1. Introduction
Mobile sensors are being deployed in many application areas to enable easier information retrieval in communication environments, from sensing and diagnostics to critical infrastructure monitoring (e.g., see [
1,
2,
3]).
The current reduction in manufacturing costs makes random deployment of the sensors more attractive. Since existing sensor deployment scenarios cannot always ensure precise placement of sensors, their initial deployment may be somewhat random. In some cases, the sensors may have drifted to new arbitrary positions over time. Even initially deterministically placed sensors may create random patterns of effectiveness due to failures.
A typical sensor is able to sense and, thus, cover a bounded region specified by its sensing radius [
4]. To monitor and protect a larger region against intruders, every point of the region has to be within the sensing range of a sensor. It is also known that proximity between sensors affects the transmission and reception of signals and causes the degradation of performance [
5]. Therefore, in order to avoid interference, a critical value, say
s, is established. It is assumed that, for a given parameter
s, two sensors interfere with each other during communication if their distance is less than
s (see [
6,
7]). However, random deployment of the sensors might leave some gaps in the coverage of the area, and the sensors may be too close to each other. Therefore, to attain coverage of the area and to avoid interference, the reallocation of sensors may be the only option. Moreover, the ability to move the mobile sensors to the final destinations is not unrealistic. Clearly, the displacement of a team of sensors should be performed in the most efficient way.
The energy consumptionfor the displacement of a set of n sensors is measured by the sum of the respective displacements and the power of the individual sensors. We define below the concept of a-total displacement.
Definition 1 (
a-total displacement).
Let be a constant. Suppose the displacement of the i-th sensor is a distance . The a-total displacement is defined as the sum .
The motivation for this cost metric arises from the fact that the parameter a in the exponents represents various conditions on the region lubrication and friction, which affect the sensor movement.
We consider n mobile sensors, which are placed independently and uniformly at random on the unit interval and on the unit square.
For the case of unit interval
, each sensor is equipped with an omnidirectional antenna of identical sensing radius
. Thus, a sensor placed at location
x on the unit interval can cover any point at a distance at most
, either to the left or the right of
x (see
Figure 1a).
For the case of unit square , each sensor has the identical square sensing radius .
Definition 2 (cf. [
8] square sensing radius)
. We assume that a sensor located in position where can cover any point in the area delimited by the square with corner points and call the square sensing radius of the sensor. The concept of the square sensing radius was introduced in the paper [
8].
Figure 1b illustrates the square sensing radius.
However, in most cases, the sensing area of a sensor is a circular disk of radius , but our upper bound result, proven in the sequel, for square sensing radius is obviously valid for a circular disk of radius equal to circumscribing the square.
The sensors are required to move from their current random locations (see
Figure 2) to new positions to satisfy the following requirement.
Definition 3 (-C&I requirement). Fix . A set of sensors placed on the m-dimensional unit cube satisfies the coverage and interference requirements:
- (a)
Every point on the m-dimensional unit cube is within the range of a sensor, i.e., the m-dimensional unit cube is completely covered.
- (b)
Each pair of sensors is placed at a Euclidean distance greater than or equal to s.
In this paper, we investigate the problem of energy-efficient displacement of the random mobile sensors.
Definition 4 (Energy efficient displacement)
. Assume that n mobile sensors are placed independently and uniformly at random on the unit interval or on the unit square. The sensors move from their initial random location to the final destination so that, in their final placement, the sensor system satisfies the -coverage and interference requirement and the a-total displacement is minimized in expectation.
In WSNs, energy consumption is the fundamental problem to study. It is known that the sensors consume much more energy during movement than during sensing or communication [
9]. The proposed solution can be widely used in border surveillance to detect intruders illegally crossing the protected area.
Throughout the paper, we use the Landau asymptotic notations:
- (i)
if there exist a constant and integer N such that for all ;
- (ii)
if there exist a constant and integer N such that for all ;
- (iii)
if and only if and .
1.1. Contribution and Outline of the Paper
Let be a constant. Assume that n mobile sensors with the identical sensing radius and square sensing radius are placed independently at random with the uniform distribution on the unit interval and on the unit square.
In this paper, we give the picture of the threshold phenomena for the coverage and interference requirement in one dimension, as well as in two dimension (see Definition 3). The a-total displacement (the energy consumption) is used to measure the movement cost (see Definition 1), while the Euclidean distance is used for the interference distance, and the sensing area of a sensor in two dimension is a square (see Definition 2). Let us also recall that, in two dimension, the sensors can move directly to the final locations via the shortest route, not only in a vertical and horizontal fashion.
