A Nonuniform Sensor Distribution Strategy for Avoiding Energy Holes in Wireless Sensor Networks

Abstract

The energy hole problem exerts great impact on the energy efficiency and lifetime of wireless sensor networks (WSNs) based on many-to-one communication model. Unequal cluster emerged in recent years is a good way to alleviate the energy hole problem by dispersing cluster heads' burden. However, it fails to address this problem fundamentally due to its inherent characteristics. The single non-uniform nodes distribution strategy can alleviate the energy hole problem well by setting more nodes in networks to achieve energy balance, yet it may result in low energy efficiency and high cost of the networks. In this paper, by analyzing and minimizing intra- and inter-cluster energy consumption, we construct a suboptimal unequal cluster for WSNs. We propose a non-uniform sensor distribution strategy based on the previous unequal cluster in accordance with the energy balance principle. Simulation results show that our proposed non-uniform sensor nodes distribution strategy can not only achieve good energy efficiency as the unequal cluster method, but also ensure the network energy consumption balance and resolve the energy hole problem completely as the non-uniform sensor distribution approach. Furthermore, our algorithm needs fewer sensors to be settled than single non-uniform node distribution.

1. Introduction

Advances in MEMS-based sensor technology and wireless communications in recent years have contributed to the development of low-cost, low-power, multifunctional sensor nodes that are small in size with short communication distance and weak computational ability. These small sensor nodes are capable of sensing the environment, storing and processing the collected sensor data, and interacting and collaborating with each other within the network [1]. Due to the characteristics of strict energy constraint and nonrechargeable energy provision, the energy resource of sensor networks should be wisely managed to extend the lifetime of sensors. Since energy consumption is of vital importance for wireless sensor networks, much attention has been paid to low-power hardware design, collaborative signal processing techniques, and energy-efficient algorithms at various WSNs [2, 3].

The main goal of WSNs is to collect useful information as much as possible in the monitoring area, which implies that energy efficiency and lifetime of networks are very important. In order to achieve high energy efficiency and increase network lifetime, sensors are often hierarchically organized into clusters. Within a cluster, each node has its own cluster head (CH) and transmits data to its CH over relatively short distance, which in turn forwards the data (or it is further aggregated) to the sink via a single-hop or a multihop path through other CHs. Previous researches have shown that multi-hop communication between a data source and a data sink is usually more energy efficient than direct transmission due to the characteristics of wireless channel. This method can ensure high energy efficiency. However, the energy hole problem arises when the multi-hop forwarding model is used in intercluster communication. As the cluster heads close to the data sink are burdened with heavy relay traffic, they will die much faster than the other cluster heads, reducing sensing coverage and causing network partitioning [2, 4–7]. Although the strategy of rotating the cluster head role ensures that sensors consume energy in a more consistent manner, the energy hole problem described previously still cannot be eliminated. Experimental results in [4] show that when the network lifetime is over, up to 90% of the total initial energy of the nodes is left unused in uniform sensor distribution WSNs. The unbalanced distribution of communication loads caused by this problem has become a key factor that affects the lifetime of the networks.

In order to solve this problem, many strategies have been proposed. Some proposed clustering algorithms to alleviate this problem by wisely arranging the cluster size. The result is that clusters closer to the base station are expected to have smaller cluster size. By this way, the CHs will consume less energy during the intracluster data processing and preserve more energy for the inter-cluster relay traffic to prevent their premature death. This kind of unequal cluster strategy can alleviate the energy hole problem and achieve better energy efficiency than equal cluster structure in WSNs. But the energy hole problem described previously still cannot be eliminated completely due to its inherent characteristics of many-to-one communication. In nonhierarchical flat structure WSNs, Lian et al. [4] explicitly proposed a nonuniform node distribution strategy to solve the energy hole problem and enhance the data capacity. In [5], the authors also argue that energy depletion balance may be achieved by using the non-uniform node distribution method. Wu et al. [8] proposed a non-uniform node distribution method to address the energy hole problem. These single nonuniform node distribution strategies can solve the energy hole problem but need more nodes to be deployed for sharing the task of data relay without any data aggregation, thus resulting in lower energy efficiency and high cost.

In this paper, by combining the advantages of both unequal cluster and non-uniform node distribution approach, we develop a non-uniform sensor distribution strategy based on unequal cluster to eliminate the energy hole problem. We adopt the corona model of WSNs and divide energy consumption in each cluster into two parts including intra- and intercluster energy consumption. By analyzing the energy consumption relationship between intra-cluster data processing and inter-cluster data forwarding, we derive the expression of the per node energy consuming rate (ECR) based on the distance from clusters to the sink. Through ECR model, we find that if the cluster size (the same as corona width) is reasonably arranged, ECR will be minimized, which means minimal average energy consumption of nodes and longer network lifetime. With the minimal ECR in each corona, we deduce the non-uniform sensor distribution method.

In our strategy, nodes with different density are placed in different coronas according to the previous non-uniform sensor distribution results. If we let   $\hspace{0.17em}{\rho }_{\min }$ denote the lowest node density that can meet the requirement of networks coverage and connectivity, the set of sensors with density higher than ${\hspace{0.17em}\hspace{0.17em}\rho }_{\min }$ in some coronas will be selected to be put into sleep mode, which means no participation in data sensing and forwarding tasks. All active nodes are organized into unequal clusters. When some nodes die prematurely due to the energy hole problem, the redundant sleeping nodes will wake up in time to ensure the normal operation of networks [9]. Our goal is to ensure that there is nearly no residual energy in networks when networks lose information collection and transmission abilities, rather than that all settled sensors use up their energy at the same time. Theoretical analysis and simulation results show that our non-uniform sensor distribution strategy based on unequal cluster can obtain both the energy efficiency of unequal hierarchy cluster and achieve the energy consumption balance of non-uniform sensor distribution for WSNs. The energy hole problem can be resolved completely using the least sensor nodes. Moreover, the lifetime of network can be longer compared with single unequal cluster protocol or non-uniform node distribution strategy.

The remainder of the paper is organized as follows. Section 2 covers related researches in this area; Section 3 introduces the system assumptions used throughout this paper; Section 4 establishes and analyzes the ECR model of sensor node with intra- and inter-cluster communication and introduces the calculation algorithm for non-uniform sensor distribution strategy; Section 5 elaborates on our simulation efforts and the analysis of the results obtained; Section 6 offers concluding remarks and points out research directions in the future.

2. Literature Review

In recent years, a large number of papers have been published on how to extend the network lifetime and solve energy hole problem for WSNs. Up till now, many algorithms about hierarchical cluster and nodes distribution have been proposed for addressing them. In the following part, we will give a brief review to the most important researches and findings related to our approach.

Li and Mohapatra [10] initiated the energy hole problem study in a large many-to-one sensor network. They described the energy hole in a corona model and defined the per node traffic load and the per node energy consuming rate (ECR), both of which are used in our paper. Based on the observation for ECR in each corona, they proved that nodes in inner coronas consume energy much faster and have shorter lifetime. They developed a mathematical model to analyze the energy hole problem and proved that hierarchical deployment and data compression have a positive effect in a uniformly distributed sensor network. Olariu and Stojmenović [11] are the first researchers to study the issue of whether energy hole can be avoided from a theoretical perspective. Assuming a wireless sensor network with uniform node distribution and uniform data reporting functions, they proposed an energy model to analyze the relationship between the network lifetime and the width of each corona in concentric corona model. By further assuming that the transmission range of sensor is adjustable, they demonstrated that when all the coronas have the same width, the energy consumed by routing can be minimized. Moreover, they points out conditions for avoiding the unbalanced energy depletion problem.

