Design of a Probability Density Function Targeting Energy-Efficient ...

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Design of a Probability Density Function Targeting Energy-Efficient Node Deployment in Wireless Sensor Networks Subir Halder and Sipra DasBit, Member, IEEE

Abstract—In wireless sensor networks the issue of preserving energy requires utmost attention. One primary way of conserving energy is judicious deployment of sensor nodes within the network area so that the energy flow remains balanced throughout the network and prevents the problem of occurrence of energy holes. Firstly, we have analyzed network lifetime, found node density as the parameter which has significant influence on network lifetime and derived the desired parameter values for balanced energy consumption. Then to meet the requirement of energy balancing, we have proposed a probability density function (PDF), derived the PDF’s intrinsic characteristics and shown its suitability to model the network architecture considered for the work. A node deployment algorithm is also developed based on this PDF. Performance of the deployment scheme is evaluated in terms of coverage-connectivity, energy balance and network lifetime. In qualitative analysis, we have shown the extent to which our proposed PDF has been able to provide desired node density derived from the analysis on network lifetime. Finally, the scheme is compared with three existing deployment schemes based on various distributions. Simulation results confirm our scheme’s supremacy over all the existing schemes in terms of all the three performance metrics. Index Terms—Energy balance, network lifetime, node deployment, probability density function, wireless sensor network.

I. I NTRODUCTION HE advent of efficient short range radio communication and advances in miniaturization of computing devices have made possible for large-volume commercial production of wireless sensor nodes as well as large-scale real-world deployment of the same to form a wireless sensor network (WSN). Such a network typically suffers from a number of unavoidable problems such as resource constrained nodes, random node deployment sometimes in an unattended open field where it is very difficult to replace/ recharge battery etc. So the network as a whole must minimize the energy usage in order to enable untethered and unattended operation for an extended period of time. Therefore, a critical consideration in designing such WSNs is conserving energy to maximize the post deployment network lifetime [1]. The rate of energy

T

Manuscript received May 11, 2013; revised October 10 and December 28, 2013. The associate editor coordinating the review of the paper and approving it for publication was I.-R. Chen. S. Halder is with the Department of Computer Science and Engineering, Dr. B. C. Roy Engineering College, Durgapur, India 713206 (e-mail: [email protected]). S. DasBit is with the Department of Computer Science and Technology, Bengal Engineering and Science University, Shibpur, India 711103 (e-mail: [email protected]). Digital Object Identifier 10.1109/TNSM.2014.031714.130583

depletion in the network primarily depends on the deployment nature of the nodes. The nature of deployment, on the other hand, mainly depends on the application environment [2]. In WSNs, nodes can be deployed either randomly or in pre-determined manner. In random deployment, nodes are deployed randomly, generally in an inaccessible terrain. For example, in the application domain of disaster recovery or in forest fire detection, nodes are dropped by helicopter in random manner [3]. On the contrary, in pre-determined deployment, the locations of the nodes are specified. This type of deployment is used in applications when sensors are expensive or when their operations are significantly affected by their positions. The applications include placing imaging and video sensors, populating an area with highly precise seismic sensors, underwater WSN applications, monitoring manufacturing plants etc. One important way of conserving energy is by uniform energy consumption or load distribution throughout the network. Non-uniform dissipation of energy in any part of the network may stop functioning of that part of the network leading to a phenomenon known as the energy hole problem [4]. Sometimes even after the network lifetime is over, due to energy hole problem a substantial amount of energy still remains in the nodes leading to significant wastage of energy. The energy hole problem arises when more data are transmitted by certain nodes of the network than the other nodes resulting in extra energy dissipation of those nodes [5]. Therefore, if any part of the network is affected by the energy hole problem, the whole network gets affected badly as uneven consumption of energy in the network leads to premature shortening of network lifetime. To avoid this, care should be taken during node deployment such that energy dissipation of all the nodes takes place uniformly ensuring load balancing throughout the network. A. Motivation Due to the nature of operation of WSN, nodes near the sink bear the major share of data forwarding compared to the nodes in rest of the network. So it is a common problem that nodes near the sink get drained off more quickly than the other nodes, thereby, creating energy holes near the sink resulting in loss of connectivity while most of the nodes are alive. Further, it is well known that sensor nodes have limited battery life and it is sometimes infeasible to replenish energy via battery replacements and therefore prolonging network

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HALDER and DASBIT: DESIGN OF A PROBABILITY DENSITY FUNCTION TARGETING ENERGY-EFFICIENT NODE DEPLOYMENT IN WIRELESS . . .

lifetime is of utmost importance. In such a network with energy constraints, in order to prolong network lifetime, nodes should be deployed with varying densities depending upon its position from the sink. One important issue that arises in such energy-constrained networks is to avoid energy holes in order to improve a network lifetime. To be more specific, we are interested in finding the answer for the following question-“Is it possible that all sensor nodes die simultaneously irrespective of their positions from the sink so that there will be no energy hole in a WSN, thereby, prolonging network lifetime?” This motivates us to explore a solution by providing varying node density in different parts of the network area based upon its proximity from the sink.

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able to provide desired network lifetime. In addition to the contribution in [6], here we show the impacts of routing and medium access control (MAC) protocols on the performance of the scheme. The rest of the paper is organized as follows. In Section II, results from the literature review are presented. The network model considered for the present work is presented in Section III. Analysis on network lifetime is done in Section IV. Section V presents the proposed node deployment scheme along with the proposed PDF based on which the scheme is developed. In Section VI, the performance of the scheme is evaluated based on both qualitative and quantitative analyses. Finally the paper is concluded with some mention about the future scope of the work in Section VII.

B. Contributions and Organization One of the solutions to avoid energy hole problem in WSN while maintaining coverage and connectivity is to deploy nodes with varying densities based on their positions from the sink. On the contrary, another promising approach may be to design a dedicated distribution function which perfectly conforms to such a need of varying densities. We have proposed a PDF based on which we have developed a node deployment strategy which not only keeps energy hole problem away from the network but also ensures enhancement of network lifetime while maintaining coverage and connectivity of the network. To the best of our knowledge we are the first to propose such a tailored-made PDF to be used in node deployment to avoid energy hole problem. The preliminary version of this work was reported in [6]. Our proposed PDF based node deployment strategy is pre-determined in nature. We extend our earlier work [6] in several aspects. The main contributions of this paper are as follows: • Unlike [6] here, we analyze the method of controlling network lifetime and found node density as a major parameter which has significant role to control network lifetime. The main question to be addressed is: How many nodes per unit area should be deployed in different parts of the sensor field in order to achieve energy balancing throughout the network and enhancement of network lifetime? The desired node densities derived out of the said analysis, guarantee that all the nodes exhaust their energy at the same time, and hence, energy balancing is achieved. • Based on the analysis, we propose a PDF and derive the PDF’s intrinsic characteristics e.g. expectation, covariance etc. • We develop a node deployment algorithm based on the PDF. It provides the node density to avoid energy hole problem, thereby, achieves enhanced network lifetime while maintaining coverage and connectivity of the network. • Performance of the scheme is evaluated both through qualitative and quantitative analyses. In qualitative analysis unlike [6], we analyze whether the PDF based scheme has been able to achieve the desired target set prior to the designing of PDF, towards energy hole elimination. • Similar to qualitative analysis, we show through simulation the extent to which our proposed PDF has been

II. L ITERATURE R EVIEW Many works have been reported so far that deal with the issue of balancing the load throughout the network with a goal to reduce the energy hole problem for prolonging network lifetime. All these works have been conducted through different approaches for achieving this goal. Each type of the above schemes has their own strengths and limitations. We have categorized the existing works based on their commonality in approaches which are presented below.

A. Transmission Paths Based Strategies In each of the works described below, nodes choose routing path differently. 1) Data Traffic and Distance Based Strategies: In these types of strategies, network lifetime is prolonged by making the nodes choose routing path judiciously considering data traffic along the path and distance. The distance considered is either transmission distance between the node and the sink or the distance between the node and neighbouring nodes towards sink. Azad and Kamruzzaman [7] have proposed energy balanced transmission range regulation policies for maximizing network lifetime in WSNs with corona based architecture. Firstly, they have analyzed and found two parameters- ring thickness and hop size - responsible for energy balancing. They have proposed a transmission range regulation scheme of each node and determined the optimal ring thickness and hop size for maximizing network lifetime. Simulation results show substantial improvements in terms of network lifetime and energy usage distribution over existing policies. However, before implementation of the proposed transmission policies, it requires significant computation to determine the optimal ring thickness and hop size. Song et al. [8] have presented two algorithms viz. centralized and distributed for the same corona based network architecture. Network lifetime has been optimized using a proposed decision factor computed by selecting the right transmission range of nodes in each corona. They have claimed that the algorithms not only reduce the complexity for searching right transmission range of node but also obtain results approximated to the optimal solution.

