Channel Allocation Algorithm Based on Swarm Intelligence for a Wireless Monitoring Network

Abstract

In wireless networks, multiple monitoring nodes are used to collect users’ transmission data in real time, which can be used for fault diagnosis and analytical feedback of the wireless network. Due to the limited number of monitoring nodes, key issues include how to optimize and schedule the channel resources of each node to cover more users, obtain more network data, and maximize the quality of network monitoring. In this paper, a channel allocation algorithm based on swarm intelligence—“discrete bacterial foraging optimization”—is proposed based on the classic bacterial foraging optimization algorithm. The position of each dimension in the iterative process is discretized to binary 0 or 1 to encode and express the channel allocation problem of wireless monitoring networks, and the channel allocation scheme is optimized by location updates guided by bacterial foraging. Many simulation and practical experiments have proved the effectiveness of the algorithm, and it also has low complexity and provable convergence. Compared with similar algorithms, this algorithm improves monitoring quality by 1.428% while boosting speed by up to 32.602%. The algorithm has lower complexity, higher performance, and can converge to the optimal solution at a faster rate.

1. Introduction

Wireless networks refer to communication networks that use radio waves as the information transmission medium. According to topology, wireless networks can be divided into infrastructure networks and infrastructure-free networks, which expand the cable network services space. Because of their low cost, short cycle, flexible structure, and many other advantages, wireless networks eliminate the limitations of the time, place, and object. In recent years, wireless network (WLAN, WiFi, WiMax, Mesh) technology has developed rapidly [1,2]. Strong openness, high dynamics, and easy scalability have resulted in many hidden dangers while promoting network construction: due to the lack of perfect security protection, these networks are prone to information security issues such as illegal information interception and unauthorized information services. The mobility of the user group makes it difficult to allocate network resources fairly and reasonably, which makes network congestion and idle conditions more common. Wireless channel characteristics are changeable and susceptible to interference, which leads to a decrease in the reliability of data transmission [3,4,5]. Wireless network monitoring technology has been applied to analyze the status of wireless networks in real time and improve the security, stability, and rationality of resource allocation. Figure 1 shows how wireless monitoring networks relate to wireless networks. Wireless networks include WLAN, WiFi, WiMax, WMN, Adhoc, etc. The object to be monitored in a wireless network might be a wireless network router, access point, or mobile user [6,7,8,9,10].

A wireless monitoring network is composed of special equipment called wireless sniffers or monitors, which can capture the activity information of users in the wireless network, save the frame information or physical layer information, and conduct distributed or centralized analysis. Real-time online or offline analysis using logs is possible, and the analytic results are mainly used to assist the fault diagnosis and resource management of the wireless network. However, it is very laborious to establish a wireless monitoring network covering an entire network. Challenges in the operation process include (a) difficulty covering the target network monitoring area due to the limited number, memory, and receiving signal range of monitoring sniffers; (b) the necessity of designing an algorithm to allocate the best channel for monitoring sniffers to improve the quality of monitoring (QoM) of the network; and (c) the inability to ensure that the monitoring network obtains all data packets transmitted in the network and the fact that data packets contain only limited information, which cannot facilitate complete performance evaluation and fault diagnosis of the network [11]. Regarding the second item, certain limitations of algorithms remain, so algorithm design is a primary research concern regarding current monitoring networks and also the topic of this paper in view of the multichannel characteristics of wireless networks.

With upgrades of technology and reductions in hardware costs, conventional single-radio monitoring sniffers cannot meet the demands of high QoM, and the deployment of a multiradio monitoring sniffer network is the inevitable trend of network monitoring. However, multiradio monitoring is currently based on single-radio sniffers as the algorithm design and does not employ the channel allocation algorithm of multiradio sniffers. In practical application, the geographical location of each radio in the multisniffer network is the same and thus they share the same adjacent sniffers and users. If the multiradio sniffers are simply mapped as multiple single-sniffer radios, the advantages of multiradio sniffer monitoring cannot be leveraged given the increased complexity of the algorithm [12,13,14,15,16].

To ensure high QoM of the monitoring network, dynamic adjustment to the working channel of the monitoring sniffers could be made according to the user’s channel allocation status. For a large-scale multiradio sniffer channel monitoring networks, channel allocation involves complex combinations and optimization. The exhaustive traversal method is very inefficient, so how to reasonably allocate channel resources for each monitoring sniffer to obtain the highest possible QoM is a key issue.

According to the inherent characteristics of wireless monitoring networks, we attempt to study the channel allocation problem of multiradio monitoring sniffers, mainly focusing on the following points.

First, the multiradio sniffer network model is extended based on the existing single-radio sniffer network model, and we propose a coding method and correction process for sniffer radios in the channel space.

Second, we propose a channel allocation algorithm of multiradio wireless monitoring sniffers based on discrete bacterial foraging optimization (DBFO). The position quantity of each dimension in the iterative process is discretized to binary 0 or 1 based on the traditional bacterial foraging optimization (BFO) algorithm with fine search characteristics and global optimization ability, thus achieving the encoding of the channel allocation of a wireless monitoring network. This channel allocation algorithm designed for multiradio monitoring sniffers can ensure the QoM of wireless monitoring networks reaches the optimal or approximately optimal stage.

Third, we carry out convergence analysis of the DBFO algorithm, and the convergence of the discrete BFO according to probability 1 is proved theoretically, and the probability of the position update for each dimension of the DBFO is analyzed. Finally, the effectiveness of the algorithm is tested by simulation and actual experiments.

In this paper, a form of swarm intelligence, DBFO, is proposed to achieve the optimal channel allocation for multiradio, multichannel wireless monitoring networks. We support this view based on theoretical arguments and experiments [17]. The structure of the remainder of this paper is as follows. We analyze related work from recent years. An explanation of the problem is provided. A channel allocation algorithm based on swarm intelligence (DBFO) is detailed. The algorithm is verified by simulations and practical network experiments. Finally, we summarize the paper and look forward to future work.

2. Related Works

The research on wireless monitoring networks has mainly addressed the development of monitoring equipment, system framework design, sniffer arrangement, sniffer channel allocation, data fusion and inference, etc. Among the many ways to improve network QoM, optimizing the channel allocation of monitoring sniffers to cover more network users is an optimization method that may require the fewest resources.

In 2009, Song et al. [17] proposed two problems of sniffer channel allocation: minimize the maximum number of channels for each sniffer and minimize the total number of channels for all sniffers. They solved both problems using IP-min–max, LP-min–max, and greedy-min–max methods. In the same year, Shin et al. [18] studied the sniffer channel allocation problem in a Mesh network to maximize the coverage of users. They described this as a maximum coverage problem based on group budget constraints and solved it using greedy and linear programming (LP) algorithms, which attained good performance.

In 2011, Arora et al. [19] studied optimal monitoring in a multichannel wireless network using the sequence learning method. Considering the activity of users, initially unknown to the sniffers, they proposed sequentially learning the users’ activity along with channel allocation decisions while maximizing the benefits of this allocation, which resulted in the fundamental tradeoff between exploration and exploitation [20]. As a distributed solution, Xia et al. [21] applied Gibbs sampler theory and proposed a novel channel allocation algorithm for sniffers to maximize the QoM of wireless monitoring networks. This method utilizes local information to allocate resources to the channel with low Gibbs energy, but in most cases, it fails to achieve the global optima.

In 2012, Shin et al. [22] proposed an online distributed algorithm, DA-OSCA, for sniffer channel allocation, which combines linear programming rounding technology with a near-end optimization algorithm to solve the channel allocation problem, and proved that the algorithm can achieve the maximum monitoring range of $1-1/e$.