Let , be arbitrary small constants independent on the number of sensors n.
Table 1 summarizes our main contribution in one dimension.
We prove the following results.
- (1)
When the sensing radius and the interference distance , the expected minimal a-total displacement for the requirement is in .
- (2)
As the sensing radius is a little above and the interference distance is a little below , the expected minimal a-total displacement for the requirement sharply declines to .
Table 2 summarizes our main contribution in two dimensions.
We prove the following results.
- (1)
When the square sensing radius and the interference distance , the expected minimal a-total displacement for the requirement is in .
- (2)
As the square sensing radius is a little above and the interference distance is a little below , the expected minimal a-total displacement for the requirement sharply declines to .
Notice that n sensors on the unit interval with sensing radius and the interference distance have to move to the anchor positions to satisfy the -coverage and interference. When and , there are no anchor positions predetermined in advance. A similar remark holds for the sensors on the unit square .
Our theoretical results imply that the expected a-total displacement is constant and independent of the number of sensors for some parameters a. Namely, we have the following upper bounds:
- (i)
For the random sensors on the unit interval, when
i.e., the sum of the sensing area of
n sensors is a little bigger than the length of the unit interval, it is possible to provide the full area coverage in
expected
a-total displacement with
.
- (ii)
For the random sensors on the unit square, when
i.e., the sum of the sensing area of
n sensors is asymptotically a little bigger than the area of unit square, the expected
a-total displacement with
to provide full area coverage is
. Obviously, this result is easily applicable to the model when the sensing area of a sensor is a circular disk of radius
by taking the circle circumscribing the square. Namely, when
then the expected two-total displacement to provide full area coverage is constant.
This constant cost seems to be of practical importance due to efficient monitoring against illegal trespassers. It is well known that intrusion detection is an important application of wireless sensor networks. In this case, it is necessary to ensure coverage with good communication.
Notice that the constant expected cost in (i) and (ii) is valid for n random sensors with the identical sensing radius on the interval of length x and for n random sensors with the identical square sensing radius on the square .
We also present three algorithms (see Algorithms 1–3). It is worthwhile to mention that, even though the algorithms are simple, the analysis is challenging. Notice that Algorithms 1–3 can be implemented by a centralized controller telling each sensor where and when to move. In
Section 2, we prove some technical properties of the Beta distribution with the special positive integer parameters needed in the current paper (see Lemmas 2 and 3).
The overall organization of the paper is as follows.
Section 1.2 briefly summarizes some related work.
Section 2 gives some properties of the Beta distribution, the results of which are used to analyse the
-C&I requirement in WSNs.
Section 3 and
Section 5 deal with sensors on the unit interval. In
Section 4 and
Section 6, we investigate sensors on the unit square, while further insights into the higher dimension are discussed in
Section 7.
Section 8 deals with the experimental evaluation of Algorithm 1.
Section 9 contains conclusions and directions for future work. Finally, for the sake of readability, certain technical proofs are deferred to the Appendices (
Appendix A,
Appendix B,
Appendix C,
Appendix D,
Appendix E,
Appendix F and
Appendix G).
Algorithm 1 moving sensors on |
Require: The initial locations of n mobile sensors, placed uniformly and independently at random on the unit interval . Ensure: The final positions of the sensors such that:
- (i)
The distance between consecutive sensors is greater than or equal to s and less than or equal to - (ii)
The leftmost sensor is at a distance less than or equal to from the origin.