The major purpose for eliminating the energy hole problem is to enhance the network energy efficiency and prolong the lifetime. In flat WSNs, many authors [9, 12, 13] adopted the strategy of adjusting the transmission power of nodes to avoid the energy hole problem. By assigning different transmission radii according to the distance from sensors to sink, energy hole problem can be alleviated. Considering the energy consumption distribution for single-hop and multihop communication to the sink, Perillo et al. [14] proposed an alternate mode between multihop and singlehop to achieve energy consumption balance. They calculated the optimization of network lifetime in a linear programming problem. But the authors only consider that nodes use up their energy simultaneously without thinking of using energy efficiently to collect useful data. For hierarchical network algorithms based on cluster architecture, Heinzelman et al. [6] proposed the LEACH algorithm. Due to the communication with the base station by single hop, lots of energy is consumed in the long-distance communication. Many other cluster-based hierarchy algorithms are improved from LEACH [15–20]. Benefiting from data aggregation for redundant sensing data and the decrease of communication distance among nodes, hierarchical network algorithms can achieve better energy efficiency and energy consumption balance than flat algorithms. The energy hole problem is alleviated to some extent but is far from being solved.

To avoid the energy hole problem in multi-hop communication WSNs, unequal cluster concept which is different from the general equal hierarchical cluster algorithms has been adopted to extend the network lifetime in these years [21–24]. Soro and Heinzelman [21] investigated firstly an unequal clustering size model (UCS) to balance the energy consumption of cluster heads in multi-hop heterogeneous WSNs. Through both theoretical and experimental analyses, they proved that unequal cluster could be useful, especially for heavy traffic applications. EEUC [24, 25] has been proved as an efficient algorithm to address energy hole problem. In EEUC, clusters closer to the base station are smaller in size than those farther away from sink; thus, cluster heads closer to the base station can save some energy for forwarding inter-cluster data. This algorithm works well in balancing the energy consumption among cluster heads and slowing down their premature death. However, due to the inherent characteristics of a many-to-one communication, energy hole problem cannot be eliminated completely.

Another strategy to avoid the energy hole is the node density control, which can balance the energy consumption effectively. Lian et al. [4] proposed the non-uniform node distribution strategy in wireless sensor networks. The energy hole is caused by massive energy consumption near the sink. The energy hole problem can be solved completely by deploying more nodes near the sink to relay the distant data. Wu et al. [26] proposed a non-uniform node distribution strategy to achieve the subbalanced energy depletion. The authors point out that if the number of nodes in every corona increases in geometric progression with a predetermined ratio, the network can achieve balanced energy depletion. Olariu and Stojmenović [11] discussed the non-uniform node distribution strategy in wireless sensor networks. Assuming an energy consumption model in which only energy consumption for data transmission is considered, they proved that balanced energy depletion can be achieved when the node density of each corona is arranged proportionally. In their scheme, nodes near the sink have to send data at a lower rate. Compared with the hierarchical cluster algorithms, single node density control strategy can guarantee simultaneous energy depletion of all sensors but will result in lower energy efficiency and high cost because of the massive redundant data transmission and redundant node distribution.

3. Preliminaries

Before elaborating on our algorithm, we will introduce the characteristics of the network model used in our study. We consider a WSN consisting of sensors and sink and make the following assumptions.

3.1. Assumptions on Node and Energy of the Network

We consider a sensor network with N nodes in a circular area within a radius of R to continuously monitor the sensing area. We denote ith sensor by $v_i$ and the corresponding sensor node set only $V=\{v_1,v_2,\dots ,v_k\}$ where $\hspace{0.17em}\hspace{0.17em}|V|=N$ . The sink is located at the center as shown in Figure 1. Our assumptions about the sensor nodes and the network model are as follows.

• (1)

Each sensor has the maximum transmission range denoted by $r_{\max }$ , and the sink node and all sensors are stationary after disposition.

(2)

Nodes can estimate the approximate distance to another node based on the received signal strength.

(3)

Nodes can use power control to adjust the transmission power according to the distance to the receiver.

(4)

The links between nodes are symmetric, and time division multiple access (TDMA) scheduled data transmission from normal nodes to its cluster head.

(5)

The network runs a periodic data gathering application. The sensor generates traffic at an average rate of λ bits/second and sends it to its CH, which in turn delivers it to the sink using multi-hop communication.

Figure 1

Sensor network area consisting of coronas.

A typical sensor node includes three basic units: sensing unit, processing unit, and transceivers. For our energy model of multi-hop forwarding scheme, we assume a free space propagation channel model [23]. We ignore the power consumption of node for sensing because it is constant at any time and cannot be reduced with whatever means. Thus, the energy model of a sensor involves the power for data aggregating, data receiving, and data transmission according to this radio hardware energy dissipation in both the free space ( $d^2\hspace{0.17em}\hspace{0.17em}$ Power loss) and the multipath fading ( $d^4\hspace{0.17em}\hspace{0.17em}$ Power loss) channel models. If the distance is shorter than the threshold $d_0$ , the free space (FS) model is used; otherwise, the multi-path (MP) model is used. If only $\hspace{0.17em}\hspace{0.17em}{E}_{\text{Tx}}$ denotes the energy consumption for transmitting data, $E_{\text{Rx}}$ the energy consumption for receiving data, and $E_{\text{Ag}}$ the energy consumption of aggregating data, the energy for transmitting, receiving, and aggregating l bits data over distance d can be calculated as follows:

$$\begin{array}{cc}& E_{\text{Tx}}=\{\begin{array}{cc}{E}_{\text{Tx}}=\left(e_1+e_2^{*}{d}^2\right)l,\hspace{1em}d<d_0,\hfill & \hfill \\ E_{\text{Tx}}=\left(e_1+e_4^{*}{d}^4\right)l,\hspace{1em}d\ge d_0,\hfill & \hfill \end{array}\\ E_{\text{Rx}}=e_{1}l,\\ E_{\text{Ag}}=e_{3}l.\end{array}$$

(1)

Here, the electronics energy $e_1$ depends on factors such as the digital coding and modulation. The amplifier energies $e_2$ and $e_4$ are the energies required for power amplification in the two radio models, respectively. $e_3$ is the energy consumption for dealing with the aggregation of unit sensor data. Some typical values for the parameters above in current sensor technologies are as follows: $\hspace{0.17em}\hspace{0.17em}{e}_1=50\hspace{0.17em}\text{nJ}/\text{bit}$ , $e_2=10\hspace{0.17em}\text{pJ}/\text{bit}/{\text{m}}^2$ , $e_3=5\hspace{0.17em}\text{nJ}/\text{bit}/\text{signal},\hspace{0.17em}\hspace{0.17em}$ and ${\hspace{0.17em}\hspace{0.17em}e}_4=0.0013\hspace{0.17em}\text{pJ}/\text{bit}/{\text{m}}^4$ .

3.2. The Sensor Network Model Based on Corona

We adopt the corona model in this paper. The area around sink is divided into coronas of a dynamic width as illustrated in Figure 1. We assume that the sink has a steady energy supply and a powerful radio that can cover the whole monitoring area. K concentric circles of radius $0<r_1<r_2<\cdots <r_k=R$ are centered at the sink with the corresponding node distribution density ${\hspace{0.17em}\hspace{0.17em}\rho }_1>{\rho }_2>\cdots >{\rho }_k={\rho }_{\min }$ . The width of corona $C_i$ is ${\mathrm{\Delta }}_i$ which is delimited by the circles of radius $r_{i-1}$ and $r_i$ (equivalent to ${\hspace{0.17em}\hspace{0.17em}r}_i-r_{i-1}$ ). Sensors can adjust their transmission ranges to save energy. All the sensors are deployed in such a way that reliable communication between sensors in adjacent coronas can be guaranteed, and the width of each corona does not exceed the sensor's maximum transmission range $\hspace{0.17em}\hspace{0.17em}{r}_{\max }$ .

