Energy Consumption Averaging and Minimization for the Software Defined Wireless Sensor Networks With Edge Computing

The software-defined (SD) and edge computing (EC) are emerging technologies that have been used to improve the network operation efficiency of wireless sensor networks (WSNs). Due to the advantages of the SD and EC technologies, the area of WSNs has achieved a new dimension and breakthrough. However, the limited energy allocation mechanism in edge-SD wireless sensor networks (ESDWSNs) makes the energy consumption of different nodes unbalanced. In this paper, we propose an energy allocation optimization (EAO) algorithm that solves the energy averaging and minimization (ECAM) problem in ESDWSNs by selecting appropriate relay nodes and de-duplicated data flows. Specifically, we first establish a novel three-layer network architecture based on the edge computing and software-defined technologies. Then we proposed the ECAM problem, which minimizes the energy consumption in ESDWSNs. Furthermore, we propose an adaptive Levenberg-Marquardt algorithm and derive the optimization value of energy cost function. The extensive simulation results based on the NS-2 simulator demonstrate the energy balance efficiency of the EAO algorithm in ESDWSNs.

SECTION I. Introduction

Wireless sensor networks (WSNs) with limited energy and computing capabilities make the scheduling strategy of data transmission less complicated. Unbalanced energy allocation method shortens the life cycle of WSNs [1], [2]. Traditional network architecture and data scheduling strategies are difficult to save limited physical resources. Therefore, the network architecture, data processing and forwarding mechanisms need to be redesigned [3].

There are some breakthroughs in the scheduling algorithms of wireless sensor networks. The proposed flow split optimization (FSO) algorithm in [4] selects the relay nodes with the smallest redundant data flow as the optimal transmission path, which can minimize the traffic load of software defined wireless sensor networks (SDWSNs). And the profit maximization multi-round auction (PMMRA) algorithm in [5] is introduced for matching users with resource providers and determining the final payment. The mechanism consists of a bidding strategy, user matching, payment determination, and online mechanism. The PMMRA mechanism can more effectively guarantee the profit of the resource provider and the benefit of the perceived device.

Although these scheduling strategies [4], [5] can improve network performance to a certain degree, but they have certain limitations in network architecture and data processing. Among the above two strategies, novel software definition and edge computing techniques are used separately. This inspired us to combine these two novel technologies to propose a new network architecture and data scheduling strategy. The software-defined technology [6] virtualizes the control plane by decoupling itself from the data plane in traditional network devices. In the virtualized control plane, the controllers can maintain a global view of the entire network and dynamically manage network devices [7]. Edge computing (EC) is such a promising technology that narrows the distance between edge devices and the clouds by turning any potential resource-abundant device into edge cloud [8]. We integrate SD and EC ideas into WSNs for forming a new network architecture, called the edge-SD wireless sensor networks (ESDWSNs), which can highlight the advantages of both SD and EC technologies in WSNs.

Compared to software-defined wireless sensor networks (SDWSNs) and edge wireless sensor networks (EWSNs), the ESDWSNs architecture can perform more detailed division of network functions, i.e., the collecting, analyzing, processing, and transmitting of data [9]. This architecture can change network functions from serial operation mode to parallel operation mode [10]. Among them, software-defined technology can grasp the network state from a global perspective, which prevents the scheduling strategy from falling into local optimum and further avoids energy imbalance [11]. On the other hand, edge computing technology can process source data in real time, which avoids transmitting redundant data to the cloud center, thereby saving network resources.

However, the FSO algorithm transmits all network status information to the software-defined controller, which may cause control link congestion due to overloaded transmission. Also, the analysis and processing of redundant data are done by software-defined controllers, and massive amounts of perceptual data can overwhelm software-defined controllers. The PMMRA mechanism does not consider the data redundancy in the same perceived region, and it sends numerous redundancy and irrelevant data to the cloud center, thus wasting valuable network resources. Therefore, these limitations motivate us to propose an effective strategy for achieving energy consumption averaging and minimization in ESDWSNs. On the other hand, the existing schedule strategies [4], [5] are limited to the counterpoise of energy consumption and data processing.

In this paper, we study the problem of energy consumption averaging and minimization (ECAM) in ESDWSNs. Our goal is to eliminate redundant packets and select the optimal data forwarding paths through the given energy cost function. Further, our data scheduling strategy can balance and save energy consumption by using the SD and EC technologies. Specifically, we first constructed the ECAM problem formally based on the network architecture characteristics of ESDWSNs. Then, the energy consumption sum of non-redundancy packets is analyzed. We also use the Lagrangian dual method and adaptive Levenberg-Marquardt algorithm to derive the optimal value of energy cost function in ESDWSNs. Afterward, we propose the energy allocation optimization (EAO) algorithm whose key parameters are the optimal determined values. Finally, the simulation results show that the EAO algorithm has a lower energy balance level and higher network throughput. The EAO algorithm can be applied to applications such as ecological environment monitoring and prediction in harsh environments, which require higher real-time data transmission requirements.

Compared to previous works, our contributions in this paper can be summarized as follows:

• We propose an energy system model to characterize the energy consumptions of different packet types in edge-SD wireless sensor networks.

• We characterize the energy cost maximization function for realizing the objective of average energy consumption in ESDWSNs, and prove the solution convergence of energy cost maximization. In addition, we take the optimal value of energy cost function as the metric for selecting the non-redundant data transmission path.

• We propose an energy allocation optimization (EAO) algorithm to remove redundant data and select appropriate relay nodes for energy balance in ESDWSNs. Our EAO algorithm not only minimizes the transmitted data but also balances the energy consumption of the entire network.

The remainder of this paper is organized as follows: Section II provides the network and system model and describes the problem of energy consumption averaging and minimization (ECAM) in Section III. The Section IV presents the energy allocation optimization (EAO) algorithm. In Section V, the scheduling efficiency of EAO algorithm is extensively appraised. In the end, Section VII concludes this paper.

SECTION II. Network and System Model

A. Network Model

We consider balancing network energy allocation and data processing in a novel network architecture that utilizes edge computing techniques in software-defined wireless sensor networks. This is a recent out-of-the-box network technology, which we call edge-SD wireless sensor networks (ESDWSNs). We choose nodes with larger residual energy as edge servers to balance energy consumption. Therefore, the life cycle of ESDWSNs will be extended. Generally, the edge servers are deployed in the area where the sensors are concentrated, thereby reducing the transmission distance of each node for sending data. In the proposed architecture of ESDWSNs, three decoupled layers - perception, edge, and control - are shown in Fig. 1. Data transmission is done at the sensing layer and the edge layer. The edge layer and the control layer are connected by a bidirectional control link. The working of these layers is discussed below.

FIGURE 1.

Edge-SD wireless sensor networks (ESDWSNs).