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2) Probability Based Strategies: These types of strategies employ probability based routing path selection with a target to balance energy consumption and enhance network lifetime. Boukerche et al. [9] have studied the problem of energybalanced data propagation in corona based WSNs for uniform and non-uniform deployments. The authors have proposed a density based probabilistic data propagation protocol towards balancing the energy consumption. In each step a node in a corona that holds data on-line, calculates the probability of data delivery either by hop-by-hop or directly to the sink based on the density information of the neighbouring coronas. Finally, the authors have shown that the proposed protocol works well for both uniform and non-uniform network deployments. Powell et al. [10] have proposed a probability based data propagation algorithm where the nodes compute off-line the probability of number of times data sent directly to the sink and data sent to a next hop neighbour. They have observed that by controlling the ratio of these two, energy consumption in each layer/slice is balanced but individual node in each slice is not well balanced. It is due to the unequal number of neighbouring nodes in the adjacent slices. Jarry et al. [11] have analyzed the data gathering and network lifetime maximization problems and based on the analysis, the authors have designed probabilistic on-line distributed routing strategies for different network structures. They have formulated the conditions to compute an energybalanced data propagation pattern for WSNs. Cheng et al. [12] have proposed a spatio-temporal compressive data gathering mechanism, where nodes send/forward sensory data to the sink as per a predefined probability. Since the data is forwarded with predefined probability, only a fraction of the readings from each node are transmitted to the sink, leading to reduced traffic and prolonged lifespan. B. Deployment Based Strategies The following works on enhancement of network lifetime are based on judicious node deployment. Wu et al. [13] have proposed a non-uniform node distribution, where, the number of nodes to be distributed in a layer is determined based on the minimum number of nodes required in the upper adjacent layer. They have concluded that only subbalanced energy depletion in the network is possible. Chang et al. [14] have proposed two node deployment strategies viz. distance-based and density-based strategies, with an objective for balanced energy consumption among the nodes. They have shown that the proposed strategies can efficiently balance energy consumption of each node and prolong network lifetime. Wang et al. [15] have given an analytical model for the coverage and network lifetime issues using a 2-D Gaussian distribution. They have proposed two deployment algorithms that achieve larger coverage and longer network lifetime using limited number of sensor nodes. They have concluded that the proposed algorithms effectively increase network lifetime with polynomial time complexity. Liu et al. [16] have proposed a node deployment algorithm for optimizing target surveillance in WSNs. The proposed deployment algorithm is designed in such a way that one can obtain predetermined network lifetime by deploying minimum number of nodes. Through

extensive simulation, the authors have shown that the proposed algorithm gives close-to-optimal solution. C. Mobile Sink/Agent Based Strategies In these types of strategies, load among all the nodes are distributed in a balanced manner by changing the position of the sink or using mobile agent so that network lifetime is enhanced. Luo and Hubaux [17] have proposed a deployment algorithm by considering mobile sink where the nodes lying nearer the sink keep on changing resulting in even distribution of load among the nodes. They have also proposed an appropriate routing protocol which supports mobility of sink. Ammari and Das [18] have also provided three different solutions for eliminating energy hole problem. In one of the three solutions, the authors have proposed a localized energy aware Voronoi diagram based data forwarding protocol considering homogeneous nodes and mobile sink. Lin et al. [19] have developed an energy balancing scheme for network lifetime maximization in WSNs using mobile agents. They have designed an energy prediction strategy by means of which mobile agents know about the remaining energy of all sensor nodes. Accordingly the mobility of agents is controlled so that the nodes with less remaining energy can communicate through mobile agent and avoid long-distance communication, thereby, evading uneven energy consumption. In the above, a number of existing works on node deployment are reviewed under different categories. Our scheme generally belongs to the category of ‘Deployment Based Strategies.’ The existing works reviewed under this category either have not used any mathematical model to implement the node deployment scheme or even if any such model is used, it has not addressed energy balance and network lifetime together. Our scheme, on the contrary, presents a node deployment strategy based on a proposed PDF and addresses energy balance and network lifetime together. III. N ETWORK M ODEL In this section, we describe the network architecture along with the assumptions. Definitions of coverage and connectivity in association with sensing and communication model respectively are described next. The energy model is presented at the end of this section. A. Architecture The authors in [20] have proposed a realistic network model for WSNs where the network area is covered by a set of concentric circles centered at the sink. They have also provided an in-depth discussion of its real life implementation. Further, in [17] it is proved that in case of the said network model, for enhancement of network lifetime, the best position for a sink is the center of the circle. Primarily these two works [17], [20] along with other relevant works [13], [15], [21] motivate us to consider present network architecture. However, in this architecture, energy consumption in annuli can be kept balanced but individual nodes in the annuli may remain imbalanced [10] and this has been mitigated by considering q-switch [13] routing policy elaborated in Section VI.B.1.

HALDER and DASBIT: DESIGN OF A PROBABILITY DENSITY FUNCTION TARGETING ENERGY-EFFICIENT NODE DEPLOYMENT IN WIRELESS . . .

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adding further complexity in theoretical analysis. However, in simulations to make the assumption realistic, additionally we consider a real MAC protocol for investigating the impact of the protocol on network lifetime.

Nr LN

a

2r

r

L1

L2

Sink

a Fig. 1.

Layered network area.

We consider a square shaped network area a × a which is covered by a set of uniform-width coronas or annuli (Fig. 1) [13], [21]. Each such annulus is designated with width r as layer. The sink is considered to be located at the center of the network area and responsible for collecting data from the sensors nodes. Nodes are placed in different layers surrounding the sink. A layer is identified as Li where i = 1, 2, . . . , N . Here i = 1 indicates the layer nearest to the sink andi = N indicates the layer farthest from the sink a . where N = 2r We assume that all the sensor nodes are homogeneous with respect to their initial energy, sensing and communication ranges while an unlimited amount of energy is set for the sink. The nodes are static and need to be distributed in each layer within the network with certain node density. The node density [5], [13] is the ratio of number of nodes in an area (layer) and the area (of the layer). It is also assumed that there is no local coordination among the nodes and therefore, the nearby nodes report same events to the sink. Periodic data gathering applications are considered where sensory data generation rate is proportional to the area (1 square unit) irrespective of the shape of the network area. Given a unit area, if the data generation rate is ρ bits per sec, it is meant that this unit area generates ρ bits/sec of data to be transmitted towards the sink. So, the data that needs to be reported from a given area a × a is ρ × a × a bits. The data is collected by the nodes and sent to the sink through multi-hop communication after a unit time-interval. A single sink is responsible for gathering the sensors’ data and controlling network operations. Further, we have assumed that both the time-intervals between two successive generations of data per area and two successive collections of data at the node as 1 sec. During theoretical analysis, for the sake of simplicity, an ideal MAC layer with no collision and retransmission is assumed and that does not lose any generality. The reason for not considering real MAC in theoretical analysis is the complexity involved in real MAC, such as need of control packet (TDMA scheduling), retransmission in case of collision (CSMA/CA) etc. Moreover, some of the overheads such as retransmission are a real time parameter which needs probabilistic formulation

B. Sensing and Communication Model 1) Sensing Model: We define a unit area to be covered, if every point in that area is within the sensing range of at least one active node. Alternatively, if each point in an area is covered by at least α nodes, it is known as α-coverage. The nodes perform observation [22] at an angle of 360◦ . The maximal circular area centered around a node ν that can be covered by it is defined as its sensing area S(ν). The radius of S(ν) is called the ν’s sensing range [14] Rs . We assume the relationship between r and Rs must satisfy the condition r ≤ 2Rs [6], [15] for covering the network area (Fig. 1). In case of 1-coverage, node density (λ) = 1/S (ν). Also, the coverage area C(X) of a set of nodes X is the union of the sensing areas covered by each node in X i.e. C(X) = ∪∀ν∈X S (ν) [22]. 2) Communication Model: We define a network as connected if any active node can communicate with any other active node, possibly using intermediate nodes as relays, so that the information collected by the nodes can be relayed back to sink [23]. We assume that two nodes can directly exchange messages if their Euclidean distance is not larger than the communication range Rc . Further, we assume that the relationship between r and Rc must satisfy the condition r ≤ Rc [6], [15] for ensuring connectivity in the network area (Fig. 1). Lemma 1: For a given network area a × a, in order to maintain connectivity of the network, the number of layers  a and to maintain (N ) stands in relation with Rc as N = 2R c coveragethe number of layers (N ) stands in relation with Rs  a . as N = 4R s Proof: If the radius of each layer in the layered architecture is r, then the distance between the center of the network area and the periphery of a layer-i is i × r (Fig. 1). Now for the farthest layer (i.e., layer-N ), the distance between the center and the periphery of layer-N is N × r and N r = a2 a (Section III.A) or, N r = 2R (putting r = Rc ). Therefore, for c a given network area a×a, in order to maintain connectivity of the network  relationship between Rc and N should stand  the a as N = 2Rc . a a Similarly, we have N = 2r = 4R (replacing r = 2Rs ). s Therefore, for a given network area a× a, in order to maintain coverage of the network  relationship between Rs and N  the a should stand as N = 4Rs . C. Energy Model We have considered the first order radio model [15] as the energy model where energy consumption of a node is dominated by its wireless transmissions and receptions; so the other energy consumption factors such as for sensing and processing are neglected. According to this radio model, energy consumed by a node for transmission and reception is as follows:

