An analytical model to estimate the performance metrics of a finite multihop network deployed in a rectangular region

Journal Pre-proof An analytical model to estimate the performance metrics of a finite multihop network deployed in a rectangular region Jaiprakash Nagar, S.K. Chaturvedi, Sieteng Soh PII:

S1084-8045(19)30326-1

DOI:

https://doi.org/10.1016/j.jnca.2019.102466

Reference:

YJNCA 102466

To appear in:

Journal of Network and Computer Applications

Received Date: 21 August 2018 Revised Date:

28 August 2019

Accepted Date: 15 October 2019

Please cite this article as: Nagar, J., Chaturvedi, S.K., Soh, S., An analytical model to estimate the performance metrics of a finite multihop network deployed in a rectangular region, Journal of Network and Computer Applications (2019), doi: https://doi.org/10.1016/j.jnca.2019.102466. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier Ltd.

An Analytical Model to Estimate the Performance Metrics of a Finite Multihop Network Deployed in a Rectangular Region Jaiprakash Nagar1, S. K. Chaturvedi2 and Sieteng Soh3 1,2 Subir Chowdhury School of Quality and Reliability, IIT Kharagpur, West Bengal, India 3 Department of Computing, Curtin University, Australia [email protected], [email protected], [email protected] Abstract- This paper presents a generalized analytical model to estimate the commonly employed network performance metrics, viz., node isolation probability, minimum node degree distribution, and the probability of -connectivity in a wireless multi-hop network deployed in a finite regular rectangular region rather than a square region of limited scope. We derive and provide closed form expressions for the effective area covered by a node positioned in various sub regions of the rectangular region. Our derivation considers the boundary effects of the region that has significant impact on the effective coverage area by a node, which is important to accurately estimate the minimum node degree distribution, -connectivity and node isolation probability. We validate and compare our proposed analytical model by performing simulation to show that the results obtained through our proposition are more plausible and also conform to the specific cases of square region considered by other researchers. Later, we also discuss the tradeoffs between the transmission range of the nodes and the number of nodes that result in a fully -connected network with finite number of nodes deployed in a rectangular region of varying dimensions. Keywords: Node isolation probability, minimum node degree distribution, -connectivity, wireless multihop networks, boundary effects, wireless nodes.

I. Introduction The popularity of wireless sensor networks (WSNs) have increased tremendously in recent past due to their relatively low costs and power requirements, ease in deployment even in a remote and/or inaccessible terrain almost in no time, and emergency and/or hazardous scenarios [1]-[2]. The consequences of such versatility not only have led to the opening new vistas for their possible deployment in several critical/non-critical areas such as in military, security, industrial automation and control, telecommunications, health sector but also fired the researchers’ interests in the area of its design and analysis [3]. Most WSNs consist of a large number of small, low power, low cost sensors performing the role of routers to transceivers utilizing the single/multi-hop communication over wireless channels, and operating in a decentralized and self-organized fashion without the need of any fixed infrastructure requirements [4]. The proper operation of wireless multi-hop networks relies on the faithfully exchanging information among the nodes or to a specified gateway, thus, making connectivity as one of the basic and necessary requirement for its judicious planning and effective operations. A WSN is fully connected if there exists at least one single/multi-hop path connecting each of the other sensor nodes. Since each node of a wireless multi-hop network contributes to its overall connectivity, full multi-hop connectivity (meaning there is at least one path between each pair of the

nodes) is a sufficient condition for the faithful exchange of information among the nodes. A multi-hop network may get disconnected due to inappropriate network topology, isolated nodes caused by small transmission range, low node density, low power and/or nodes failure upon deployment. Therefore, in order to design a multi-hop resilient network, researchers have been investigating and evaluating the connectivity - one of the most basic and pertinent network performance metric of these networks by proposing their models. The -connectivity ensures the availability of -mutually independent paths between any pair of network forming nodes. In other words, a -connected network would remain 1connected even after the failure of  − 1 network forming nodes, thus, making it robust against the nodes failure [5]. Literature indicates that the minimum node degree distribution and node isolation probability of a multi-hop network are the essential properties in checking a network’s overall connectivity [6], [7]. The minimum node degree distribution of a network is defined as the probability that each node of the network has at least  distinct neighbors [8], and an arbitrary node of the network is said to be isolated if it has no connection to any other node of the network [9]. In the remainder of this paper a network means a wireless multi-hop network. In most of the available analytical models, researchers have considered uniform node density and node degree throughout the whole network region to investigate the connectivity properties of a network deployed over a two-dimensional region. However, simulation and theoretical results have a wide discrepancy due to the non-consideration of boundary effects phenomenon, i.e., nodes positioned near the physical boundaries of the region cover an effective area relatively smaller than the area covered by the nodes positioned at the middle of the region. The boundary effects cause nonlinearity in the degrees of nodes deployed near the borders of the network region, hence, there are more chances for a node to get isolated near the physical boundaries of the region. Therefore, nodes deployed near the physical boundaries of the region play a critical role in the overall connectivity of the network, besides, due to boundary effects, the asymptotic connectivity results available for large-scale networks are not scalable for the finite networks [10]-[12]. The analytical models incorporating boundary effects do exist in the literature to evaluate the topological characteristics of the network, however, with some limitations, e.g., they are usually valid either for low [13]-[16], or high node density [17], [19]-[20], [29] because by increasing the network size in terms of number of nodes from a small to a large scale can cause breakdowns of linearity and dependencies between variables, and lead to non-linear behavior of the network dynamics. Therefore, the analytical models applicable for low node density may not be applicable at high node density and vice-versa [10]. It is worth to mention that there is no standard definition of node density in the available literature indicates, e.g., the authors in [21] deploy a network with 40 nodes with large transmission ranges inside the Cambridge University campus area and assume it a high node density network. Similarly, for a connected network, the network is considered with high node density if the probability that the network is 1-connected ( ) ≥ 0.95 and otherwise with low node density [22]. Likewise, in [23], if all nodes in the group are within one hop of each other, then it is taken as a high node density network. On the other hand, if the number of nodes, within one hop of each other, is up to 20% of the remainder of the group, then it is considered as low node density network. In our work, we consider a network is of low node density if the number of nodes per unit area is less than or equal to one; otherwise the network would be a high node density network.

The authors of [14] and [24] have shown that the shape and dimensions of the region of the network deployment have a significant impact on the coverage and the mean effective degree of the network nodes; hence, it becomes pertinent to incorporate the impact of shape and dimensions of the network region in the analytical framework. The work done in [13], [16]-[17], [19], [25]-[26], [29] considered a square region to investigate topological properties such as the minimum degree distribution and hence the connectivity of the network. The analytical framework presented in [13], [16] consider a finite square region to investigate the topological properties at low node density only, e.g., 0.00035 nodes/m2. Moreover, two models proposed to estimate the connection probability in [16] have their own limitations. For instance, the proposed Corner Border Dominance model for the connectivity analysis is only valid for high transmission range (at low node density). More specifically, the model provides a good approximation only for high connection probability requirements (greater than or equal to 95%). On the other hand, the Border Effects Avoidance model [16] eliminates the border effects by considering the square region with side length ℒ+2 , i.e., the model increases the side length by 2 from ℒ - for transmission range  - to nullify the border effects, and to keep the network node density constant. Thus, the latter model requires deploying extra sensor nodes. The analytical results in [16] approximate the simulation results at higher transmission ranges only whereas the analytical results for smaller range do not fully conform to the simulation results. The mathematical model derived in [17], [19]-[20], [29] assume a unit square region at high node density, e.g., in 10 to 50 nodes/m2 [20], to investigate the minimum node degree distribution or/and connectivity of the network. But these analytical frameworks are not appropriate to analyze the network properties of a network deployed in a rectangular region - a more general and practical region than the unit square one. Load-balancing is another major issue in wireless sensor networks and can be achieved by deploying the total number of nodes in a given region of interest in such a way that every node process/transmit almost equal number of data packets to the next hop neighbor or to the sink node. In this way, every node consumes almost an equal amount of energy which in turn increases the network lifetime. In [30], the authors have proposed a non-uniform node distribution strategy which ascertains coverage with load balance by routing packets via minimum hop paths that results in substantial increment in network lifetime. Similarly, in [31], authors have proposed a Node Deployment Scheme based on Path Creation (NDSPC) for wireless sensor networks in the mountain-road environment. In this scheme, they proposed to deploy some extra transmission paths on appropriate locations to significantly lessen the energy holes around the sink, thus, reducing the transmission delay without noticeable increase in the deployment cost. It was shown that the network lifetime achieved through the NDSPC deployment scheme would have a substantial improvement over the other deployment methods. The work in [32] also addressed the load balancing problem and proposed a node deployment algorithm named APOLLO to minimize the maximum sensing range of nodes to achieve load balancing -coverage-often applied to enhance the fault tolerance in the event of node failures. They pointed out that the results would have been different had the boundary effects also considered. The authors in [33] propose a load balancing strategy for data transmission of WSNs, called super link-based data drainage (SLDD) to transfer data from the locations far away from the sink with high data traffic to the locations close to the sink with low data traffic. They further prove that the proposed SLDD algorithm outperforms ACO-Greedy [34] and NDSBA [35] node deployment methods in terms of transmission

