Virtual movement of relay nodes for two-tier wireless sensor networks in tunnels

Abstract

In wireless sensor networks, node placement plays a significant role in terms of meeting design goals, such as cost effectiveness, connectivity, lifetime, and data latency. In tunnel environments, the energy consumption of sensor nodes and relay nodes is imbalanced because of the long and narrow shape of tunnels. In this article, to reduce the use of extra batteries and the number of forwarding times for a message, we propose a new approach to assist in the placement of relay nodes for two-tier wireless sensor networks in a tunnel environment. The approach proposed is based on the principle that working relay nodes can change approximately at the same time during network operation. It seems the relay nodes move to a new place when they are instead by dormant relay nodes. The approach provides a novel method that has been proved to find optimal places to deploy relay nodes in tunnel environment. Moreover, the approach balances the energy consumption of entire sensor nodes and prolongs the lifetime of entire networks significantly, which is shown in simulations.

Introduction

Wireless multiple-sensor networks are used in a wide variety of applications. In most cases, wireless networks are used on the earth’s surface,^1,2 but certain cases require sensors to be deployed underground or in tunnels,^3,4 especially at coal mine. In wireless sensor networks (WSNs), critical research problems, such as energy consumption, network capacity planning, and routing efficiency, may be different in mining tunnels, as well. ³ Node propagation and lifetime are influenced by tunnel shape, gas, humidity, and cleanliness of the environment.^3–5 Because of the special environment, sink nodes are generally distant from the data sources, and direct transmission from data sources to sinks is usually not practical. Therefore, multi-hop routing is appropriate for data transmission, and a clustering topology is a good choice to achieve network scalability.^3,5,6

A real deployment of operational WSNs is a challenging task. During the process of placement, several aspects—such as routing, fault tolerance, scalability, data integrity, and network lifetime—must be considered. In addition to these aspects, coal safety criteria should be included. Careful node placement is fundamental for successful deployment of WSNs while meeting Quality of Service (QoS) requirements. The network is organized in a multi-layer architecture for complex management. If one relay layer is added, we called it a two-tier network. The relay layer is an important layer in the organization. They relay signals to sink nodes to save sensor nodes’ (SNs) energy or balance the energy of whole WSNs. Some methods have been proposed to improve the efficiency of relays’ communication to sink. ⁷ Moreover, many approaches discuss how to deploy the relay nodes (RNs) to improve the lifetime and QoS of WSNs.^8,9 However, few studies have considered organization of the relay layer to improve the performance of the SN layer such that the entire network is improved. The organization is to deploy RNs in order to improve a whole network performance.

In China, most SNs are deployed in a long lane at coal mine. These RNs near the sink would consume more energy. ⁹ An easy way to solve it is to add extra battery on these nodes. Considering coal mine safety requirements, the capacity of a single battery installed on any wireless nodes must be less than 60 Ah. ¹⁰ Most SNs need 1–3 Ah. However, some RNs close to the sink node need more than 60 Ah. How to deal with it is one motivation of our research.

However, when we performed simulations of multi-layer WSNs, we observed that certain SNs die easily if the relay layer is static upon initialization. If we changed the position of some RNs, the network was sometimes affected significantly. The motivation of this article is to determine how to organize an RN layer to achieve an optimal result. The contributions of this article are as follows:

1. Based on the multi-hop network model, an approach for RNs in two-tier WSNs is proposed to save the energy of SNs and balance their energy consumption in a tunnel environment. Based on a result that a mobile sink can save SNs’ energy, ² the approach makes fixed RNs work in turns to obtain the virtual movement of relay nodes (VMRN).

2. To provide a method for finding optimal placement for RNs deployment in tunnels and present a simple proof of the result.

The remainder of this article is organized as follows. In section “Related work,” we review several existing related works. Section “Primary node clusters in tunnels” outlines the primary conceptions of a system model for linear WSNs. We discuss the system model for the VMRN in section “Problem and system model.” The energy consumption of deployment and a solution to find the optimal result is given in section “Solution and energy efficiency analyses.” Finally, the conclusions are presented, and future work is suggested in section “Conclusion.”

Related work

There are many approaches for node placement to improve the performance of WSNs. Two-tier WSNs have been studied in the literature.^3–6,11,12 These excellent studies try to improve network lifetime and communication quality. Node deployment optimization is the first concept that should be considered. Hou et al. proposed that the nodes in WSNs be divided into three layers: microsensor nodes (MSNs), aggregation and forwarding nodes (AFNs), and a base station (BS). MSNs can be application-specific SNs and constitute the lower tier of the network. AFNs are used for data aggregation and forwarding (or relaying) the information aggregated over the hop to the next AFN toward the BS. The number and position of SNs are known prior to deployment, and the network lifetime can be extended by placing additional RNs into the network in order to provide extra energy to the existing AFNs that lie in the upper (second) tier of the network. Hou et al. assumed that the working environment of these nodes is open and that the AFNs can find a new way to forward information. However, in our application, mining tunnels have a special shape, thus influencing the deployment.

Among these approaches, prior researchers have proposed models of WSNs for mining tunnels (mtWSNs) in the form of linear or cylindrical models.^3–9 The WSNs in tunnels are considered to be linear models, which provide an easier topology to compute the cost of energy consumption.^12,13 In the works by Yang and Xiao ⁶ and Wu et al., ¹¹ a cylindrical model of mtWSNs is built and adjusts the distance between the RNs to save energy within the whole network. They adopted Zimmerling et al.’s result ¹⁴ to find an appropriate place for RNs. The distances between these RNs are the same, so the RNs that are close to sink have more batteries to carry the messages of other nodes forward to the sink base. The nearest RN to the sink has to carry too many batteries if the tunnel is long, which is banned in coal mines.

Wu et al. ¹¹ built a cylindrical model and proposed a routing algorithm, bounce routing in tunnels (BRIT), based on the model. They assumed that the connection of the entire RNs is similar to a string of beads and built an equation of extra energy based on the assumption that the energy of each node is used up at the same time. They also discussed a hybrid signal propagation process in three-dimensional underground tunnels and attempted to implement an assortment of sensor deployment strategies in tunnels. However, they did not consider the case in which sensors near the sink node live shortly when they forward information on multi-paths. In addition, their work did not consider clusters. Liu et al. ¹² considered clusters by cutting the cylinder into a flat and trying to place these nodes on the flat. They assumed that each SN has covered a region in terms of communication range, which leads to many overlapping regions among nodes. They searched for the minimum number of overlapping regions that covered all the SNs to place the RNs. In the work by Chen et al., ¹⁵ the shape of the tunnel is considered, and the nodes are arranged on the wall of the tunnel carefully to cover the tunnel. The routing topology can be controlled by the power level of the transmitter because it directly controls the distance coverage of RNs. However, this work merely provides several criteria to build the networks.

