Strip-based deployment with cooperative communication to achieve connectivity and information coverage in wireless sensor networks

Abstract

Coverage and connectivity in wireless sensor network have been studied extensively in existing research works with physical and information coverage. The optimal deployment to achieve both information coverage and connectivity, on arbitrary values of the ratio of r_c and r_s, has been studied in previous work; meanwhile, the extended strip-based deployment based on information coverage is also studied. Either information coverage or cooperative communication could exploit collaboration of sensor nodes to improve the efficiency of deployment, while how good is strip-based deployment with both information coverage and cooperative communication is worth to be measured when the value of r_c/r_s is varied. In this article, the relationship between the density of sensors needed to achieve physical or information coverage and connectivity and the variety of r_c/r_s is derived in closed form for strip-based deployment of wireless sensor networks with cooperative communication. Then, a summary of different combinations of coverage and connectivity is provided, that physical or information coverage with or without cooperative communication could be employed to achieve full coverage and connectivity for strip-based deployment. Finally, some new strategy could be proposed based on the fusion of physical and information coverage to improve strip-based deployment. Some numerical results are provided to show the efficiency of all schemes to help researchers design more effective deployment schemes.

Keywords

Information coverage connectivity strip-based deployment cooperative communication WSNs

Introduction

Coverage and connectivity are critical issues for research on wireless sensor network (WSN).^1–8 The coverage issue normally depends on the sensing model of sensors, which has been studied in existing research works. One commonly used model is physical coverage, in which a sensor would cover a disk region centered on it with a radius as its sensing range.^1–3 Furthermore, the other type is that sensing accuracy might be dependent on the distance,⁹ whereas each point in the field could be determined by the sensing intensity that could be measured by its nearby sensors. Every point could be deemed to be covered with different levels of sensing intensities based on this sensing model. As a result, a new sensing model based on distributed estimation theory was proposed by Wang and colleagues.^9–11 Based on different detection techniques and scenarios, many probabilistic coverage models have also been proposed in past research work.^12–17

In a WSN, each sensor usually has limited communication range r_c, which might be different from the sensing range r_s.⁴ For coverage and connectivity in deployment of WSNs, some useful results could be found in existing works. If r_c is twice r_s at least, full coverage implies the full connectivity for WSNs.⁶ When $r_{c} \geq \sqrt{3} r_{s}$ , a regular triangle deployment is the most efficient scheme to achieve coverage and connectivity. When r_c = r_s, strip-based deployment can be near-optimal.⁷ Moreover, there are few results for arbitrary value of r_c/r_s, while it could be any specific value in practice.⁸

In recent years, coverage and connectivity in WSNs have been studied extensively in existing research works with physical coverage,^1–8 probabilistic coverage,^18–24 and coverage with data fusion.^25–27 However, r_c/r_s is usually regarded as a fixed value. Some regular deployments for coverage and connectivity with arbitrary values of r_c/r_s are studied recently,^1,8 while most of these are based on physical coverage model. The strip-based deployment is proposed to achieve coverage and 1-connectivity or 2-connectivity in Bai et al.⁸ Moreover, the deterministic deployment and random deployment with information coverage are proposed in Wang et al.^9,10 Furthermore, the regular deployments to achieve both information coverage and connectivity, on arbitrary values of r_c/r_s, are studied in our previous work;²⁸ meanwhile, the extended strip-based deployment based on information coverage is also studied in Wei et al.²⁸

Cooperative communication (CC)^29,30 has emerged in WSN, and sensors with a single antenna could share the antennas of others that have spatial diversity, such as the MIMO (multiple input, multiple output) system. The source sensor node and helper sensor nodes could transmit independent copies of sensing data simultaneously to the destination sensor node, which can combine partial signals of sensor nodes and decode them.^31–33 Any one-hop neighbors within the transmission range of the source sensor node can be regarded as potential helper sensor nodes. There are extensive works on the physical layer of CC,^25,34,35 and the research on higher layer is also being studied increasingly, since it could be used in coverage control,²⁶ broadcasting,^31,32 and routing.^27,33 Regular deployment with CC model for two-node cooperation to achieve K-connectivity and full coverage in WSNs is studied in Lu et al.³⁶

Although cooperative coverage, such as information coverage, and cooperative connectivity, such as CC, have been studied for coverage and connectivity in WSN, there are a few results for considering them both so as to improve the performance of deployment. In response to this, we focus on strip-based deployment for information coverage and cooperative connectivity in WSNs, with arbitrary values of r_c/r_s, and propose some enhanced scheme for strip-based deployment.

Some related works are summarized in Table 1. It can be found that there are many works on coverage and connectivity in WSNs, yet there are still some open issues that need to be addressed. Our work is motivated by these problems:

Either information coverage or CC could exploit collaboration of sensors to improve the efficiency of deployment. However, when both of these are considered, the arbitrary values of r_c/r_s could affect the strip-based deployment rather than regular deployment, since all the communication range should be identical in the latter. Hence, the question that needs to be answered is, how good is strip-based deployment with both information coverage and CC while the value of r_c/r_s is varied?

There is a small issue on deployment with information coverage while the value of r_c/r_s is varied; we will propose a new strategy of deployment based on fusion of physical and information coverage to enhance the sensor deployment in WSN.

Table 1.

Related works on connectivity and coverage for WSNs.

References	Issue	Coverage connectivity (model)	r_c/r_s
1, 8	Connectivity, coverage	Physical, no CC	Variety
3, 5 –7, 19 –24	Connectivity, coverage	Physical, no CC	Fixed
9 –11, 14	Connectivity, coverage	Information, no CC	Fixed
12, 13	Connectivity, coverage	Probabilistic, no CC	Fixed
15	Connectivity, barrier coverage	Information, no CC	Fixed
16	Coverage	Probabilistic, no CC	Fixed
17	3D coverage	Probabilistic, no CC	Fixed
2, 37, 38	Connectivity, coverage	Heterogeneous disks, common	Variety
39 –41	Multiple connectivity, Multiple coverage	Physical, no CC	Fixed
18	Connectivity, coverage	Probabilistic, no CC	Fixed
42 –44	Connectivity, coverage Connectivity, coverage	Data fusion, no CC Physical, CC	Fixed Variety

CC: cooperative communication.

The main contributions of this article are the following:

For strip-based deployment of WSNs with CC, the relation between r_c/r_s and density of sensors to achieve physical or information coverage and connectivity is derived in closed form, compared with that without CC.

A summary of different combinations of coverage and connectivity is provided, that physical or information coverage with or without CC could be employed to achieve full coverage and connectivity for strip-based deployment of WSN.

Strip-based deployment with information coverage is improved so that some results are derived for all strip-based deployment with this improved information coverage and CC.

The organization of this article is as follows: section “Coverage model” gives preliminary information about coverage models. Strip-based deployment is analyzed with both physical and information coverage in section “Strip-based deployment.” In section “Deployment with CC,” the strip-based deployment is extended in connectivity by CC, and different combinations of coverage and connectivity are provided. Section “Extension with information coverage and CC” provides some discussion and improvement on information coverage of strip-based deployment. Then, the performance of the deployment scheme is evaluated by numeric simulations in section “Numerical results.” Finally, conclusions are presented in section “Conclusion.”

Coverage model

Physical coverage

The classical disk sensing model has been widely used in numerous studies of WSNs.^{1–8,12–14} Any sensor could only sense the event that happens within its sensing range r_s. Based on this model, a point in the area is said to be “covered” physically by a sensor when the distance between the sensor’s position and this point is less than the sensing range r_s. This disk sensing model is widely accepted due to its simplicity and convenience for modeling and analysis. Thus, the probability that a point p is covered physically by a sensor node s_i can be modeled as

\Pr {p \in D (s_{i})} = {\begin{matrix} 1, d (s_{i}, p) \leq r_{s} \\ 0, d (s_{i}, p) > r_{s} \end{matrix}

(1)

where $d (s_{i}, p)$ is the Euclidean distance between the sensor s_i and point p.

Information coverage

The following illustration is defined in Wang et al.:^9,10 Consider K sensors, in which each sensor could obtain its locations by some methods and could acquire the measurements with unknown parameter θ of an target or event at some location. Let ${\hat{θ}}_{K}$ and ${\tilde{θ}}_{K} = {\hat{θ}}_{K} - θ$ denote the parameter estimate by K sensors and its estimation error, respectively, and d_i, i = 1, 2, …, K, denote the normalized distance between the location of event or target and sensor i. The parameter θ is assumed to be attenuated as the distance increases, and the measurement with the parameter θ, at a sensor, may be disturbed by an additive noise n, that is, $z_{i} = θ / d_{i}^{α} + n_{i}, α > 0$ . Thus, the objective of the parameter estimator is to estimate θ from the corrupted measurements z_i. Let $\hat{θ}$ and $\tilde{θ} = \hat{θ} - θ$ denote the parameter estimate and its estimation error, respectively. While there are K available measurements, some classical estimators such as best linear unbiased estimator can be adopted to estimate θ. If all noises are independent and are Gaussian noises with zero mean and standard deviation σ_i, then the estimate ${\hat{θ}}_{K}$ based on the best linear unbiased estimator could be expressed as

{\hat{θ}}_{K} = \frac{\sum_{i = 1}^{K} d_{i}^{- α} σ_{i}^{- 2} z_{i}}{\sum_{i = 1}^{K} d_{i}^{- 2 α} σ_{i}^{- 2}}

(2)

Thus, this could achieve minimum mean squared error (MSE) on ${\hat{θ}}_{K}$ , which could be expressed as

{\tilde{θ}}_{K} = \frac{\sum_{i = 1}^{K} d_{i}^{- α} σ_{i}^{- 2} n_{i}}{\sum_{i = 1}^{K} d_{i}^{- 2 α} σ_{i}^{- 2}}

(3)

When some event or target that has appeared at a given location could be estimated with a guaranteed estimation error by K distributed sensors, this point could be regarded as “information covered” by these K sensors. It is noteworthy that the estimation error ${\hat{θ}}_{K}$ is a random variable with zero mean and variance $σ_{i}^{2}$ . A point is called “information covered” by K sensors when it can be (K,ε)-covered at this point, if there are K distributed sensors to estimate the parameter cooperatively to achieve $\Pr {| {\tilde{θ}}_{K} | \leq A} \geq ε$ . This implies that if this probability is not less than ε, the parameter could be estimated by K sensors effectively at this point. Therefore, a region is said to be (K,ε)-covered fully if each point in this region is (K,ε)-covered without any doubt.

