# An Energy-Aware Data Transmission Scheme under the Guarantee of Reliability for 3D WSNs

Authors: J. H. Zhang; Z. X. Yi; C. Y. Peng
Journal: Journal of Sensors
 (2020)

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## Abstract

Three-dimensional wireless sensor networks (3D WSNs) play an important role to provide data collection services for Internet of things (IoT) in the real applications. However, many of the existing WSN data collection researches are based on a relatively simple linear or plane network model. The three-dimensional space problems are simplified to two-dimensional plane, which limits the applicability. In this paper, the data collection in 3D WSN is studied. In the three-dimensional space, we firstly analyze the data loads, energy consumption, and end-to-end (E2E) delay of each node when the network is following the shortest path routing. The mathematical analysis of data loads and E2E delay of each node are presented. Based on the analysis of data loads and energy consumption, an energy-ware data transmission scheme is proposed to achieve the trade-off optimization between the E2E delay and network lifetime under the guarantee of the transmission reliability. The key point of the proposed scheme is to make fully use of the unbalanced energy consumption of the 3D WSN. The performance of the proposed scheme is discussed, analyzed, and evaluated. The theoretical analysis and simulation results show that the E2E network delay and energy efficiency can be improved under the constraint of transmission reliability.


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## Body

## 1. Introduction

Three-dimensional (3D) data collection is one of the key contents of Internet of things (IoT) [1–3]. Three-dimensional wireless sensor network (3D WSN), which is composed of many low-cost and low-power sensor nodes, is an important bridge of the integration of the physical world and the information world. Thus, 3D WSN is an important part of the data acquisition in IoT [4, 5]. For example, many different types of sensors are widely deployed in different fields of environmental and ecological monitoring [6–9], health monitoring [10–12], home automation [13, 14], traffic control [15, 16], and coal mine production security [17], etc. They are applied to collect many data from the environment, traffic, terminals, and industry production for IoT.However, data collection of 3D WSN faces the following challenges.(1)
Many of the existing WSN data collection researches are based on a relatively simple linear or plane network model. The research on 3D spatiotemporal data collection is mainly by extending the two-dimensional conclusion directly to the three-dimensional space. Or the three-dimensional space problems are simplified to two-dimensional plane. This limits the applicability [18–20]. Besides, the difficulty and computational complexity of data collection in 3D space of sensor network are much higher than that in 2D plane [21, 22]. For example, the problem of network topology and geometry is extremely complex in 3D space. Thus, the computational complexity of 3D space algorithm is increased greatly. And, the problem of data collection with optimal solution in 2D plane may be a nondeterministic polynomial (NP) problem in 3D space(2)
To reduce the end-to-end (E2E) delay and energy consumption under the guarantee of the transmission reliability is challenging in 3D wireless sensor networks. The data collection of 3D WSN is complicated because of the large-scale deployment, the limited energy of sensor nodes, and the unreliability of wireless communication. Some studies focus on improving the network transmission reliability, or reducing transmission delay, or improving the energy efficiency and prolonging the network lifetime. How to reduce the energy consumption of nodes and prolong the network lifetime while reducing the delay and ensuring the reliability of data transmission is an important problem to be effectively solved in 3D WSN data collectionTo address the above issues, the mathematical expression of the data loads and E2E delay of each node in 3D WSN is presented. Then, an energy-aware data collection scheme for 3D WSN is proposed in the paper. The trade-off optimization of reducing energy consumption, reducing time delay, and improving reliability is obtained in the proposed scheme. The main contributions of the paper are as follows:(1)
The data loads and E2E delay of each node in 3D WSN are analyzed. In the paper, a 3D WSN is considered, which is a more realistic and suitable assumption in the real applications. In the three-dimensional space, we analyze the data loads, energy consumption, and E2E delay of each node. The mathematical model of data loads and E2E delay of each node is statistically presented(2)
An energy-aware data transmission scheme achieving the trade-off optimization between the E2E delay and the network lifetime with the guarantee of transmission reliability is proposed for 3D WSN. The key point of the proposed scheme is to make fully use of the unbalanced energy consumption of the 3D WSN. Specifically, for the nodes with more residual energy, they send multiple-packet copies consecutively to ensure transmission reliability, and they do not need to wait for the returned acknowledgement (ACK) to reduce the waiting time and transmission delay of each hop. For the nodes with more data loads and energy consumption, they adopt the overtime retransmission scheme to reduce the energy consumption. That is, the node sends the packet and then waits for the ACK from the receiver, and it will retransmit the packet when the transmitted packet is lost and the time is out(3)
The validity of the mathematical analysis is demonstrated through comparison between theoretical and simulation results. And the performance of the proposed scheme is discussed, analyzed, and evaluated. The theoretical analysis and simulation results show that trade-off optimization of energy and E2E network delay under the constraint of transmission reliability can be achievedThe structure of this paper is organized as follows: the related works are reviewed in Section2. Section 3 depicts the system model and assumptions adopted in the paper. The mathematical model of data loads and E2E delay of each node in 3D WSN are presented in Section 4. Based on the analysis, an energy-aware data transmission scheme to achieve the trade-off optimization with the guarantee of transmission reliability is proposed for 3D WSN in Section 5, including performance analysis. The simulation results are described and analyzed in Section 6 to evaluate the theoretical analysis and the proposed scheme. Finally, the paper concludes in Section 7.


## 2. Related Works

Several studies have already been conducted to analyze and characterize the energy consumption and energy hole phenomena for 2D WSN. The energy consumption and energy hole phenomena are characterized for 2D WSN by applying theoretical analysis in Refs. [23, 24]. The analytical results are applied to 2D WSN routing and nodes deployment in order to balance the energy consumption and improve the network lifetime in Ref. [25]. However, 3D WSN is more realistic and suitable in the real applications, such as in coal mine and under water applications. As far as we know, there are no energy consumption analysis for 3D WSN.For reducing the cost energy, there are many existing studies, such as compressed sensing (CS), data aggregation, and sleep/awake scheduling. Compressed sensing theory is widely used to reduce the communication cost of wireless sensor nodes and realize information reconstruction [26, 27]. The matrix completion technique is applied to realize low redundancy data collection [28, 29]. In the aspect of cluster-based data aggregation, a variety of data aggregation optimization algorithms for cluster head selection have been proposed successively to prolong the network lifetime, such as Refs. [30, 31]. By considering energy consumption and constraints of communication, Ref. [31] presented an optimal topology of sensors algorithm by applying a new mathematical model for fuzzy clustering in 3D WSN. Data aggregation can effectively reduce the energy consumption of nodes, but most of the existing energy-efficient data aggregation research focuses on minimizing the energy consumption of data collection protocol. The network lifetime may not be improved by reducing the energy consumption of the whole protocol [25]. It is interesting and effective to improve the network performance by decreasing the energy consumption of hotspots and making full use of the extra energy consumption of nonhotspots. Sleep/wake-up scheduling is also a typical mechanism in wireless sensor networks to optimize the energy consumption. In Ref. [32], a self-adaptive sleep/wake-up scheduling approach is proposed based on the reinforcement learning technique. In the scheme, each node can decide its own operation mode of sleep, listening, or transmission in each time slot.There are some studies which focus on reducing transmission delay or ensuring transmission reliability besides reducing the cost energy in data collection of 3D WSN. Ref. [33] proposed a fault-tolerant and energy-efficient broadcast routing protocol by partitioning the 3D network into a set of easily manageable clusters. In Ref. [34], the authors proposed a dynamic three-dimensional fuzzy routing based on traffic probability to enhance network lifetime and increase packet delivery ratio. To ensure the packet transmission reliability, timeout retransmission, packet reproduction, and broadcast schemes are widely adopted in packet transmission to the destination. However, packet reproduction and broadcast schemes may cause exhausted energy consumption. And timeout retransmission scheme requires extra time for ensuring packet transmission reliability. Therefore, to reduce the E2E delay and energy consumption under the guarantee of the transmission reliability is a challenging issue for data gathering in 3D wireless sensor networks.In summary, there is no theoretical analysis to characterize the energy consumption for 3D WSN. Besides, how to achieve the trade-off optimization between the E2E delay and the network lifetime under the requirements of transmission reliability based on low hardware system configuration is one of the hot issues to be solved in the field.


## 3. System Model and Problem Statements

### 3.1. System Model

In the paper, we consider a WSN withN sensor nodes uniformly and randomly scattered in the sphere. The radius and node density are assumed R (m) and ρ (m−3), respectively. The sink node is placed at the centre of the sphere. Besides, the node transmission radius is assumed r (m). The transmission radius is unchanged. And nodes are no longer moving once deployed. The data is collected round by round. In one round of data collection, each node periodically generates one packet and transmitted it to the relaying node following the shortest path routing. Then, the packet is transmitted to the sink hop by hop. Due to the unreliability of wireless link in 3D WSN, the packet transmission is lossy, and we assume the successful probability of one-hop transmission is p. All the nodes except the sink have the same initial energy. The energy of sink is assumed enough to gather all the data. Each node consumes energy for sending or receiving packets. For sending or receiving one-bit data, the cost energy is calculated as follows.
(1)Et=Eμ+εϵdα,Er=Eμ,whered represents the distance between sending and relaying nodes. If it is less than the threshold, α=2. Otherwise, α=4. The other parameters’ means and values can be found in Ref. [35].


### 3.2. Problem Statements

3D WSN is more realistic and suitable in the real applications. However, many of the existing WSN data collection researches are based on a relatively simple linear or plane network model. As far as we know, there are no energy consumption analyses for 3D WSN. In addition, to reduce the E2E delay and energy consumption under the guarantee of the transmission reliability is a challenging issue for data gathering in 3D WSN. To address the issues, the meaning of the parameters is defined firstly.In this paper, the end-to-end network transmission delay is named E2E delay. For a source node, it is defined as the time required from the packet initiated by the node to be successfully transmitted to the sink. For example, the source nodei with h hops to the sink, if τm denotes the time delay of the mth hop, the E2E delay of node i is represented by ti, and ti=∑m=1hτm. Therefore, different source nodes may have different E2E delay. For the whole network, the E2E delay is defined as the maximum E2E delay of the source nodes in the 3D WSN. The objective of reducing E2E delay is to reduce the maximum E2E delay of the nodes in the 3D WSN. Thus, the mathematical expression is
(2)T=minmaxti0<i≤n.The network transmission reliability is defined as the proportion of the number of source nodes with generated packets successfully transmitted to the sink to the total number of nodes in the network. It is formulated as follows.(3)ξ=∑i=1NμiN,whereμi denotes whether the packet initiated by the source node i successfully transmitted to the sink. When it succeeds, μi=1. Otherwise, μi=0. If ζ represents the network transmission reliability constraint to be satisfied, ξ≥ζ is required.For the nodes with less data loads, the energy consumption is relatively small. To fully utilize the residual energy of these nodes is meaningful while ensuring the network lifetime. In the paper, the network lifetime is defined as the time from the beginning of the network operation to the death of the first node. Thus, it depends on the node with the maximal energy cost in the network. To prolong the network lifetime means to reduce the maximal energy cost. Thus, we have the following equation.(4)η=minmaxEcosti,whereEcosti represents the energy consumption of node.Therefore, the trade-off problem could be expressed by the following equation.(5)T=minmaxti0<i≤n,η=minmaxEcosti,i=1,2,⋯,N,ξ=∑i=1NμiN≥ζ.To summarize, the main purpose of the paper is to statistically analyze the data loads of each node in 3D WSN. Based on the analysis results, we want to achieve the trade-off optimization between the E2E network transmission delay and the energy efficiency under the constraint of the network reliability.