Protocols are another good method to improve the performance of linear networks.^16,17 Protocols can rebuild the topology of WSNs and evenly distribute the load of the data integration and transmission to all nodes. For instance, transmission power level (TPL) is defined to evaluate the costs of energy consumption in a network for each sensor or relay. ¹⁶ Then, a novel method attempts to find a route with minimum energy cost by a heuristic process called the Lagrangian method. The nodes in the nearest cluster to the sink should have sufficient energy if this protocol is adopted in mtWSNs. It is difficult to decide which nodes belong to the cluster in this case.

If we want to avoid addressing the volume of traffic in the network and focus on the neighborhood of the sinks, an alternative method for optimizing linear WSNs is to adopt mobile sink nodes.^18,19 It is easy to recharge batteries in mobile sink nodes. The mobile sink moves along the road, similar to a pipeline or tunnel, and collects the data from the SNs that are fixed in place. However, this is difficult to use in underground mine tunnels for two main reasons. The first is that it is not real-time, and the other is that it is difficult to set up mobile sink gear in a complex environment. Some prior works are compared in Table 1.

Table 1.

Comparison of some prior works based on different design criteria.

Solution name	Algorithm	Approach	Application environment	Architecture	Involved layer
DyLS 3	Latin square	To adjust RNs in sequence	Tunnel	Dynamic, zig-zag	MAC and transport
SPINDS 5	Heuristic	To adjust whole RNs	Plane	Dynamic, flat	Transport
MIP 8	QuickXplain	Extra RNs for robustness	Plane	Dynamic, flat	Transport
BRIT 11	Iterative	A time metric for adjusting	Tunnel	Cylindrical	MAC and transport
Approach based on QoS 12	Iterative	A QoS metric for adjusting	Tunnel	Cylindrical	Transport
Chain WSN 15	Iterative	3 criterions for adjusting	Tunnel	Chain	Transport

RNs: relay nodes; MAC: media access control; QoS: Quality of Service; WSN: wireless sensor network; BRIT: bounce routing in tunnels; DyLS: dynamic multiple access scheduling based on Latin squares; SPINDS: smart pairing and intelligent disc search; MIP: mixed-integer programming.

Our work is motivated by the results presented by Wu ³ where the authors deployed SNs in tunnels. Note the idea of dormant on the SN layer in energy-constrained wireless networks; we try to apply dormant strategies on relay layer. Combined with the idea of mobile sink nodes, a novel solution for energy efficiency is proposed.

Primary node clusters in tunnels

In a cluster-based WSN, a two-tier architecture is usually formed. We focus on the two-tier architecture of WSNs that is motivated by distributed source coding. In two-tier WSNs, there are three types of nodes in the network: SNs, RNs, and a BS. The BS is the sink node for data collection and receives all data from the network; thus, it is assumed to have sufficient power to operate. Therefore, power of this node dissipation is not considered in this article.

SNs constitute the lower tier of the networks. Each of the SNs is often small and low-cost. The SNs are densely deployed in a specific place for monitoring applications, and they are assumed to survive a long time to collect information on the monitored object. Using this information, we can obtain a comprehensive view of the monitored area by analyzing the correlated data. The SNs capture the information at regular intervals and directly send it to the local RN in one hop. If we consider the multi-hop among SNs, there would be extra layers in the network architecture, which increases the complexity of the networks as well as the difficulty to control and install them. Furthermore, when one SN is depleted, it will cease to function, thus perhaps leading to a monitoring hole in the area. We call the network dead if this situation occurs.

The SNs form clusters, consisting of a leader called cluster head (CH) in the upper tier of the architecture. The SNs of a cluster forward the sensed data to CHs, which then aggregate and route them to the BS directly or via other CHs. Because one of the main functions of these CHs is to relay the signal, we call them RNs. Certainly, we can appoint some nodes as RNs and attach extra batteries to them. RNs and SNs rarely have replenished power supplies, especially in a coal mine.

However, it is quite obvious that these RNs near the BS would die first if all relays were to have the same energy. The imbalanced energy consumption of the CHs may lead some of the CHs to quickly die, thus reducing the lifetime of the WSN significantly. In tunnels, this situation is even more serious. Therefore, it is necessary to rearrange these RNs for better performance. From the perspective of energy density, it is easy to attempt to add extra batteries to prolong the lifetime of RNs adjoining the sink, but how many batteries we should add is optimal.

Problem and system model

In the first subsection, we discuss the relay layer of two-tier network. In the second part, the modules that consume energy, including processor module, sensor module, and wireless communication module, are considered. Based on them, we discuss our model of virtual movement (VM) in the third section.

Relay model of network

In the case of a long tunnel, the shape of a coal lane mapped to a flat is a thin rectangle. Therefore, the link of RNs on one route is linear. The topology of the link is a one-dimensional chain labeled by i = 1,…, M, as depicted in Figure 1. The only sink is located at the end of the line. We assume that all sensors are homogeneous and have an ID number.

Figure 1.

Linear model of WSNs in tunnels.

Assume that we place N nodes on the monitoring path with a constant hop distance and that the number of RNs is M; that is to say, the number of clusters is M. The ith cluster has N_i nodes $(\sum _{i=1}^{N_i}=N)$ . The entire length of the tunnel is λ, and the distance between the RNs of the ith cluster and the (i − 1)th cluster is d_i. We assume a data collection application in which the sensors organized as clusters are required to send their sensing data to a CH at a certain rate. The CH nodes would transfer data to the sink node by a multi-hop route. When we look at a single part of the route, an RN is the CH of one cluster. SNs in the cluster send messages to an RN in one hop.

To facilitate the placement of nodes, we imagine that the tunnel is mapped into a thin two-dimensional (2D) rectangle. The actual distance between a pair of nodes will be slightly shorter than that in a real tunnel. The distance is assumed to be approximately the same as that in an actual environment. Table 2 lists the notations used in this article.

Table 2.

Notations and variables.

