)2/A. The
distributions of sensor locations are identical and independent.
Hence,
A
D’
Figure 5. An Example of Non-convex ‘Polygon’
r
N
2
2

 

p D   1   r  d    1   r  d   . □
A
A
 

i 1 
N
d
Figure 4. An Example of Uncovered Circle
then a being out of any sensor region will affect the probability
that b is uncovered. In other words, these two events are not
independent.
Theorem 2. In a uniformly distributed sensor network, there
exist two points a and point b. The probability pb|a that point b is
out of any sensor region given point a is not in any sensor region
is not equal to pb, i.e., pb≠pb|a .
PROOF. From Theorem 1, we obtain
 r 2
p a  pb  1 
A




2N
.
However, we know p a b
 2r 2
 1 
A

N

  p a  pb in

certain scenario (Figure 3). This is equivalent to pb≠pb|a. □
We are ready to compute the probability of an uncovered
disk. Note that we cannot directly apply the result of Theorem 1
to our next problem. A disk contains an infinite number of
points and the events that these points are not outside of any
sensor region are dependent. Instead, we recycle the technique
used in proving Theorem 1.
Theorem 3. In a uniformly distributed sensor network, the
probability that a disk, say D, does not intersect with any of the
sensor regions is denoted by pD. And pD = (1-r+d)2/A)N
where d is the radius of D.
As we will explain later, we believe Theorem 3 provides an
upper bound for the more general question about the probability
of an uncovered area. That is, if we have N sensors in the
network of sensing field A, what is the worst probability that 
percent of area is not covered? We will provide a conjecture
and theorem to support our argument. Firstly, we will discuss
that the disk/circle yields the worst (or largest) probability
among all shapes of the same size. Secondly, we will prove that
under some criterion the probability of a continuous uncovered
disk is always larger than that of a set of disjoint ones with the
same total size.
If area D does not intersect with any sensor region, then it
must exhibit the following property. Any arc/curve connecting
two adjacent vertices of D is non-convex. This property holds
because each arc of D belongs to a circular sensor region of
radius r. The only exception, which we don’t consider in this
paper, occurs when D is at the brink of the sensing field.
Non-convex ‘polygons’ are used to describe figures with this
property. Figure 5 shows a typical D with five vertices. In
addition, we consider D has no inside hole. Of course, D may
contain a number of sensor regions that form one or multiple
inside holes. The upper bound provided in Theorem 3 will still
hold because the inside holes will reduce the probability of an
uncovered area.
Now, we will explain why the circle figure yields the worst
(or largest) probability of an uncovered area among all shapes
of the same size. It is a well known geometric fact that the
v-sided regular polygon, which has v equal sides, exhibits the
largest area of all v-sided polygons with a given perimeter. An
example is that the square has the largest area among all 4-sided
polygons with a given perimeter. Another well know fact is that
the circle is the plane figure with highest ratio of area to
1.
r
v1
v2
D
2.
v3
D’
v5
3.
v4
A non-convex ‘polygon’ seems to have worse area to
perimeter ratio than the corresponding regular polygon of
the same size has. Figure 7 shows a regular triangle and a
non-convex ‘polygon’ of the same size. We believe the
non-convex figure has larger forbidden region than the
regular triangle does.
A regular polygon has worse area to perimeter ratio than the
circle of the same size has.
A figure with better area to perimeter ratio seems to result
in a smaller forbidden region where no sensor will be
found.
Let show the example of Figure 7 to support conjecture 4. It
is trivial to show that the forbidden region for the regular
triangle is r2+lr, where l is the perimeter of the triangle. The
forbidden region for the non-convex ‘polygon’ is derived in the
following.
Figure 6. Forbidden Region Resulting in
Uncovered Area D
3    2
3
a

r  r 2     ,
2
2
r

r
where a is the length of the arc in the non-convex ‘polygon’.

a

r
2   
 
Figure 7. Comparing the Forbidden Regions
perimeter. That is, given a specific perimeter, the circle will
result in the largest area. Or equivalently, to encompass a
desired area, the circle requires the minimal perimeter when
compared with any convex polygon.
In Figure 6, the dotted lines form a convex polygon whose
area is larger than D. Furthermore, the total length of these
dotted lines is shorter than D’s perimeter. We conclude that any
uncovered area in a sensor network with v vertices has lower
ratio of area to perimeter than the v-sided regular polygon of the
same perimeter. In summary, the circle must have a shorter
perimeter than the uncovered area given the same area. If D is
an uncovered area, then no sensor will be found in either D or
the surrounding circular stripe of width r (Figure 6). Our goal is
to show that given an arbitrary D, the circle of the same area of
D will always result in smaller area where no sensor can be
found.
Conjecture 4. A non-convex ‘polygon’ D and a circle E are of
the same size. A closed contour CD encompassing D and the
distance between any point a on CD and D is r. The circular area
between the contours CD and D is denoted as D*. Similarly, CE
is another contour containing E and the distance between any
point on CE and E is r. The circular area between CE and E is
denoted as E*. D* > E*.
Although we are not able to provide a solid proof of this
conjecture at the writing of this paper, we believe it is valid
based on the following remarks.

