Study on Gibbs Optimization-Based Resource Scheduling Algorithm in Data Aggregation Networks

Abstract

In data aggregation networks (WSNs, ad hoc, mesh, etc.), it is key to schedule the network resources, such as channels and TDMA time slots, to minimize the communication conflict and optimize the network data-gathering performance. In this paper, the resources scheduling problem is formulated as a vertex coloring problem in graph theory. Then, a multi-channel TDMA scheduling algorithm based on the Gibbs optimization is proposed. By defining the Gibbs energy expression according to the objective function of the problem, the joint probability of channel and time slot can be computed for the optimized selection of channels and time slots. This algorithm is low-complexity and its convergence performance can be proven. Experiments with different network parameters demonstrate that the proposed algorithm can reduce the communication conflict, improve the network throughput, and effectively reduce the network transmission delay and scheduling length for the data aggregation networks.

1. Introduction

In data aggregation networks (WSNs, ad hoc, mesh, etc.), data is transmitted by a series of common nodes, forwarded and aggregated to the source node, which is called data aggregation. In this process, it is necessary to ensure the integrity of the data, as well as to minimize the transmission delay (shorten the convergence time) [1,2]. In a high-density data aggregation network, the concurrent data transmission of a large number of nodes will bring constant channel conflicts and packet loss, which will seriously affect the performance of the data acquisition system [3,4]. Among many MAC protocols, the time-division multiple address (TDMA) scheduling algorithm is a classic and effective method used to solve the above problems [5]. Compared with the scheduling algorithm of a competitive mechanism, TDMA can be used to better eliminate transmission conflicts between data blocks, reduce the number of retransmissions, and improve network efficiency. With TDMA, nodes can be made to stay dormant in non-working time slots, reducing the energy consumption of nodes and thus extending the working time of the entire network. On this basis, the introduction of multi-channel technology can enable multiple nodes to work on different channels at the same time, thereby further improving the network’s data throughput and information transmission capabilities.

In the data aggregation network, there are three main factors that limit the efficiency of data collection: (1) the conflict in the wireless network, (2) the half-duplex communication method of the sensor node, and (3) the topology of the network. The half-duplex working mode of the node and the topology of the network are not the focus of this paper. This paper investigates the multi-channel TDMA scheduling problem in data aggregation networks, with the aim of reducing the number of network conflicts, increasing the throughput of the entire network, and reducing transmission delays through rational scheduling with a fixed number of time slots and channels, thus achieving the most efficient data collection and aggregation.

The rest of the paper is organized as follows. The related works are introduced in Section 2. The network model and problem definition are described in Section 3. The multi-channel TDMA scheduling algorithm and theoretical analysis are given in Section 4. An experimental analysis is provided in Section 5, and in Section 6, a summary of the full text is given.

2. Related Works

In recent years, TDMA scheduling algorithms have been widely studied in data aggregation networks [6]. Ying et al. proposed a spectrum resource scheduling model based on an adaptive chaotic-distributed differential artificial bee colony algorithm. The algorithm uses chaotic distribution to generate an initial solution. In the search phase, the differential evolution strategy and artificial bee colony algorithm are used to work in parallel to find the optimal local solution. The selection operator is used to calculate the probability of selecting new bee sources, which improves the reliability and convergence speed of the algorithm [7]. The literature in [8] proposes a dynamic time slot allocation algorithm (ED-TDMA) focusing on maximizing the network capacity. The proposed algorithm adjusts the time slot length according to the bit error rate (BER) of the system. Alghamdi et al. proposed a dynamic slot scheduling for tasks having different time constraints executed over wireless sensor networks (WSN). These tasks may have different priorities. They are usually repeated periodically. They present an algorithm that is able to provide a dynamic scheduling procedure for all tasks [9]. The above algorithms are all applicable in the case of a single channel; however, a lot of research indicates that a multiplexing multi-channel can be used to greatly improve the throughput of the network. In recent years, many multi-channel transmission protocols such as DCAS [10], TMMAC [11], MMSN [12], etc., have been proposed, and Chen Hua et al. proposed a distributed scheduling protocol suitable for a multi-channel [13] and proved that the usage of this protocol can greatly improve the information transmission efficiency of the entire network. In the above work, time slots and channels have been allocated for all links in the network so that they do not interfere with each other. The research object of this paper is the multi-channel TDMA scheduling problem in the data aggregation network, that is, to allocate the time slots and channels to links in the network, i.e., the links on the constructed routing tree.

Although network-wide data collection and data aggregation are related to some extent [14], they are not exactly the same. The goal of the entire network data collection algorithm is to collect the original data of all nodes in the network, whereas the goal of the data aggregation algorithm is to collect the aggregation results.

In terms of the multi-channel, Kori et al. proposed a type 2 fuzzy logic system (T2FLS)-based resource scheduling technique that uses a software agent approach to solve the scheduling problem by considering the energy of the WSN, the number of neighboring nodes, and the available bandwidth [15]. Nguyen et al. studied the minimum time-aggregation scheduling problem in duty-cycled sensor networks, and a collision-resistant dynamic (CORD) scheduling approach has been proposed with an aim to provide fresh data for emerging IoT applications. The proposed approach is applicable to any initial routing structure and it dynamically changes the receiver of a transmission whenever this change can reduce the aggregation time. Nguyen et al. provided a data collection and scheduling algorithm to optimize energy consumption and transmission reliability [16]. Meirui et al. propose a cluster-based distributed data-aggregation scheduling algorithm, with distributed multi-power and multi-channel (DMPMC) features that can minimize the data aggregation latency in multi-channel and multi-power WSNs. To save energy, low-transmission power is used for packet transmissions inside a cluster and high power is used for packet transmissions among clusters [17]. Lenka et al. proposed a hybrid approach, known as distributed hybrid slot scheduling, for slot scheduling that prepares a feasible schedule in a distributed manner and at the same time reduces the number of feasible schedules to achieve optimality [18]. Durmaz et al. also proposed a multi-channel data collection protocol known as the ENHANCE method [19]. Firstly, this method is used to establish a routing tree and secondly, a channel is allocated to each node so that all child nodes use the channel to send data, and which allocate time slots for the links in the tree so that they do not interfere with each other. In the process of allocating channels, the algorithm is used to prioritize channels to the nodes with the most interference. However, this method is a centralized algorithm and ignores adjacent channel interference, the orthogonality of frequencies depends on the transceiver characteristics, and adjacent channel interference can also cause serious collisions. Uyanik et al. proposed TDMA-based multi-channel scheduling algorithms for multi-channel wireless sensor networks with spatial reuse of channels and time slots. They decrease the required amount of time slots for a round of data gathering by effectively assigning channels and time slots to sensor nodes. The node- and level-based algorithms proposed in the paper color a conflict graph of the original network to determine the time slots that the nodes will use. Then, the original network is scheduled for transmission [20]. Again, the method does not take into account the correlation of channel interference.