Symbol	Definition
λ	The whole length of tunnel.
m	A variable of cluster number, M is the total number of cluster.
rm,n	Relay nodes n in cluster m. Rm is the relay nodes set of cluster m.
ni	The number of nodes in cluster i.
Ci	Cluster i.
di	Distance between the relay nodes of ith cluster and (i − 1)th cluster. d0 is an optimal relay distance.
Es,0	Initial energy of sensor node s.
si	A node s in the ith cluster.
Ek,s	Energy of a node s in the kth cluster.
Etx(i,k)	The power dissipated at node i while transmitting a one-bit message to node k. Ei−1, tx (i) is that (i − 1) transmitting all messages to node i. Ei,tx(Rk) is the power dissipated at node i while transmitting messages to an RN set Rk.
Erx(i)	The power dissipated at node i while receiving a one-bit message. Ei−1, rx (i) is that (i − 1) receives all messages from cluster of node i. Ei,rx(Rk) is the power dissipated at node i while receiving messages from an RN set Rk.
εrx,εele	Distance-independent constant terms, εele = 50 nj/bit.
εamp	A coefficient associated with the distance-dependent term, εamp = 0.0013 pj/bit/m4.
α	Path loss index, 2 ≤ α ≤ 4.
d 0	The optimal relaying distance by Wu et al., 11 shown as formula (3).
$P_{s,r_m}$	The number of bits that rm relays from node s. Ps is the whole number of bits that rm relays.

RN: relay node.

Power dissipation

Because of their rough surface and complex geometry, underground tunnels are often hostile radio frequency (RF) environments. To maintain reliable communication, we must consider the energy of each node. We refer to the radio model used in the works by Wang et al. ²⁰ and Heinzelman et al. ²¹ in this article. Considering the hardware circuit, the modules that consume energy include processor module, sensor module, and wireless communication module. With rapid development of integrated circuit technology, low energy consumption and high performance of processors and sensors are more widely available. ²² Therefore, in this article, we ignore these two parts, processor module and sensor module, of energy consumption calculation in the nodes and focus on the energy consumption of wireless communication modules. For the sake of simplicity, the general assumptions of the model are as follows:

• 1. SNs generate and send one bit of data per unit time. The initial energy of each SN is E_s,₀ > 0, but the sink has no energy limitation. Each node has the same initial energy, that is, $E_{s_a,0}=E_{s_b,0}(s_a\ne s_b;s_a,s_b\in C_i)$ .

• 2. We consider the energy related to data transmission, receiving, and aggregating. The power consumed by data communication, including receiving and transmitting, is the dominant factor. ²³ The power dissipated at node s when transmitting a one-bit message to node i is given by formula (1)

$$E_{\mathrm{tx}}(s,i)={\epsilon }_{\mathrm{ele}}+{\epsilon }_{\mathrm{amp}}{d}_i^{\alpha }$$ (1)

and the receiving formula is given by formula (2)

$$E_{\mathrm{rx}}(s,k)={\epsilon }_{\mathrm{rx}}$$ (2)

where ε_rx and ε_ele are distance-independent constant terms; ε_amp is a coefficient associated with the distance-dependent term, which accounts for the energy dissipated in the transmitter amplifier; α is the path loss index, whose value is 2 ≤ α ≤ 4. ²⁴ Typical values of these parameters are ε_ele = 50 nj/bit and ε_amp = 0.0013 pj/bit/m⁴.

• 3. In this article, it is also assumed that the channel is symmetric; therefore, the energy spent on transmitting a packet from node i to j is the same as that from node j to i for any given value of signal-to-noise ratio (SNR). In the approaches of Wang et al., ²⁰ Heinzelman et al., ²¹ and Acharya et al., ²⁵ they used both the free space (distance 2 power loss) and multi-path fading (distance 4 power loss) channel modes. According to Zimmerling et al., ¹⁴ total relaying energy is minimized when hop distances are equal to a optimal relaying distance. The optimal relaying distance d₀ is given as formula (3)

$$d_0=\sqrt[\alpha ]{\frac{{\epsilon }_{\mathrm{ele}}}{({\epsilon }_{\mathrm{amp}}(\alpha -1))}}$$ (3)

where if d_k ≤ d₀, α = 2; otherwise, α = 4. The energy for receiving a bit is E_rx(i), whose value is typically 50 nj/bit. Assume that the relay distance is the optimal value d₀ for saving energy.

• 4. For the linear model, in addition to the data in the cluster of RN i, i would receive and relay the data from an RN of the (i + 1)th cluster. To complete this task, the node should have more energy for each bit message as formula (4)

$$E_{i,\mathrm{tx}}(i+1)=({\epsilon }_{\mathrm{ele}}+{\epsilon }_{\mathrm{amp}}{d}_i^{\alpha })\sum _{k=i+1}^M{N}_k$$ (4)

Therefore, the first RN nearest to the BS should be equipped with many batteries if the tunnel is very long.

Problem and VM

For a WSN, when one node dies, the whole network may die because a sensing hole is created. To increase the performance and lifetime of a network, it is expected that all nodes are exhausted at the same time. However, because of different distances among SNs and RNs, this expectation is difficult to achieve. For example, if there are just two nodes in a place, and one is relaying data from the other, it is easy to compute the batteries they require. However, each cluster contains many nodes. Thus, it is difficult to know the exact number for each cluster because of complex geometry. Moreover, there are some differences in tunnel models relative to models in open space. Owing to the length of tunnels, SNs in the area (Figure 2, labeled A) that are far away from RNs still die faster than those closer to the RNs, as is shown in Figure 2. This is easy to understand in terms of the formula in which the transmitting energy consumption varies directly with distance to the αth power. Figure 3 shows that the energy consumption of nodes after several cycles is significantly different if α = 2. Figure 3 suggests that more than one-third of the power in the SNs close to the RNs may remain alive when the power of the SNs far from the RNs is depleted. Thus, there is a significant imbalance of energy consumption in SNs for WSNs in tunnels. This is the reason why fixed RNs shorten the lifespan of WSNs.

Figure 2.

Area where nodes consume energy faster.

Figure 3.

Theoretical differences of energy consumption. (Each vertical line paralleled to the Y-axis indicates the energy consumption of the node at the intersection point of the line cutting the X-axis.)

To save energy in the SNs and design an energy-load balanced model, mobile sinks in previous research works, such as Jawhar, ¹⁸ Luo et al., ²⁶ provide a wise way to solve this problem. Each of the SNs never sends its messages to the sink over a long distance when the sink moves from one side to the other. In terms of this, we hope to move the RNs in each cluster. However, it is difficult to design and arrange mobile nodes in tunnels.

