Cognitive routing optimization protocol based on multiple channels in wireless sensor networks

Abstract

With the development of modern communication, available spectrum resources are becoming increasingly scarce, which reduce network throughput. Moreover, the mobility of nodes results in the changes of network topological structure. Hence, a considerable amount of control information is consumed, which causes a corresponding increase in network power consumption and exerts a substantial impact on network lifetime. To solve the real-time transmission problem in large-scale wireless mobile sensor networks, opportunistic spectrum access is applied to adjust the transmission power of sensor nodes and the transmission velocity of data. A cognitive routing and optimization protocol based on multiple channels with a cross-layer design is proposed to study joint optimal cognitive routing with maximizing network throughput and network lifetime. Experimental results show that the cognitive routing and optimization protocol based on multiple channels achieves low computational complexity, which maximizes network throughput and network lifetime. This protocol can be also effectively applied to large-scale wireless mobile sensor networks.

Introduction

With the development of modern communication, available spectrum resources are becoming increasingly scarce, which reduce network throughput. Moreover, the mobility of nodes results in the changes of network topological structure. Hence, a large amount of control information is consumed, which causes a corresponding increase in network power consumption and exerts a substantial impact on network lifetime. Recent studies have showed that the network lifetime is one of the most essential issues in large-scale wireless mobile sensor networks, whereas network throughput is considered the most important resource in such networks. ¹

The extensive use of wireless communication technology has promoted the development of cognitive radio (CR) technology ² and the dynamic spectrum access mechanism. Moreover, opportunistic spectrum access (OSA) can effectively improve spectrum efficiency and communication quality. ³

The latest researches show that the dynamic adjustment of node transmission power and data transmission rate can increase the lifetime and throughput of wireless mobile sensor networks, and shorten the delay of end-to-end communication. Moreover, a suitable OSA can effectively solve the conflicts caused by a large number of nodes in the network.^4,5 However, in these scenarios, multiple users shared the same subcarriers, which made the traditional orthogonal frequency division multiplexing (OFDM) inferior to CR technology. ⁶ Solving conflicts among channels is becoming increasingly popular. ⁷ Interestingly, spectral selection may effectively improve the transmission performance of a neighbor sensor node. ⁴

The distributed subcarriers and power control algorithms aim to minimize the power consumption of each bit information from subcarriers,^8,9 and the simulation results demonstrated that the performance of the proposed approach is close to that of the centralized optimal solution. However, the authors only considered the data transmission rate and power consumption of a neighbor sensor node and disregarded the performance of the entire route. Inspired by biological systems, Son et al. ¹⁰ proposed a bio-inspired scheduling algorithm that reduces the energy consumption and delay for wireless sensor networks (WSNs), in which the energy-efficient routing path and the energy consumption are investigated using multiple channels for data transmission. Simulation experiment showed the effectiveness of the proposed method.

Several OSAs based on MAC (medium access control) schemes in CR networks are discussed in detail in Sultana et al. ¹¹ The differences between the conventional MAC protocols and OSA-based MAC protocols were investigated. Palma ¹² proposed a new communication protocol, named as energy efficiency protocol (EFP), is based on a hop-by-hop transport scheme and is especially devised to simultaneously solve the network energy consumption and the performance of the closed-loop system. The proposed protocol can be implemented by means of three heuristics, basically using distinct rules to control the maximum number of retransmissions allowed in terms of the voltage level of the batteries of the network nodes. El Mougy et al. ¹³ presented two routing approaches for WSNs; these approaches apply the concepts of node cooperation and information exchange to achieve cognition across multiple network layers. Hanefi ¹⁴ proposed a new multichannel allocation approach, named hybrid multichannel allocation for WSNs, named as HMCA in our stduy, based on hybrid time division multiple access (TDMA) and frequency division multiple access (FDMA) techniques and using dual radio with multichannel communication. Simulation experiment showed the proposed method assured steady and high packet delivery ratios in large-scale networking environments even with hundreds of sensor nodes. Spachos and Hatzinakos ¹⁵ presented a real-time cognitive WSN for carbon dioxide monitoring at a complex indoor environment. Experimental results validated the effectiveness of the proposed method.

Although many studies have considered route protocols based on spectrum choice, certain disadvantages require attention. First, some studies cannot be applied to large-scale wireless mobile sensor networks. Second, network lifetime and throughput are not considered (or only one of them is considered) when making route choices. To address these issues, a multiple-channel cognitive routing optimization protocol is proposed in the current work for selecting spectrum while maximizing network lifetime and throughput in large-scale wireless mobile sensor networks. Our proposed method based on the model of signal-to-interference plus noise ratio (SINR) can effectively solve the dynamic spectrum allocation, data transmission rate, and power control problems. The rest of this article is organized as follows. Section “Related work” summarizes system model, protocol assumption, performance analysis, and so on. Section “Results and discussion” is about simulation parameters, simulation results, and experimental analysis. Finally, conclusion of this study along with the future work is mentioned in section “Conclusion.”

Related work

A three-layer network topology is typically used in large-scale wireless mobile sensor networks, in where the information collected by a cluster head will be transmitted to the base station in multiple hops. ¹⁶ In this study, a corresponding clustering process of Deng et al. ¹⁷ and Hadi et al. ¹⁸ is adopted to analyze multiple hops transmission between cluster heads and base stations.

System model

Assume that primary users (PUs) and secondary users (SUs) represent cluster nodes in large-scale wireless mobile sensor networks. The multiple address access technology of OFDM is used in PUs, and they possess user authority. Moreover, PUs can only transmit in their allocated spectra. This access technology is only controlled by the destination nodes and is not affected by the non-authoritative users. However, SUs have not any authorized spectrum and only transmit data with the help of the idle spectrum of PUs.