1) Perceptual Layer

All data acquisition devices (DADs) are located in this layer. All these DADs behave according to the work decisions made by the edge server residing in the edge layer [12]. The work decisions determine the operating status of the DADs, and all collected data is transmitted to the edge server. The perceptual layer is only responsible for the data acquisition on the source side, thus saving the energy consumption of the sensors in the ESDWSNs.

2) Edge Layer

The edge layer consists of edge servers deployed in different perceptual areas. Each edge server is responsible for the analysis, processing, and transmission of all collected data in the current sensing area [13]. Then, each edge server determines the data that needs to be collected according to the needs of the cloud center and eliminates redundant data and irrelevant data. Specifically, the edge server first sends a data transmission requirement to the software-defined controller in the control layer and performs a corresponding transmission mechanism for scheduling the data flow. Then, the edge layer reduces the amount of data collected across the network transferred to the cloud center.

3) Control Layer

The control layer is the core of the ESDWSNs architecture and acts as the character of decision making. This layer works according to the control logic provided to the software-defined controller. All the forwarding strategies are decided by the controller and added to the instruction set in this layer. One of the most important characteristics of SDN is that the control logic can be programmed and reconfigured according to different network environments. Therefore, based on the software-defined approach in ESDWSNs, we present a flow management scheme that optimizes the trade-offs between energy efficiency and data processing.

The ESDWSNs performs energy consumption balancing under the premise of efficient data transmission by using effective links. Therefore, the main purpose of this paper is to use the scheduling algorithm to reasonably distribute the data flow to the optimal link. The control layer divides the perceptual layer into several edge regions according to the network status. The sensors with the most energy remaining in each region are selected as edge servers, which form the edge layer. The perceptual data is first preprocessed by the edge server, and then the control layer determines the transmission path of non-redundant perceptual data.

B. System Model

An edge-SD wireless sensor networks (ESDWSNs) composed of a SDN controller, $m$ edge servers and a sensor-set, $\mathcal {N}=\{N_{11}, N_{12},\cdots, N_{1k}, N_{21},\cdots, N_{mk}\}$ , with $n =|\mathcal {N}|$ . Sensors collect data from the environment and then transmit it to the edge server in the current edge region. The edge server is responsible for processing and forwarding the collected data. The edge server manages all the sensors in the edge region. The ESDWSNs is divided into some regions based on the transmission radius of edge server $R$ , and the area of an edge region is $R\times R\,\,m^{2}$ .

In a similar environment, a large amount of redundant data is collected, which cannot satisfy the user’s needs and produce irrelevant data. Therefore, we should remove redundant and extraneous data as much as possible to reduce the energy consumption of data transmission. Then, the edge server processes the collected data and notifies the SDN controller to plan the paths for them. Based on the consideration, we establish the energy cost model of packet types in the following.

Different sensors need to transmit all types of data packets collected to the sink node through relay nodes. We assume that the maximum amount of data collected in an ESDWSNs is $C$ , and the minimum amount of data $c$ must be transferred to the sink node. We set the packet type set as $L$ , the packet type set $L$ is equally divided into $l$ parts, $l\in L, |L|=\lceil \frac {C}{c}\rceil$ . For example, the maximum amount of data processed in an ESDWSNs is $500M$ , and the minimum amount of data with effective forwarding is $30M$ , the number of packet types $|L|=\lceil \frac {C}{c}\rceil =17$ .

We define $\mathcal {N}^{(l)}$ as the valid sensor set with packet type $l$ , $l\in L$ , and any type-$l$ packet will be processed by at least one sensor from the set $\mathcal {N}^{(l)}$ . For any packet type $l$ , the set $\mathcal {N}^{(l)}$ is assumed to be non-empty. We set $N_{i}^{(l)}$ as the sum of packet type $l$ in the $i$ -th sensor, and $e_{i}^{(l)}$ is the energy consumption set of type-$l$ packet routed through the sensor $i$ . $e^{(l)}$ denotes the energy consumption set of type-$l$ packet in ESDWSNs. We define the energy consumption as $$\begin{equation*} e=(e_{i}^{(l)}; i\in \mathcal {N}, l\in L)\tag{1}\end{equation*}$$

$E$ denotes the energy consumption set of all packet types in ESDWSNs, where $E=\bigcup \limits _{l\in L}\bigcup \limits _{i\in \mathcal {N}}\{e_{i}^{(l)}\}$ . $\mathcal {E}_{i}$ indicates the energy consumption of the $i$ -th sensor, where $\mathcal {E}_{i}=\|E_{i}^{(l)}\|_{1}, i\in \mathcal {N}$ . Let $\mathcal {E}$ be the energy consumption sum of total network. Thus we can get $$\begin{equation*} \mathcal {E}=\sum \limits _{i\in \mathcal {N}}\sum \limits _{l\in L}e_{i}^{(l)}\tag{2}\end{equation*}$$

We define packet similarity to distinguish the data collected from the two sensors.

Definition 1 (Packet Similarity):

The packet similarity $S$ is defined as the percentage between the type-consistent packets and all packet types among sensor nodes, i.e., $$\begin{equation*} S(\varepsilon)=\frac {|\mathcal {N}^{(l)}_{1}\bigcap \mathcal {N}^{(l)}_{2}\bigcap \cdots \bigcap \mathcal {N}^{(l)}_{n}|}{|\mathcal {N}^{(l)}_{1}\bigcup \mathcal {N}^{(l)}_{2}\bigcup \cdots \bigcup \mathcal {N}^{(l)}_{n}|},\quad 0 \leq S(\varepsilon)\leq 1\tag{3}\end{equation*}$$ where $n$ indicates the number of sensors in ESDWSNs, $k\in \mathcal {K}$ . $|\mathcal {N}^{(l)}_{1}\bigcap \mathcal {N}^{(l)}_{2}\bigcap \cdots \bigcap \mathcal {N}^{(l)}_{n}|$ is the type-consistent packets, and $|\mathcal {N}^{(l)}_{1}\bigcup \mathcal {N}^{(l)}_{2}\bigcup \cdots \bigcup \mathcal {N}^{(l)}_{n}|$ denotes the total packet types in adjacent sensors.

The packet similarity $S$ denotes the redundancy degree of packets collected among all sensors. The smaller the similarity $S(\varepsilon)$ is, the less the data redundancy is. We classify the packets into $l$ types according to similarity $S$ , i.e., the similarity $S$ must belong to a certain interval of the average $l$ aliquot. Therefore, a data flow composed of $l$ different types of data packets needs to be sent to sink node. The energy consumption required for transmitting the data flow is shown in Fig. 2.

FIGURE 2.

The energy consumption of packet types.

We assume that the number of edge region is $m$ , and $a$ is the number of sensors in a edge region. The energy consumption status of all non-redundancy packets is defined as the event $\varepsilon$ . $\mathcal {E}_{\max }$ indicates the node of maximum energy consumption in ESDWSNs.

Theorem 1:

In the ESDWSNs, the energy consumption sum of non-redundancy packets is no greater than $ae_{\max }^{m}$ .