There is an alternative way to move RNs if we consider the node movement from the perspective of node position shifting. We select some nodes as RNs for every cluster, but only one of them in each cluster is working at a time. These RNs change in sequence; that is to say, an RN changes in a cluster from one frontier to the other along the tunnel. At the same time, all clusters’ RNs change according to the first one. This provides the appearance that the RNs are moving. We call it “virtual movement of RNs” (VMRN), and the solution is called VMRNS.

It may seem as if this solution merely changes the CHs in clusters, similar to the low-energy adaptive clustering hierarchy (LEACH) protocol. ²⁷ But the difference from LEACH is each RNs in a cluster works in a sequence and attached with extra battery according to formula (4). Figure 4 shows the process of selected RNs as the sequence. First, the sequence {r_1,1, r_2,1,…, r_M,1} works until one of their batteries is depleted. Then, a new sequence {r_1,j, r_2,j,…, r_M,j} is selected instead of them where r_i,j ∈ C_i. It seems the sequence {r_1,1, r_2,1,…, r_M,1} moves to a new position. It is called VM. With the VM, two new problems come out. One is where these RNs should be placed when we select n nodes as RNs. The other is how many nodes we should select as RNs in one cluster.

Figure 4.

Virtual movement model of relay nodes.

Solution and energy efficiency analyses

To solve the problems mentioned above, we consider energy dissipation in the case where an RN is placed in a straight tunnel first. Because the tunnel width is much shorter than the length by up to one or two orders of magnitude, we assume that the tunnel is a line. If an RN moves along the line of a tunnel, its speed is an important factor. Assume the speed is a constant value. If all the RNs were moving along the same direction at the same speed, we can find that the energy consumption is not related to distance.

Assume that there are M clusters on a straight line and the number of RNs in cluster k is n^r_k. RNs in cluster k are r_k,i (k ∈ {1, M}, i ∈ {1,…, n^r_k}), all RNs is a set R_k. r_k,i is at a point x_k,i such that d₀ = |x_k+1,i − x_k,i| = |x_k,i − x_k−1,i|. s_a and s_b are two nodes that belong to the kth cluster at the beginning and are on the route of an RN as it moves. s_a and s_b send messages continuously. s_a and s_b will select the nearest RN as its relay.

Lemma 1

If all RNs r_k,i move from point x_k,i to x_k+1,i with constant speed v at the same time and the energy dissipation of node s as it transmits messages during the RNs moving process is $E_{s_a,\mathrm{tx}}(R_k)$ , for any nodes $s_a,s_b\in C_k,E_{s_a,\mathrm{tx}}(R_k)=E_{s_b,\mathrm{tx}}(R_k)$ .

Proof

According to assumptions, the distance between two RNs in adjoining clusters is the optimal relaying distance d₀; that is, the range of a cluster is d₀. We prove the case N = 1 first. An RN r_k,0 moves from x_k,0 to x_k+1,0, as shown in Figure 5. When a node in cluster k changes its owner during the process of RN movement, the variation range of the node’s transmit distance is [0, d₀/2], we can find point x_a,1 and x_a,2 such that d₀/2 = |x_a,2 − x_sa| = |x_sa − x_a,1|. That is to say, when r_k,0 moves at the right of x_a,2, s_a may change r_k−1,0 as its RN. When r_k,0 is in [x_k,0, x_a,2], the energy dissipation of node s_a is denoted by $E_{s_a,\mathrm{tx}}(r_k,0)$

$$E_{s_a,\mathrm{tx}}\left(r_{k,0}\right)=\underset{x_{k,0}}{\overset{x_{a,2}}{\int }}({\epsilon }_{\mathrm{ele}}+{\epsilon }_{\mathrm{amp}}{x}^{\alpha })\mathrm{dx}$$ (5)

Figure 5.

Distance changes when RN moves.

Then, s_a changes its RN. When r_k−1,0 is in [x_a,1, x_k,0], the energy dissipation of node s_a is $E_{s_a,\mathrm{tx}}(r_{k-1,0})$

$$E_{s_a,\mathrm{tx}}\left(r_{k-1,0}\right)=\underset{x_{a,1}}{\overset{x_{k,0}}{\int }}({\epsilon }_{\mathrm{ele}}+{\epsilon }_{\mathrm{amp}}{x}^{\alpha })\mathrm{dx}$$ (6)

Then, we can determine the energy dissipation of node s_a in cluster m when RNs are moving, as shown in formula (7)

$$\begin{array}{c}\multicolumn{1}{c}{E_{s_a,\mathrm{tx}}\left(R_k\right)=E_{s_a,\mathrm{tx}}\left(r_{k,0}\right)+E_{s_a,\mathrm{tx}}\left(r_{k-1,0}\right)}\\ \multicolumn{1}{c}{=\underset{x_{a,1}}{\overset{x_{k,0}}{\int }}({\epsilon }_{\mathrm{ele}}+{\epsilon }_{\mathrm{amp}}{x}^{\alpha })\mathrm{dx}+\underset{x_{k,0}}{\overset{x_{a,2}}{\int }}({\epsilon }_{\mathrm{ele}}+{\epsilon }_{\mathrm{amp}}{x}^{\alpha })\mathrm{dx}}\\ \multicolumn{1}{c}{=\underset{x_{a,1}}{\overset{x_{a,2}}{\int }}({\epsilon }_{\mathrm{ele}}+{\epsilon }_{\mathrm{amp}}{x}^{\alpha })\mathrm{dx}=\underset{0}{\overset{d_0/2}{\int }}{E}_{s_a,\mathrm{tx}}\left(R_k\right)\mathrm{dx}}\end{array}$$ (7)

We can get the same result to s_b, and then $E_{s_a,\mathrm{tx}}(R_k)=E_{s_b,\mathrm{tx}}(R_k)$ . We can get the same result when 2 ≤ n^r_k.

Lemma 1 suggests that the energy consumption of all SNs in a cluster is balanced when RN of the cluster moves to the position of its adjacent RN’s. The lifetime of the SNs is prolonged.

However, in Lemma 1, the environment is ideal because some conditions are difficult to meet. For instance, “all RNs move with constant speed at the same time” is an unsuitable condition because it is hard to set synchronized clocks. So, we attempt to find a suboptimal result to balance energy dissipation among SNs.