A spectrum is divided into two separate channels: data channel (DC) and common control channel (CCC). DC consists of a series of discrete sub-bands, which are marked as $\{f_{min},f_{min+1},...,f_{max-1},f_{max}\}$ . The band of each sub-band is marked with a discrete subscript $(\omega )$ . $\mathrm{CCC}(f_{3c})$ is used to consult with the spectrum access of SUs. $f_{3c}\notin \{f_{min},f_{max}\}$ is generated by the interaction information from the control channels.

Destination nodes are assumed to be fixed cognitive infrastructure that permits access from PUs and SUs. Transmitters are adjusted to a series of discrete bands according to a variable carrier set by all users including PUs and SUs. PUs directly communicate with SUs with a single hop through the base stations, and SUs transmit sensor data to the base stations with multiple hops. A multiple wireless network is typically modeled as a direct connected graph $G=(C,E)$ , where $C=\{c_1,...,c_{M+N}\}$ denotes the limited set of PUs, $\left|C\right|=M+N$ ; $\mathrm{PU}=\{c_1,...,c_M\}$ and $\mathrm{SU}=\{c_{M+1},...,c_{M+N}\}$ represent the sets of PUs and SUs, respectively; and $(c_i,c_j)\in E$ indicates the directed wireless links between $c_i$ and $c_j$ .

Protocol assumptions

For convenience, the following protocol assumptions are used in this study:

1. All SUs have the same physical characteristics.

2. The location and velocity of all SUs are known.

3. All SUs can calculate the time of packet transmission, which is defined as the ratio of the size of a packet to the transmission rate.

4. To satisfy the requirements of different communication power and data transmission rates, each SU may be permitted to select multiple subcarriers. Data transmission rate, communication power, and subcarrier may be assigned with the corresponding algorithms.

Definitions

Prior to designing the cognitive routing and optimization protocol based on multiple channels (CROMC) in large-scale mobile WSNs, we first present the relevant definitions used in this study.

Spectrum hole

Spectrum hole is an important parameter in cognitive wireless technology. It represents the opportunity possessed by a spectrum. The frequency point is adopted as the spectrum hole of a user $c_i$ . The communication power of $c_i$ should ensure the need of the bit error rate (BER) from the receiving terminal and the interference excluded from those of the PUs and SUs in this work. This formula is mathematically described as follows

$$\frac{P_{\mathrm{ij}}\left(f\right)A_{\mathrm{ij}}\left(f\right)G}{N_j\left(f\right)+\sum _{v_k\in C,k\ne i}{P}_k\left(f\right)L_{\mathrm{kj}}\left(f\right)}\ge {\lambda }_{\mathrm{SU}}$$

(1)

$$\begin{array}{cc}\frac{P_l^R\left(f\right)}{N{I}_l\left(f\right)+\mathrm{\Delta }{I}_{\mathrm{il}}\left(f\right)}\ge \lambda ,\hfill & \forall v_l\in C\hfill \end{array}\mathrm{ongoing}$$

(2)

In equation (1), $P_{\mathrm{ij}}(f)$ means the communication power of the link $(c_i,c_j)$ within the frequency of $f$ , $A_{\mathrm{ij}}\left(f\right($ means the transmission loss from point $c_i$ to point $c_j$ , $G$ means the processing gain, $N_j(f)$ presents the noise of the receiver within the frequency of $f$ , and ${\lambda }_{\mathrm{SU}}$ represents the threshold of SINR that ensures the BER of SU destination. In equation (2), $P_l^R(f)$ means the signal power received from user $c_l$ , $N{I}_l(f)$ represents the noise and interference of $c_l$ before transmitting from the user of $c_i$ , $\mathrm{\Delta }{I}_{\mathrm{il}}(f)$ means the interference generated from the user of $c_i$ , and $\lambda$ presents the threshold of SINR derived from the PUs or SUs according to the type of $c_1$ .

According to equations (1) and (2), we can calculate the available minimum and maximum transmission power of every frequency point from each SU, respectively. The corresponding formulas are defined as follows

$$P_{\mathrm{ij}}^{min}\left(f\right)=\frac{{\lambda }_{\mathrm{SU}}·(N_j\left(f\right)+\sum _{k\in C,k\ne i}{P}_k\left(f\right)L_{\mathrm{kj}}\left(f\right))}{A_{\mathrm{ij}}\left(f\right)G}$$

(3)

$$P_{\mathrm{ij}}^{max}\left(f\right)=\underset{l\in C}{min}\left\{\frac{P_l^R\left(f\right)/\lambda -N{I}_l\left(f\right)}{A_{\mathrm{il}}\left(f\right)}\right\}$$

(4)

where $P_{\mathrm{ij}}^{min}(f)=\{p_{\mathrm{ij}}^{min}(f)\}$ and $P_{\mathrm{ij}}^{max}(f)=\{p_{\mathrm{ij}}^{max}(f)\}$ , which represent the sets of minimum and maximum communication power in the link of $(c_i,c_j)$ respectively. If $P_{\mathrm{ij}}^{min}(f)<P_{\mathrm{ij}}^{max}(f)$ , then the frequency point $f$ may be regarded as a spectrum hole by the SU of $c_i$ .

Latency

Latency is a time delay between the source node and the destination node in a WSN. It is presented as follows

$$T_{\mathrm{av}}=T_t+T_p+T_q$$

(5)

where $T_{\mathrm{av}}$ means the average latency, $T_t$ presents the transmission latency, $T_p$ denotes the process latency, and $T_q$ means the queuing delay. The data $N_0$ is assumed to be divided into $K$ packets, where $K$ is a parameter determined from the transmission rate, transmission power, and route. Hence, every factor in equation (5) is represented as follows

$$T_t=\sum _{i=1}^n\frac{N_0/K}{R_{i-1,i}}$$

(6)

$$T_p=\sum _{i=1}^n\frac{N_0/K}{R_p}$$

(7)

$$T_q=\sum _{i=1}^n(\frac{N_0(K-1)}{2\times K}+Q_i)\times (\frac{1}{R_{i-1,i}}+\frac{1}{R_p})$$

(8)

In equations (6)–(8), $N_0$ denotes the data in the source node, $K$ denotes the number of data packets, $Q_i$ represents the data that will be sent to the queue, $R_{i-1,i}$ represents the transmission rate in the link $(l_{i-1},l_i)$ , $R_p$ denotes the data process rate, and $n$ means the number of hops between the source node and the destination node.