Initialization: Sort the initial locations of n sensors with respect to the origin of the interval, the location of sensors after sorting - 1:
Let - 2:
forndo - 3:
if then - 4:
move left to right the sensor to the new position - 5:
else if then - 6:
move right to left the sensor to the new position - 7:
else - 8:
do nothing; - 9:
end if - 10:
end for - 11:
ifthen - 12:
- 13:
for n do - 14:
move right to left the sensor to the new position - 15:
end for - 16:
end if
|
Algorithm 2 for the -coverage and interference requirement on when provided that and are fixed and independent of |
Require: The initial locations of n mobile sensors with the identical sensing radius , placed uniformly and independently at random on the unit interval Ensure: The final positions of sensors the to satisfy the -coverage and interference requirement on the interval Initialization: Apply Algorithm for and the random sensors Let be the location of n sensors after Algorithm - 1:
switch () - 2:
case A - 3:
do nothing; - 4:
case B - 5:
move the sensor to the new position - 6:
while do - 7:
move the sensor to the new position ; - 8:
end while - 9:
case C - 10:
for to n do - 11:
move the sensor to the position - 12:
end for - 13:
end switch
|
Algorithm 3 for the -coverage and interference requirement on the when and provided that and are fixed and independent of |
Require: The initial locations of n mobile sensors with the identical square sensing radius , placed uniformly and independently at random on the unit square Ensure: The final positions of the sensors satisfying the -coverage and interference requirement on the square Initialization: Choose sensors at random; Sort the initial locations of sensors according to the second coordinate; let the sorted locations be
- 1:
fortodo - 2:
for to do - 3:
move sensor to position - 4:
end for - 5:
end for - 6:
forto do - 7:
apply Algorithm for and sensors - 8:
end for
|
1.2. Related Work
There are extensive studies dealing with both the coverage (e.g., see [
12,
13,
14,
15,
16]) and interference problems (e.g., see [
17,
18,
19,
20]). Closely related to barrier and area coverage, the matching problem is also of interest in the research community (e.g., see [
11,
21,
22,
23]).
An important setting in considerations of the coverage of a domain is when the sensors are initially placed at random with a uniform distribution. Some authors proposed using several rounds of random displacement to achieve complete coverage of a domain [
24,
25]. Another approach is to have the sensors relocate from their initial position to a new position to achieve the desired coverage [
26,
27].
In this article, we present a novel mathematical theory of the Beta distribution. As an application to sensor networks, we study the most important and difficult cases for the threshold phenomena:
On the unit interval when the sensing radius is close to and the interference distance s is close to , i.e., and ;
On the unit square when the square sensing radius is close to and the interference distance s is close to , i.e., and .
for coverage and interference (see Definition 3), provided that and are arbitrary small constant independent of the number of sensors n.
Compared to the coverage problem, the requirement not only ensures coverage, but also avoids interference and is more reasonable in order to provide reliable communication within the network.
It is worth mentioning that, in this paper, in two dimensions, the sensors can move directly to the final locations with a shortened distance, not only in a vertical and horizontal fashion, as in [
28] for the unit square. Hence, our analysis in the current paper when the sensors can move directly to the final locations via the shortest route not only in a vertical and horizontal fashion completes the picture of the threshold phenomena.
More importantly, our investigation is closely related to the papers [
28,
29] with respect to the analysis of the expected
a-total displacement for the coverage problem where the sensors are randomly placed on the unit interval [
29] and at a higher dimension [
28]. Both papers study performance bounds for some algorithms, using Chernoff’s inequality. The methods used in these papers have limitations—the most important and difficult cases when the sensing radius
is close to
and the square sensing radius
is close to
were not included in [
28,
29]. Moreover, in the paper [
28], the sensors can move only parallel to the axes. Hence, the analysis of the coverage problem in [
28] is incomplete.
Therefore, it is natural to investigate the general case when the sensor can move directly to the final locations via the shortest route not only in a vertical and horizontal fashion.
Finally, it is worth mentioning that our work is related to the series of papers [
6,
30,
31,
32]. In [
6,
31], the author investigated the maximum of the expected sensor’s displacement (the time required) for coverage and interference. In [
6,
31], it was assumed that the
n sensors are initially deployed on
according to the arrival times of the Poisson process with arrival rate
, and coverage (connectivity) is in the sense that there are no uncovered points from the origin to the last rightmost sensor. The work by [
30] investigated the expected minimal
a-total displacement for the interference–connectivity requirement when the
n sensors are initially placed on
according to
d identical and independent Poisson processes, each with arrival rate
. It is worth pointing out that the
d-dimensional model in [
30] is only the direct extension of the interference–connectivity requirement from one dimension to the
d-dimensional space and the sensors move only parallel to the axes (see
Table 3).
2. Results on the Beta Distribution
In this section, we provide three lemmas about the Beta distribution pertinent for the -C&I requirement in WSNs. We also introduce some basic concepts and notations that will be used in the sequel.
In this paper, in the one-dimensional scenario, the
n mobile sensors are thrown independently at random following a uniform distribution in the unit interval
. Let
be the position of the
ℓ-th sensor after sorting the initial random locations of
n sensors with respect to the origin of the interval
i.e., the
ℓ-th-order statistics of the uniform distribution in the unit interval. It is known that the random variable
obeys the Beta distribution with parameters
(see [
33], p. 13).