First, the signals that s_a sends are discrete; we assume that s_a sends Ω bits to RNs. According to Figures 5 and 6, each SN passes through two phases as an RN moves from x_k,0 to x_k+1,0. One of the two phases is a segment in which the node belongs to cluster k, and the other is to join cluster k − 1. Thus, the energy consumption comprises two parts. Assume that s_a sends p₁ bits to r_k,0 and p₂ to r_k−1,0. Then, we have formula (8)

$$\begin{array}{c}\multicolumn{1}{c}{E_{s_a,\mathrm{tx}}=E_{s_a,\mathrm{tx}}\left(r_{k,0}\right)+E_{s_a,\mathrm{tx}}\left(r_{k-1,0}\right)}\\ \multicolumn{1}{c}{=\sum _{s_a\in C_k}^{p_1}({\epsilon }_{\mathrm{ele}}+{\epsilon }_{\mathrm{amp}}{\left(|x_{s_a}-x_{k,0}|\right)}^{\alpha })+}\\ \multicolumn{1}{c}{\sum _{s_a\in C_{k-1}}^{p_2}({\epsilon }_{\mathrm{ele}}+{\epsilon }_{\mathrm{amp}}{\left(|x_{s_a}-x_{k-1,0}|\right)}^{\alpha })}\end{array}$$ (8)

where k ∈ {1, M} and p₁ + p₂ = Ω.

Figure 6.

Different distances of the substituted RN.

Second, we consider the VMRN. In this situation, it is difficult for the movement of RNs to be continuous because they are fixed and not deployed without gaps. We consider a case in which an extra RN sequence RQ = {r_k,1, r_k,2,…, r_k,q} is inserted between r_k,0 and r_k+1,0. These RNs in RQ will work in turns, which seems r_k,0 changes its position. Therefore, the energy dissipation of s_a changes when an RN r_k,j substitutes for r_k,j−1, j ∈ {1,…, q}. Assume that r_k,j will relay $P_{s_a,R_{k,j}}$ bits for s_a. Formula (8) should change as formula (9)

$$E_{s_a,\mathrm{tx}}(\mathrm{RQ})=\sum _{j=0}^q{P}_{s_a,r_{k,j}}({\epsilon }_{\mathrm{ele}}+{\epsilon }_{\mathrm{amp}}{\left(|x_{s_a}-x_{k,j}|\right)}^{\alpha }),s_a\in C_k$$ (9)

where k ∈ {1, M}, j ∈ {0, q}, $\Omega =\sum _{j=0}^q{P}_{s_a,R_{k,j}}$ . Formula (9) would be consistent with $E_{s_a,\mathrm{tx}}$ in Lemma 1 if q→∞ and $P_{s_a,r_{k,i}}=P_{s_a,r_{k,j}}(i,j\in \{0,q\})$ . $P_{s_a,r_{k,j}}$ denotes the number of bits that r_k,j relays from node s_a.

Nevertheless, it is not sufficient for maximizing the lifetime of WSN if $E_{s_a,\mathrm{tx}}(\mathrm{RQ})$ is minimum. We must solve the problem of energy balance in a cluster. For simplicity, we consider a case in which one extra RN (r_k,1) is inserted between two RNs, that is, each interval would be inserted one RN. The position of r_k,0, x_k,0, is variable, which affects the energy consumption of the nodes in its working range. There is a problem to identify the proper position to place r_k,0 such that the node whose energy consumption is maximum would consume the least amount of energy in cluster k. This optimization problem is described by the following

$$\mathrm{Min}\{\mathrm{Max}(|E_{s_b,\mathrm{tx}}(\{r_{k-1,1},r_{k,0},r_{k,1}\})-E_{s_a,\mathrm{tx}}(\{r_{k-1,1},r_{k,0},r_{k,1}\})|)\}$$

$$\mathrm{s}.\mathrm{t}.E_{s_a,\mathrm{tx}}(r_{k,j})>0,E_{s_b,\mathrm{tx}}(r_{k,j})>0,s_a,s_b\in C_k,k\in \{1,M\}$$

$$\begin{array}{c}|x_s-x_{i,j}|\le \frac{d_0}{2},s\in \{s_a,s_b\};x_s,x_{i,j}\in C_i;j\in \{0,1\}\hfill \\ P_{s_a}=P_{s_b},s_a,s_b\in C_k\hfill \\ x_{k,0}<x_{k,1}<x_{k+1,0},k\in (1,M-1)\hfill \end{array}$$ (10)

where $P_{s_a}=P_{s_a,r_{k-1,1}}+P_{s_a,r_{k,0}}+P_{s_a,r_{k,1}}$ , r_k−1,1 is an RN instead of r_k−1,0, $P_{s_a,r_{k,0}}$ is the number of bits that r_k,0 relays from node s_a. For simplicity, we assume that all nodes transmit messages for the same amount of time. It is intuitive that the best positions for r_k,1 would be the middle of the intervals between r_k,0 and r_k+1,0. We will prove it.

Proposition 1

Assume that there are M clusters on a straight line. Suppose that RN in cluster k is R_k,0 (k ∈ {1, M}). All RNs {R_k} perform a VM. The location of a node s is x_s, and the location of r_k,0 is x_k,0. d₀ = |x_k,0 − x_k−1,0|. For each interval of sequence {r_k,0}, one node r_k,1 situated on x_k,1 would be selected as an RN such that |x_k,1 − x_k,0| = |x_k+1,0 − x_k,1|. Then, the additional sequence {r_k,0, r_k,1} is one optimal solution of 1-inserted case to problem (10) if $P_{s,r_{k,0}}=P_{s,r_{k,1}}$ .