## 3.1. System Model

In the paper, we consider a WSN withN sensor nodes uniformly and randomly scattered in the sphere. The radius and node density are assumed R (m) and ρ (m−3), respectively. The sink node is placed at the centre of the sphere. Besides, the node transmission radius is assumed r (m). The transmission radius is unchanged. And nodes are no longer moving once deployed. The data is collected round by round. In one round of data collection, each node periodically generates one packet and transmitted it to the relaying node following the shortest path routing. Then, the packet is transmitted to the sink hop by hop. Due to the unreliability of wireless link in 3D WSN, the packet transmission is lossy, and we assume the successful probability of one-hop transmission is p. All the nodes except the sink have the same initial energy. The energy of sink is assumed enough to gather all the data. Each node consumes energy for sending or receiving packets. For sending or receiving one-bit data, the cost energy is calculated as follows.
(1)Et=Eμ+εϵdα,Er=Eμ,whered represents the distance between sending and relaying nodes. If it is less than the threshold, α=2. Otherwise, α=4. The other parameters’ means and values can be found in Ref. [35].


## 3.2. Problem Statements

3D WSN is more realistic and suitable in the real applications. However, many of the existing WSN data collection researches are based on a relatively simple linear or plane network model. As far as we know, there are no energy consumption analyses for 3D WSN. In addition, to reduce the E2E delay and energy consumption under the guarantee of the transmission reliability is a challenging issue for data gathering in 3D WSN. To address the issues, the meaning of the parameters is defined firstly.In this paper, the end-to-end network transmission delay is named E2E delay. For a source node, it is defined as the time required from the packet initiated by the node to be successfully transmitted to the sink. For example, the source nodei with h hops to the sink, if τm denotes the time delay of the mth hop, the E2E delay of node i is represented by ti, and ti=∑m=1hτm. Therefore, different source nodes may have different E2E delay. For the whole network, the E2E delay is defined as the maximum E2E delay of the source nodes in the 3D WSN. The objective of reducing E2E delay is to reduce the maximum E2E delay of the nodes in the 3D WSN. Thus, the mathematical expression is
(2)T=minmaxti0<i≤n.The network transmission reliability is defined as the proportion of the number of source nodes with generated packets successfully transmitted to the sink to the total number of nodes in the network. It is formulated as follows.(3)ξ=∑i=1NμiN,whereμi denotes whether the packet initiated by the source node i successfully transmitted to the sink. When it succeeds, μi=1. Otherwise, μi=0. If ζ represents the network transmission reliability constraint to be satisfied, ξ≥ζ is required.For the nodes with less data loads, the energy consumption is relatively small. To fully utilize the residual energy of these nodes is meaningful while ensuring the network lifetime. In the paper, the network lifetime is defined as the time from the beginning of the network operation to the death of the first node. Thus, it depends on the node with the maximal energy cost in the network. To prolong the network lifetime means to reduce the maximal energy cost. Thus, we have the following equation.(4)η=minmaxEcosti,whereEcosti represents the energy consumption of node.Therefore, the trade-off problem could be expressed by the following equation.(5)T=minmaxti0<i≤n,η=minmaxEcosti,i=1,2,⋯,N,ξ=∑i=1NμiN≥ζ.To summarize, the main purpose of the paper is to statistically analyze the data loads of each node in 3D WSN. Based on the analysis results, we want to achieve the trade-off optimization between the E2E network transmission delay and the energy efficiency under the constraint of the network reliability.


## 4. Analysis of Data Loads and E2E Delay for 3D WSN

In the section, the data loads, energy consumption, and E2E delay of each node in the 3D WSN is statistically analyzed. In the analysis, the packets are forwarded following the shortest path routing. They are relayed hop by hop from the source nodes to the sink. Thus, the mathematical analysis is carried out in the hop-by-hop transmission mode.


### 4.1. Data Loads and E2E Delay of Each Node without Packet Loss

We firstly analyze the simplest case, that is, there is no packet loss in transmission. Following the shortest path routing, nodei with distance l to the sink in 3D WSN will forward the packets from the source nodes with distance of l+r, l+2r,…, l+zr to the sink, where l+zr≤R. Figure 1 shows the sketch of data forwarding.Theorem 1.
For the nodei with distance l to the sink in a 3D WSN, the number of sending and receiving data packets is denoted, respectively, by ϵrl and ϵtl. They are mathematically represented by the following equation.
(6)ϵrl=∑n=1zl+nr2l2,ifl+zr≤R,0,ifl+r>R,ϵtl=ϵrl+1.Proof.
The sphere is dived into many microelements, such as microhexahedrons. Considering the microhexahedronClwith the distance of ll=hr+x,x<r to the sink, the microhexahedron is illustrated in Figure 2. As shown in Figure 2, the length of longitude direction is lsinφ, the width of latitude direction is lsinφdθ, and the radial height is dl, so the volume of the microhexahedron is
(7)dv=l2sinφdldφdθ.Figure 1
Sketch of data forwarding in 3D WSN.Figure 2
Illustration of the microhexahedron.Considering the node densityρ, the number of nodes in Cl is
(8)NCl=dv∙ρ=ρl2sinφdldφdθ.The nodes inCl need to receive and forward packets from the source nodes with distance of l+r, l+2r,…, l+zrl+zr≤R. Similarly, for the microhexahedron Cl+krwith the distance of l+krk=1,2,⋯,z, the number of nodes is NCl+kr=ρl+kr2sinφdldφdθ. Therefore, the total number of receiving packets for the microhexahedron Cl is
(9)ℵl=ρl+r2+l+2r2+⋯+l+zr2sinφdldφdθ.We can get the following equation.(10)ℵl=ρ∑k=1zl+kr2sinφdldφdθ.Thus, for the nodes at the distancel, the average number of packets received by each node is
(11)ϵrl=ℵlNCl=ρl+r2+l+2r2+⋯+l+zr2sinφdldφdθρl2sinφdldφdθ.After simplification, we have(12)ϵrl=∑n=1zl+nr2l2,ifl+zr≤R,0,ifl+r>R.Because each node generates one data packet in one round of data gathering, the number of transmitted packets of a node at the distancel is
(13)ϵtl=l+r2+l+2r2+⋯+l+zr2l2+1.That is,ϵtl=ϵrl+1.From the above equations, we can see that the data loads are smaller with the biggerl. Based on the above analysis, the energy cost of the node with distance l to the sink is calculated by
(14)El=ϵrl∗Eelec+ϵtl∗Eelec+ϵtl∗εfs∗x2,ifx<d0andz=0,El=ϵrl∗Eelec+ϵtl∗Eelec+ϵlt∗εfs∗x4,ifx≥d0andz=0,El=ϵrl∗Eelec+ϵtl∗Eelec+ϵtl∗εfs∗r2,ifr<d0andz≠0,El=ϵrl∗Eelec+ϵtl∗Eelec+ϵtl∗εfs∗r4,ifr<d0andz≠0,where l=hr+x,x<r.If we assume the one-round transmission time for one hop isτ, for the node with distance of l to the sink, the E2E delay is th=hτ/2; therefore, the maximal network delay is t=maxhτ/2.


### 4.2. Data Loads and E2E Delay of Each Node in 3D WSN When Sending One Data Packet and One ACK Returned in Hop-by-Hop Mode

Because of the unreliable wireless links, there are some packets lost in the transmission. To ensure the transmission reliability, the sender needs to wait for the returned ACK from the receiver to ensure the packets successfully transmitted. When the time is out, the sender does not receive the returned ACK; it will retransmit the packet.whereσh+k,th+kζ=σh+k,thζ=1−1−ph+kqh+kγh+kζ/ph+kqh+k and γh+kζ=log1−ζ1/h+k/log1−ph+kqh+k. φtl and φrl, respectively, represent the sending and receiving number of ACK packets.Theorem 2.
Under the required network transmission reliabilityζ, one-hop transmission reliability ph, and the returned ACK transmission reliability qh, the number of sending and receiving data packets and ACK packets is, respectively
(15)ϵtl=∑k=0zσh+k,thζl+kr2l2,ϵrl=ϵtl+rph+1,φtl=ϵrl,φrl=ϵtlphqh−1,Proof.
The maximum transmission number of packets for nodes atl=hr+x is denoted by γhζ. And the required one-hop transmission reliability is ζ1/h. We have ζ1/h=1−1−phqhγhζ. Hence, the following is obtained.
(16)γhζ=log1−ζ1/hlog1−phqh.According to Ref. [36], we get the number of transmitted packets statistically as the following equation.
(17)σh,thζ=1−1−phqhγhζphqh.Similarly, for the nodes withh+j hops to the sink, the maximum transmission number and the sending number of packets are statistically obtained as follows.
(18)γh+jζ=log1−ζ1/h+jlog1−ph+jqh+j,σh+j,th+jζ=σh+j,thζ=1−1−ph+jqh+jγh+jζph+jqh+j.According to Section4.1, for the node with distance of l to the sink, it needs to send its own generated packet and forward the packets from the source nodes at the position with distance of l+r, …, l+zr to the sink. Therefore, the statistical sending packet number is ϵtl=σh,thζ+σh+1,thζl+r2/l2+σh+2,thζl+2r2/l2+⋯+σh+z,thζl+zr2/l2. After simplification, we get
(19)ϵtl=∑k=0zσh+k,thζl+kr2l2,ϵrl=ϵtl+rph+1.That isϵrl=σh+1,th+1ζph+1l+r2/l2+σh+2,th+1ζph+1l+2r2/l2+⋯+σh+z,th+1ζph+1l+zr2/l2. The sending number of ACKs equals to the number of receiving data packets, which is φtl=ϵrl. And the number of receiving ACKs equals the number of receiving data packets of its next hop multiplied by qh−1 considering the ACK loss in the transmission. It is φrl=ϵtlphqh−1.If we assume the one-round transmission time for one hop isτ, the time interval between each sending packet copies is χ, and the one round trip time out is ι, the E2E delay for the node with distance of l to the sink is calculated by the following.
(20)ti=hτ+σh,thδ−1ι.