In many other unequal cluster algorithms like EEUC, the cluster's size far from the sink is always bigger than the cluster's size near the sink, so it is assumed that a sensor in $C_i$ corona uses a transmission radius of ${\mathrm{\Delta }}_i$ to reach a sensor in $C_{i-1}$ corona. This assumption can ensure effective communication between adjacent cluster heads with the cluster sizes being strictly reduced. In this paper, we do not define such a prerequisite for transmission radius of cluster head. Furthermore, we find that the cluster size will not always decrease as the distance from cluster to the sink is shortened under the premise that ECR of each corona is minimized. To ensure effective data transmission in hop-by-hop communication, for instance, in C _i corona in our model, we select the maximum width between corona $C_i$ and corona $C_{i-1}$ as the sensor transmission radius to ensure effective communication with each sensor in corona $C_{i-1}$ . Cosider

$$\begin{array}{cc}& t_{ix}=\max \left\{{\mathrm{\Delta }}_i,{\mathrm{\Delta }}_{i-1}\right\}.\end{array}$$

(2)

We have the following assumptions about the clustering approach in each corona: sensors whose distance to the sink is in $(r_{i-1},r_i]$ are organized into clusters to cover the corona $C_i$ . A sensor located in the corona $C_i$ is assigned to the nearest CH in the same corona. All sensors are organized into clusters, and their data is relayed by the closest CH in the adjacent corona to the sink in multi-hop communication. Such a corona-based model enables us to analyze theoretically the relationship among ECR, the traffic volume relayed by CH, and the distance from cluster to the sink.

4. Nonuniform Sensors Distribution Strategy

As mentioned previously, when WSNs are sensing and collecting data, the redundant distribution sensor nodes are in the sleep mode. They turn off most of their components and only run a timer circuitry to listen to the channel. Hence, energy consumption of sensors in the sleep mode can be ignored. The active nodes are distributed uniformly with the minimal node density ${\hspace{0.17em}\hspace{0.17em}\rho }_{\min }$ in the network. Based on this, the optimal ECR of each corona is calculated in uniform node distribution network. In the following part of this chapter, we will analyze the energy consumption for communication in cluster and construct the optimal unequal cluster network. Then, we will create a mathematical model for ECR and calculate its minimal value. Finally, we will derive the non-uniform node distribution strategy based on unequal cluster according to the optimal ECR in each corona.

4.1. ECR Model Based on Intra- and Intercluster Energy Consumption

In this section, we will analyze the energy consumption in cluster and deduce the mathematical model among the ECR, the cluster size, and the distance from node to sink. We formulate the suboptimal cluster size calculation approach in each corona based on the ECR. Instead of considering the energy consumption balance in every corona directly [3], we optimize the ECR in each corona. Let $N_i$ denote the number of nodes in corona $C_i$ and $E_i$ , the energy consumed per unit time by all nodes in it. We can divide the energy consumption in corona $C_i$ into two parts: the intra- and inter-cluster energy consumption. Let $E_{\text{intra}\_i}$ denote the intra-cluster energy consumption for local sensing, information processing, and communicating, and   $E_{\text{inter}\_i}$ the inter-cluster energy consumption used for relaying the traffic data from the outside coronas. According to the assumptions and previous network model, $E_i$ can be calculated by

$$\begin{array}{cc}& E_i=E_{\text{intra}\_i}+E_{\text{inter}\_i}.\end{array}$$

(3)

Let $p_i$ denote the probability that the nodes become CHs in corona $C_i$ . Like many distance-based cluster formation algorithms, we assume that each CH is located at the center of its cluster. Then, we can calculate $E_{\text{intra}\_i}$ as below:

$$\begin{array}{cc}& E_{\text{intra}\_i}=N_i{p}_i{E}_{\text{Cluster}\_i}.\end{array}$$

(4)

In this equation, $E_{\text{Cluster}\_i}$ is the energy consumption per second for one cluster in corona $C_i$ and can be calculated as follows [27]:

$$\begin{array}{cc}& E_{\text{Cluster}\_i}=E_{\text{CH}\_i}+E_{\text{nonCH}\_i}.\end{array}$$

(5)

In the equation, $E_{\text{CH}\_i}$ is the energy used by CH to receive data from the member nodes, aggregate the data, and transmit the aggregate data to CHs in the next corona. $E_{\text{nonCH}\_i}$ is the energy used by each noncluster head node to transmit its data to the cluster head during a unit time. $E_{\text{CH}\_i}$ and $E_{\text{nonCH}\_i}$ can be given by

$$\begin{array}{cc}& E_{\text{CH}\_i}=\left(\frac{1}{p_i}-1\right)e_1\lambda +\frac{e_3\lambda }{p_i}+\frac{\alpha \lambda \left({\mathrm{\Delta }}_i^2{e}_2+e_1\right)}{p_i},\\ E_{\text{nonCH}\_i}=\left(e_1+{\mathrm{\Delta }}_i^2\mathrm{\hspace{0.17em}\hspace{0.17em}}{e}_2\right)\lambda ,\end{array}$$

(6)

where α denotes the aggregation coefficient for all CHs. Substituting (5), (6) into (4), the expected power consumption for intracluster of all CHs in the corona $C_i$ can be given by

$$\begin{array}{cc}& E_{\text{intra}\_i}=N_i\lambda \left\{\left[2\left(1-p_i\right)+\alpha \right]e_1+\left(1-p_i+\alpha \right){\mathrm{\Delta }}_i^2\mathrm{\hspace{0.17em}\hspace{0.17em}}{e}_2+e_3\right\}.\end{array}$$

(7)

As the sensing data relay from the outside coronas is done hop by hop, the total traffic load carried by the CHs in the corona $C_i$ is equal to the total traffic volume originating from all clusters in corona $i+1$ to K. So $E_{\text{inter}\_i}$ for all nodes in the corona $C_i$ can be given approximately by:

$$\begin{array}{cc}& E_{\text{inter}\_i}={\rho }_{\min }\pi \left(R^2-r_i^2\right)\left(2e_1+e_2{\mathrm{\Delta }}_1^2\right)\alpha \lambda .\end{array}$$

(8)

Let $E_{\text{node}\_i}$ denote the ECR in the corona $C_i$ ; it can be given by

$$\begin{array}{cc}& E_{\text{node}\_i}=\frac{E_i}{N_i}.\end{array}$$

(9)

Accordingly, the number of active nodes in corona $C_i$ can be expressed by ${\hspace{0.17em}\hspace{0.17em}\rho }_{\min }\pi (2r_i{\mathrm{\Delta }}_i-{\mathrm{\Delta }}_i^2)$ ; substituting (7), (8), and (3) in (9), $E_{\text{node}\_i}$ is more specifically given by

$$\begin{array}{l}{E}_{\text{node}\_i}\\ \hspace{1em}=\lambda \left\{\left[2\left(1-p_i\right)+\alpha \right]e_1+\left(1-p_i+\alpha \right){\mathrm{\Delta }}_i^2\mathrm{\hspace{0.17em}\hspace{0.17em}}{e}_2+e_3\right\}\\ \phantom{\text{hhhh}}+\alpha \lambda \mathrm{\hspace{0.17em}\hspace{0.17em}}\frac{\left(R^2-r_i^2\right)\left(2e_1+e_2{\mathrm{\Delta }}_i^2\right)}{2r_i\mathrm{\hspace{0.17em}\hspace{0.17em}}{\mathrm{\Delta }}_i-{\mathrm{\Delta }}_i^2\mathrm{\hspace{0.17em}\hspace{0.17em}}}.\end{array}$$

(10)