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Energy consumption for transmitting n-bits data over a distance d is et (n, d) = eelec n + eamp nd2 . Since we have assumed the transmission range of a node as Rc , the above can be rewritten by replacing d with Rc : et (n, Rc ) = eelec n + eamp nRc2 = et n

(1)

where et = eelec + eamp Rc2 and et is energy required to transmit one bit of data. Energy consumption for receiving n-bits data is er (n) = eelec n = er n

(2)

where er is energy required to receive one bit of data and er = eelec . IV. A NALYSIS ON N ETWORK L IFETIME This section presents an analysis on network lifetime with an objective to find out the parameter(s) which have significant influence on network lifetime so that the lifetime can be extended by controlling the parameter values. In presence of several existing state-of-the-art definitions of network lifetime [1] and lifetime of a node, the present work considers the following definitions throughout the paper. Definition 1. The node lifetime (measured in unit time) is defined as the time when the initial energy of a node is completely drained out so that it is neither able to transmit its own data nor able to forward any data. It is measured as the ratio of the initial energy of a node and the energy consumption rate of the node. Considering initial energy of each sensor node as ε0 and the energy consumption rate of a node in layer-i as ECRi , the lifetime of a node in layer-i is ε0 . LTi = ECR i Definition 2. The network lifetime is defined as the time interval from the beginning of the network operation until the proportion of dead nodes exceeds a certain threshold, which may result in loss of coverage of a certain region, and/or network partitioning [5]. If the total number of nodes in layer-i is Ti , the lifetime of a layer-i in the network is ε0 ×Ti ε0 Ti ×ECRi = ECRi . It is same as the lifetime of a node in layer-i as defined previously. If any of the layer’s lifetime terminates which causes loss of coverage within the layer, it results in termination of network lifetime. Therefore, the network lifetime can be determined by the  shortest  lifetime of a layer and it is ε0 . expressed as min∀i ECR i We assume that a node requires minimum energy consumption for transmitting/forwarding data to the sink. Therefore, the node should transmit data to the sink via the shortest path i.e. a node in layer-i (Section III.A) requires i hops to transmit data to the sink. This routing policy is basically a simplified version of the q-switch, shortest path routing policy proposed in [13] which is used in our scheme and briefed in Section VI.B.1. In this analysis the simplification is made by considering a node in layer-i finds only one node in layer-(i − 1) to forward data towards sink. Further, we assume that area of a layer-i (Fig. 1) is Ai = π(2i − 1)r2 , where r is the width of a layer and λi is the node density of layer-i. Therefore, the total number of nodes in layer-i is given as Ti = λi × Ai for i = 1, 2, . . . , N . The nodes of all the layers except the farthest layer from the

sink spend their energy by transmitting their own sensory data, receiving data from the nodes of adjacent layers farther away from the sink and forwarding the received data. Nodes in the farthest layer spend energy only for transmitting their own data. Therefore, the data transmission rate of a node (for transmitting its own sensory data) in layer-i is ρ ρ × Ai = . λi × Ai λi

(3)

Further, a node in layer-i receives data from the nodes of layer(i + 1) i.e. the nodes in layer-(i + 1) transmit data towards the nodes of layer-i to forward the same towards the sink. So, the average data transmission rate of layer-(i + 1) towards a node in layer-i is λi+1 π (2i + 1) r2 ρ λi+1 (2i + 1) ρ λi+1 Ai+1 ρ . = = λi Ai λi π (2i − 1) r2 λi (2i − 1) The rate of data relayed by a node of layer-i is the sum of the above quantity for all the farther layers i.e., N h=i+1 λh (2h − 1) ρ . (4) λi (2i − 1) Therefore, the total data transmission rate per node in layer-i (mi ) can be obtained from (3) and (4) as follows: N ρ h=i+1 λh (2h−1)ρ + for i = 1, 2, . . . , (N − 1) λ λi (2i−1) i mi = ρ for i = N λN (5) The first component of the above expression of mi i.e. ρ/λi is for transmitting the node’s sensory data and the second component is for forwarding the outward adjacent layers’ data. Now the energy consumption rate of a node in layer-i for transmission is: For transmitting the node’s own data: ρ × et λi

(6)

where et is energy required to transmit one bit of data. For transmitting the relay data received from the farther adjacent layers: N h=i+1 λh (2h − 1) ρ × et . (7) λi (2i − 1) So the energy consumption  rate of each node in layer-i due

Tx is computed from (6) and (7) as to transmission ECRi (8), at the top of the next page. Similarly, we calculate the energy consumption rate of each  node in layer-i for receiving ECRiRx data from the farther layers as follows:

N h=i+1 λh (2h − 1) ρ Rx er ECRi = (9) λi (2i − 1) where i = 1, 2, . . . , (N − 1) and er is the energy required to receive one bit of data. Hence the total energy consumption rate of each node in a layer-i is ECRi = ECRiTx + ECRiRx .

HALDER and DASBIT: DESIGN OF A PROBABILITY DENSITY FUNCTION TARGETING ENERGY-EFFICIENT NODE DEPLOYMENT IN WIRELESS . . .

ECRiTx =

⎧  ⎨ ⎩

ρ λi + ρ λN et

N

λh (2h−1)ρ λi (2i−1)

h=i+1

209

 et

for i = 1, 2, . . . , (N − 1) for i = N

(8)

So, the total energy consumption rate of the nodes of layer-i is



ECRi =

ECRiTx + ECRiRx Tx ECRN

for i = 1, 2, . . . , (N − 1) . for i = N (10)

We know, energy depletion across the network is balanced [13] when all the nodes of the network exhaust their energy at the same time. To be more specific, if balanced energy depletion is attained in the network then all nodes located in any layer have the same lifetime. Alternatively, all the nodes exhaust their energy at the same time. Therefore, for energy balancing, the following condition must be satisfied-ECRi = ECRi+1 = · · · = ECRN . Now rewriting (10) with the help of (8) and (9), we have the equation at the top of the next page. After simplification and basic transformations, we obtain (11), at the top of the next page. Equation (11) implies that the ratio of node density between two consecutive layers depends on layer number i and the total number of layers N . Further, the node density in a layer is uniform but this node density varies in different layers. The nature of variation is such, that the node density is maximum at the layer nearest to the sink and it decreases in the layers farther away form the sink i.e., λ1 > λ2 > · · · > λN . Now, (11) is a non-linear equation and computation of λi is fairly complex. However, it suggests that one can compute λi , for i = 1, 2, . . . , (N − 1), if λN is known. Considering 1-coverage (Section III.B.1), λN = 1/S (ν) where S (ν) = πRs2 . Moreover, the balanced energy consumption can be obtained in different layers of the network if the nodes are distributed in accordance with the desired node density given in (11). When the nodes are distributed according to (11) to get balanced energy consumption, we can ensure that all the nodes deployed in the sensor field completely deplete their energy at the same time. Now, from the Definition 1/2, the lifetime of a node/the network lifetime (LT ) can be expressed as: ε0 for i = 1, 2, . . . , N . Replacing the denominator LTi = ECR i with the help of (10), (8) and (9) we have LTi in (12), at the top of the next page. As we have assumed earlier that the nodes in a layer report data to the sink in minimum hops, therefore the derived network lifetime (see (12)) provides the upper bound of the network lifetime. Also we can say that the upper bound of the network lifetime is achievable by controlling node density λi in each layer, as given in (11). V. P ROBABILITY D ENSITY F UNCTION BASED N ODE D EPLOYMENT (PDFND) From the analysis of network lifetime provided in the previous section, we have found that the ratio of node density between two consecutive layers depends on layer number i and the total number of layers N . Further, for balanced

Fig. 2.