In 2013, we proposed a Monte Carlo enhanced particle swarm optimization algorithm MC-PSO [23]. The algorithm designs a two-dimensional mapping particle coding and its movement scheme and uses the Monte Carlo method to correct it. In addition, in 2015, Xia et al. [24] proposed a Multiple Quantum Immune Clone Algorithm (MQICA) algorithm that takes advantage of the parallel nature of quantum computing (QC). Compared to the traditional Quantum Immune Clone Algorithm (QICA), MQICA has many characteristics inherited from immune and evolutionary algorithms. At the same time, alleles and Gaussian mutations were introduced in MQICA to further improve the performance of the algorithm.

In 2019, Shungo et al. [25] proposed a multiarmed bandit (MAB) algorithm, which is a promising method to solve the tradeoff between channel exploration and improving communication quality. With the chaotic oscillation time series generated by a semiconductor laser, the ultrafast solution of the MAB problem was proved.

In 2020, Prativa et al. [26] proposed a channel allocation algorithm based on game theory (GT). When designing the channel allocation game, the choice of utility function plays an important role in meeting the higher performance level of the channel allocation algorithm. The influence of various utility functions that may be used in the GT-based channel allocation algorithm was discussed.

Although many studies have validated the importance of sniffer channel allocation in wireless monitoring networks, little research has focused on the multiradio, multichannel allocation problem [25,27,28]. In the literature [18,21], a model for multiradio, multichannel allocation was discussed, although it was simplified to a single-radio, multichannel solution without considering the significant differences between multiradio sniffers and single-radio sniffers. Thus, the channel allocation problem of multiradio sniffers in a monitoring network has not been solved.

3. Problem Description

3.1. Network Model

Basic wireless monitoring networks include three entities: wireless monitoring sniffers, wireless users, and wireless channels. We assume that a wireless monitoring network is composed of $m$ monitoring sniffers, $n$ users, and $q$ channels, and each sniffer is equipped with $p$ radios, which can be different for heterogeneous sniffers. To simplify the description, this article assumes that all sniffers are isomorphic; that is, $p$ is the same. This article considers the sniffers’ channel allocation when multiple radios $\left(p\ge 2\right)$ are available in the sniffer and the number of radios is lower than the number of channels.

Assume a wireless monitoring network consisting of $m$ monitoring sniffers, $n$ users, $q$ optional channels, and $p$ radios per sniffer $\left(p\ge 2,p<q\right)$. $S=\left\{s_1,s_2,\dots ,s_m\right\}$ indicates the set of sniffers, $U=\left\{u_1,u_2,\dots ,u_n\right\}$ indicates the set of users, $C=\left\{c_1,c_2,\dots ,c_q\right\}$ indicates the set of channels, $R=\left\{r_1,r_2,\dots ,r_p\right\}$ indicates the set of radios, and $S^R=\left\{s_1^{r_1},s_1^{r_2},\dots ,s_1^{r_p},\dots ,s_m^{r_1},s_m^{r_2},\dots ,s_m^{r_p}\right\}$ indicates the set of radios of all sniffers, where $s_i^{r_w}$ is the $w$-th radio of sniffer $s_i$. When radio $s^r\in S^R$ selects the same channel as its neighboring user $u\in U$, $s^r$ can receive the transmitted data sent by $u$.

The mapping of multiradio sniffers and users in a wireless monitoring network is illustrated by an undirected bipartite graph $G^R=\left(S^R,U,E\right)$, shown in Figure 2. $E$ represents the all-edges set. The condition for sniffer radio $s^r$ and user $u$ to generate an edge $e=\left(s^r,u_i\right)\in E$ is that $s^r$ can capture the information transmitted by $u$, or that $u$ is in the monitoring range of $s^r$.

Because the amount of monitoring information obtained by a sniffer is the sum of the information captured by all $p$ radios by sniffing user channels, $G^R$ can be expressed as $G=\left(S,U,E\right)$, which is shown in Figure 3. This means the condition for $G$ to produce an edge is that $s$ can monitor the activity information of $u$; in other words, $u$ is monitored by $s$. $N\left(u\right)$ represents the set of adjacent sniffers of user $u$, and $N\left(s\right)$ represents the set of adjacent users of sniffer $s$.

We assume that the communication radius of a sniffer is twice that of its monitoring radius. When the monitoring ranges of two sniffers intersect, it means that these two sniffers are adjacent to each other, or that one sniffer is a neighbor sniffer of the other, and $V\left(s\right)$ represents the set of neighbor sniffers of a given sniffer. The neighbor users and neighbor sniffer information of sniffers in the network are a priori, and before the sniffer allocates channels, the user’s channel allocation can also be obtained through full radio frequency scanning.

3.2. Problem Formulation

The channel allocation problem of multiradio sniffers is essentially that each radio $s_i^{r_w}\left(i=1,\dots ,m;w=1,\dots ,p\right)$ of all sniffers in the monitoring network searches for channels in the channel set $C=\left\{c_1,c_2,\dots ,c_q\right\}$. The optimal allocation plan is to find channels for each sniffer radio through a search to achieve the best QoM. Channel allocation plan $a:S\to C$ can be represented by the search results of the sniffer radio in the channel space. The most important feature of multiradio sniffing is that different radios of the same sniffer are in the same position and their adjacent sniffer and neighbor user information is shared.

Assuming a monitoring network with $m$ nodes, each node is equipped with $p$ radio stations and $q$ channels. The searching space of the multiradio, multichannel allocation problem is shown in Figure 4, where $m\times p$ columns denote $m$ sniffers, each sniffer has $p$ radios, and $q$ rows denote the number of channels. Each radio $s_i^{r_w}$ searches in the channel space $\left\{c_1,c_2,\dots ,c_q\right\}$. The shaded area in Figure 4 indicates the channel allocation scheme for this wireless monitoring network.

When the sniffer radio searches for a certain channel, there are two possibilities for the corresponding relationship between the sniffer radio and channel. One is that the sniffer radio allocates the channel, and the other is that the sniffer radio does not allocate the channel. Given the binary problem of channel allocation, we use a binary mode to perform 0–1 coding on the search status of the channel space and obtain the channel allocation plan through the coding result. This binary coding method not only clearly indicates the allocation status of sniffer radios but also flexibly promotes the combination of algorithms and channel allocation modes.

Figure 5 is a coding table of sniffer radio and channel mapping that corresponds to the sniffer radio channel search result in Figure 4, where $i\left(i=1,\dots ,m\right)$ represents the sniffer number, $w\left(w=1,\dots ,p\right)$ represents the radio number of each sniffer, and $k\left(k=1,\dots ,q\right)$ represents the channel number. Taking the first sniffer as an example, its first, second, and p-th radios are assigned to channel 2, $q-1$, and 1, respectively.

We know from the coding table that each sniffer radio has $q$ coding possibilities. If sniffers allow different radios to reuse channels, there are $q^{m\times p}$ types of optional channel allocation schemes for the entire monitoring network. If the problem of duplication of collected information is considered, after the constraint of sniffers prohibiting different radios from multiplexing channels, there are ${\left(C_q^p\right)}^m$ kinds of optional channel allocation schemes for the monitoring network. These two situations are discussed separately.