In [27], the author has proved that approximate $2\pi r_i/{\mathrm{\Delta }}_i$ CHs are needed to cover corona $C_i$ at least in corona model. Then we can get the CH election probability $p_i$ as follows:

$$\begin{array}{cc}& p_i=\frac{2r_i\mathrm{\hspace{0.17em}\hspace{0.17em}}}{{\rho }_{\min }\left(2r_i{\mathrm{\Delta }}_i^2-{\mathrm{\Delta }}_i^3\right)}.\end{array}$$

(11)

For simplicity, we set λ = 1 bit/second which will not affect the results of our analysis. Substituting (11) into (10), we can simplify $E_{\text{node}\_i}$ as follows:

$$\begin{array}{l}{E}_{\text{node}\_i}=[\phantom{\frac{4e_1{r}_i}{{\rho }_{\min }\left(2r_i{\mathrm{\Delta }}_i^2-{\mathrm{\Delta }}_i^3\right)}}\left(2e_1+\alpha e_1+e_3\right)+\left(e_2+\alpha e_2\right){\mathrm{\Delta }}_i^2\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-\frac{4e_1{r}_i}{{\rho }_{\min }\left(2r_i{\mathrm{\Delta }}_i^2-{\mathrm{\Delta }}_i^3\right)}-\frac{2e_2\mathrm{\hspace{0.17em}\hspace{0.17em}}{r}_i}{{\rho }_{\min }\left(2r_i-{\mathrm{\Delta }}_i\right)}]\\ \hspace{1em}\hspace{1em}\hspace{1em}+\alpha \mathrm{\hspace{0.17em}\hspace{0.17em}}\frac{\left(R^2-r_i^2\right)\left(2e_1+e_2{\mathrm{\Delta }}_i^2\right)}{2r_i{\mathrm{\Delta }}_i-{\mathrm{\Delta }}_i^2}.\end{array}$$

(12)

In order to further study the relationship of ECR between intra- and inter-cluster communications, we will split $E_{\text{node}\_i}$ into two parts. Let $E_{\text{ave}\_\text{intra}\_i}$ denote the node's average energy consumption rate in intra-cluster communication and $E_{\text{ave}\_\text{inter}\_i}$ the node's average energy consumption rate in inter-cluster communication. Without loss of generality, we assign a reasonable value to α, for example α = 0.1. Obviously, we have ${\mathrm{\Delta }}_i\le r_i$ ( ${\mathrm{\Delta }}_i=r_i$ if and only if $i=1$ ). According to (5), (10), and (11), $E_{\text{ave}\_\text{intra}\_i}$ and $E_{\text{ave}\_\text{inter}\_i}$ can be calculated as folows:

$$\begin{array}{l}{E}_{\text{ave}\_\text{intra}\_i}=2e_1+\alpha e_1+e_3+\left(e_2+\alpha e_2\right){\mathrm{\Delta }}_i^2\\ \phantom{\text{aaaaaaaaaaa}}-\frac{4e_1{r}_i}{{\rho }_{\min }\left(2r_i{\mathrm{\Delta }}_i^2-{\mathrm{\Delta }}_i^3\right)}-\frac{2e_2{r}_i}{{\rho }_{\min }\left(2r_i-{\mathrm{\Delta }}_i\right)}\end{array}$$

(13)

$$\begin{array}{l}{E}_{\text{ave}\_\text{inter}\_i}=\alpha \frac{\left(R^2-r_i^2\right)\left(2e_1+e_2{\mathrm{\Delta }}_i^2\right)}{2r_i{\mathrm{\Delta }}_i-{\mathrm{\Delta }}_i^2}.\end{array}$$

(14)

4.2. The ECR Model Analysis

It can be seen from (10) that the data aggregation coefficient α and the cluster head probability $p_i$ always affect the node's average energy consumption rate. When α varies from 1 to 0 and cluster heads compress data more efficiently, the node's average energy consumption gets smaller and the energy hole of network can be alleviated as proved by many researchers. Equivalently, when the probability of being a cluster head $p_i$ increases, the energy used for intra-cluster communication will decrease due to shorter communication distance between nodes and their cluster heads. The data aggregation and unequal hierarchical cluster protocol can effectively reduce the ECR in WSNs, which can lead to higher energy efficiency and longer network lifetime. This is why non-uniform sensor distribution strategy based on unequal cluster is better than general non-uniform sensor distribution method.

Next, we will analyze the relationship among $E_{\text{node}\_i}$ , ${\mathrm{\Delta }}_i$ , and $r_i$ . To enhance the energy efficiency and prolong the network lifetime, we need to define the optimal ${\mathrm{\Delta }}_i$ to minimize ECR in corona $C_i$ . We assume that a WSN covers a circular sensing region with R = 200 meters and $P_{\min }$ = 0.00318, which implies that 400 sensors are uniformly distributed in this monitoring area. By studying the nodes' energy consumption rate with different cluster radius ${\mathrm{\Delta }}_i$ based on fixed $r_i$ through simulations, we find that ECR can be minimized for each constant distance from cluster to the sink when an appropriate value is assigned to cluster size ${\mathrm{\Delta }}_i$ . Our analytical findings can be found in the following simulation tests.

Figures 2, 3, 4, and 5 demonstrate the changes for ECR, average energy consumption rate for intra-cluster data processing and inter-cluster communication versus cluster size ${\mathrm{\Delta }}_i$ when $r_i$ is, respectively, defined as 35 meters, 60 meters, 90 meters, and 150 meters. As shown in Figures 2 and 3, when the upper bound of corona $C_i$ is closer to sink ( $r_i$ is small), $E_{\text{node}\_i}$ will be dominated by $E_{\text{ave}\_\text{inter}\_i}$ as ${\mathrm{\Delta }}_i$ increases. The reason is that the nodes' energy is mainly used for relaying the traffic data coming from the outside coronas, and inter-cluster communication is the key factor to cause the energy hole in this scenario. On the contrary, $E_{\text{node}\_i}$ expands rapidly as ${\mathrm{\Delta }}_i$ decreases. This result proves that it is not always the best option to reduce the sizes of clusters that are closer to the sink, even though it has been adopted in many existing unequal cluster protocols by many researchers. The cluster size cannot be defined as too small when the cluster is close to sink.

Figure 2

Node energy consumption rate versus cluster radius ${\hspace{0.17em}\hspace{0.17em}\mathrm{\Delta }}_i$   ( $r_i$ = 35 meters).

Figure 3

Node energy consumption rate versus cluster radius   ${\mathrm{\Delta }}_i$ ( $r_i$ = 60 meters).

Figure 4

Node energy consumption rate versus cluster radius    ${\mathrm{\Delta }}_i$   ( $r_i$ = 90 meters).

Figure 5

Node energy consumption rate versus cluster radius    ${\mathrm{\Delta }}_i$   ( $r_i$ = 150 meters).

As shown in Figures 4 and 5, when the upper bound of corona $C_i$ is away from sink ( $r_i$ is big), we can see that $E_{\text{ave}\_\text{intra}\_i}$ will determine the value of $E_{\text{node}\_i}$ as ${\mathrm{\Delta }}_i$ increases. This phenomenon can be explained as follows: when corona $C_i$ is away from sink, the data traffic that needs to be relayed will decrease, which means that the energy consumption for inter-cluster communication is reduced. The energy used for intra-cluster communication becomes the major factor to determine the value of $E_{\text{node}\_i}$ . But it is worth mentioning that the cluster size increase in these coronas will not always reduce the value of ECR. This result shows that it is not always the best way to save energy by simply increasing the size of clusters away from the sink. It is of no help to alleviate the energy consumption rate in inner coronas but increases their own energy consumption. The author has proved [28] that the network lifetime is determined by nodes in the innermost corona in multi-hop communication no matter how the clusters in outer corona in a uniform network are adjusted and organized. Furthermore, we can see that there is an optimal cluster size ${\mathrm{\Delta }}_i$ for each $r_i$ to minimize ECR as indicated in Figures 2–5.