Proof 1:

We indicate the event $\varepsilon$ in the following by considering events $N_{ij}$ ’s. $$\begin{align*}&\hspace{-0.5pc}\varepsilon =(N_{11}\bigcup N_{12}\bigcup \cdots \bigcup N_{1a})\bigcap \cdots \bigcap \\&\qquad\qquad\qquad\qquad\qquad\displaystyle {(N_{m1}\bigcup N_{m2}\bigcup \cdots \bigcup N_{ma}) } \tag{4}\end{align*}$$ From (4) and by De Morgan’s laws, the event $\varepsilon$ is given by $$\begin{align*} \mathcal {E}(\varepsilon)=&\sum \limits _{i=1}^{a}\mathcal {E}(N_{1i}\bigcap N_{2i}\bigcap \cdots \bigcap N_{mi}) \\\leq&\sum \limits _{i=1}^{a}e_{max_{i}}^{m} \\=&ae_{\max }^{m}\tag{5}\end{align*}$$ where $\mathcal {E}\mathcal {}(\varepsilon)=\sum \limits _{l=1}^{(L)}e(\mathcal {N}^{(l)})$ . This completes the proof.

Note that any edge server can communicate freely with other edge servers through the available scheduling of SDN controller, and energy consumption adjustment between sensors inside the edge layer does not influence the effective transmission of data packets. Also, at least one valid path can satisfy the transmission of the packet requirement $\omega$ . Table 1 lists some notations that will be used in this paper.

TABLE 1 Notations Table

SECTION III. Problem Formulation

For the purpose of minimizing the redundancy and irrelevance of transmitted data, the ESDWSNs attempts to optimize different objectives. However, wireless sensor networks have limited computing, storage, and energy resources and cannot support complex computation and path scheduling optimization strategies. In this paper, we divide the network into three layers and put the optimization calculation and data transmission methods on the edge layer. The optimal path selection by the control layer will make the minimum resource consumptions of the perception layer. Under the above works, the traffic needs to be distributed and balanced among the available paths in ESDWSNs, which is essential for maintaining the average energy consumption of all sensors.

In this part, we formulate the problem of energy consumption averaging and minimization (ECAM) in ESDWSNs, which minimizes redundant data transmitted. We will propose a computing and routing strategy that enables the efficient transmission of data flows and average energy consumption in ESDWSNs. If the optimal path selected are relay nodes with a relatively large ratio of energy consumption to residual energy, the energy consumption of ESDWSNs can be balanced. In addition, the transmitting packets with minimal redundancy will minimize energy consumption. Therefore, we will establish the following optimization function of ECAM problem based on the above analysis. $$\begin{align*}&\hspace {-2pc}OPT-1 ~\max \sum \limits _{i\in \mathcal {K}}\sum \limits _{l\in L}\rho ^{(l)}_{i}F\left({\frac {e^{(l)}_{i}}{Z_{\max }}}\right) \tag{6}\\&\qquad \text {subject to} ~\forall l\in L, \\&\hphantom {\qquad \text {subject to} ~}C1:~\sum \limits _{i\in \mathcal {K}}\sum \limits _{l\in L}e^{(l)}_{i}\leq kZ_{\min }, \\&\hphantom {\qquad \text {subject to} ~}C2:~\sum \limits _{l=1}^{(L)}e(\mathcal {N}^{(l)})\leq ae_{\max }^{m}, \\&\hphantom {\qquad \text {subject to} ~}C3:~\rho _{k}\in \{0,1\},\tag{7}\end{align*}$$ where $Z_{\max }$ indicates the maximum value of remaining energy in ESDWSNs. We define $F(x)$ as energy cost function, and the function $F(x)$ reflects the degree of energy consumption balance. We consider the polynomial function of energy cost as $$\begin{equation*} F(x_{i})=\frac {1}{2}\log _{2}^{(1+x_{i}\xi _{i}^{1+S(\varepsilon)})},\quad 0 < S(\varepsilon)\leq 1,\tag{8}\end{equation*}$$ where $\xi$ is the irrelevant data factor, and the size of $\xi$ value is determined by the irrelevant degree of the data in the network. It is clear that the energy cost function $F$ is increasing in terms of $e^{(l)}_{i}$ , and the maximization function (OPT-1) is strictly convex.

The remaining energy of any active sensor must satisfy the energy requirements of the current data transmission, therefore, constraint $C1$ is the energy consumption constraint of a transmission link, which specifies that the total energy consumption of all the relay nodes in transmission link is bounded by $k$ times the minimum remaining energy of relay nodes $Z_{\min }$ , the collected packets can be efficiently forwarded to the cloud server through the edge server, $k$ denotes the number of the relay nodes in a transmission link. The packet similarity level in ESDWSNs is displayed in constraint $C2$ . Constraint $C3$ is selection criteria for effective relay nodes, and the constraint indicates that each relay node can handle the processing of multiple packets, but each data packet can only be assigned to one relay node.

The optimization function (OPT-1) is to balance and save energy consumption of all sensors throughout the ESDWSNs by transmitting valid packets and selecting the appropriate transmission path, thereby extending the life cycle of the entire network. Constraint $C3$ is a discrete integer optimization variable. Therefore, we relax the constraint on it and rewrite constraint $C3$ to the following conditions of energy consumption. $$\begin{equation*} \sum \limits _{i\in \mathcal {K}}\rho _{i}e_{i}\leq ke_{\max }\tag{9}\end{equation*}$$ where $k$ is the number of relay nodes through which a valid data transmission path passes.

If the optimization function (OPT-1) directly uses the traditional lagrange-dual algorithm to find the optimal value, then the value of $Z_{\max }$ will be repeated multiple times, which will increase the running time of scheduling algorithm. Therefore, we set $x^{(l)}=\frac {e^{(l)}}{Z_{\max }}$ for reducing the computing time. Afterward, the maximization function of energy cost (OPT-1) can reformulate into the following convex optimization function (OPT-2) for concerning to $x$ . $$\begin{align*}&\hspace {-2pc}OPT-2 ~\max \sum \limits _{i\in \mathcal {K}}\sum \limits _{l\in L}\rho ^{(l)}_{i}F(x^{(l)}_{i}) \tag{10}\\&\qquad \text {subject to} ~ \forall l\in L, \\&\hphantom {\qquad \text {subject to} ~}\sum \limits _{i\in \mathcal {K}}\sum \limits _{l\in L}x^{(l)}_{i}\leq kZ_{\min }, \tag{11}\\&\hphantom {\qquad \text {subject to} ~}\sum \limits _{i\in \mathcal {N}}\sum \limits _{l\in L}x^{(l)}_{i}\leq ax_{\max }^{m} \tag{12}\\&\hphantom {\qquad \text {subject to} ~}\sum \limits _{i\in \mathcal {K}}\rho _{i}x_{i}\leq kx_{\max }\tag{13}\end{align*}$$

The problem (OPT-2) is an energy cost maximization problem with energy consumption constraints, and the Lagrangian dual method [14] and adaptive Levenberg-Marquardt (ALM) method [15] are adopted to find the optimal value $x^{*}$ . The optimal value $F(x^{*})$ is a metric for selecting the transmission paths with average energy consumption in ESDWSNs.