3-D graph (surface plot) of the PDF.

energy consumption, the required node density is maximum in the layer nearest to the sink and it decreases in the layers farther away from the sink. Considering these observations and taking guidelines from the analysis, in this section, we have designed a PDF targeting its implementation in lifetimeenhancing node distribution in WSNs. Also, we have presented a node deployment algorithm based on the proposed PDF [6]. A. Proposed Probability Density Function [6] The mathematical domain under consideration is divided into a number of concentric circles having radii increasing arithmetically from r to (N × r ) with a difference of r . In the mathematical domain, if (x, y) be a point and it lies between circles (i − 1) and i, then the probability density at that point is   2  k (2i − 1)  2 f x, y; N, i, r  = , ∀ (i − 1)2 r  < x2 + y 2 ≤ ir  N 2 i4 (13)

where i = 1, 2, . . . , N and k is a constant as follows:   −1 2 52 (2N − 1) 32 2  2 k = N π (r ) 1 + 4 + 4 + · · · + . 2 3 N4 Fig. 2 is the 3-D graph of the proposed PDF. The characteristics of the PDF show decrease of the functional value with increase in the value of i implying lower probability and vice versa. Theorem 1: The value of constant k is:   −1 2 52 (2N − 1) 32 2  2 k = N π (r ) 1 + 4 + 4 + · · · + . 2 3 N4 Theorem 2: If the random variables X and Y follow a proposed PDF with parameters N and i, then the cumulative distribution function (CDF) of X and Y is given as

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ρ et + λi

N



N λh (2h − 1) ρ ρ h=i+2 λh (2h − 1) ρ (et + er ) = (et + er ) et + λi (2i − 1) λi+1 λi+1 (2i + 1)

h=i+1

 λi = λi+1 LTi =

F [X ≤ x, Y ≤ y] =

 2

kπ (r ) N2



i   j=1

2i + 1 2i − 1



et (2i − 1) + (et + er ) et (2i + 1) + (et +

λi (2i−1)ε0  ρ(2i−1)et +(et +er ) N h=i+1 λh (2h−1)ρ λN ε0 ρ(2i−1)et

2

2

2

(2j − 1) η −i + j4 i4

 .

2

We choose (x, y) such that 0 ≤ x2 + y 2 ≤ η 2 (r ) , where i ≤ η ≤ i + 1. The proofs of Theorem 1 and Theorem 2 are omitted due to space limitations. We refer the reader to Theorem 1 and Theorem 2 in [6] for the detail proof. Theorem 3: If the two random variables X and Y follow the proposed PDF with parameters N and i, then the expectation of X and Y is given as  2 N  2 3 1 k (r )  2 + − − . E [XY ] = N 2 i=1 i i3 i2 2i4 Proof: The expectation of the two variables X and Y with parameters N and i can be given as E [XY ] = E1 [XY ] + E2 [XY ] + · · · + EN [XY ] =

i  i=1

Ei [XY ] (14)

where Ei [XY ] is the expectation of X and Y for a given value of i. Now, Ei [XY ]

Ei [XY ]

Ei [XY ]

=

=

=

k N 2 i4 4k N 2 i4



ir



xy ⎡ ⎢ x⎢ ⎣ 

dy dx  (ir  )2 − x2 

((i − 1) r  )2 − x2  4  2 k (r  ) 2 3 1 + − − . N2 i i3 i2 2i4

⎤ y

⎥ dy ⎥ ⎦ dx

0

(15)

So, replacing Ei [XY ] in (14) with the value of (15) we get  4 N  k (r )  2 2 3 1 E [XY ] = + 3− 2− 4 . (16) N 2 i=1 i i i 2i Theorem 4: If the two random variables X and Y follow a proposed PDF with parameters N and i, then the covariance of X and Y is given as the equation at the top of the next page. Proof: From the definition of covariance [24] we know that Cov (X, Y ) = E [XY ] − E [X] E [Y ]. We can find the covariance of the two discrete and random variables X and Y for a particular value i in the domain. The

N

h=i+1 N er ) h=i+2

λh (2h − 1)

 (11)

λh (2h − 1)

for i = 1, 2, . . . , (N − 1)

(12)

for i = N

covariance of the two discrete and random variables X and Y for the entire domain is obtained by summing different values of the parameter i, where i = 1, 2, . . . , N Cov (X, Y ) = Cov (X, Y ) =

N  i=1 N 

Covi (X, Y ) [Ei [XY ] − Ei [X] Ei [Y ]] .

i=1

We get the expectation of X and Y for a particular value of i, i.e. Ei [XY ] from (15) 4  2 k (r ) 2 3 1 Ei [XY ] = + 3− 2− 4 . N2 i i i 2i The expectation of X for a particular value of i, i.e. Ei [X] can be calculated as   k Ei [X] = x dy dx N 2 i4 ⎡  ⎤  2  2 ir  (ir ) − x ⎢ ⎥  4k ⎢ dy ⎥ Ei [X] = x ⎣ ⎦ dx 2 4  N i 2  2 0 ((i − 1) r ) − x   3 3 4k (r ) 3 1 + − Ei [X] = . 3N 2 i2 i3 i4 From the above equations, the expectation of X in the entire domain is  3 N  3 1 4k (r )  3 + 3− 4 . E [X] = (17) 3N 2 i=1 i2 i i As our network model is symmetric, so the expectation of the random variables X and Y in the entire domain is same. We can say that Ei [X] = Ei [Y ]. The covariance of random variables X and Y is N

2 N   Ei [XY ] − Ei [X] Cov (X, Y ) = i=1

Cov (X, Y )

i=1 2

= E [XY ] − [E [X]] .

(18)

In (18), replacing by (16) and (17) we get covariance of random variables X and Y as the second equation at the top of the next page. Although in preliminary version [6] of this work, the PDF was proposed, the proofs of the Theorem 3 & 4 are provided in this present work only.

HALDER and DASBIT: DESIGN OF A PROBABILITY DENSITY FUNCTION TARGETING ENERGY-EFFICIENT NODE DEPLOYMENT IN WIRELESS . . .

211

  2  N 2 k (r )3   2 3 1 3 1 4 3 + 3− 2− 4 − Cov (X, Y ) = + 3− 4 r N 2 i=1 i i i 2i 3 i2 i i   2  N 2 k (r )3   2 3 1 3 1 4 3 + 3− 2− 4 − Cov (X, Y ) = + 3− 4 r N 2 i=1 i i i 2i 3 i2 i i

B. Proposed PDF-based Node Deployment The PDF proposed in the previous section is discrete in nature. Our objective is to deploy sensor nodes in the layered network area (Fig. 1) with the proposed PDF. The PDF is mapped with the node deployment in a layered network area as follows: the parameter i represents the layer number for both the proposed PDF and layered network area (see Fig. 1) where i = 1, 2, . . . , N ; the parameter r of the proposed PDF corresponds to the width r of the annuli/layer. Therefore, the relationship between r and Rc is r ≤ Rc , whereas, between r and Rs is r ≤ 2Rs . The density function is designed as per the analysis in Section IV. It is a non-uniform one i.e. the value of PDF is higher for the nodes deployed around the sink whereas the value is lower as one moves away from the sink. The PDF of deploying a sensor node at point f (x, y) located in layer-i is given as follows: k (2i − 1) /N 2 i4 where i = 1, 2, . . . , N and k is the proportionality constant. Further, the probability for the nodes deployed within a layer is k (2i − 1) Ai /N 2 i4 , where Ai is the area of layeri and k is as follows:  −1 N 2 2 2 4 (2i − 1) /i (19) k = N πr i=1

where r is the width of the layer-i. The area of layer-i is given as " # ! $ 2 Ai = πi2 r2 − π (i − 1) r2 = π (2i − 1) r2 . By replacing the value of Ai , the probability (pi ) of deploying nodes at a layer-i is given as # $−1 2 pi = k (2i − 1) πr2 N 2 i4 . (20) The number of nodes in a layer-i (Ti ) is equal to the probability (pi ) of deployment of nodes at layer-i multiplied by the total number of nodes (Ttotal ) i.e. Ti = pi × Ttotal . Therefore, the node density of a layer-i (for i = 1, 2, . . . , N ) according to our proposed PDF is given as N pi pi × Ttotal = × λ × Aj λi = j=1 j Ai Ai k (2i − 1) πr2 N λi = (2j − 1) λj . (21) j=1 N 2 i4 The above expression implies that the node density determined by the proposed PDF is controlled by the parameters i and N and that conforms to the guideline provided in Section IV. Further, the node density in a layer is uniform and probability of deploying nodes is equal, but this node density as well as probability varies in different layers. The nature of variation is that the node density is maximum in the layer nearest to

the sink and it decreases in the layers farther away form the sink. Therefore, we have λ1 > λ2 > · · · > λN . C. Algorithm for Node Deployment Algorithm 1 Input: a, r, Ttotal ; Output: λi // area parameter, width of layer, and total number of nodes to be deployed compute maximum number of layers N using Lemma 1 compute constant k using Theorem 1 for i = N to 1 do compute pi using (20) compute λi using (21) end for