Figure 6 illustrates the change of $E_{\text{node}\_i}$ versus the distance $r_i$ from sink to coronas and unequal cluster size ${\mathrm{\Delta }}_i$ in the whole sensing area. In practical applications, we have the limited condition ${\mathrm{\Delta }}_i\le r_i$ . From the simulation result, we can see that the curved surface of $E_{\text{node}\_i}$ is given with an irregular concave surface. There is an extreme small point for ECR with a group of corresponding $r_i$ and ${\mathrm{\Delta }}_i$ . At the same time, with $r_i$ increasing from 0 to 200 meters, the unequal cluster's size ${\mathrm{\Delta }}_i$ will increase firstly and then decrease when $E_{\text{node}\_i}$ is minimized. It proves that the cluster will not always become smaller in size as its distance to the sink decreases, under the precondition that ECR in unequal clusters is minimized. Based on such results, we can calculate a group of optimal ${\mathrm{\Delta }}_i$ and divide the monitoring area into several coronas in which ECR is minimized, under the premise that the network parameters are fixed. In other words, the unequal cluster can also been constructed. With the minimal ECR in each corona, the network can obtain longer lifetime and high energy efficiency, with the energy hole problem being further alleviated.

Figure 6

Node energy consumption rate versus the distance form sink $r_i$ and cluster size ${\mathrm{\Delta }}_i$ .

4.3. The Computation for Sub-optimal Unequal Cluster and Minimal ECR

From the previous analysis, we can calculate a group of optimal ${\mathrm{\Delta }}_i$ to achieve minimal ECR. In the following section, we will give the optimal algorithm to determine the minimal ECR and the optimal ${\mathrm{\Delta }}_i$ of each corona. Within each corona, we can construct the unequal hierarchical cluster. Then, we can calculate the number of sensors that need to be distributed in every corona, by following the principle that all nodes will use up their energy when network is no longer able to collect and transmit information.

We notice that the cluster size is equal to the corona width in corona $C_1$ ; hence ${\mathrm{\Delta }}_1=r_1$ . Then, according to (11), the ECR in corona $C_1$ can be calculated with the following formula:

$$\begin{array}{l}{E}_{\text{node}\_1}=2e_1+\alpha e_1+e_3-\frac{2e_2}{{\rho }_{\min }}+\left(e_2+\alpha e_2\right){\mathrm{\Delta }}_1^2\\ \phantom{\text{aaaaaaaa}}-\frac{4e_1}{{\rho }_{\min }{\mathrm{\Delta }}_1^2}+\alpha \frac{\left(R^2-{\mathrm{\Delta }}_1^2\right)\left(2e_1+e_2{\mathrm{\Delta }}_1^2\right)}{{\mathrm{\Delta }}_1^2}.\end{array}$$

(15)

In (15), ${\mathrm{\Delta }}_1$ is the only unknown parameter for the function $E_{\text{node}\_1}$ if the monitoring area radius R is predetermined. Figure 7 illustrates the change of $E_{\text{node}\_1}$ with cluster size ${\mathrm{\Delta }}_1$ when $R=200\hspace{0.17em}\text{m}$ . As can be seen from this curve, there is an optimal cluster size to minimize the per node energy consumption rate. If $R=200$ meters, we can find that ECR in corona $C_1$ gets the minimal value when ${\mathrm{\Delta }}_{\text{opt}\_1}$ $=32.16\hspace{0.17em}\text{m}$ , according to the numeric computation in (15). Through further analysis, we can find that once the radius of monitoring area is defined, the corresponding optimal cluster size ${\mathrm{\Delta }}_{\text{opt}\_1}$ can be obtained. With this optimal ${\mathrm{\Delta }}_{\text{opt}\_1}$ , the ECR in innermost corona is minimized, and the network lifetime can be maximized.

Figure 7

Node energy consumption rate in corona $C_1$ versus cluster size ${\mathrm{\Delta }}_i$ .

After ${\mathrm{\Delta }}_1$ is obtained, we begin to calculate other corona width (cluster size) through the following iterative algorithm. We have the relation

$$\begin{array}{cc}& r_i=r_{i-1}+{\mathrm{\Delta }}_i.\end{array}$$

(16)

Submitting (16) into (12), we have

$$\begin{array}{l}{E}_{\text{node}\_i}=\left(2e_1+\alpha e_1+e_3\right)+\left(e_2+\alpha e_2\right){\mathrm{\Delta }}_i^2\\ \hspace{1em}\hspace{1em}\hspace{1em}-\frac{4e_1\left(r_{i-1}+{\mathrm{\Delta }}_i\right)}{{\rho }_{\min }\left(2r_{i-1}{\mathrm{\Delta }}_i^2+{\mathrm{\Delta }}_i^3\right)}-\frac{2e_2\left(r_{i-1}+{\mathrm{\Delta }}_i\right)}{{\rho }_{\min }\left(2r_{i-1}+{\mathrm{\Delta }}_i\right)}\\ \hspace{1em}\hspace{1em}\hspace{1em}+\alpha \frac{\left[R^2-{\left(r_{i-1}+{\mathrm{\Delta }}_i\right)}^2\right]\left(2e_1+e_2{\mathrm{\Delta }}_i^2\right)}{2r_{i-1}{\mathrm{\Delta }}_i+{\mathrm{\Delta }}_i^2}.\end{array}$$

(17)

Differentiating (17) for ${\mathrm{\Delta }}_i$ , $E_{\text{node}\_i}$ is minimized by the value of ${\mathrm{\Delta }}_i$ that is a solution of

$$\begin{array}{cc}& A{\mathrm{\Delta }}_i^6+B{\mathrm{\Delta }}_i^5+C{\mathrm{\Delta }}_i^4+D{\mathrm{\Delta }}_i^3+E{\mathrm{\Delta }}_i^2+F{\mathrm{\Delta }}_i+G=0.\end{array}$$

(18)

In (18)

$$\begin{array}{c}A=-6\alpha {\rho }_{\min }{e}_2,\\ B=\alpha {\rho }_{\min }\left(-10e_2{r}_{i-1}-10e_3\right),\\ D=\alpha {\rho }_{\min }\left(4R^2{e}_2-4e_2{r}_{i-1}^2-16e_3{r}_{i-1}-8e_1\right)-4e_2,\\ C=\alpha {\rho }_{\min }\left(6R^2{e}_2{r}_{i-1}-6e_2{r}_{i-1}^3-12e_1{r}_{i-1}-12e_1\right)\\ \phantom{\text{llllll}}-6e_2{r}_{i-1},\\ E=\alpha {\rho }_{\min }\left(2R^2-2r_{i-1}^2-16e_1{r}_{i-1}\right)-16e_1,\\ F=\alpha {\rho }_{\min }\left(2R^2{r}_{i-1}-2r_{i-1}^3\right)-36e_1{r}_{i-1},\\ G=-16e_1{r}_{i-1}^2.\end{array}$$

(19)

Obviously, if $r_{i-1}$ is a known number, ${\mathrm{\Delta }}_i$ will be the only unknown parameter in (19). According to the Galois theory [28], the root of (18) cannot be obtained by elementary algebra. However, we can use numeric solutions to calculate the roots of general polynomial equation. Define

$$\begin{array}{cc}& f\left(x\right)=A{x}^6+B{x}^5+C{x}^4+D{x}^3+E{x}^2+Fx+G.\end{array}$$

(20)

Then,

$$\begin{array}{cc}& f^{\prime }\left(x\right)=6A{x}^5+5B{x}^4+4C{x}^3+3D{x}^2+Ex+F,\\ f^{\mathrm{\prime \prime }}\left(x\right)=30A{x}^4+20B{x}^3+12C{x}^2+6Dx+E.\end{array}$$