Assume that $K$ packets are under the same condition. Then, the total latency transmitting $N_0$ bits may be represented as follows

$$\begin{array}{c}\multicolumn{1}{c}{T_{\mathrm{total}}=K\times (T_t+T_p+T_q)}\\ =K\times \left(\begin{array}{c}\sum _{i=1}^n\frac{N_0/K}{R_{i-1,i}}+\sum _{i=1}^n\frac{N_0/K}{R_p}+\hfill \\ \multicolumn{1}{c}{\sum _{i=1}^n(\frac{N_0(K-1)}{2\times K}+Q_i)\times (\frac{1}{R_{i-1,i}}+\frac{1}{R_p})}\end{array}\right)\hfill \end{array}$$

(9)

In general, when a vast amount of data are required to be transmitted in the large-scale mobile senor networks, the value of $K$ is extremely large. Then, equation (9) is typically approximated as follows

$$T_{\mathrm{total}}=K\times \sum _{i=1}^n(\frac{N_0}{2}+Q_i)\times (\frac{1}{R_{i-1,i}}+\frac{1}{R_p})$$

(10)

Network lifetime

The maximization of network lifetime can be divided into the following aspects: the minimization of path energy consumption and the realization of load balancing, which needs to consider the residual energy of each node and the amount of queuing data, that is, energy standard deviation. In this section, we discuss only the power consumption in spectrum allocations, no more network lifetime. The load balancing of route is discussed in the following sections.

Power consumption is modeled as the sum of the consumption of transmission and processing. For an $H$ hop route with $N_0$ data, the total power consumption is described as follows

$$E_{\mathrm{total}}=K\times \sum _{i=1}^H(\frac{P_{i-1,i}\times {(N_0/K)}^2}{R_{i-1,i}}+\frac{E_p\times N}{K})$$

(11)

where $P_{i-1,i}$ denotes the power consumed by 1 bit data and $E_p$ represents the average power required to process 1 bit data, including decoding and coding.

Route capability

The topology of an SU is usually changed because of the following reasons: First, PU access forces the spectrum withdrawal of an SU. Second, the mobility of an SU also changes its topological structure. That is, if the two nodes associated with a link are in their transmission range without affecting the communication of PUs in the network, then the link is accessible. However, changes in topology will result in broken links and packet loss. Therefore, only a limited number of data can be successfully transmitted over a limited route lifetime. Notably, route capability means that those data can be transmitted in links, and route lifetime represents the time that those data can be successfully transmitted.

Recent research shows several spectrum prediction methods based on history information, which can not only provide effective spectrum utilization but also predict the spectrum stability of links.^19,20 However, Hanefi ¹⁴ believed that route capability is more important than simple route stability for on-demand routing. For example, when the data are transmitted to node $c_i$ , the links before $c_i$ are broken, but data remain accessible. Although spectrum prediction methods are used to predict spectrum stability, broken links with spectrum withdrawal should still be addressed. For simplicity, when analyzing route capability and lifetime, the mobility of SUs is only considering. The route capability of $N_{\psi }$ may be expressed as

$$\sum _{i=1}^{h+1}(N_{\psi }+Q_i)(\frac{1}{R_{i-1,i}}+\frac{1}{R_p})\le t_{h-1,h},1\le h\le H$$

(12)

From equation (12), it can be obtained that

$$N_{\psi }=\underset{1\le h\le H}{min}2\times \left\{\frac{\frac{t_{h-1,h}}{\mathrm{K}}-\sum _{i=1}^{h+1}{Q}_i(\frac{1}{R_{i-1,i}}+\frac{1}{R_p})}{\sum _{i=1}^{h+1}(\frac{1}{R_{i-1,i}}+\frac{1}{R_p})}\right\}$$

(13)

where $t_{h-1,h}$ denotes the prediction connection time of the link $(c_{h-1},c_h)$ , which satisfies the following inequation

$$\begin{array}{c}\multicolumn{1}{c}{{\left(\begin{array}{c}{x}_{h-1}\left(t_1\right)+v_{h-1}\left(t_1\right)·\cos {\theta }_{h-1}\left(t_1\right)·t_{h-1,h}\hfill \\ \multicolumn{1}{c}{-x_h\left(t_1\right)-v_h\left(t_1\right)·\cos {\theta }_h\left(t_1\right)·t_{h-1,h}}\end{array}\right)}^2}\\ \multicolumn{1}{c}{+{\left(\begin{array}{c}{y}_{h-1}\left(t_1\right)+v_{h-1}\left(t_1\right)·\sin {\theta }_{h-1}\left(t_1\right)·t_{h-1,h}\hfill \\ \multicolumn{1}{c}{-y_h\left(t_1\right)-v_h\left(t_1\right)·\sin {\theta }_h\left(t_1\right)·t_{h-1,h}}\end{array}\right)}^2}\\ \multicolumn{1}{c}{\le {\left(R_{h-1,h}^{max}\right)}^2}\end{array}$$

(14)

where $t_1$ represents the reference start time and $R_{h-1,h}^{max}$ indicates the maximum connected distance between $c_{h-1}$ and $c_h$ , which is determined with transmission power, total noise interference, and SINR threshold value.

