Mobility Improves Coverage of Sensor Networks
Benyuan Liu
Dept. of Computer Science
University of
Massachusetts-Lowell
Lowell, MA 01854
Peter Brass
Dept. of Computer Science
City College of New York
New York, NY 10031
Olivier Dousse
School of Computer and
Communication Sciences
EPFL
Lausanne, Switzerland
Philippe Nain
INRIA
06902 Sophia Antipolis
France
Don Towsley
Dept. of Computer Science
University of Massachusetts
Amherst, MA 01002
ABSTRACT
Previous work on the coverage of mobile sensor networks fo-
cuses on algorithms to reposition sensors in order to achieve
a static configuration with an enlarged covered area. In this
paper, we study the dynamic aspects of the coverage of a
mobile sensor network that depend on the process of sensor
movement. As time goes by, a position is more likely to be
covered; targets that might never be detected in a stationary
sensor network can now be detected by moving sensors. We
characterize the area coverage at specific time instants and
during time intervals, as well as the time it takes to detect a
randomly located stationary target. Our results show that
sensor mobility can be exploited to compensate for the lack
of sensors and improve network coverage. For mobile tar-
gets, we take a game theoretic approach and derive optimal
mobility strategies for sensors and targets from their own
perspectives.
Categories and Subject Descriptors
C.2.1 [Computer-Communication Networks]: Network
Architecure and Design—Network Communications, Wire-
less Communication.
General Terms
Design, Performance.
Keywords
Coverage, Sensor Network, Mobility.
Permission to make digital or hard copies of all or part of this work for
personal or classroom use is granted without fee provided that copies are
not made or distributed for profit or commercial advantage and that copies
bear this notice and the full citation on the first page. To copy otherwise, to
republish, to post on servers or to redistribute to lists, requires prior specific
permission and/or a fee.
MobiHoc’05, May 25–27, 2005, Urbana-Champaign, Illinois, USA.
Copyright 2005 ACM 1-59593-004-3/05/0005 ...$5.00.
1. INTRODUCTION
Recently, there has been substantial research in the area of
sensor network coverage. The coverage of a sensor network
represents the quality of surveillance that the network can
provide, for example, how well a region of interest is moni-
tored by sensors, and how effectively a sensor network can
detect intruders (targets). It is important to understand
how the coverage of a sensor network depends on various
network parameters in order to better design and use sensor
networks for different application scenarios.
In many applications, sensors are not mobile; they remain
stationary after their initial deployment. The coverage of
such a stationary sensor network is determined by the ini-
tial network configuration. Once the deployment strategy
and sensing characteristics of the sensors are known, the
network coverage can be computed and remains unchanged
over time. Recently, there has been a strong desire to de-
ploy sensors mounted on mobile platforms such as mobile
robots. Such mobile sensor networks are extremely valuable
in situations where traditional deployment mechanisms fail
or are not suitable, for example, a hostile environment where
sensors cannot be manually deployed or air-dropped. Also,
in application scenarios such as atmosphere and ocean envi-
ronment monitoring, sensors move with the surrounding air
or ocean currents. The coverage of a mobile sensor network
now depends not only on the initial network configurations,
but also on the mobility behavior of the sensors.
While the coverage of a sensor network with immobile
sensors has been extensively explored and is relatively well
understood [12, 8, 9, 3, 7], reseachers have only recently
started to study the coverage of mobile sensor networks.
Most of this work focuses on algorithms to reposition sen-
sors in desired positions in order to enhance network cov-
erage [5, 1, 10, 15, 14]. More specifically, these proposed
algorithms strive to spread sensors in the field so as to max-
imize the covered area. The main differences among these
works are how exactly the desired positions of sensors are
computed. Although the algorithms can adapt to changing
environments and recompute the sensor locations accord-
ingly, sensor mobility is exploited essentially to obtain a
new stationary configuration that improves coverage after
the sensors move to their desired locations.