The following assumptions could be made in this article similar to that in Wang et al.9,10 for simplicity: all noises are Gaussian; thus, the summation of these noises is still Gaussian with zero mean. Furthermore, assuming that all noises have the same variance $σ_{i}^{2} = σ^{2}$ , k = 1,2,…, K, then we have

\Pr {| {\tilde{θ}}_{K} | \leq A} = 1 - 2 Q (\frac{A}{σ \sqrt{{(\sum_{i = 1}^{K} d_{i}^{- 2 α})}^{- 1}}})

(4)

where $Q (x) = (1 / \sqrt{2} π) \int_{x}^{\infty} \exp (- t^{2} / 2) dt$ . For comparison with the physical coverage, all distance D_i would be transformed into normalized distance d_i = D_i/r_s, where r_s is the sensing range of physical coverage, which is set to be the same as the corresponding distance that enables the estimation error reach threshold ε for information coverage whereupon there is $Q (A / d_{1}^{α} σ) = (1 / 2) (1 - ε)$ that implies the (1,ε)-covered sensing range. For simplicity, assuming all sensors choose the same threshold as all noises have the same variance $σ_{i}^{2} = σ^{2}$ , k = 1,2,…, K, then A = σ. Besides, assuming α = 1 as a reference attenuation coefficient, information coverage could be achieved by a single sensor within its sensing range, that is, (1,ε)-covered. While d_i = 1, that is, D_i = r_s, the threshold ε could be computed by equation (4) where ε = 0.683. In this article, we define ${\hat{r}}_{s}$ as the sensing range for (1,ε) coverage, and it can be calculated as ${\hat{r}}_{s} = r_{s} / (Q^{- 1} ((1 - ε) / 2))$ , where Q⁻¹(x) is the inverse of the Q-function. Theoretically, all D_i in information coverage should be normalized by ${\hat{r}}_{s}$ ; however, when ε = 0.683, their two sensing ranges are identical.

Strip-based deployment

The original strip-based deployment is extended in this section, from the physical coverage model to the information coverage model, and some results are presented based on the theoretical analysis.

Strip-based deployment with physical coverage

The original strip-based deployment is proposed to achieve coverage and 1-connectivity or 2-connectivity in Bai et al.⁸ (as in Figure 1).

Figure 1.

Strip-based deployment with coverage and (a) 1-connectivity and (b) 2-connectivity, $r_{c} / r_{s} < \sqrt{3}$ . White-filled circles form the horizontal strips, while dark-filled circles indicate the positions of sensors that form (a) single vertical strip or (b) two vertical strips. Here, $d_{α} = \min {r_{c}, \sqrt{3} r_{s}}$ and $d_{β} = r_{s} + \sqrt{r_{s}^{2} - d_{α}^{2} / 4}$ . The vertical strips could be removed when $r_{c} / r_{s} < \sqrt{3}$ .

Considering a Euclidean plane as the sensing field, horizontal strips of sensors are shaped by placing sensors with a regular separation of $d_{α} = \min {r_{c}, \sqrt{3} r_{s}}$ . These horizontal strips are made up of horizontally deployed sensors, with rows of sensor shifted to the right and left alternately by a separation of $d_{α} / 2$ . The separation between adjoining horizontal strips is $d_{β} = r_{s} + \sqrt{r_{s}^{2} - d_{α}^{2} / 4}$ , while $r_{c} / r_{s} < \sqrt{3}$ , and two adjoining horizontal sensor strips are not connected. As a result, some additional sensors are needed in the middle just for 1-connectivity or in the left- and right-hand sides for 2-connectivity, which are in the sensing region, represented by dark-filled circles in Figure 1.

Denote the distance between sensors of adjoining horizontal strips as δ, then $δ = \sqrt{{(d_{α} / 2)}^{2} + d_{β}^{2}}$ . Thus, N_ad sensors are needed to connect sensors in adjoining horizontal strips to achieve 1-connectivity or 2-connectivity; hence, N_ad could be calculated as in Bai et al.⁸

N_{ad} = {\begin{matrix} ⌈ δ / r_{c} - 1 ⌉ 1 - connectivity \\ 2 (⌈ δ / r_{c} - 1 ⌉) 2 - connectivity \end{matrix}

(5)

Some definitions are also provided that are the same as in Bai et al.⁸

Definition 1 [Connection Chord, Connection Angles]

Assuming two sensors are connected, that are deployed at x and y, there is a common chord within these two sensing disks $D_{r_{s}} (x)$ and $D_{r_{s}} (y)$ . This could be indicated by the line segment AB in Figure 2. As the location of these sensors changes, the length of their common chord changes simultaneously. When the specified values of r_s and r_c are given, the shortest common chord which could achieve connectivity is defined as connection chord, and its length is denoted by l(r_s, r_c). Furthermore, the central angles associated with the shortest connection chord in $D_{r_{s}} (x)$ and $D_{r_{s}} (y)$ are addressed as connection angles, which are denoted by ϕ(r_s, r_c). It should be noted that l(r_s, r_c) = 0 if r_c ≥ 2r_s.

Figure 2.

Two sensors located at x and y are connected under the situation that d(x, y) = r_c. Thus, chord AB is the connection chord, while $∠ AxB$ and $∠ AyB$ are the connection angles.

When N(r_s, r_c) is addressed as the minimum quantity of sensors for full coverage in strip-based deployment, this can be divided into two portions—N_h(r_s, r_c), which denotes the quantity of sensors required in horizontal strips, and N_v(r_s, r_c), which denotes the number of sensors required in vertical strips. The latter could be 0 in some situation as in Lemma 1, which is also defined in Wei et al.²⁸ Overall, N(r_s, r_c) can be expressed as

N (r_{s}, r_{c}) = N_{h} (r_{s}, r_{c}) + N_{v} (r_{s}, r_{c})

(6)

Lemma 1

When $r_{c} / r_{s} \geq \sqrt{3}$ , the sensors in adjoining horizontal strips are connected natively, and there is no need to add any sensors in vertical strips, that is, N_v(r_s, r_c) = 0.²⁸

Proof

When $r_{c} / r_{s} \geq \sqrt{3}$ , $d_{α} = \min {r_{c}, \sqrt{3} r_{s}} = \sqrt{3} r_{s} \leq r_{c}$ , and the vertical separation between adjoining horizontal strips is $d_{β} = r_{s} + \sqrt{r_{s}^{2} - d_{α}^{2} / 4} = 3 r_{s} / 2$ . Hence, the distance between two sensors in adjoining horizontal strips is $d_{s} = \sqrt{{(d_{α} / 2)}^{2} + d_{β}^{2}}$ , and there is

\frac{d_{s}}{r_{c}} = \frac{\sqrt{{(\frac{d_{α}}{2})}^{2} + {d_{β}}^{2}}}{r_{c}} = \frac{\sqrt{3} r_{s}}{r_{c}} \leq 1

(7)

Thus, $d_{s} \leq r_{c}$ , and sensors in the adjoining horizontal strip could be natively connected; therefore, N_v(r_s, r_c) could be 0.

It could be found that the Euclidean plane in two dimensions can be tiled by non-overlapping Voronoi polygons indicated by the dashed lines in Figure 3. It could be observed that the area of this polygon is area per node (APN) as in Definition 1 in Bai et al.⁸

Figure 3.

Non-overlapping polygons cover the two-dimensional plane; Voronoi polygons are formed by the dashed line.