delay and network lifetime. Similarly, authors in [36] have proposed a node deployment strategy based on their proposed probability density function (pdf) for wireless multi-hop networks to achieve energy balancing to enhance the network lifetime while maintaining the coverage and connectivity criteria. The presented results showed that the network lifetime of the proposed PDFND was much better than that of NNDS [37], NDGD [38] and NDUD [39] respectively for 5-layer network. In this work, however, we have confined ourselves to study the boundary effects phenomenon to estimate the network performance metrics. The study of load-balancing in presence of the boundary effects is outside the scope of this work and would be our future work. Besides, the related work is classified in Table I. The motivational factors for our proposed framework are to overcome the aforementioned limitations and considerations of border effects in the model. Firstly, our work provides a generalized analytical framework to investigate and estimate the network metrics (i.e., minimum node degree distribution, node isolation probability, and the -connectivity) for a network deployed in a more practical and scalable rectangular region. More specifically, our results for rectangular region should also be applicable for a square region of finite or unit area without any alterations. Thus, our work generalizes the existing solutions that are applicable only for a finite square region and a unit square region. Secondly, our framework can be used for networks with both high and low node density. Simulations in Section IV show the accuracy of the framework on networks with node density ranges, e.g., from 0.0001-0.03 nodes/m2 for finite to 10-50 nodes/m2 for a unit square, respectively. Note that accurate analytical framework for rectangular region enables us to discuss the tradeoffs between the transmission range of the nodes and the number of nodes that result in a fully -connected network with finite number of nodes deployed in a rectangular region of different dimensions. In our opinion, this discussion is necessary and pertinent to observe the veracity of the claim made by earlier researches that the probability of -connectivity is a linear function of the nodes’ transmission range rt and the number of nodes deployed in a given region [18]. Analytical framework for the more general shaped network, i.e., rectangular region, consequently requires more mathematical rigor as compared to solutions for the square regions. Among others, as later described in Section IV, firstly, the rectangular region includes more number of transmission range scenarios than a square region. More specifically, our solution for the rectangular region includes 13 transmission range scenarios, as compared to the seven scenarios required in [20] for unit square region. Note that several of these scenarios in a square region are the subset of the scenarios generated in a rectangular region. Secondly, while each transmission range scenario in square shaped networks also contains several sub-regions, a scenario in rectangular region, in general includes more sub-regions than in a unit or finite square region. More specifically, our framework in total considers 83 sub-regions as compared to only 34 subregions for the unit square region in [20]. Generating the 83 sub-regions for rectangular shaped network is imperative to identify and compute all effective coverage areas of each node lying in one of these subregions so as to produce more plausible and accurate estimation of network metrics at hand (i.e., minimum node degree distribution, node isolation probability and -connectivity). The proposed framework can be utilized to predict the performance metrics of the multi-hop networks in various applications such as surveillance and intrusion detection in border areas, environmental monitoring, telecommunication systems, and fault tolerant applications with low as well as high node density.

Table 1: Related work

Reference #

Work and Limitations

[4]

Related to a space decomposition method to analyze the probability distribution of the distance between nodes in a wireless ad-hoc network in a finite rectangular or hexagonal region. Useful to calculate the node degree and max-flow capacity of the network, however, did not examine the connectivity performance of the network.

[10]

Derives analytical expressions to calculate the effective coverage area of nodes lying in lateral and corner borders of the square region and proposes a heuristic correction function that includes the border effects to the simulation results. Only applicable for a square shaped network deployment area, besides, does not analyze the influence of shape & size of deployment region and boundary effects on the connectivity. Examines the impact of boundary effects on the network coverage fraction for a probabilistic sensing model and deduces that the boundary effects reduce the coverage fraction. But lacks the quantification of boundary effects on connectivity performance of the network.

[12]

[13]-[14], [16]

Estimate the multi-hop network metrics such as average node degree and connectivity for a 2 finite square region at low node density (<1 nodes/m )

[15]

Examines the influence of node failure probability and sensing models on the coverage of a network deployed in a circular region of interest. But does not analyze the connectivity performance of the network and the results are obtained at a low node density (e.g., 0.000637 nodes/m ).

[17], [19][20], [29]

Examine the connectivity metrics of high node density multi-hop networks deployed in unit square regions only. For example, the model derived in [17] is valid for a finite square only and the results are obtained at a very high node density (25-256 nodes/m ). Have different assumptions for low or high node density networks. For instance, in [22], a network is considered with high node density if the probability that the network is 1connected ( ) ≥ 0.95 and otherwise with low node density.

[21]-[23]

[13],[16]-[17], [19],[25]-[26], [29] [30]-[39]

Consider a square region to investigate topological properties such as the minimum degree distribution and hence the connectivity of the network. Address a major issue in wireless sensor networks known as load-balancing to enhance the network lifetime by proposing different node deployment schemes which may results in a substantial increment in network lifetime.

To propose our model, we consider a network consisting of  sensor nodes with equal transmission range r meters, independently and uniformly deployed in a rectangular region of length ℒ meters and width ω meters. The authors in [20] argued that estimating the effective coverage area of a node lying inside the region of interest is a simple yet powerful method to determine the minimum node degree distribution, node isolation probability and the -connectivity of a network. Therefore, we divide the rectangular network region into various sub-regions to derive analytical expressions for the effective area covered by a node lying in any of these sub-regions formed in the middle or close to the boundary regions at a given transmission range scenario. The main features of the paper are summarized here below: •

We have derived a generalized analytical framework incorporating boundary effects to compute the effective coverage area of a node in boundary regions to evaluate the performance metrics of wireless multi-hop networks.

•

•

The proposed analytical model can be used to investigate the performance metrics of wireless multi-hop networks deployed in a rectangular shaped area of any dimension at low as well as high node density. The analytically derived results can be very useful to predict the connectivity metrics such as node isolation probability, minimum node degree distribution and -connectivity of wireless multi-hop networks before their actual deployments.

The rest of this paper is organized as follows: Section II, presents the network model, communication model and connectivity related definitions and key challenges in the formulation of the proposed model. We present analytical formulation to incorporate the boundary effects and to obtain the closed form expressions for the minimum node degree distribution, node isolation probability and hence the probability of -connectivity of a network deployed in a rectangular region in section III. The effective area covered by a node lying in various sub-regions at a given transmission range scenario is presented in section IV. Section V provides analytical and simulation results and discusses their comparison for the minimum node degree distribution and node isolation probability, thereby connectivity. Finally, section VI concludes the paper.

II. Preliminaries In this section, we briefly discuss the network model in terms of node distribution, transmission range, connectivity related relevant terms and some key issues in evaluating the minimum node degree distribution, the node isolation probability and -connectivity.

A. Nodes Distribution and Communication Model Let there be a network consisting of finite number of nodes - distributed independently and uniformly-inside a finite rectangular region ℜ - a 2-D Real Euclidian domain. The sides and vertices of the rectangular region are represented by ℓ and ℓ respectively, and are numbered in anticlockwise direction, for ℓ ∈ {1,2,3,4}. Without loss of generality, we take the first vertex of the rectangular region to be located at the origin (0, 0). Let the position of an arbitrary node inside the rectangular region be denoted by ' = (), *). Similar to in [20], the rectangular region of length ℒ meters and width + meters can be defined as, ℜ = {' = (), *) ∈ ℜ |0 ≤ ) ≤ ℒ, 0 ≤ * ≤ +}.

(1)

Further, the node distribution probability density function can be expressed as ./ (') = 0

1, ' ∈ ℜ 0, ' ∉ ℜ.

(2)

The physical measure of the rectangular region is defined by |ℜ| = 2 ds ('), where ds(') = d)dy, and the integration is performed over the 2-D rectangular region ℜ. we note that our model is the generalization of that in [20] that considers a unit square, i.e., we have|ℜ| = ℒ × ω, in contrast to |ℜ| = 1 in [20]. The coverage area of a node positioned at ' can be taken to be equal to the area of a disc denoted by |A('; r )| = πr  , having radius equal to the transmission range (r ) and is centered at the node.