SECTION IV. Energy Allocation Optimization Algorithm

In this section, the Karush-Kuhn-Tucker (KKT) condition [16] of maximization function (OPT-2) is converted into the equations without any constraint in the variable space through utilizing the Lagrangian dual method, afterward, we use the ALM to find the optimal value $F(x^{*})$ . Finally, a novel schedule algorithm is proposed for solving the ECAM problem.

A. Lagrangian Dual Method

Because the optimization function (OPT-2) is strictly convex, therefore, the Lagrangian dual decomposition approach [17] is adopted for converting (OPT-2) to an unconstrained optimization function. We first introduce Lagrangian multipliers $\alpha$ , $\beta$ and $\gamma$ for relaxing constraints (11), (12) and (13). Further, the partial Lagrangian of maximization function (OPT-2) can be expressed as follows: $$\begin{align*} L(x, \alpha, \beta, \gamma)=&\sum \limits _{i\in \mathcal {K}}\sum \limits _{l\in L}\rho ^{(l)}_{i}F(x^{(l)}_{i}) \\&+\,\alpha \left({kZ_{\min }-\sum \limits _{i\in \mathcal {K}}\sum \limits _{l\in L}x^{(l)}_{i}}\right) \\&+\,\beta \left({ax_{\max }^{m}-\sum \limits _{i\in \mathcal {N}}\sum \limits _{l\in L}x^{(l)}_{i}}\right) \\&+\,\gamma \left({kx_{\max }-\sum \limits _{i\in \mathcal {K}}\rho _{i}x_{i}}\right)\tag{14}\end{align*}$$

Equation (14) can be solved in the following by the Lagrangian dual method. $$\begin{equation*} \min \limits _{\alpha \geq 0, \beta \geq 0, \gamma \geq 0}\tilde {d}(\alpha, \beta, \gamma)\tag{15}\end{equation*}$$

The dual function of equation (14) can be expressed as $$\begin{equation*} \tilde {d}(\alpha, \beta, \gamma)\equiv \max \mathcal {(L)}(x^{(l)}_{i},\alpha, \beta, \gamma)\tag{16}\end{equation*}$$

Substituting the formula (14) into the formula (16), so we can get $$\begin{equation*} \tilde {d}(\alpha, \beta, \gamma)=\max [\tilde {d}_{0}+\tilde {d}^{(l)}_{i}(x^{(l)}_{i},\alpha, \beta, \gamma)]\tag{17}\end{equation*}$$ where, $$\begin{align*} \tilde {d}_{0}=&\alpha kZ_{\min }+\beta ax_{max}^{m}+\gamma kx_{\max }\tag{18}\\ \tilde {d}_{i}(x_{i}^{(l)},\eta, \psi, \upsilon)=&\sum \limits _{i\in \mathcal {K}}\sum \limits _{l\in L}\rho ^{(l)}_{i}F(x^{(l)}_{i}) \\&-\,\alpha \sum \limits _{i\in \mathcal {K}}\sum \limits _{l\in L}x^{(l)}_{i} \\&-\,\beta \sum \limits _{i\in \mathcal {N}}\sum \limits _{l\in L}x^{(l)}_{i} \\&-\,\gamma \sum \limits _{i\in \mathcal {K}}\rho _{i}x_{i}\tag{19}\end{align*}$$

It is clear that the equation (17) can be solved by the following derivation method. $$\begin{equation*} \frac {\partial (\tilde {d}_{i}(x_{i}^{(l)},\alpha, \beta, \gamma))}{\partial x_{i}^{(l)}}=0\tag{20}\end{equation*}$$

Combining equations (19) and (20) gives the optimal energy allocation solution, i.e., $$\begin{equation*} {x_{i}^{(l)}}^{*}=\left[{\rho ^{(l)}_{i}\left({\frac {1}{2\ln 2\mathcal {A}}-\frac {1}{\xi _{i}^{1+S(\varepsilon)}}}\right)}\right]\tag{21}\end{equation*}$$ where $\mathcal {A}=\alpha +\beta +\gamma \rho _{i}$ . The Lagrange multiplier optimal value can be iteratively updated by the ALM algorithm, and its recursive form as $$\begin{align*} \alpha ^{(l)}(t+1)=&\left[{\alpha ^{(l)}(t)-d_{1}\left({kZ_{\min }-\sum \limits _{i\in \mathcal {K}}\sum \limits _{l\in L}x^{(l)}_{i}}\right)}\right]^{+} \qquad \tag{22}\\ \beta ^{(l)}(t+1)=&\left[{\beta ^{(l)}(t)-d_{2}\left({ax_{\max }^{m}-\sum \limits _{i\in \mathcal {N}}\sum \limits _{l\in L}x^{(l)}_{i}}\right)}\right]^{+} \tag{23}\\ \gamma ^{(l)}(t+1)=&\left[{\gamma ^{(l)}(t)-d_{3}\left({kx_{\max }-\sum \limits _{i\in \mathcal {K}}\rho _{i}x_{i}}\right)}\right]^{+}\tag{24}\end{align*}$$ where $t$ is the number of iterations, $d_{1}$ , $d_{2}$ and $d_{3}$ are iteration step sizes.

The Lagrange multiplier will converge to the optimal value $(\alpha ^{*}, \beta ^{*}, \gamma ^{*})$ by calculating the appropriate step size by the Algorithm 1 and repeating the iterative process described above. We will adopt the ALM algorithm to search for the most appropriate number of iterations and iteration step sizes.

Algorithm 1 Adaptive Levenberg-Marquardt (ALM) algorithm

1:

Require: Given $\vartheta _{1}\in \mathbb {R}^{n},\mu _{\min }\geq 1$ .

2:

Set $G_{1}=J_{1}, \lambda _{1}=\mu _{1}\|W\|^\delta, t:=1, m:=1$ .

3:

while $\|G_{t}W_{t}\|\neq 0$ do

4:

Calculate $d_{t}$ and $r_{t}$ by solving equations (26) and (27), respectively.