### 4.3. Data Loads and E2E Delay of Each Node When Sending One Data Packet and Returningn-ACKs in Hop-by-Hop Mode

In order to reduce the unnecessary energy cost caused by the ACK lost in packet retransmission scheme. Ref. [36] proposed the sending one packet and returning multi-ACKs transmission scheme. In the section, the data loads in 3D WSN adopting the sending one packet and returning multi-ACKs transmission scheme are statistically analyzed.where σh+k,thζ=1−1−ph1−1−qhnγh+kζ/ph1−1−qhn and γh+kζ=log1−ζ1/h+k/log1−ph+k1−1−qh+kn.Theorem 3.
For the nodes with distancel=hr+x to the sink, the one-hop transmission reliability ζ1/h needs to be satisfied. Under the condition that the one-hop transmission reliabilities are, respectively ph and qh for data packet and ACK packet, and there are n returned ACKs, the number of sending and receiving data and ACK packets is, respectively
(21)ϵtl=∑k=0zσh+k,thζl+kr2l2,ϵrl=∑k=1zσh+k,thζph+1l+kr2l2,φtl=n∙ϵrl,φrl=n∙ϵtlphqh−1,Proof.
Considering the maximum number of transmitting packetsγhζ, the following equation is obtained.
(22)ζ1/h=1−1−ph1−1−qhnγhζ.And we get(23)γhζ=log1−ζ1/hlog1−ph1−1−qhn.Therefore, the statistical number of the packets sent by the source node to ensure the one-hop transmission reliability is calculated by(24)σh,thζ=1−1−ph1−1−qhnγhζph1−1−qhn.Similarly, for the source nodes withh+j hops to the sink, the transmission number of their own generated packet is
(25)σh+j,thζ=σh+j,th+jζ=1−1−ph+j1−1−qh+jnγh+jζph+j1−1−qh+jn.Besides, the data loads transmitted by nodes with distancel to the sink also include the generated packets from the source nodes located in the area with the distances of l+r, l+2r,…, l+zr to the sink. Therefore, the number of transmitted and received packets is mathematically described as follows.
(26)ϵtl=σh,thζ+σh+1,thζl+r2l2+σh+2,thζl+2r2l2+⋯+σh+z,thζl+zr2l2,ϵrl=σh+1,th+1ζph+1l+r2l2+σh+2,th+1ζph+1l+2r2l2+⋯+σh+z,th+1ζph+1l+zr2l2.And becausen-ACKs are returned for one successful received data packet, the number of sent and received ACKs is obtained.
(27)φtl=n∙ϵrl,φrl=n∙ϵtlphqh−1.If we assume the one-round transmission time for one hop isτ, the time interval between each sent packet copy is χ, and the time out for retransmitting is ι, the E2E delay for the node with distance of l to the sink is
(28)hσh,thζ−1ι+τ2+σh,thζph+1n−1χ+τ2=hτ+σh,thζ−1ι+σh,thζphn−1χ.


### 4.4. Data Loads and E2E Delay of Each Node by Sending Multiple-Packet Copies Consecutively without ACK in Hop-by-Hop Mode

Theorem 4.
For ensuring the transmission reliability, the number of transmitted packets for nodes with distancel to the sink is calculated by the following equation.
(29)ϵtl=∑k=0zλh+kζl+kr2l2,ϵrl=ϵtl+rph+1,
whereλh+kζ=log1−ζ1/h+k/log1−ph+k.Proof.
To ensure the transmission reliability, the number of sending packet copies for nodes with distancel meets the following equation.
(30)ζ1/h=1−1−phλζ.Therefore, the number of packet copies sent consecutively without the returned ACK by source nodes with the distancel to the sink are mathematically formulated as:
(31)λhζ=log1−ζ1/hlog1−ph.It means that each forwarding node transmitsλhζ packet copies to ensure the one-hop transmission reliability in the routing path to the sink. Thus, the number of packets forwarded and received by nodes with distance l to the sink is obtained, respectively, as follows.
(32)ϵtl=λhζ+λh+1ζl+r2l2+λh+2ζl+2r2l2+⋯+λh+zζl+zr2l2,ϵrl=ϵtl+rph+1.If there are multiple-packet copies transmitted to the next hop, the transmission is successful when the next hop successfully receives the first arrived packet. For the number ofλhζ packet copies, the probability of the first, second, and third packet copy successfully transmitted is Pμ=1=ph, Pμ=2=1−phph, and Pμ=3=1−ph2ph, respectively. In the same way, the probability of the λhζth packet copy successfully transmitted is Pμ=λhζ=1−phλhζ−1ph. As described in the above section, τ denotes the hop transmission round time, and χ represents the interval time of transmission between packets or packet copies. For the source node with h hops to the sink, the E2E delay is represented as the following equation.
(33)th=h∑i=1λhζ1−phi−1phτ2+i−1χ.


## 4.1. Data Loads and E2E Delay of Each Node without Packet Loss

We firstly analyze the simplest case, that is, there is no packet loss in transmission. Following the shortest path routing, nodei with distance l to the sink in 3D WSN will forward the packets from the source nodes with distance of l+r, l+2r,…, l+zr to the sink, where l+zr≤R. Figure 1 shows the sketch of data forwarding.Theorem 1.
For the nodei with distance l to the sink in a 3D WSN, the number of sending and receiving data packets is denoted, respectively, by ϵrl and ϵtl. They are mathematically represented by the following equation.
(6)ϵrl=∑n=1zl+nr2l2,ifl+zr≤R,0,ifl+r>R,ϵtl=ϵrl+1.Proof.
The sphere is dived into many microelements, such as microhexahedrons. Considering the microhexahedronClwith the distance of ll=hr+x,x<r to the sink, the microhexahedron is illustrated in Figure 2. As shown in Figure 2, the length of longitude direction is lsinφ, the width of latitude direction is lsinφdθ, and the radial height is dl, so the volume of the microhexahedron is
(7)dv=l2sinφdldφdθ.Figure 1
Sketch of data forwarding in 3D WSN.Figure 2
Illustration of the microhexahedron.Considering the node densityρ, the number of nodes in Cl is
(8)NCl=dv∙ρ=ρl2sinφdldφdθ.The nodes inCl need to receive and forward packets from the source nodes with distance of l+r, l+2r,…, l+zrl+zr≤R. Similarly, for the microhexahedron Cl+krwith the distance of l+krk=1,2,⋯,z, the number of nodes is NCl+kr=ρl+kr2sinφdldφdθ. Therefore, the total number of receiving packets for the microhexahedron Cl is
(9)ℵl=ρl+r2+l+2r2+⋯+l+zr2sinφdldφdθ.We can get the following equation.(10)ℵl=ρ∑k=1zl+kr2sinφdldφdθ.Thus, for the nodes at the distancel, the average number of packets received by each node is
(11)ϵrl=ℵlNCl=ρl+r2+l+2r2+⋯+l+zr2sinφdldφdθρl2sinφdldφdθ.After simplification, we have(12)ϵrl=∑n=1zl+nr2l2,ifl+zr≤R,0,ifl+r>R.Because each node generates one data packet in one round of data gathering, the number of transmitted packets of a node at the distancel is
(13)ϵtl=l+r2+l+2r2+⋯+l+zr2l2+1.That is,ϵtl=ϵrl+1.From the above equations, we can see that the data loads are smaller with the biggerl. Based on the above analysis, the energy cost of the node with distance l to the sink is calculated by
(14)El=ϵrl∗Eelec+ϵtl∗Eelec+ϵtl∗εfs∗x2,ifx<d0andz=0,El=ϵrl∗Eelec+ϵtl∗Eelec+ϵlt∗εfs∗x4,ifx≥d0andz=0,El=ϵrl∗Eelec+ϵtl∗Eelec+ϵtl∗εfs∗r2,ifr<d0andz≠0,El=ϵrl∗Eelec+ϵtl∗Eelec+ϵtl∗εfs∗r4,ifr<d0andz≠0,where l=hr+x,x<r.If we assume the one-round transmission time for one hop isτ, for the node with distance of l to the sink, the E2E delay is th=hτ/2; therefore, the maximal network delay is t=maxhτ/2.


## 4.2. Data Loads and E2E Delay of Each Node in 3D WSN When Sending One Data Packet and One ACK Returned in Hop-by-Hop Mode

Because of the unreliable wireless links, there are some packets lost in the transmission. To ensure the transmission reliability, the sender needs to wait for the returned ACK from the receiver to ensure the packets successfully transmitted. When the time is out, the sender does not receive the returned ACK; it will retransmit the packet.whereσh+k,th+kζ=σh+k,thζ=1−1−ph+kqh+kγh+kζ/ph+kqh+k and γh+kζ=log1−ζ1/h+k/log1−ph+kqh+k. φtl and φrl, respectively, represent the sending and receiving number of ACK packets.Theorem 2.
Under the required network transmission reliabilityζ, one-hop transmission reliability ph, and the returned ACK transmission reliability qh, the number of sending and receiving data packets and ACK packets is, respectively
(15)ϵtl=∑k=0zσh+k,thζl+kr2l2,ϵrl=ϵtl+rph+1,φtl=ϵrl,φrl=ϵtlphqh−1,Proof.
The maximum transmission number of packets for nodes atl=hr+x is denoted by γhζ. And the required one-hop transmission reliability is ζ1/h. We have ζ1/h=1−1−phqhγhζ. Hence, the following is obtained.
(16)γhζ=log1−ζ1/hlog1−phqh.According to Ref. [36], we get the number of transmitted packets statistically as the following equation.
(17)σh,thζ=1−1−phqhγhζphqh.Similarly, for the nodes withh+j hops to the sink, the maximum transmission number and the sending number of packets are statistically obtained as follows.
(18)γh+jζ=log1−ζ1/h+jlog1−ph+jqh+j,σh+j,th+jζ=σh+j,thζ=1−1−ph+jqh+jγh+jζph+jqh+j.According to Section4.1, for the node with distance of l to the sink, it needs to send its own generated packet and forward the packets from the source nodes at the position with distance of l+r, …, l+zr to the sink. Therefore, the statistical sending packet number is ϵtl=σh,thζ+σh+1,thζl+r2/l2+σh+2,thζl+2r2/l2+⋯+σh+z,thζl+zr2/l2. After simplification, we get
(19)ϵtl=∑k=0zσh+k,thζl+kr2l2,ϵrl=ϵtl+rph+1.That isϵrl=σh+1,th+1ζph+1l+r2/l2+σh+2,th+1ζph+1l+2r2/l2+⋯+σh+z,th+1ζph+1l+zr2/l2. The sending number of ACKs equals to the number of receiving data packets, which is φtl=ϵrl. And the number of receiving ACKs equals the number of receiving data packets of its next hop multiplied by qh−1 considering the ACK loss in the transmission. It is φrl=ϵtlphqh−1.If we assume the one-round transmission time for one hop isτ, the time interval between each sending packet copies is χ, and the one round trip time out is ι, the E2E delay for the node with distance of l to the sink is calculated by the following.
(20)ti=hτ+σh,thδ−1ι.