The above studies either mainly focus on the infinite number of time slots or channels or ignore the correlation of adjacent channel interference. In a real network, the time slots or channels are limited. Therefore, it is necessary to consider an algorithm to reduce the number of collisions in the entire network through efficient scheduling when the number of time slots and channels is fixed. In addition, the correlation of the interference between channels also needs to be considered.

In this paper, the resource scheduling problem is formulated as a vertex coloring problem in graph theory and a distributed multi-channel TDMA scheduling algorithm based on Gibbs optimization is proposed. By defining the Gibbs energy expression in terms of the objective function of the problem, the joint probability of channels and time slots can be calculated to optimize the selection of the channels and time slots. Based on the annealing Gibbs sampler theory design, the local energy of a node is linked to the distribution function in the Gibbs sampler theory and the channel time slot selection result is represented by vertex coloring. The node calculates the local energy of each color combination with the information of the neighboring nodes. If the local energy of a color combination is lower, then the node is more likely to select it. After several channel selections are performed in the above way, all nodes will select the color combination with the lower energy, thus obtaining a selection scheme with lower communication conflicts. Based on the complete Gibbs optimization theory, the algorithm is proven to be of convergence and the effectiveness of the algorithm has been verified by a large number of experiments. Compared with some centralized methods, although it cannot reach the global optimum, it has good achievability and better application value.

3. Problem Definition

Assuming that the network is static, the nodes in the network periodically perceive the environmental conditions and send information to the sink node in a multi-hop manner. Time is divided into time slots of the same size and multiple time slots together form a frame. All nodes in the network except for the sink node will send the information they perceive to its upper node and the upper node will package all this information into data packets and send it to the sink node. If the collected information cannot be sent with a data packet, multiple packets are needed. Assuming that each time slot is long enough, all data packets could be sent. Because the amount of data perceived by the node is small, the above assumption is credible.

For easy reference, all notations are described in

Table 1. Description of notations.

Notation	Description
$G=\left(V,E\right)$	graph G is an undirected graph, V is the set of vertices, E is the set of edges
$E_{ij}$	link between node i and node j
$d_{\min }$	number of hops between the vertices of the link
SINR	signal to interference and noise ratio
$P_{ij}$	the arrival power of node i detected on node j
$N_j$	the noise interference near node j
deg(Si)	the number of neighbor nodes of node Si
N1(Si)	the neighbor node set of node Si
$T=\left(V,E_t\right)$	the tree generated on the basis of the undirected graph G with S as the root node
Children(Si)	the number of child nodes of node Si in the tree
$t_i$	the allocated time slot
$f_i$	the allocated channel
$Ge=\left(V^{\prime },E^{\prime }\right)$	$\mathrm{interference} \mathrm{graph} \mathrm{based} \mathrm{on} \mathrm{the} \mathrm{undirected} \mathrm{graph} G \mathrm{and} \mathrm{the} \mathrm{routing} \mathrm{tree} T$.
$\mathit{S}$	the ratio of rectangle length to width
$\mathit{C}$	$\mathrm{the} \mathrm{set} \mathrm{of} \mathrm{all} \mathrm{vertices} \mathrm{with} \mathrm{a} \mathrm{combination} \mathrm{of} \mathrm{the} \mathrm{main} \mathrm{colors} {\mathit{C}}_1 \mathrm{and} \mathrm{secondary} \mathrm{colors} {\mathit{C}}_2$
a	a node’s choice of main and secondary colors in the interference graph
$Q_{s_i}\left(a\right)$	$\mathrm{the} \mathrm{communication} \mathrm{conflict} \mathrm{of} \mathrm{the} \mathrm{node} S_i$
$1(•)$	indicator function
$W_{s_i}$	the interference degree of the node Si
$P{s}_i\left(s_j\right)$	$\mathrm{when} \mathrm{node} s_i \mathrm{receives} \mathrm{messages} \mathrm{from} \mathrm{node} s_j, \mathrm{the} \mathrm{power} \mathrm{perceptron} \mathrm{will} \mathrm{sense} \mathrm{the} \mathrm{power} \mathrm{of} \mathrm{node} s_j$
${\pi }_T\left(\mathit{X}\right)$	distribution functions for Gibbs sampling
$\mathit{X}$	a solution vector of the system
$\epsilon \left(\mathit{X}\right)$	$\mathrm{the} \mathrm{energy} \mathrm{of} \mathrm{solution} \mathrm{vector} \mathit{X}$
V(L)	non-negative expression
${\mu }_{s_i}\left(x_{s_i}\right)$	the probability of $s_i$ picking the color combination $x_{s_i}$
$\Delta G$	degree of graph

.

3.1. Interference Model

In the process of assigning time slots and channels to links in the network, the following two types of conflicts must be taken into consideration, as shown in Figure 1.

Direct conflict: Two links with common nodes are assigned the same time slot. Because the node is working in half-duplex mode, it can neither send nor receive at the same time nor can it receive messages sent by multiple nodes at the same time.

Indirect conflict: Two links are assigned the same time slot and channel and the receiving end of either link is within the sending range of the other link.