(21)

It is obvious that $f(x)$ , $\hspace{0.17em}\hspace{0.17em}{f}^{\prime }(x)$ , and $f^{\mathrm{\prime \prime }}(x)$ are continuous functions in the whole real domain. Assume that there is a number $q\in [\mathrm{0,1}]$ , where $f(q)=0$ . According to Newton-Raphson theorem, if $f^{\prime }(q)\ne 0$ , there is a $\delta >0$ and for any initial approximation $q_0$ , $q_0\in [q-\delta ,q+\delta ]$ , the sequence $\{q_k{\}}_{k=0}^{\infty }$ defined by the iteration

$$\begin{array}{cc}& q_{k+1}=q_k-\frac{f\left(q_k\right)}{f^{\prime }\left(q_k\right)}\hspace{1em}\left(k=\mathrm{0,1},2,\dots \right)\end{array}$$

(22)

will converge to q. Thus, given an initial value for $x_0$ , for example, $x_0=0.01$ , we start the iteration based on $x_n=x_{n-1}-f(x_{n-1})/f^{\prime }(x_{n-1})$ to update $x_n$ until the difference between $x_n$ and $x_{n-1}$ is smaller than the certain reservation threshold. Consequently, we can get the approximate optimal cluster size ${\mathrm{\Delta }}_{\text{opt}\_i}=x_n$ . Because all sensors have the maximum transmission range $r_{\max }$ and the restriction of ${\mathrm{\Delta }}_i\le r_{\max }$ , we have ${\mathrm{\Delta }}_i=r_{\max }$ when ${\mathrm{\Delta }}_{\text{opt}\_i}\ge r_{\max }$ . Substituting ${\mathrm{\Delta }}_{\text{opt}\_i}$ and $r_i$ into (11), we can get the minimal value of $E_{\text{node}\_i}$ in corona $C_i$ .

The values of ${\mathrm{\Delta }}_{\text{opt}\_i}$ and $r_1$ in corona $C_1$ that we have obtained are the initial conditions for our iterative algorithm. For the relation $r_2=r_1+{\mathrm{\Delta }}_{\text{opt}\_2}$ , if we put it into (17), we can obtain ${\mathrm{\Delta }}_{\text{opt}\_2}$ through the previous iterative algorithm. Repeat the same procedure and submit $r_3=r_2+{\mathrm{\Delta }}_{\text{opt}\_2}$ into (17), we can obtain ${\mathrm{\Delta }}_{\text{opt}\_3}$ . Repeating this iterative process (16)–(22) constantly, we can calculate all the optimal cluster size in every corona $\mathrm{\hspace{0.17em}\hspace{0.17em}}{\mathrm{\Delta }}_{\text{opt}\_1},{\mathrm{\Delta }}_{\text{opt}\_2},\dots ,{\mathrm{\Delta }}_{\text{opt}\_k}$ and $\mathrm{\hspace{0.17em}\hspace{0.17em}}{r}_1,r_2,\dots ,r_k$ .

From (13), in the outermost corona $C_k$ , ${\hspace{0.17em}\hspace{0.17em}E}_{\text{ave}\_\text{inter}\_k}=0$ and $E_{\text{ave}\_\text{intra}\_k}$ is an increasing function of ${\mathrm{\Delta }}_i$ when ${\mathrm{\Delta }}_i<r_i$ . When ECR is minimized, cluster size will be close to zero, which means that cluster is degraded into one node. In order to guarantee the energy efficiency of the cluster, we require that iterative process be terminated when $R-r_{i-1}\le r_1$ or $R-r_{i-1}\le R\sqrt[4]{(R^2{e}_2-2e_1)/N{e}_2{R}^2}$ and define the cluster size of outermost corona $C_k$ as $R-r_{i-1}$ . $R\sqrt[4]{(R^2{e}_2-2e_1)/N{e}_2{R}^2}$ is the optimized cluster size to ensure the coverage and connectivity for circle model of WSNs in hierarchical cluster structure [29]. Since the optimal size of the outermost cluster cannot be determined, our algorithm is only a suboptimal solution. Fortunately, the average energy consumption rate in the outermost corona is always the smallest in the entire network, thus exerting impact on our algorithm for eradicating the energy hole problem.

4.4. The Nonuniform Nodes Distribution Strategy

After we have obtained $\mathrm{\hspace{0.17em}\hspace{0.17em}}{\mathrm{\Delta }}_{\text{opt}\_1},{\mathrm{\Delta }}_{\text{opt}\_2},\dots ,{\mathrm{\Delta }}_{\text{opt}\_k}$ and $\mathrm{\hspace{0.17em}\hspace{0.17em}}{r}_1,r_2,\dots ,r_k$ , the per node energy consuming rate (ECR) in each corona $E_{\text{node}\_1},E_{\text{node}\_2},\dots ,E_{\text{node}\_k}$ can be obtained as well. Since the sink node is not limited to energy, the iterative algorithm can run on it. In the following part, we will firstly introduce node deployment redundancy rate coefficient ${\psi }_i$ , the ratio of the nodes density in corona $C_i$ , and the nodes density in the outmost corona $C_k$ . We assume that there is no sleeping node in corona $C_k$ and the node density is ${\rho }_{\min }$ Consider

$$\begin{array}{cc}& {\psi }_i=\frac{{\rho }_i}{{\rho }_k}=\frac{{\rho }_i}{{\rho }_{\min }}.\end{array}$$

(23)

To put it simple, we assume that each sensor has the power of $\tau$ joule and use $S_i$ to denote the area of corona $C_i$ . Then, there are ${\rho }_{\min }{S}_i$ active nodes to participate in the network operation simultaneously in this corona. When all the distributed nodes use up all their energy, their survival time can be calculated as follows:

$$\begin{array}{cc}& T_i=\frac{{\rho }_i{S}_i\tau }{{\rho }_{\min }{S}_i{E}_{\text{node}\_i}}=\frac{{\psi }_i\tau }{E_{\text{node}\_i}}.\end{array}$$

(24)

Ideally, if there is no residual energy for all nodes when network loses its function, the network lifetime and the energy efficiency are maximized. That is,

$$\begin{array}{cc}& \frac{{\psi }_1\tau }{E_{\text{node}\_1}}=\frac{{\psi }_2\tau }{E_{\text{node}\_2}}=\cdots =\frac{\tau }{E_{{\text{node}}_k}}.\end{array}$$

(25)

By (25), we can obtain

$$\begin{array}{cc}& {\psi }_i=\frac{E_{{\text{node}}_i}}{E_{{\text{node}}_k}}\hspace{1em}\left(1\le i<k\right).\end{array}$$

(26)

According to (23) and (23), the optimal sensor distribution density and the number of sensors in $C_i$ corona can be given by

$$\begin{array}{cc}& {\rho }_i=\frac{E_{\text{node}\_i}}{E_{\text{node}\_k}}{\rho }_{\min },\\ N_i={\rho }_{\min }\pi {\mathrm{\Delta }}_i\left(2r_i-{\mathrm{\Delta }}_i\right)\frac{E_{\text{node}\_i}}{E_{\text{node}\_k}}.\end{array}$$

(27)

As the minimal values of $E_{\text{node}\_1}$ , $\hspace{0.17em}\hspace{0.17em}{E}_{\text{node}\_2},\dots ,E_{{\text{node}}_i},\dots ,E_{\text{nod}e\_k}$ have been determined as above, we can obtain the sensor distribution density ${\rho }_1,{\rho }_2,\dots ,{\rho }_{k-1}$ and the sensors distribution number $\mathrm{\hspace{0.17em}\hspace{0.17em}}{N}_1$ , ${\hspace{0.17em}\hspace{0.17em}N}_2,\dots ,N_k$ in each corona. ${\rho }_{\min }$ is the preliminary parameter of WSNs. As a result, energy hole problem of the network is resolved by using this non-uniform sensor distribution strategy based on unequal cluster.