5:

if $r_{t}\geq p_{0}$ then

6:

$\vartheta _{t+1}=\vartheta _{t}+d_{t}$ ;

7:

else

8:

$\vartheta _{t+1}=\vartheta _{t}$ ;

9:

end if

10:

if $p_{2}\leq r_{t}\leq p_{3}$ then

11:

$\mu _{t+1}=\mu _{t}$ ;

12:

else

13:

if $r_{t}\geq p_{1}$ and $m\leq m_{\max }$ then

14:

$G_{t+1}=G_{t}$ ;

15:

$\mu _{t+1}=\max \{c_{2}\mu _{t},\mu _{\min }\}$ ;

16:

$\lambda _{t+1}=\lambda _{t}$ ;

17:

else

18:

$G_{t+1}=J_{t}$ ;

19:

$\mu _{t+1}=c_{1}\mu _{t}$ ;

20:

$\lambda _{t+1}=\mu _{t+1}\|W_{t+1}\|^\delta$ ;

21:

end if

22:

end if

23:

if $G_{t+1}$ is updated then

24:

$m:=m+1$ ;

25:

else

26:

$m:=1$ ;

27:

end if

28:

$t:=t+1$ ;

29:

end while.

B. Adaptive Levenberg-Marquardt Algorithm

In this subsection, we first translate the equation (20) into the equality $W(\vartheta) = 0$ [15], where $\vartheta =[x^{T},\alpha ^{T},\beta ^{T},\gamma ^{T}]^{T}$ . And then we set $\phi (\vartheta)=\frac {1}{2} \|W(\vartheta)\|^{2}$ and rewrite the function (OPT-2) as follows: $$\begin{equation*} \min \phi (\vartheta)=\frac {1}{2} \|W(\vartheta)\|^{2}.\tag{25}\end{equation*}$$

We can not solve equation (25) directly by using the optimal equivalent method. Therefore, we record each exact Levenberg-Marquardt (LM) step and each approximate LM step calculation as an iteration and test each iteration as a test step. We define test step $d_{t}$ $$\begin{equation*} (G_{t}^{T}G_{t}+\lambda _{t}I)d_{t}=-G_{t}^{T}W(\vartheta).\tag{26}\end{equation*}$$ where $G_{t}$ is the current step Jacobian matrix $J_{t}$ or the Jacobian matrix used in the previous iteration, and its value depends on the update quality of the previous step. $m_{\max }$ is the upper limit of the number of $J_{t}$ iterations of a single Jacobian matrix and $I$ is a unit matrix. $J(\vartheta)$ denotes the Jacobian matrix of $W(\vartheta)$ [15].

To determine whether $d_{t}$ is acceptable and how to update $\lambda _{t}$ , we define the ratio between the actual drop in $t$ iterations and the estimated drop in the following function. $$\begin{equation*} r_{t}=\frac {\|W_{t}\|^{2}-\|W(\vartheta _{t}-d_{t})\|^{2}}{\|W_{t}\|^{2}-\|W(\vartheta _{t}-G_{t}d_{t})\|^{2}}\tag{27}\end{equation*}$$ where $\|.\|$ is 2 norm. We set $c_{1}>1>c_{2}>0, 0 < p_{0} < p_{2} < p_{1} < p_{3} < 1, 1 < \delta \leq 2, \mu _{1}>\mu _{\min }>0$ . $\mu _{\min }$ is a sufficiently small normal number, so that the parameter $\mu _{t}$ is not too small. We first propose adaptive Levenberg-Marquardt (ALM) algorithm. Secondly, the convergent of ALM algorithm is proved under certain conditions.

To illustrate the stability of ALM algorithm, we demonstrate the convergence proof of the ALM algorithm in the following.

Theorem 2:

Assume that $\{\vartheta _{t}\}$ is the sequence generated by using the ALM algorithm, the $\{dist(\vartheta _{t}, \vartheta ^\ast)\}$ converges to 0.

Proof 2:

The proved conditions in [18] as follows: $$\begin{align*} \|d_{t}\|\leq&c'_{1}dist(\vartheta _{t}, \vartheta ^\ast) \tag{28}\\ \|W(\vartheta _{t}+d_{t})\|\leq&c'_{2}dist(\vartheta ^{t}, \vartheta ^\ast)^{\delta } \\&+\,o(dist(\vartheta _{t}, \vartheta ^\ast)) \\=&o(dist(\vartheta _{t}, \vartheta ^\ast))\tag{29}\end{align*}$$ where $c'_{1}>0, c'_{2}>0, 1 < \delta \leq 2$ .

The following condition is deduced in view of ALM algorithm. $$\begin{align*} \|\vartheta _{t}+d_{t}-\vartheta ^\ast \|\leq&\|\vartheta _{t}-\vartheta ^\ast \|+\|d_{t}\| \\\leq&\|\vartheta _{t}-\vartheta ^\ast \|+c'_{1}dist(\vartheta _{t}, \vartheta ^\ast) \\\leq&\|(c'_{1}+1)\vartheta _{t}-\vartheta ^\ast \| \\\leq&b'\tag{30}\end{align*}$$ where $\vartheta _{t+1}\in N(\vartheta ^\ast,b), b'\leq b$ , and $N(\vartheta ^\ast,b)$ denoted the neighbourhood of optimal value $\vartheta ^\ast$ .

Owing to $W(\vartheta _{t+1})=W(\vartheta _{t}+d_{t})$ , $c''$ indicated a constant, so we can get $$\begin{equation*} W(\vartheta _{t+1})\leq \|W(\vartheta _{t}+d_{t})\|+c''\|d_{t}\|^{2}\tag{31}\end{equation*}$$ Blending the equation (28) to (31), thus we have $$\begin{align*} dist(\vartheta _{t+1}, \vartheta ^\ast)\leq&c'W(\vartheta _{t+1}) \\\leq&\|c'W(\vartheta _{t}+d_{t})\|+c'c''\|d_{t}\|^{2} \\\leq&c'o(dist(\vartheta _{t}, \vartheta ^\ast))+c'c''c_{1}^{,2}dist(\vartheta _{t}, \vartheta ^\ast)^{2} \\=&o(dist(\vartheta _{t}, \vartheta ^\ast))\tag{32}\end{align*}$$ where $c'>0, c''>0$ .

Comprehensive proof of the above, $dist(\vartheta _{t}, \vartheta ^\ast)$ converge to 0. This completes the proof.

C. Energy Allocation Optimization Algorithm

In this subsection, we propose a novel energy allocation optimization (EAO) algorithm for achieving the average energy consumption based on the calculation results of Algorithm 1 in ESDWSNs. The algorithm aims to find a data flow transmission path with maximum energy cost, and ensure the ESDWSNs with the lowest energy consumption. Therefore, the EAO algorithm takes into account both the remaining energy and the data redundancy degree.

The first step of the EAO algorithm determines the optimal number of edge regions. To establish the efficient three-layer architecture of ESDWSNs for data processing and forwarding, we use Theorem (3) to find the optimal number of edge regions. Therefore, the number of sensors in each edge region does not exceed $\lfloor \frac {n}{\mathcal {O}_{opt}}\rfloor$ , $e_{\omega }$ is the minimal energy consumption for satisfying the transmission of data request $\omega$ . In this step, the nodes send the remaining energy status $Z_{ij}$ to SDN controller, then the controller puts the nodes for satisfying the energy requirement into the valid node-set $T_{i}$ . After that, we divide the effective nodes in the set $T_{i}$ equally into $\mathcal {O}_{opt}$ edge regions. Assume that there is at least one valid link for ensuring the effective transmission of data flow in ESDWSNs.