B. Connectivity Related Definitions In this paper, following connectivity properties pertinent to our proposed work are considered and are taken from [16], [20] for the sake of completeness. Definition 1 - Node degree: The number of nodes lying inside coverage area of a particular node is known as its degree. Definition 2 - Conditional Probability of Connectivity: Conditional probability of connectivity that an arbitrarily located node according to a uniform probability density function (pdf) is connected to a node positioned at ' is assumed to be represented by the cumulative distribution function (cdf) ℱ(';  ). ℱ(';  ) ≜

|<(';=> ) ⋂ ℜ | ℒ∗A



(3)

Definition 3 - Node Isolation Probability: A node is a said to be isolated when it has no node in its coverage area. Considering that the node isolation probability ΡCDE is independent for every node, probability that a particular node located at ' is isolated is obtained by (1 − ℱF (';  )) and can be averaged over all possible positions of ℜ to estimate ΡCDE as ΡFG ≜ 2H1 − ℱF (';  )I



= 2H1 − ℱF (';  )I



./ (')JK(') JK(')

(' ∈ ℜ)

(4)

Definition 4 - Minimum Node Degree: When  numbers of nodes are uniformly distributed within a rectangular region, the minimum number of neighbors of any node inside the region is known as minimum node degree of the region denoted by D. The related pdf, which is known as minimum node degree distribution, is given by .L (;  ) = Ρ(M = )

Q Q  O − 1P × ≜ (1 − ∑QRS JK('))  2HℱF (';  )I H1 − ℱF (';  )I J

(5)

Definition 5 - Connected Network: A network is said to be connected (or 1-connected) if for each pair of nodes there exists at least one connecting path. Definition 6 - T-Connected Network: A network is said to be -connected ( = 1,2, …  − 1) if there exist at least  disjoint paths between any arbitrary pair of nodes. Putting differently, we can say that a network is -connected if the deletion of any arbitrary ( − 1) nodes does not disconnect the network. Probability that the network is -connected is represented by Ρ V . In [27], the author models the network with a geometric random graph and states that when the number of nodes  in the network is large enough, the network becomes -connected at the same time it attains the minimum node degree equal to  with high probability. On the basis of above mentioned arguments and the discussion in [28], it can be said that fX (; r ) functions as an upper limit on Ρ VYEZ and gets closer as both Ρ VYEZ and fX (; r ) reaches one or the number of nodes reaches infinity. It can be expressed as .L (;  ) ≥ Ρ  ( )

.L (;  ) = [  , [  ( ) → 1 (6)

Here, we find that the minimum node degree distribution is a fundamental property of multi-hop networks as it is essential to estimate the node isolation probability ΡFG ( ) and network connectivity [ V . The mathematical relation between ΡFG and .L (;  ) is given by .L (1;  ) = (1 − [FG ( ))

(7)

In order to estimate the node isolation probability ΡCDE (r ) and to measure the minimum node degree distribution fX (; r ), equation (4) and (5), respectively, following issues are stated in [20]. • •

The first issue is to measure the overlap area |^(';  ) ⋂ ℜ |⁄(ℒ ∗ +) to find the cdf in (3). The second issue is to average the cdf given in (3) over the region to measure the minimum node degree distribution .L (;  ) in (5) and to estimate the node isolation probability ΡFG ( ) in (4).

The above issues raised in [20] are aggravated on considering the region as a scalable rectangular one and need further investigations, and mathematical rigor for the region other than just a unit square. Our proposed formulations for a rectangular region not only cover the above issues but also provide solution in a general way.

III. Proposed Model Here, we present our proposed general model to calculate the effective area covered by a node lying near the boundary regions, middle region and in the corner of a scalable rectangular network region. The network’s metrics stated above are computed and compared thereafter following the model’s results.

A. Computing Effective Coverage Area Considering Border Effects The boundary effects are characterized by the circular segment areas formed outside every side and the corner overlap areas between two circular segments formed at every vertex (corner effects). As defined earlier, the overlap area |^(';  ) ⋂ ℜ |⁄(ℒ ∗ +), is conditional probability of connectivity that a randomly and uniformly distributed node is connected to a node positioned at '. The area of the circular section lying outside side  is represented by ℬ (';  ), as shown in Figure 1 and is calculated by subtracting the area of the triangular portion from the area of the circular sector, thus we get   () ⁄ )  − ) d  − )  , ∆G (',  ) = f <  ℬ = a  cos 0, hiℎklmKk

(8)

The Euclidean distance between side ℓ and ' is represented by ∆G (', ℓ ) for ℓ = 1, 2, 3, 4. Likewise, the areas of the circular sections lying outside sides  , n , and o, respectively, are represented as p

ℬ (';  ) =   cos  O P − * d  − *  ℬn (';  ) =   cos  O ℬo (';  ) =   cos  O

=>

ℒ) =>

Ap =>

(9)

P − (ℒ − )) d  − (ℒ − ))

(10)

P − (+ − *) d  − (+ − *)

(11)

The area of the corner overlap region between two circular sectors at vertex  is represented by q (';  ) as shown in Figure 1 and is calculated by subtracting the area of the two triangular portions from the area of the circular sector, thus we get

Figure 1: Effective area covered by a node near boundary and in a corner of a rectangular region. r = 2  − 2)d  − *  − 2*d  − ) 

s =

vwGd xy 2 sin ( ) => 

(13)





q (';  ) =   s − (d  − *  − ))* − (d  − )  − *)) 

(12)





(14)

Likewise, the areas of the corner overlapping parts formed at vertices  , n , and o, respectively are represented as r = 2  − 2(ℒ − ))d  − *  − 2*d  − (ℒ − )) s = 2 sin ( 

vwGdxz =>



)

(16) 

q (';  ) =    s −  (d  − *  − (ℒ − )))* −  (d  − (ℒ − )) − *)(ℒ − ))

For vertex n

rn = 2  − 2(ℒ − ))d  − (+ − *) − 2(+ − *)d  − (ℒ − )) sn = 2 sin (



vwGdx{



=>

)

For vertex o



 

ro = 2  − 2)d  − (+ − *) − 2(+ − *)d  − )  so = 2 sin ( 

vwGdx|



=>

)



(18)

(20)

(21) (22)



qo (';  ) =   so − (d  − (+ − *) − ))(+ − *) − (d  − )  − (+ − *))) 

(17)

(19)

qn (';  ) −   sn − (d  − (+ − *) − (ℒ − )))(+ − *) − (d  − (ℒ − )) − (+ − *))(ℒ − )) 

(15)



(23)

It is worth to note that the formulas for qℓ are valid only when the Euclidean distance between ' and vertex ℓ represented by ∆G (', ℓ ) is less than  . The cdf ℱ(';  ) in (3) can be represented in a closed form by using (8)-(11), (14), (17), (20), and (24).

B. Network Metrics Formulations Here, we discuss how to find the average overlap area over the whole rectangular region. First of all, the whole rectangular region is divided into various non-overlapping sub-regions based on the various corner and boundary effects that occur in that region. Because of the symmetry of the rectangle, some sub-regions have the same count of the corner and boundary effects which are used for averaging. Various types of non-overlapping sub-regions are represented by ℜ, ℜ,…,ℜ}, and nC , i ∈ {1, 2, … , Z} represents the number of sub-regions of type ℜC . Here the conditional probability of connectivity for a node positioned at ' ∈ ℜC is represented by ℱC ('; r ), and then the node isolation probability is estimated by ΡFG ( ) = ∑€FR F 2(1 − ℱF (';  ))  JK(')

(24)

and the minimum node degree distribution is represented as  ∑€ .L (;  ) = Ρ(M = ) = (1 − ∑QRS FR F O

Q Q −1 P × 2HℱF (';  )I H1 − ℱF (';  )I JK(')) (25) J

 can also be calculated using this framework as The expected node degree represented by M  = ( − 1) 2 ℱ(';  ) JK(') M

= ( − 1) ∑€FR F 2 ℱ(';  ) JK(')

(26)

The given framework gives us a closed form expression to calculate the minimum node degree distribution and hence the node isolation probability and the probability of -connectivity.

IV. Scenarios: Boundary Effects Versus Transmission Range As stated earlier, there exists a total of 13 different scenarios of transmission range with respect to rectangular dimensions of the region, as opposed to seven scenarios for a unit square region [20]. The criterion of dividing the system into 13 scenarios comes out from the fact that ℱ('; r ) is a function of

both node location ' and transmission range r. For a rectangular region, if r > √ℒ  + ω, then the disc A('; r ) would cover the whole rectangular region ℜ, and hence, ℱ('; r ) = 1, irrespective of the

node location. For intermediate values of the node range 0 ≤ r ≤ √ℒ  + ω, both ' and r need to be taken into account in determining ℱ('; r ). The whole rectangular region is divided into various nonoverlapping sub-regions based on the various corner and boundary effects because these nonoverlapping sub-regions change with the transmission range r. It is obvious that the transmission range of nodes for a rectangular region will lie in the range 0 ≤ r ≤ √ℒ  + ω. But, when the number of non-overlapping sub-regions for a transmission range r changes, then the values of ℒ, and ω will change for the next transmission range scenario; else it will remain same until the number of nonoverlapping sub-regions remains same. The 13 scenarios are explained as follows.