Thus, the maximum APN of this sensor deployment is expressed as

γ_{\max} = n [\frac{1}{2} r_{s}^{2} \sin (φ)] + (k - n) \frac{1}{2} r_{s}^{2} \sin (\frac{2 π - n φ}{k - n})

(8)

where $φ = 2 \arccos (d_{α} / 2 r_{s})$ ; the polygon can be divided by some triangles and then k = 6 and n = 2; thus, there is

γ_{\max} = {r_{s}}^{2} (\sin (φ) + 2 \sin (\frac{π - φ}{2}))

(9)

When S_A is the area of the sensing region, without considering the boundary effect, the minimum quantity of sensors in all horizontal strips to obtain coverage and connectivity could be determined by

N_{h} (r_{s}, r_{c}) = \frac{S_{A}}{{r_{s}}^{2} (\sin (φ) + 2 \sin (\frac{π - φ}{2}))}

(10)

Strip-based deployment with information coverage

The strip-based deployment with information coverage is defined in Wei et al.,²⁸ such as (2,ε) coverage, that points in the sensing region could be covered by two sensors cooperatively. It can be expressed as

\Pr {| {\tilde{θ}}_{K} | \leq A} = 1 - 2 Q (\frac{A}{σ \sqrt{\frac{1}{d_{1}^{- 2 α} + d_{2}^{- 2 α}}}}) \geq ε

(11)

Without loss of generality, there is A = σ, α = 1, and d₁ = d₂ = d; thus, the maximum distance between the points in the midline of two sensors and the centers of two sensing disks could be figured out, while (2,ε) information coverage can be achieved opportunely for this distance. This normalized distance ${\hat{d}}_{\max}$ could be figured out as follows: when this normalized distance is more than ${\hat{d}}_{\max},$ $(2, ε)$ coverage cannot be achieved by two neighboring sensors

{\hat{d}}_{\max} = \frac{\sqrt{2}}{Q^{- 1} (\frac{1 - ε}{2})}

(12)

In strip-based deployment with the physical coverage model, the maximum distance between two neighboring sensors in horizontal strips is $\sqrt{3} r_{s}$ , depending on its definition: $d_{α} = \min {r_{c}, \sqrt{3} r_{s})$ . When separation of sensors in horizontal strips is $d_{α} = \sqrt{3} r_{s}$ , there is least overlap by two sensing disks, while the length of this common connection chord l is

l = 2 \sqrt{{r_{s}}^{2} - {(\frac{d_{α}}{2})}^{2}} = r_{s}

(13)

Definition 2 [Generalized Connection Chord, Generalized Connection Angle]

An extension of connection chord, Generalized Connection Chord, could be defined based on information coverage,²⁸ that is, a line segment on midline of two sensors. Then, the length of this line segment is equal to ${\hat{r}}_{s}$ similar to common connection chord in physical coverage. Similarly, Generalized Connection Angle could be redefined from Connection Angle. The horizontal and vertical separation ${\hat{d}}_{α}$ and ${\hat{d}}_{β}$ could be figured out with this Generalized Connection Chord to achieve (2,ε) coverage.

As shown in Figure 4, the position O (0,0) is set as the center point of adjoining four sensors in our coordinate, and the radius r of sensing disk is equal to ${\hat{r}}_{s}$ . Two sensors in horizontal strip are located at positions (–x₀,0) and (x₀,0); here, x₀ > 0 and the length of AB is ${\hat{r}}_{s}$ , while x₀ is the candidate x-coordinate of the sensors’ position. In this case, two neighboring sensors could be deployed as far as it could. The x-coordinate could also be determined by r_c; in this way, sensors would be deployed on (±x₀,0) or (±r_c/2,0). The coordinates of two endpoints of generalized connection chord are set as (0,y₀) and (0,–y₀), here y₀ > 0. Thus, the maximum distance between endpoints of generalized connection chord and the center of sensing disk is ${\hat{D}}_{\max}$ , where

{\hat{D}}_{\max} = \bar{CA} = \bar{CB} = \bar{DA} = \bar{DB}

Figure 4.

The connected sensors located at positions (–x₀,0) and (x₀,0). Generalized Connection Chord AB is the extended connection chord with $d (x_{0}, A) = d (x_{0}, B) \leq {\hat{D}}_{\max}$ , and $l (A, B) = {\hat{r}}_{s}$ .

and there is ${\hat{D}}_{\max} = {\hat{d}}_{\max} {\hat{r}}_{s}$ by denormalization. When ${\hat{d}}_{α} = 2 x_{0}$ and $y_{0} = \sqrt{{\hat{D}}_{\max}^{2} - {({\hat{d}}_{α} / 2)}^{2}} = \sqrt{{\hat{D}}_{\max}^{2} - x_{0}^{2}}$ , then $x_{0} = \sqrt{{\hat{D}}_{\max}^{2} - y_{0}^{2}}$ . As the length of generalized connection chord $l = {\hat{r}}_{s}$ , there are

y_{0} = \frac{l}{2} = \frac{{\hat{r}}_{s}}{2} = \frac{r_{s}}{2 Q^{- 1} (\frac{1 - ε}{2})}

(14)

\begin{matrix} x_{0} & = \sqrt{{\hat{D}}_{\max}^{2} - {y_{0}}^{2}} \\ = \sqrt{{(\frac{\sqrt{2} r_{s}}{Q^{- 1} (\frac{1 - ε}{2})})}^{2} - {(\frac{r_{s}}{2 Q^{- 1} (\frac{1 - ε}{2})})}^{2}} = \frac{\sqrt{7} r_{s}}{2 Q^{- 1} (\frac{1 - ε}{2})} \end{matrix}

(15)

Therefore, the final deployment can be described as

{\hat{d}}_{α} = \min {r_{c}, 2 x_{0}} = \min {r_{c}, \frac{\sqrt{7} r_{s}}{Q^{- 1} (\frac{1 - ε}{2})}}

(16)

{\hat{d}}_{β} = \frac{l}{2} + {\hat{r}}_{s} = \frac{3 {\hat{r}}_{s}}{2} = \frac{3 r_{s}}{2 Q^{- 1} (\frac{1 - ε}{2})}

(17)

The distance between two sensors in adjoining horizontal strips with (2,ε) coverage is

{\hat{d}}_{s} = \sqrt{{(\frac{{\hat{d}}_{α}}{2})}^{2} + {\hat{d}}_{β}^{2}} = \sqrt{{(\frac{{\hat{d}}_{α}}{2})}^{2} + {(\frac{3 r_{s}}{2 Q^{- 1} (\frac{1 - ε}{2})})}^{2}}

(18)

While ε = 0.683, ${\hat{d}}_{α} = \min {r_{c}, 2 x_{0}} = \min {r_{c}, \sqrt{7} r_{s}}$ and ${\hat{d}}_{β} = (l / 2) + {\hat{r}}_{s} = 3 r_{s} / 2$ .

Then, the Generalized Connection Angle can be expressed as $\hat{φ} = 2 \arccos ({\hat{d}}_{α} / 2 {\hat{d}}_{l})$ , where ${\hat{d}}_{l} = \sqrt{{(l / 2)}^{2} + {(d_{α} / 2)}^{2}}$ .

Definition 3 [Generalized APN]

The extended concepts of APN could be defined based on information coverage.²⁸Generalized APN indicates the area of a Voronoi polygon, which is similar to that in the definition of APN. It could be used for any sensor node to represent the average contribution on quality of service (QoS) of WSNs, such as coverage or communication, as shown in Figure 5.

Figure 5.

Non-overlapping polygons that cover plane in two dimensions; the Voronoi polygon is the hexagon formed by the blue line.

To derive the expression of ${\hat{N}}_{h} (r_{s}, r_{c})$ , the number of sensors in horizontal strips for (2,ε) information coverage, it could also be found that the area of this hexagon as in Figure 5 is APN in this deployment. Thus, a similar way just as in physical coverage could be used; as a result, maximum APN in this deployment is

{\hat{γ}}_{\max} = {\hat{d}}_{l}^{2} \sin (\hat{φ}) + 2 {\hat{d}}_{l} {\hat{r}}_{s} \sin (\frac{π - \hat{φ}}{2})

(19)

Then, the following results could be obtained on sensor density requirement for strip-based deployment to achieve (2,ε) information coverage and connectivity

{\hat{ρ}}_{cov, con} = \frac{1}{{\hat{γ}}_{\max}} = \frac{1}{{\hat{d}}_{l}^{2} \sin (\hat{φ}) + 2 {\hat{d}}_{l} {\hat{r}}_{s} \sin (\frac{π - \hat{φ}}{2})}

(20)

When ${\hat{N}}_{A}$ is the area of sensing region to achieve (2,ε) information coverage and connectivity, while the boundary effect is not considered, there is

{\hat{N}}_{h} (r_{s}, r_{c}) = {\hat{N}}_{A} {\hat{ρ}}_{cov, con} = \frac{1}{{\hat{d}}_{l}^{2} \sin (\hat{φ}) + 2 {\hat{d}}_{l} {\hat{r}}_{s} \sin (\frac{π - \hat{φ}}{2})}

(21)

Since the total number $\hat{N} (r_{s}, r_{c})$ of sensors needed are divided into two parts— ${\hat{N}}_{h} (r_{s}, r_{c})$ , the number of sensors in all horizontal strips, and ${\hat{N}}_{v} (r_{s}, r_{c})$ , the number of sensor in all vertical strips—there is

\hat{N} (r_{s}, r_{c}) = {\hat{N}}_{h} (r_{s}, r_{c}) + {\hat{N}}_{v} (r_{s}, r_{c})

(22)

{\hat{N}}_{v} (r_{s}, r_{c}) = {\begin{matrix} N_{ad} r_{c} < \frac{\sqrt{7} r_{s}}{Q^{- 1} (\frac{1 - ε}{2})} \\ 0 r_{c} \geq \frac{\sqrt{7} r_{s}}{Q^{- 1} (\frac{1 - ε}{2})} \end{matrix}

(23)

This can be explained in Lemma 2, which was also defined in Wei et al.²⁸ as the same deployment with the physical coverage model.