## 4.3. Data Loads and E2E Delay of Each Node When Sending One Data Packet and Returningn-ACKs in Hop-by-Hop Mode

In order to reduce the unnecessary energy cost caused by the ACK lost in packet retransmission scheme. Ref. [36] proposed the sending one packet and returning multi-ACKs transmission scheme. In the section, the data loads in 3D WSN adopting the sending one packet and returning multi-ACKs transmission scheme are statistically analyzed.where σh+k,thζ=1−1−ph1−1−qhnγh+kζ/ph1−1−qhn and γh+kζ=log1−ζ1/h+k/log1−ph+k1−1−qh+kn.Theorem 3.
For the nodes with distancel=hr+x to the sink, the one-hop transmission reliability ζ1/h needs to be satisfied. Under the condition that the one-hop transmission reliabilities are, respectively ph and qh for data packet and ACK packet, and there are n returned ACKs, the number of sending and receiving data and ACK packets is, respectively
(21)ϵtl=∑k=0zσh+k,thζl+kr2l2,ϵrl=∑k=1zσh+k,thζph+1l+kr2l2,φtl=n∙ϵrl,φrl=n∙ϵtlphqh−1,Proof.
Considering the maximum number of transmitting packetsγhζ, the following equation is obtained.
(22)ζ1/h=1−1−ph1−1−qhnγhζ.And we get(23)γhζ=log1−ζ1/hlog1−ph1−1−qhn.Therefore, the statistical number of the packets sent by the source node to ensure the one-hop transmission reliability is calculated by(24)σh,thζ=1−1−ph1−1−qhnγhζph1−1−qhn.Similarly, for the source nodes withh+j hops to the sink, the transmission number of their own generated packet is
(25)σh+j,thζ=σh+j,th+jζ=1−1−ph+j1−1−qh+jnγh+jζph+j1−1−qh+jn.Besides, the data loads transmitted by nodes with distancel to the sink also include the generated packets from the source nodes located in the area with the distances of l+r, l+2r,…, l+zr to the sink. Therefore, the number of transmitted and received packets is mathematically described as follows.
(26)ϵtl=σh,thζ+σh+1,thζl+r2l2+σh+2,thζl+2r2l2+⋯+σh+z,thζl+zr2l2,ϵrl=σh+1,th+1ζph+1l+r2l2+σh+2,th+1ζph+1l+2r2l2+⋯+σh+z,th+1ζph+1l+zr2l2.And becausen-ACKs are returned for one successful received data packet, the number of sent and received ACKs is obtained.
(27)φtl=n∙ϵrl,φrl=n∙ϵtlphqh−1.If we assume the one-round transmission time for one hop isτ, the time interval between each sent packet copy is χ, and the time out for retransmitting is ι, the E2E delay for the node with distance of l to the sink is
(28)hσh,thζ−1ι+τ2+σh,thζph+1n−1χ+τ2=hτ+σh,thζ−1ι+σh,thζphn−1χ.


## 4.4. Data Loads and E2E Delay of Each Node by Sending Multiple-Packet Copies Consecutively without ACK in Hop-by-Hop Mode

Theorem 4.
For ensuring the transmission reliability, the number of transmitted packets for nodes with distancel to the sink is calculated by the following equation.
(29)ϵtl=∑k=0zλh+kζl+kr2l2,ϵrl=ϵtl+rph+1,
whereλh+kζ=log1−ζ1/h+k/log1−ph+k.Proof.
To ensure the transmission reliability, the number of sending packet copies for nodes with distancel meets the following equation.
(30)ζ1/h=1−1−phλζ.Therefore, the number of packet copies sent consecutively without the returned ACK by source nodes with the distancel to the sink are mathematically formulated as:
(31)λhζ=log1−ζ1/hlog1−ph.It means that each forwarding node transmitsλhζ packet copies to ensure the one-hop transmission reliability in the routing path to the sink. Thus, the number of packets forwarded and received by nodes with distance l to the sink is obtained, respectively, as follows.
(32)ϵtl=λhζ+λh+1ζl+r2l2+λh+2ζl+2r2l2+⋯+λh+zζl+zr2l2,ϵrl=ϵtl+rph+1.If there are multiple-packet copies transmitted to the next hop, the transmission is successful when the next hop successfully receives the first arrived packet. For the number ofλhζ packet copies, the probability of the first, second, and third packet copy successfully transmitted is Pμ=1=ph, Pμ=2=1−phph, and Pμ=3=1−ph2ph, respectively. In the same way, the probability of the λhζth packet copy successfully transmitted is Pμ=λhζ=1−phλhζ−1ph. As described in the above section, τ denotes the hop transmission round time, and χ represents the interval time of transmission between packets or packet copies. For the source node with h hops to the sink, the E2E delay is represented as the following equation.
(33)th=h∑i=1λhζ1−phi−1phτ2+i−1χ.


## 5. An Energy-Aware Data Transmission Scheme under the Guarantee of Delay and Reliability

From the statistical analysis, the overall trend of data loads is increasing with the nodes located in the region closer to the sink. Thus, nodes far from the sink have more residual energy. To fully use the residual energy to reduce the E2E delay and ensure the transmission reliability, an energy-aware data transmission scheme is proposed. The proposed energy-aware data transmission scheme is aimed at improving the energy utilization of the nodes far from the sink and achieves the trade-off optimization between the E2E delay and the energy under the constraint of the network reliability.


### 5.1. The Overall Scheme

The overall of the proposed scheme is illustrated in Figure3. As shown in the figure, the network is divided into near and far areas. For the source or relay nodes in the far area, there are multiple-packet copies which are transmitted consecutively without waiting for ACK. For nodes in the near area, they follow the overtime retransmission mechanism to conserve energy. The one-hop transmission time is reduced compared with the timeout retransmission scheme, because there is no need to wait for ACK or even retransmit the packet when timeout. Besides, the energy utilization of the nodes far from the sink is improved by sending multiple-packet copies consecutively.Figure 3
Sketch of the proposed scheme.


### 5.2. The Design of the Proposed Scheme

The parameterδ is used to represent the partition of the network into the far and near regions. If the source nodes with distance l to the sink are bigger than δR, the source nodes are in the far area and send multiple-packet copies consecutively without waiting for the return of ACK. The number of packet copies λhζ is calculated following equation (31).The multiple-packet copies are transmitted consecutively with the interval timeχ. When one of the packet copies is successfully transmitted to the next hop node, the one-hop transmission succeeds. Then, the packet is forwarded with the same number of copies until the next hop node with the distance is smaller than or equal to δR. If the relay nodes with the distance is smaller than δR, the packets are transmitted following the overtime retransmission scheme. The maximum number of transmissions is also λhζ for forwarding nodes in near area. If the distance l of the source nodes is smaller than δR, the source nodes adopt the overtime retransmission scheme. The maximum number of transmissions γhζ is calculated as equation (23). After the source node sends one data packet to its next hop, it waits for the returned ACK packet. If it receives the ACK, the data packet is successfully transmitted to its next hop. If it does not receive the returned ACK caused by data packet loss or ACK loss, the source node will retransmit the lost packet after the one round trip timeout to. The retransmission is implemented until the data packet successfully transmitted to the next hop and the ACK received under the constraint of the maximum number of transmissions γhζ.The detailed implementation is described in Algorithm1.Algorithm 1.
An energy-aware transmission scheme for 3D WSN.1Begin2//δ the network boundary3  //N:the total number of nodes4  //ζ: The required network reliability5for each source nodei=1:N6   djuge the node in near area or far area7if node i in far area8    calculatesλhζ following equation (31)9   initiatesλhζ packet copies and sends them per interval χ to the next hop10if the packet successfully transmitted to the next hop and the next hop in far area11go to step 912else if the successfully relayed next hop in near area13μ=114whileμ<=λhζ15    sends one data packet and waits for an ACK16if node i does not receive the returned ACK17μ=μ+1;18else19     break;20end21end22else record the packet is lost23end24else25  calculatesγhζ following equation (23)26μ=127whileμ<=γhζ28    sends one data packet and wait for an ACK29if node i does not receive the returned ACK30μ=μ+1;31else32     break;33end34end35end36end for37end


### 5.3. Performance Analysis and Discussion

In the section, the performance of the proposed scheme for 3D WSN is statistically analyzed.


#### 5.3.1. Data Loads

If the source node with distancel is bigger than δR, the node sends packet copies consecutively with the number of λhζ without waiting for the returned ACK. The source nodes in the location with distance of l+r, l+2r,…,l+zr adopt the same transmission scheme. They relay their generated data packet copies to the node with distance l hop by hop. Therefore, the data load of the node with distance l is obtained.
(34)ϵtl=λhζ+λh+1ζl+r2l2+λh+2ζl+2r2l2+⋯+λh+zζl+zr2l2,ϵrl=ϵtl+rph+1,φtl=φrl=0.If the nodes with distancel is less than δR, and we assume l+ϑr≥δR, we know that the nodes located in the region with the distance between l and l+ϑ−1r adopt the overtime retransmission scheme which is sending one data packet and returning n-ACKs. If the node is a source node, it transmits its generated packet. In addition, it needs to forward data packets from the nodes in the regions with the distance of l+r, l+2r,…, l+ϑ−1r, l+ϑr,…, l+zr. Therefore, the statistical number of data packets needs to be transmitted and received is, respectively, as follows.
(35)ϵtl=σh,thζ+σh+1,thζl+r2l2+σh+ϑ−1,thζl+ϑ−1r2l2+λh+ϑζl+ϑr2l2+⋯+λh+zζl+zr2l2,ϵrl=σh+1,th+1ζph+1l+r2l2+σh+2,th+1ζph+1l+2r2l2+⋯+σh+ϑ−1,th+1ζph+1l+ϑ−1r2l2+ϵtl+rph+1,where γhζ=log1−ζ1/h/log1−ph1−1−qhn and σh,thζ=1−1−ph1−1−qhnγhζ/ph1−1−qhn.The number of sent and received ACK packets is(36)φtl=n∙ϵrl,φrl=n∙ϵtlphqh−1.For a node withh hops to the sink, we get the following comparison from the above results,
(37)λhζ=log1−ζ1/hlog1−ph>γhζ=log1−ζ1/hlog1−ph1−1−qhnn≥1.γhζ is the maximum number of one data packet transmissions to ensure the one-hop transmission reliability. This means that the node following the multiple-packet copy transmission has more data loads than that adopting the timeout retransmission. Therefore, for the nodes located in the far area, the multiple-packet copy transmission could improve the residual energy utilization under the guarantee of the transmission reliability and without influence on the network lifetime. And the timeout retransmission scheme adopted for the nodes located in near area could reduce energy cost compared with the application of multiple-packet copy transmission.


#### 5.3.2. End-to-End Delay

If the source node with distancel is bigger than δR, the E2E delay is
(38)th=ω∑i=1λhζ1−phi−1phτ2+i−1χ+h−ωτ+σh,thζ−1ι+σh,thζphα−1χ,where ω denotes the hops of the data packets needed to be relayed to the next relay node in the region with distance less than δR. If the node with distance l is less than δR, the E2E delay is
(39)th=hτ+σh,thζ−1ι+σh,thζphα−1χ.For the E2E delay of a node, we need to compare the one-hop transmission time required between multiple-packet copy transmission scheme and the timeout retransmission scheme. We haveτ+σh,thζ−1ι+σh,thζphα−1χ−∑i=1λhζ1−phi−1phτ/2+i−1χ>τ/2+σh,thζ−1ι+σh,thζphα−1χ+λhζ−1χ>0.


## 5.1. The Overall Scheme

The overall of the proposed scheme is illustrated in Figure3. As shown in the figure, the network is divided into near and far areas. For the source or relay nodes in the far area, there are multiple-packet copies which are transmitted consecutively without waiting for ACK. For nodes in the near area, they follow the overtime retransmission mechanism to conserve energy. The one-hop transmission time is reduced compared with the timeout retransmission scheme, because there is no need to wait for ACK or even retransmit the packet when timeout. Besides, the energy utilization of the nodes far from the sink is improved by sending multiple-packet copies consecutively.Figure 3
Sketch of the proposed scheme.