Proof (reference to Figure 6)

When collecting data ranges from x_k−1,1 to x_k,1, node r_k,0 is working as a relaying node of cluster k. Assume that x_k,1 is the middle of x_k,0 and x_k+1,0; that is, |x_k,1 − x_k,0| = |x_k+1,0 − x_k,1| = |x_k+1,0 − x_k,0|/2. When the working time of r_k,0 is running out, r_k,1 will relay data instead of r_k,0. E_s,tx({r_k−1,1, r_k,0, r_k,1}) is a piecewise function of x_s as follows

$$E_{s,tx}\left(\left\{r_{k-1,1},r_{k,0},r_{k,1}\right\}\right)=\{\begin{array}{c}{P}_{s,R_{k,0}}\left({\epsilon }_{ele}+{\epsilon }_{amp}{\left(\left|x_s-x_{k,0}\right|\right)}^{\alpha }\right)+P_{s,R_{k-1,1}}\left({\epsilon }_{ele}+{\epsilon }_{amp}{\left(\left|x_s-x_{k-1,1}\right|\right)}^{\alpha }\right),x_s\in \left[x_{k-1,1},x_{k,0}\right]\\ P_{s,R_{k,0}}\left({\epsilon }_{ele}+{\epsilon }_{amp}{\left(\left|x_s-x_{k,0}\right|\right)}^{\alpha }\right)+P_{s,R_{k,1}}\left({\epsilon }_{ele}+{\epsilon }_{amp}{\left(\left|x_s-x_{k,1}\right|\right)}^{\alpha }\right),x_s\in \left[x_{k,0},x_{k,1}\right]\end{array}$$ (11)

To prove Proposition 1 by contradiction.

Suppose that $s_a,s_b\in C_k$ , such that $E_{s_b,\mathrm{tx}}(\{r_{k-1,1},r_{k,0},r_{k,1}\})$ is the maximum value and $E_{s_a,\mathrm{tx}}(\{r_{k-1,1},r_{k,0},r_{k,1}\})$ is the minimum. If we cannot obtain $\mathrm{Min}\{\mathrm{Max}(|E_{s_b,\mathrm{tx}}(\{r_{k-1,1},r_{k,0},r_{k,1}\})-E_{s_a,\mathrm{tx}}(\{r_{k-1,1},r_{k,0},r_{k,1}\})|)\}$ when r_k,1 is at x_k,1, there must be another position x′_k,1 to satisfy this. Assume that r′_k,1 is set on a position x′_k,1, such that $E_{s_b,\mathrm{tx}}(\{r{\prime }_{k-1,1},r_{k,0},r{\prime }_{k,1}\})$ is the max value and $E_{s_a,\mathrm{tx}}(\{r{\prime }_{k-1,1},r_{k,0},r{\prime }_{k,1}\})$ is the min. Then, for the sequence {r′_1,1,…, r′_k,1,…, r′_M,1}, |x′_k,1 − x′_k−1,1| = d₀. Suppose there are two points x_k,c and x_k,d such that |x′_k,1 − x_k,c| = |x_k,d − x′_k,1| = d₀/2 if r′_k,1 works. Without loss of generality, we assume x′_k,1 > x_k,1.

According to assumption, the RNs work for the same amount of time; we have $P_{s_b,r_{k,0}}=P_{s_b,r_{k,1}}=P_{s_b,{r\prime }_{k,1}}=P_{s_a,r_{k,0}}=P_{s_a,r_{k,1}}=P_{s_a,{r\prime }_{k,1}}$ . The function $E_{s,\mathrm{tx}}(\{r_{k-1,1},r_{k,0},r_{k,1}\})$ is continuous when x_s ∈ [x_k−1,1, x_k,0] or x_s ∈ [x_k,0, x_k,1]. According to Fermat’s theorem, let the two-order derivative of $E_{s,\mathrm{tx}}(\{r_{k-1,1},r_{k,0},r_{k,1}\})$ equal zero, and we can get the min extreme points (x_k−1,1 + x_k,0)/2 or (x_k,0 + x_k,1)/2, respectively. Comparing the two extreme values, we get the same least value. We also get the global maximum value on the point x_sb = x_k,1 or x_sb = x_k,0 or x_sb = x_k−1,1. On the other side, E_s,tx({r′_k−1,1, r_k,0, r′_k,1}) is continuous when x_s ∈ [x_k−1,1, x′_k−1,1] or x_s ∈ [x′_k−1,1, x_k,0] or x_s ∈ [x_k,0, x_k,1], respectively. Then, according to Fermat’s theorem, E_s,tx({r′_k−1,1, r_k,0, r′_k,1}) has its minimum value on $x{\prime }_{s_a}=(x_{k,0}+x{\prime }_{k,1})/2$ and maximum on $x{\prime }_{s_b}=x_{k,0}$ or $x{\prime }_{s_b}=x{\prime }_{k-1,1}$ .

According to the hypothesis, if the proposition is mistakes we have $|E_{s_b,\mathrm{tx}}(\{r_{k-1,1},r_{k,0},r_{k,1}\})-E_{s_a,\mathrm{tx}}(\{r_{k-1,1},r_{k,0},r_{k,1}\})|-(|E_{{s\prime }_b,\mathrm{tx}}(\{r{\prime }_{k-1,1},r_{k,0},r{\prime }_{k,1}\})-E_{{s\prime }_a,\mathrm{tx}}(\{r{\prime }_{k-1,1},r_{k,0},r{\prime }_{k,1}\})|)>0$ . For E_s,tx({r_k−1,1, r_k,0, r_k,1}) is a piecewise function, we have to consider two parts of them. So, the range [x_k−1,1, x_k,1] is divided into four parts [x_k−1,1, x′_k−1,1], [x′_k−1,1, x_k,0], [x_k,0, x_k,c], and [x_k,c, x_k,1].

First, when x_s ∈ [x_k,c, x_k,1], we have

$$\begin{array}{c}{\epsilon }_{\mathrm{amp}}{\left(|x_{s_b}-x_{k,0}|\right)}^{\alpha }+{\epsilon }_{\mathrm{amp}}{\left(|x_{s_b}-x_{k,1}|\right)}^{\alpha }\hfill \\ \multicolumn{1}{c}{-{\epsilon }_{\mathrm{amp}}{\left(|x_{s_a}-x_{k,0}|\right)}^{\alpha }-{\epsilon }_{\mathrm{amp}}{\left(|x_{s_a}-x_{k,1}|\right)}^{\alpha }}\\ -({\epsilon }_{\mathrm{amp}}{\left(|x_{{s\prime }_b}-x_{k,0}|\right)}^{\alpha }+{\epsilon }_{\mathrm{amp}}{\left(|x_{{s\prime }_b}-{x\prime }_{k,1}|\right)}^{\alpha }-{\epsilon }_{\mathrm{amp}}{\left(|x_{{s\prime }_a}-x_{k,0}|\right)}^{\alpha }-{\epsilon }_{\mathrm{amp}}{\left(|x_{{s\prime }_a}-{x\prime }_{k,1}|\right)}^{\alpha })>0\hfill \end{array}$$ (12)