Protocol design

Given the current spectrum environment and hardware constraints, CROMC aims to maximize network throughput and lifetime. Figure 1 shows the overall frame of CROMC. Hence, the following assumptions are presented. First, the node with the largest residual energy in the cluster is selected as the cluster head. Second, information in the network is transmitted through every cluster head with multiple hops. Therefore, the network lifetime is maximized through path loss and energy balance according to equations (3) and (4). The network throughput is inversely proportional to path transmission delay when data are transmitted simultaneously. Therefore, maximizing network lifetime and throughput is equivalent to minimizing path loss and delay and comprehensively considering the load balancing problem. To minimize path power and delay, we first select the spectrum $F_{\mathrm{ij}}$ and the corresponding transmission power of $P_{\mathrm{ij}}=\{P_{\mathrm{ij}}(f){\}}_{f\in F_{\mathrm{ij}}}$ .

Figure 1.

The overall frame of CROMC.

Therefore, the multi-objective optimization expression is established and load balancing is discussed in subsection “Routing.” For simplicity, we assume that the spectrum of $F_{\mathrm{ij}}$ may be discrete sub-band sets and the transmission power of $P_{\mathrm{ij}}$ at each frequency point exhibits no difference. The mathematical expression is described as follows:

P1:

Given: $(c_{i,}{c}_j)$ , ${\mathrm{I}}_j,{\mathrm{N}}_j,{\mathrm{A}}_{\mathrm{ij}},P_{\mathrm{ij}}^{min},P_{\mathrm{ij}}^{max},p^{\mathrm{Bgt}},\mathrm{\Delta }{f}^{\mathrm{Bgt}}$

Find: $F_{\mathrm{ij}},P_{\mathrm{ij}},\mathrm{\psi },K$

Minimize: $T_{\mathrm{\Psi }},E_{\mathrm{\Psi }}$

$$\begin{array}{c}\mathrm{s}.\mathrm{t}.max\left\{P_{\mathrm{ij}}^{min}\left(f\right)\right\}\le P_{\mathrm{ij}}\left(f\right)\le min\left\{P_{\mathrm{ij}}^{max}\left(f\right)\right\},\forall f\in F_{\mathrm{ij}}\hfill \\ P_{\mathrm{ij}}=P_{\mathrm{ij}}\left(f\right)w\mathrm{\Delta }{f}_{\mathrm{ij}}\le p^{\mathrm{Bgt}}\hfill \\ \mathrm{\Delta }{f}_{\mathrm{ij}}\le \mathrm{\Delta }{f}^{\mathrm{Bgt}}\hfill \\ N_{\mathrm{\Psi }}\ge N_0\hfill \end{array}$$

(15)

where $\mathrm{\Delta }{f}_{\mathrm{ij}}=\left|F_{\mathrm{ij}}\right|$ means the number of sub-bands, ${\mathrm{I}}_j=[I_j(f_{min}),I_j(f_{min+1}),\dots ,I_j(f_{max})]$ denotes the interference vector at the receiver of $c_j$ , ${\mathrm{N}}_j=[N_j(f_{min}),N_j(f_{min+1}),\dots ,N_j(f_{max})]$ represents the noise vector at the receiver of $c_j$ , ${\mathrm{A}}_j=[A_j(f_{min}),A_j(f_{min+1}),\dots ,A_j(f_{max})]$ means the transmission loss vector, and $c_i,c_j\in \mathrm{SU}$ . The first constraint condition shows the existence of spectrum holes, whereas the other two conditions indicate the constraint from the user’s hardware. The last constraint emphasizes that routing capability should be larger than the amount of data to be transferred. However, such routes are occasionally not found in networks, thereby showing that the constraint conditions are too strict to satisfy. Therefore, when we solve the aforementioned optimization problem, the constraint condition can be ignored and applied only to the route selection standard.

Evidently, the solution for $\mathrm{P}1$ is the minimization of finite multiple target vectors. Therefore, the multi-objective optimization problem can be transformed into a single-objective optimization problem by using the weight factor, which is described as follows:

P2:

Given: $(c_{i,}{c}_j)$ , ${\mathrm{I}}_j,{\mathrm{N}}_j,{\mathrm{A}}_{\mathrm{ij}},P_{\mathrm{ij}}^{min},P_{\mathrm{ij}}^{max},p^{\mathrm{Bgt}},\mathrm{\Delta }{f}^{\mathrm{Bgt}}$

Find: $F_{\mathrm{ij}},P_{\mathrm{ij}},\psi ,K$

Minimize: ${\mathrm{\Phi }}_{\mathrm{\Psi }}={\lambda }_1{T}_{\mathrm{\Psi }}+{\lambda }_2{E}_{\mathrm{\Psi }}$

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{s}.\mathrm{t}.max\left\{P_{\mathrm{ij}}^{min}\left(f\right)\right\}\le P_{\mathrm{ij}}\left(f\right)\le min\left\{P_{\mathrm{ij}}^{max}\left(f\right)\right\},\forall f\in F_{\mathrm{ij}}}\\ \multicolumn{1}{c}{P_{\mathrm{ij}}=P_{\mathrm{ij}}\left(f\right)w\mathrm{\Delta }{f}_{\mathrm{ij}}\le p^{\mathrm{Bgt}}}\\ \multicolumn{1}{c}{\mathrm{\Delta }{f}_{\mathrm{ij}}\le \mathrm{\Delta }{f}^{\mathrm{Bgt}}}\end{array}$$

(16)

where ${\lambda }_1\in \mathrm{\Lambda }$ means the importance of each target, and

$${\lambda }_i>0,\sum _i{\lambda }_i=1$$

(17)

${\mathrm{\Phi }}_{\mathrm{\Psi }}$ is described in detail as follows

$$\begin{array}{c}{\mathrm{\Phi }}_{\mathrm{\Psi }}={\lambda }_{1}K·\sum _{i=1}^H(N_0/2+Q_i)(\frac{1}{R_{i-1,i}}+\frac{1}{R_{\mathrm{p}}})+\hfill \\ {\lambda }_{2}K·\sum _{i=1}^H(\frac{P_{i-1,i}{(N_0/K)}^2}{R_{i-,i}}+E_{\mathrm{p}}{N}_0/K)\hfill \end{array}$$