Theorem 3:

In the ESDWSNs, the optimal number of edge regions $\mathcal {O}_{opt}$ is equal to $\sqrt {\frac {n\tau }{2\pi \varphi }}\frac {D}{d^{2}}$ .

Proof 3:

See the proof at the end of the paper.

The non-redundant packets will be found from the valid sensors in this second step. Each sensor will identify packets that are different from each other by calculating $\mathcal {N}^{(l)}_{j}=\mathcal {N}^{(l)}_{j}-\bigcup \limits _{j=j+1}^{\lfloor \frac {n}{\mathcal {O}_{opt}}\rfloor }\mathcal {N}_{T_{j+1}}$ . Each relay node only forwards data packets with non-redundant and satisfying user needs in ESDWSNs. For instance, assume that the source node $n_{1}$ sends the non-redundant packets $\mathcal {N}^{(l)}_{1}$ to the relay node $n_{2}$ , afterward, the relay node $n_{2}$ will forward the data packets of $\mathcal {N}_{1}^{(l)}$ and $\mathcal {N}_{2}^{(l)}$ to the next relay node $n_{3}$ , $\mathcal {N}^{(l)}_{2}=\mathcal {N}^{(l)}_{2}-\bigcup \limits _{i=3}^{\lfloor \frac {n}{\mathcal {O}_{opt}}\rfloor }\mathcal {N}^{(l)}_{T_{j+1}}$ .

The third step is to choose the routing paths with the maximum function of energy cost $F(x^{\ast })$ in the set $\mathcal {T}_{j}$ . The bigger the value of energy cost function $F(x^{\ast })$ is, the greater the get energy of sensor is, thereby making the transmission delay lower. $x^{\ast }$ is the optimal energy requirement for satisfying the constraints in ESDWSNs, the energy cost function $F(x^{\ast })$ and constraints will concentrate data transmission on nodes with the larger remaining energy. In the end, the SDN controller will transmit the operation result of EAO algorithm to the edge servers, and the edge servers will update the node status in the edge regions, respectively.

The processes of energy allocation optimization algorithm is summarized in Algorithm 2.

Algorithm 2 Energy Allocation Optimization (EAO) Algorithm

Require:

Collected data flow $\omega$ , the energy information $Z_{i}$ in the $i$ -th sensor.

Ensure:

The optimal transmission paths.

/*Step 1: determine the edge regions*/

1:

for $i=1\,\,to\,\, \mathcal {O}_{opt}$ do

2:

for $j=1\,\,to\,\, \lfloor \frac {n}{\mathcal {O}_{opt}}\rfloor$ do

3:

Nodes send energy information $Z_{ij}$ to SDN controller;

4:

Calculate the $\mathcal {O}_{opt}$ by Theorem (3);

5:

if $Z_{i}\geq e_{\omega }$ then

6:

Placing node $n_{ij}$ into set $T_{i}$ ;

7:

end if

8:

$j=j+1$ ;

9:

Select the node with the highest energy $e_{\max }$ from the set $T_{i}$ as the edge server;

10:

end for

11:

$i=i+1$ ;

12:

end for

/*Step 2: select the non-redundant data packets */

13:

Send data flow information $L_{i}$ to the edge server of current region $M_{j}$ ;

14:

for $j=1\,\,to\,\, \lfloor \frac {n}{\mathcal {O}_{opt}}\rfloor$ do

15:

$\mathcal {N}^{(l)}_{j}=\mathcal {N}^{(l)}_{j}-\bigcup \limits _{j=j+1}^{\lfloor \frac {n}{\mathcal {O}_{opt}}\rfloor }\mathcal {N}^{(l)}_{T_{j+1}}$ ;

16:

$j=j+1$ ;

17:

end for

18:

Send the non-redundant packet information in each node to the edge server $M_{j}$ ;

/*Step 3: select the optimal paths with energy balance*/

19:

for $i=1\,\,to\,\, \mathcal {O}_{opt}$ do

20:

for $j=1\,\,to\,\, \lfloor \frac {n}{\mathcal {O}_{opt}}\rfloor$ do

21:

Find all the nodes in $T_{i}$ that need to transmit packets;

22:

Establish a data flow transmission path and calculate optimization function $F(x^{\ast })$ by Algorithm (1);

23:

if the number of valid transmission paths is greater than or equal to 2 then

24:

$j=j+1$ ;

25:

Compare the $F(x^{\ast })$ and select the path in $\mathcal {T}_{j}$ with the maximum energy cost function $F(x^{\ast })$ ;

26:

else

27:

Deliver the path to set $\mathcal {T}_{j}$ ;

28:

end if

29:

end for

30:

Transfer data flow to the edge server $M_{i}$ of the current region;

31:

Edge server $M_{i}$ forwards data flow to the edge server $M_{i}+1$ ;

32:

$i=i+1$ ;

33:

end for

34:

The SDN controller updates the current network status and sends the control commands of data transmission.

D. Time Complexity Analysis

In this subsection, the time complexity of EAO algorithm will be analyzed in the following.

Firstly, the EAO algorithm will judge the number of edge regions, the process needs $O\left({\mathcal {O}_{opt}\lfloor \frac {n}{\mathcal {O}_{opt}}\rfloor }\right)$ time, where $\mathcal {O}_{opt}\lfloor \frac {n}{\mathcal {O}_{opt}}\rfloor \leq n$ . Secondly, it takes time $O\left({\lfloor \frac {n}{\mathcal {O}_{opt}}\rfloor }\right)$ for calculating the sent non-redundant packets in each valid node. In the end, the EAO algorithm needs to select the optimal transmission path with maximum energy cost function $F(x^{\ast })$ . The transmission path needs to compare the energy cost function $F(x^{\ast })$ of available paths in every edge region, and it has $\mathcal {O}_{opt}$ edge regions in ESDWSNs, thus the procedure consumes $O\left({\mathcal {O}_{opt}\lfloor \frac {n}{\mathcal {O}_{opt}}\rfloor }\right)$ time. And then the EAO algorithm will send operation results and non-redundant packets through employing $O\left({\mathcal {O}_{opt}\lfloor \frac {n}{\mathcal {O}_{opt}}\rfloor }\right)$ time, and this step takes $O\left({\mathcal {O}_{opt}\lfloor \frac {n}{\mathcal {O}_{opt}}\rfloor +\mathcal {O}_{opt}\lfloor \frac {n}{\mathcal {O}_{opt}}\rfloor }\right)$ time. In short, the total time required to execute EAO algorithm as follows $$\begin{align*}&\hspace {-2pc}O\left({\mathcal {O}_{opt}\lfloor \frac {n}{\mathcal {O}_{opt}}\rfloor +\lfloor \frac {n}{\mathcal {O}_{opt}}\rfloor +\mathcal {O}_{opt}\lfloor \frac {n}{\mathcal {O}_{opt}}\rfloor +\mathcal {O}_{opt}\lfloor \frac {n}{\mathcal {O}_{opt}}\rfloor }\right) \\=&O\left({3\mathcal {O}_{opt}\lfloor \frac {n}{\mathcal {O}_{opt}}\rfloor +\lfloor \frac {n}{\mathcal {O}_{opt}}\rfloor }\right) \\\leq&O\left({3n+\lfloor \frac {n}{\mathcal {O}_{opt}}\rfloor }\right)\tag{33}\end{align*}$$