Lemma 2

When $r_{c} \geq (\sqrt{7} r_{s} / (Q^{- 1} ((1 - ε) / 2)))$ , the sensors in adjoining horizontal strips are connected natively to achieve (2,ε) coverage, without additional sensors in vertical strips, that is, ${\hat{N}}_{v} (r_{s}, r_{c}) = 0$ .²⁸

Proof

The minimum distance between the two sensors in adjoining horizontal strips to achieve (2,ε) coverage is ${\hat{d}}_{s}$ , when $r_{c} \geq (\sqrt{7} r_{s} / (Q^{- 1} ((1 - ε) / 2)))$ ; as ${\hat{d}}_{α} = \min {r_{c}, (\sqrt{7} r_{s} / (Q^{- 1} ((1 - ε) / 2)))}$ , there is ${\hat{d}}_{α} = \sqrt{7} r_{s} / (Q^{- 1} ((1 - ε) / 2)) \leq r_{c}$ , and then there is

\begin{matrix} {\hat{d}}_{s} & = \sqrt{{(\frac{{\hat{d}}_{α}}{2})}^{2} + {\hat{d}}_{β}^{2}} \\ = \sqrt{{(\frac{\sqrt{7} r_{s}}{2 Q^{- 1} (\frac{1 - ε}{2})})}^{2} + {(\frac{3 r_{s}}{2 Q^{- 1} (\frac{1 - ε}{2})})}^{2}} = \frac{2 r_{s}}{Q^{- 1} (\frac{1 - ε}{2})} \end{matrix}

(24)

Thus, there is

r_{c} \geq \frac{\sqrt{7} r_{s}}{Q^{- 1} (\frac{1 - ε}{2})} > \frac{2 r_{s}}{Q^{- 1} (\frac{1 - ε}{2})} = {\hat{d}}_{s}

(25)

So in this case, sensors in adjoining horizontal strips are connected natively, without sensors needed in vertical strips.

As shown in Figure 4, the position of sensor E is

(0, d_{β}) = (0, - y_{0} - {\hat{r}}_{s}) = (0, \frac{- 3 r_{s}}{2 Q^{- 1} (\frac{1 - ε}{2})})

(26)

Then the distance between sensor C and sensor E is

\begin{matrix} D_{CE} & = \sqrt{{(\frac{d_{α}}{2})}^{2} + {d_{β}}^{2}} = \sqrt{{(0 - x_{0})}^{2} + {(0 - y_{0} - {\hat{r}}_{s})}^{2}} \\ = \sqrt{{(\frac{\sqrt{7} r_{s}}{2 Q^{- 1} (\frac{1 - ε}{2})})}^{2} + {(\frac{3 r_{s}}{2 Q^{- 1} (\frac{1 - ε}{2})})}^{2}} \\ = \frac{2 r_{s}}{2 Q^{- 1} (\frac{1 - ε}{2})} = 2 {\hat{r}}_{s} = 2 r \end{matrix}

(27)

This implies that the distance between sensor C and sensor E is twice of radius r of sensing disk, so there are two tangent sensing disks. In this case, the Generalized Connection Chord could reach its lower bound in terms of length.

Lemma 3

The sensing region is fully information covered with no gap, under the deployment which is defined in Figure 4

Proof

It would be shown that if the union of (2,ε) coverage of sensors C and D and (1,ε) coverage of sensor E could fully cover the region defined by triangle CDE, then the total region, which can be divided by many triangles similar to triangle CDE, could be fully covered by another set of three sensors. Thus, the total sensing region is fully (2,ε)-covered with no gap. First, the curve with identical detection probability is drawn up, which could be used to determine the edge of coverage by two cooperative sensors. Based on $1 / (\sqrt{(1) / (d_{1}^{- 2} + d_{2}^{- 2}})) = Q^{- 1} ((1 - ε) / 2)$ , and the coordinate in Figure 4, there is

\begin{matrix} \frac{{r_{s}}^{2}}{{(x + \frac{\sqrt{7} r_{s}}{2 Q^{- 1} (\frac{1 - ε}{2})})}^{2} + y^{2}} + \frac{{r_{s}}^{2}}{{(x - \frac{\sqrt{7} r_{s}}{2 Q^{- 1} (\frac{1 - ε}{2})})}^{2} + y^{2}} \\ = {(Q^{- 1} (\frac{1 - ε}{2}))}^{2} \end{matrix}

(28)

This is the curve with identical detection probability to determine the edge of (2,ε) coverage (sensors C and D), and the curve with identical detection probability to determine the edge of (1,ε) coverage (sensor E) could also be determined by $(x - 0)^{2} + (y + y_{0} + {\hat{r}}_{s})^{2} = {\hat{r}}_{s}^{2}$ , and then there is

x^{2} + {(y + \frac{3 r_{s}}{2 Q^{- 1} (\frac{1 - ε}{2})})}^{2} = {(\frac{r_{s}}{Q^{- 1} (\frac{1 - ε}{2})})}^{2}

(29)

Without loss of generality, we could carry out normalization $r_{s} / (Q^{- 1} ((1 - ε) / 2)) = 1$ , as $r_{s} = 1$ and $ε = 0.683$ . Then, the above two curves could be expressed as

{\begin{matrix} x^{2} + {(y + \frac{3}{2})}^{2} = 1 \\ \frac{1}{{(x + \frac{\sqrt{7}}{2})}^{2} + y^{2}} + \frac{1}{{(x - \frac{\sqrt{7}}{2})}^{2} + y^{2}} = 1 \end{matrix}

(30)

The solutions of the above two equations could be worked out, and since y is always negative in the region defined by triangle CDE, the sensing region is fully (2,ε)-covered with no gap

{\begin{matrix} y_{1} = \sqrt{1 - x^{2}} - \frac{3}{2} \\ y_{2} = - \frac{1}{2} \sqrt{4 \sqrt{7 x^{2} + 1} - 4 x^{2} - 3} \end{matrix}

(31)

where $y_{1}$ and $y_{2}$ are solutions of the above two equations, and the tangency points are the middle points of lines CE and DE, such that the horizontal ordinate of tangency points are $\pm (1 / 2) x_{0} = \pm (\sqrt{7} r_{s} / (4 Q^{- 1} ((1 - ε) / 2))) = \pm (\sqrt{7} / 4))$ , which is indicated by the “range” line in Figure 6.

Figure 6.

The relation between two curves with range line of tangency points.

We will check the relation between $y_{1}$ and $y_{2}$ in the range determined by $[- (1 / 2) x_{0}, (1 / 2) x_{0}]$ since there may be a gap in this range for coverage. As in Figure 6, $y_{2}$ is always under $y_{1}$ in this range $[- (1 / 2) x_{0}, (1 / 2) x_{0}]$ , so there is always overlap in this region; then the sensing region is fully information covered with no gap

Deployment with CC

CC model

In Yu et al.,³⁰ the CC model was described as follows: Traditional connectivity without CC assumes that node A with transmission range R is connected to node B, which is in a disk with radius R and center A. Generally, if the received average signal-to-noise ratio (SNR) of node B from node A is not less than a given threshold $τ$ , the link from A to B can be regarded as available. Adopting this model, the transmission range with CC could be determined. Each node has a maximum limit $P_{MAX}$ on transmission power, while $P_{i}$ is the transmission power of node i. $τ$ is the threshold of SNR required by the receiver to decode received data with a given channel model. The channel coefficient $h_{ij}$ , which is from node i to node j, can be generated by a Rayleigh distribution. $d_{ij}$ is the distance between node i and node j, $α$ is the path loss exponent, and N is the noise power. For the direct communication from node i to node j, the average SNR of node j from node i, $γ_{ij}$ , should meet the following condition

\frac{P_{i} E [{| h_{ij} |}^{2}]}{{(d_{ij})}^{α} N} = γ_{ij} \geq τ, P_{i} \leq P_{MAX}

(32)

Multiple nodes transmit the same data packet to destination node j simultaneously by CC, while the multipath signals are then received by node j. The total SNR of the output with maximal-ratio combining (MRC) on node j could be expressed as the summation of received average SNRs:⁴⁵ $γ_{j} = \sum_{i \in Ω} γ_{ij}$ , where $Ω$ is the set of source node and its helper nodes. When the nodes in $Ω$ transmit data packet simultaneously, by CC, the following condition must be satisfied to decode data at destination node j

\sum_{i \in Ω} \frac{P_{i} E [{| h_{ij} |}^{2}]}{{(d_{ij})}^{α} N} = \sum_{i \in Ω} γ_{ij} \geq τ, P_{i} \leq P_{MAX}

(33)

As in equation (33), CC could extend transmission coverage. The source node could use CC only after transmitting the request for cooperation and source data to neighbor nodes by direct communication. Without loss of generality, noise N can be normalized as 1 in equations (32) and (33), and channel gain could also be normalized as $E [| h_{ij} |^{2} = 1$ . When the source node transmits data to neighbor nodes by direct communication, that is, traditional connectivity without CC, there is

\frac{P_{i}}{{(d_{ij})}^{α}} \geq τ

(34)

In the coverage and connectivity models, $r_{c}$ could be defined as follows: when $P_{i}$ is set to a given value, there is a minimum $d_{ij}$ that the average SNR of node j from node i could reach the lower bound $τ$ . Thus

r_{c} = {(\frac{P_{i}}{τ})}^{\frac{1}{α}}

(35)

Under the same model, when two neighbor nodes use CC to connect to the other node, there is

\frac{P_{1}}{{(d_{1 j})}^{α}} + \frac{P_{2}}{{(d_{2 j})}^{α}} \geq τ

(36)

When the destination node is deployed on the middle line of two cooperative nodes, that is, exactly strip-based deployment, all nodes use the same transmit power for traditional connectivity without CC, which can guarantee the basic connectivity between the source node and the helper node. Thus, there is $P_{1} = P_{2} = P_{i}$ , and the extended transmit range $r_{c}^{c}$ by CC is $r_{c}^{c} = d_{1 j} = d_{2 j}$ .