Definition 1.

The distance between two links. For any two links sum$E_{ij}=(v_i,v_j)$and$E_{km}=(v_k,v_m)$in the graph$G=\left(V,E\right)$, the distance between the two links is defined as the minimum number of hops between the vertices of the link, which is recorded as$d_{\min }$. If the two links have a common node, then$d_{\min }=0$.

Theorem 1.

The sufficient non-essential condition for the collision of two links is$d_{\min }\le 1$.

Proof. 

Sufficiency: If there is a direct conflict between two links, according to the definition of direct conflict, there would be a common node between the two links, then $d_{\min }=0$; if the two links conflict indirectly, the receiving end of one link must be in the sending range of the other and the two vertices remain within the range of one hop, so $d_{\min }=1$. □

When the distance between the two links $d_{\min }\le 1$, by allocating different time slots, the two links can completely avoid conflicts. It can be seen that the sufficient non-essential condition for collisions between two links is the distance between the links $d_{\min }\le 1$.

The channel interference model SINR [21] (signal to interference and noise ratio) is used in this paper. For node j to successfully receive the data packet sent by node i, SINR_ij must be greater than a specific threshold β and the calculation formula of SINR_ij is as follows:

$$SIN{R}_{ij}=\frac{P_{ij}}{N_j+{\displaystyle \sum _{k\ne i}{P}_{kj}}}$$

(1)

where $P_{ij}$ represents the arrival power of node i detected on node j, and $N_j$ represents the noise interference near node j. Although the calculation of SINR is rather complicated and requires statistics and calculation of a large number of parameters, the use of the SINR interference model can help to achieve better network performance in theory and in practice.

3.2. Network Model

Assuming that there is one sink node and n sensors in the network as sensing nodes, based on the SINR interference model, an undirected graph $G=\left(V,E\right)$ is used to represent the entire network, where V represents the set of all nodes, $s\in V$ represents the sink node, and E is the set of all edges, representing the collection of all links in the network. In graph G, deg(S_i) represents the number of neighbor nodes of node Si, and N1(S_i) represents the neighbor node set of node S_i. We use $T=\left(V,E_t\right)$ to represent the tree generated on the basis of the undirected graph G with S as the root node; $E_t\in E$ represents the set of all edges. Children(S_i) represents the number of child nodes of node S_i in the tree. Each node in the tree is equipped with a frequency sensor to detect the received power and each node has the same transmit power. In the convergence process, the convergence data is sent to the sink node through the routing tree. The data of the child nodes are collected for aggregation by the node and sent to the upper layer node and this repeats until it reaches the sink node.

Take Figure 2 as an example. The left half of Figure 2 is an undirected graph $G=\left(V,E\right)$ and the right half is a routing tree $T=\left(V,E_t\right)$ constructed by the algorithm. It can be seen that the complex network has become a simpler routing tree. The scheduling algorithm in this paper is based on the routing tree. The specific method of constructing the routing tree will be introduced below.

3.3. Multi-Channel Scheduling

Multi-channel scheduling can effectively improve the reusability of time, thereby improving the throughput of the entire network. The use of multiple channels can effectively avoid the impact of indirect conflicts, as shown in Figure 3. The solid line in the figure represents the actual communication link, the dashed line represents the interference link, $t_i$ is the allocated time slot, and $f_i$ is the allocated channel. If there is only one channel, the network needs six time slots to avoid collisions, as shown in Figure 3a. However, if there are three channels, the network only needs three time slots to avoid collisions, as shown in Figure 3b. It can be drawn from the above-mentioned that direct conflicts between links can only be resolved by allocating different time slots and the two that may have indirect conflicts can only communicate at the same time when there are conditions as follows: (1) different allocated time slots, and (2) the allocated channels are different.

3.4. Interference Graph

In order to intuitively reflect the interference relationship between the links in the routing tree T, we build an interference graph $Ge=\left(V^{\prime },E^{\prime }\right)$ based on the undirected graph $G$ and the routing tree $T$. Each vertex $Ge$ represents a link in the routing tree $T$. If the link represented by the two vertices in $Ge$ directly conflicts in the undirected graph $G$, then the two vertices are connected by a solid line. The two vertices indirectly conflict in $Ge$ and are connected by a dashed line.

Figure 4 shows the interference graph $Ge=\left(V^{\prime },E^{\prime }\right)$ transformed from the undirected graph $G=\left(V,E\right)$ and the routing tree $T=\left(V,E_t\right)$ in Figure 2. The solid line represents the occurrence of a direct conflict and the dotted line represents the occurrence of an indirect conflict. For example, the node $V_{12}$ of the interference graph represents a link between node $V_1$ and node $V_2$ in the routing tree, and the solid line between node $V_{12}$ and node $V_{24}$ indicates that the link between $V_1$ and $V_2$ has a direct conflict with the link between $V_2$ and $V_4$ in the routing tree.

3.5. Interference Graph

The coloring problem originated from the famous “four-color problem”. To put it simply, it is to color a map on a flat or spherical surface and only four colors can make any two adjacent countries of different colors. The coloring problem of graphs can be divided into vertex coloring, edge coloring, and full coloring problems [22,23]. Normally, graph coloring refers to the vertex coloring of the graph, which means that all vertices in the graph are colored to ensure that the colors of adjacent vertices are different. Since both edge coloring and full coloring can be transformed into vertex coloring of graphs, the problem of vertex coloring of graphs has always been a hot issue of research. The problem of the N-fold vertex coloring of graphs is mainly to use the least colors to color the vertices of the graph, requiring each vertex to have N different colors, and the 2N colors of adjacent vertices are different [24].

In the data aggregation network, the allocation of channels and time slots to the communication link can be regarded as the problem of the edge coloring of the routing tree. By transforming the routing tree T into the interference graph Ge, the edge coloring problem is transformed into the vertex coloring problem of the interference graph. Since each link is allocated to time slots and channels, it is the same as assigning two colors to the vertices in the interference graph Ge. These two colors are referred to as the main color and the secondary color of the vertex. The main color represents the time slot allocated to the link, and the secondary color represents the allocated channel. In the process of coloring the interference graph, we define the constraints as follows:

Definition 2.