Assume that
are positive integers. The Beta distribution
(see [
34]) with parameters
is the continuous distribution on
with the probability density function
given by
The cumulative distribution function of the Beta distribution with parameters
is given by the incomplete Beta function:
Moreover, the incomplete Beta function is related to the binomial distribution by
(see [
34], Identity 8.17.5, for
, and
) and the binomial identity:
The following inequality, which relates the binomial and Poisson distribution, was discovered by Yu. V. Prohorov (see [
35], Theorem 2; [
36]).
where
is some integer that satisfies
.
We also use the classical Stirling’s approximation for the factorial (see [
37], p. 54):
We use the following notation for the positive parts of .
We are now ready to give some useful properties of the Beta distribution in the following sequences of lemmas.
Lemma 1. Let . Assume that n is a positive integer. Then, Lemma 2. Let be a constant. Fix independently of n. Let . Assume that are positive integers and . Then, Lemma 3. Let be a constant. Fix independently of n. Let . Assume that are positive integers and . Then, Lemma 4. Fix . Assume that the sensor movement M is the finite sum of movements for i.e., . Then,where is some constant, which depends only on fixed a and ℓ. 3. Coverage and Interference Requirement When the Sensing Radius and the Interference Distance
In this section, we recall the known results about the expected
a-total displacement to fulfil the
requirement when
n mobile sensors with the identical sensing radius
are distributed uniformly at random and independently on the unit interval
. That is, the sum of the sensing area of
n sensors is equal to the length of the unit interval. Observe that, in the case when the sensing radius
and the interference distance
, the only way to achieve the
-coverage and interference requirement on the unit interval
is for the sensors to occupy the equidistant anchor positions
, for
(see
Figure 3a). The following exact asymptotic result was proven in [
29].
Theorem 1 ([
29])
. Let a be an even positive natural number. Assume that n mobile sensors are thrown uniformly and independently at random on the unit interval . The expected a-total displacement of all n sensors when the i-th sensor is sorted in increasing order moves from its current random location to the equidistant anchor location , for , respectively, is . In [
10], Theorem 1 was extended to all real-valued exponents
.
Theorem 2 ([
10])
. Fix . Assume that n mobile sensors are thrown uniformly and independently at random on the unit interval . The expected a-total displacement of all n sensors, when the i-th sensor sorted in increasing order moves from its current random location to the equidistant anchor location , for , respectively, is The gamma function
is defined to be an extension of the factorial to real number arguments. It is related to the factorial by
provided that
. It is also worthwhile to mention that the extension of the direct combinatorial method from [
29] leads to the exact asymptotic result in Theorem 2 only when
a is an odd natural number (see [
38], Theorem 2).
4. Coverage and Interference Requirement When the Square Sensing Radius and the Interference Distance
In this section, we analyse the expected a-total displacement to achieve the requirement when n mobile sensors with the identical square sensing radius are thrown uniformly at random and independently on the unit square provided that n is the square of a natural number. That is, the sum of the sensing area of n sensors is equal to the area of the unit square.
Observe that, to fulfil the
-coverage and interference requirement, the sensors have to occupy the following anchor positions
where
and
n must be the square of a natural number (see
Figure 3b).
It is known that the expected one-total displacement in this case is
Namely, the following theorem about the optimal transportation cost for random matching was obtained in [
11], a book related to these problems, which developed modern methods to bound stochastic processes.
Theorem 3 ([
11], Chapter 4.3)
. Let for some Assume that n mobile sensors are thrown uniformly and independently at random on the unit square Consider the non-random points evenly distributed as follows: where Then,where the infimum is over all permutations of and where d is the Euclidean distance. We are now ready to extend Theorem 3 to the displacement to the power a provided that
Theorem 4. Fix Let for some Assume that n mobile sensors are thrown uniformly and independently at random on the unit square Consider the non-random points evenly distributed as follows:
where Then,where the infimum is over all permutations of and where d is the Euclidean distance. 5. Coverage and Interference Requirement When the Sensing Radius and the Interference Distance
In this section, we analyse the expected a-total displacement to fulfil the requirement when n mobile sensors with the identical sensing radius are distributed uniformly at random and independently on the unit interval That is, the sum of sensing area of n sensors is greater than the length of the unit interval.