D. Illustrative Example Let us consider a 200 × 200 sq unit area where 100 nodes with Rc = 25 units are deployed employing the proposed probability density function. The number of layers N is: % & % & a 200 N= = = 4 (using Lemma 1). 2 × Rc 2 × 25 Now replacing the values of N and Rc in (19), the value of k can be computed as   −1 52 72 32 2 2 k = 4 π (25) 1 + 4 + 4 + 4 2 3 4 k = 0.004 [where r = Rc ]. Now the probability of deploying nodes in each of the four layers is obtained by replacing k by 0.004, r by 25 and N by 4 in (20). For example, pi is obtained as, 2 pi = 0.49 (2i − 1) /i4 . Probability of deployment of nodes in layer-1, p1 = 0.49. Similarly p2 = 0.27, p3 = 0.15 and p4 = 0.09. Using (21), node density in each of the 4 layers is as follows: in layer-1, λ1 = 0.00024 × 100 = 0.024, in layer-2, λ2 = 0.0045, in layer-3, λ3 = 0.0015, in layer-4, λ4 = 0.00065. We observe that the node density in each layer obtained from the algorithm conforms to the non-uniform nature of the PDF. Therefore, it fulfils our objective of deploying more number of nodes towards the sink and decreasing the number of nodes as the distance from the sink increases. Finally, we claim that the proposed deployment is feasible. As reported in state-of-the-art works [25], [26] on design and deployment of WSN, air-dropped deployment in a controllable manner is feasible even in an inaccessible terrain. We propose to compute the node density in each part (layer/annuli) of the

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Node density (node/sq. m)

network off-line prior to the actual deployment. At last, the nodes are to be dropped (e.g. from helicopter) using a point (sink) as the center following the pre-computed node densities of the proposed PDF. One important application of PDFND based node deployment is battlefield surveillance where more detailed information is needed around the headquarters and the sink may be placed at the headquarters. Unlike the preliminary version [6], in the proposed node deployment (Sections V.B, V.C, and V.D) we have considered node density instead of number of nodes in each layer as the parameter of concern for making the scheme energy balanced, thereby, getting enhanced network lifetime. This is as per the guideline of the analysis done in Section IV.

Now replacing the value of et and using (5), we have (using energy model in Section III.C) N ' ( λh (2h − 1) ρ mi et + h=i+1 er = mN eelec + eamp Rc20 . λi (2i − 1) After rearranging the above equation, we have the equation at the top of the next page. Similar to the farthest layer, for the rest of the layers we can have the equation at the top of the next page. Let us assume that the network lifetime under the regulated  transmission policy is LT . The lifetime maximization thus can be formulated as the following optimization problem:  max LT Subject to (22)–(26) at the top of the next page. The objective function is the lifetime of the network. The constraints (22) and (23) specify the transmission range of the nodes for balanced energy consumption where the nodes are located in the farthest layer and in the other layers respectively. The constraint (24) limits the maximum transmission range of

Achieved node density

1.5

Desired node density

1 0.5 0 -0.5 -1 1

2

3

Layer number

4

5

Node density (node/sq. m)

(a) 5-layer network.

E. Formulating Network Lifetime Maximization as an Optimization Problem In addition to solving network lifetime maximization by varying node density in each layer, in this section we show that the same can be achieved by regulating transmission range of nodes in each layer. Precisely, we formulate network lifetime maximization as transmission range regulation based optimization problem. We assume that the nodes in the farthest layer have the longest transmission range Rc0 (i = N ) and the nodes in the other layers (i = 1, 2, . . . , (N − 1)) have transmission range Rci in decreasing order. As mentioned in Section IV the nodes in all the layers i (i = 1, 2, . . . , (N − 1)) consume energy for transmitting their own sensory data and for carrying relay data. However, the nodes in the farthest layer (i = N ) from the sink do not carry any relay data. So, let us assume that the energy required to transmit one bit of data in the farthest layer is et = eelec + eamp Rc20 . Further, in order to obtain energy balancing the average energy consumption  rate of each layer must be same i.e., ECRi = ECRN , where  ECRN is energy consumption rate of the nodes of layer-N under the transmission range regulation policy. Now using (8), (9), and (10) we get the following: N λh (2h − 1) ρ ρ ρ  (et + er ) = et + h=i+1 e. λi λi (2i − 1) λN t

-3 2 ×10

-3 11 ×10

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9

Desired node density

7 5 3 1 -1 1

2

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Layer number (b) 10-layer network. Fig. 3.

Node densities in each layer for various network sizes.

a node whereas constraint (25) specifies that the total energy consumed by each node should not exceed its initial energy reserve. VI. P ERFORMANCE A NALYSIS Performance of the present node deployment strategy is measured based on two parameters such as energy balance and network lifetime. Both qualitative and quantitative analyses are presented here. A. Qualitative Analysis In this section we have analyzed the performance of the proposed PDF qualitatively to show in which extent our node deployment scheme (Section V) is close to fulfill the desired objective (Section IV). 1) Energy Balance: The network is said to be energy balanced when the nodes located in any layer have the same lifetime. Based on this condition we have derived the desired node density (λi ) of a layer using (11) (see Section IV) as follows: λi = (2i + 1) λi+1 (2i + 1)



et (2i − 1) + (et + er ) et (2i + 1) + (et +

N

h=i+1  er ) N h=i+2

λh (2h − 1) λh (2h − 1)

 .

On the other hand, in our node deployment strategy, nodes are deployed in different layers with varying node density (λi )

HALDER and DASBIT: DESIGN OF A PROBABILITY DENSITY FUNCTION TARGETING ENERGY-EFFICIENT NODE DEPLOYMENT IN WIRELESS . . .



Rc0

1 = mN eamp

Rc0

1 = mN eamp

1 Rci = mi eamp



N

ρ  e − λN t



N

h=i+1 λh (2h − 1) ρ er − mi eelec λi (2i − 1)

N

h=i+1 λh (2h − 1) ρ er − mN eelec λi (2i − 1)

mi e t +

ρ  e − λN t

λh (2h − 1) ρ er − mN eelec λi (2i − 1)

h=i+1

mi e t +



1 Rci = mi eamp



N

h=i+1 λh (2h − 1) ρ er − mi eelec λi (2i − 1)

213

 12

 12

 12 , ∀TN

(22)

, ∀Ti , i < N

(23)

 12

(N × r) ≥ Rc0 > Rci > 0 ∀Ti , 1 ≤ i ≤ N

  ECRi × LTi − ε0 ≤ 0 ∀i ∈ {1, 2, . . . N }

(24) (25)



ECRi , LTi ≥ 0 ∀i ∈ {1, 2, . . . N }

which is achieved node density and is given as (Section V.B, see (21)) λi =

k (2i − 1) πr2 N (2j − 1) λj . j=1 N 2 i4

It is observed from both the above equations that node density among the layers is non-uniform in nature but uniform within the layers. Moreover, as A1 < A2 < · · · < AN , the nature of variation of node density is that it is maximum in the layer nearest to the sink and it decreases in the layers farther away form the sink. To see the effectiveness of the proposed scheme, Fig. 3 plots both the desired and achieved node densities for two different network sizes. After deriving (see Section IV) the node density in the farthest layer for both the cases, desired and achieved node densities of the remaining layers are calculated iteratively using (11) and (21) respectively. It is clear from the plot, in all the cases desired and achieved node densities are almost same and that indicates the proposed scheme has been able to achieve energy balancing. 2) Network Lifetime: The desired lifetime, according to (12), is as the equation at the top of the next page. Now following the method for calculating LTi (Section IV, see (12)) we derive LTi as the equation at the top of the next page. If we compare LTi (desired) and LTi (achieved), as λi and λi are found almost same, LTi and LTi are also same. 3) Coverage and Connectivity: In addition to the energy balance and network lifetime, we have also measured coverage and connectivity to show the extent of maintaining coverage and connectivity by the proposed node deployment strategy. This section formulates necessary constraints to be satisfied for maintaining coverage and connectivity. It also contains a couple of Lemmas along with the proofs with an objective to show the extent of maintaining coverage and connectivity by the PDF. To measure the coverage, the concept of coverage

(26)

density (Ci ) [27] has been used. If the sensing area S(ν) (refer to Section III.B.1) of each node is mutually exclusive, the coverage density of layer-i Ci is defined as Ci =

Ti × S(ν) . Ai

If Ci = 1 i.e., 1-coverage (Section III.B.1), we say that Ai is covered by minimum number of nodes and coverage area of each node is mutually exclusive. If Ci > 1 i.e., αcoverage, we say that Ai is covered by more than the minimum number of nodes and therefore, coverage area of a node is overlapped with the coverage area of the other nodes in the area. The sensing accuracy would increase proportionally with the overlapping of coverage area, thus making the scheme more robust against sensing failure. Lemma 2: For a given network area, the proposed PDF gives the coverage density of a layer-i as Ci = Ti /4 (2i − 1). Proof: Let us consider Ti numbers of nodes are deployed in layer-i. The S(ν) of each sensor is calculated as πRs2 where Rs is the sensing radius of each sensor. So, C (Ti ) = ∪∀ν∈Ti S(ν) = Ti × S(ν) = Ti × πRs2 . From the definition of Ci we have Ci =

Ti × S(ν) T  × πRs2 = i . Ai (2i − 1) πr2 T  ×πR2

T

i s i Replacing r by 2Rs , we have Ci = (2i−1)π4R 2 = 4(2i−1) . s From the above expression it is observed that, the coverage density of layer-i depends on the number of nodes deployed in layer-i and it is inversely proportional to layer number i, which suits the requirement for energy balancing (Section IV). Also, one can achieve the desired coverage density by controlling the number of deployed nodes Ti in various layers within the network.