Coloring constraints: In the interference graph, if the solid line between the two vertices is connected, the main colors of the two vertices are required to be different; if the two vertices are connected by a dashed line, the main color and the secondary color of the two vertices are required to be different.

Based on the definition mentioned above, we take the vertex coloring problem of interference graphs as the vertex double coloring problem with constraints.

In accordance with the condition of coloring, we perform a vertex double coloring of Figure 4, as shown in Figure 5. The solid line represents the occurrence of a direct conflict and the dotted line represents the occurrence of an indirect conflict. The color of the left half of the vertex represents the main color, and the color of the right half of the vertex represents the secondary color. By the physical meaning of the primary and secondary colors, it is certain that the primary and secondary colors of the vertex can be the same. It can be seen from the figure that we only need three main colors and two secondary colors to complete the vertex coloring of Figure 4.

3.6. Minimization of Communication Conflict

Assuming that the network has k assignable time slots and m assignable channels, the network model shows that the interference graph has $n$ vertices. $\mathit{S}=\left\{S_1,S_2,\cdots ,S_n\right\}$ represents the set of all vertices, ${\mathit{C}}_1=\left\{C_{11},C_{12},\cdots ,C_{1k}\right\}$ is the set of all vertices in the main colors in the interference graph, ${\mathit{C}}_2=\left\{C_{21},C_{22},\cdots ,C_{2m}\right\}$ is the set of all vertices in the secondary colors. $\mathit{C}=\left\{\left(C_{11},C_{21}\right),\left(C_{11},C_{22}\right),\cdots ,\left(C_{1k},C_{2m}\right)\right\}$ is the set of all vertices with a combination of the main and secondary colors. Use a to represent a node’s choice of main and secondary colors in the interference graph and use A to represent all possible choices. The vector a can be represented by $\mathit{a}=\left\{a\left(s_1\right),a\left(s_2\right),\cdots ,a\left(s_n\right)\right\}$, where $a\left(s_i\right)\in C$ represents a combination of the main color and the secondary color selected by the node $S_i$ from the set C.

Definition 3.

A direct collision domain is a collection of nodes connected to a neighbor of one hop with a solid line in the interference graph.

Definition 4.

An indirect collision domain is a collection of one-hop nodes connected by the dashed line with a neighbor node in the interference graph.

Definition 5.

The interference degree is the number of nodes included in the link branch represented by the node in the interference graph in the routing tree.

According to Definitions 3–5, we can obtain the following definition:

Definition 6.

Communication Conflict of Node (CCN). When the vertices of the interference graph Ge choose a certain combination scheme$\mathit{a}\in A$for coloring, the communication conflict of the node$S_i$is defined as

$$Q_{s_i}\left(a\right)=({\displaystyle \sum _{s_j\in I^{\prime }{s}_i}1(c_1(s_j)=c_1(s_i))}+{\displaystyle \sum _{s_j\in I^{″}{s}_i}1(c_1(s_j)=c_1(s_i),c_2(s_j)=c_2(s_i))})*W_{s_i}$$

(2)

where$1(•)$is the indicator function; when the condition is met, it is equal to 1, otherwise, it is 0.$I^{\prime }{s}_i$represents the direct conflict domain of the node$S_i$,$I^{″}{s}_i$is the indirect conflict domain of the node, and$W_{s_i}$is the interference degree of the nodeS_i.$c_1(s_j)$is expressed as the color assigned by the node$S_j$in the color set$C_1$and$c_2(s_j)$is expressed as the color assigned by the node$S_j$in the color set$C_2$.

Through Formula (2), we can see that the communication conflict of a node is not only related to the neighbors of the node but also closely related to the position of the node in the network. When the node is closer to the sink node, if a conflict occurs, the greater the impact on the data convergence of the entire network.

Given the combination option a, the communication conflict of the data aggregation network is

$$Q\left(\mathit{a}\right)={\displaystyle \sum _{s_i\in \mathrm{S}}Q{s}_i\left(\mathit{a}\right)}$$

(3)

It can be seen that the better the combination selection scheme, the smaller the communication conflict of the data aggregation network. According to the above definition, the minimization of network communication conflicts in the data aggregation problem is defined as follows:

Definition 7.

Given an undirected graph$G=\left(V,E\right)$, k time slots, and m channels, the problem of minimizing network communication conflicts is to build a routing tree T in the undirected graph G and assign channels and time slots to the links in T so that it meets the constraints on the interference graph Ge and minimizes the communication conflict of the entire network.

Theorem 2.

The problem of minimizing network communication conflicts is NP complete.

Proof. 

(1) From the definition, the problem of minimizing network communication conflicts is a kind of combinatorial explosion problem, which cannot be solved by a polynomial algorithm. It is an NP problem.

(2)

The problem of minimizing the vertex coloring of graphs is a special case of minimizing network communication conflicts and is an NP-complete problem [25]. The problem of minimizing vertex coloring is described as follows: Given an undirected graph $G_c=\left(V_c,E_c\right)$ and a non-negative integer K. Whether there is a non-negative integer $k\le K$, divided by $V_c$ into k disjoint subsets (each subset represents a specific color) $\{V_{c_1},V_{c_2},\cdots ,V_{c_k}\}$ so that the nodes in the same subset are not adjacent in the graph $G_c$. We can see that when the number of channels m = 1, $Q\left(\mathit{\alpha }\right)=0$, the problem of minimizing network communication conflicts is a typical problem of minimizing vertex coloring. □

It can be seen in Equations (1) and (2) that the problem of minimizing network communication conflicts is an NP-complete problem. This is proven above.