5.1. Analysis of Algorithm 1
Fix Let and be arbitrary small constants independent of the number of sensors n, and let
This subsection is concerned with reallocating the n random sensors within the unit interval to achieve only the following property:
We present a basic and energy-efficient algorithm
(see Algorithm 1). To illustrate Algorithm 1, let us consider the following simple example. We consider the initial location
of four sensors on the unit interval such that
(see
Figure 4).
Firstly, Algorithm 1 moves the sensor right-to-left at the position and the sensor does not move. Then, Algorithm 1 moves the sensor left-to-right at the position and the sensor left-to-right at the position
Theorem 5 states that the expected a-total displacement of algorithm is in when and Algorithm 1 is very simple, but the asymptotic analysis is not totally trivial. We note that the asymptotic analysis of Algorithm 1 is crucial in deriving the threshold phenomena.
In the proof of Theorem 5, we combine combinatorial techniques with the properties of the Beta distribution (see Equation (
10) in Lemma 2 and Equation (
11) in Lemma 3). The estimations for the Beta distribution with special positive integer parameters in Lemma 2 and Lemma 3 are new to the best of the author’s knowledge.
We now briefly discuss one technical issue in Steps (3)–(4) of Algorithm 1. It may happen that, for some initial random location of n sensors , Algorithm 1 moves some sensors to the right endpoint of the interval Namely, there exists with the following property: moves to some point in for all , and moves to the right endpoint of the interval for all Let be the location of n sensors after Algorithm 1. Then, to avoid interference to achieve the property that the distance between consecutive sensors is greater than or equal to s, we have to deactivate some sensors. Namely:
If , then for all , the sensors will no longer sense;
If then for all , the sensors will no longer sense.
Theorem 5. Let be a constant. Fix and independently of the number of sensors Assume that n mobile sensors are thrown uniformly and independently at random on the unit interval Then, Algorithm 1 for and reallocates the random sensors within the unit interval so that:
- (i)
The distance between consecutive sensors is greater than or equal to s and less than or equal to
- (ii)
The leftmost sensor is at a distance less than or equal to from the origin.
- (iii)
The expected a-total displacement is
Notice that Theorem 5 is valid regardless of the sensing radius; it depends only on the fact that the relocated sensors are not too far.
Finally, the following lemma will be helpful in the proof of the main results in
Section 5.2 for the sensors on the unit interval. In the proof of Lemma 5, we combine probabilistic techniques together with Estimation (
9) in Lemma 2 for the Beta distribution from
Section 2.
Lemma 5. Let be a constant. Fix and independently of the number of sensors Let and Let be the location of the n-th sensor after algorithm Then, 5.2. Analysis of Algorithm 2
Let us recall that is fixed and and are arbitrary small constants independent of the number of sensors In this subsection, we present algorithm (see Algorithm 2) for the requirement. We prove that the expected a-total displacement of algorithm is in when and Notice that our Algorithm 2 consists of two phases. During the first phase (see Initialization), we apply Algorithm 1. Then, in the second phase (see Case B and Case C), we add the additional sensors’ movement. Let be the location of sensors after Algorithm 2. The additional movement depends on the position of sensor in the interval
We now briefly explain the ideas behind the proof of Theorem 6 and the correctness of Algorithm 2:
- (i)
We have initially n random sensors on the unit interval with the identical sensing radius Firstly, we apply Algorithm 1 for and to achieve only the following properties:
- -
The distance between consecutive sensors is greater than or equal to and less than or equal to
- -
The first leftmost sensor is at a distance less than or equal to from the origin.
Applying Theorem 5, we deduce that the expected a-total displacement in the Initialization of Algorithm 2 is
- (ii)
Since the sensors have sensing radius and the distance between consecutive sensors is less than or equal to the -coverage and interference requirement is solved in expected a-total displacement in Case B of Algorithm 2. In this case, only a fraction of of rightmost sensors can move. We upper-bound the movement to the power a of each these sensors by (see Case 2 in the proof of Theorem 6).
- (iii)
In Case C, we move the sensors to equidistant anchor locations in expected a-total displacement. However, we can upper-bound the probability with which Case C occurs (see Lemma 5) to achieve the desired expected a-total displacement.
We are now ready to prove the main theorem for the sensors on the unit interval.