• Case 1: Sniffers Allow Different Radios to Reuse Channels

In this allocation scheme, the sniffer code table has multiple 1s in the same row. For example, in Figure 5, for sniffer $m$, two 1s are in row $q$, which means the sniffer’s radios 2 and $p$ are both assigned to channel $q$. If we accumulate the elements in the same row for each sniffer, the 2D binary coding will be altered to 2D accumulation coding, as shown in Figure 6. Each column represents the channel usage situation for a sniffer, represented by $A\left(s_i\right)=({\sigma }_{i,1},{\sigma }_{i,2},\dots ,{\sigma }_{i,q}$), where ${\sigma }_{i,k}\left(0\le {\sigma }_{i,k}\le p\right)$ indicates the number of times channel $k$ has been assigned to sniffer $i$. For example, in Figure 6, the m-th column $A\left(S_m\right)=\left(0,1,\dots ,0,2\right)$, where ${\sigma }_{m,2}=1,$ means channel 2 has been assigned to a radio of sniffer $m,$ and ${\sigma }_{m,q}=2$ denotes that channel $q$ has been assigned to two radios of sniffer $m$.

Although this can cause duplicate reception of user data by monitoring nodes, which may result in wasted resources, this improves the reliability of the data. It can provide multiple guarantees for data transmission, prevent data loss, improve the robustness of data transmission, and adapt to network applications that require high accuracy in data transmission.

Definition 1. 

Quality of monitoring (QoM) of a sniffer in Case 1. When the wireless monitoring network adopts channel allocation scheme $\mathit{a}$, the monitoring quality of sniffer $s$ can be defined as follows:

$$Q_s\left(\mathit{a}\right)={\displaystyle \sum }_{u\in N\left(s\right)}\frac{p_{u}F\left[A\left(s\right)A{\left(u\right)}^T\right]}{F\left[A\left(s\right)A{\left(u\right)}^T\right]+{{\displaystyle \sum }}_{s^{\prime }\in V\left(s\right)}F\left[A\left(s^{\prime }\right)A{\left(u\right)}^T\right]}$$

(1)

$$F\left(N\right)={\displaystyle \sum }_{i=1}^N\frac{1}{k^{i-1}}$$

(2)

where $u$ represents the adjacent users of $s$; $N\left(s\right)$ represents the adjacent user set of $s$; $p_u$ represents the transmission probability of $u$; $s^{\prime }$ represents the adjacent sniffer of $s$; $V\left(s\right)$ represents the set of adjacent sniffers of $s$; and $A\left(s\right)$ represents the channel coding of $s$. The number of radios that $s$ and $u$ work on the same channel can be calculated by $A\left(s\right)A{\left(u\right)}^T$, and the same form can be deduced by analogy. Because multiple radios multiplexing a channel will result in repeated capture of user information, this type of situation can be processed as in Equation (2), where $k\left(k\ge 1\right)$ is a constant and the value of $k$ can be set according to the requirements of network monitoring reliability. As the number increases, the amount of information will increase exponentially. For example, a sniffer $s$ has three radios working on the same channel $c$ at the same time to monitor the activity information of user $u$, then $A\left(s\right)A{\left(u\right)}^T$ calculates that $N=3$, and the amount of information captured by sniffer $s$ for $u$ is $F\left(3\right)=1+1/k+1/k^2$. It is worth noting that when $N=0$, $F\left(N\right)=0$, meaning that the amount of information captured for $u$ is 0.

• Case 2: Sniffers Prohibit Different Radios from Multiplexing Channels

Under this constraint condition, different radios of a sniffer must operate on different channels to avoid collecting repeat information for a sniffer.

In case 2, the sniffer encoding table is added with the constraint that multiple 1s are not allowed for the same sniffer. If there is a case where the constraint is not satisfied, the assignment scheme is modified. For example, in the coding table in Figure 5, there is a situation where the constraint is not satisfied for radio $q$ of sniffer $m$. Therefore, this coding should be modified so that the station appearing to multiplex the channel is moved to another channel, as shown in Figure 7. The revision scheme is repeated until the constraint is satisfied. The coding table of Figure 5 is modified into the coding table of Figure 7, where radios $2$ and $p$ of sniffer m are assigned to channels $q$ and $q-1$, respectively.

Each sniffer in the 2D binary encoding is accumulated, and the accumulation result is binarized so that the non-zero element is 1. The final 2D accumulation encoding is obtained as shown in Figure 8.

Each column in Figure 8 represents the coding of the corresponding channel of each sniffer, which is still represented by $A\left(s_i\right)=\left({\sigma }_{i,1},{\sigma }_{i,2},\dots ,{\sigma }_{i,q}\right)$, but at this time there are only two values for ${\sigma }_{i,k}$ to indicate whether channel $k$ is allocated to sniffer $i$; if it is not allocated, it is 0, otherwise it is 1. Taking sniffer 1 as an example, the channel allocation result can be obtained from the code $A\left(s_1\right)=\left(1,1,\dots ,1,0\right)$, which means channel 1, 2, … $q-1$ have been allocated to the radio of sniffer 1.

Definition 2. 

QoM of a sniffer in Case 2. When the wireless monitoring network adopts channel allocation scheme $\mathit{a}$, the monitoring quality of sniffer $s$ can be defined as follows:

$$Q_s\left(\mathit{a}\right)={\displaystyle \sum }_{u\in N\left(s\right)}\frac{p_{u}F\left[A\left(s\right)A{\left(u\right)}^T\right]}{1+{{\displaystyle \sum }}_{s^{\prime }\in V\left(s\right)}A\left(s^{\prime }\right)A{\left(u\right)}^T}$$

(3)

The meanings of parameters such as $u,N\left(s\right),V\left(s\right)$, etc. are same as those in Case 1. Because multiple channels are forbidden, different radios of the sniffer work on different channels and the active users that can be monitored are different, so there is no need to consider processing repeated information captured by the sniffer radio.

The formula expression of sniffer QoM shows that whether it is QoM in Case 1 or Case 2, when more neighbor users of $s$ and radio $s$ work on the same channel, the transmission probability increases, and when the number of neighbor sniffer radios that use the channel allocated by $s$ to monitor the same user is lower, the QoM of $s$ increases.

Definition 3. 

QoM of the network. When the wireless monitoring network adopts channel allocation scheme $\mathit{a}$, the QoM of the wireless monitoring network can be defined as follows:

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

(4)

where $S$ represents the collection of all sniffers in the monitoring network. Equation (4) shows that the better the QoM, the more active users that can be monitored by the wireless network, the greater the amount of user information captured, and the higher the quality of service provided by the wireless monitoring network. The research goal of this paper is to find a channel allocation scheme to maximize the network QoM under the condition of limited sniffers and users. The optimal channel allocation scheme is as follows:

$${\mathit{a}}^{*}=\mathrm{argmaxQ}\left(\mathit{a}\right)$$

(5)

This channel coding and QoM function expression show that in the channel allocation process of multiradio sniffers, each sniffer radio does not simply optimize its channel allocation to find the best objective function but also considers the channel conditions of adjacent users and sniffers. In short, the channel allocation of each sniffer is not independent. For the entire monitoring network, QoM is a multidimensional combination optimization problem.

4. Design of Hardware and Software

In this paper, we introduce a form of swarm intelligence [29,30,31,32,33,34,35,36,37,38,39], the DBFO algorithm, to solve the multiradio, multichannel allocation problem in wireless monitoring networks [40].

To increase the readability of the algorithm, the parameters and their meanings in the algorithm are presented in

Table 1. Algorithm parameters and their meanings.