(18)

Given that the condition of equality is obtained from an inequality, we set

$$K=\sqrt{\frac{{\lambda }_2·\sum _{i=1}^H\left(\frac{P_{i-1,i}{N}_0^2}{R_{i-1,i}}\right)}{{\lambda }_1·\sum _{i=1}^H(N_0/2+Q_i)(\frac{1}{R_{i-1,i}}+\frac{1}{R_{\mathrm{p}}})}}$$

(19)

Then, ${\mathrm{\Phi }}_{\mathrm{\Psi }}$ may achieve the minimization value, it is that

$$\begin{array}{c}\multicolumn{1}{c}{{\mathrm{\Phi }}_{\mathrm{\Psi }}=}\\ \multicolumn{1}{c}{2\sqrt{{\lambda }_1{\lambda }_2\sum _{i=1}^H(N_0/2+Q_i)(\frac{1}{R_{i-1,i}}+\frac{1}{R_{\mathrm{p}}})\sum _{i=1}^H\left(\frac{P_{i-1,i}{N}_0^2}{R_{i-1,i}}\right)}}\\ \multicolumn{1}{c}{+{\lambda }_{2}H{E}_{\mathrm{p}}{N}_0}\end{array}$$

(20)

Furthermore, considering the condition of the Cauchy–Schwarz inequality, we set

$$\frac{{\lambda }_2{N}_0^2{P}_{i-1,i}}{{\lambda }_1(N_0/2+Q_i)(1+R_{i-1,i}/R_{\mathrm{p}})}=\eta$$

(21)

where $\eta$ is a given constant, of which the dimension with $P_{i-1,i}$ can be defined as the Cauchy–Schwarz power. We further obtain the minimization of ${\mathrm{\Phi }}_{\mathrm{\Psi }}$ , and the corresponding mathematical expression is presented as follows

$${\mathrm{\Phi }}_{\mathrm{\Psi }}=2·\sum _{i=1}^H\sqrt{{\lambda }_1{\lambda }_2{N}_0^2(N_0/2+Q_i){\rho }_{i-1,i}}+{\lambda }_{2}H{E}_{\mathrm{p}}{N}_0$$

(22)

where ${\rho }_{i-1,i}=\frac{P_{i-1,i}}{R_{i-1,i}}(\frac{1}{R_{i-1,i}}+\frac{1}{R_{\mathrm{p}}})$ .

Spectrum allocation and routing

To obtain the maximum network lifetime and throughput according to the goal of the protocol design, we achieve only the optimization problem of $\mathrm{P}2$ and consider the energy equilibrium and the fourth constraint condition (i.e. routing capability) in $\mathrm{P}1$ . According to equations (21) and (22), the process may be executed in two steps: minimizing the spectrum allocation of ${\rho }_{i-1,i}$ and the routing of ${\mathrm{\Phi }}_{\mathrm{\Psi }}$ .

Spectrum allocation

The spectrum allocation algorithm is implemented by every distributed SU with the given spectrum environment, which is described as follows:

P3:

Given: $(c_{i,}{c}_j)$ , ${\mathrm{I}}_j,{\mathrm{N}}_j,{\mathrm{A}}_{\mathrm{ij}},P_{\mathrm{ij}}^{min},P_{\mathrm{ij}}^{max},p^{\mathrm{Bgt}},\mathrm{\Delta }{f}^{\mathrm{Bgt}}$

Find: $F_{\mathrm{ij}}$ , $P_{\mathrm{ij}}$

Minimize: ${\rho }_{\mathrm{ij}}$

$$\begin{array}{c}\mathrm{s}.\mathrm{t}.max\left\{P_{\mathrm{ij}}^{min}\left(f\right)\right\}\le P_{\mathrm{ij}}\left(f\right)\le min\left\{P_{\mathrm{ij}}^{max}\left(f\right)\right\}\hfill \\ \forall f\in F_{\mathrm{ij}}\hfill \\ P_{\mathrm{ij}}=P_{\mathrm{ij}}\left(f\right)w\mathrm{\Delta }{f}_{\mathrm{ij}}\le p^{\mathrm{Bgt}}\hfill \\ \mathrm{\Delta }{f}_{\mathrm{ij}}\le \mathrm{\Delta }{f}^{\mathrm{Bgt}}\hfill \\ \frac{{\lambda }_2{N}_0^2{P}_{\mathrm{ij}}}{{\lambda }_1(N_0/2+Q_i)(1+R_{\mathrm{ij}}/R_p)}=\eta \hfill \end{array}$$

(23)

From the fourth constraint condition, we can obtain

$$P_{\mathrm{ij}}=\frac{{\lambda }_1\eta (N_0/2+Q_i)(1+R_{\mathrm{ij}}/R_p)}{{\lambda }_2{N}_0^2}$$

(24)

We set $R_{\mathrm{ij}}=\mathrm{Mw}\mathrm{\Delta }{f}_{\mathrm{ij}}$ , where the value of $M$ is determined with the selected modulation and encoding. According to equations (23) and (24), we can obtain the following expression

$$P_{\mathrm{ij}}\left(f\right)=\frac{{\lambda }_1\eta M(N_0/2+Q_i)(1+R_{\mathrm{ij}}/R_p)}{{\lambda }_2{N}_0^2{R}_{\mathrm{ij}}}$$

(25)