Theorem 6. Let be a constant. Fix and independently of the number of sensors Let Assume that n mobile sensors with the identical sensing radius are thrown uniformly and independently at random on the unit interval Then, Algorithm 2 solves the -coverage and interference requirement and has expected a-total displacement
Proof. There are three cases to consider:
Case 1: The algorithm terminates after Step 3. This case adds nothing to the expected a-total displacement.
Case 2: The algorithm terminates after Step 8. Then,
Let us recall that
and the distance between consecutive sensors is less than or equal to
Hence, we upper-bound the movement to the power
a of the
-th sensor for
as follows:
Observe that the movement of the
-th sensor is positive only when
From this, we see that only
sensors can move.
Observe that the movement to the power a of the n-th sensor is also less than
Hence, this adds to the
a-total displacement:
Case 3: The algorithm terminates after Step 12. Then,
In this case, we upper-bound the expected
a-total displacement in Steps (5)–(7) of algorithm
by
Then, by Lemma 5, the probability that this case can occur is
, and this adds to the expected
a-total displacement at most:
Finally, combining together the estimation from the Initialization (see Theorem 5), Case 1, Case 2, Case 3, as well as Lemma 4, we conclude that the expected a-total displacement of algorithm is at most This is enough to prove Theorem 6. □
6. Coverage and Interference Requirement for Square Sensing Radius and Interference Distance
In this section, we analyse the expected a-total displacement to achieve the requirement when n mobile sensors with the identical square sensing radius are thrown uniformly at random and independently on the unit square That is, the sum of the sensing area of n sensors is greater than the area of the unit square.
Let us recall that is constant and are fixed arbitrary small constant independent of the number of sensors
We prove that the expected a-total expected displacement of the algorithm (see Algorithm 3) is in when and
Notice that our Algorithm 3 is in two phases. During the first phase (see Steps (1)–(7)), we use a greedy strategy and move all the sensors only according to the second coordinate. As a result of the first phase, we obtain
lines, each with
random sensors. For the second phase, the main result from
Section 5 (see Theorem 6) is applicable.
It is worth pointing out that the first phase of Algorithm 3 reduces the a-total displacement on the unit square to the a-total displacement on the unit interval. Obviously, Algorithm 3 moves sensors only in a vertical and horizontal fashion, but it is powerful enough to derive the desired threshold.
We are now ready to prove the main result for the sensor on the unit square.
Theorem 7. Let be a constant. Fix and as arbitrary small constants independently of the number of sensors Let Assume that n mobile sensors with the identical square sensing radius are thrown uniformly and independently at random on the unit square Then, Algorithm 3 solves the -coverage and interference requirement and has expected a-total displacement in
Proof of Theorem 7. Firstly, we look at the expected
a-total displacement in the first phase of the algorithm (see Steps (1)–(7)). It was proven in [
28] that the expected
a-total displacement in Steps (1)–(7) of Algorithm 3 is in
(see the estimation of
for
in the proof of [
28], Theorem 5, Formulas (8) and (10), p. 41).
Observe that, in the second phase of Algorithm 3 (see Steps (8)–(10)), we have lines each with random sensors with the identical sensing radius According to Theorem 6, the expected a-total displacement is This together with Lemma 4 completes the proof of Theorem 7. □
7. Sensors in Higher Dimensions
In this section, we discuss the expected a-total displacement for the -coverage and interference requirement in higher dimensions, when
Let us recall that the proposed Algorithm 3 moves the sensors only in a vertical and horizontal fashion and reduces the a-total displacement on the unit square to the a-total displacement on the unit interval.
Hence, Algorithm 3 can be extended for the random sensors on the m-dimensional cube when We can, similar to the square sensing radius (see Definition 2) define an m-dimensional cube sensing radius, move the sensors only according to the axes, and reduce the a-total displacement on the unit cube to the a-total displacement on the unit interval.
Namely, for the sensors with the identical m-cube sensing radius (the sum of the sensing area of n sensors is greater than the area of the unit cube) and the interference distance , it is possible to give an algorithm with expected a-total displacement for all powers However, even though Theorem 7 can be generalized for the random sensors with the identical m-cube sensing radius on the m-dimensional cube, when the proposed generalization is weak.
Notice that Theorem 3 is closely related to the main result of paper [
21]. Namely, consider two sequences
of points that are independently uniformly distributed and the non-random points
are evenly distributed, i.e.,
where
on the unit square
, then
where
ranges over all permutations of
and
for some
On the other hand, there is a difference between
(the two-dimensional case) and
(the case of dimension at least three). Namely, for two sequences
of points that are independently uniformly distributed on the
m-dimensional cube
when
, we have
provided that
ranges over all permutations of
(see [
39] for details).