When it takes the lower bound $τ$ , there is $(P_{i} / (r_{c}^{c})^{α}) + (P_{i} / (r_{c}^{c})^{α}) = τ$ , that is, $(2 P_{i} / (r_{c}^{c})^{α}) = τ$ , then the extended transmit range $r_{c}^{c}$ by CC can be expressed as

r_{c}^{c} = 2^{\frac{1}{α}} r_{c}

(37)

where path loss factor $α$ is set to 2 and 4. For simplicity, the SNR threshold $τ$ is set to 1; thus, it could be found that the transmit range is extended by CC since $2^{(1 / α)} > 1$ .

Strip-based deployment with CC

There should be information transfer without CC between the source node and the helper node to guarantee the basic connectivity, so the CC could not be employed effectively in triangle, square, and hexagon deployments that distances between all neighboring nodes are equal. However, the distances between neighboring nodes in vertical and horizontal strips are different; it could be employed to extend the range of connectivity by CC.

First, we recall the deployment of physical coverage without CC; there is $d_{α} = \min {r_{c}, \sqrt{3} r_{s}}$ and $d_{β} = r_{s} + \sqrt{r_{s}^{2} - d_{α}^{2} / 4}$ for horizontal and vertical strips; then the distance of two sensors in adjoining horizontal strips is $d_{s} = \sqrt{{(d_{α} / 2)}^{2} + d_{β}^{2}}$ .

In Bai et al.,⁸ there is proof that when $r_{c} / r_{s} \geq \sqrt{3}$ , the nodes in adjoining horizontal strips could connect naturally, that is, $d_{s} \leq r_{c}$ . If $r_{s}$ is fixed while $r_{c}$ is increased to this threshold, then there is no need for additional nodes in vertical strips,

Next, it would be shown how deployment could be improved when CC is adopted. Since two adjoining nodes in horizontal strip take advantage of CC, the transmit range in vertical direction could be extended, as shown in equation (37), and we need to examine how this extension would affect the strip-based deployment.

The separation of sensors in a horizontal strip is still $d_{α} = \min {r_{c}, \sqrt{3} r_{s}}$ . Since $r_{c}^{c} = 2^{(1 / α)} r_{c}$ , the extended $r_{c}^{c}$ would affect whether the additional nodes in vertical strip are needed, that is, the threshold of $r_{c} / r_{s}$ is changed. As this threshold comes from when the separation of adjoining horizontal strips is $d_{s} = r_{c}$ , then there is $r_{c} = \sqrt{3} r_{s}$ correspondingly, which is proved in Wei et al.²⁸

When CC is adapted to the strip-based deployment, then the separation of adjoining horizontal strips meets $d_{s} = r_{c}^{c} = 2^{(1 / α)} r_{c}$ . However, there is a new threshold of $r_{c} / r_{s}$ rather than $r_{c} / r_{s} \geq \sqrt{3}$ , which implies that $r_{c} / r_{s}$ does not need to be increased beyond $\sqrt{3}$ to achieve connectivity between adjoining horizontal strips. Therefore, this result could be used to reduce the sensor s in vertical strips in some case that the new threshold is achieved while $r_{c} / r_{s} < \sqrt{3}$ . At this time, there is $d_{α} = r_{c}$ , $d_{β} = r_{s} + \sqrt{{r_{s}}^{2} - {r_{c}}^{2} / 4}$ ; thus, the distance between adjoining horizontal strips is

d_{s} = \sqrt{{(\frac{d_{α}}{2})}^{2} + d_{β}^{2}} = \sqrt{(\frac{{r_{c}}^{2}}{2}) + {(r_{s} + \sqrt{{r_{s}}^{2} - {r_{c}}^{2} / 4})}^{2}}

(38)

Lemma 4

If there is $(r_{c} / r_{s}) \geq ((2 \sqrt{2^{(1 / α)} - (1 / 4)}) / 2^{(2 / α)})$ , the sensors in adjoining horizontal strips could be connected natively with CC, and there is no need to add any sensors in vertical strips, in other words, N_v(r_s, r_c) = 0.

Proof

$d_{s}$ is the distance of two sensors in adjoining horizontal strips, and its maximum could affect the new threshold. If there is $d_{s} = r_{c}^{c} = 2^{(1 / α)} r_{c}$ , the adjoining horizontal strip could be connected natively with CC; as a result, the new threshold could be expressed as $r_{c} / r_{s} = ((2 \sqrt{2^{(1 / α)} - (1 / 4)}) / 2^{(2 / α)})$ .

If there is $(r_{c} / r_{s}) \geq ((2 \sqrt{2^{(1 / α)} - (1 / 4)}) / 2^{(2 / α)})$ , then $(r_{s} / r_{c}) \leq (2^{(2 / α)} / (2 \sqrt{2^{(1 / α)} - (1 / 4)}))$ and

\begin{matrix} \frac{{d_{s}}^{2}}{{r_{c}}^{2}} & = \frac{1}{4} + {(\frac{r_{s}}{r_{c}} + \sqrt{{(\frac{r_{s}}{r_{c}})}^{2} - \frac{1}{4}})}^{2} \\ \leq \frac{1}{4} + {(\frac{2^{\frac{2}{α}}}{2 \sqrt{2^{\frac{1}{α}} - \frac{1}{4}}} + \sqrt{\frac{2^{\frac{4}{α}}}{4 (2^{\frac{1}{α}} - \frac{1}{4})} - \frac{1}{4}})}^{2} = 2^{\frac{2}{α}} \end{matrix}

(39)

As a result, there is $d_{s} \leq 2^{(1 / α)} r_{c} = r_{c}^{c}$ , so the sensors in adjoining horizontal strips could connect to each other natively by CC.

Strip-based deployment with information coverage and CC

In our previous article,²⁸ we proved that if $r_{c} \geq (\sqrt{7} r_{s} / (Q^{- 1} ((1 - ε) / 2)))$ , the sensors in adjoining horizontal strips are connected natively for the (2,ε) coverage, without any additional sensors needed in vertical strips. However, it could be found that there is “greater than” rather than “no less than” in the derivation, which is in equation (44) in Wei et al.,²⁸ so it is not a tight lower bound. In fact, when $r_{c}$ increases to some threshold which is less than $\sqrt{7} r_{s} / (Q^{- 1} ((1 - ε) / 2))$ , the sensors in adjoining horizontal strips could be connected natively. We would derive this threshold in this section.

When the full coverage is based on (2,ε) coverage model for strip-based deployment, there is $d_{α} = \min {r_{c}, (\sqrt{7} r_{s} / (Q^{- 1} ((1 - ε) / 2)))}$ and ${\hat{d}}_{β} = (l / 2) + {\hat{r}}_{s} = (3 {\hat{r}}_{s} / 2) = (3 r_{s} / (2 Q^{- 1} ((1 - ε) / 2)))$ as separation parameter in strip-based deployment. At this time, since the threshold is less than $\sqrt{7} r_{s} / (Q^{- 1} ((1 - ε) / 2))$ , the distance between sensors in adjoining horizontal strips is $d_{s} = \sqrt{{(d_{α} / 2)}^{2} + d_{β}^{2}} = \sqrt{{(r_{c} / 2)}^{2} + {(3 r_{s} / (2 Q^{- 1} ((1 - ε) / 2)))}^{2}}$ . When $d_{s} = r_{c}$ , the sensors in adjoining horizontal strips could connect each other natively; thus, there is $(r_{c} / 2)^{2} + (3 r_{s} / (2 Q^{- 1} ((1 - ε) / 2)))^{2} = {r_{c}}^{2}$ . Then, we could derive the threshold as

\frac{r_{c}}{r_{s}} = \frac{\sqrt{3}}{Q^{- 1} (\frac{1 - ε}{2})}

(40)

Finally, we could improve Lemma 2 in subsection “Strip-based deployment with information coverage,” which is also Lemma 3 in Wei et al.²⁸

Lemma 2a

For the (2,ε) coverage model, if $r_{c} \geq (\sqrt{3} r_{s} / (Q^{- 1} ((1 - ε) / 2)))$ , the sensors in adjoining horizontal strips are connected natively, without any additional sensor in vertical strips, that is, ${\hat{N}}_{v} (r_{s}, r_{c}) = 0$ .