In this paper, we study the coverage of a mobile sensor
network from a different perspective. Instead of trying to
achieve an improved network configuration as the end re-
sult of sensor movement, we identify and characterize the
dynamic aspects of network coverage that depend on the
movement of sensors.
Specifically, we are interested in the coverage resulting
from the continuous movement of sensors. This coverage
is not available if the sensors stop moving. We now briefly
describe the coverage provided by the sensor movement, and
the related research problems.
First, previously uncovered areas become covered as sen-
sors move through them and covered areas become uncov-
ered as sensors move away. As a result, the locations covered
by sensors change over time, and a greater area will be cov-
ered over time than in the case where sensors are stationary.
Also, a location is now not always covered. It alternates be-
tween being covered and not being covered. This raised
the following question: what is the area coverage at a given
time instant? what is the area coverage over a time interval?
what are the durations of time that a location is covered and
not covered?
Second, note that an initially undetected intruder will
never be detected in a stationary sensor network if the in-
truder remains stationary or moves along an uncovered path.
In a mobile sensor network, an intruder is more likely to be
detected as the moving sensors patrol the field. Thus, sen-
sor mobility provides a time-varying coverage not available
in a sensor network with stationary sensors. This can signif-
icantly improve the intrusion detection capability of a sensor
network. This raises the additional questions: how quickly
can the sensors detect an intruder? how does the detection
time depend on the searching strategy of the sensors? and
what are the optimal mobility strategies for the sensors and
the intruder?
The main contributions of our work are:
First, we characterize the fraction of the area covered by
sensors for a randomly-deployed stationary sensor network.
This characterization shows how the covered area depends
on the density and sensing characteristics of the sensors.
It provides a baseline for comparison with a mobile sensor
network.
We then consider a random mobility model for sensors
and study the effect of sensor mobility on various aspects of
network coverage. We show that, while the fraction of the
covered area at any given time instant remains unchanged,
the area being covered during a time interval is improved as
sensors move around. As a result, intruders and events that
will never be detected in a stationary sensor network can
now be detected in a mobile sensor network. This scenario
is of great importance for applications that do not require
simultaneous coverage of all locations at specific time in-
stants.
Unlike a stationary sensor network where a location al-
ways remains either covered or not covered, due to sensor
movement, a location is now only covered part of the time,
alternating between being covered and not being covered.
We characterize this coverage and covering time tradeoff by
the fraction of time a location is covered, which is deter-
mined by the density and sensing range of the sensors, and
does not depend on the sensor mobility. While this time av-
erage characterization shows, to a certain extent, how well
a point is covered, it does not reveal the duration of the
time that a point is covered and uncovered. The time scales
of such durations are very important for network planning;
they present the time granularity of the intrusion detection
capability that a mobile sensor network can provide. To this
end, we further characterize the time durations that a point
is covered and not covered.
In some applications, it is important to detect intruders
in the field of interest as quickly as possible. To this end,
we study the detection time of an intruder, which is defined
to be the time elapsed before the intruder is first detected.
Intruders that will never be detected in a stationary sensor
network can now be detected by moving sensors. We obtain
the distribution of the detection time for a randomly located
stationary intruder. The results suggest that sensor mobility
can be exploited to effectively reduce the detection time of
a stationary intruder when the number of sensors is limited.
For mobile intruders, the detection time depends on both
the sensor and intruder mobility strategies. We take a game
theoretic approach and study the best worst-case perfor-
mance of a mobile sensor network in terms of the intruder
detection time. For a given sensor mobility behavior, we
assume that an intruder can choose its mobility strategy so
as to maximize its detection time (its lifetime before be-
ing detected). On the other hand, sensors choose a mobility
strategy that minimizes the maximum detection time result-
ing from the intruder’s mobility strategy. We prove that the
optimal sensor mobility strategy is for each sensor to choose
its direction uniformly at random. The corresponding in-
truder mobility strategy is to remain stationary in order to
maximize the time before it is detected.