To obtain the solution for $\mathrm{P}3$ , the spectrum environment should be evaluated to find the optimal transmission power at each frequency point. Then, the transmission sub-band sets are selected based on the optimization results. From ${\rho }_{i-1,i}=P_{i-1,i}/P_{i-1,i}{R}_{i-1,i}{R}_{i-1,i}(1/1R_{i-1,i}{R}_{i-1,i}+1/1R_{\mathrm{p}}{R}_{\mathrm{p}})$ , the optimization of $\mathrm{P}3$ is equivalent to the maximization of $R_{\mathrm{ij}}$ , that is, $\mathrm{\Delta }{f}_{\mathrm{ij}}$ . Therefore, we can obtain $R_{\mathrm{ij}1}$ , $R_{\mathrm{ij}2}$ , and $R_{\mathrm{ij}3}$ from the last three constraints of $\mathrm{P}3$ . The corresponding mathematical expression is defined as follows

$$R_{\mathrm{ij}1}=(\frac{{\lambda }_2{N}_0^2{P}^{\mathrm{Bgt}}}{{\lambda }_1\eta (N_0/2+Q_i)}-1)R_p$$

(26)

$$R_{\mathrm{ij}2}=M·w\mathrm{\Delta }{f}^{\mathrm{Bgt}}$$

(27)

$$R_{\mathrm{ij}3}=\frac{{\lambda }_1\eta M(N_0/2+Q_i)}{{\lambda }_2{N}_0^{2}max\left\{P_{\mathrm{ij}}^{min}\left(f\right)\right\}-{\lambda }_1\eta M(N_0/2+Q_i)/R_p}$$

(28)

Then, the transmission sub-band sets of $\mathrm{\Delta }{f}_{\mathrm{ij}}(f)$ and the optimal transmission power of $p_{\mathrm{ij}}(f)$ are obtained at each frequency point, which are

$$\mathrm{\Delta }{f}_{\mathrm{ij}}(f)=\{\begin{array}{cc}\lfloor \frac{min(R_{\mathrm{ij}1},R_{\mathrm{ij}2},R_{\mathrm{ij}3})}{\mathrm{Mw}}\rfloor ,\hfill & \begin{array}{c}max\left\{P_{\mathrm{ij}}^{min}\left(f\right)\right\}\hfill \\ \multicolumn{1}{c}{\ge \frac{{\lambda }_1\eta M(N_0/2+Q_i)}{{\lambda }_2{N}_0^2{R}_{\mathrm{proc}}},}\end{array}\hfill \\ \multicolumn{1}{c}{\lfloor \frac{min(R_{\mathrm{ij}1},R_{\mathrm{ij}2})}{\mathrm{Mw}}\rfloor ,}& \begin{array}{c}max\left\{P_{\mathrm{ij}}^{min}\left(f\right)\right\}\hfill \\ \multicolumn{1}{c}{\le \frac{{\lambda }_1\eta M(N_0/2+Q_i)}{{\lambda }_2{N}_0^2{R}_{\mathrm{p}}}}\\ \multicolumn{1}{c}{\le min\left\{P_{\mathrm{ij}}^{max}\left(f\right)\right\},}\end{array}\hfill \\ \multicolumn{1}{c}{\mathrm{null},}& \begin{array}{c}min\left\{P_{\mathrm{ij}}^{max}\left(f\right)\right\}\hfill \\ \multicolumn{1}{c}{\le \frac{{\lambda }_1\eta M(N_0/2+Q_i)}{{\lambda }_2{N}_0^2{R}_{\mathrm{p}}}}\end{array}\hfill \end{array}$$

(29)

$$\begin{array}{c}{P}_{\mathrm{ij}}\left(f\right)=\frac{{\lambda }_1\eta M(N_0/2+Q_i)}{{\lambda }_2{N}_0^2}(\frac{1}{\mathrm{Mw}\mathrm{\Delta }{f}_{\mathrm{ij}}\left(f\right)}+\frac{1}{R_p})\hfill \\ min\left\{P_{\mathrm{ij}}^{max}\left(f\right)\right\}\ge \frac{{\lambda }_1\eta M(N_0/2+Q_i)}{{\lambda }_2{N}_0^2{R}_{\mathrm{p}}}\hfill \end{array}$$

(30)

where ⌊•⌋ indicates the lower bound integer of the number of sub-bands. According to equations (29) and (30), the selection of the spectrum of $F_{\mathrm{ij}}$ should satisfy the following sub-band sets. First, the optimization transmission power of each sub-band should be as small as possible. Second, the number of sub-bands should match the optimization of the transmission power thereby determining the spectrum allocation and the number of sub-bands of each link. Finally, the optimal transmission power of each link can be calculated using equation (30). If no spectrum resource based on the negotiation result is available, then the message will not be transmitted temporarily. The corresponding algorithm is described in Table 1.

Table 1.

Algorithm 1.

sort all frequency in ascending order of minimum transmission power
$f_{\mathrm{ij}}^{*}=\varphi ,\mathrm{\Delta }{f}_{\mathrm{ij}}^{*}=0,\mathrm{\Delta }{f}_{\mathrm{ij}}=\mathrm{\Delta }{f}^{\mathrm{Bgt}},f_{\mathrm{ij}}=1$
while $\mathrm{\Delta }{f}_{\mathrm{ij}}>0$ do
get the $max\{P_{\mathrm{ij}}^{min}(f)\}$ of the first $\mathrm{\Delta }{f}_{\mathrm{ij}}$ frequencies
calculate $\mathrm{\Delta }{f}_{\mathrm{ij}}\_\mathrm{tmp}$ using Eq.(29)
if $\mathrm{\Delta }{f}_{\mathrm{ij}}\_\mathrm{tmp}\ge \mathrm{\Delta }{f}_{\mathrm{ij}}$ then
$f_{\mathrm{ij}}^{*}=$ the first $\mathrm{\Delta }{f}_{\mathrm{ij}}$ frequencies
$\mathrm{\Delta }{f}_{\mathrm{ij}}^{*}=\mathrm{\Delta }{f}_{\mathrm{ij}}$
Break
Else
$\mathrm{\Delta }{f}_{\mathrm{ij}}=\mathrm{\Delta }{f}_{\mathrm{ij}}-1$
end if
end while