Next, we consider the threshold when the sensors in adjoining horizontal strips are connected natively with CC for the (2,ε) coverage. In this situation, the deployment parameters are still as

$d_{α} = \min {r_{c}, (\sqrt{7} r_{s} / (Q^{- 1} ((1 - ε) / 2)))}$ and ${\hat{d}}_{β} = (l / 2) + {\hat{r}}_{s} = (3 {\hat{r}}_{s} / 2) = 3 r_{s} / (2 Q^{- 1} ((1 - ε) / 2))$ . Since CC could extend the range of communication, this threshold would be less than the threshold defined in equation (40) intuitively, and $r_{c}$ is also less than $\sqrt{7} r_{s} / (Q^{- 1} ((1 - ε) / 2))$ ; thus, the distance between sensors in adjoining horizontal strips is also $d_{s} = \sqrt{{(r_{c} / 2)}^{2} + {(3 r_{s} / (2 Q^{- 1} ((1 - ε) / 2)))}^{2}}$ . At this time, if $d_{s} = r_{c}^{c} = 2^{(1 / α)} r_{c}$ , the sensors in adjoining horizontal strips could be connected natively with CC; therefore, $(r_{c} / 2)^{2} + (3 r_{s} / (2 Q^{- 1} ((1 - ε) / 2)))^{2} = 2^{(2 / α)} {r_{c}}^{2}$ .

Then, we could derive the threshold as

\frac{r_{c}}{r_{s}} = \frac{3}{2 \sqrt{2^{\frac{2}{α}} - \frac{1}{4}} Q^{- 1} (\frac{1 - ε}{2})}

(41)

Lemma 5

For the (2,ε) information coverage model, if $r_{c} / r_{s} \geq (3 / (2 \sqrt{2^{(2 / α)} - (1 / 4)} (Q^{- 1} ((1 - ε) / 2))))$ , the sensors in adjoining horizontal strips are connected natively with CC, without any additional sensor in vertical strips, that is, ${\hat{N}}_{v} (r_{s}, r_{c}) = 0$ .

Proof

If there is $\sqrt{7} / (Q^{- 1} ((1 - ε) / 2)) \geq (r_{c} / r_{s}) \geq (3 / (2 \sqrt{2^{(2 / α)} - (1 / 4)} (Q^{- 1} ((1 - ε) / 2))))$ , then

\frac{{d_{s}}^{2}}{{r_{c}}^{2}} = \frac{{(\frac{d_{α}}{2})}^{2} + d_{β}^{2}}{{r_{c}}^{2}} = \frac{1}{4} + \frac{9}{4 {(Q^{- 1} (\frac{1 - ε}{2}))}^{2}} {(\frac{r_{s}}{r_{c}})}^{2}

(42)

Since $r_{c} \geq (3 r_{s} / (2 \sqrt{2^{(2 / α)} - (1 / 4)} (Q^{- 1} ((1 - ε) / 2))))$ , then $(r_{s} / r_{c})^{2} \leq ((4 (Q^{- 1} ((1 - ε) / 2))^{2}) (2^{2 / α} - (1 / 4))) / 9)$ , thus there is

\begin{matrix} \frac{{d_{s}}^{2}}{{r_{c}}^{2}} & \leq \frac{1}{4} + \frac{9}{4 {(Q^{- 1} (\frac{1 - ε}{2}))}^{2}} \frac{4 {(Q^{- 1} (\frac{1 - ε}{2}))}^{2} (2^{\frac{2}{α}} - \frac{1}{4})}{9} \\ = \frac{1}{4} + 2^{\frac{2}{α}} - \frac{1}{4} = 2^{\frac{2}{α}} \end{matrix}

(43)

Then, there is $d_{s} \leq 2^{(1 / α)} r_{c}$ or $d_{s} \leq r_{c}^{c}$ , which implies that the sensors in adjoining horizontal strips are connected natively with CC.

On the other hand, when $(r_{c} / r_{s}) \geq (\sqrt{7} / (Q^{- 1} ((1 - ε) / 2)))$ , the distance between sensors in adjoining horizontal strips is $d_{s} = \sqrt{(\sqrt{7} r_{s} / {(2 Q^{- 1} ((1 - ε) / 2))}^{2} + {(3 r_{s} / (2 Q^{- 1} ((1 - ε) / 2)))}^{2}} = (2 r_{s} / (Q^{- 1} ((1 - ε) / 2)))$ then

\frac{d_{s}}{r_{c}} = \frac{2}{Q^{- 1} (\frac{1 - ε}{2})} \frac{r_{s}}{r_{c}} \leq \frac{2}{Q^{- 1} (\frac{1 - ε}{2})} \frac{Q^{- 1} (\frac{1 - ε}{2})}{\sqrt{7}} = \frac{2}{\sqrt{7}}

(44)

As a result, because $d_{s} \leq (2 / \sqrt{7}) r_{c} < r_{c}$ , sensors in adjoining horizontal strips can be connected natively even without CC.

Summary of different combinations

We summarize the different situations in which sensors in adjoining horizontal strips are connected natively:

1. If all sensors are deployed with physical coverage and without CC, then when $r_{c} / r_{s} \geq \sqrt{3}$ , the sensors in adjacent horizontal strip can be natively connected, which was proved by Bai et al.⁸

2. If all sensors are deployed with physical coverage and CC, then when $r_{c} / r_{s} \geq (2 \sqrt{2^{(1 / α)} - (1 / 4)}) / (2^{(2 / α)})$ , the sensors in adjacent horizontal strip can be natively connected, which is proved in subsection “Strip-based deployment with CC,” and this threshold is about 1.079 while $α = 2$ .

3. If all sensors are deployed with information coverage and without CC, then when $r_{c} / r_{s} \geq (\sqrt{3} / (Q^{- 1} ((1 - ε) / 2)))$ , sensors in adjacent horizontal strip can be natively connected.

Proof

When $\sqrt{7} / (Q^{- 1} ((1 - ε) / 2))) \geq (r_{c} / r_{s}) \geq (\sqrt{3} / (Q^{- 1} ((1 - ε) / 2)))$ , the deployment parameters are denoted as ${\hat{d}}_{α} = \min {r_{c}, (\sqrt{7} r_{s} / (Q^{- 1} ((1 - ε) / 2)))} = r_{c}$ and ${\hat{d}}_{β} = (l / 2) + {\hat{r}}_{s} = (3 {\hat{r}}_{s} / 2) = 3 r_{s} / (2 Q^{- 1} ((1 - ε) / 2))$ . And then there is $d_{s}^{2} / r_{c}^{2} = (1 / 4) + (9 / (4 (Q^{- 1} ((1 - ε) / 2))^{2})) (r_{s} / r_{c})^{2}$ . Since $r_{c} \geq (\sqrt{3} r_{s} / (Q^{- 1} ((1 - ε) / 2)))$ and $(r_{s} / r_{c})^{2} \leq (Q^{- 1} ((1 - ε) / 2))^{2} / 3$ , then

\begin{matrix} \frac{{d_{s}}^{2}}{{r_{c}}^{2}} & = \frac{1}{4} + \frac{9}{4 {(Q^{- 1} (\frac{1 - ε}{2}))}^{2}} {(\frac{r_{s}}{r_{c}})}^{2} \leq \frac{1}{4} \\ + \frac{9}{4 {(Q^{- 1} (\frac{1 - ε}{2}))}^{2}} \frac{{(Q^{- 1} (\frac{1 - ε}{2}))}^{2}}{3} = 1 \end{matrix}

(45)

Therefore, $d_{s} \leq r_{c}$ , so sensors in adjacent horizontal strips can be natively connected.

On the other hand, when $(r_{c} / r_{s}) \geq (\sqrt{7} / (Q^{- 1} ((1 - ε) / 2)))$ , this situation was proved in our last article;²⁸ if $r_{c} \geq (\sqrt{7} r_{s} / (Q^{- 1} (1 - ε) / 2))$ , the sensors in adjoining horizontal strips are connected natively for the (2,ε) coverage, without any additional sensors needed in vertical strips.

4. If all sensors are deployed with information coverage and CC, then what is the threshold of $r_{c} / r_{s}$ above which the sensors in adjacent horizontal strips can be natively connected?

For this configuration, the deployment parameters are still denoted as $d_{α} = \min {r_{c}, (\sqrt{7} r_{s} / (Q^{- 1} ((1 - ε) / 2)))}$ and ${\hat{d}}_{β} = (l / 2) + {\hat{r}}_{s} = (3 {\hat{r}}_{s} / 2) = (3 r_{s} / (2 Q^{- 1} ((1 - ε) / 2)))$ . Since this is a relax definition which is extended from the definition of Connection Chord and the horizontal separation for strip-based deployment by physical coverage, the threshold on this situation is slightly larger than the threshold of deployment with physical coverage and CC.

As the threshold could be reduced by the function of CC, while it is still smaller than $\sqrt{7} / (Q^{- 1} ((1 - ε) / 2))$ , there is $d_{s} = \sqrt{{(r_{c} / 2)}^{2} + (3 r_{s} / {(2 Q^{- 1} ((1 - ε) / 2))}^{2}}$ . When $r_{c}^{c} = 2^{(1 / α)} r_{c} = d_{s}$ , the adjacent horizontal strips of sensors are natively connected. Then there is

\frac{r_{c}}{r_{s}} = \frac{3}{2 \sqrt{2^{\frac{2}{α}} - \frac{1}{4}} Q^{- 1} (\frac{1 - ε}{2})}

(46)

Therefore, if $(r_{c} / r_{s}) \geq (3 / (2 \sqrt{2^{(2 / α)} - (1 / 4)} (Q^{- 1} ((1 - ε) / 2))))$ , the adjacent horizontal strips of sensors are natively connected, which is 1.1339, slightly larger than 1.079, while ε = 0.683 and $α = 2$ .