A. Transmission Range: … ≤ †‡ ≤ ˆ⁄‰ In this scenario, the whole rectangular region can be divided into four quadrants. For the given transmission range  , each quadrant can be partitioned into five (Š = 5) separate sub-regions ℜ, ℜ, ℜn , ℜn , ℜo , and ℜ‹ as shown in Figure 2. Here, we notice that the area of each sub-region is limited by either one side or two sides, e.g., if an arbitrary node is located in sub-region ℜ‹ , the overlap area of the

ℜF

F

ℱF (';  

4

Π



4

Π  `

n

4

Œ   &` „ ` 

o

4

Œ   &` „ `  q 

‹

4



Π  &`

Figure 2: Various sub-regions for … - †‡ - ˆ⁄‰. Ž &; †‡  and sub-regions count Ž for each sub-region.

node is confined by one side only. Therefore, we estimate :F &';   only for the following sub-regions:  ( {) ∈ & , 0.5, * ∈ & , 0.5+}  ( {) ∈ &0,  , * ∈ & , 0.5+}

n ( {) ∈ &0,  , * ∈ &d   )  ,  } o ( {) ∈ &0,  , * ∈ &0, d   )  } ‹ ( {) ∈ & , 0.5, * ∈ &0,  }

We also notice that when an arbitrary node is placed anywhere in sub-region  the whole overlap area ^&';   of the node lies inside the rectangle , i.e., the node suffers no boundary or corner effects. Therefore, : &';   ( Œ  . Further, when an arbitrary node is placed anywhere in subregion , the overlap area ^&';   of the node is limited by side  , i.e., a circular segment is made outside side  only. Thus, : &';   ( Œ   ` &';  . Again, when an arbitrary node is placed anywhere in sub-region n , the overlap area ^&';   of the node is limited by side  and , i.e., two circular segments are made outside sides  and  , but there is no corner overlap between them. Hence, :n &';   ( Œ   &` &';   „ ` &';  . In addition, when an arbitrary node is placed anywhere in sub-region o , the overlap area ^&';   of the node is limited by side  and  and vertex , i.e., two circular segments are made outside sides  and  and there exists a corner overlap between them. Therefore, :o &';   ( Œ   H` &';   „ ` &';    q &';  I. Lastly, when an arbitrary node is placed anywhere in sub-region ‹, the overlap area ^&';   of the node is limited by side  only, i.e., a circular segments is made outside side  and there is no corner overlap. Therefore, :‹ &';   ( Œ   H` &';  I. Figure 3 shows the number of sub-regions F of each type and their corresponding closed-form expressions. Here, `ℓ and qℓ represents `ℓ &';   and qℓ &';   respectively. It has been observed that the sub-regions and  get smaller with the increase in  from 0 to +⁄2 and become zero for  ( + ⁄2. On the other hand, sub-regions n , o and ‹ get larger when  is increased from 0 to + ⁄2.

B. Transmission Range: ˆ⁄‰ - †‡ - ‘⁄‰ Let us consider the case in which transmission range of the nodes lies in the interval +⁄2 -  - ⁄2. In this range, we get seven (Š ( 7 separate sub-regions as shown in Figure 4 and can be represented as  ( {) ∈ H0, d+&2  + I, * ∈ &0, +   } ⋃ {) ∈ &d+&2  +,  , * ∈ &0, d   )  }

F

F

  n o ‹ ” •

4 4 4 4 4 4 4

:F &';   Œ   &` „ `  q  Œ   &` „ `  Œ   &` „ ` „ `o  q  qo  Œ   &` „ ` „ `o  q  Œ   &` „ ` „ `o  Œ   &` Œ   &` „ `o 

Figure 5: Various sub-regions for –⁄‰ - —˜ - ‘⁄‰. ™ &; —˜  and sub-regions count š™ for each sub-region.  ( {) ∈ Hd+&2  +,  I, * ∈ &d   )  , +    }

n ( {) ∈ H0, d   &0.5+ I, * ∈ &+  d   )  , 0.5+}

o ( {) ∈ H0, d   &0.5+ I, * ∈ &+   , +  d   )  }⋃ {) ∈ &d   &0.5+ , d+&2  +, * ∈ &+   , d   )  }

‹ ( {) ∈ Od   &0.5+ , d+&2  + P , * ∈ Od   )  , 0.5+P}⋃ {) ∈ &d+&2  + ,  , * ∈ &+   , 0.5+} ” ( {) ∈ & , 0.5, * ∈ &0, +   } • ( {) ∈ & , 0.5, * ∈ &+   , 0.5+}

For the given interval,  ⁄2 is taken as the upper limit of transmission range because we get same types of sub-regions up to ⁄2 and types of sub-regions change beyond ⁄2. In this case, one can observe that the sub-regions ” and • get smaller with the increase in  from +⁄2 to ⁄2 , and for  (  ⁄2 they become zero. Figure 3 shows the number of sub-regions F of each type and their corresponding closed-form expressions.

C. Transmission Range: ‘⁄‰ - †‡ - ‘ „ ˆ  √‰‘ˆ Now, we take the case of transmission range lying in the interval ⁄2 -  -  „ +  √2+. In this interval, we get nine (Š ( 9 separate sub-regions as shown Figure 4; they can be represented as  ( {) ∈ H0, d+&2  + I, * ∈ &0, +   } ⋃ {) ∈ &d+&2  +,    , * ∈ &0, d   )   }

 ( {) ∈ &   , 0.5, * ∈ &0, d   &)   }

n ( {) ∈ &   , 0.5, * ∈ &d   &)   , d   )  }

o ( {) ∈ &   , 0.5, * ∈ &d   )  , +   }

‹ ( {) ∈ &d+&2  +,    , * ∈ &d   )  , +   } ” ( {) ∈ &    , 0.5, * ∈ &+   , 0.5+ }

• ( {) ∈ Hd   &0.5+ , d+&2  + I, * ∈ &d   )  , 0.5+} ⋃

{) ∈ &d+&2  + ,    , * ∈ &+   , 0.5+}

› ( {) ∈ &0, d   &0.5+, * ∈ &+   , +  d   )  } ⋃ {) ∈ &d   &0.5+ , d+&2  +, * ∈ &+   , d   )  } œ ( { ) ∈ &0, d   &0.5+ , * ∈ &+  d   )  , 0.5+ }

F

ℜ ℜ ℜn ℜo ℜ‹ ℜ” ℜ• ℜ› ℜœ

F 4 4 4 4 4 4 4 4 4

:F &';  )

Œ  − (ℬ + ℬ − q ) Œ  − (ℬ + ℬ + ℬn − q − q ) Œ  − (ℬ + ℬ + ℬn − q ) Œ  − (ℬ + ℬ + ℬn ) Œ  − (ℬ + ℬ ) Œ  − (ℬ + ℬ + ℬn + ℬo ) Œ  − (ℬ + ℬ + ℬo ) Œ  − (ℬ + ℬ + ℬo − q ) Œ  − (ℬ + ℬ + ℬo − q − qo )

Figure 4: Various sub-regions for ‘⁄‰ ≤ —˜ ≤ ‘ + – − √‰‘–. ™ (; —˜ ) and sub-regions count š™ for each sub-region.

The upper limit of transmission range for this case, i.e., ℒ + + − √2ℒ+, is calculated as the value of  for which circle )  + *  =   , lines ) = ℒ −  , and * = + −  intersect. Here, the subregion ℜ‹ gets smaller with the increase in  and becomes zero for  = ℒ + + − √2ℒ+. Figure 4 shows the number of sub-regions F of each type and their corresponding closed-form expressions.

D. Transmission Range: ‘ + ˆ − √‰‘ˆ ≤ †‡ ≤ ( ‘‰ + ˆ‰ )⁄žˆ

Here, we consider the case of transmission range lying in the interval ℒ + + − √2ℒ+ ≤  ≤ (ℒ  + 4+ )⁄8+. In this interval, we get nine (Š = 9) separate sub-regions as shown in Figure 5, which can be represented as ℜ = {) ∈ (0, ℒ −  ), * ∈ (0, + −  )}

ℜ = {) ∈ ( ℒ −  , 0.5ℒ), * ∈ H0, d  − () − ℒ) I}

ℜn = {) ∈ ( ℒ −  , 0.5ℒ ), * ∈ (d  − () − ℒ) , d  − )  )} ⋃ {) ∈ (ℒ −  , d+(2 − +)), * ∈ (d  − () − ℒ) ,0.5+)}

⋃ {) ∈ (d+(2 − +), 0.5ℒ), * ∈ (d  − () − ℒ) , d  − )  )}

ℜo = {) ∈ (d+(2 − +), 0.5ℒ), * ∈ (d  − )  , + −  ) } ℜ‹ = {) ∈ (ℒ −  , d+(2 − +)), * ∈ (+ −  , d  − )  ) }

ℜ” = {) ∈ (ℒ −  , d+(2 − +)), * ∈ (d  − )  , 0.5+)}

⋃ {) ∈ (d+(2 − +), 0.5ℒ), * ∈ (+ −  , 0.5+) }

ℜ• = {) ∈ (d  − (0.5+) , ℒ −  ), * ∈ (d  − )  , 0.5+)}

ℜ› = {) ∈ H0, d  − (0.5+) I, * ∈ H+ −  , + − d  − )  I}

⋃ {) ∈ Hd  − (0.5+), ℒ −  I, * ∈ (+ −  ), d  − )  }

ℜœ = {) ∈ H0, d  − (0.5+) I, * ∈ H+ − d  − )  I, 0.5+)}

For this case, the upper limit of transmission range, i.e., ( ℒ  + 4ω )⁄8ω, is calculated as the value of r for which circle )  + y  = r , lines ) = ℒ ⁄2, and y = ω − r intersect. We observe that the subregion ℜo gets smaller with the increase in r and it becomes zero for r = ( ℒ  + 4ω )⁄8ω. Figure 5 shows the number of sub-regions nC of each type and their corresponding closed-form expressions.