Figure 6 shows the impact of different channel and time slot scheduling schemes on the entire network communication conflict in the same network environment. From Figure 2 and Figure 4, we have learned the network topology and interference diagram. Assuming that the number of time slots and the number of channels are both two, the channel time slot allocation scheme in Figure 6a makes the communication conflict for the entire network also two. In addition, the channel time slot allocation of Figure 6b makes the communication collision of the whole network five. It can be seen that for the same network environment, the network communication conflicts caused by different channel time slot scheduling schemes are different. Our goal is to find a channel time slot scheduling scheme to minimize network communication conflicts.

4. Multi-Channel TDMA Scheduling Algorithm

The coloring problem in graphs is an NP-complete problem [26]. The Gibbs optimization-based scheduling algorithm is proposed in this paper to minimize the communication conflict. The following two steps are included in the scheduling algorithm:

• Constructing routing tree T on the basis of undirected graph G, and constructing the interference graph Ge based on graph G and routing tree T.
• Performing conditional double coloring for the vertex in the interference graph Ge to minimize the communication conflict.

4.1. Construction of Routing Trees

Definition 8.

Power Priority. When node$s_i$receives messages from node$s_j$, the power perceptron will sense the power of node$s_j$(recorded as$P{s}_i\left(s_j\right)$), based on a determination of the priority of the response to the messages of$s_j$. The higher the power of$P{s}_i\left(s_j\right)$, the higher the priority of node$s_j$for node$s_i$.

Constructing degree-constrained BFS routing trees is similar to the existing algorithms. The only difference is the maximum number of child nodes in the trees is limited in the BFS method. The construction of a degree-constrained BFS routing tree starts from the sink node and works its way down. Initially, it is assumed that all nodes are synchronized and that each node starts a timer for a sufficiently long period of time; all nodes have the same transmission power and carry a power sensor. Then, a routing tree is constructed in the network with the following steps.

(1)

Nodes in the tree check their number of child nodes and broadcast the construction message if the number is less than d. Otherwise, the broadcast stops.

(2)

Upon receiving multiple construction messages, the node with the highest priority is selected for all the nodes that are not in the tree as the father node based on the power priority of the broadcasting node, which sends the father node a message requesting to join.

(3)

When receiving multiple request messages from nodes outside the tree, nodes in the tree select the node with higher priority based on its own number of child nodes and power priority and send an ACK message to the node with the higher priority.

(4)

A node outside the tree joins in the tree only after receiving an ACK message from its father node. Otherwise, the node with the higher priority is selected as the father node, and a message requesting to join is sent to it.

(5)

The above process is repeated until no nodes join the routing tree. When the timer is triggered, if the node has not yet joined the routing tree, it needs to broadcast to look for the message of the father node. The node in the tree receives the message and replies with its own number of child nodes. Based on the messages, the node selects a node in the tree with the number of child nodes less than d as its own father node. If there are no nodes with a number of child nodes less than d, the node with the least number of child nodes is selected.

This construction method cannot guarantee that the maximum number of child nodes in the routing tree does not exceed d. However, generally, the maximum number of child nodes in the routing tree constructed with this method is close to d, as long as d is reasonable. The construction of the BFS routing tree is described in Algorithm 1:

Algorithm 1. Construction of routing trees.

Input: G = (V, E), the max number of children d, the sink node S

Output:$T=\left(V,E_t\right)$

01. Initialization: $V_1$ = [1], $E_1=\varnothing$;

02. do while

03. for i 1 to |$V_1$|

04. if (children($s_i$) < d)

05.  for j 1 to deg ($s_i$)

06.   if ($P{s}_i\left(s_j\right)$ == max(P$s_i$(N1($s_i$))))

07.    $V_1$ = $V_1$ + {$s_j$};

08.    $E_1$ = $E_1$ + {($s_i$, $s_j$)};

09.    children ($s_i$) = children ($s_i$) + 1;

10. N1($s_i$) = N1($s_i$) − {$s_j$};

11. end if

12. end for

13. end if

14. end for

15. $V_2$ = $V$ − $V_1$;

16. for i 1 to |$V_2$|

17. for j 1 to |$V_1$|

18.  if (children ($s_j$) == min (children ($V_1$)))

19.   $V_1$ = $V_1$ + {$s_i$};

20.   $E_1$ = $E_1$ + {($s_i$, $s_j$)};

21.   children ($s_j$) = children ($s_j$) + 1;

22.  end if

23. end for

24. end for

25. end while ($V_2$ == $\varnothing$)

26. V = $V_1$;

27. $E_t$ = $E_1$;

Figure 7 presents the construction process of the routing tree in Figure 2 and d = 2. Node $V_1$ is aggregated to broadcast the message of construction, and $V_2$, $V_3$ join in the tree, as shown in Figure 7a. Node $V_2$, $V_3$ keep broadcasting, and node $V_4, V_5$, and $V_6$ join in the tree, as shown in Figure 7b,c. Since then, all nodes in the network are in the routing tree, which means that the construction of the routing tree is finished.

From routing tree T and undirected graph G, the interference graph Ge can be easily obtained. Initially, all the vertices in the interference graph Ge are in the uncolored state.

4.2. Vertex Coloring

The Gibbs optimized node coloring algorithm (GSNC) is put forward. Then the related Gibbs optimized theories will be introduced.

4.2.1. Gibbs Optimization

Gibbs optimization is a Markov chain Monte Carlo (MCMC) algorithm, first put forward by physicist J. W. Gibbs in 1902, mainly for counting direct sampling of random sequences in physical phenomena. In 1984, German and Stuart developed the Gibbs optimized theory on this basis. The most prominent property of the Gibbs optimization is the construction of its Markov chains with the method of constructing conditional distributed sequences along with a series of complementary directions. It is suitable for solving multi-dimensional optimized problems that are discrete and have correlations between the dimensions.