3
    
2
 
3
3    2
3
2a 
 2
2
r  r 2 

  r  3ar
2
2
r 
 3
Comparing the two forbidden areas, we found they exhibit
identical form. That is, the forbidden region is equal to the
sensor region plus the multiplication of the perimeter and the
radius of the sensor region. Since we know if they have the
same area, then 3a must be larger than l. Consequently, this
demonstration supports our remark 1 listed above.
Next we examine the case of disjoint uncovered areas. We
prove that the probability upper holds if the disjoint uncover
areas satisfy certain condition.
Theorem 5. In a uniformly distributed sensor network, the
probability that a disk D of radius d does not intersect with any
sensor region is denoted by pD. Assume there are a set S of
disjoint disks, D1, D2, …, Dk. Their respective radiuses are d1,
d2, …, dk. and their respective centers are c1, c2, …, ck. (Figure
8(a) shows an example of two disjoint uncover areas.)
Moreover, the cumulative area of the elements of S is equal to
the area of D. We denote the probability that the elements of S
do not intersect with any sensor region pS. If |ci-cj| - di - dj > 2r,
for any i, j = 1..k, i≠j, then we have pD > pS.
k
PROOF. We know
d
i 1
2
i
 d 2 . Using Theorem 3, we have
2


p D  1   r  d 

A

 .
N
Since |ci-cj| - di - dj > 2r, for any i, j = 1..k, i≠j, we have
formula for this computation.
r
d1
A  d 2  d 
d2

 r  d 2
1 

A

(a)



 k

2


π  r  d
i 

i  1



1 

A






N



k



 kr 2  2r  d i  d 2  


i 1

1 


A




In this section, we calculate the probability upper bound using
equation 1. In both Figures 8 and 9, we show that the
probability upper bound changes against the ratio of the sensor
region to the sensing field. If the sensor region is 2% of the
sensing field, then we need at least 50 sensors to cover the entire
sensing field. The three curves in each graph represent the
upper bounds of various ’s. For example, the diamond-dash
line plots the probability upper bound for 0.1% of area being
uncovered.
k
i 1
k
 ( k  1) r 2  2r ( d i  d )
i 1
2
k


  di   d 2  2  di d j  d 2
i , j 1
 i 1

k
N=200
1.00E-01
k
d
i 1
i
d
k
 ( k  1)r  2r ( d i  d )  0
2
i 1
We conclude pD > pS.
□
Probability upper bound
i j

N
IV. NUMERICAL RESULTS
N
kr 2  2r  d i  d 2  r 2  2rd  d 2




Conceptually, the upper bound probability is calculated as if
the uncovered area were a circle of A in size. Based on
theorem 5 and conjecture 6, we know this probability is larger
than that for the disjoint circles of the same aggregated area, A.
For each disjoint circle, we apply theorem 3 and conjecture 4
and obtain that each disjoint circle results in higher probability
than the corresponding non-convex ‘polygon’. Thus, we
conclude that equation 1 is the upper bound for the probability
of an uncovered area of size A.
Figure 8. Disjoint Uncovered Areas


r 2  2rd  d 2
 1 
A

N
(b)
pS
N


r 2

 1 
 2r
   (Equation 1)
A
A


d2
d1




A

1.00E-03
1.00E-05
1.00E-07
0.001
1.00E-09
0.005
1.00E-11
0.01
1.00E-13
1.00E-15
Theorem 5 verifies the upper bound when the distance
between each pair of disjoint uncovered areas is far enough.
However, we believe this condition can be relaxed without
affecting the validity of the theorem. As an example shown in
Figure 8(b), the disjoint areas will still result in larger marginal
stripe.
Conjecture 6. With the rest being the same as stated in
Theorem 5, if |ci-cj| - di - dj > 0, for any i, j = 1..k, i≠j, then we
have pD > pS.
With theorems 3 and 5, and conjectures 4 and 6, we are ready
to calculate the upper bound for the probability that  portion of
the sensing field A is uncovered given N sensors and sensor
region radius r. Specifically, equation 1 provides the exact
1.00E-17
0.01%
0.05%
0.20%
1.00%
5.00%
T he ratio of sensor region to sensing field
Figure 9. Probability for Uncovered Areas, N=200
It should be noted that not all data points in these two figures
can be interpreted as the upper bound probability of certain
uncover area. For example, it is evident that when the sensor
region is 0.01% of the sensing field, then it is certain that 0.01%
of sensing field is uncovered. Therefore, when the aggregated
sensor region is smaller than the sensing field, equation 1 simply
provides the probability that no sensor is found in a circle with
radius of r+d. As N increases and the aggregated sensor region
grows larger than the sensing field, equation 1 starts to describe
the upper bound probability of certain uncovered area.
Probability upper bound
N=500
0.01
1E-06
1E-10
1E-14
1E-18
1E-22
1E-26
1E-30
1E-34
1E-38
1E-42
0.01%
0.001
0.005
0.01
0.05%
0.20%
1.00%
5.00%
The ratio of sensor region to sensing field
Figure 10. Probability of Uncovered Areas, N=500
The data show that when the aggregated sensor region is
about 10-20 times of the sensing field, the probability upper
bound for various uncover areas is in smaller than 10-5.
V. CONCLUSION
We derive an upper bound for the probability that  percent of
the sensing field is not covered by any sensors. We illustrate
how the bound changes as the system parameters vary. It is
shown that the probability for uncovered area is not trivial even
in some cases where the sensor density seems satisfactory. Next,
we would like to investigate how various boundary conditions
affect our result and how tight the calculated bound is. Using
the result of this paper, we can compute the bound for the
average size of the uncovered areas. We can compute it against
the random sensor networks and find out the discrepancy. We
also plan to apply the result to design economical algorithms for
minimum sensor set selection for detection and/or routing
purposes.
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