4.5. The Data Transmission Mechanism

In order to get higher data transmission efficiency, we design an energy-balancing layered data transmission mechanism based on the previous non-uniform sensor distribution algorithm. Firstly, the sink constructs the unequal coronas, and sensor node can be deployed according to the results derived in the previous section. Sensor nodes select CHs and join the adjacent CH based on its distance to sink. Upon completion of the suboptimal unequal clusters, CHs are able to deliver their data to the sink. Each CH firstly aggregates the data received from its cluster members and then transfers them to the sink node via a multi-hop path through other intermediate CHs. The organization of intra-cluster data transmission is similar to LEACH, so we will not elaborate it again in this section. The pseudocode of the inter-cluster data relay algorithm is presented in Algorithm 1. In order to achieve balanced energy depletion among the CHs, they firstly exchange their energy and position information CHMessage to maintain a real-time table CoronaList about the neighbor cluster head. By this algorithm, CH can select one relay CH with maximum energy resource. At the same time, it has to exchange the residual energy message with all candidate relay CHs in lines 4–12 of Algorithm 1. After selecting the relay CH with the maximum residual energy, the CH can forward its own data and the data coming from its upper corona. The process of selecting relay CH and forwarding data will be repeated until the data arrive at sink node in lines 13–17 of Algorithm 1. If there is more than one candidate with the same maximum residual energy, choose one of them randomly. When SystemMessage is received, for instance, instructions for completing data transfer or reelecting cluster head, and another interrupt instruction is trigged by sink node, the network will terminate the data transmission process.

Algorithm 1: The data transmission mechanism.

(1) CalculateCHMessage(ID, Layer_ID, ResEnergy)

(2) Exchange(CHMessage) in adjacent corona

(3) Rec_Message(CHMessage) from Neighbor_CH

(4) if (Neighbor_CH.Layer_ID = CHMessage.Layer_ID + 1)

(5)  add node Neighbor_CH to CH.Up_CoronaList

(6)  Return (CHMessage(ID, Layer_ID, ResEnergy)) to Neighbor_CH

(7) end if

(8) if (Neighbor_CH.Layer_ID = CHMessage.Layer_ID − 1)

(9)  add node Neighbor_CH to CH.Down_CoronaList

(10)  Request (CHMessage(ID, Layer_ID, ResEnergy)) from Neighbor_CH

(11)  Update (CH.Down_CoronaList)

(12) end if

(13) while (Data_arrival = TRUE)

(14)  {  if (Rec_Message(Neighbor_CH.Data) from CH.Up_CoronaList)

(15)   {CH.Down = MaxResEnergy(CH.Down_CoronaList)

(16)   Transfer_Data(CH.Down, Neighbor_CH.Data)}

(17)  else{ if (SystemMessage = TRUE)

(18)   EXIT}

(19)  }

On the other hand, as indicated in Algorithm 1, since the non-uniform sensor distribution strategy based on unequal cluster can be calculated by the sink node, the complexity of network cluster protocol is determined by the clustering algorithm and inter-cluster data transmission mechanism. The non-uniform sensor distribution strategy based on unequal cluster does not increase the complexity of network.

5. Simulation Results

In this section, both the numerical results and the performance results of our strategy will be presented via simulation by MATLAB. At the beginning, we will calculate the optimal corona width and build the unequal hierarchical cluster network based on network parameters. Then we calculate the node distribution density in each corona. As the node distribution algorithm is based on unequal cluster, the performance of unequal clustering will directly influence the energy efficiency of network. Therefore, we firstly examine the performance of our unequal cluster algorithm, by comparing with two typical cluster protocols EEUC and LEACH. Then, we verify the effectiveness of our algorithm in eliminating the energy hole problem. At the same time, we prove that our distribution strategy has better energy efficiency and network lifetime with less sense nodes compared with the single node distribution strategy [26].

In order to conduct the experiments, proper parameters for both the sensor nodes and the network should be defined. We assign ${\rho }_{\min }$ with the initial value 0.00318, which is the minimum node density to ensure the effective coverage and data collection in the monitoring area. Once sensor nodes have been settled, the active nodes in each corona are organized into clusters and the other redundant nodes are kept in sleep-listening model. Then, each ordinary node forwards certain bits of data to its cluster head, which in turn aggregates and forwards the received data to sink by multi-hop communication. When some nodes die prematurely, nodes in sleep model will wake up and join the nearest cluster to fill in the vacancy. We adopt the STEM [30] sleep-listening mechanism in this paper. Every sensor node will keep a table of neighboring nodes in its competing range. Because the cluster head node always dies earlier than general nodes, it can select the nearest node to wake up according to STEM mechanism. The node's energy consumption for wakening, listening, and detection in sleep model is much lower than that in active model, so for simplicity, we do not consider it. The simulation parameters for our proposed mechanism are defined in Table 1.

Table 1

Parameters and characteristics of the network.

Parameter	Value
Network size (circle)	R = 200 meters
Sink location	(0, 0)
Data packet size	256 bytes
Initial energy	0.5 J
Initial ${\rho }_{\min }$	0.00318
Aggregation ratio	0.1

5.1. Calculation for Nonuniform Sensor Distribution Based on Unequal Cluster

According to our non-uniform sensor distribution strategy, the circle sensing area is partitioned into unequal clusters firstly based on the parameters in Table 1. The size of the innermost corona will be determined firstly, followed by size of outermost corona until the conditions of terminating iteration is triggered. Then, the corresponding non-uniform distribution sensors density with minimal ECR in each corona can be achieved as indicated in Table 2.

Table 2

Optimal nonuniform sensor distribution density and unequal clusters.

Corona	Cluster size (m)	Per node energy consuming rate (ECR) $(\mu \text{J}/\text{s})$	Redundancy rate coefficient ${\psi }_i$	Nonuniform sensor distribution density ${\rho }_i$	The number of sensors in each corona
$C_1$	32.16	518.73	4.761	0.01514	55
$C_2$	51.22	204.72	1.879	0.00598	115
$C_3$	47.95	156.81	1.439	0.00458	150
$C_4$	38.35	131.74	1.209	0.00385	139
$C_5$	29.32	108.95	1	0.00318	107

As mentioned in the previous section, the energy hole problem may arise in multi-hop wireless sensor networks if the cluster heads close to the data sink are burdened with heavy relay traffic. Different from the general opinions on unequal cluster, for instance, EEUC which argues that clusters closer to sink should be smaller in size, we find that cluster will not always be downsized as the distance from clusters to sink decreases so long as ECR in each corona obtains the minimal value. Downsizing the cluster will result in cluster increase in inner corona. Though it can alleviate relay traffic burden on the CH nodes, ECR in the whole cluster will increase dramatically, which will exacerbate the energy hole problem because of the decrease of cluster members. This result serves as an important guideline for the design of unequal cluster. In the following simulation test, we will prove that our unequal cluster can achieve better performance than EEUC.

5.2. Performance of Nonuniform Sensor Distribution Strategy Based on Unequal Cluster

Since the sensor distribution strategy is based on unequal clusters, the performance of the unequal cluster based on our optimal algorithm will exert great impact on network. We will firstly review the characteristics of the unequal cluster by comparing with cluster algorithms such as LEACH and EEUC. Simulation results show that our unequal cluster can achieve better energy consumption balance between cluster heads and minimal ECR in the whole network. Secondly, we will look into our non-uniform sensor distribution strategy in terms of the residual energy of nodes, the number of nodes that need to be distributed, and the network lifetime.