Routing

The goal of spectrum allocation is the minmization of ${\rho }_{i,j}$ . After determining spectrum and transmission power among all links, a route is selected to miminize ${\mathrm{\Phi }}_{\mathrm{\Psi }}$ in equation (20). Then, the solution for $\mathrm{P}2$ is completed. Each link should be assigned a link cost of ${\omega }_{\mathrm{ij}}$ according to the system model described in subsection “System model.” Therefore, ${\mathrm{\Phi }}_{\mathrm{\Psi }}$ denotes the total cost of the routing plus factor of ${\lambda }_{2}H{E}_p{N}_0$ . The link cost of ${\omega }_{\mathrm{ij}}$ can be defined as follows

$${\omega }_{\mathrm{ij}}=\sqrt{{\lambda }_1{\lambda }_2{N}_0^2(N_0/2+Q_i){\rho }_{\mathrm{ij}}}$$

(31)

Obviously, ${\omega }_{\mathrm{ij}}$ always non-negative. Thus, a non-negative weight graph can be obtained in this network. If the link is inaccessible after its negotiation, that is, the spectrum is unavailable, then the weight of the link is set to positive infinity. Therefore, the goal of the protocol design is to find the path (i.e. the shortest path) with the minimum weight sum from the source node to the destination node in the graph.

As indicated in subsection “Spectrum allocation and routing,” energy balance and routing capability among cluster heads should be considered when selecting routes to obtain maximum network lifetime and throughput. Energy balance among cluster heads may be adjusted with link cost. In equation (31), the linear combination of residual energy among cluster head nodes is realized by fixed constants, $k_1\mathrm{and}{k}_2$ . The adjusted link cost is described as follows

$${\omega }_{\mathrm{ij}}^{*}={\kappa }_1·\frac{{\omega }_{\mathrm{ij}}}{{\left({\omega }_{\mathrm{ij}}\right)}_{min}}+{\kappa }_2·\frac{e_{j\_\mathrm{init}}}{e_{j\_\mathrm{resi}}}$$

(32)

where $e_{j\_\mathrm{resi}}$ means the residual energy of the node of $c_j$ and $e_{j\_\mathrm{init}}$ means the initial energy. The variables of $({\omega }_{\mathrm{ij}}{)}_{min}$ and $e_{j\_\mathrm{init}}$ are used to normalize ${\omega }_{\mathrm{ij}}$ . The ${\kappa }_i\in \mathrm{\Lambda }$ denotes the importance of each factor, and

$$\begin{array}{cc}{\kappa }_i>0,\hfill & \sum _i{\kappa }_i=1\hfill \end{array}$$

(33)

when $Q_i=0,\mathrm{\Delta }{f}_{\mathrm{ij}}=\mathrm{\Delta }{f}^{\mathrm{Bgt}}$ , the minimum value of ${\omega }_{\mathrm{ij}}$ may be obtained using equation (31), that is

$${\left({\omega }_{\mathrm{ij}}\right)}_{min}=\frac{1}{2}{\lambda }_1{N}_0(\frac{1}{\mathrm{Mw}\mathrm{\Delta }{f}^{\mathrm{Bgt}}}+\frac{1}{R_{\mathrm{proc}}})·\sqrt{\eta }$$

(34)

For route capability, we should select the route that comprises the link with the maximum route capability and can guarantee $N_{\psi }\ge N_0$ . From equation (13), we can draw the following conclusion. For a given possible route, that is, $t_{h-1,h},Q_i,h,\mathrm{and}{R}_p$ , the maximization of routing capability is equivalent to the maximization of each $R_{i-1,i}$ . Therefore, route capability will not be handled, only when $N_{\psi }\ge N_0$ . However, if no link exists in which route capability is more than $N_0$ , then the source node can only transfer data in real time by using the current optimal routing.

In conclusion, for given the non-negative weight subgraph $G_0$ , the positive constant $\xi$ , and the node pair (s, t), the problem becomes seeking the shortest route from s to t, which are arranged in ascending order according to total weight.

The $\xi$ shortest path problem is a classic shortest path problem that seeks multiple paths based on weight in ascending order. Many researchers have proposed a series of algorithms, which uses the distributed algorithm to search for a route. That is, each SU can update and send route information to its neighbor nodes in the spectrum negotiation stage. Then, the base station makes a decision based on the received routing information according to the sum weight in ascending order based on the received routing information. If the negotiation indicates that no link exists between two given users, then the users will neither update nor transmit routing information. The existence and uncertainty of the shortest path is described as follows.

Lemma 1 (existence)

The shortest route exists in a feasible route that satisfies the route selection method.

Proof

From the preceding description, SUs update and transmit routing information only in negotiations with available links. Therefore, we can draw the following conclusions. First, the route information reaching the base station represents a feasible route. Second, the transmission time is exceedingly long for all available information to reach the base station. The shortest route is a feasible route with the sum of the minimum weight, which satisfies the route selection method. Thus, Lemma 1 is proven.

Lemma 2 (deterministic)

The shortest route can be determined.

Proof

All feasible routes have the unique sum of the non-negative weights, assuming that the total weight of available route $i$ is ${\mathrm{\Phi }}_i>0(i=1,\dots ,\xi )$ . For graphs with a limited node, the available links must be a finite value, that is, $\xi <\infty$ . Therfore, all available route weights compose a finite element set, which is called as $\mathrm{\Phi }={\left\{{\mathrm{\Phi }}_i\right\}}_{i=1}^{\xi }$ . The elements in the set are arranged in ascending order, that is, ${\mathrm{\Phi }}_1\le {\mathrm{\Phi }}_2\le \dots \le {\mathrm{\Phi }}_i<{\mathrm{\Phi }}_{i+1}\le \dots \le {\mathrm{\Phi }}_{\xi }$ . Therefore, Lemma 2 is proven.