Parameters	Meaning
$S$	a population of bacteria
$N_c$$,N_{re}$$,N_{ed}$	the rated number of chemotaxis, replication, and migration operations
$N_s$	the maximum amount of swimming in one direction
$P_{ed}$	the probability of migration
${\theta }^i\left(j,k,l\right)$	$\mathrm{the}\mathrm{position}\mathrm{of}\mathrm{bacterium}i$$\mathrm{when}\mathrm{it}\mathrm{completes}\mathrm{the}j\mathrm{th}$$\mathrm{chemotaxis},k\mathrm{th}$$\mathrm{replication},\mathrm{and}l\mathrm{th}$ migration
${\theta }_d^i$	$\mathrm{the}\mathrm{position}\mathrm{component}\mathrm{of}\mathrm{bacterium}i$$\mathrm{in}\mathrm{the}d\mathrm{th}$ dimension
${\Delta }_d^i$	$\mathrm{the}{d}^{\mathrm{th}}$$\mathrm{dimensional}\mathrm{randomly}\mathrm{selected}\mathrm{chemotactic}\mathrm{direction}\mathrm{of}\mathrm{bacterium}i$
${\varphi }_d^i$	the unit directional vector of each dimension
${\lambda }^i$	the length of a swimming step
${\rho }_d^i\left(j,k,l\right)$	$\mathrm{stochastic}\mathrm{serials}\mathrm{of}\mathrm{bacterium}i$$\mathrm{when}\mathrm{it}\mathrm{completed}j\mathrm{th}$$\mathrm{chemotaxis},k\mathrm{th}$$\mathrm{replication},\mathrm{and}l\mathrm{th}$ migration
$p_{0\leftrightarrow 1}$	$\mathrm{bacterium}i$$\mathrm{performs}\mathrm{chemotaxis}j$ operation, the probability of a certain dimension position changing
$\Theta$	the bacterial population
$\Omega$	the state space of the bacterial position
$S$	the population size
$u_x$	$\mathrm{the}xth$$\mathrm{state}\mathrm{in}{\Omega }^S$
$\delta$	$\mathrm{is}\mathrm{the}\mathrm{minimum}\mathrm{algebra}\mathrm{generated}\mathrm{by}\mathrm{all}\mathrm{cylinder}\mathrm{sets}\mathrm{of}\Omega$
$\mathrm{P}$	$\mathrm{a}\mathrm{real}\mathrm{value}\mathrm{measurement}\mathrm{function}\mathrm{defined}\mathrm{as}\left(\Omega ,\delta \right)$
$p_{xy}\left(n\right)$	$\mathrm{the}\mathrm{transition}\mathrm{probability}\mathrm{of}\left\{\Theta \left(n\right)\right\}$
$B^{*}$	the optimal solution set
$Q\left(\mathit{a}\right)$	the fitness function
$\mathrm{A}$	coding table coding table

.

4.1. Basic BFO

The BFO algorithm is a metaheuristic algorithm proposed by Passino when he studied the behavior of bacteria [27]. As a population intelligence optimization method, it is suitable for solving NP-hard optimization problems. As a bacterial population-based search technique, its solution process is independent of initial values, reaches global optimum through convergence, and does not require gradient information of the optimization object during the optimization process. Because of its advantages of fast convergence and simple operation, it is widely used in many fields.

The BFO algorithm simulates the process of the bacterial population including three steps of chemotaxis, reproduction, and elimination dispersal to update the colony position to obtain the optimal solution.

Suppose there is a population of $S$ bacteria, with $N_c$, $N_{re}$, $N_{ed}$ denoting the respective rated number of chemotaxis, replication, and migration operations. $N_s$ represents the maximum amount of swimming in one direction, and $P_{ed}$ indicates the probability of migration.

Chemotaxis Operation. Bacterial chemotaxis features two basic movements: flipping and swimming. The algorithm defines flipping as the unit step of a bacterium’s movement in any direction. If the fitness function value improves after flipping, then movement is continued in the same direction until the fitness function value no longer improves or reaches the number of mobile steps preset by $N_s$, which is called the swimming process [27].

Replication Operation. Bacterial populations start the replication operation after $N_c$ chemotaxis operations. All fitness function values of bacteria in the process of chemotaxis accumulate to obtain a bacterial healthy function value, and these function values are sorted to eliminate the smaller half and reproduce the larger half, which can maintain the same population in the process of elimination and replication.

Migration Operation. After $N_{re}$ replication operations, bacterial swarming behavior may occur, which undermines the population diversity. To improve the global optimization capability, the BFO algorithm introduces a mutation operation. A probability $P_{ed}$ is set, which performs the migration operation for the probability of bacterial compliance, moving to an arbitrary position in the search space randomly. The migration operation is conducive to escaping the local optima to find the global optima.

4.2. Process of DBFO

Assuming that there are $S$ bacteria in the population, ${\theta }^i\left(j,k,l\right)$ represents the position of bacterium $i$ when it completes the j-th chemotaxis, k-th replication, and l-th migration operations, abbreviated as ${\theta }^i\in R^D$, where $D$ represents the problem dimension. $D=m\times p\times q$, in a monitoring network with $m$ monitoring sniffers, $p$ radios, and $q$ channels. ${\theta }^i=\left[{\theta }_1^i,{\theta }_2^i,\dots ,{\theta }_D^i\right]$, where ${\theta }_d^i$ represents the position component of bacterium $i$ in the d-th dimension. ${\Delta }^i=\left[{\Delta }_1^i,{\Delta }_2^i,\dots ,{\Delta }_D^i\right]$ represents the directional vector of bacterium $i$ when performing the chemotaxis operation, where ${\Delta }_d^i\in \left[-1,1\right]$ represents the d-th dimensional randomly selected chemotactic direction of bacterium $i$. ${\Phi }^i={\Delta }^i/\sqrt{{\left({\Delta }^i\right)}^T\times {\Delta }^i}$ represents the unit vector of ${\Delta }^i$. ${\Phi }^i=\left[{\varphi }_1^i,{\varphi }_2^i,\dots ,{\varphi }_D^i\right]$, where ${\varphi }_d^i$ represents the unit directional vector of each dimension.

The sigmoid function is used in the BFO algorithm, and its dimensions are discretized into binary values as follows.

$$\Delta {\theta }_d^i\left(j+1,k,l\right)=\Delta {\theta }_d^i\left(j,k,l\right)+{\lambda }^i{\varphi }_d^i$$

(6)

$${\theta }_d^i\left(j+1,k,l\right)=\left\{\begin{array}{l}0, otherwise\hfill \\ 1,{\rho }_d^i\left(j+1,k,l\right)<sigmoid\left(\Delta {\theta }_d^i\left(j+1,k,l\right)\right)\hfill \end{array}\right.$$

(7)

where $\Delta {\theta }_d^i\left(j,k,l\right)$ indicates in the d-th dimension the moving distance of bacterium $i$ when it completed j-th chemotaxis, k-th replication, and l-th migration operations; ${\lambda }^i$ indicates the length of a swimming step; ${\theta }_d^i$ indicates the unit directional vector in the d-th dimension; ${\theta }_d^i\left(j+1,k,l\right)$ indicates the d-th dimensional position of bacterium $i$ when it completes $\left(j+1\right)$-th chemotaxis, k-th replication, and l-th migration operations; ${\rho }_d^i\left(j+1,k,l\right)$ indicates stochastic serials of bacterium $i$ when it completes $\left(j+1\right)$-th chemotaxis, k-th replication, and l-th migration operations. The sequence follows a uniform distribution, ${\rho }^i\sim U\left(0,1\right)$, $sigmoid\left(x\right)=1/\left(1+exp\left(-x\right)\right)$.