F

ℜ

F

ℜ ℜn ℜo ℜ‹ ℜ” ℜ• ℜ› ℜœ

4 4 4 4 4 4 4 4 4

:F &';  )

Œ  − (ℬ + ℬ − q )

Œ  − (ℬ + ℬ + ℬn − q − q ) Œ  − (ℬ + ℬ + ℬn − q ) Œ  − (ℬ + ℬ + ℬn )

Œ  − (ℬ + ℬ + ℬn + ℬo − q ) Œ  − (ℬ + ℬ + ℬn + ℬo ) Œ  − (ℬ + ℬ + ℬo )

Œ  − (ℬ + ℬ + ℬo − q )

Œ  − (ℬ + ℬ + ℬo − q − qo )

Figure 5: Various sub-regions for ‘ + – − √‰‘– ≤ —˜ ≤ ( ‘‰ + –‰ )⁄ž–. ™ (; —˜ ) and sub-regions count š™ for each sub-region.

E. Transmission Range: ( ‘‰ + ˆ‰ )⁄žˆ ≤ †‡ ≤ ( ‘‰ + ˆ‰ )⁄ž‘

This scenario considers the case of transmission range lying in the interval ( ℒ  + 4+ )⁄8+ ≤  ≤ (4ℒ  + + )⁄8ℒ . In this interval, we get nine (Š = 9) separate sub-regions as shown in Figure 6, and they can be represented as ℜ = { ) ∈ (0, ℒ −  ), * ∈ (0, + −  )}

ℜ = {) ∈ Hℒ −  , ℒ − d+(2 − +)I, * ∈ (0, d  − () − ℒ) )} ⋃

{) ∈ Hℒ − d+(2 − +), 0.5ℒI, * ∈ (0, + −  )}

ℜn = {) ∈ Hℒ −  , ℒ − d+(2 − +)I, * ∈ (+ −  , d  − )  )} ⋃

{) ∈ (ℒ − d+(2 − +), 0.5ℒ ), * ∈ (d  − () − ℒ) , d  − )  ) }

ℜo = {) ∈ (ℒ −  , 0.5ℒ), * ∈ (d  − )  , 0.5+)}

ℜ‹ = {) ∈ (d  − (0.5+) , ℒ −  ), * ∈ (d  − )  , 0.5+)}

ℜ” = {) ∈ H0, d  − (0.5+) I, * ∈ H+ −  , + − d  − )  I)} ⋃ {) ∈ (d  − (0.5+) , ℒ −  I, * ∈ (+ −  , d  − )  )}

ℜ• = {) ∈ (0, d  − (0.5+)) , * ∈ (+ − d  − )  , 0.5+)}

ℜ› = {) ∈ (ℒ −  , ℒ − d+(2 − +)), * ∈ (d  − () − ℒ), + −  )} ℜœ = {) ∈ (ℒ − d+(2 − +), 0.5ℒ), * ∈ (+ −  , d  − () − ℒ) )}

The upper limit of transmission range in this case is (4ℒ  + +  )⁄8ℒ. The limit is calculated as the value of  for which circle )  + *  =  , lines ) = ℒ −  , and * = + ⁄2 intersect. For this case, we notice that the sub-regions ℜo and ℜ‹ get smaller with the increase in  and become zero for  = (4ℒ  + + )⁄8ℒ . Figure 6 shows the number of sub-regions F of each type and their corresponding closed-form expressions.

F. Transmission Range: (‘‰ + ˆ‰)⁄ž‘ ≤ †‡ ≤ …. ¡√‘‰ + ˆ‰

This scenario considers the case of transmission range lying in the interval (4ℒ  + + )⁄8ℒ ≤  ≤ 0.5√ℒ  + +  . In this interval, we get nine (Š = 9) separate sub-regions as shown in Figure 7. The subregions can be represented as

ℜ = { ) ∈ (0, ℒ −  ), * ∈ (0, + −  )}

F ℜ ℜ ℜn ℜo ℜ‹ ℜ” ℜ• ℜ› ℜœ

:F &';  ) Œ  − (ℬ + ℬ − q ) Œ  − (ℬ + ℬ + ℬn − q − q ) Œ  − (ℬ + ℬ + ℬn + ℬo − q ) Œ  − (ℬ + ℬ + ℬn + ℬo ) Œ  − (ℬ + ℬ + ℬo ) Œ  − (ℬ + ℬ + ℬo − q ) Œ  − (ℬ + ℬ + ℬo − q − qo ) Œ  − (ℬ + ℬ + ℬn − q ) Œ  − (ℬ + ℬ + ℬn + ℬo − q − q )

F 4 4 4 4 4 4 4 4 4

Figure 6: Various sub-regions for ( ‘‰ + –‰ )⁄ž– ≤ —˜ ≤ ( ‘‰ + –‰ )⁄ž‘. ™ (; —˜ ) and sub-regions count š™ for each sub-region. ℜ = {) ∈ Hℒ −  , ℒ − d+(2 − +)I, * ∈ (0, d  − () − ℒ))} ⋃ {) ∈ Hℒ − d+(2 − +), 0.5ℒI, * ∈ (0, + −  )}

ℜn = {) ∈ Hℒ −  , ℒ − d+(2 − +)I, * ∈ (+ −  , + − d  − )  )} ⋃

{) ∈ (ℒ − d+(2 − +), d  − (0.5+) ), * ∈ (d  − () − ℒ) , + − d  − )  )}

⋃{) ∈ (d  − (0.5+) ,0.5ℒ), * ∈ (d  − () − ℒ) , d  − )  ) ℜo = {) ∈ ( d  − (0.5+) ,0.5ℒ ), * ∈ (d  − )  ,0.5+)}

ℜ‹ = {) ∈ (ℒ −  , d  − (0.5+) ), * ∈ (+ − d  − )  ,0.5+)}

ℜ” = {) ∈ (0, ℒ −  ), * ∈ ( + −  , + − d  − )  )} ℜ• = {) ∈ (0, ℒ −  ), * ∈ (+ − d  − )  ,0.5+)}

ℜ› = {) ∈ (ℒ −  , ℒ − d+(2 − +) ) , * ∈ (d  − () − ℒ) , + −  )}

ℜœ = {) ∈ Hℒ − d+(2 − +) , 0.5ℒI, * ∈ (+ −  , d  − () − ℒ) )}

In this case, the upper limit of transmission range, i.e., 0.5√ℒ  + +  . We use the value of  for which circle )  + *  =   , lines ) = ℒ⁄2, and * = +⁄2 intersect to calculate the limit. For this case,

the sub-regions ℜo and ℜ‹ get smaller with the increase in  and become zero for  = 0.5√ℒ  + + . Figure 7 shows the number of sub-regions F of each type and their corresponding closed-form expressions. ℜF F ℜ ℜ ℜn ℜo ℜ‹ ℜ” ℜ• ℜ› ℜœ

4 4 4 4 4 4 4 4 4

ℱF (';  )

Œ  − (ℬ + ℬ − q ) Œ  − (ℬ + ℬ + ℬn − q − q ) Œ  − (ℬ + ℬ + ℬn + ℬo − q ) Œ  − (ℬ + ℬ + ℬn + ℬo ) Œ  − (ℬ + ℬ + ℬn + ℬo − q − qo ) Œ  − (ℬ + ℬ + ℬo − q ) Œ  − (ℬ + ℬ + ℬo − q − qo ) Œ  − (ℬ + ℬ + ℬn − q )

Œ  − (ℬ + ℬ + ℬn + ℬo − q − q )

Figure 7: Various sub-regions for (‘‰ + –‰ )⁄ž‘ ≤ —˜ ≤ …. ¡√‘‰ + –‰ . ™ (; —˜ ) and sub-regions count š™ for each sub-region.

F

ℜ ℜ ℜn ℜo ℜ‹ ℜ” ℜ• ℜ›

F :F &';  ) 4 4 4 4 4 4 4 4

ℜœ

4

ℜS

4

Œ  − (ℬ + ℬ − q ) Œ  − (ℬ + ℬ + ℬo − q ) Œ  − (ℬ + ℬ + ℬo − q − qo ) Œ  − (ℬ + ℬ + ℬn − q − q ) Œ  − (ℬ + ℬ + ℬn − q ) Œ  − (ℬ + ℬ + ℬn + ℬo − q − q ) Œ  − (ℬ + ℬ + ℬn + ℬo − q ) Œ  − (ℬ + ℬ + ℬn + ℬo − q − q − qn − qo ) Œ  − (ℬ + ℬ + ℬn + ℬo − q − q − qo ) Œ  − (ℬ + ℬ + ℬn + ℬo − q − qo )

Figure 8: Various sub-regions for …. ¡√‘‰ + –‰ ≤ —˜ ≤ –. ™ (; —˜ ) and sub-regions count š™ for each subregion.