The remainder of the paper is structured as follows. The
network model and coverage measures are presented in Sec-
tion 2. In Section 3, we study the fraction of the area being
covered at specific time instants and during a time interval.
In Section 4, we study the detection time for both stationary
and mobile intruders. In Section 5, we review related work
on the coverage of sensor networks. Finally, we summarize
the paper in Section 6.
Throughout the paper, shorthand X ∼ exp(µ) will stand
for P (X < x) = 1 − exp(−µx), namely, the random variable
X is exponentially distributed with parameter µ. Also, we
will use the words intruder and target interchangeably.
2. NETWORK AND MOBILITY MODEL
In this section, we describe the network and mobility
model used in this study, and define three coverage mea-
sures of a mobile sensor network.
2.1 Sensing Model
We assume that each sensor has a sensing radius, r. A
sensor can only sense the environment and detect events
within its sensing area, which is the disk of radius r centered
at the sensor. A point is said to be covered by a sensor if
it is located in the sensing area of the sensor. The sensor
network is thus partitioned into two regions, the covered
region, which is the region covered by at least one sensor,
and the uncovered region, which is the complement of the
covered region. An intruder is said to be detected if it lies
within the covered region.
2.2 Coverage measures
To study the coverage of a sensor network, we define the
following three coverage measures.
Definition 1. Area coverage: The area coverage of a
sensor network at time t, f
a
(t), is the fraction of the geo-
graphical area covered by one or more sensors at time t.
Definition 2. Area coverage over a time interval:
The area coverage of a sensor network during time interval
[0, t), f
i
(t), is the fraction of the geographical area covered
by at least one sensor at some point of time within [0, t).
Definition 3. Detection time: Consider a target lo-
cated at a random position outside of the covered area of a
sensor network at time t = 0. The detection time of the
target, X, is defined to be the time at which the target first
enters the sensing area of a sensor, i.e., the target is first
detected by the sensor.
All three coverage measures depend not only on static
properties of the sensor network (sensor density and sens-
ing range), but also on the sensor mobility behavior. The
characterization of area coverage at specific time instants
is important for applications that require simultaneous cov-
erage of the network, while the area coverage over a time
interval is appropriate for applications that do not require
simultaneous coverage of all locations at specific time in-
stants, but rather prefer to cover the network within some
time interval. The detection time measures how quickly a
sensor network can detect a randomly located target that is
not initially covered.
2.3 Location and Mobility Model
We consider a network consisting of a large number of
sensors placed in a vast two-dimensional geographical re-
gion. For the initial configuration, we assume that, at time
t = 0, the locations of these sensors are uniformly and inde-
pendently distributed in the region. Such a random initial
deployment is desirable in scenarios where prior knowledge
of the region of interest is not available. Also, random de-
ployment can be the result of certain deployment strategies.
For example, sensors may be air-dropped or launched via
artillery in battlefields or unfriendly environments. Under
this assumption, the sensor locations can be modeled by a
stationary two-dimensional Poisson point process. Denote
the density of the underlying Poisson point process as λ.
The number of sensors located in a region R, N(R), fol-
lows a Poisson distribution of parameter λkRk, where kRk
represents the area of the region.
P (N (R) = k) =
e
−λkRk
(λkRk)
k
k!
. (1)
Since each sensor covers a disk of radius r, the initial
configuration of the sensor network can be described by a
Poisson Boolean model B(λ, r). In a stationary sensor net-
work, sensors do not move after being deployed and network
coverage remains the same as that of the initial configura-
tion. In a mobile sensor network, depending on the mobile
platform and application scenario, sensors can choose from
a wide variety of mobility strategies, from passive movement
to highly coordinated and complicated motion. Sensors de-
ployed in the air or ocean move passively according to exter-
nal forces such as air or ocean currents; simple robots may
have a limited set of mobility patterns, and advanced robots
can navigate in a more complicated fashion.
In this work, we consider the following simple sensor mo-
bility model. We assume sensors move independently of each
other and with coordination among them. The movement
of a sensor is characterized by its speed and direction. A
sensor randomly chooses a direction θ ∈ [0, 2π) according to
some distribution with probability density function f
s
Θ
(θ).