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LTi

=

λi (2i−1)ε0 N

[ρ(2i−1)et +(et +er )

LTi =

λN ε0 ρ(2N −1)et

⎧ ⎨ ⎩

h=i+1

λh (2h−1)ρ]

ε0 (2i−1)λi   ρ(2i−1)et +(et +er ) N h=i+1 λi (2h−1)ρ  ε0 λN ρ(2N −1)et

Lemma 3: For a network, if coverage is ensured, connectivity of the network is also ensured. Proof: We have discussed that when the network area is covered by minimum number of nodes, then Ci = 1. Now when the network is covered by minimum number of nodes, the maximum distance between the two nodes is 2Rs . In our communication model we have assumed that two nodes can communicate with each other if the Euclidean distance between them is less or equal to Rc and we have considered Rc ≤ 2Rs . As the maximum distance between the nodes is 2Rs and Rc ≤ 2Rs , so we can say that the connectivity is guaranteed if coverage is ensured. B. Quantitative Analysis The effectiveness of the proposed node deployment scheme, reported in Section V.B is evaluated through simulation. Moreover, all the theoretical claims made through qualitative analysis presented in Section VI.A are justified by simulation results. Simulation results of our scheme PDFND are compared with three existing node deployment schemes namely non-uniform node distribution strategy (NNDS) [13], node deployment with Gaussian distribution (NDGD) [15] and node deployment with Uniform distribution (NDUD) [28]. 1) Simulation Environment: The simulation is performed using MATLAB (version 7.1). We have done qualitative analysis considering simplified q-switch routing [13]. However, in simulation we have used the same routing protocol with no simplification for all the three schemes. This routing protocol is briefed along with the scheme [13] in the next paragraph. In NNDS the authors have proposed a non-uniform node distribution strategy for the uniform-width corona model. Here the nodes are deployed in such a way that the node densities in a corona increases in geometric proportion with common ratio q(> 1) from corona (N − 1) to corona 1. We assume that the number of nodes deployed at the farthest corona and the common ratio are known a priori. Once the number of nodes deployed at the farthest corona and the common ratio are known, nodes for the remaining coronas are computed and these computed numbers are exponentially increasing function of the common ratio q. The NNDS uses q-switch routing where the source node always selects one reachable relay node with maximum remaining energy in its subsequent inner layer to forward data. If there is more than one relay node with the same maximum remaining energy, one of them is chosen randomly. Once the source node selects the relay node, it forwards the data of its own as well as those received from the nodes of adjacent layers farther away from the sink. The selected relay node repeats this process until the data

for i = 1, 2, . . . , (N − 1) for i = N

for i = 1, 2, . . . , (N − 1) for i = N

arrives at a node in layer-1, after which the data is sent to the sink. Hence, the routing itself takes care of individual node’s load balancing and that eliminates the problem [10] of annuli architecture as stated in Section III.A. In NDGD, the authors have considered that the nodes are deployed using two dimensional Gaussian distribution and node density function at point f (xi , yi ) as − 1 f (x, y) = e 2πσx σy



(x−xi )2 2 2σx

+

(y−yi )2 2 2σy



,

where σx and σy are the standard deviations for x and y dimensions. Further, the authors have considered two deployment types: σx = σy and σx = σy . However, during simulation we have considered σx = σy = σ, which conforms to a disk model and that is similar to our network model. So, the probability density function of deploying a sensor node for point (x, y) is: 1 − x2 +y2 2 e 2σ . 2πσ 2 It is evident from the above equation that any two points in the disk having same distance from the center-point, have the same deployment probability. In NDUD, nodes are uniformly and independently distributed in the layered network area, the probability fa that a point is covered by sensor nodes is f (x, y) =

2

fa = 1 − e−λπRs

where Rs is the sensing range of the nodes and λ is the node density. We simulate our work both under ideal scenario and realistic scenario. Here, by ideal scenario we mean the scenario considered during theoretical analysis (Section IV and Section VI.A) i.e., simplified q-switch routing protocol, ideal MAC layer and the energy consumption only for transmission and reception. On the other hand, in realistic scenario we consider q-switch routing protocol and real MAC protocol which includes idle/sleep schedule of the nodes. Moreover, unlike ideal scenario, in realistic scenario energy consumption is considered for idle, sleeping and sensing in addition to transmission and reception. The real MAC protocol has been implemented by funneling-MAC [29]. The funneling-MAC is a hybrid MAC protocol where TDMA (schedule-based) is used in nodes located within a few hops from the sink whereas CSMA/CA (contention-based) is used in nodes located far away from the sink. The sink broadcasts a beacon for nodes located within a smaller number of hops by controlling the transmission power of the beacon. The nodes which receive the beacon are considered as f -nodes and perform TDMA

HALDER and DASBIT: DESIGN OF A PROBABILITY DENSITY FUNCTION TARGETING ENERGY-EFFICIENT NODE DEPLOYMENT IN WIRELESS . . .

TABLE I PARAMETER VALUES U SED IN S IMULATION

Average residual energy per node (Avg RE per node): It is defined as the residual energy in a node of a particular layer after the network lifetime ends. It is evaluated as follows: Avg RE per node =

while the nodes that do not receive the beacon perform CSMA/CA. During simulation we have considered the nodes located within layer-2 use TDMA schedule whereas nodes beyond layer-2 use CSMA/CA. We have considered energy consumption rates for sensing, remaining idle and remaining sleeping are 20%, 5%, 2.5% of the energy consumption rate of reception respectively. Further, in simulation, all the funnelingMAC implementing parameter (e.g. slot size, superframe size, moving average factor) values are considered same as in [29]. During implementation of all the schemes, we have deployed 500 and 2000 nodes for network with 5 and 10 layers respectively. For all the schemes, in order to have an integer number of sensor nodes for each layer, the upper ceil function is employed. We consider the energy cost to run the transmitter/receiver radio circuitry per bit processed (eelec ) as 50 nJ/bit. Also, we consider the energy used by the transmitter amplifier (eamp ) to achieve an acceptable signal to noise ratio (30 dB) as 10 pJ/bit/m2 . This setting requires receiver sensitivity -52 dBm. Furthermore, we consider data ready delay/measurement delay as 1 sec in our simulation. Our routing protocol [13] and MAC protocol [29] does not rely on any synchronization protocol and therefore, the synchronization parameters are not considered during simulation. For NNDS, the common ratio q is considered as 2 and for NDGD, standard deviation σ is considered as 70. All the parameters and their corresponding values used for simulation are listed in Table I. Extensive simulation has been performed with a confidence level of 95% and 5% accuracy. To achieve this, we collect results in groups of 2,000 observations per group. In one group, we obtain a mean out of 2,000 observations collected. We run at least 5 groups to get a minimum of 5 means from which we calculate the grand mean and estimate the difference of the grand mean from the true mean with 95% confidence. If the accuracy obtained is greater than 5%, we run more groups and collect more observations until the specified 5% accuracy requirement is achieved. 2) Simulation Metrics: Similar to qualitative analysis, (Section VI.A) energy balance, network lifetime and coverageconnectivity have been considered as performance metrics in simulation. We define two more parameters namely energy consumption rate per node and average residual energy per node for evaluating the extent of energy balance in the network. Further, we evaluate coverage-connectivity using the parameter coverage density (Section VI.A.3). Energy consumption rate per node (ER): It is defined as energy consumption of a node per unit time.