## 5.2. The Design of the Proposed Scheme

The parameterδ is used to represent the partition of the network into the far and near regions. If the source nodes with distance l to the sink are bigger than δR, the source nodes are in the far area and send multiple-packet copies consecutively without waiting for the return of ACK. The number of packet copies λhζ is calculated following equation (31).The multiple-packet copies are transmitted consecutively with the interval timeχ. When one of the packet copies is successfully transmitted to the next hop node, the one-hop transmission succeeds. Then, the packet is forwarded with the same number of copies until the next hop node with the distance is smaller than or equal to δR. If the relay nodes with the distance is smaller than δR, the packets are transmitted following the overtime retransmission scheme. The maximum number of transmissions is also λhζ for forwarding nodes in near area. If the distance l of the source nodes is smaller than δR, the source nodes adopt the overtime retransmission scheme. The maximum number of transmissions γhζ is calculated as equation (23). After the source node sends one data packet to its next hop, it waits for the returned ACK packet. If it receives the ACK, the data packet is successfully transmitted to its next hop. If it does not receive the returned ACK caused by data packet loss or ACK loss, the source node will retransmit the lost packet after the one round trip timeout to. The retransmission is implemented until the data packet successfully transmitted to the next hop and the ACK received under the constraint of the maximum number of transmissions γhζ.The detailed implementation is described in Algorithm1.Algorithm 1.
An energy-aware transmission scheme for 3D WSN.1Begin2//δ the network boundary3  //N:the total number of nodes4  //ζ: The required network reliability5for each source nodei=1:N6   djuge the node in near area or far area7if node i in far area8    calculatesλhζ following equation (31)9   initiatesλhζ packet copies and sends them per interval χ to the next hop10if the packet successfully transmitted to the next hop and the next hop in far area11go to step 912else if the successfully relayed next hop in near area13μ=114whileμ<=λhζ15    sends one data packet and waits for an ACK16if node i does not receive the returned ACK17μ=μ+1;18else19     break;20end21end22else record the packet is lost23end24else25  calculatesγhζ following equation (23)26μ=127whileμ<=γhζ28    sends one data packet and wait for an ACK29if node i does not receive the returned ACK30μ=μ+1;31else32     break;33end34end35end36end for37end


## 5.3. Performance Analysis and Discussion

In the section, the performance of the proposed scheme for 3D WSN is statistically analyzed.


### 5.3.1. Data Loads

If the source node with distancel is bigger than δR, the node sends packet copies consecutively with the number of λhζ without waiting for the returned ACK. The source nodes in the location with distance of l+r, l+2r,…,l+zr adopt the same transmission scheme. They relay their generated data packet copies to the node with distance l hop by hop. Therefore, the data load of the node with distance l is obtained.
(34)ϵtl=λhζ+λh+1ζl+r2l2+λh+2ζl+2r2l2+⋯+λh+zζl+zr2l2,ϵrl=ϵtl+rph+1,φtl=φrl=0.If the nodes with distancel is less than δR, and we assume l+ϑr≥δR, we know that the nodes located in the region with the distance between l and l+ϑ−1r adopt the overtime retransmission scheme which is sending one data packet and returning n-ACKs. If the node is a source node, it transmits its generated packet. In addition, it needs to forward data packets from the nodes in the regions with the distance of l+r, l+2r,…, l+ϑ−1r, l+ϑr,…, l+zr. Therefore, the statistical number of data packets needs to be transmitted and received is, respectively, as follows.
(35)ϵtl=σh,thζ+σh+1,thζl+r2l2+σh+ϑ−1,thζl+ϑ−1r2l2+λh+ϑζl+ϑr2l2+⋯+λh+zζl+zr2l2,ϵrl=σh+1,th+1ζph+1l+r2l2+σh+2,th+1ζph+1l+2r2l2+⋯+σh+ϑ−1,th+1ζph+1l+ϑ−1r2l2+ϵtl+rph+1,where γhζ=log1−ζ1/h/log1−ph1−1−qhn and σh,thζ=1−1−ph1−1−qhnγhζ/ph1−1−qhn.The number of sent and received ACK packets is(36)φtl=n∙ϵrl,φrl=n∙ϵtlphqh−1.For a node withh hops to the sink, we get the following comparison from the above results,
(37)λhζ=log1−ζ1/hlog1−ph>γhζ=log1−ζ1/hlog1−ph1−1−qhnn≥1.γhζ is the maximum number of one data packet transmissions to ensure the one-hop transmission reliability. This means that the node following the multiple-packet copy transmission has more data loads than that adopting the timeout retransmission. Therefore, for the nodes located in the far area, the multiple-packet copy transmission could improve the residual energy utilization under the guarantee of the transmission reliability and without influence on the network lifetime. And the timeout retransmission scheme adopted for the nodes located in near area could reduce energy cost compared with the application of multiple-packet copy transmission.


### 5.3.2. End-to-End Delay

If the source node with distancel is bigger than δR, the E2E delay is
(38)th=ω∑i=1λhζ1−phi−1phτ2+i−1χ+h−ωτ+σh,thζ−1ι+σh,thζphα−1χ,where ω denotes the hops of the data packets needed to be relayed to the next relay node in the region with distance less than δR. If the node with distance l is less than δR, the E2E delay is
(39)th=hτ+σh,thζ−1ι+σh,thζphα−1χ.For the E2E delay of a node, we need to compare the one-hop transmission time required between multiple-packet copy transmission scheme and the timeout retransmission scheme. We haveτ+σh,thζ−1ι+σh,thζphα−1χ−∑i=1λhζ1−phi−1phτ/2+i−1χ>τ/2+σh,thζ−1ι+σh,thζphα−1χ+λhζ−1χ>0.


## 5.3.1. Data Loads

If the source node with distancel is bigger than δR, the node sends packet copies consecutively with the number of λhζ without waiting for the returned ACK. The source nodes in the location with distance of l+r, l+2r,…,l+zr adopt the same transmission scheme. They relay their generated data packet copies to the node with distance l hop by hop. Therefore, the data load of the node with distance l is obtained.
(34)ϵtl=λhζ+λh+1ζl+r2l2+λh+2ζl+2r2l2+⋯+λh+zζl+zr2l2,ϵrl=ϵtl+rph+1,φtl=φrl=0.If the nodes with distancel is less than δR, and we assume l+ϑr≥δR, we know that the nodes located in the region with the distance between l and l+ϑ−1r adopt the overtime retransmission scheme which is sending one data packet and returning n-ACKs. If the node is a source node, it transmits its generated packet. In addition, it needs to forward data packets from the nodes in the regions with the distance of l+r, l+2r,…, l+ϑ−1r, l+ϑr,…, l+zr. Therefore, the statistical number of data packets needs to be transmitted and received is, respectively, as follows.
(35)ϵtl=σh,thζ+σh+1,thζl+r2l2+σh+ϑ−1,thζl+ϑ−1r2l2+λh+ϑζl+ϑr2l2+⋯+λh+zζl+zr2l2,ϵrl=σh+1,th+1ζph+1l+r2l2+σh+2,th+1ζph+1l+2r2l2+⋯+σh+ϑ−1,th+1ζph+1l+ϑ−1r2l2+ϵtl+rph+1,where γhζ=log1−ζ1/h/log1−ph1−1−qhn and σh,thζ=1−1−ph1−1−qhnγhζ/ph1−1−qhn.The number of sent and received ACK packets is(36)φtl=n∙ϵrl,φrl=n∙ϵtlphqh−1.For a node withh hops to the sink, we get the following comparison from the above results,
(37)λhζ=log1−ζ1/hlog1−ph>γhζ=log1−ζ1/hlog1−ph1−1−qhnn≥1.γhζ is the maximum number of one data packet transmissions to ensure the one-hop transmission reliability. This means that the node following the multiple-packet copy transmission has more data loads than that adopting the timeout retransmission. Therefore, for the nodes located in the far area, the multiple-packet copy transmission could improve the residual energy utilization under the guarantee of the transmission reliability and without influence on the network lifetime. And the timeout retransmission scheme adopted for the nodes located in near area could reduce energy cost compared with the application of multiple-packet copy transmission.


## 5.3.2. End-to-End Delay

If the source node with distancel is bigger than δR, the E2E delay is
(38)th=ω∑i=1λhζ1−phi−1phτ2+i−1χ+h−ωτ+σh,thζ−1ι+σh,thζphα−1χ,where ω denotes the hops of the data packets needed to be relayed to the next relay node in the region with distance less than δR. If the node with distance l is less than δR, the E2E delay is
(39)th=hτ+σh,thζ−1ι+σh,thζphα−1χ.For the E2E delay of a node, we need to compare the one-hop transmission time required between multiple-packet copy transmission scheme and the timeout retransmission scheme. We haveτ+σh,thζ−1ι+σh,thζphα−1χ−∑i=1λhζ1−phi−1phτ/2+i−1χ>τ/2+σh,thζ−1ι+σh,thζphα−1χ+λhζ−1χ>0.


## 6. Simulation Results and Analysis

In the section, the mathematical analysis of data loads and E2E delay in 3D WSN are evaluated by comparison with the simulation results, and performance of the proposed scheme is validated. The evaluation on the performance of the proposed scheme includes the influence of parameters and comparison with the existed schemes. The simulation results are analyzed in detail.


### 6.1. Parameter Settings

In the simulation, a 3D spherical WSN withN=5000 nodes and R=50m is considered. In the network, sink is located at the center. Each node except sink with the same initial energy 1 (J) and the transmission radius is assumed r=10m. Other parameters are shown in Table 1. The simulation is implemented hundreds of times. The average simulation result of one round of data collection is adopted to evaluate the performance.Table 1
Parameters’ means and values.
ParameterMeansValued0 (m)Distance threshold87Eelec (nJ/bit)Transmitting circuit loss50εfs (pJ/bit/m2)Power amplification energy10εamp (pJ/bit/m4)Power amplification energy0.0013lp (bit)Data packet length100la (bit)ACK packet length10τ (ms)Round-trip time50ι (ms)Round-trip time to out60χ (ms)The interval time for next packet transmission10


### 6.2. Evaluation on the Mathematical Analysis

In the section, the mathematical analysis of data loads in 3D WSN is evaluated firstly. The validity is proved by comparing with the simulation results. The correlation coefficient between the theoretical and experimental results is applied to the quantitative evaluation on the fitting degree.