The speed of the sensor, V
s
, is randomly chosen from a fi-
nite range v
s
∈ [0, V
max
s
], according to a distribution density
function of f
s
V
(v).
Throughout the rest of this paper, we will refer to the
initial sensor network configuration as a random sensor net-
work B(λ, r), and the above mobility model as random mo-
bility model. The coverage measures defined above are func-
tions of the actual locations of sensors, which vary for dif-
ferent realizations. In this work, we will study the expected
value of the coverage measures.
3. AREA COVERAGE
In this section, we study and compare the area coverages
of both stationary and mobile sensor networks. We analyt-
ically characterize the area coverage. We then discuss the
implications of our results on network planning and show
that sensor mobility can be exploited to compensate for the
lack of sensors to increase the area being covered during a
time period. Finally, we point out, due to the sensor mo-
bility, a point is only covered part of the time; we further
characterize this effect by determining the fraction of time
that a point is covered.
Theorem 1. At any given time instant t > 0, the area
coverage of a stationary sensor network B(λ, r) is
f
a
(t) = 1 − e
−λπr
2
. (2)
Proof. This is a result from stochastic geometry [4]. Here
we present the key arguments of the proof.
Consider a bounded region R; the vacancy V within R is
defined to be the area in R not covered by sensors.
V =
R
χ(x)dx.
where
χ(x) =
1 x is not covered
0 otherwise.
Using Fubini’s theorem we have
E(V ) =
R
E{χ(x)}dx.
Consider an arbitrary point x in region R and denote the
number of sensors which cover the point as N. Point x is
covered by sensors located within distance r. It follows im-
mediately from the Poisson point process assumption that N
has a Poisson distribution with parameter λπr
2
. Therefore,
we have
E{χ(x)} = P (x is not covered) = P (N = 0) = e
−λπr
2
.
and
E(V ) =
R
E{χ(x)}dx = kRke
−λπr
2
.
Note that the above derivation is independent of R. The
area coverage can thus be obtained as follows.
f
a
= 1 −
E(V )
kRk
= 1 − e
−λπr
2
.
time 0
time t
Figure 1: Coverage of mobile sensor network: the
left figure depicts the initial network configuration
at time 0 and the right figure illustrates the effect of
sensor mobility during time interval [0, t). The solid
disks constitutes the area being covered at the given
time instant, and the union of the shaded region
and the solid disks represents the area being covered
during the time interval.
2
This formula characterizes the dependence of area cover-
age on the network properties. It allows us to compute the
fraction of the area being covered for a given sensor den-
sity and sensing range. For example, in order to achieve a
desired area coverage f
a
(0 < f
a
< 1) almost surely, the
density required is given by
λ = − ln(1 − f
a
)/πr
2
.
In a stationary sensor network, a location always remains
either covered or not covered. The area coverage does not
change over time. The following theorem characterizes the
effect of sensor mobility on network coverage.
Theorem 2. Consider a sensor network B(λ, r) at time
t = 0, with sensors moving according to the random mobility
model.
1. At any time instant t, the fraction of area being covered
is
f
a
(t) = 1 − e
−λπr
2
, ∀t ≥ 0. (3)
2. The fraction of area that has been covered at least once
during time interval [0, t) is
f
i
(t) = 1 − e
−λ(πr
2
+2rE[V
s
]t)
. (4)
3. The fraction of the time a point is covered is
f
t
= 1 − e
−λπr
2
. (5)
Proof. Given the initial node placement and the random
mobility model, at any time instant t, the locations of the
sensors still form a two dimensional Poisson point process of
the same density [11, Theorem 9.14]. Therefore, the fraction
of the area covered at time t remains the same as in the
initial configuration, f
a
(t) = 1 − e
−λπr
2
.