After each flip operation in the convergence process, the results are verified by the QoM of the fitness function. If $Q\left({\theta }^i\left(j+1,k,l\right)\right)>Q\left({\theta }^i\left(j,k,l\right)\right)$, swimming in the current direction continues until the fitness function value QoM does not increase or the maximum number of swimming $N_s$ is reached. The count of chemotaxis updates only when flipping and remains the same when swimming. To facilitate understanding, the swimming count $w$ is introduced.

$$\Delta {\theta }_d^i\left(j,w+1,k,l\right)=\Delta {\theta }_d^i\left(j,w,k,l\right)+{\lambda }^i{\varphi }_d^i$$

(8)

$${\theta }_d^i\left(j,w+1,k,l\right)=\left\{\begin{array}{l}1,{\rho }_d^i\left(j,w+1,k,l\right)<\hfill \\ sigmoid\left(\Delta {\theta }_d^i\left(j,w+1,k,l\right)\right)\hfill \\ 0, otherwise\hfill \end{array}\right.$$

(9)

Except for swimming count $w$, other parameters’ interpretations are the same as those in Equations (6) and (7). In the swimming operation, each dimensional chemotactic direction ${\varphi }_d^i$ does not change. The BFO algorithm can be discrete in binary space based on Equations (7) and (9). Chemotaxis is the most basic and important operation. Therefore, discretization is mainly carried out for chemotaxis.

By converting continuous quantities into discrete binary numerical space search problems through the BFO algorithm, we can use the discrete BFO algorithm to solve the wireless monitoring network channel allocation problem.

4.2.1. Analysis of Positional Changing Probability

The literature [12] has proposed an idea of positional changing probability for discrete particle swarm optimization. Here, we also analyze each dimension positional changing probability for DBFO. The process of bacteria updating their position is somewhat a probability change; for a single dimension, the probability of swimming distance variation is 1.

Suppose bacterium $i$ doing chemotaxis $j$ operation; the probability of a certain dimension assuming 1 is $sigmoid\left(\Delta {\theta }_d^i\left(j,k,l\right)\right)$, then the probability assuming 0 is $1-sigmoid\left(\Delta {\theta }_d^i\left(j,k,l\right)\right)$. If the dimension is 0 before bacterium $i$ undergoes chemotaxis $j-1$ operation, then the changing probability of this dimension is sigmoid $\left(\Delta {\theta }_d^i\left(j,k,l\right)\right)$; if the dimension is 1 before bacterium $i$ undergoes chemotaxis $j-1$ operation, then the changing probability of this dimension is $1-sigmoid\left(\Delta {\theta }_d^i\left(j,k,l\right)\right)$. After bacterium $i$ performs chemotaxis $j-1$ operation, the probability of a certain dimension assuming 0 is $1-sigmoid\left(\Delta {\theta }_d^i\left(j-1,k,l\right)\right)$, and the probability it takes 1 is $sigmoid\left(\Delta {\theta }_d^i\left(j-1,k,l\right)\right)$. Therefore, when bacterium $i$ performs chemotaxis $j$ operation, the probability of a certain dimension position changing is

$$\begin{array}{l}{p}_{0\leftrightarrow 1}= \hfill \\ sigmoid\left(\Delta {\theta }_d^i\left(j-1,k,l\right)\right)\times \left[1-sigmoid\left(\Delta {\theta }_d^i\left(j,k,l\right)\right)\right]\hfill \\ +sigmoid\left(\Delta {\theta }_d^i\left(j,k,l\right)\right)\times \left[1-sigmoid\left(\Delta {\theta }_d^i\left(j-1,k,l\right)\right)\right]\hfill \end{array}$$

(10)

This formula shows that the probability of a position change in a dimension can be obtained from the distance of the position component moving in chemotaxis $j-1$ and $j$ operations. Integrating Equation (10) together with the sigmoid function creates the following formula:

$$\begin{gathered} P_{0\leftrightarrow 1}=\frac{1}{1+\exp \left(-V{\theta }_d^i\left(j-1,k,l\right)\right)}\times \left[1-\frac{1}{1+\exp \left(-V{\theta }_d^i\left(j,k,l\right)\right)}\right] \\ +\frac{1}{1+\exp \left(-V{\theta }_d^i\left(j,k,l\right)\right)}\times \left[1-\frac{1}{1+\exp \left(-V{\theta }_d^i\left(j-1,k,l\right)\right)}\right] \end{gathered}$$

(11)

Formula (11) is the relationship between the distance moved by a 1D position component and the change of position, as shown in Figure 9. It can be seen from the figure that the probability of $0\leftrightarrow 1$ change is related to the distance of two chemotactic operations, but the greatest probability change is 0 in both chemotactic operations, and its maximum value is 0.5.

In a particular case, when the two chemotactic operations are at the same distance, that is $\Delta {\theta }_d^i\left(j,k,l\right)=\Delta {\theta }_d^i\left(j-1,k,l\right)$, the variable is $\Delta {\theta }_d^i\left(j,k,l\right)$ only. The functional relationship is shown in Figure 10, and the graph is similar to a parabola. The vertices are in the place of $\Delta {\theta }_d^i\left(j,k,l\right)=0$ and the maximum value is 0.5, which means the bacteria do not move. The probability of bacteria changing position from 0 and 1 is

$$P_{0\leftrightarrow 1}=2\times \frac{1}{1+\exp \left(-V{\theta }_d^i\left(j,k,l\right)\right)}\times \left[1-\frac{1}{1+\exp \left(-V{\theta }_d^i\left(j,k,l\right)\right)}\right]$$

(12)

We can see from Figure 10 that the greater the distance the bacterium is from the original position, the smaller the probability of bacterial positions changing from 0 and 1. The final global convergence is when positions 0 and 1 are unchanged; the position turbulence effect will not occur.

4.2.2. Discrete BFO Convergence Analysis

Theorem 1. 

The bacterial D-dimensional position solution space ${\theta }^i=\left({\theta }_1^i,{\theta }_2^i,\dots ,{\theta }_D^i\right)$ produced by the DBFO algorithm is a discrete random process and constitutes a finite homogeneous Markov chain.

Proof. 

Suppose that $\Omega$ is the state space of the bacterial position, $\Theta =\left\{{\theta }^1,{\theta }^2,\dots ,{\theta }^S\right\}$ represents the bacterial population, $S$ represents the population size, and ${\theta }^i$ represents the position of bacterium $i$ in the bacterial population, ${\theta }^i\in \Omega$, $\Theta \in {\Omega }^s$. The bacterial positions are discrete in the process of chemotaxis, the discrete probability 0 or 1 of the convergence process is obtained by the change of distance in each dimension of the bacteria. □

$${\theta }_d^i\left(j+1,k,l\right)=\left\{\begin{array}{l}1,{\rho }_d^i\left(j+1,k,l\right)<sigmoid\left(\Delta {\theta }_d^i\left(j+1,k,l\right)\right)\hfill \\ 0, otherwise\hfill \end{array}\right.$$

(13)

$\because$ Bacterial position is a D-dimensional state space.

$\therefore \Omega \in {\Re }^D$.

$\therefore$ The position state space dimension of the bacterial population is $\left|{\Omega }^S\right|=D^S$, and the number of binary states is ${\left(2^D\right)}^S=2^{kmS}$.

$\therefore$ The state space of the bacterial population is finite. In [41], suppose $\delta$ is the minimum algebra generated by all cylinder sets of $\Omega$, then P is a real value measurement function defined as $\left(\Omega ,\delta \right)$. Therefore, the probability space of the bacterial position solution can be indicated as $\left(\Omega ,\delta ,P\right)$, and the population probability space of DBFO can be indicated as $\left({\Omega }^S,\delta ,P\right)$.

$\therefore {\theta }^i$ is the random solution defined by $\left(\Omega ,\delta ,P\right)$, and its value is discrete in iterations. Obviously, $\Theta$ constitutes the discrete random process in $\left({\Omega }^S,\delta ,P\right)$.