#### 6.2.1. Evaluating the Mathematical Analysis of Data Loads and E2E Delay When Sending One Data Packet without Packet Loss

The mathematical analysis of data loads in 3D WSN is evaluated when following the sending one data packet without packet loss. Figures4(a) and 4(b) show the comparison between the simulation results and the theoretical analysis. As shown in Figure 4(a), the overall trend of the number of the sending packet is increasing with the nodes closer to the sink, which is reflected by both the simulation and theoretical results. In addition, we can see that the number of sending packet of each node in theory is almost median line of the simulation. The difference between them is caused by the assumption in theory analysis that the nodes in 3D WSN are dense enough. The correlation coefficient between the theoretical and experimental results is 0.6306. Therefore, the theoretical results are in reasonable agreement with the simulation results. The comparison of E2E delay in mathematical analysis and simulation is shown in Figure 4(b). The average error between the theoretical and simulation results is nearly 0. The correlation coefficient between the theoretical and simulation results is nearly 1. Therefore, the mathematical analysis of E2E delay in the 3D WSN is verified when sending one data packet without packet loss. Figure 5 describes the simulation results of energy cost of each node in the 3D WSN. The result shows that the energy cost increase as the nodes are closer to the sink, and the changing trends with distance to the sink is almost the same as that of the data loads.Figure 4
Comparison of the simulation results and the theoretical analysis in a 3D WSN. (a) The number of sending packet of each node. (b) The E2E delay of each node.
(a)(b)Figure 5
Energy cost in the 3D WSN.


#### 6.2.2. Evaluating the Mathematical Analysis of Data Loads and E2E Delay When Sending One Data Packet with One ACK Returned in Hop-by-Hop Mode

The comparison of the simulation results and the theoretical results of data loads and E2E delay in the 3D WSN when sending one data packet with one ACK returned are shown in Figures6(a) and 6(b). As shown in Figure 6(a), the overall increasing trend of the number of the sending packet is reflected by both the theoretical results and the simulation results. From the figure, we can see that the theoretical result of each node is the median line of the simulation results. The correlation coefficient between the theoretical and experimental results is 0.6190. The mathematical results agree with the simulation results to a certain extent. In Figure 6(b), the comparison of E2E delay between mathematical analysis and the simulation results is shown. The correlation coefficient between the theoretical and experimental results is 0.9898. It shows that the simulation results are consistent with the theoretical results.Figure 6
Comparison of the simulation results and the theoretical analysis in a 3D WSN. (a) The number of sending packet of each node. (b) The E2E delay of each node.
(a)(b)


#### 6.2.3. Evaluating the Mathematical Analysis of Data Loads and E2E Delay in 3D WSN When Sending One Data Packet and Returningn-ACKs

To evaluate the validity of the mathematical analysis of data loads and E2E delay in 3D WSN when sending one data packet and returningn-ACKs, we assume n=2 randomly. When sending one data packet and returning 2-ACKs, the validity of the theoretical analysis is proved by comparing with the simulation results. Figures 7(a) and 7(b) show the comparison of the simulation and theoretical results in the 3D WSN. In Figure 7(a), both the theoretical results and the simulation results demonstrate the overall increasing trend of the number of the sending packet with the nodes closer to the sink. The correlation coefficient between the theoretical and experimental results is 0.6232. The difference between them is because that the nodes in 3D WSN are assumed dense enough in the mathematical characterizing. The comparison of E2E delay between the mathematical analysis and the simulation is shown in Figure 7(b). As can be seen from the figure, the mathematical analysis of E2E delay in 3D WSN is in consistent with the simulation results. The correlation coefficient between the theoretical and experimental results is 0.9929.Figure 7
Comparison of the simulation results and the theoretical analysis in a 3D WSN. (a) The number of sending packet of each node. (b) The E2E delay of each node.
(a)(b)


#### 6.2.4. Evaluating the Mathematical Analysis of Data Loads and E2E Delay in 3D WSN When Sending Multiple-Packet Copies without Returning ACK

When sending multiple-packet copies without returning ACK, the comparison of the simulation and the theoretical results in the 3D WSN is shown in Figures8(a) and 8(b). As can be seen from Figure 8(a), the changing trend of the number of the sending packet is increasing with the nodes closer to the sink, which is reflected by both the theoretical analysis and the simulation results. From the figure, the theoretical result of each node is the median line of the simulation result. The correlation coefficient between the theoretical and experimental results is 0.6217. The mathematical results are consistent with the simulation results. The comparison of E2E delay between mathematical analysis and simulation is demonstrated in Figure 8(b). The correlation coefficient between the theoretical and experimental results is 0.9992. They are found to be in reasonable line with the simulation results.Figure 8
Comparison of the simulation results and the theoretical analysis in a 3D WSN. (a) The number of sending packet of each node. (b) The E2E delay of each node.
(a)(b)


### 6.3. Performance Evaluation on Different Parameters

In the part, the parameters influence on the proposed scheme are evaluated and analyzed. The performance of the E2E delay, the energy consumption, and the network reliability are estimated under the differentδ, α, p, and r.


#### 6.3.1. Different Network Partitionδ

As described above, the parameterδR is used to divide the network into two partitions. Under different δ, the network performance of reliability, energy cost, and E2E delay are presented. The results are demonstrated in Table 2. From this table, we can see that the proposed energy-aware transmission scheme can meet the required transmission reliability ζ=0.8 whether δ=0.4 or δ=0.2. There is more energy consumption for δ=0.2 shown in Table 2. And there is less energy cost for δ=0.4. This is because that there are less nodes adopting the multiple-packet copy transmission and more nodes applying the timeout retransmission scheme when δ has bigger value. The energy cost of timeout retransmission scheme is less than that of multiple-packet copy transmission scheme under the same network conditions. Therefore, there are less energy cost when δ=0.4 compared with when δ=0.2. For this reason, the E2E delay and the maximal network E2E delay when δ=0.4is bigger than when δ=0.2 as shown in the table.Table 2
Simulation results under differentδ when p=0.7, α=1, r=10, and ζ=0.8.
δReliabilityThe maximal energy cost (J)The maximal E2E delay(ms)δ=0.285.95%0.0043307.5556δ=0.485.15%0.0037486.1250


#### 6.3.2. Different Number of Returned ACKsα

Under differentα, the network performance of reliability, energy cost, and E2E delay is evaluated. In Table 3, the network reliability under different α is shown. From the table, the required transmission reliability is satisfied. And with the increasing number of ACK, the transmission reliability is improved, and the maximal energy cost of the nodes in one round of data gathering is reduced. The E2E delay is also reduced. This is because that the ACK loss will lead to packet retransmission. Thus, reducing the ACK loss will reduce the unnecessary packet retransmission. Therefore, a reasonable number of returned ACK will decrease the unnecessary packet retransmission caused by ACK loss.Table 3
Simulation results under differentαwhen p=0.7, δ=0.4, r=10, and ζ=0.8.
αReliabilityThe maximal energy cost (J)The maximal E2E delay (ms)α=185.15%0.0037486.1250α=289.64%0.0031419.4186α=391.76%0.0031379.7674


#### 6.3.3. Different Hop Transmission Reliabilityp

The network performance results of reliability, energy cost, and E2E delay are presented in Table4 under different p. In Table 4, the transmission reliability is ensured under different one-hop transmission reliability p whether δ=0.2 or δ=0.4. However, the energy cost is bigger with the worse link. The E2E delay is also bigger compared with high reliable transmission link as shown in the table. It is obvious that to meet the transmission reliability constraint, the worse link needs a greater number of packet retransmissions, which cause more energy cost and larger E2E delay.Table 4
Simulation results under differentpwhen α=1, r=10, and ζ=0.8.
pδReliabilityThe maximal energy cost (J)The maximal E2E delay (ms)p=0.7δ=0.285.95%0.0043307.5556δ=0.485.15%0.0037486.1250p=0.6δ=0.285.06%0.0053380.2439δ=0.484.37%0.0047670.7317


#### 6.3.4. Different Transmission Radiusr

Under differentr, the network performance of reliability, energy cost, and E2E delay are evaluated. Table 5 shows the results. From the table, the transmission reliability is ensured under different r. Besides, the results show the overall changing trend of the energy cost of the nodes in one round of data gathering. The overall trend is increasing with smaller distance to the sink. This is because that the nodes near the sink need to forward more packets. Besides, as can be seen from the table, the E2E delay is reduced with the bigger transmission radius. This is because that the bigger radius could reduce the transmission hops to the sink. Also, the number of sending packet copies or the maximal retransmission number is reduced. Therefore, the E2E delay is decreased.Table 5
Simulation results under differentrwhen α=1, δ=0.2, p=0.7, and ζ=0.8.
rReliabilityThe maximal energy cost (J)The maximal E2E delay (ms)r=1085.95%0.0043307.5556r=1287.55%0.0054262.5000r=1588.07%0.0037226.9048


#### 6.3.5. Different Network Reliabilityζ

Under different requirements of network reliabilityζ, the network performance of achieved reliability, energy cost, and E2E delay are presented in the part. Table 6 shows the network actual reliability achieved under different ζ. From the table, the results show the proposed transmission scheme could meet the different transmission reliability requirements. As can be seen from the table, the energy cost is bigger when the bigger required reliability under different transmission reliability constraints. It is obvious that higher transmission reliability needs the greater number of packet retransmissions under the same wireless link lost. Similarly, the E2E delay also gets bigger as can be seen from the table.Table 6
Simulation results under differentζwhen α=1, δ=0.2, p=0.7, and r=10.
ζReliabilityThe maximal energy cost (J)The maximal E2E delay (ms)ζ=80%85.95%0.0043307.5556ζ=70%81.49%0.0040299.0244ζ=60%64.25%0.0028290.4176


### 6.4. Performance Comparison

In the section, the performance of the proposed scheme is estimated by comparing with the existing schemes of multiple-packet copies without ACK returned transmission (MP) and timeout retransmission scheme represented typically by sending one data packet with one ACK returned (ODOA).In Figure9(a), the results demonstrate that the proposed energy-aware transmission scheme can meet the required transmission reliability ζ=0.8. As can be seen from Figures 9(b) and 9(c), there is more energy consumption for MP scheme. And there is less energy cost for ODOA scheme. The energy cost of the proposed transmission scheme is less than MP and bigger than ODOA from the overall perspective. From Figures 9(d) and 9(e), we can see that the delay is bigger with the greater distance of the nodes to the sink overall. The maximum delay of the proposed transmission scheme is bigger than the MP and smaller than the ODOA. Compared with MP, the maximum energy cost is reduced by 29.13% and 17.23%, respectively, for the proposed scheme when δ=0.2 and δ=0.4. Comparing the proposed scheme with the ODOA, the maximum delay is, respectively, reduced by 43.60% and 64.32% when δ=0.2 and δ=0.4. Therefore, the proposed scheme can realize the trade-off optimization between energy efficiency and E2E delay.Figure 9
Comparison between schemes underα=1, p=0.7, and r=10. (a) The network reliability. (b) The energy cost of each node. (c) The maximal energy consumption. (d) The E2E delay of each node. (e) The maximal E2E delay.
(a)(b)(c)(d)(e)


## 6.1. Parameter Settings

In the simulation, a 3D spherical WSN withN=5000 nodes and R=50m is considered. In the network, sink is located at the center. Each node except sink with the same initial energy 1 (J) and the transmission radius is assumed r=10m. Other parameters are shown in Table 1. The simulation is implemented hundreds of times. The average simulation result of one round of data collection is adopted to evaluate the performance.Table 1
Parameters’ means and values.
ParameterMeansValued0 (m)Distance threshold87Eelec (nJ/bit)Transmitting circuit loss50εfs (pJ/bit/m2)Power amplification energy10εamp (pJ/bit/m4)Power amplification energy0.0013lp (bit)Data packet length100la (bit)ACK packet length10τ (ms)Round-trip time50ι (ms)Round-trip time to out60χ (ms)The interval time for next packet transmission10


## 6.2. Evaluation on the Mathematical Analysis

In the section, the mathematical analysis of data loads in 3D WSN is evaluated firstly. The validity is proved by comparing with the simulation results. The correlation coefficient between the theoretical and experimental results is applied to the quantitative evaluation on the fitting degree.