G. Transmission Range: …. ¡√‘‰ + ˆ‰ ≤ †‡ ≤ ˆ

The transmission range for this case lies in the interval 0.5√ℒ  + +  ≤  ≤ +. In this interval, we get ten (Š = 10) separate sub-regions as shown in Figure 8. They can be given as ℜ = {) ∈ (0, ℒ − r ), y ∈ (0, ω − r )}

ℜ = {) ∈ (0, ℒ − r ), y ∈ ( ω − r , ω − dr  − )  )} ℜn = {) ∈ (0, ℒ − r ), y ∈ (ω − dr  − )  ,0.5ω)}

ℜo = {) ∈ Hℒ − r , ℒ − dω(2r − ω)I, y ∈ (0, dr  − () − ℒ) )} ⋃ {) ∈ Hℒ − dω(2r − ω), 0.5ℒI, y ∈ (0, ω − r )}

ℜ‹ = {) ∈ (ℒ − r , ℒ − dω(2r − ω) ), y ∈ (dr − () − ℒ) , ω − r )}

ℜ” = {) ∈ (ℒ − dω(2r − ω) , 0.5(ℒ − ω⁄(ω + ℒ  )d(4r  − (ℒ  + ω))(ℒ  + ω ) )),

y ∈ (ω − r , dr  − () − ℒ) )} ⋃ {() ∈ (0.5(ℒ − ω⁄(ω + ℒ  )¢H4r  − (ℒ  + ω )I(ℒ  + ω) ),0.5ℒ),

y ∈ (ω − r , ω − dr  − )  )}

ℜ• = {) ∈ Hℒ −  , ℒ − d+(2 − +) I, * ∈ (+ −  , + − d  − )  )}

⋃ {) ∈ (ℒ − d+(2 − +), 0.5(ℒ − +⁄(+ + ℒ  )¢H4  − (ℒ  + + )I(ℒ  + + ) ))),

* ∈ (d  − () − ℒ), + − d  − )  )}

ℜ› = {) ∈ (ℒ − d  − (0.5+) ,0.5ℒ), * ∈ (+ − d  − () − ℒ) ,0.5+)}

ℜœ = {) ∈ (0.5(ℒ − +⁄(+ + ℒ  )¢H4  − (ℒ  + + )I(ℒ  + + ) ), ℒd  − (0.5+))), * ∈ H+ − d  − )  , d  − () − ℒ), I} ⋃ {) ∈ Hℒ − d  − (0.5+), 0.5ℒI,

* ∈ (+ − d  − )  , + − d  − () − ℒ) )}

ℜS = {) ∈ (ℒ −  , 0.5(ℒ − +⁄(+ + ℒ  )d(4  − (ℒ  + + )(ℒ  + + )) ), * ∈ (+ − d  − )  ,0.5+)}

⋃ {) ∈ H0.5(ℒ − (+⁄(+ + ℒ  )I¢H4  − (ℒ  + +  )I(ℒ  + + ) , ℒ − d  − (0.5+)), * ∈ (d  − () − ℒ), 0.5+)}

In this case, the upper limit of transmission range, i.e., +, is calculated as the value of  for which circle )  „ *  (  , and * ( + intersect. The sub-regions  , ℜo and ℜ‹ for this case get smaller with the increase in  and become zero for  = +. Figure 8 shows the number of sub-regions F of each type and their corresponding closed-form expressions.

H. Transmission Range: ˆ ≤ †‡ ≤ (‘‰ + ˆ‰ )⁄‰‘

Now, we take the case of transmission range lying in the interval + ≤  ≤ (ℒ  + +  )⁄2ℒ. In this interval, we get six (Š = 6) separate sub-regions as shown in Figure 9 and can be represented as ℜ = {) ∈ (d  − +  , ℒ −  ), * ∈ (0, + − d  − )  )}

ℜ = {) ∈ (0, ℒ −  ), * ∈ (+ − d  − )  ,0.5+)} ℜn = {) ∈ (ℒ −  , 0.5ℒ), * ∈ (0, + − d  − )  )}

ℜo = {) ∈ H0.5(ℒ − (+⁄(+  + ℒ  )Id(4  − (ℒ  + + ))(ℒ  + + ) ), ℒd  − (0.5+) ), * ∈ (+ − d  − )  , d  − () − ℒ) }⋃{) ∈ Hℒ − d  − (0.5+), 0.5ℒI, * ∈ (+ − d  − )  , + − d  − () − ℒ))}

ℜ‹ = {) ∈ (ℒ − d  − (0.5+) ,0.5ℒ), * ∈ (+ − d  − () − ℒ) ,0.5+)}

ℜ” = {) ∈ (0.5(ℒ − (+⁄(+ + ℒ  ))d(4  − (ℒ  + + ))(ℒ  + + ) ), ℒ − d  − (0.5+) ),

* ∈ (d  − () − ℒ) , 0.5+)}

In this case, the upper limit of transmission range, i.e., (ℒ  + + )⁄2ℒ , is calculated as the value of  for which circles )  + *  =   , and () − ℒ) + (* − +) =   and line ) =  intersect. Here, it has been observed that the sub-region ℜ gets smaller with the increase in  and becomes zero for  = (ℒ  + + )⁄2ℒ . Figure 9 shows the number of sub-regions F of each type and their corresponding closed-form expressions.

ℜF F ℱF (';  ) ℜ 4

Œ  − (ℬ + ℬ + ℬo − q )

ℜn 4

Œ  − (ℬ + ℬ + ℬn + ℬo − q − q )

ℜ 4 ℜo 4 ℜ‹ 4 ℜ” 4

Œ  − (ℬ + ℬ + ℬo − q − qo )

Œ  − (ℬ + ℬ + ℬn + ℬo − q − q − qo ) Œ  − (ℬ + ℬ + ℬn + ℬo − q − q − qn − qo ) Œ  − (ℬ + ℬ + ℬn + ℬo − q − qo )

Figure 9: Various sub-regions for – ≤ —˜ ≤ (‘‰ + –‰ )⁄‰‘. ™ (; —˜ ) and sub-regions count š™ for each sub-region.

F

ℜ ℜ

F 4 4

ℜn

4

ℜo

4

ℜ‹

4

:F &';  )

Œ  − (ℬ + ℬ + ℬo − q − qo ) Œ  − (ℬ + ℬ + ℬn + ℬo − q − q ) Œ  − (ℬ + ℬ + ℬn + ℬo − q − q − qn − qo ) Œ  − (ℬ + ℬ + ℬn + ℬo − q − q − qo ) Œ  − (ℬ + ℬ + ℬn + ℬo − q − qo )

Figure 10: Various sub-regions for (‘‰ + –‰ )⁄‰‘ ≤ —˜ ≤ d‘‰ ⁄ + ¡–‰ ⁄ž + (£– ⁄¤‘‰ ). ™ (; —˜ ) and subregions count š™ for each sub-region.

I. Transmission Range: (‘‰ + ˆ‰ )⁄‰‘ ≤ †‡ ≤ d‘‰⁄ + ¡ˆ‰ ⁄ž + (£ˆ ⁄¤‘‰ )

For the next scenario, we take the case of transmission range lying in the interval (ℒ  + + )⁄2ℒ ≤  ≤

dℒ  ⁄4 + 5+ ⁄8 + (9+ o⁄64ℒ  ). In this interval, we get five (Š = 5) separate sub-regions as shown in Figure 10, which can be represented as ℜ = {) ∈ (0, ℒ −  ), * ∈ (0 , 0.5+)}

ℜ = {) ∈ (d  − + , 0.5ℒ), * ∈ (0, + − d  − )  )}

ℜn = {) ∈ (ℒ − d  − (0.5+) , 0.5ℒ), * ∈ (+ − d  − () − ℒ) , 0.5+)}

ℜo = {) ∈ (ℒ −  , d  − + ), * ∈ (0 , d  − () − ℒ) )} ⋃

{) ∈ (d  − + , ℒ − d  − (0.5+)), * ∈ (+ − d  − )  , d  − () − ℒ) )} ⋃

{) ∈ (ℒ − d  − (0.5+), 0.5ℒ), * ∈ (+ − d  − )  , + − d  − () − ℒ) )} ℜ‹ = {) ∈ (ℒ −  , ℒ − d  − (0.5+)), * ∈ (d  − () − ℒ), 0.5+)}

In this case, the upper limit of transmission range, i.e., dℒ  ⁄4 + 5+ ⁄8 + (9+ o⁄64ℒ  ), is calculated as the value of  for which circles () − ℒ) + *  =  , and () − ℒ) + (* − +) =  

and line ) = d  − +  intersect. Notice that the area covered under the second limit of sub-region ℜo

gets smaller with the increase in  and becomes zero for  = dℒ  ⁄4 + 5+ ⁄8 + (9+ o⁄64ℒ  ). Figure 10 shows the number of sub-regions F of each type and their corresponding closed-form expressions.

J.