$\because$ Every swimming operation advances in the same direction based on current position in chemotaxis operation; that is, ${\theta }^i\left(j,w+1,k,l\right)={\theta }^i\left(j,w,k,l\right)+{\lambda }^i{\Phi }^i$. Flipping is associated with the current state; that is, ${\theta }^i\left(j+1,k,l\right)={\theta }^i\left(j,k,l\right)+{\lambda }^i{\Phi }^i$. Replication operations reproduce or eliminate the current state of bacteria based on the overlapping fitness value after the chemotaxis operation. The migration operation determines whether the current bacteria change their state based on the threshold.

$\therefore$ In summary, bacteria change their position through three steps: convergence, replication, and migration, which make the next state of BFO relate only to the current state, which is consistent with the definition of a Markov chain. Meanwhile, the transfer probability is only related to the position deviation of the current generation, so the Markov chain can be regarded as flanking. Therefore, it can be concluded that $\Theta$ constitutes a finite chi-square Markov chain in the state space ${\Omega }^S$. It can be proved by Theorem 1.

The goal of replication and migration operations is to speed up convergence and help jump out of the local optimum, promoting the optimization process. The core of the DBFO algorithm is the chemotactic operation. By proving that in the swimming condition, the chemotaxis process is convergent in probability 1, the BFO algorithm is proven to be convergent in probability 1, which is mentioned in the literature [40]. Suppose $\Theta \left(n\right)=\left\{{\theta }^1,{\theta }^2,L,{\theta }^S\right\}$, where $n$ indicates the evolution iteration, $\Theta \left(n\right)$ indicates population position in n iteration, and ${\theta }^i$ indicates individual position of bacterium $i$. Suppose $Q\left(\mathit{a}\right)$ is the QoM value in channel allocation scheme $\mathit{a}$; $Q\left(\mathit{a}\right)$ is the fitness function of $B^{*}=\left\{\theta \right|\max \left(Q\left(\theta \right)\right)=Q^{*}\}\left(\theta \in \Theta \left(n\right)\right)$, called the optimal solution set, where $Q^{*}$ is the global optimal solution in the best channel distribution.

Definition 4. 

Suppose ${\mathit{a}}^{*}={\theta }^{*}$ is the best channel allocation scheme, ${\theta }^{*}=argmaxQ\left(\mathit{a}\right)=arg{Q}^{*}$, if and only if $P\left\{\underset{n\to \infty }{\mathit{lim}}{Q}_n\left(\theta \right)=Q^{*}\right\}=1$, then the algorithm converges to $Q^{*}$ under probability 1. That is, after sufficient iterations, the probability of the optimal solution in a population approaches 1.

Theorem 2. 

The DBFO algorithm converges in probability 1.

Proof. 

Suppose the size of population $S$ is a point in the state space ${\Omega }^S$ and $u_x\in {\Omega }^S$ indicates $u_x$ is the x-th state in ${\Omega }^S$. Accordingly, $u_x=\left\{{\theta }^1,{\theta }^2,\dots ,{\theta }^S\right\}$ in DBFO and ${\Theta }_x^n$ indicates the population $\Theta \left(n\right)$ is in the state of $u_x$ after the x-th chemotactic operation. We know that from Theorem 1 that $\Theta$ is a discrete random process; thus, the transition probability of $\left\{\Theta \left(n\right)\right\}$ is $p_{xy}\left(n\right)$, and then $p_{xy}\left(n\right)=p\left\{{\Theta }_{n+1}^y/{\Theta }_n^x\right\}$. Before every replication operation, the best individual reservation operation is performed; therefore, for any $n\ge 0$, there is $f\left(\Theta \left(n+1\right)\right)\ge f\left(\Theta \left(n\right)\right)$. Supposing $X=\left\{x\right|u_x\cap B^{*}\ne \varnothing \}$, there are two states as follows. □

When $x\in X,y\notin X,$

$\because$ The optimal solution occurs after $n$ iterations and will not degenerate.

$\therefore p_{xy}\left(n\right)=0$

When $x\notin X,y\in X$, ∵ $f\left({\Theta }_{n+1}^y\right)\ge f\left({\Theta }_n^x\right)$,

$\therefore p_{xy}\left(n\right)>0$.

Supposing $p_x\left(n\right)$ is the probability of population $\Theta \left(n\right)$ in state $u_x$, then $p\left(n\right)={\displaystyle \sum }_{x\notin X}{p}_x\left(n\right)$, and we know from the characteristics of a Markov chain that

$$p\left(n+1\right)={\displaystyle \sum }_{u_x\in {\Omega }^S}{\displaystyle \sum }_{y\notin X}{p}_x\left(n\right)p_{xy}\left(n\right)=={\displaystyle \sum }_{x\in X}{\displaystyle \sum }_{y\notin X}{p}_x\left(n\right)p_{xy}\left(n\right)+{\displaystyle \sum }_{x\notin X}{\displaystyle \sum }_{y\notin X}{p}_x\left(n\right)p_{xy}\left(n\right)$$

Because

$${\displaystyle \sum }_{x\notin X}{\displaystyle \sum }_{y\in X}{p}_x\left(n\right)p_{xy}\left(n\right)+{\displaystyle \sum }_{x\notin X}{\displaystyle \sum }_{y\notin X}{p}_x\left(n\right)p_{xy}\left(n\right)={\displaystyle \sum }_{x\notin X}{p}_x\left(n\right)=p\left(n\right),$$

Then, the upper formula can be written as

$$p\left(n+1\right)={{\displaystyle \sum }}_{x\in X}{{\displaystyle \sum }}_{y\notin X}{p}_x\left(n\right)p_{xy}\left(n\right)+[p\left(n\right)-{{\displaystyle \sum }}_{x\notin X}{{\displaystyle \sum }}_{y\in X}$$

$$p_x\left(n\right)p_{xy}\left(n\right)]<{\displaystyle \sum }_{x\in X}{\displaystyle \sum }_{y\notin X}{p}_x\left(n\right)p_{xy}\left(n\right)+p\left(n\right)$$

$\because$ When $x\in X$,$y\notin X$,$p_{xy}\left(n\right)=0$,

then $0\le p\left(n+1\right)<p\left(n\right)$. So,$\underset{n\to \infty }{lim}p\left(n\right)=0$.

$\because P\left\{\underset{n\to \infty }{lim}{Q}_n\left(\theta \right)=Q^{*}\right\}=1-\underset{n\to \infty }{lim}{\displaystyle \sum }_{x\notin X}{p}_x\left(n\right)$= $1-\underset{n\to \infty }{lim}p\left(n\right)$

$\therefore P\left\{\underset{n\to \infty }{lim}{Q}_n\left(\theta \right)=Q^{*}\right\}$=$1$.

According to Definition 4, the algorithm converges to the optimal solution in probability 1. Theorem 2 is certified.

4.3. Update and Revision of the Channel Coding Table

Assuming there are $m$ sniffers, each sniffer has $p$ radios, and there are $q$ channels, a sniffer channel coding table $A={\left({\sigma }_{i,w,k}\right)}_{m\times p\times q}$ is formed by $m\times p\times q$ binary numbers, where ${\sigma }_{i,w,k}$ represents the coding of radio $w$ in sniffer $i$ on channel $k$, ${\displaystyle \sum }_{w=1}^p{\sigma }_{i,w,k}={\sigma }_{i,k}$. The coding table represents the current position of the bacteria, and the DBFO algorithm is used to update and revise the channel coding table.

Initialize all elements of the code table to 0, indicating that the current position of each bacterium is at point 0.