215

Sum of residual energy of nodes in a layer . Number of nodes in the layer

We have conducted two sets of experiments. One set of experiment is to compare simulation results with analytical results considering network lifetime as the parameter. In the other set of experiments, our scheme is compared with three other competing schemes considering both ideal and realistic scenarios. In this set of experiments again energy balance, network lifetime and coverage-connectivity are considered as performance metrics. 3) Comparison of Results (Analytical vs Simulated): In this section, the analytical (Section IV) performance of the scheme in terms of network lifetime is compared with the simulated (Section V) performance and the results are plotted in Fig. 4. Here both the set of results consider ideal scenario including ideal MAC. To be more specific, to make the comparison platform at par, both set of results are plotted considering ideal MAC. We observe from the plot that the nature of graph for the analytical result is perfectly straight irrespective of network sizes whereas the simulation result is fairly straight and that indicates the algorithm has been able to provide almost perfect energy-balanced network lifetime as desired by the theoretical analysis. We also observe that the network lifetime decreases with the increase of network sizes. This is because the data traffic increases while the network size increases, especially for the layers nearer the sink. Finally, the most important observation is, for both the 5-layer (Fig. 4(a)) and 10-layer (Fig. 4(b)) network sizes, analytical results and the average of simulation results are almost same. The slight differences between the analytical and simulated results are due to the minute variation of desired and achieved node densities (refer Fig. 3). 4) Comparison of Results (Competing schemes): This section compares our scheme’s performance with the three competing schemes considering both ideal and realistic scenarios. a) Energy balancing [6] In this section energy balancing of the scheme is evaluated in terms of the following two parameters. (1) ER (Energy consumption rate per node) Fig. 5 shows the ER for different network sizes. We observe that in PDFND, for both ideal and realistic scenarios, the ER for a particular network size is constant for all the layers and this rate varies with network sizes. Precisely, ER increases with increase in network size. For example, in case of ideal scenario, the ER is 1.01 mJ/sec for 5-layer network whereas for 10-layer network it is 1.19 mJ/sec. Similarly, in realistic scenario, these values are 1.21 mJ/sec and 1.45 mJ/sec respectively. On the contrary, in NNDS, NDGD and NDUD it is observed that the ER varies in different layers for a given network size. Further, in NNDS, NDGD and NDUD, irrespective of network size, nodes in layer-1 have maximum ER and nodes in the farthest layer have the lowest ER. Therefore, nodes deployed in the layers nearer the sink drain

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9.5

816.23

For λ' (Simulated)

8.5

For λ (Analytical)

7.5

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816.21 816.19 816.17

P DFND (Ideal) NNDS (Ideal) NDGD (Ideal) NDUD (Ideal)

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P DFND (Real) NNDS (Real) NDGD (Real) NDUD (Real)

683.102 683.098

P DFND (Ideal) NNDS (Ideal) NDGD (Ideal) NDUD (Ideal)

P DFND (Real) NNDS (Real) NDGD (Real) NDUD (Real)

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Fig. 4. Network lifetime for various network sizes (analytical vs simulated).

Fig. 5.

out their energy much more quickly in comparison to nodes deployed in layers farther away from the sink. This justifies our claim that PDFND is relatively more energy balanced compared to all the competing schemes NNDS, NDGD and NDUD. Now for all the schemes, if we compare the results of ideal and realistic scenario, it is observed that ER (realistic) in all the cases is higher compared to ER (ideal). The additional energy usage for realistic scenario is due to the implementation of MAC protocol. Another important observation in realistic scenario is that, ER nearer the sink is less compared to ER away from the sink. As CSMA/CA is used in nodes away from the sink, unlike TDMA, number of collisions, however infrequent, is non-zero and this justifies the above result. In numerical value, for PDFND irrespective of network sizes, the average increase in ER, in realistic scenario compared to the ideal scenario are 19% and 20%, for the nodes nearer the sink and far away from the sink respectively. Similarly these values are 20% and 22% for NNDS, 22% and 24% for NDGD, 5% and 10% for NDUD.

(Fig. 6) nodes in each of the two layers viz. layer-5 and layer6, have drained off completely, though the nodes of other layers in the network have sufficient energy for carrying out normal network operation, causing the phenomenon known as energy hole. Similarly, in NDUD, the energy in nodes of layer-1 has drained off completely though the nodes of other layers in the network have adequate energy for normal network operation. So, both NDGD and NDUD suffer from energy hole problem. In NNDS, the plots upto certain layers starting from the nearest layer from the sink are relatively flat compared to the results in rest of the layers and that implies energy wastage caused by imbalance in energy consumption among the layers. Therefore, NNDS also suffer from energy imbalance problem affecting network lifetime. However, the PDFND plot is almost a straight line indicating that all the nodes in each layer exhaust energy almost completely ending the network lifetime. For example, in case of ideal scenario, it leaves less than 0.2 nJ energy for 5-layer network whereas for 10-layer network it is 0.32 nJ. Similarly, in case of realistic scenario, it leaves less than 0.25 nJ energy for 5-layer network whereas for 10-layer network it is 0.29 nJ. Therefore, we can say that PDFND is energy balanced and utilizes energy, the scarcest resource, more efficiently than the other deployment schemes.

(2) Avg RE per node (Average residual energy per node) Fig. 6 illustrates the comparison considering avg RE per node as a performance metric. We observe that node deployment using NDGD or NDUD results in relatively abrupt change in avg RE per node in each layer and this nature remains independent of network size. For example, in NDGD

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In this section network lifetime is evaluated for various network sizes. The graphs illustrated in Fig. 7 represent the network lifetime for two different network sizes. For ideal scenario, it is observed that the network lifetime of PDFND is 18.28%, 48.40%, and 350% more than that of NNDS, NDGD and NDUD respectively for 5-layer network. For 10-layer network it is 19.83%, 42.30%, and 380% more than that of NNDS, NDGD and NDUD respectively. It is also observed that with increase in network size network lifetime decreases, e.g. for 5layer network it is 816.21 mins whereas for 10-layer network it is 683.06 mins. This is due to the fact that with increase in network size, the nodes in the innermost layer need to relay increased volume of data from the outer layers, thereby, causing higher energy consumption. Moreover, in PDFND the flat nature of the plot ensures that in all the layers, network lifetime terminates in more or less same time as compared to NNDS, NDGD and NDUD. This ensures that energy in PDFND is balanced to a greater extent than all the competent schemes. Now if we compare the simulation results of network lifetime, both for ideal and realistic scenarios, network lifetime is reduced in realistic scenario, as there is additional energy consumption due to the implementation of MAC protocol. Further, in realistic scenario, irrespective of network sizes, the

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reduction in network lifetime is less near the sink compared to other parts of the network. To be more specific, in PDFND when reduction is 19% near the sink, it is 20% in rest of the network. Similarly, in NNDS, NDGD and NDUD these values are 20% & 22%, 22% & 24% and 4% & 10% respectively. As CSMA/CA is used in the entire network area except near the sink, due to collision and retransmission additional energy is consumed compared to near the sink where TDMA is used. From the above observations, it is also revealed that the impact caused by inclusion of realistic scenario on network lifetime

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is the highest in NDGD and the least in NDUD. Although a subset of results on energy balancing and network lifetime was presented in [6], here the entire result set is compared (Figs. 5, 6, 7) with one additional competing scheme NNDS [13]. Moreover, for all the competing schemes including ours, an additional set of results are plotted using real MAC. c) Coverage and connectivity In Fig. 8, we plot coverage density in all the layers for 5layered network. Our primary observation is that except the scheme NDUD, in all the other schemes i.e., PDFND, NDGD and NNDS, coverage density reduces in layers as the distance of layers from the sink increases fulfilling the objective of deploying more nodes near the sink. The next observation is NDGD gives more overlapping sensing coverage in layer-2, and NDUD in layer-5 but NDGD fails to give any overlapping sensing coverage (see Section VI.A.3) in layer-5. To be more specific, in layer-5, coverage density is less than one implying NDGD’s incapability of providing coverage. On the other hand, NDUD provides uniform coverage density in all the layers but that does not provide energy balancing requirement. However, NNDS provides coverage density as per the requirement of energy balancing. Finally, if we compare PDFND with the most competent scheme NNDS, we find that in PDFND coverage density of layers 1, 4, 5 are 74%, 58.36%, 2% more, respectively, than NNDS. However, in layer 2, 3 these values are marginally less (10.66%, 4.86%) than the NNDS. Therefore, we claim that our scheme PDFND not only provides coverage density as per the requirement of energy balancing but also provides higher average coverage density throughout the network compared to the most competing scheme NNDS. VII. C ONCLUSION AND F URTHER W ORK In this work we have proposed a node deployment scheme for multi-hop WSNs using a PDF defined by us. The target of the scheme is to achieve energy balancing and enhancing network lifetime while maintaining coverage and connectivity. To start with, we have analyzed network lifetime and identified node density as a parameter which has significant influence on network lifetime. Then, theoretical formulation of node density for balanced energy consumption is presented. Based on the analysis of network lifetime we have designed a PDF targeting its implementation in lifetime-enhancing node distribution in WSNs. Intrinsic characteristics of the PDF and its suitability for modeling the network architecture of this work are discussed. A node deployment algorithm is also developed based on the proposed PDF to implement the scheme. Further, we have provided theoretical formulation of coverage-connectivity, energy balancing, network lifetime and have derived certain constraints, involving some important network parameters, to be satisfied to achieve the target. We claim that our scheme successfully achieves the target. The claims are substantiated by performing both qualitative and quantitative analyses. Finally, the results of quantitative analysis are compared with three existing works [13], [15], [28] on node deployment and that clearly demonstrates our scheme’s dominance over the existing works.