Transmission Range: d‘‰ ⁄ + ¡ˆ‰ ⁄ž + (£ˆ ⁄¤‘‰ ) ≤ †‡ ≤ dˆ‰ + (…. ¡‘)‰

Here, we take the case of transmission range lying in the interval dℒ  ⁄4 + 5+ ⁄8 + (9+ o⁄64ℒ  ) ≤

 ≤ d+  + (0.5ℒ) . In this interval, we get five (Š = 5) separate sub-regions as shown in Figure 11 and can be represented as ℜ = {) ∈ (0, ℒ −  ), * ∈ (0,0.5+)}

ℜ = {) ∈ Hd  − +  , 0.5ℒI, * ∈ (0, + − d  − )  )}

ℜn = {) ∈ (ℒ − d  − (0.5+), 0.5ℒ), * ∈ (+ − d  − () − ℒ) ,0.5+)}

ℜo = {) ∈ (ℒ −  , ℒ − d  − (0.5+)), * ∈ (0 , d  − () − ℒ) )}

F ℜ ℜ

F 4 4

ℜn

4

ℜo

4

ℜ‹

4

:F &';  ) Œ  − (ℬ + ℬ + ℬo − q − qo ) Œ  − (ℬ + ℬ + ℬn + ℬo − q − q ) Œ  − (ℬ + ℬ + ℬn + ℬo − q − q − qn − qo ) Œ  − (ℬ + ℬ + ℬn + ℬo − q − q − qo ) Œ  − (ℬ + ℬ + ℬn + ℬo − q − qo )

Figure 11: Various sub-regions for d‘‰ ⁄ + ¡–‰ ⁄ž + (£– ⁄¤‘‰ ) ≤ —˜ ≤ d–‰ + (…. ¡‘)‰ . ™ (; —˜ ) and subregions count š™ for each sub-region. ⋃{) ∈ Hℒ − d  − (0.5+), d  − + I, * ∈ (0, + − d  − () − ℒ))} ⋃{) ∈ Hd  − + , 0.5ℒI, * ∈ (+ − d  − )  , + − d  − () − ℒ) )}

ℜ‹ = {) ∈ (ℒ −  , ℒ − d  − (0.5+)), * ∈ (d  − () − ℒ) , 0.5+)}

In this case, the upper limit of transmission range, i.e., d+  + (0.5ℒ) , is calculated as the value of  for which circles )  + (* − +) =  , and () − ℒ) + (* − +) =   and line ) = ℒ ⁄2 intersect. For this case, when  increases, the sub-region ℜ gets smaller and it becomes zero

for  = d+  + (0.5ℒ) . Figure 11 shows the number of sub-regions F of each type and their corresponding closed-form expressions.

K. Transmission Range: dˆ‰ + (…. ¡‘)‰ ≤ †‡ ≤ ‘

This case considers the case when transmission range is in the interval d+  + (0.5ℒ) ≤  ≤ ℒ. In this interval, there are four (Š = 4) separate sub-regions as shown in Figure 12. The four sub-regions can be represented as ℜ = {) ∈ (0, ℒ −  ), * ∈ (0 ,0.5+)}

ℜ = {) ∈ (ℒ −  , ℒ − d  − (0.5+)), * ∈ (0 , d  − () − ℒ) )}

⋃ {) ∈ (ℒ − d  − (0.5+) , ℒ − d  − + ), * ∈ (0, + − d  − () − ℒ) )}

ℜn = {) ∈ (ℒ − d  − (0.5+), ℒ − d  − + ), * ∈ (+ − d  − () − ℒ) , 0.5+) }

⋃ {) ∈ (ℒ − d  − +  , 0.5ℒ), * ∈ (0, 0.5+)}

ℜo = {) ∈ (ℒ −  , ℒ − d  − (0.5+) ), * ∈ (d  − () − ℒ) ,0.5+)}

We calculate the upper limit of transmission range, i.e., ℒ, as the value of  for which circle )  + *  =   , and line ) = ℒ intersect. For this case, when  increases, the sub-region ℜ gets smaller and becomes zero for  = ℒ. Figure 12 shows the number of sub-regions F of each type and their corresponding closed-form expressions.

F F :F &';  ) ℜ 4 ℜ 4

Œ  − (ℬ + ℬ + ℬo − q − qo ) Œ  − (ℬ + ℬ + ℬn + ℬo − q − q − qo ) Œ  − (ℬ + ℬ + ℬn + ℬo − q − q − qn − qo ) Œ  − (ℬ + ℬ + ℬn + ℬo − q − qo )

ℜn 4 ℜo 4

Figure 12: Various sub-regions for d–‰ + (…. ¡‘)‰ ≤ —˜ ≤ ‘. ™ (; —˜ ) and sub-regions count š™ for each sub-region.

L. Transmission Range: ‘ ≤ †‡ ≤ d‘‰ + (…. ¡ˆ)‰

Let us now consider the case of transmission range in the interval ℒ ≤  ≤ dℒ  + (0.5+) . As shown in Figure 13, for this interval, we get three (Š = 3) separate sub-regions that can be represented as ℜ = {) ∈ (0, ℒ − d  − (0.5+)), * ∈ (0, d  − () − ℒ) )} ⋃

{) ∈ (ℒ − d  − (0.5+) , ℒ − d  − + ), * ∈ (0, + − d  − () − ℒ))}

ℜ = {) ∈ (ℒ − d  − (0.5+), ℒ − d  − + ), * ∈ (+ − d  − () − ℒ) , 0.5+)}⋃ {) ∈ (ℒ − d  − +  , 0.5ℒ), * ∈ (0, 0.5+)}

ℜn = {) ∈ (0, ℒ − d  − (0.5+)), * ∈ (d  − () − ℒ) , 0.5+)}

The upper limit of transmission range in this case, i.e., dℒ  + (0.5+) , is calculated as the value of  for which circles )  + *  =   and )  + (* − +) =  , and line * = +⁄2 intersect. The sub-

regions ℜ and ℜn get smaller with the increase in  and become zero for  = dℒ  + (0.5+) . Figure 13 shows the number of sub-regions F of each type and their corresponding closed-form expressions.

ℜF

ℜ

F 4

ℜ

4

ℜn

4

ℱF (';  )

Œ  − (ℬ + ℬ + ℬn + ℬo − q − q − qo ) Œ  − (ℬ + ℬ + ℬn + ℬo − q − q − qn − qo ) Œ  − (ℬ + ℬ + ℬn + ℬo − q − qo )

Figure 13: Various sub-regions for‘ ≤ —˜ ≤ d‘‰ + (…. ¡–)‰ . ™ (; —˜ ) and sub-regions count š™ for each subregion.

F

ℜ ℜ

F 4 4

:F &';  )

Œ  − (ℬ + ℬ + ℬn + ℬo − q − q − qo ) Œ  − (ℬ + ℬ + ℬn − q − q − qn − qo )

Figure 14: Various sub-regions for d‘‰ + (…. ¡–)‰ ≤ —˜ ≤ √‘‰ + –‰ . ™ (; —˜ ) and sub-regions count š™ for each sub-region.

M. Transmission Range: d‘‰ + (…. ¡ˆ)‰ ≤ †‡ ≤ √‘‰ + ˆ‰

Finally, for the last scenario, we take the case of transmission range lying in the interval ℒ ≤  ≤

dℒ  + (0.5+) . In this interval, we get two (Š = 2) separate sub-regions as shown in Figure 14 and represent the two sub-regions as

ℜ = {) ∈ (0, ℒ − d  − + ), * ∈ (0, + − d  − () − ℒ) )} ℜ = {) ∈ (0 , ℒ − d  − + ), * ∈ (+ − d  − () − ℒ) , 0.5+)} ⋃{) ∈ (ℒ − d  − +  , 0.5ℒ), * ∈ (0,0.5+)}

For this last case, it has been observed that ℱ(';  ) = 1 for the transmission range  ≥

√ℒ  + +  . Figure 14 shows the number of sub-regions F of each type and their corresponding closedform expressions.

V. Results: Analytical, Simulated and Comparison In this section, we provide an exhaustive comparison of the theoretical and simulation results to substantiate our proposed analytical model. We provide the performance of the proposed model under various network scenarios, viz., different values of , various length and width of the rectangular region, and different node density from 0.0001-0.03 nodes/m2 to in a finite square to 10-50 nodes/m2 in a unit square region. We show with our exhaustive simulations that the rendered analytical results are consistent with the simulation results for each scenario. In addition, we also show that proposal in [20] is a special case of our proposed model when the length and width of the rectangular region is set to be unity. We implement all the proposed equations and algorithm in MATLAB® 2017a. Each Monte Carlo simulation result presented in this section is the average of performing 10000 iterations.

A. Node Isolation Probability We analyze the impact of boundary regions on the node isolation probability ΡFG ( ) w.r.t. transmission range  for a finite network deployed in a rectangular region. Here, we evaluate the node isolation probability at  = 20, 50, 100, 200, and 300 for different dimensions of the rectangular region assuming constant node density ¥ =  ⁄(ℒ × +) nodes per meter square.

Figure 15: Node isolation probability ¦™§¨ &—˜  w.r.t. transmission range —˜ of a rectangular region &‘ ( ©…… ª, – ( ž… ª ) for « ( ‰…, ¡…, ©……, ‰……, ¬…… nodes.

Figure 16: Node isolation probability ¦™§¨ (—˜ )w.r.t. transmission range —˜ of a rectangular region (‘ = ¬…… ª, – = ‰… ª ) for « = ‰…, ¡…, ©……, ‰……, ¬…… nodes.

Figure 17: Node isolation probability ¦™§¨ (—˜ ) w.r.t. transmission range —˜ of a rectangular region (‘ = ¡…… ª, – = …… ª ) for « = ‰…, ¡…, ©……, ‰……, ¬…… nodes.

Figure 18: Node isolation probability ¦™§¨ (—˜ ) w.r.t. transmission range —˜ of a rectangular region (‘ = © ª, – = © ª ) for « = ©…, ‰…, ¡… nodes.