The position of the bacterium is updated through chemotaxis, replication, and migration operations of the DBFO algorithm. In the process of chemotaxis, bacterial flipping positions update mainly by changing moving distance of each dimension in Equation (6) and the dimensional position of discretization in Equation (7); bacterial swimming positions update mainly through Equations (8) and (9). In the process of replication, take the bacterial positions of the entire process of chemokines, for instance, in formula QoM, sort the front half position results after the cumulative sum of all bacteria, copy them, and remove half of the remaining bacteria. In the process of migration, each bacterium updates its position according to a certain probability occurrence.

The channel coding table is updated by transforming the bacterial position, and the coding table as a channel allocation scheme needs to comply with the following constraints: (1) since the node radio can only be assigned one channel at the same time, each node radio in the coding table can only have one element coded as “1” in the corresponding channel; (2) in order not to cause radio waste, each radio must have a corresponding channel coded as “1”; (3) for the case of “prohibiting multiplexing of channels by different radio stations”, the maximum column vector in the cumulative coding table is coded as “1”.

In the initialization stage, all elements in the coding table are 0. At this time, the coding table cannot be brought into the QoM formula for the solution. It needs to flip its direction to generate a new position based on the initial one and then be used to solve the solution. Due to the random generation of different position components, it is difficult to guarantee fully meeting the channel allocation principle after discretization, so revision of the binary coding table needs to occur after each bacterial position update. The revision steps are as follows:

• Step 1: Select a column of elements in the coding table for which no operation is performed $\left({\sigma }_{i,w,1},{\sigma }_{i,w,2},\dots ,{\sigma }_{i,w,q}\right)$; if only one of the selected columns is coded as 1,—that is, ${\displaystyle \sum }_{k=1}^q{\sigma }_{i,w,k}=1$—go to Step 4.
• Step 2: If there are multiple elements encoded as 1 in the selected column—that is, ${\displaystyle \sum }_{k=1}^q{\sigma }_{i,w,k}>1$—for each $k\left({\sigma }_{i,w,k}=1\right)$, count the number of users monitored when the radio selects the channel: $Q_{i,w}\left(k\right)={\displaystyle \sum }_{u\in N\left(s_i\right)}{p}_u\cdot A_{k}A{\left(u\right)}^T$, where $A_k$ is a $q$ dimensional vector; except for the k-th dimension being 1, the rest are 0. Sort the calculated values, allocate the channel corresponding to $max\left(Q_{i,w}\left(k\right)\right)$ and set its code to 1, and clear the rest to 0.
• Step 3: If all column elements are encoded as 0—that is, ${\displaystyle \sum }_{k=1}^q{\sigma }_{i,w,k}=0$—for each $k\left(k=1,\dots ,q\right)$, calculate the number of active users that the sniffer radio is monitoring on the channel, allocate the channel corresponding $max\left(Q_{i,w}\left(k\right)\right)$, and code it as 1.
• Step 4: If the coding table does not satisfy constraint (1)—that is, ${\displaystyle \sum }_{k=1}^q{\sigma }_{i,w,k}=1\left(i=1,\dots ,m;w=1,\dots ,p\right)$—return to Step 1, otherwise, continue to Step 5.
• Step 5: Enter the code collection state. In Case 1, generate the cumulative coding table such that ${\sigma }_{i,k}={\displaystyle \sum }_{w=1}^p{\sigma }_{i,w,k}$; in Case 2, determine whether the column elements exceed 1 after generating the cumulative coding table; if so, go to step 6.
• Step 6: If there is an element greater than 1 in the accumulation table—that is, ${\sigma }_{i,k}>1$—first, set the element to 1. Then count the channels coded as 0 in this sniffer, assign the radios that repeatedly occupy the channel to the empty channel, and sort the calculated results to balance the coding of 1.

Figure 11 shows a revised channel allocation process after updating encodings for four sniffers, two radios per sniffer, and 10 channels in a wireless monitoring network. Figure 11a indicates the coding after updating bacterial positions at a certain time, where the column element “(1)” violates the constraint of channel allocation (1); column element “(2)” violates the constraint of channel allocation (2); column element “(3)” violates the constraint of channel allocation (3); and other column elements satisfy the constraints of channel allocation. Figure 11b corresponds to the cumulative coding table after Step 2 correction. Figure 11c corresponds to the coding table after the Step 3 correction. Figure 11d represents the cumulative coding table in Case 1. Figure 11e represents the cumulative coding table in Case 2. Figure 11f represents the coding table after the reassignment of radios for the multiplexed channel in Figure 11e.

4.4. Pseudo Code of the DBFO Channel Algorithm

Based on the prior description, we demonstrate the complete DBFO-based channel allocation algorithm for wireless monitoring networks in Algorithm 1. The pseudo-code of the algorithm shows that its complexity is only related to the rated number of convergence, replication, and migration operations: $N_c$, $N_{re}$, and $N_{ed}$.

Algorithm 1. DBFO-Based Channel Allocation Algorithm.
	Input: /*The parameters of the wireless monitoring network and DBFO algorithm */
		Sniffer set $S=\left\{s_1,s_2,\dots ,s_m\right\}$; Radio set $R=\left\{r_1,r_2,\dots ,r_p\right\}$; User set $U=\left\{u_1,u_2,\dots ,u_n\right\}$;
		$A\left(u_j\right)$ and $P_{u_j}\left(j=1,2,\dots ,n\right)$; Channel set $C=\left\{c_1,c_2,\dots ,c_q\right\}$.
		Population size $N$. Chemotaxis number $N_c$; Replication number $N_{re}$;
		Migration number $N_{ed}$; Migration probability $P_{ed}$.
	Output: Channel allocation vector $\mathit{a}$ and QoM.
1.	Set ${\theta }^i=0\left({\theta }^i\in \theta ,1\le i\le N\right)$;/*Generate the initial bacterial encoding*/
2.	for $l\leftarrow 1$ to $N_{ed}$ do
3.		for $k\leftarrow 1$ to $N_{re}$ do
4.			for $j\leftarrow 1$ to $N_c$ do
5.					$\Delta {\theta }^i\left(j,k,l\right)\leftarrow$ The displacement of each bacterium;
6.					${\theta }^i\left(j,k,l\right)\leftarrow$ based on sigmoid function discrete $\Delta {\theta }^i\left(j,k,l\right)$ to 0–1;
7.					Mapping ${\theta }^i\left(j,k,l\right)$ to the bacterial encoding table;
8.					Revise the updated bacteria to satisfy singularity;
9.					$Q_i\leftarrow$function($\Delta {\theta }^i\left(j,k,l\right)$); /*compute the target function value of each bacterial position*/
10.					$Q^{*}=max\left(Q_i\right)$/*choose the maximum value*/
11.			end
12.			Generate a new bacterial population based on ${\displaystyle \sum }_{j=1}^{N_e+1}{Q}_i\leftarrow$
13.			reproduce N/2 premium bacteria, eliminate N/2 poor bacteria;
14.		end
15.		if $P_{ed}>rand$ do
16.				${\theta }^i\leftarrow$ Generate new bacterial positions
17.		end
18.	end

5. Experimental Results

5.1. Simulations

The hardware environment for our simulation experiments is an Intel(R) Core(TM) i5-8300H CPU @ 2.30GHz, NVIDIA GeForce GTX 1050 Ti with 6 GB of video memory and 8 GB of RAM. The CPU manufacturer is Intel Corporation, originated from Kulim, Malaysia, and the GPU manufacturer is NVIDIA Corporation, originated from California, USA. The computer system is Windows 7.