The main formulation of the Gibbs optimization with an annealing mechanism is:

$${\pi }_T\left(X\right)=\frac{1}{Z_T}{e}^{-\frac{1}{T}\epsilon \left(X\right)}$$

(4)

where $\mathit{X}$ is the solution vector of the system,$\Lambda$ is the collection of all solution energies, so $\mathit{X}\in \Lambda$, $T>0$ is the annealing temperature, and energy expression $\epsilon \left(\mathit{X}\right)$ is the energy of solution vector $X$. Because $0<\epsilon \left(\mathit{X}\right)<+\infty$, $Z_T={\displaystyle \sum _{X^{\prime }\in \wedge }{e}^{-\frac{1}{T}\epsilon (X^{\prime })}}$ is a normalized constant, ${\pi }_T\left(\mathit{X}\right)\in [0,1]$.

In the Gibbs optimization, each factor in the sample is selected and updated to obtain a new sample, and the latter operation needs to be based on the results of the previous selection [27,28,29,30]. Supposing that solution vector $\mathit{X}$ has the value of $d$ dimension, $X=\left(x_1,\cdots ,x_d\right)$. Assuming that the solution vector in the calculation for the $t$ time is $X^{\left(t\right)}=\left(x_1^{\left(t\right)},\cdots ,x_d^{\left(t\right)}\right)$, then for every $i\left(i=1,2,\cdots ,d\right)$ in the calculation of the $t+1$ time, $x_i^{\left(t+1\right)}$ is selected in the solution space according to the value of the following probability $p\left(x_i|x_1^{\left(t+1\right)},\cdots , x_{i-1}^{\left(t+1\right)}, x_{i+1}^{\left(t\right)}, \cdots , x_d^{\left(t\right)}\right)$. The above operations are repeated to complete the optimization of the solution vector and achieve the solution $X^{\left(0\right)}$, $X^{\left(1\right)}$, …, $X^{\left(1\right)}$, $X^{\left(t+1\right)}$, … in the Gibbs steady state.

Theorem 3.

A stochastic solution vector must converge to a steady state after finite times of Gibbs-optimized iterations of the computation.

The concepts of subgroup and potential are also defined in the Gibbs optimization. A set [1] consisting of independent elements is called a subgroup; a non-empty set L, whose elements are adjacent to each other, is also called a subgroup. A subgroup L is called the largest subgroup if it ceases to be a subgroup when any of its elements are added to it. As Figure 8 shows, potential connects a non-negative expression V(L) with L, whose value only relates to the value of elements in L. When L is not a subgroup, the value of V(L) constantly is zero.

If the following equation holds, then the energy expression can be expressed by a potential V(L):

$$\epsilon \left(X\right)={\displaystyle \sum _{L}V\left(L,X\right)}$$

(5)

Supposing that interference graph Ge has n nodes $s_1,s_2,\cdots ,s_n$, (i = 1, 2, …, n) in a solution $x_{s_i}$, i.e., the color the node selects, all solutions constitute set $C$, and the solution of Ge can be expressed as a solution vector $X=\left(x_{s_1},x_{s_2},\cdots ,x_{s_m}\right)$ $x_{s_i}\in C,i=1,2,\cdots ,n$. Therefore, the local energy of $s_i$ is the sum of potentials of subgroups about $s_j$, and the equation is as follows:

$${\epsilon }_{s_i}\left(x_{s_i}\right)={\displaystyle \sum _{L:s_i\in L}V(L})$$

(6)

Substituting the local energy ${\epsilon }_{s_i}$ of $s_i$ into the Gibbs-optimized distribution equation, the probability of $s_i$ picking the color combination $x_{s_i}$ can be deduced:

$${\mu }_{s_i}\left(x_{s_i}\right)=\frac{\exp \left(-\frac{1}{T}{\epsilon }_{s_i}\left(x_{s_i}\right)\right)}{{\displaystyle \sum _{x_{s_i}^{\prime }\in C}\exp \left(-\frac{1}{T}{\epsilon }_{s_i}\left(x_{s_i}^{\prime }\right)\right)}}$$

(7)

The solutions of $s_i$ are calculated based on the solutions of the neighboring nodes, that is, based on the coloring of the neighboring nodes, $s_i$ picks its own color combinations in the set C according to Equation (7) to update the solution once. It can be seen from Equation (7) that nodes select their own color combinations using roulette gambling. The lower the local energy of a color combination, the higher the probability of it being selected. $s_i$ requires only the information of the neighboring nodes to update the solution based on the local energy, and after the continuous solution of the algorithm, the coloring scheme of the vertices in the interference graph Ge will converge to the solution vector with lower global energy.

4.2.2. Methodology

The optimized goal in this paper is to minimize the number of communication conflicts of the nodes (CCN) in the interference graph. The equation is as follows:

$$Q(a)={\displaystyle \sum _{s_i}\left(Q_{s_i}\left(a_i\right)\right)}={\displaystyle \sum _{s_i}\left(\left({\displaystyle \sum _{s_j\in I^{\prime }{s}_i}1(c_1(s_j)=c_1(s_i))+{\displaystyle \sum _{s_j\in I^{″}{s}_i}1(c_1(s_j)=c_1(s_i),c_2(s_j)=c_2(s_i))}}\right)*W_{s_i}\right)}$$

(8)

That is, looking for a combination scheme a, which can minimize $Q\left(\mathit{a}\right)$. Since it is not a centralized solution, Equation (8) needs to be altered to the communication conflict of the nodes:

$$Q_{s_i}\left(a\right)=({\displaystyle \sum _{s_j\in I^{\prime }{s}_i}1(c_1(s_j)=c_1(s_i))}+{\displaystyle \sum _{s_j\in I^{″}{s}_i}1(c_1(s_j)=c_1(s_i),c_2(s_j)=c_2(s_i))})*W_{s_i}$$

(9)