Hence, it seems that Theorem 3 together with Theorem 4 can be generalized for n random mobile sensors on the m-dimensional cube when , and the following result should hold.
Assume that
n random variables
are independently uniformly distributed and the non-random points
evenly distributed at the the positions
for
and
on the unit
m-dimensional cube
, then
for all powers
where
ranges over all permutations of
and
for some
Therefore, it is an open problem to prove that the -coverage and interference requirement for an m-cube sensing radius (the sum of the sensing area of n sensors is equal to the area of unit cube) and the interference distance can be solved in and to study the expected a-total displacement for the -coverage and interference requirement, when and
8. Experimental Results
In this section, we provide a set of experiments to confirm the discovered theoretical threshold for the expected a-total displacement. Wolfram Mathematica was used for our experiments when , and We distinguish two cases:
Case 1: sensing radius and interference distance
In this case, we conduct Algorithm 4.
Algorithm 4 Realisation of Algorithm 1 |
- 1:
- 2:
while n ≤ 5000 do - 3:
Generate independently and uniformly n random points on the unit interval ; - 4:
Calculate according to Algorithm 1 for and - 5:
Insert the points into the chart; - 6:
- 7:
end while
|
Notice that the experimental a-total displacement of Algorithm 4 is constant and independent of the number of sensors for is for , and is for Therefore, the carried out experiments confirm very well our theoretical upper bound estimation for for , and for (see Theorem 5 for , and ).
Case 2: sensing radius and interference distance
In this case, we conduct Algorithm 5.
Algorithm 5 Realisation of Theorem 2 |
- 1:
- 2:
while n ≤ 60 do - 3:
for to 200 do - 4:
Generate independently and uniformly random points on the unit interval ; - 5:
Calculate according to Theorem 2; - 6:
end for - 7:
for to 20 do - 8:
Calculate the average ; - 9:
Insert the points into the chart; - 10:
end for - 11:
- 12:
end while
|
In
Figure 8,
Figure 9 and
Figure 10, the black points represent the numerical results of the conducted experiments. The additional lines
are the plots of a function, which is the theoretical estimation (see the leading term in the asymptotic result of Theorem 2 for
, and
). It is worth pointing out that numerical results are situated near the theoretical line.
It is also possible to repeat the experiments to all exponents as well as Algorithms 2 and 3.
9. Conclusions and Future Direction
In this paper, the following natural problem was investigated: given n uniformly random mobile sensors in an m-dimensional unit cube, where what is the minimal energy consumption to move them so that they are pairwise at an interference distance at least s apart and so that every point of the m-dimensional unit cube is within the range of at least one sensor?
As the energy consumption measure for the displacement of n sensors, we considered the a-total displacement defined as the sum where is the distance sensor i has been moved and The main findings can be summarized as follows:
For the sensors placed on the unit interval, sensing radius , and interference distance , the expected minimal a-total displacement is of order When and provided that and are arbitrary small constants independent of the number of sensors then there is an algorithm with expected a-total displacement for all powers
For the case of the unit square and , square sensing radius , and interference distance , the expected minimal a-total displacement is at least of order provided that n is the square of a natural number. When and provided that and are arbitrary small constants independent of the number of sensors then there is an algorithm with expected a-total displacement for all powers
This paper opens several research directions.
First, it would be interesting to know what happens if and depend on n and decrease to This would give the complete picture of the threshold phenomena for the coverage and interference requirement.
Second, in this paper, we investigated the coverage and interference requirement only for one- and two-dimensional networks. It is an open problem to generalize this study to higher dimensions and investigate threshold phenomena for the m-dimensional cube, similar to 1- and 2-dimensional cubes.
Additionally it would be interesting for future research to study the coverage and interference requirement for a non-uniform displacement of sensors, on other domains, as well for some real-life sensor displacement.
We proved that the energy consumption for the coverage and interference requirement is constant and independent of the number of sensors for some parameters (see Equations (
1) and (
2)). While we discussed the practical importance of this constant energy consumption, an open problem for future study is the experimental evaluation of energy consumption for some real-life sensor displacement. However, this experimental evaluation for some real-life sensor deployment may be rather expensive due to the large number of sensors that would be required.