The OPNET Modeler uses a three-layer modeling mechanism: the top layer is the net-work model, which customizes the topology of the network; the second layer is the node model, which consists of the corresponding protocol model and defines the device characteristics; and the bottom layer is the process model, which describes the protocols and algorithms with a finite state machine model. In the network-level model, nine monitoring nodes with 200 users are configured, each with two radios and six available channels, and each monitoring node is equipped with a single RF IEEE 802.11.b interface card with a data rate of 11 Mbps and a node monitoring radius of 120 m. The bandwidth of each orthogonal channel is assumed to be the same. The experimental parameters of the DBFO algorithm are set in

Table 2. Experimental parameter settings.

$\mathit{S}$	${\mathit{N}}_{\mathit{c}}$	${\mathit{N}}_{\mathit{r}\mathit{e}}$	${\mathit{N}}_{\mathit{e}\mathit{d}}$	${\mathit{N}}_{\mathit{s}}$	${\mathit{P}}_{\mathit{e}\mathit{d}}$	${\mathit{d}}_{\mathit{a}\mathit{t}\mathit{t}\mathit{r}\mathit{a}\mathit{c}\mathit{t}}$	${\mathit{w}}_{\mathit{a}\mathit{t}\mathit{t}\mathit{r}\mathit{a}\mathit{c}\mathit{t}}$	${\mathit{h}}_{\mathit{r}\mathit{e}\mathit{p}\mathit{e}\mathit{l}\mathit{l}\mathit{a}\mathit{n}\mathit{t}}$	${\mathit{w}}_{\mathit{r}\mathit{e}\mathit{p}\mathit{e}\mathit{l}\mathit{l}\mathit{a}\mathit{n}\mathit{t}}$
10	50	4	2	4	0.2	0.05	0.1	0.05	0.05

.

The high-performance DBFO parameters are obtained through a large number of experiments as shown in Table 2. The convergence experiments conducted in Case 1 (the sniffer allows different radios to allocate the same channel) and Case 2 (the sniffer forbids different radios to allocate the same channel) are shown in Figure 12 and Figure 13.

InitQom in Figure 12 and Figure 13 represents the monitoring quality obtained after the first chemotactic operation of the algorithm; different InitQom values indicate that bacteria in the algorithm started the chemotactic process in different directions.

The iterations of the abscissa represents the number of chemotactic operations. Case 1 and Case 2 achieve almost the same QoM, and the InitQom of Case 2 is generally greater than that of Case 1. Theoretically, Case 2 can be seen as a special case of 1; the range of solutions when searching the channel allocation schemes will greatly be reduced and thus, it is easier to reach convergence and the complexity of the curve is reduced. Figure 14 shows more experiments to reduce the experimental error and reflect the experimental effect more accurately.

In a comparative experiment, we compare the method of this paper with existing representative allocation algorithms. These algorithms include the LP algorithm, the MAB algorithm [25], and GT-based channel allocation [26].

LP: rounding to solve the optimization problem through linear programming relaxation.

MAB: a promising approach that resolves the tradeoff between channel exploration and the exploitation of enhanced communication quality.

GT: a channel allocation optimization algorithm based on GT.

We conducted the comparative experiments in a network with nine monitoring sniffers, six channels, and 200 users. The QoM of every algorithm is shown in Figure 15.

As depicted in Figure 15, after 69 iterations of chemotactic processes, the DBFO algorithm converges to the extremely optimal solution (QoM = 8.78884). The LP algorithm is a deterministic algorithm, so it only needs to run once during the experiment, and it converges to a QoM of 8.7929. The MAB algorithm basically converges at the 101st generation, and the QOM obtained is 8.69417. The GT algorithm converges at about 90 generations, and the QoM is 8.70121. We found little difference between the MAB and GT algorithms. Although the QoM value of MAB in early iterations is greater than that of the GT algorithm, the QoM value obtained is almost the same after they reach stability.

We compared the four algorithms by changing the number of channels and synthesized the experimental results for statistical analysis, as shown in

Table 3. Statistical results of four algorithms with different channel numbers.

Number of Channels	DBFO	LP	MAB	GT
	QoM T(s)	QoM T(s)	QoM T(s)	QoM T(s)
3	8.595 4.176	8.474 6.196	8.127 4.723	8.211 4.364
6	7.758 4.845	7.562 7.352	7.078 5.093	7.117 5.377
9	7.263 5.315	7.081 9.619	6.324 6.216	6.407 5.749

. Figure 16 shows the specific process of the iterations of the three algorithms.

The DBFO, MAB, and GT algorithms are run 200 times in each set of experiments, and the final converged QoM value and running time are statistically averaged, which are represented by QoM and $\overline{T}$, respectively. LP only runs once because it is a deterministic method. The statistical results indicate that the DBFO algorithm has a certain superiority in the quality of the solution compared to other algorithms in solving the channel allocation problem. From Table 3, it can be seen that the performance of the algorithm proposed in this paper is also improved by 2.57% when the number of channels is 9 relative to LP, which has the highest monitoring quality, and by 7.549% relative to GT, which has the fastest convergence speed. Collectively, the DBFO proposed in this paper outperforms the other three algorithms both in terms of monitoring quality and convergence speed. With the increase in the number of channels, the monitoring quality and convergence speed of DBFO do not change much, which indicates that DBFO has higher robustness compared with the other three algorithms.

5.2. Practical Network Experiment

It is impossible to verify the actual effectiveness of the DBFO algorithm applied to the channel allocation problem of wireless monitoring networks only from simulation experiments. Therefore, we use observation and statistics to test the actual application performance of the algorithm by arranging multiple monitoring nodes on campus. The practical network experiment scenario in this paper is a campus, based on the campus wireless network (IEEE 802.11.b WLAN) to evaluate the proposed DBFO algorithm. A total of 21 monitoring points are deployed in a school building of the university to collect user information during working hours (9:00 a.m. to 5:00 p.m.). During 8 h, the sniffers monitored a total of 556 users, and the number of users monitored on the three different channels was 267, 138, and 151. The information transmission probabilities of these users are shown in

Table 4. Transmission probability statistics of users.

Information Transmission Probability	0–0.01	0.01–0.02	0.02–0.04
Numbers of users	481	34	41

.

Figure 17 depicts the comparison of the experimental results of DBFO and the other three algorithms in the practical network experiment. It can be seen from the figure that QoM (number of monitored active users) advances with the increase in the number sniffers (from five to 21). In addition to the experiment with 21 sniffers, the other set of experiments are conducted repeatedly with different sniffers randomly allocated from the 21 sniffers. Because the average activity probability is 0.0029, the largest number of active users is less than 1.6 during each timeslot. Through comparison with the three other algorithms, we found that the experimental results of the DBFO algorithm are significantly better than those of LP, MAB, and GT.

6. Conclusions

This research focused on how to optimize channel allocation to maximize QoM in multichannel, multi-radio wireless networks, converting the channel allocation problem into a 2D channel coding problem. In the two cases of whether different radios of the same sniffer can multiplex a channel, we defined sniffer and wireless monitoring network QoM values and expressed them in a formulated manner; furthermore, we adopted a form of swarm intelligence, the DBFO algorithm, as a solution. The BFO algorithm is first discretized into binary space, and then the channel allocation problem of the wireless monitoring network is coded, and the coding table is modified by three steps, i.e., chemotaxis, replication, and migration, to finally obtain an optimal allocation of channels. After extensive experimental verification, the algorithm proposed in this paper is applicable to the channel allocation problem of multi-radio wireless monitoring networks. The algorithm achieves higher performance and faster convergence than other algorithms, with a high-quality solution, and also makes up for the defects of prior research lacking the exploration of multiradio channel allocation.