### 6.2.1. Evaluating the Mathematical Analysis of Data Loads and E2E Delay When Sending One Data Packet without Packet Loss

The mathematical analysis of data loads in 3D WSN is evaluated when following the sending one data packet without packet loss. Figures4(a) and 4(b) show the comparison between the simulation results and the theoretical analysis. As shown in Figure 4(a), the overall trend of the number of the sending packet is increasing with the nodes closer to the sink, which is reflected by both the simulation and theoretical results. In addition, we can see that the number of sending packet of each node in theory is almost median line of the simulation. The difference between them is caused by the assumption in theory analysis that the nodes in 3D WSN are dense enough. The correlation coefficient between the theoretical and experimental results is 0.6306. Therefore, the theoretical results are in reasonable agreement with the simulation results. The comparison of E2E delay in mathematical analysis and simulation is shown in Figure 4(b). The average error between the theoretical and simulation results is nearly 0. The correlation coefficient between the theoretical and simulation results is nearly 1. Therefore, the mathematical analysis of E2E delay in the 3D WSN is verified when sending one data packet without packet loss. Figure 5 describes the simulation results of energy cost of each node in the 3D WSN. The result shows that the energy cost increase as the nodes are closer to the sink, and the changing trends with distance to the sink is almost the same as that of the data loads.Figure 4
Comparison of the simulation results and the theoretical analysis in a 3D WSN. (a) The number of sending packet of each node. (b) The E2E delay of each node.
(a)(b)Figure 5
Energy cost in the 3D WSN.


### 6.2.2. Evaluating the Mathematical Analysis of Data Loads and E2E Delay When Sending One Data Packet with One ACK Returned in Hop-by-Hop Mode

The comparison of the simulation results and the theoretical results of data loads and E2E delay in the 3D WSN when sending one data packet with one ACK returned are shown in Figures6(a) and 6(b). As shown in Figure 6(a), the overall increasing trend of the number of the sending packet is reflected by both the theoretical results and the simulation results. From the figure, we can see that the theoretical result of each node is the median line of the simulation results. The correlation coefficient between the theoretical and experimental results is 0.6190. The mathematical results agree with the simulation results to a certain extent. In Figure 6(b), the comparison of E2E delay between mathematical analysis and the simulation results is shown. The correlation coefficient between the theoretical and experimental results is 0.9898. It shows that the simulation results are consistent with the theoretical results.Figure 6
Comparison of the simulation results and the theoretical analysis in a 3D WSN. (a) The number of sending packet of each node. (b) The E2E delay of each node.
(a)(b)


### 6.2.3. Evaluating the Mathematical Analysis of Data Loads and E2E Delay in 3D WSN When Sending One Data Packet and Returningn-ACKs

To evaluate the validity of the mathematical analysis of data loads and E2E delay in 3D WSN when sending one data packet and returningn-ACKs, we assume n=2 randomly. When sending one data packet and returning 2-ACKs, the validity of the theoretical analysis is proved by comparing with the simulation results. Figures 7(a) and 7(b) show the comparison of the simulation and theoretical results in the 3D WSN. In Figure 7(a), both the theoretical results and the simulation results demonstrate the overall increasing trend of the number of the sending packet with the nodes closer to the sink. The correlation coefficient between the theoretical and experimental results is 0.6232. The difference between them is because that the nodes in 3D WSN are assumed dense enough in the mathematical characterizing. The comparison of E2E delay between the mathematical analysis and the simulation is shown in Figure 7(b). As can be seen from the figure, the mathematical analysis of E2E delay in 3D WSN is in consistent with the simulation results. The correlation coefficient between the theoretical and experimental results is 0.9929.Figure 7
Comparison of the simulation results and the theoretical analysis in a 3D WSN. (a) The number of sending packet of each node. (b) The E2E delay of each node.
(a)(b)


### 6.2.4. Evaluating the Mathematical Analysis of Data Loads and E2E Delay in 3D WSN When Sending Multiple-Packet Copies without Returning ACK

When sending multiple-packet copies without returning ACK, the comparison of the simulation and the theoretical results in the 3D WSN is shown in Figures8(a) and 8(b). As can be seen from Figure 8(a), the changing trend of the number of the sending packet is increasing with the nodes closer to the sink, which is reflected by both the theoretical analysis and the simulation results. From the figure, the theoretical result of each node is the median line of the simulation result. The correlation coefficient between the theoretical and experimental results is 0.6217. The mathematical results are consistent with the simulation results. The comparison of E2E delay between mathematical analysis and simulation is demonstrated in Figure 8(b). The correlation coefficient between the theoretical and experimental results is 0.9992. They are found to be in reasonable line with the simulation results.Figure 8
Comparison of the simulation results and the theoretical analysis in a 3D WSN. (a) The number of sending packet of each node. (b) The E2E delay of each node.
(a)(b)


## 6.2.1. Evaluating the Mathematical Analysis of Data Loads and E2E Delay When Sending One Data Packet without Packet Loss

The mathematical analysis of data loads in 3D WSN is evaluated when following the sending one data packet without packet loss. Figures4(a) and 4(b) show the comparison between the simulation results and the theoretical analysis. As shown in Figure 4(a), the overall trend of the number of the sending packet is increasing with the nodes closer to the sink, which is reflected by both the simulation and theoretical results. In addition, we can see that the number of sending packet of each node in theory is almost median line of the simulation. The difference between them is caused by the assumption in theory analysis that the nodes in 3D WSN are dense enough. The correlation coefficient between the theoretical and experimental results is 0.6306. Therefore, the theoretical results are in reasonable agreement with the simulation results. The comparison of E2E delay in mathematical analysis and simulation is shown in Figure 4(b). The average error between the theoretical and simulation results is nearly 0. The correlation coefficient between the theoretical and simulation results is nearly 1. Therefore, the mathematical analysis of E2E delay in the 3D WSN is verified when sending one data packet without packet loss. Figure 5 describes the simulation results of energy cost of each node in the 3D WSN. The result shows that the energy cost increase as the nodes are closer to the sink, and the changing trends with distance to the sink is almost the same as that of the data loads.Figure 4
Comparison of the simulation results and the theoretical analysis in a 3D WSN. (a) The number of sending packet of each node. (b) The E2E delay of each node.
(a)(b)Figure 5
Energy cost in the 3D WSN.


## 6.2.2. Evaluating the Mathematical Analysis of Data Loads and E2E Delay When Sending One Data Packet with One ACK Returned in Hop-by-Hop Mode

The comparison of the simulation results and the theoretical results of data loads and E2E delay in the 3D WSN when sending one data packet with one ACK returned are shown in Figures6(a) and 6(b). As shown in Figure 6(a), the overall increasing trend of the number of the sending packet is reflected by both the theoretical results and the simulation results. From the figure, we can see that the theoretical result of each node is the median line of the simulation results. The correlation coefficient between the theoretical and experimental results is 0.6190. The mathematical results agree with the simulation results to a certain extent. In Figure 6(b), the comparison of E2E delay between mathematical analysis and the simulation results is shown. The correlation coefficient between the theoretical and experimental results is 0.9898. It shows that the simulation results are consistent with the theoretical results.Figure 6
Comparison of the simulation results and the theoretical analysis in a 3D WSN. (a) The number of sending packet of each node. (b) The E2E delay of each node.
(a)(b)


## 6.2.3. Evaluating the Mathematical Analysis of Data Loads and E2E Delay in 3D WSN When Sending One Data Packet and Returningn-ACKs

To evaluate the validity of the mathematical analysis of data loads and E2E delay in 3D WSN when sending one data packet and returningn-ACKs, we assume n=2 randomly. When sending one data packet and returning 2-ACKs, the validity of the theoretical analysis is proved by comparing with the simulation results. Figures 7(a) and 7(b) show the comparison of the simulation and theoretical results in the 3D WSN. In Figure 7(a), both the theoretical results and the simulation results demonstrate the overall increasing trend of the number of the sending packet with the nodes closer to the sink. The correlation coefficient between the theoretical and experimental results is 0.6232. The difference between them is because that the nodes in 3D WSN are assumed dense enough in the mathematical characterizing. The comparison of E2E delay between the mathematical analysis and the simulation is shown in Figure 7(b). As can be seen from the figure, the mathematical analysis of E2E delay in 3D WSN is in consistent with the simulation results. The correlation coefficient between the theoretical and experimental results is 0.9929.Figure 7
Comparison of the simulation results and the theoretical analysis in a 3D WSN. (a) The number of sending packet of each node. (b) The E2E delay of each node.
(a)(b)


## 6.2.4. Evaluating the Mathematical Analysis of Data Loads and E2E Delay in 3D WSN When Sending Multiple-Packet Copies without Returning ACK

When sending multiple-packet copies without returning ACK, the comparison of the simulation and the theoretical results in the 3D WSN is shown in Figures8(a) and 8(b). As can be seen from Figure 8(a), the changing trend of the number of the sending packet is increasing with the nodes closer to the sink, which is reflected by both the theoretical analysis and the simulation results. From the figure, the theoretical result of each node is the median line of the simulation result. The correlation coefficient between the theoretical and experimental results is 0.6217. The mathematical results are consistent with the simulation results. The comparison of E2E delay between mathematical analysis and simulation is demonstrated in Figure 8(b). The correlation coefficient between the theoretical and experimental results is 0.9992. They are found to be in reasonable line with the simulation results.Figure 8
Comparison of the simulation results and the theoretical analysis in a 3D WSN. (a) The number of sending packet of each node. (b) The E2E delay of each node.
(a)(b)


## 6.3. Performance Evaluation on Different Parameters

In the part, the parameters influence on the proposed scheme are evaluated and analyzed. The performance of the E2E delay, the energy consumption, and the network reliability are estimated under the differentδ, α, p, and r.