Note that the complexity of the EAO algorithm is linear to the number of sensor $n$ and the number of edge region $\mathcal {O}_{opt}$ . Its time complexity is low, when the values of $n$ and $\mathcal {O}_{opt}$ are small. Therefore, the EAO algorithm is energy-efficient owing to lower time complexity in ESDWSNs.

SECTION V. Simulation Results

In this section, we use the NS-2 [19] to evaluate the performance of energy allocation optimization (EAO) algorithm. Also, we employ the energy balance level (EBL) to analyze the level of average energy consumption, and the total network throughput demonstrates the ability of the ESDWSNs for transmitting data flow.

A. Simulation Settings

The simulated monitoring region is set to 500m by 500m with 300 to 500 perceptual nodes. A SDN controller is responsible for all edge servers and 100 nodes deployed in each edge region. The edge region is expressed by a $R\times R\,\,2\text{D}$ region. We run ten rounds of simulation for each type of parameter setting. Assume that 10 to 50 percent of the collected data is irrelevant. Table 2 lists the network parameter settings in the simulation experiment.

TABLE 2 Parameter Setting

B. Comparison of the Energy Balance Levels

In this subsection, we first define the energy balance level (EBL), then contrast the EBL between the EAO algorithm and other baseline algorithms for the irrelevant data proportions.

Definition 2 (Energy Balance Level):

The energy balance level $EBL$ is defined as the ratio between the sum of the absolute value of energy consumption difference in different nodes and the total energy consumption in ESDWSNs, i.e., $$\begin{equation*} EBL=\frac {\|\big(\bigcup \limits _{i=1}^{\mathcal {O}_{opt}}\left({\bigcup \limits _{j=1}^{\lfloor \frac {n}{\mathcal {O}_{opt}}\rfloor }\mid E^{(l)}_{ij}-E^{(l)}_{i(j+1)}\mid }\right)\|_{1}}{\|\bigcup \limits _{i=1}^{n}E^{(l)}_{i}\|_{1}}\tag{34}\end{equation*}$$ $\lfloor \frac {n}{\mathcal {O}_{opt}}\rfloor$ denotes the optimal number of edge regions, $E^{(l)}_{i}$ is the energy consumption set of the $i$ -th sensor.

EBL reflects the level of average energy consumption in ESDWSNs. The bigger the EBL is, the lower the level of average consumption is, vice versa.

Fig. 3 depicts the developing of EBLs in ESDWSNs as the proportion of irrelevant data $P$ grows where $P = \xi -1$ . When the average single-hop transmission distance of the path $R$ is 50 meters, the energy balance level (EBL) is lowest. Moreover, the EBL decreases remarkably as the proportion of irrelevant data $P$ increases. This reason is that the relatively shorter transmission distance $R$ enables the energy consumption for transferring data to be lower, thereby making the energy consumption difference with path scheduling by using the EAO algorithm to smaller. According to equation (12), less data transfer can greatly saves energy consumption, so the EBLs become smaller. Therefore, the energy balance level (EBL) and the average transmission distance of $R$ and the proportion of irrelevant data $P$ are inversely proportional. The simulation result manifests that the appropriate values of $R$ and $P$ can improve the energy balance level.

FIGURE 3.

The energy balance level (EBL) comparison.

We define the EAO algorithm that does not adopt the energy cost maximization method as Stochastic Strategy. Accordingly, the EAO algorithm with the energy cost maximization method is Optimal Strategy. The Fig. 3(b) shows that the EBL of the optimal strategy is smaller than the EBL of stochastic strategy as the proportion of irrelevant data $P$ increases. This is because the EAO algorithm fully considers the remaining energy of all effective nodes and the redundancy of the data packets. However, the stochastic strategy does not effectively select the relay nodes and non-redundant data packets transmitted, thereby resulting in the energy consumption of the effective nodes being in an unbalanced status.

Furthermore, we consider the different distributions of the remaining energy of the two types of nodes, i.e., uniform distribution and uneven distribution. Fig. 3(c) shows that the EBL of uniform distribution is lower than the uneven distribution, however, there is not a big gap between these two EBLs. The reason is that the energy deployment by uniform distribution makes it easier to achieve efficient energy distribution, as the proportion of irrelevant data $P$ increases. Uneven energy distribution status requires the selection of appropriate relay nodes and the node status with data collected is random. Therefore, it is difficult to achieve the balanced distribution of energy in a short time. Also, the EAO algorithm constantly balances the energy consumption of each node in ESDWSNs, so that these two types of energy status do not influence greatly on the scheduling process of EAO algorithm.

C. Comparison of Network Throughput

In this subsection, we consider network throughput, which is defined as the total bandwidth of all flows in the ESDWSN to the cloud center. We assume proportional fairness when calculating flow bandwidth, i.e., each contending flow on an oversubscribed wireless link gets the bandwidth proportional to its demand. We compare the network throughput for the different transmission radii and different strategies as simulation time increases. Furthermore, we compare the network throughput of our algorithm, flow split optimization (FSO) algorithm [4] and profit maximization multi-round auction (PMMRA) algorithm [5].

Fig. 4 demonstrates the comparison of network throughput for different transmission radii $R$ . We can observe that the network throughput is maximized when $R$ equals to $50m$ . The larger the transmission radius, the fewer the number of appropriate relay sensors, and the more packets each node forwards. This results in a higher packet loss rate for data transmission between relay nodes, which further leads to a decrease in network throughput. In addition, the network throughput tend to stabilize as simulation time increases gradually. The reason is that the relay nodes in ESDWSNs are selected according to the energy consumption optimization with the continuous operation of the EAO algorithm, so that the data transmission of ESDWSNs becomes gradually balanced.

FIGURE 4.

Network throughput comparison.

Furthermore, we set that the distribution of sensors is disorganized. We experimented the throughput results of Stochastic Strategy and Optimal Strategy in ESDWSNs as the simulation time prolonged. We can find from Fig. 4(b) that the optimal strategy has a higher network throughput as the simulation time increases gradually, and the network throughput results of both algorithms have jitters to a certain degree. This reason is that the optimal strategy can select appropriate relay nodes in SDWSNs, which makes the ESDWSNs have more bandwidth and lower packet loss ratio. However, the stochastic strategy always selects the relay nodes with uneven energy consumption, which causes the data scheduling strategy to fail. Also, the uneven energy distribution of sensors has a certain impact on the data scheduling of EAO algorithm in SDWSNs, but the EAO algorithm still maintains the relatively higher network throughput.