Next, utilizing Equation (9) to design the energy expression:

$$\begin{array}{ll}\epsilon (x)& ={\displaystyle \sum _{s_i} \frac{Q_{s_i}(a(s_i))}{N_{s_i}}}={\displaystyle \sum _{s_i}\left(\frac{{\displaystyle \sum _{s_j\in I^{\prime }{s}_i}1(c_1(s_j)=c_1(s_i))+{\displaystyle \sum _{s_j\in I^{″}{s}_i}1(c_1(s_j)=c_1(s_i),c_2(s_j)=c_2(s_i))}}}{N_{s_i}}\right)}*W_{s_i}\\ & ={\displaystyle \sum _{s_i}{\displaystyle \sum _{s_j\in I^{\prime }{s}_i}\left(\frac{1(c_1(s_j)=c_1(s_i))}{N_{s_i}}\right)}}*W_{s_i}+{\displaystyle \sum _{s_i}{\displaystyle \sum _{s_j\in I^{″}{s}_i}\left(\frac{1(c_1(s_j)=c_1(s_i),c_2(s_j)=c_2(s_i))}{N_{s_i}}\right)*W_{s_i}}}\end{array}$$

(10)

where $N{s}_i$ is the number of the one-hop neighbor of $s_i$, so the energy expression of the node is

$${\epsilon }_{s_i}(a(s_i))=\left({\displaystyle \sum _{s_j\in I^{\prime }{s}_i}\frac{1(c_1(s_j)=c_1(s_i))}{N_{s_i}}}+\sum _{s_j\in I^{″}{s}_i}\frac{1(c_1(s_j)=c_1(s_i),c_2(s_j)=c_2(s_i))}{N_{s_i}}\right)*W_{s_i}$$

(11)

V is the potential about L by Equation (12):

$$V(L)=\{\begin{array}{cc}\left(\frac{1(c_1(s_j)=c_1(s_i),s_j\in I^{\prime }{s}_i)+1(c_1(s_j)=c_1(s_i),c_2(s_j)=c_2(si),s_j\in I^{″}{s}_i)}{N_{s_i}}\right)*W_{s_i}& L=\{s_i,s_j\}\\ 0& else\end{array}$$

(12)

Thus, the energy expression can be expressed as a sum of potentials, which satisfies the conditions for the application of the Gibbs optimization theory.

According to the method of the Gibbs-optimized theory, the probability distribution of $s_i$ selecting color combinations $x_{s_i}$ is obtained by substituting Equation (11) for Equation (7). After iterative computation, each node picks the color combination with the lower energy, and the wireless aggregation network converges to the color combination with the lower energy, i.e., the time slot and channel selection scheme with the lower communication conflict of the node.

To facilitate the graph vertex coloring method based on the Gibbs optimization to converge faster to the most optimized color combination globally, we incorporate an annealing mechanism:

$$T=\frac{T_0}{\log (2+t)}$$

(13)

where $T_0$ is the initial value, and t is the number of method iterations. The whole process of the coloring algorithm is described in Algorithm 2:

Algorithm 2. Graph vertex coloring based on the Gibbs optimization.

Input: The graph Ge = (V, E), the number of main color k, the number of secondary color m, the max number of iterations $q$, Initial temperature $T_0$

Output: Coloring result of every node in Ge

01. Initialization: count = 0;

02. do while

03. count = 0;

04. T = $T_0$/log(2 + t)

05. for $i$ 1 to |V|

06. for $j$ 1 to k

07.  for z 1 to m

08.   Energy ($s_i$, $j$, z);

09.  end for

10. end for

11. for $j$ 1 to k DO

12.  for z 1 to m DO

13.   Probability ($s_i$, T, Energy ($s_i$, $j$, z)); //Probability of the node selecting the present color combination

14.  end for

15. end for

16. count = count + Conflict ($s_i$); //Total conflicts in the present network

17. end for

18. end while (count = 0 or t > $q$)

5. Experiments and Analysis

The time slot and channel-scheduling algorithm based on the Gibbs optimization in the paper is compared to the Greedy algorithm, the ENHANCE [19] scheme, and LCA [20] scheme to analyze the feasibility of the algorithm in this paper. The Greedy algorithm is a distributed algorithm that selects color combinations for each vertex in the interference graph and minimizes the communication conflicts of the vertices. Both the ENHANCE and LCA are centralized algorithms.

Table 2. The tag recognition effect at 50 m.

${\mathit{T}}_0$	Maximum Number of Iterations	Degree of the Routing Tree
0.005	1000	$\Delta G/3$

demonstrates the settings of the parameters of the algorithm in this paper.

$T_0$ is an important parameter for annealing Gibbs. If $T_0$ is large, the algorithm converges slowly in execution, which is beneficial for searching for a global optimal solution; if $T_0$ is small, the algorithm converges fast in execution but may achieve the local optimal solution. In the application experiments of the paper, the algorithm has better solution performances when $T_0$ is taken between 0.001 and 0.01 and the maximum number of iterations is 1000 generations.

In this paper, the algorithm experiments were completed on the MATLAB R2018b(China) platform. The nodes were randomly distributed in an area of 250 × 250 m². In the process of constructing the routing tree, the node with ID 1 is selected as the aggregation node by default. Figure 9 displays the network topology graph and the routing tree constructed when the number of nodes is 25 and the communication radius is 100.

The Greedy algorithm and ENHANCE algorithm are deterministic algorithms, so they only need to be run once to obtain a deterministic result; the experimental result of this algorithm is the average value after running it 20 times.

5.1. Effects of Network Parameters on Scheduling Algorithms

This experiment verifies the effects of the network parameters on the scheduling algorithm by making comparisons in terms of the number of nodes, the communication radius, and the number of assignable time slots. This is shown in Figure 10.

Figure 10a presents the effects of node changes in the network on the scheduling algorithm. The number of nodes in the network changes from 50 to 400 with an increase of 50; the communication radius of the nodes is 100 m; the number of assignable time slots is 25; and the number of assignable channels is 3. In the case of a small number of nodes in the network, all four algorithms can obtain a better solution. It can be seen that when the number of nodes is less than 100, an optimal solution is found in both the algorithm of this paper and the LCA algorithm, making the conflicts of the whole network zero. As the number of nodes and the network complexity increase, the number of network communication conflicts in this algorithm decreases significantly compared to the other three algorithms and the difference is a significant increase in the process.