### 6.3.1. Different Network Partitionδ

As described above, the parameterδR is used to divide the network into two partitions. Under different δ, the network performance of reliability, energy cost, and E2E delay are presented. The results are demonstrated in Table 2. From this table, we can see that the proposed energy-aware transmission scheme can meet the required transmission reliability ζ=0.8 whether δ=0.4 or δ=0.2. There is more energy consumption for δ=0.2 shown in Table 2. And there is less energy cost for δ=0.4. This is because that there are less nodes adopting the multiple-packet copy transmission and more nodes applying the timeout retransmission scheme when δ has bigger value. The energy cost of timeout retransmission scheme is less than that of multiple-packet copy transmission scheme under the same network conditions. Therefore, there are less energy cost when δ=0.4 compared with when δ=0.2. For this reason, the E2E delay and the maximal network E2E delay when δ=0.4is bigger than when δ=0.2 as shown in the table.Table 2
Simulation results under differentδ when p=0.7, α=1, r=10, and ζ=0.8.
δReliabilityThe maximal energy cost (J)The maximal E2E delay(ms)δ=0.285.95%0.0043307.5556δ=0.485.15%0.0037486.1250


### 6.3.2. Different Number of Returned ACKsα

Under differentα, the network performance of reliability, energy cost, and E2E delay is evaluated. In Table 3, the network reliability under different α is shown. From the table, the required transmission reliability is satisfied. And with the increasing number of ACK, the transmission reliability is improved, and the maximal energy cost of the nodes in one round of data gathering is reduced. The E2E delay is also reduced. This is because that the ACK loss will lead to packet retransmission. Thus, reducing the ACK loss will reduce the unnecessary packet retransmission. Therefore, a reasonable number of returned ACK will decrease the unnecessary packet retransmission caused by ACK loss.Table 3
Simulation results under differentαwhen p=0.7, δ=0.4, r=10, and ζ=0.8.
αReliabilityThe maximal energy cost (J)The maximal E2E delay (ms)α=185.15%0.0037486.1250α=289.64%0.0031419.4186α=391.76%0.0031379.7674


### 6.3.3. Different Hop Transmission Reliabilityp

The network performance results of reliability, energy cost, and E2E delay are presented in Table4 under different p. In Table 4, the transmission reliability is ensured under different one-hop transmission reliability p whether δ=0.2 or δ=0.4. However, the energy cost is bigger with the worse link. The E2E delay is also bigger compared with high reliable transmission link as shown in the table. It is obvious that to meet the transmission reliability constraint, the worse link needs a greater number of packet retransmissions, which cause more energy cost and larger E2E delay.Table 4
Simulation results under differentpwhen α=1, r=10, and ζ=0.8.
pδReliabilityThe maximal energy cost (J)The maximal E2E delay (ms)p=0.7δ=0.285.95%0.0043307.5556δ=0.485.15%0.0037486.1250p=0.6δ=0.285.06%0.0053380.2439δ=0.484.37%0.0047670.7317


### 6.3.4. Different Transmission Radiusr

Under differentr, the network performance of reliability, energy cost, and E2E delay are evaluated. Table 5 shows the results. From the table, the transmission reliability is ensured under different r. Besides, the results show the overall changing trend of the energy cost of the nodes in one round of data gathering. The overall trend is increasing with smaller distance to the sink. This is because that the nodes near the sink need to forward more packets. Besides, as can be seen from the table, the E2E delay is reduced with the bigger transmission radius. This is because that the bigger radius could reduce the transmission hops to the sink. Also, the number of sending packet copies or the maximal retransmission number is reduced. Therefore, the E2E delay is decreased.Table 5
Simulation results under differentrwhen α=1, δ=0.2, p=0.7, and ζ=0.8.
rReliabilityThe maximal energy cost (J)The maximal E2E delay (ms)r=1085.95%0.0043307.5556r=1287.55%0.0054262.5000r=1588.07%0.0037226.9048


### 6.3.5. Different Network Reliabilityζ

Under different requirements of network reliabilityζ, the network performance of achieved reliability, energy cost, and E2E delay are presented in the part. Table 6 shows the network actual reliability achieved under different ζ. From the table, the results show the proposed transmission scheme could meet the different transmission reliability requirements. As can be seen from the table, the energy cost is bigger when the bigger required reliability under different transmission reliability constraints. It is obvious that higher transmission reliability needs the greater number of packet retransmissions under the same wireless link lost. Similarly, the E2E delay also gets bigger as can be seen from the table.Table 6
Simulation results under differentζwhen α=1, δ=0.2, p=0.7, and r=10.
ζReliabilityThe maximal energy cost (J)The maximal E2E delay (ms)ζ=80%85.95%0.0043307.5556ζ=70%81.49%0.0040299.0244ζ=60%64.25%0.0028290.4176


## 6.3.1. Different Network Partitionδ

As described above, the parameterδR is used to divide the network into two partitions. Under different δ, the network performance of reliability, energy cost, and E2E delay are presented. The results are demonstrated in Table 2. From this table, we can see that the proposed energy-aware transmission scheme can meet the required transmission reliability ζ=0.8 whether δ=0.4 or δ=0.2. There is more energy consumption for δ=0.2 shown in Table 2. And there is less energy cost for δ=0.4. This is because that there are less nodes adopting the multiple-packet copy transmission and more nodes applying the timeout retransmission scheme when δ has bigger value. The energy cost of timeout retransmission scheme is less than that of multiple-packet copy transmission scheme under the same network conditions. Therefore, there are less energy cost when δ=0.4 compared with when δ=0.2. For this reason, the E2E delay and the maximal network E2E delay when δ=0.4is bigger than when δ=0.2 as shown in the table.Table 2
Simulation results under differentδ when p=0.7, α=1, r=10, and ζ=0.8.
δReliabilityThe maximal energy cost (J)The maximal E2E delay(ms)δ=0.285.95%0.0043307.5556δ=0.485.15%0.0037486.1250


## 6.3.2. Different Number of Returned ACKsα

Under differentα, the network performance of reliability, energy cost, and E2E delay is evaluated. In Table 3, the network reliability under different α is shown. From the table, the required transmission reliability is satisfied. And with the increasing number of ACK, the transmission reliability is improved, and the maximal energy cost of the nodes in one round of data gathering is reduced. The E2E delay is also reduced. This is because that the ACK loss will lead to packet retransmission. Thus, reducing the ACK loss will reduce the unnecessary packet retransmission. Therefore, a reasonable number of returned ACK will decrease the unnecessary packet retransmission caused by ACK loss.Table 3
Simulation results under differentαwhen p=0.7, δ=0.4, r=10, and ζ=0.8.
αReliabilityThe maximal energy cost (J)The maximal E2E delay (ms)α=185.15%0.0037486.1250α=289.64%0.0031419.4186α=391.76%0.0031379.7674


## 6.3.3. Different Hop Transmission Reliabilityp

The network performance results of reliability, energy cost, and E2E delay are presented in Table4 under different p. In Table 4, the transmission reliability is ensured under different one-hop transmission reliability p whether δ=0.2 or δ=0.4. However, the energy cost is bigger with the worse link. The E2E delay is also bigger compared with high reliable transmission link as shown in the table. It is obvious that to meet the transmission reliability constraint, the worse link needs a greater number of packet retransmissions, which cause more energy cost and larger E2E delay.Table 4
Simulation results under differentpwhen α=1, r=10, and ζ=0.8.
pδReliabilityThe maximal energy cost (J)The maximal E2E delay (ms)p=0.7δ=0.285.95%0.0043307.5556δ=0.485.15%0.0037486.1250p=0.6δ=0.285.06%0.0053380.2439δ=0.484.37%0.0047670.7317


## 6.3.4. Different Transmission Radiusr

Under differentr, the network performance of reliability, energy cost, and E2E delay are evaluated. Table 5 shows the results. From the table, the transmission reliability is ensured under different r. Besides, the results show the overall changing trend of the energy cost of the nodes in one round of data gathering. The overall trend is increasing with smaller distance to the sink. This is because that the nodes near the sink need to forward more packets. Besides, as can be seen from the table, the E2E delay is reduced with the bigger transmission radius. This is because that the bigger radius could reduce the transmission hops to the sink. Also, the number of sending packet copies or the maximal retransmission number is reduced. Therefore, the E2E delay is decreased.Table 5
Simulation results under differentrwhen α=1, δ=0.2, p=0.7, and ζ=0.8.
rReliabilityThe maximal energy cost (J)The maximal E2E delay (ms)r=1085.95%0.0043307.5556r=1287.55%0.0054262.5000r=1588.07%0.0037226.9048


## 6.3.5. Different Network Reliabilityζ

Under different requirements of network reliabilityζ, the network performance of achieved reliability, energy cost, and E2E delay are presented in the part. Table 6 shows the network actual reliability achieved under different ζ. From the table, the results show the proposed transmission scheme could meet the different transmission reliability requirements. As can be seen from the table, the energy cost is bigger when the bigger required reliability under different transmission reliability constraints. It is obvious that higher transmission reliability needs the greater number of packet retransmissions under the same wireless link lost. Similarly, the E2E delay also gets bigger as can be seen from the table.Table 6
Simulation results under differentζwhen α=1, δ=0.2, p=0.7, and r=10.
ζReliabilityThe maximal energy cost (J)The maximal E2E delay (ms)ζ=80%85.95%0.0043307.5556ζ=70%81.49%0.0040299.0244ζ=60%64.25%0.0028290.4176


## 6.4. Performance Comparison

In the section, the performance of the proposed scheme is estimated by comparing with the existing schemes of multiple-packet copies without ACK returned transmission (MP) and timeout retransmission scheme represented typically by sending one data packet with one ACK returned (ODOA).In Figure9(a), the results demonstrate that the proposed energy-aware transmission scheme can meet the required transmission reliability ζ=0.8. As can be seen from Figures 9(b) and 9(c), there is more energy consumption for MP scheme. And there is less energy cost for ODOA scheme. The energy cost of the proposed transmission scheme is less than MP and bigger than ODOA from the overall perspective. From Figures 9(d) and 9(e), we can see that the delay is bigger with the greater distance of the nodes to the sink overall. The maximum delay of the proposed transmission scheme is bigger than the MP and smaller than the ODOA. Compared with MP, the maximum energy cost is reduced by 29.13% and 17.23%, respectively, for the proposed scheme when δ=0.2 and δ=0.4. Comparing the proposed scheme with the ODOA, the maximum delay is, respectively, reduced by 43.60% and 64.32% when δ=0.2 and δ=0.4. Therefore, the proposed scheme can realize the trade-off optimization between energy efficiency and E2E delay.Figure 9
Comparison between schemes underα=1, p=0.7, and r=10. (a) The network reliability. (b) The energy cost of each node. (c) The maximal energy consumption. (d) The E2E delay of each node. (e) The maximal E2E delay.
(a)(b)(c)(d)(e)


## 7. Conclusions

In the paper, the data loads of each node in 3D WSN are analyzed in mathematical model. The energy cost is unbalanced. For the nodes far from the sink, the energy cost is less. However, there are more energy costs for the nodes near the sink. Based on the analysis, an energy-aware transmission scheme is proposed for 3D WSN to achieve trade-off optimization between E2E delay and energy efficiency under the guarantee of transmission reliability. For nodes far from the sink, they send or forward multiple-packet copies consecutively without waiting for ACK. This could reduce the one-hop transmission delay and improve the energy efficiency under the guarantee of the network transmission reliability. To save energy cost, the nodes near the sink send one packet and then wait for the returned ACK. If the source or relay nodes receive the ACK, the packet is successfully transmitted to the next hop. Otherwise, the nodes will retransmit the packet when timeout.However, the correlation efficient between the simulation and theoretical results in the mathematical analysis is not high enough. And the contention model is not considered in the proposed scheme. To improve the theoretical analysis accuracy and combine the contention model to reduce the energy cost and E2E delay under the transmission reliability constraint is our further work.



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*Source: 8855073-2020-09-28.xml*