As a future extension of our work, the deployment strategy may be made more realistic by considering 3-D environment. Moreover, the scheme may be analyzed with a target to obtain optimal node density by considering various QoS parameters. R EFERENCES [1] I. Dietrich and F. Dressler, “On the lifetime of wireless sensor networks,” ACM Trans. Sensor Netw., vol. 5, pp. 1–39, Feb. 2009. [2] S. Halder, A. Ghosal, and S. DasBit, “A pre-determined node deployment strategy to prolong network lifetime in wireless sensor network,” Comput. Commun., vol. 34, pp. 1294–1306, July 2011. [3] M. Younis and K. Akkaya, “Strategies and techniques for node placement in wireless sensor networks: a survey,” Ad Hoc Netw., vol. 6, pp. 621–655, June 2008. [4] A. Liu, X. Jin, G. Cui, and Z. Chen, “Deployment guidelines for achieving maximum lifetime and avoiding energy holes in sensor network,” Inf. Sci., vol. 230, pp. 197–226, May 2013. [5] J. Li and P. Mohapatra, “Analytical modeling and mitigation techniques for the energy hole problem in sensor networks,” Pervasive Mobile Comput., vol. 3, pp. 233–254, June 2007. [6] S. Halder, A. Ghosal, A. Chaudhuri, and S. DasBit, “A probability density function for energy-balanced lifetime-enhancing node deployment in WSN,” in Proc. 2011 LNCS Int. Conf. Computational Sci. Appl., vol. 6018, pp. 472–487. [7] A. K. M. Azad and J. Kamruzzaman, “Energy-balanced transmission policies for wireless sensor networks,” IEEE Trans. Mobile Comput., vol. 10, pp. 927–940, July 2011. [8] C. Song, M. Liu, J. Cao, Y. Zheng, H. Gong, and G. Chen, “Maximizing network lifetime based on transmission range adjustment in wireless sensor networks,” Comput. Commun., vol. 32, pp. 1316–1325, July 2009. [9] A. Boukerche, D. Efstathiou, S. Nikoletseas, and C. Raptopoulos, “Exploiting limited density information towards near-optimal energy balanced data propagation,” Comput. Commun., vol. 35, pp. 2187–2200, Nov. 2012. [10] O. Powell, P. Leone, and J. Rolim, “Energy optimal data propagation in wireless sensor networks,” J. Parallel Distrib. Comput., vol. 67, pp. 302–317, Mar. 2007. [11] A. Jarry, P. Leone, S. Nikoletseas, and J. Rolim, “Optimal data gathering paths and energy-balance mechanisms in wireless networks,” Ad Hoc Netw., vol. 9, pp. 1036–1048, Aug. 2011. [12] J. Cheng, Q. Ye, H. Jiang, D. Wang, and C. Wang, “STCDG: an efficient data gathering algorithm based on matrix completion for wireless sensor networks,” IEEE Trans. Wireless Commun., vol. 12, pp. 850–861, Feb. 2013. [13] X. Wu, G. Chen, and S. K. Das, “Avoiding energy holes in wireless sensor networks with nonuniform node distribution,” IEEE Trans. Parallel Distrib. Syst., vol. 19, pp. 710–720, May 2008. [14] C. Y. Chang and H. R. Chang, “Energy-aware node placement, topology control and MAC scheduling for wireless sensor networks,” Comput.Netw., vol. 52, pp. 2189–2204, Aug. 2008. [15] D. Wang, B. Xie, and D. P. Agrawal, “Coverage and lifetime optimization of wireless sensor networks with Gaussian distribution,” IEEE Trans. Mobile Comput., vol. 7, pp. 1444–1458, Dec. 2008. [16] H. Liu, X. Chu, Y. Leung, and R. Du, “Minimum-cost sensor placement for required lifetime in wireless sensor-target surveillance networks,” IEEE Trans. Parallel Distrib. Syst., vol. 24, pp. 1783–1796, Sep. 2013. [17] J. Luo and J. P. Hubaux, “Joint sink mobility and routing to maximize the lifetime of wireless sensor networks: The case of constrained mobility,” IEEE/ACM Trans. Netw., vol. 18, pp. 871–884, June 2010. [18] H. Ammari and S. Das, “Promoting heterogeneity, mobility, and energyaware Voronoi diagram in wireless sensor networks,” IEEE Trans. Parallel Distrib. Syst., vol. 19, pp. 995–1008, July 2008. [19] K. Lin, M. Chenb, S. Zeadally, and J. J. P. C. Rodrigues, “Balancing energy consumption with mobile sgents in wireless sensor networks,” Future Generation Comput. Syst., vol. 28, pp. 446–456, Feb. 2012. [20] S. Olariu, A. Wadaa, L. Wilson, and M. Eltoweissy, “Wireless sensor networks: leveraging the virtual infrastructure,” IEEE Netw., vol. 18, pp. 51–56, July 2004. [21] F. Barsi, A. A. Bertossi, C. Lavault, A. Navarra, S. Olariu, M. C. Pinotti, and V. Ravelomanana, “Efficient location training protocols for heterogeneous sensor and actor networks,” IEEE Trans. Mobile Comput., vol. 10, pp. 377–391, Mar. 2011. [22] D. Tian and N. D. Georganas, “Connectivity maintenance and coverage preservation in wireless sensor networks,” Ad Hoc Netw., vol. 3, pp. 744–761, Nov. 2005.

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[23] M. Cardei and J. Wu, “Energy-efficient coverage problems in wireless ad-hoc sensor networks,” Comput. Commun., vol. 29, pp. 413–420, Feb. 2006. [24] Available: http://en.wikipedia.org/wiki/Covariance [25] M. Bernard, K. Kondak, I. Maza, and A. Ollero, “Autonomous transportation and deployment with aerial robots for search and rescue missions,” J. Field Robotics, vol. 28, pp. 914–931, Nov. 2011. [26] W. Z. Song, R. Huang, M. Xu, B. A. Shirazi, and R. LaHusen, “Design and deployment of sensor network for real-time high-fidelity volcano monitoring,” IEEE Trans. Parallel Distrib. Syst., vol. 21, pp. 1658–1674, Nov. 2010. [27] X. Han, X. Cao, E. L. Lloyd, and C. C. Shen, “Deploying directional sensor networks with guaranteed connectivity and coverage,” in Proc. 2008 IEEE Int. Conf. Sensing Commun. Netw., pp. 153–160. [28] B. Liu and D. Towsley, “A study of the coverage of large-scale sensor networks,” in Proc. 2004 IEEE Int. Conf. Mobile Ad hoc Sensor Syst., pp. 475–483. [29] G. S. Ahn, E. Miluzzo, A. T. Campbell, S. G. Hong, and F. Cuomo, “Funneling-MAC: a localized, sink-oriented MAC for boosting fidelity in sensor networks,” Tech. Rep. CU/EE/TAP-TR-200608-003. Available: http://www.cs.dartmouth.edu/~sensorlab/funnelingmac/TAPTR-2006-08-003.pdf

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Subir Halder received the B.Tech. degree in electronics and communication engineering and the M.Tech. degree in computer science and engineering from Kalyani Government Engineering College, Kalyani, India, in 2003 and 2006, respectively. He is an Assistant Professor of Computer Science and Engineering at Dr. B. C. Roy Engineering College, Durgapur, India. Currently, he is pursuing the Ph.D. in computer science and technology at Bengal Engineering and Science University, India. His current research interests include network modeling and analysis, performance evaluation and optimization, and wireless sensor networks. He has published research works in reputed conference proceedings and journals in his field. Sipra DasBit is a Professor and Head of the Department of Computer Science and Technology, Bengal Engineering and Science University, Shibpur, West Bengal, India. A recipient of the Career Award for Young Teachers from the All India Council of Technical Education (AICTE), she has more than 20 years of teaching and research experience. She has published many research papers in reputed journals and refereed international conference proceedings. She also has two books and one book chapter on mobile computing to her credit. Her current research interests include wireless sensor networks and mobile computing.