Let $x{\prime }_{s_b}=x{\prime }_{k,1}$ and $x_{s_b}=x_{k,1}$ , we have

$${\left(|x_{s_b}-x_{k,0}|\right)}^{\alpha }-{\left(|x_{s_a}-x_{k,0}|\right)}^{\alpha }-{\left(|x_{s_a}-x_{k,1}|\right)}^{\alpha }-({\left(|x_{{s\prime }_b}-x_{k,0}|\right)}^{\alpha }-{\left(|x_{{s\prime }_a}-x_{k,0}|\right)}^{\alpha }-{\left(|x_{{s\prime }_a}-{x\prime }_{k,1}|\right)}^{\alpha })>0$$ (13)

when x_s ∈ [x_k,c, x_k,1], $x{\prime }_{s_a}=(x_{k,0}+x{\prime }_{k,1})/2$ and $x_{s_a}=(x_{k,0}+x_{k,1})/2$ , substitute them into formula (13), we have

$${\left(|x_{k,1}-x_{k,0}|\right)}^{\alpha }-2{\left(\left|\frac{x_{k,1}-x_{k,0}}{2}\right|\right)}^{\alpha }-({\left(|{x\prime }_{k,1}-x_{k,0}|\right)}^{\alpha }-2{\left(\left|\frac{{x\prime }_{k,1}-x_{k,0}}{2}\right|\right)}^{\alpha })>0$$ (14)

However, according to the hypothesis, x′_k−1,1 > x_k−1,1 and x′_k,1 > x_k,1. Because 2 ≤ α ≤ 4 and the function x^α − 2(x/2) ^α is non-decreasing when x_s ∈ [x_k,c, x_k,1], we have

$${\left(|x_{k,1}-x_{k,0}|\right)}^{\alpha }-2{\left(\left|\frac{x_{k,1}-x_{k,0}}{2}\right|\right)}^{\alpha }\le ({\left(|{x\prime }_{k,1}-x_{k,0}|\right)}^{\alpha }-2{\left(\left|\frac{{x\prime }_{k,1}-x_{k,0}}{2}\right|\right)}^{\alpha })$$ (15)

According to formula (15), $|E_{s_b,\mathrm{tx}}(\{r_{k-1,1},r_{k,0},r_{k,1}\})-E_{s_a,\mathrm{tx}}(\{r_{k-1,1},r_{k,0},r_{k,1}\})|-|E_{{s\prime }_b,\mathrm{tx}}(\{r{\prime }_{k-1,1},r_{k,0},r{\prime }_{k,1}\})-E_{{s\prime }_a,\mathrm{tx}}(\{r{\prime }_{k-1,1},r_{k,0},r{\prime }_{k,1}\})|\le 0$ , which contradicts the assumption. We can get the same result when x_s ∈  [x_k−1,1, x′_k−1,1] or [x′_k−1,1, x_k,0] or [x_k,0, x_k,c]. We also encounter the same contradiction when x′_k,1,1 < x_k,1.

The above proposition leads us to an extended result: the lifetime of SNs is maximized if these substituted RNs owning the same working time obey uniform distribution throughout the cluster. This is formalized in the following proposition.

Proposition 2

Assume that there are M clusters on a straight line. Suppose that RN in cluster k is R_k (k ∈ {1, M}). All RNs {R_k} perform a VM. The location of node s in the kth cluster is x_s. For each interval of sequence {r_k,0}, n nodes in sequence {r_k,1,…, r_k,n} would be selected as RN in the kth cluster such that |x_k,i − x_k,i−1| = |x_k,j − x_k,j−1|, where r_k,i situated on x_k,i, ∀i, j ∈ {1,n} and i ≠ j. The number of bits relayed by r_k,i from node s is $P_{S,r_{k,i}}$ . The energy dissipation of node s while transmitting messages during an RN switching process is E_s,tx. Then, the sequence {r_1,0,…, r_1,n,…, r_k0n,…, r_M,n} is a solution to problem (10) if $P_{s,r_{k,i}}=P_{s,r_{k,j}},i,j\in \{1,n\}$ .

According to the above proposition, we have some optimal position to deploy the RNs. We can add extra batteries to the nodes that perform the RNs’ VM to prolong the lifetime of normal SNs in WSN.

Moreover, we have to solve the problem of synchronization. Here, we refer to gossip methods. If the voltage of a working RN node is below a preset level, the conditions-satisfied RN would notify its neighbors and transmits the VM signal over the networks by gossips. Then, the RNs will be substituted in the whole networks. We describe an algorithm for the solution as follows.

VMRN Gossip Algorithm (VMRN-GA) pseudocode for RN r k,i .

Initial: Eflagid = 1, Clusterid = k, RNid = i, MVflag = 0, urgent bit bu = 0, Feedback = 0;

// For RNk,0 Eflagid = 0, MVflag is a public variable.

void ReceiveMSG_VM (Message msg){ // Receive messages with an urgent bit msg.bu from Ck and rk+1,0.

1. Read the urgent bit msg.bu;

2. If(msg.bu) {

3.  Read msg.MVflag; //A bit denotes the message is a MV command.

4.  If(msg.MVflag){ //MV starts.

5.   Read msg.Feedback bit;

6.   If(msg.Feedback == 1){ Eflagid = −1; MVflag = 0;} //RN becomes a normal sensor nodes if feedback msg received.

7.   Else{

8.    Read msg.Clusterid bit and msg.RNid bit;

9.    If(Clusterid == msg. Clusterid && msg.RNid == RNid){ // It is the RN named RNid whose turns to do VM.

10.     Eflagid = 1; // The RN r k,i begin to work.

11.     SendMsg_VM (Feedback, 1,0,currenttime); //To tell the RN who start VM that MV msg was received.

12.    }

13.    ElseIf(abs(Clusterid- msg. Clusterid) == 1){ //The VM message is from adjoining cluster

14.     If(!MVflag){ //To start VM and relay it

15.      MVflag = 1; SendMsg_VM(Null, MVflag,1,currenttime); //Node becomes a RN and disperses the MV gossip}

16.    }

17.   }

18.  } Else{

19.   Read other urgent bit and goto its handler; }

20. }Else{

21.  Calculate its own energy level Eremains and compare the result with thresholds th.

22.  If(Eremains<th){

23.   If(!MVflag){

24.    MVflag = 1; Sendmsg_VM(Null, MVflag,1,currenttime); }

25.   }Else SendMsg_VM(msg, 0,0,0);

26.  }

}

Int SendMsg_VM(Message msg, int MVflag, int count, Time time){

27.  If(!Eflagid){ //If a RN node is not a working RN it only sends what it sensed.

28.   Send sensing msgs; return;}

29.  If(MVflag){ //To start VM messages

30.   New a message msgn in which msgn.bu == 1 && msgn.MVflag == 1

31.   && msgn.Clusterid == Clusterid && msgn. RNid == msg.RNid+count;

32.   Wait(period);

33.   If(Compare(currenttime, time) > time requirement) count++; //RN decides whether the time is out.

34.   If(count > MaxRNid) send error msg.err;

35.   Else SendMsg_VM(Null, MVflag, count, time);

36.  }

37. }Else{ Relay msg; Send sensing msgs; } //If a RN node is a working RN it relays what it received.