5.2.1. The Performance of Unequal Cluster

In our scenarios, we use the same parameters for EEUC mechanism [24] with the node density being ${\rho }_{\min }$ . We also conduct lots of experiments to determine the optimal number of clusters for LEACH. Figure 8 shows the total energy consumed by all cluster heads in three algorithms after thirty rounds of simulations. The energy consumed by cluster heads per round in our unequal cluster and EEUC is much lower than that of LEACH. Due to the need for sending their packets to sink via single hop, the energy consumption of cluster heads is much higher in LEACH. Moreover, the CHs' energy consumption in our unequal cluster network is slightly higher than the CHs' energy consumption in EEUC because there are more cluster heads sharing the tasks of data forwarding in EEUC than in our unequal cluster network. If evaluated only from the perspective of energy consumption on cluster head, EEUC can balance the energy consumption of cluster head better than our approach. However, if the energy consumption of all nodes in the cluster is taken into account, our unequal cluster approach consumes less energy than EEUC.

Figure 8

Total energy consumption of cluster heads.

Generally speaking, in accordance with the basic principle to avoid energy hole problem using unequal cluster algorithm, more nodes need to participate in traffic data relay, in inner corona in particular. However, to ensure longer network lifetime, we should not only consider the energy balance among the cluster head nodes but also consider the energy balance among all cluster members. ECR is an important parameter to measure energy consumption of all cluster members. In EEUC, the balance of energy consumption for cluster heads is the only factor that is taken into consideration, with ECR which has direct bearings on network lifetime being ignored. Figure 9 shows the change of ECR based on the distance from nodes to sink for three algorithms. It can be seen that our algorithm is better than EEUC in terms of nodes' average energy consumption rate. For LEACH, the average energy consumption of network nodes will increase sharply when nodes are away from sink because of the long distance in single-hop communications.

Figure 9

The energy consumption per node.

5.2.2. Energy Efficiency and Lifetime

In this part, we will analyze the energy efficiency and lifetime of our strategy. Firstly, we will study the residual energy of each node when the network lifetime ends. Then, we compare the sensor density of our strategy with that of other non-uniform sensor distribution methods. Finally, we verify that our non-uniform sensor distribution strategy based on unequal cluster can obtain longer network lifetime and better energy efficiency.

Firstly, we examine residual energy of nodes when the network lifetime ends. We define the network lifetime as the time period until WSNs cannot guarantee the effective coverage and data sensing and collection for the monitoring area. And sensor death in the network means the sensor loses the ability to sense data or send data to its cluster head. Figure 10 shows the cumulate residual energy of nodes in each corona when the network lifetime ends. The fitting fragments of line indicating the total energy of the nodes belong to the five coronas C ₁–C ₅ from right to left. It can be seen that when the network lifetime ends, there is nearly no residual energy in the network since only few sensors have residual energy. Through further analysis, we can find that the little residual energy is the result of energy consumption unbalance in the cluster topology maintenance and node sleep-listening mechanism in different coronas. For data relay, unbalanced energy consumption is prevented, or the energy consumption balance of the entire network is achieved indirectly, and the energy hole problem is solved completely. On the other hand, it also proves that the nonoptimal ECR in the outermost corona in our strategy exerts no adverse impact on the elimination of energy hole problem.

Figure 10

The nodes' residual energy when WSNs lose function.

Secondly, we compare the deployed sensors' density of our strategy with that of other non-uniform sensor distribution methods. In [26], we propose a non-uniform node distribution strategy to achieve the balance of energy depletion. We also point out that if the density of nodes in coronas increases in geometric progression with common ratio, energy depletion balance in the network can be achieved and the energy hole problem could be solved.

In this distribution strategy, sensor nodes are not organized into cluster hierarchy structure, and the monitoring area is divided into coronas with the same width. Each node in the network has several candidate relay nodes in the next inner corona. All experiments are done with the same minimized node density ${\rho }_{\min }$ in the outmost corona to ensure the effective network coverage and data collection in monitoring area.

In Figure 11, we can see that node distribution density is much lower in our non-uniform sensor distribution strategy. Figure 12 shows the specific number of distribution nodes in each corona for the two non-uniform sensor distribution strategies. It is obvious that few sensors are needed to effectively monitor the sensing area and eliminate the energy hole problem if our strategy is adopted. This is mainly due to the inherent advantages of the unequal cluster, including data aggregating, efficient data routing mechanism, and lower radio communication conflicts. Data aggregating can significantly reduce the frequency of data forwarding, and cluster hierarchy can be used to select efficient routing path more easily.

Figure 11

Node distribution redundancy rate coefficient.

Figure 12

The number of distributed nodes in each corona.

Finally, we verify the network lifetime and energy efficiency for our non-uniform sensor distribution strategy based on the unequal cluster. In order to estimate the lifetime of the WSNs, the metrics of First Node Dies (FND), Percent of the Nodes Alive (PNA), and Last Node Dies (LND) are always used [11]. FND is useful in sparsely deployed WSNs. However, PNA is more suitable to measure the network lifetime in densely deployed WSNs. LND does not have much practical value. FND and LND are not suitable for our node distribution strategy. When WSNs lose its ability to collect and transmit data, the network is regarded as dead.

Figure 13 shows the number of sensor nodes that are still alive during simulation test, and Figure 14 illustrates the rounds of data transmission for the four strategies during their lifetime. For EEUC, as more cluster heads participate in the traffic relay in inner corona, it can balance the energy consumption between cluster heads. Moreover, it can achieve better energy efficiency and longer lifetime than LEACH with the FND metric. But the energy hole problem which is inevitable in uniform sensor distribution WSNs, even in cluster hierarchy structure, will cause much residual energy when the network lifetime ends. For single non-uniform node distribution strategy, energy consumption balance of sensors can be ensured, and all the sensor nodes will die nearly at the same time. Therefore, the energy hole problem has been resolved completely. But consider the nodes that need to be settled, we will find that this strategy may result in very low energy efficiency and high cost for WSNs.

Figure 13

The network lifetime.

Figure 14

The rounds of data transmission.

In our non-uniform sensor distribution strategy based on unequal cluster, we adjust the node distribution density according to the minimal value of ECR in each corona while taking into account the energy consumption of cluster heads and cluster members. From Figures 13 and 14, we can see that the FND occurs early in our strategy, but the network still works well in network coverage and data collection through sleep-listening mechanism of redundant nodes. When the number of survival nodes in the network approaches 400, all the remaining nodes will use up their residual energy almost at the same time, and then the network lifetime ends. In this way, the energy consumption balance for the entire network is achieved, and the energy hole problem is eliminated completely. From the simulation results, it can be seen that our strategy can ensure longer network lifetime and better energy efficiency compared with the other three strategies.

6. Conclusion

In this paper, we have proposed a nonuniform sensor distribution strategy based on unequal cluster for WSNs. The major objective of our algorithm is to resolve the energy hole problem and improve the performance of WSNs based on multi-hop communication.

By focusing on intra- and intercluster energy consumption and using the ECR model, we calculate the minimal value of ECR according to the distance to the base station and deduce the non-uniform node distribution strategy. Our non-uniform sensor distribution mechanism works well in eliminating the energy hole problem by balancing average energy consumption speed of cluster nodes. Through sleep-listening mechanism of redundant nodes and suboptimal unequal cluster protocol, our non-uniform sensor distribution strategy can ensure better energy efficiency and longer network lifetime. Moreover, we have proved that our strategy ensures better performance than LEACH, EEUC, and other single non-uniform sensor distribution algorithms, as evidenced by the simulation results.

We also find that minor unbalance in energy consumption will be caused by cluster topology maintenance and node sleep-listening mechanism in different coronas if our strategy is adopted. In this paper, we only use a simple sleep-listening schedule approach. How to get the efficient cluster topology maintenance approach and node sleep-listening mechanism for our algorithm are the major issues we need to solve in the future.