As illustrated in Figure 1, during time interval [0, t), each
sensor covers a shape of a racetrack whose expected area is
α = E[πr
2
+ 2rV
s
t] = πr
2
+ 2rE[V
s
]t.
where E[V
s
] =
V
max
s
0
f
s
V
(V )dV represents the expected sen-
sor speed.
As pointed out in [4], area coverage depends on the dis-
tribution of the random shapes only through its expected
area. Therefore, we have
f
i
(t) = 1 − e
−αλ
= 1 − e
−λ(πr
2
+2rE[V
s
]t)
.
While an uncovered location will be covered when a sensor
moves within distance r of the location, a covered location
becomes uncovered as sensors covering it move away. As
a result, a location is only covered part of the time. More
specifically, a location alternates between being covered and
not being covered, which can be modeled as an alternating
renewal process. We use the fraction of time that a location
is covered to measure this effect. The fraction of time that a
location is covered equals the probability that it is covered at
any given time instant, f
t
= 1 − e
−λπr
2
. In the next section,
we further characterize the time durations of a point being
covered and not being covered. 2
At any specific time instant, the fraction of the area being
covered in a mobile sensor network model described above
is the same as in a stationary sensor network. This is be-
cause at any time instant, the positions of the sensor are
still described by a Poisson Boolean model with the same
parameters as in the initial configuration. Unlike in a sta-
tionary sensor network, areas initially not covered can now
be covered as sensors move around in a mobile sensor net-
work. Consequently, targets in the initially uncovered areas
can be detected by the moving sensors.
Figure 1 illustrates the effect of sensor mobility on area
coverage. The union of the solid disks constitutes the area
coverage at given time instants. The area that has ever been
covered during time interval [0, t) is depicted as the union
of the shaded region and the solid disks, occupying a larger
portion of the total area.
Due to sensor mobility, the fraction of the area that has
ever been covered increases and approaches one as time goes
to infinity. The rate at which the covered area increases over
time depends on the expected speed of sensor mobility. The
faster sensors move, the more quickly the area is covered.
Therefore, sensor mobility can be exploited to compensate
for the lack of sensors to improve the area coverage over
an interval of time. Note that the area coverage during
a time interval does not depend on the distribution of the
sensors’ movement directions. Based on (4), we can compute
the expected sensor speed required to ensure that a certain
fraction of the area (f
0
) is covered within a time interval of
length t
0
.
E[V
s
] = −
λπr
2
+ log(1 − f
0
)
t
0
, for f
0
≥ 1 − e
λπr
2
.
In a stationary sensor network, a location is either always
covered or not covered, as determined by its initial config-
uration. In a mobile sensor network, as a result of sensor
mobility, a location is only covered part of the time, alter-
nating between being covered and not being covered. The
fraction of time that a location is covered corresponds to
the probability that it is covered, as shown in (5). Note
that this probability is determined by the static properties
of the network configuration (density and sensing range of
the sensors), and does not depend on sensor mobility. This
coverage-delay tradeoff can be exploited by applications that
do not require simultaneous coverage of all locations at spe-
cific time instants.
4. DETECTION TIME
In the previous section, we characterized the fraction of
area being covered at any given time and over a time in-
terval. However, these measures do not reveal how quickly
mobile sensors detect targets in the field. The time it takes
to detect an intruding target is of great importance in many
security-related applications. In this section, we study the
detection time for both stationary and mobile targets. For
mobile targets, we take a game theoretic approach and ex-
plore the best worst-case performance of the sensor network.
We will derive optimal mobility strategies for targets and
sensors that maximize or minimize the detection time.
To facilitate analysis and illustrate the effect of sensor mo-
bility on the detection time, we make the assumption that
all sensors move at a constant speed v
s
. More general speed
distributions can be approximated using the fixed speed sce-
nario.
4.1 Stationary Target
We first consider the scenario where targets are station-
ary and do not initially fall into the coverage area of any
sensor. Obviously, a stationary sensor network will never
detect these targets; the detection times of these targets are
infinite.