Proof

When $\sqrt{7} / (Q^{- 1} ((1 - ε) / 2)) \geq (r_{c} / r_{s}) \geq (3 / (2 \sqrt{2^{(2 / α)} - (1 / 4)} (Q^{- 1} ((1 - ε) / 2))))$ , the deployment parameters are denoted as $d_{s}^{2} / r_{c}^{2} = (1 / 4) + (9 / (4 (Q^{- 1} ((1 - ε) / 2))^{2}) (r_{s} / r_{c})^{2}$ . Since $r_{c} \geq 3 r_{s} / (2 \sqrt{(2^{2 / α} - (1 / 4))} (Q^{- 1} ((1 - ε) / 2))$ , then $(r_{s} / r_{c})^{2} \leq (((4 (Q^{- 1} ((1 - ε) / 2))^{2}) (2^{(2 / α)} - (1 / 4))) / 9)$ ; thus, from equation (43), there is $(d_{s}^{2} / r_{c}^{2}) \leq 2^{(2 / α)}$ , so $d_{s} \leq 2^{(1 / α)} r_{c}$ , that is, $d_{s} \leq r_{c}^{c}$ , confirming adjoining horizontal strips are connected natively with CC.

On the other hand, when $(r_{c} / r_{s}) \geq (\sqrt{7} / (Q^{- 1} ((1 - ε) / 2)))$ , the distance between sensors in adjoining horizontal strips is $d_{s} = 2 r_{s} / (Q^{- 1} ((1 - ε) / 2))$ ; then, from equation (43), there is $(d_{s} / r_{c}) \leq (2 / \sqrt{7)} < 1$ . Thus, adjoining horizontal strips are connected natively without CC

Extension with information coverage and CC

Improvement of information coverage

Now we focus on some new issue which could be found in our previous article.²⁸ When $r_{c} / r_{s}$ is in the low region, it seems that the efficiency of information coverage is worse than that of the physical coverage, which is shown in Figure 12 of Wei et al.²⁸ In fact, this is caused by the definition of strip-based deployment for information coverage, which is quite naïve.

For the strip-based deployment for physical coverage, the parameter of deployment is $d_{α} = \min {r_{c}, \sqrt{3} r_{s}}$ and $d_{β} = r_{s} + \sqrt{r_{s}^{2} - d_{α}^{2} / 4}$ . Furthermore, the parameter of that for information coverage is ${\hat{d}}_{α} = \min {r_{c}, (\sqrt{7} r_{s} / (Q^{- 1} ((1 - ε) / 2))}$ and ${\hat{d}}_{β} = (l / 2) + {\hat{r}}_{s} = 3 r_{s} / (2 Q^{- 1} ((1 - ε) / 2))$ . In fact, there is a relax definition on ${\hat{d}}_{β}$ such that Generalized Connection Chord is always $l = {\hat{r}}_{s}$ . The reason of this relax definition is extended from the definition of strip-based deployment by physical coverage, in which there is Connection Chord and the horizontal separation is $d_{α} = \min {r_{c}, \sqrt{3} r_{s}}$ . When $r_{s}$ is fixed and $r_{c}$ is increased, if $r_{c} \leq \sqrt{3} r_{s}$ , the horizontal separation depends on $r_{c}$ , with larger $r_{c}$ resulting in smaller Connection Chord. If $r_{c} > \sqrt{3} r_{s}$ , the horizontal separation does not depend on $r_{c}$ , and larger $r_{c}$ cannot reduce the length of Connection Chord any more. As a result, the smallest length of Connection Chord is $r_{s}$ at this time. Meanwhile, we set the smallest length of Generalized Connection Chord to ${\hat{r}}_{s}$ for information coverage and then proved that this deployment could fully cover the region in subsection “Strip-based deployment with information coverage.” We proved that the horizontal separation with this length of Generalized Connection Chord could achieve full information coverage, so any other horizontal separation, with the parameter of deployment as ${\hat{d}}_{β} = ((l / 2) + {\hat{r}}_{s})$ , could achieve full information coverage obviously.

What was previously ignored was that Generalized Connection Chord is not required to calculate ${\hat{d}}_{β}$ when $r_{c} / r_{s}$ is very small for information coverage, since it is too relaxed for coverage and just physical coverage is enough. However, this could result in coverage efficiency of information coverage even smaller than that of physical coverage while $r_{c} / r_{s}$ is in the low region. Therefore, we redefine the strip-based deployment for information coverage as follows

{\hat{d}}_{α} = \min {r_{c}, \frac{\sqrt{7} r_{s}}{Q^{- 1} (\frac{1 - ε}{2})}}

(47)

{\hat{d}}_{β} = {\begin{matrix} {\hat{r}}_{s} + \sqrt{{\hat{r}}_{s}^{2} - {r_{c}}^{2} / 4} r_{c} \leq \sqrt{3} {\hat{r}}_{s} \\ \frac{l}{2} + {\hat{r}}_{s} = \frac{3 {\hat{r}}_{s}}{2} = \frac{3 r_{s}}{2 Q^{- 1} (\frac{1 - ε}{2})} r_{c} > \sqrt{3} {\hat{r}}_{s} \end{matrix}

(48)

It was shown that there is a threshold which defined two regions, in that the Connection Chord and Generalized Connection Chord are employed to calculate ${\hat{d}}_{β}$ separately.

Deployment with improved information coverage and CC

If all sensors are deployed with this improved information coverage and CC, then what is the threshold of $r_{c} / r_{s}$ above which the sensors in adjacent horizontal strips are natively connected?

Since this threshold is smaller than $\sqrt{7} / (Q^{- 1} ((1 - ε) / 2))$ as the help of CC, and when $r_{c} > \sqrt{3} {\hat{r}}_{s}$ , there is

$d_{s} = \sqrt{{(r_{c} / 2)}^{2} + {(3 r_{s} / (2 Q^{- 1} ((1 - ε) / 2)))}^{2}}$ ; when $r_{c}^{c} = 2^{(1 / α)} r_{c} = d_{s}$ , the adjacent horizontal strips of sensors are natively connected. Then, there is $r_{c} / r_{s} = (3 / (2 \sqrt{2^{(2 / α)} - (1 / 4)} (Q^{- 1} ((1 - ε) / 2))))$ . Therefore, if $(r_{c} / r_{s}) \geq (3 / (2 \sqrt{2^{(2 / α)} - (1 / 4)} (Q^{- 1} ((1 - ε) / 2))))$ , the adjacent horizontal strips of sensors are natively connected.

Proof

When $(\sqrt{7} / (Q^{- 1} ((1 - ε) / 2))) \geq (r_{c} / r_{s}) \geq (3 / (2 \sqrt{2^{(2 / α)} - (1 / 4)} (Q^{- 1} ((1 - ε) / 2))))$ , the deployment parameters are denoted as $d_{s}^{2} / r_{c}^{2} = (1 / 4) + (9 / (4 (Q^{- 1} ((1 - ε) / 2))^{2})) (r_{s} / r_{c})^{2}$ ; since $r_{c} \geq (3 r_{s} / (2 \sqrt{2^{(2 / α)} - (1 / 4)} (Q^{- 1} ((1 - ε) / 2))))$ , then $(r_{s} / r_{c})^{2} \leq ((4 (Q^{- 1} ((1 - ε) / 2))^{2}) (2^{(2 / α)} - (1 / 4))) / 9$ ; thus $({d_{s}}^{2} / {r_{c}}^{2}) = 2^{(2 / α)}$ , so $d_{s} \leq 2^{(1 / α)} r_{c}$ , that is, $d_{s} \leq r_{c}^{c}$ , confirming adjoining horizontal strips are connected natively with CC.

As this is in the region $r_{c} > \sqrt{3} {\hat{r}}_{s}$ , the relation $r_{c} \geq (3 r_{s} / (2 \sqrt{2^{(2 / α)} - (1 / 4)} (Q^{- 1} ((1 - ε) / 2))))$ is definitely met.

On the other hand, when $r_{c} \leq \sqrt{3} {\hat{r}}_{s}$ , in fact, there is a physical coverage issue with the help of CC. Based on the conclusion we derived in section “Strip-based deployment with CC,” adjoining horizontal strips are connected natively with CC, where ${\hat{r}}_{s} = r_{s} / (Q^{- 1} ((1 - ε) / 2))$

Finally, when $(r_{c} / r_{s}) \geq (\sqrt{7} / (Q^{- 1} ((1 - ε) / 2)))$ , the distance between sensors in adjoining horizontal strips is $d_{s} = 2 r_{s} / (Q^{- 1} ((1 - ε) / 2))$ ; then, $d_{s} / r_{c} = (2 / (Q^{- 1} ((1 - ε) / 2))) (r_{s} / r_{c}) \leq (2 / (Q^{- 1} ((1 - ε) / 2))) ((Q^{- 1} ((1 - ε) / 2)) / \sqrt{7}) = 2 / \sqrt{7} < 1$ . Thus, adjoining horizontal strips are connected natively without CC.

Overall, we complete the discussion on different cases that $r_{c} / r_{s}$ is in different regions.

Numerical results

In this section, the aforementioned deployments are compared in terms of the number of sensors needed to achieve full coverage and full connectivity, with different models. The region of sensor deployment is a two-dimensional square region that is 1000 m × 1000 m, and the sensing range r_s is set as 30 m for the physical coverage model, while the communication range is defined as 20 m ≤ r_c ≤ 120 m for physical connectivity. As the deployment area is large enough relative to the sensing range r_s, the boundary effect is not a major factor in deployment performance; therefore, the effect of $r_{c} / r_{s}$ can be focused on in our works.