Fig. 4(c) illustrates the comparison of network throughput among EAO, FSO, and PMMRA algorithms. Our EAO algorithm has higher network throughput than other two scheduling algorithms. This is because the FSO algorithm transfers the calculation and scheduling tasks to the SDN controller, so that the path selected by the SDN controller has the poor performance in real-time, which causes network congestion and forms packet loss to a certain degree. Also, the PMMRA algorithm only favors the calculation of the network status of perceptual layer, although the PMMRA algorithm improves the efficiency of the network operation to a certain extent, its scheduling inevitably causes the data transmission to local optimum, and the packet loss phenomenon between different edge regions will be common.

D. Comparison of Average Delay

The end-to-end average delay is the time of a packet transmission process from the perceptual node to being successfully received by the cloud center. The end-to-end average delay influences the timeliness of packet transmission and is a very important indicator in complex ecological environment monitoring and others delay-sensitive applications. In this subsection, we compare the average delay for the different strategies and EAO algorithm, FSO algorithm, and PMMRA algorithm in ESDWSNs as simulation time increases.

Fig. 5(a) shows the comparison of average delays between the optimization strategy and the stochastic strategy. The average delay is relatively smaller when the simulation time is greater than the cross point. Moreover, the average transmission delay of data packets is gradually stabilized when the simulation time is longer than 3 minutes. The optimization strategy will select the optimal transmission path by calculating the energy cost maximization function according to the current status of ESDWSNs. Therefore, this process will prolong the packet transmission delay through the EAO algorithm in ESDWSNs when the simulation time is less than the cross point.

FIGURE 5.

Average delay comparison.

Furthermore, Fig. 5(b) demonstrates the comparison of average delays among EAO algorithm, FSO algorithm, and PMMRA algorithm. The proposed EAO algorithm has lowest average delay than the FSO algorithm and PMMRA algorithm. The FSO algorithm has a jitter of transmission average delay when the simulation is at the jitter phase. This is because that the FSO algorithm will reorganize the split data flows according to the real-time status of SDWSNs, which will slow the data transmission delay. In addition, the PMMRA algorithm add edge servers for collecting information, matching and deciding the final scheduling strategy. However, the transmission load of all relay nodes is too heavier since the PMMRA algorithm does not remove the redundant data, so that the data transmission delay of PMMRA algorithm is too higher. This simulation further verifies the superiority of EAO algorithm over the performance of the others two algorithms.

SECTION VI. Proof of Theorem 3

The edge server needs to transfer data to the sink node, therefore, it needs to consume more energy than the normal perceived node. The energy consumption formula of the edge server is as follows [20]. $$\begin{equation*} e_{h}=\frac {n}{\mathcal {O}}\omega \bar {e}+\omega \bar {e}+\omega \varphi \hat {d}^{4}\tag{35}\end{equation*}$$ where $\omega$ is traffic flow request, and $\bar {e}$ represents the energy consumed in the electronic circuit, $\varphi$ indicates the power dissipation factor of the power amplifier circuit under the channel propagation model.

The energy consumption of data transmission among the nodes of edge region can be expressed by the following formula. $$\begin{equation*} \mathcal {E}=\omega \bar {e}+\omega \tau {d^{2}}\tag{36}\end{equation*}$$ where $d^{2}$ represents the distance from the node to the edge server. The area of each edge region is approximately $\frac {D^{2}}{\mathcal {O}}$ , $M$ denotes the length of ESDWSNs, the distribution density of the edge server is $\rho (x,y)$ . Then the energy consumption of all nodes within the square distance from the node to the edge server can be expressed as follows. $$\begin{align*} e(d^{2})=&\int \int (x^{2}+y^{2})\rho (x,y)dxdy \\=&r^{2}\rho (r,\theta)drd\theta\tag{37}\end{align*}$$

Assume that the radius of each edge region is $r=\frac {D}{\sqrt {\pi \mathcal {O}}}$ , $\rho =\frac {1}{\left({\frac {D^{2}}{\mathcal {O}}}\right)}$ , then formula (37) can be simplified as: $$\begin{align*} e(d^{2})=&\rho \int ^{2\pi }_{\theta =0}\int ^{\frac {D}{\sqrt {\pi \mathcal {O}}}}_{r=0}r^{2}drd\theta \\=&\frac {\rho D^{4}}{2\pi \mathcal {O}^{2}} \\=&\frac {D^{2}}{2\pi \mathcal {O}}\tag{38}\end{align*}$$

Therefore, Equation (36) can be converted into the following formula. $$\begin{equation*} e=\omega \bar {e}+\omega \tau e(d^{2})\tag{39}\end{equation*}$$

Then the energy consumed by the entire network is as follows: $$\begin{align*} \mathcal {E}_{all}=&\mathcal {O}E \\=&\omega \left({n\bar {e}+n\varphi d^{4}+n\bar {e}+n\tau \frac {D^{2}}{2\pi \mathcal {O}}}\right)\tag{40}\end{align*}$$

We find partial derivatives for $\mathcal {O}$ and get the optimal number of edge region. $$\begin{equation*} \mathcal {O}_{opt}=\sqrt {\frac {n\tau }{2\pi \varphi }}\frac {D}{d^{2}}\tag{41}\end{equation*}$$

This completes the proof.

SECTION VII. Conclusion and Future Work

The main challenge of energy consumption averaging and minimization (ECAM) problem in WSNs depends on the selection of available data packets and the optimal allocation of remaining energy in effective relay nodes. Coping with these challenges, we firstly establish the architecture of edge-SD wireless sensor networks (ESDWSNs) by introducing the concept of the SD and EC technologies. Then we formulate the ECAM problem as the energy cost function, which is constrained by the energy consumption in different network status. Furthermore, we utilize the ALM algorithm to find the optimal function value of energy cost. Afterward, we present energy allocation optimization (EAO) algorithm to settle the ECAM problem in ESDWSNs. An optimal transmission path with the lowest energy consumption and averaging is found. Finally, we use simulations to prove the performance of EAO algorithm in terms of the energy balance level and total network throughput.

Although our paper has better efficiency in solving the energy hole problem of ESDWSNs, the energy hole problem will be inevitable when the network running time is extended, which can reduce network running time and data transmission efficiency. Therefore, in the future work, we consider that the energy holes influence the normal operation of ESDWSNs because the hole numbers exceeds a certain threshold. Afterward, we will reorganize the edge regions for avoiding the energy hole through the SDN controller based on mastering the global network status. In addition, the data scheduling strategy of shortest transmission path will be designed and carried out experimental monitors in the actual complex ecological environment.