However, in a mobile sensor network, a target can be de-
tected by any sensor passing within distance r of it, where
r is the common sensing range of the sensors. The detec-
tion time of a stationary target characterizes how quickly
the sensors can detect a randomly located target previously
not detected.
Theorem 3. Consider a sensor network B(λ, r) at time
t = 0, with sensors moving according to the random mobility
model at a fixed speed v
s
. Let X be the detection time of a
randomly located stationary target, we have
X ∼ exp(2λrv
s
). (6)
Proof. We divide the space evenly around point p in
k directions (k → ∞). We now have k sensor classes and
sensors of class i move in the direction θ
i
=
2πi
k
. Since each
sensor independently chooses its moving direction according
to the same distribution, each sensor class is a thinning of
the original point process. Therefore, sensor class i forms a
Poisson point process with density λ
i
= λf
s
Θ
(θ
i
)∆θ, where
∆θ = 2π/k.
Point p will be detected when a sensor moves within dis-
tance r from the point. Now let X
i
(first hit time) be the
time that point p is first detected by a sensor of class i, the
detection time X is the minimum of all the first hit times,
X = min X
i
.
Since all sensors of class i move in the same direction θ
i
at the same speed v
s
, it is more convenient to consider the
framework where sensors are relatively stationary and the
target moves in the opposite direction (2π − θ
i
) at the sen-
sor’s speed v
s
. The distance from point p to the perimeter
of the first sensor to contact it in direction (2π − θ
i
) is called
the linear contact distance of that particular direction, and
we denote it as Y
i
. From [13, page 80], we know that Y
i
fol-
lows an exponential distribution with parameter 2λ
i
r, i.e.,
Y
i
∼ exp(2λ
i
r).
Since X
i
= Y
i
/v
s
, the first hit time in direction i follows
an exponential distribution with parameter 2λ
i
rv
s
,
X
i
∼ exp(2λ
i
rv
s
).
Now the minimum of these exponential distributed first hit
times is again an exponential distribution, with a param-
eter equal to the sum of the parameters of the individual
exponential distributions:
lim
k→∞
k
i=1
2λ
i
rv
s
= lim
k→∞
k
i=1
2λf
s
Θ
(θ
i
)∆θrv
s
= 2λrv
s
2π
0
f
s
Θ
(θ)dθ
= 2λrv
s
.
Thus, we have X ∼ exp(2λrv
s
).
2
Compared to the case of stationary sensors where an un-
detected target always remains undetected, the probability
that the target is not detected in a mobile sensor network
decreases exponentially over time,
P (X ≥ t) = e
−2λrv
s
t
.
The expected detection time of a randomly located tar-
get is E[X] =
1
2λrv
s
, which is inversely proportional to the
density of the sensors (λ), the sensing range of each sensor
(r), and the speed of the sensor movement (v
s
). Note that
the expected target detection time is independent of the sen-
sor movement direction distribution density function, f
Θ
(θ).
Therefore, in order to quickly detect a stationary target, one
can add more sensors, use sensors with larger sensing ranges,
or increase the speed of the mobile sensors.
Assuming there is a requirement that the expected time
to detect a randomly located stationary target be smaller
than a specific value T
0
, we have
1
2λrv
s
≤ T
0
or equivalently,
λv
s
≥
1
2rT
0
.
Assuming the sensing range of each sensor is fixed, the
above formula presents the tradeoff between sensor density
and sensor mobility to ensure certain target detection time
requirement. The product of the sensor density and sensor
speed should be larger than a constant. Therefore, sensor
mobility can be exploited to compensate for the lack of sen-
sors, and vice versa.
In Theorem 2, we pointed out that a location alternates
between being covered and not being covered, and then de-
rived the fraction of time that a point is covered. While
the time average characterization shows, to a certain extent,
how well a point is covered, it does not reveal the duration
of the time that a point is covered and uncovered. The
time scales of such time durations are also very important
for network planning; they present the time granularity of
the intrusion detection capability that a mobile sensor net-
work can provide. Having Theorem 3, we now characterize