}

The solution includes two functions. The ReceiveMSG_VM function analyzes received messages. If RN is a working RN, it will calculate its own remaining energy and decide whether to start a VM process. Eflagid belongs to an RN and denotes whether an RN is used. On the contrary, there are some bits in a message to show its status. The VM message is signed as an urgent message to avoid delaying and to spread across all network. A substituted RN will become active and relay messages after receiving the VM signal, MVflag == 1, for its turns. After the prior RN received a Feedback message from the newer, it becomes a normal SN. The SendMsg_VM function shows that RN would relay messages if no VM signal received or stops sending VM signal if a Feedback received. There is a deadline time for a signal feeding back. If no feedback signal received before the deadline time, the working RNs will select another substituted RN to work until all substituted RNs have been tested.

Simulation results

This section presents the results of simulations based on the problem framework. We implement the system model explained in section “Problem and system model” and show the optimal result of VMRNS in section “Solution and energy efficiency analyses” by conducting simulations in a MATLAB environment. First, we simulate a situation in which the nodes die fast in a tunnel. Then, we demonstrate the performance of VMRNS for a general network configuration and compare it to the normal arrangement of relay nodes (NARNS).

The 2D representation of a tunnel is a 5 m × 500 m grid network. We place 200 SNs randomly and all RNs are staggered along the two sides of the tunnel. In this simulation, we do not consider the curved shape or other complex conditions. Without a loss of generality, we assume that the BS is at one end of the tunnel, at point (602, 2.5) (in meters). We control each distance among all nodes to be longer than 2 m to better simulate reality. The relays are hanged on the wall in mines and shown in Figure 7 crosses. The values of other parameters are listed in Table 2. Figure 7 shows a situation in which several nodes die at the same turns. From Figure 7, we can see these nodes shown as filled dots, which are at the boundary of two clusters and consume more energy than others. It is easy to understand because they are far away from the relays.

Figure 7.

Positions at which nodes die quickly.

When VMRNS is introduced, the energy consumption of the SNs changes. Figure 8 shows the energy consumption of the SNs between two RNs. We achieve this result under an ideal condition with an even distribution of SNs. To obtain a more obvious result, the simulation runs for 120 turns. From Figure 8, we can see that the energy consumption of nodes is seriously imbalanced when there is no VM. The curve indicates better balance when more SNs are added as virtual moving nodes. With an increase in number of virtual moving SNs, the energy consumption line would be flat according to Lemma 1. The nodes selected as virtual moving SNs are distributed evenly in Figure 8. If not, as shown in Figure 9, the energy consumption is also imbalanced. For a better comparison, we add one RN with a random position to illustrate the imbalance.

Figure 8.

Differences of energy consumption between virtual movement and normal two-layer approach.

Figure 9.

Comparison of energy consumption of selected nodes between middle and random position.

Although the total energy consumption of the entire network rises with an increase in number of additional RNs, the peak value of one node’s consumed energy declines. The total battery capacity given to the RNs of each cluster does not change because the batteries are evenly divided into each extra RN for VM.

From the result mentioned above, it is easy to see that the difference between the maximum and minimum values of consumed energy decreases when VMRNS is adopted. This suggests that the entire network has a longer lifetime. To evaluate the result of our work, the next simulation, shown in Figure 10, shows curves which illustrate that VMRN prolongs the lifetime of the entire network with an increase in the number of additional RNs. The upper curve is computed theoretically, and the lower curve is obtained in our simulation. In the simulation, the theoretical value of the lifetime of the entire network is difficult to achieve because not all selected nodes are at their proper positions. Virtually, most nodes selected for VMRNS do not stay at exactly the right place. This case approximates the real situation. From Figure 10, we can see that the lifetime increases significantly, nearly 15%, when one extra node is added to each cluster. As the number of added nodes increases, the lifetime curve rises slowly. This is because the difference among the average transmission distance of each node decreases with an increase in the number of RNs. The optimal value is not more than three. This result is ideal and easy to realize.

Figure 10.

Network running turns for different numbers of nodes added when VMRN is used.

According to the result, at the VMRN position, we attempted to deploy more RNs at the same time and expected to achieve the same results. However, because these extra RNs had to relay messages during the whole working time, we must attach more batteries to them. Thus, the improvement in network to save energy is negligible compared with the case of less RNs.

Conclusion

A two-tier structure can improve the performance of WSNs in tunnels. However, when we consider the case in which extra batteries are attached to RNs, the imbalance of energy consumption among SNs is omitted. We determined that the energy dissipation of SNs at the frontier of two clusters is faster than those near RNs in a tunnel environment. If we add more RNs to change this condition, we require more batteries on the RNs for the sake of more turns through which a message is relayed. We investigated and discussed such a condition and proposed a novel approach called “virtual movement of RNs” based on the concept of changing RNs in each cluster at the same time. The research results indicated the following:

1. In a tunnel environment, the proposed method based on the VMRN prolongs the lifetime of entire WSNs without extra batteries relative to conventional two-tier WSNs. This method depends on the headers of all clusters changing at the same time, which is different from other approaches based on route algorithms.

2. This method would cause higher energy consumption in entire networks but balance the energy dissipation of the networks. This would reduce deviating values among the sensors’ energy consumption.

3. This method theoretically provides proper placement of extra RNs for VMRN in linear WSNs. We proved that this is an optimal result if RNs have the same load in WSNs. However, we found that the number of added RNs between two original RNs should be not more than three.

Our work is based on the assumption that the data conform to a given distribution. This assumption is ideal and not achievable in a practical application. Future research will further improve the dynamic conditions using swarm artificial algorithms.