It can be observed from Figures 15 to 18 that the node isolation probability decreases with increase in transmission range  of the nodes as with larger transmission range, it is less likely for a node to get isolated. It can also be seen that there is almost a perfect match between our proposed analytical model and the simulation results, which proves that our model captures the boundary effects precisely to measure (24) as shown in explicable Figure 15, Figure 16, and Figure 17. Note that the networks in the three figures have low node density between 0.0001 nodes/m2 and 0.0375 nodes/m2. In contrast, a large deviation exists in node isolation probability as compared to the probability in an infinite homogeneous Poisson point process networks [18], computed as ΡFG ( ) = k ­®=>

z

(27)

The ΡFG &  computed using our model is larger than using the Poisson model in (27) because the latter ignores the boundary effects, unlike ours. We also analyze the performance of our analytical model for a unit square region. As shown in Figure 18, our analytical results perfectly match the simulation results. Note that Figure 18 considers networks with high node density, i.e., from 10 to 50 nodes/m2. The results show the generality of our method as compared to the method in [20] that can be used only for a unit square region.

B. Minimum Node Degree Distribution and T-Connectivity

This section analyzes the impact of boundary regions on the minimum node degree distribution fX &; r ) and -connectivity for  = 1, 2, 3 by varying the number of nodes ( = 20, 50, 100, 200, 300). The probability of the minimum number of neighbors of an arbitrary node, , is also determined for different values of  deployed in a rectangular region of various dimensions. Here again, the simulation results for fX (; r ) perfectly match with theanalytical results of our proposed work obtained by (25) and are presented in Figure 19 to Figure 21. Note that except for networks in Figure 19(f), Figure 20(f) and Figure 21(f) that have high node density of 10 nodes/m2, 20 nodes/m2, 50 nodes/m2, each other figure considers networks with low density from 0.0001 nodes/m2 to 0.0375 nodes/m2. Thus the results show the applicability of our framework for networks with both high and low node density. It can be observed from these figures that the probability that a node has at least  number of neighbors increases with the increase in the transmission range  of the node because larger transmission range allows the nodes to communicate with the nodes lying even at greater distances. Figure 19 (f), Figure 20 (f) and Figure 21(f), respectively show the minimum node degree distribution results for a unit square through our analytical model. It is evident that our framework exactly replicates, thus, generalizes the model of [20]. It is also worth to note that the minimum node degree distribution .L (;  ) limits the probability of -connectedness because a -connected network is obtained at the same time we obtain a network with minimum node degree , both with and without boundary effects. Our simulation results authenticate the argument that the minimum node degree distribution .L (;  ) acts an upper limit for Ρ V ( ) and gets tightens as Ρ V ( ) → 1, as shown in Figure 19, Figure 20, and Figure 21.

C. Connectivity We derive a closed form analytical expression incorporating boundary effects for the minimum node degree distribution .L (;  ), which serves as an upper limit for the -connectivity of a multi-hop network deployed in a rectangular region. In order to validate our analytical results, we simulate the network for the probability that the network is -connected Ρ V ( ) for  = 1, 2, 3) versus node transmission range  by deploying  number of nodes inside the rectangular region. Note that the probability of -connectivity increases with the increase in the transmission range of the nodes because larger transmission ranges allow nodes to communicate with the nodes located at farther distances, resulting in increased connectivity. The curve for  = 1 (black in color) in each subplot of Figure 19, Figure 20, and Figure 21 shows the probability that the network is connected with respect to transmission range  of the nodes.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 19: Minimum node degree distribution ¯° &T; —˜ ) and probability of T-Connectivity ¦T±¨š (—˜ ) for T = ©, ‰, ¬ versus range —˜ for (a) « = ‰…, (b) « = ¡…, (c) « = ©……, (d) « = ‰……, and (e) « = ¬…… in a rectangular region having dimensions (‘ = ©…… ª, – = ž… ª) and (f) shows results for a unit square having « = ©….

(a)

(b)

(c)

(d)

(e)

(f)

Figure 20: Minimum node degree distribution ¯° &T; —˜ ) and probability of T-Connectivity ¦ T±¨š (—˜ ) for T = ©, ‰, ¬ versus range —˜ for (a) « = ‰…, (b) « = ¡…, (c) « = ©……, (d) « = ‰……, and (e) « = ¬…… in a rectangular region having dimensions ( ‘ = ¬…… ª, – = ‰… ª), and (f) shows results for a unit square having « = ‰….

(a)

(b)

(c)

(d)

(e)

(f)

Figure 21: Minimum node degree distribution ¯° &T; —˜ ) and probability of T-Connectivity ¦ T±¨š (—˜ ) for T = ©, ‰, ¬ versus range —˜ for (a) « = ‰…, (b) « = ¡…, (c) « = ©……, (d) « = ‰……, and (e) « = ¬…… in a rectangular region having dimensions (‘ = ¡…… ª, – = …… ª), and (f) shows results for a unit square having « = ¡….

Furthermore, the probability of -Connectivity obtained through our framework approximates simulation results more accurately than the Weighed Border Effects (WBE) model given in [16] for the whole range of transmission. It is observed that analytical results using the CBD model in [16] underestimate -connectivity when the node density is high, whereas the value is overestimated when the node density is low. In addition, it is worth mentioning that our framework is superior to the Corner Border Dominance (CBD) model given in [16] for the higher connection probability requirements. We also obtain the network -Connectivity in a unit square region using our framework and found that our analytical model reproduces the results of [20] as shown in Figure 19 (f), Figure 21 (f), and Figure 21 (f).

VI. Conclusion Many researchers have analyzed WSN considering the boundary effects phenomenon either for an infinite or a finite network (in terms of number of nodes ) in a square region. In this work, we have presented an analytical framework, which not only has incorporated the boundary effects but also provided a more generalized approach on WSN deployed in a finite rectangular region rather than just in a square region considered by the earlier researchers. We have also derived general expressions for pertinent metrics of multi-hop WSN deployed in a rectangular region by computing the effective coverage areas of a node arbitrary positioned at various sub regions of boundary and non-boundary region(s). We investigated the topological and connectivity properties of WSN such as the minimum node degree distribution .L &;  ), node isolation probability ΡFG ( ) and the probability of connectivity Ρ V ( ) with respect to the transmission range  of the nodes, number of nodes , and varying physical dimensions of the rectangular region. Through simulation and under various dimensions of rectangular regions, we have shown that the proposed closed form expressions for node isolation probability and minimum node degree distribution obtained through our proposed analytical expressions & model are more plausible, accurate and precise as compared to [20]. The simulation results have also suggested that the minimum node degree distribution .L (;  ) acts as an upper limit for the -connection probability, which is very tight when the required probability of -connectivity is greater than 95%. The analytical results obtained through our model for node isolation probability and minimum node degree distribution for a finite network deployed in a finite rectangular/square region have found to be an agreement with the results obtained by simulation under varying scenarios of nodes for  = 20, 50, 100, 200, and 300. Our general framework also conforms to the specific cases of square region dealt by earlier researchers. Furthermore, the -connectivity results from our model are consistent rather than those obtained by [16], which underestimated it on high node density and vise-versa. Through an exhaustive analysis, we have shown that the nodes present in the boundary regions of a network in a finite region have significant impact on the minimum node degree distribution, connectivity, and the node isolation probability. We believe that the results obtained through our proposed analytical framework would be of practical use for researchers & developers, and would make a good contribution to investigate the connectivity metrics of practical wireless multi-hop networks. The natural extension of this work could be a polygon or a circular region with consideration of channel fading/interference/obstacles in the transmission model.

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Jaiprakash Nagar is currently pursuing his Ph.D. at Subir School of Quality and Reliability, Indian Institute of Technology Kharagpur, West Bengal, India. He received the integrated B. Tech & M. Tech degree in electronics and communication engineering from Gautam Buddha University Greater Noida, India, in 2015. His research interest includes wireless communication, intrusion detection in wireless sensor networks, Reliability aspects in wireless multihop networks.

S. K. Chaturvedi is currently working as a Professor at Subir Chowdhury School of Quality and Reliability, Indian Institute of Technology, Kharagpur (WB) India. He received his Ph. D. degree from Reliability Engineering Centre, IIT, Kharagpur (India) in year 2003. He did his Bachelor’s degree in electrical engineering and Master’s degree in system engineering and operations research, both from Indian Institute of Technology, Roorkee (erstwhile University of Roorkee), U. P., India in 1988 and 1990 respectively. He is on the review panel of several prominent international journals in his area of research such as IEEE transactions on Reliability, IJPE, IJQRM, IJ System Science, and IJ Failure Analysis. He has research interest in the area of reliability modeling and analysis, network reliability, life-data analysis, maintenance and optimization.

Sieteng Soh currently working as a Senior Lecturer in the Department of Computing, Curtin University of Technology, Perth, WA, Australia. He received his Ph.D. and M.S. degrees from Louisiana State University, Baton Rouge, LA, USA and B.S. degree from the University of Wisconsin-Madison, Madison, WI, USA all in electrical engineering. From 1993 to 2000, he was a Faculty Member at Tarumanagara University, Jakarta, Indonesia, where he was the Director of the Research Institute from 1998 to 2000. His research interests include network reliability, and parallel and distributed processing.

The authors declare that there is no conflict of interest.