Information coverage or CC

The comparison of different strip-based deployments with physical coverage, information coverage, and CC is shown in Figure 7. There are different vertical drops in different strip-based deployments in Figure 7, which indicate the thresholds of $r_{c} / r_{s}$ when adjoining horizontal strips are connected natively. From then on, there is no need to add any sensors in vertical strips. As a result, there are vertical drops on these curves. For physical coverage, the threshold is $r_{c} / r_{s} = \sqrt{3}$ , which is proved in Lemma 1, so there is a vertical drop here on the curve “strip1.” Similarly, it can be found that the thresholds for information coverage and physical coverage with CC are $\sqrt{3} / (Q^{- 1} ((1 - ε) / 2))$ and $(2 \sqrt{2^{(1 / α)} - (1 / 4)}) / 2^{(2 / α)}$ , which are proved in Lemma 2a and Lemma 4, respectively. $\sqrt{3} / (Q^{- 1} ((1 - ε) / 2)) = \sqrt{3}$ when ε = 0.683 and $(2 \sqrt{2^{(1 / α)} - (1 / 4))} / 2^{(2 / α)}$ is about 1.079 when $α = 2$ , which are summarized in subsection “Summary of different combinations.” It could be found that information coverage outperforms physical coverage in terms of the minimum amount of sensors needed while r_c/r_s is at a low level. On the other hand, CC could reduce the minimum amount of sensors needed with physical coverage, since the CC could extend the communication range and adjoining horizontal strips could be connected natively in some condition. However, if r_c/r_s is at a low level, there is a region where physical coverage outperforms information coverage, as there are some omissions on the original definition of deployment with information coverage. Thus, it could be improved in section “Extension with information coverage and CC.”

Figure 7.

Number of sensors needed in different strip-based deployments for 1-connectivity (strip1: physical coverage, strip1info: information coverage, stripc1: physical coverage with CC) to achieve full coverage and connectivity for various values of r_c/r_s (ε = 0.683, $α = 2$ ).

Figure 8 shows the comparison of different strip-based deployments with information coverage, CC, and both information coverage and CC. In Figure 8, there are different vertical drops in different strip-based deployments, which indicate the thresholds of $r_{c} / r_{s}$ when adjoining horizontal strips are connected natively. It can be found that the thresholds for information coverage and physical coverage with CC are $\sqrt{3} / (Q^{- 1} ((1 - ε) / 2))$ and $(2 \sqrt{2^{(1 / α)} - (1 / 4)}) / 2^{(2 / α)}$ , respectively. For information coverage with CC, the threshold is $3 / (2 \sqrt{2^{(2 / α)} - (1 / 4)} (Q^{- 1} ((1 - ε) / 2)))$ , which is proved in Lemma 5. It is about 1.1339 when ε = 0.683 and $α = 2$ .

Figure 8.

Number of sensors needed in different strip-based deployments for 1-connectivity (stripc1: physical coverage with CC, strip1info: information coverage, strip1infoc: information coverage with CC) to achieve full coverage and connectivity for various values of r_c/r_s (ε = 0.683, $α = 2$ ).

First, the minimum amount of sensors needed for two deployments with information coverage is very close no matter with or without CC. It could also be found that when $r_{c} / r_{s} \geq (\sqrt{3} / (Q^{- 1} ((1 - ε) / 2)))$ , deployment with information coverage outperforms the corresponding scheme with both physical coverage and CC in terms of minimum amount of sensors needed. This implies that information coverage is more attractive than CC for strip-based deployments while r_c/r_s is at a high level. The reason for this is if r_c/r_s is in a high level, the bottleneck of the deployment depends on sensing range; thus, data fusion could be utilized cooperatively to improve the effectiveness. However, if r_c/r_s is at a low level, there is also a region where physical coverage outperforms information coverage in terms of two schemes (strip1info and strip1infoc). From Figures 8 and 9, it seems that CC may be more useful to increase the efficiency of strip-based deployment; however, it could be improved in section “Extension with information coverage and CC,” and we will show this later.

Figure 9.

Number of sensors needed in different strip-based deployments for 1-connectivity (strip1infoc: information coverage with CC, strip1infoxz: improved information coverage without CC, strip1infoxzc: improved information coverage with CC) to achieve full coverage and connectivity for various values of r_c/r_s (ε = 0.683, $α = 2$ ).

Strip deployment with information coverage

Figure 9 shows the comparison of different deployments based on information coverage, including the redefined strip-based deployment with improved information coverage and CC. In Figure 9, there are different vertical drops in different strip-based deployments, which indicate the thresholds of $r_{c} / r_{s}$ when adjoining horizontal strips are connected natively. It can be found that the threshold for information coverage with CC is $3 / (2 \sqrt{2^{(2 / α)} - (1 / 4)} (Q^{- 1} ((1 - ε) / 2)))$ . For improved information coverage without CC, the threshold is $\sqrt{3} / (Q^{- 1} ((1 - ε) / 2))$ , which is also the threshold of information coverage since the improvement is based on this original version. For improved information coverage with CC, the threshold is $3 / (2 \sqrt{2^{(2 / α)} - (1 / 4) (} Q^{- 1} ((1 - ε) / 2)))$ , which is discussed in subsection “Deployment with improved information coverage and CC.”

As in Figure 9, it can be seen that the redefined deployment with information coverage and CC outperforms all other schemes with information coverage overall

When $r_{c} / r_{s} \leq ((2 \sqrt{2^{(1 / α)} - (1 / 4)}) / 2^{(2 / α)})$ , the redefined deployment with information coverage and CC coincides with the scheme without CC, as r_c/r_s is too small so that CC cannot play a role in collaborative connectivity. However, when $r_{c} / r_{s} \geq (\sqrt{3} / (Q^{- 1} ((1 - ε) / 2)))$ , redefined deployment with improved information coverage and CC becomes the scheme with information coverage without CC or the original deployment with information coverage and CC, since when r_c/r_s is large enough, the deployment bottleneck depends on the sensing range.

Deployment under different path loss factors

For applications in real world, the path loss factor $α$ in CC cannot be easily determined, so a more relaxed threshold is usually adopted to ensure the connectivity for CC. Some simple simulations could be done for this situation, where different path loss factors would be taken with CC to achieve the full connectivity. Figures 10 and 11 show the comparison of different deployments with various values of path loss factor. In Figures 10 and 11, there are different vertical drops in different strip-based deployments, which indicate the thresholds of $r_{c} / r_{s}$ when adjoining horizontal strips are connected natively. It can be found that thresholds for all schemes with CC would be affected by the path loss factor $α$ in CC since there is variable $α$ in threshold $(2 \sqrt{2^{(1 / α)} - (1 / 4)}) / 2^{(2 / α)}$ and $3 / (2 \sqrt{2^{(2 / α)} - (1 / 4) (} Q^{- 1} ((1 - ε) / 2)))$ . It can be found that with the increase in the path loss factor, the deployments with CC become more closed compared to that without CC. Since the CC could extend the communication range as $r_{c}^{c} = 2^{(1 / α)} r_{c}$ , the more the $α$ , the less this extension.

Figure 10.

Number of sensors needed in different strip-based deployments for 1-connectivity (strip1: physical coverage, stripc12: physical coverage with CC where $α = 2$ ; stripc14: physical coverage with CC where $α = 4$ ) to achieve full coverage and connectivity for various values of r_c/r_s (ε = 0.683).

Figure 11.

Number of sensors needed in different strip-based deployments for 1-connectivity (strip1: physical coverage, strip1infocxz2: improved information coverage with CC where $α = 2$ ; strip1infocxz4: improved information coverage with CC where $α = 4$ ) to achieve full coverage and connectivity for various values of r_c/r_s (ε = 0.683).

Conclusion

In this article, the relation between the ratio of r_c and r_s and the density of sensors required to achieve physical or information coverage and connectivity is derived in closed form so as to judge how good is strip-based deployment with both information coverage and CC for strip-based deployment of WSNs with CC. Then, a summary of different combinations of coverage and connectivity is presented, that physical or information coverage with or without CC could be employed to achieve full coverage and connectivity for strip-based deployment. Finally, the strip-based deployment is improved based on fusion of physical and information coverage, and its characteristics are derived in closed form. Also, the efficiencies of all schemes are analyzed and compared by some numerical results, and it is worth noting that the following remarks can be deduced:

Information coverage is more attractive than CC for strip-based deployments while r_c/r_s is in the high level, because if r_c/r_s is in a high level, the bottleneck of the deployment depends on the sensing range; data fusion could be utilized cooperatively to increase the effectiveness of this situation.

Redefined strip-based deployment with information coverage and CC outperforms all the other strip-based deployments with information coverage overall. With the increase in the path loss factor, the deployments with CC become more closed compared to that without CC. Since communication range is extended by CC as $r_{c}^{c} = 2^{(1 / α)} r_{c}$ , the more the $α$ , the less this extension.

Footnotes

Handling Editor: Masayoshi Aritsugi

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is supported by the National Natural Science Foundation of China under Grants No. 61401360 and Fundamental Research Funds for the Central Universities (3102017zy026).

ORCID iD

Wei Cheng

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