Figure 10b manifests the effect of the node communication radius on the scheduling algorithm. The number of nodes in the network is 200; the number of assignable time slots is 25; the number of assignable channels is 3; and the communication radius of the nodes changes from 80 m to 120 m, with an increase of 5 m. In the case of a smaller communication radius, i.e., less interference between nodes, the Greedy algorithm is slightly better than this algorithm; as the communication radius increases and the environment of the network becomes increasingly complex, the network communication conflicts and the increase of this algorithm are smaller than those of the other three algorithms. The above two experiments show that this algorithm is more suitable for a network with a much more complex environment, which is of more practical value in application.

Figure 10c,d indicate the effects of the number of time slots on the scheduling algorithm when the number of channels is 3 and 12. The number of nodes in the network is 200, the communication radius is 100 m, and the number of assignable time slots changes from 15 to 45, with an increase of 5. When the number of time slots is small, the number of conflicts of this algorithm is much smaller than that of the other two algorithms, and as the number of time slots increases, this algorithm always outperforms the Greedy and ENHANCE algorithms. As shown in Figure 10c, when the number of time slots is 45, the number of conflicts of the Greedy algorithm is 0 and that of this algorithm is 0.8. It can be seen from Figure 10d that when the number of channels is 12, the algorithm basically maintains the same solution effect as the LCA algorithm and is slightly better than the Greedy algorithm. With the increase in allocatable channel resources in the whole network, the complexity of resource allocation increases significantly, and the advantages of the Greedy algorithm are more obvious. In this case, the algorithm can still achieve relatively good performance. It is fully verified that the approximate optimal solution of the algorithm is not much different from the optimal solution of this experiment, which proves the effectiveness of the algorithm.

5.2. Effect of Routing Tree’s Degree on Scheduling Algorithm

This set of experiments proves the effect of the routing tree’s degree on the signal conflict, in which the amount of distributable time slots is 25 and the communication range of nodes is 100 m. The construction of the BFS routing tree with a limited degree is adopted in this paper. Figure 11 shows the number of signal conflicts when the number of channels is 3 and 12 and the degree of the BFS routing tree is $\Delta G/2$ and $\Delta G/4$. As shown in Figure 11a,b, when the degree of the routing tree decreases from $\Delta G/2$ to $\Delta G/4$, the signal conflicts of the three algorithms demonstrate an obvious decline. However, the proposed algorithm has a larger magnitude of decline, whose solution is closer to the optimal solution during the process of the degree decline. Comparing Figure 11a,c, when the number of channels is 3 and 12, this algorithm’s signal conflict remains constant, whereas the other two algorithms’ signal conflicts drop apparently as the channel number increases, which demonstrates that this algorithm reaches a better solution with limited resources. In Figure 11b, as the routing tree’s degree reduces to $\Delta G/4$, three channels cannot make the number of signal conflicts decrease for too many indirect conflicts. Therefore, the conclusion is that when the degree of the routing tree reduces, more channels are necessary to reduce the signal conflict as indicated in Figure 11. In addition, the proposed algorithm has better performance than the other two algorithms.

Figure 12 indicates the effects of different $T_0$ on the algorithm performance, which can be seen that when $T_0$ is big, the iteration process became slow and more volatile, and when $T_0$ is small, the algorithm soon iterated an optimum value and had low volatility. By comparison, it is found that the algorithm solution is only partially optimal. After a series of experiments, it can be concluded that the algorithm performance is better when is in the range of 0.001~0.01.

5.3. Network Throughput and Transmission Delay

This set of experiments evaluates the network throughput and network transmission delay of the algorithm. The detailed parameters are as follows:

• Network throughput: the number of packets sent in nodes of each time slot network.
• Transmission delay: the average waiting time between two successful transmissions.

In the simulation experiment, the amount of distributable time slots is 25 and the communication range of the nodes is 100 m. As shown in Figure 13, with fewer nodes, the transmission delay is shorter, which indicates that the amount of signal conflicts is lower and the nodes that are sent successfully in each time slot are fewer.

Therefore, it is still a low network throughput. With the addition of nodes, the transmission delay becomes longer. However, nodes that are sent successfully in each time slot are rapidly increasing, leading to gradually improved network throughput. Because of the fixed number of distributable time slots, when nodes keep being added, there is an increase in signal conflicts but a gradual decrease in network throughput. By reading Figure 13, it can be seen that when the channel number is 3 and the number of nodes is around 180, the network throughput of this algorithm reaches the maximum (0.18); when the channel number is 12 and the number of nodes is in the range of 100 to 200, the network throughput of this algorithm is more than 0.18. Therefore, it can be concluded that as the number of nodes reduces, the indirect conflict is the major factor limiting network throughput, and direct conflict has a greater impact on network throughput as the number of nodes increases. In this paper, the Gibbs energy function is constructed based on the objective function and a multi-channel TDMA scheduling algorithm based on the Gibbs sampling is used to optimize the selection of channels and time slots, which can handle the correlation of the interference between channels well and thus improve the throughput of the network. Figure 13 and Figure 14 show better performance in terms of network throughput and transmission delay than the other three algorithms.

6. Conclusions

Data aggregation networks are always a hot issue in the study of wireless sensor networks. Focused on scheduling channels and TDMA time slots, a scheduling algorithm based on the Gibbs optimization is designed. In this paper, the scheduling problem is transformed into a vertex coloring problem, the node communication collision is defined, and the minimum signal conflict is described. In addition, the Gibbs optimization is introduced. The best combination of channel and time slots is selected by probability distribution to solve the real scheduling problem. The design of this algorithm, in accordance with the necessary and sufficient conditions, is featured as a distributed implement ability. The experimental results show that the algorithm in this paper is effective and has a quick convergence rate and a higher quality of iteration solutions.

In the future, with the difference between the nodes’ communication and disturbance ranges under consideration, under the premise of maximizing the nodes’ life cycles using power control, we will accomplish the solution of channel and TDMA scheduling in data aggregation networks.