Protocol description

Various on-demand routes have been proposed. For example, dynamic source routing (DSR) ²¹ and ad hoc on-demand distance vector routing (AODV) ²² are classic on-demand route protocols when users need to transmit data. Similar to the DSR and AODV protocols, the CROMC protocol is also divided into two stages: the establishment stage of route and the steady state of data transmission to the base station. Figure 2 presents the procedure of CROMC protocol.

Figure 2.

The procedure of CROMC protocol.

During the establishment phase, source nodes broadcast a routing request (RREQ), and SUs negotiate the spectrum, data transmission, and transmission power with each neighbor node. Then, the negotiation results and route information are transmitted to the next hop until they reach the base station. With the aid of the received information, the base station selects the route with the smallest weight that satisfies the routing capacity. Then, parameters, such as spectrum, data transmission rate, and transmission power, are transmitted to all the nodes along the selected route. Each node receives the corresponding parameters and adjusts its transmission and receiving channels, data transmission rate, and transmission power. Finally, each node performs data transmission. In the steady state, the source node transfers packets to the base station using the allocated spectrum, data transmission rate, and transmission power. It also boots the routing maintenance mechanism when transmission error occurs. In the aforementioned stages, all SUs must maintain an open state to receive information from parent nodes.

Route establishment

When a request for data transmission is generated from a source node to a destination node, the source node broadcasts a route and establish the RREQ, which includes the following information: source node ID, base station ID, data delivery rate, transmission power, mobile velocity, and the location of every node. In this process, the route weight of RREQ transmitted from the source node is set to 0. The mechanism of carrier-sense multiple access with collision avoidance MAC is adopted to broadcast RREQ to the neighbor nodes. Each node is unable to obtain the information of its neighbors. Thus, all nodes transfer RREQ to their neighbors with the highest power, which informs SUs as many as possible.

When the other SUs receive the RREQ, a spectrum negotiation with the parent node will be initiated. In this negotiation process, the nodes may apperceive the results according to the spectrum generated from the local and parent nodes. From equations (29) and (30), the frequency spectrum $F_{\mathrm{ij}}$ , $\mathrm{\Delta }{f}_{\mathrm{ij}}$ , and $P_{\mathrm{ij}}(f)$ are seeked based on the aforementioned spectrum selection and sub-band determination algorithm. If a spectrum is available, then the frequency of the parent, data transmission rate, and transmission power of SUs are added to the RREQ; meanwhile, the spectrum’s address is added to the routing records. After calculating the link weight using equation (32), each node updates the route weights and broadcasts them. If no negotiation spectrum is available, then the RREQ is discarded.

When the base station receives all route information, a suitable route that satisfies the routing capability and possesses the sum of the minimum routing weight will be selected as described in subsection “Routing.” A routing response will also be sent to the routing initiator. The routing response is composed of the following information: the selected route records, data transmission rate, transmission power, and packet number. After receiving the response, the root initiator will send the packet series via the selected route and channel in the steady transmission stage. All negotiation information and control packets are transmitted using CCC.

Steady transmission

The routing establishment stage, which is simpler than the steady-state stage, has two parts: data transmission and routing maintenance. All packets in the steady-state transmission phase are transmitted through DC.

Prior to transmitting data, the source node first divides the data into K packets and then sends the packets according to the allocated spectrum, data transmission rate, and transmission power. If the data cannot be received completely, then the packet is considered lost due to the delay in packet transmission. The possible reasons include a change in node speed or direction, PU access, network congestion, and hardware failure in the course of transmission. Therefore, confirmation information should be added during the data transfer process. When a packet is successfully received, the node will send an acknowledgment (ACK) message to the parent node, which confirms whether the packet has been sent successfully. After completing the data transmission process, the link is disconnected and the spectrum resource is released. Subsequently, the packets are transmitted through each node according to the selected routing, determined spectrum, data transmission rate, and transmission power until the base station.

When a route is broken, route maintenance is activated and the corresponding node acts as a new source node and rebuilds the route according to the aforementioned method, thereby continuing data transmission. All negotiation information and the control packet are retransmitted via CCC.

Performance analysis of CROMC

Computational complexity analysis

We analyze the proposed protocol to evaluate the computational complexity of CROMC. The SUs and the base station play different roles. Hence, computational complexity should be discussed separately.

For each SU, spectrum negotiation, including determining the optimal transmission power and the number of sub-bands of the available spectrum for every frequency point in the spectrum environment, will be executed among its neighbor nodes. For a given spectrum, the computational complexity of the optimal transmission power is $\mathrm{\Theta }(1)$ if the number of subcarriers is $\left|F\right|$ . Meanwhile, on the basis of the spectrum selection and sub-band determination algorithm, we can draw the conclusion that the computational complexity of the available spectrum depends mainly on the variety of selected algorithms. The computational complexity of the fast classification algorithm is $\mathrm{\Theta }(\left|F\right|·\log (|F|))$ . If each SU possesses neighbor node, then the total computational complexity is $\mathrm{\Theta }(S·\left|F\right|·\log (|F|))$ .

After receiving route information, the base station should rank the routing weights in ascending order and select a suitable route as the data transmission path. Therefore, in determining the appropriate route, the computational complexity of the base station is also depended on the complexity of the classification algorithm. Assume that $\eta$ reachable routes are available. Then, the computational complexity of the base station is also $\mathrm{\Theta }(\eta \log \eta )$ .

In summary, the computational complexity of the CROMC protocol is effective in polynomial time.

Effect of mobility

The impact of node mobility is inevitable in large mobile WSNs. For the proposed protocol, node mobility will affect the change in link weights. Consequently, the sum of possible route weights obtained during the route establishment stage will change over time. In this study, we analyze the impact of mobility on the change in link weight between two points.

Given a reference moment $t_0$ , the weight of link $(l_i,l_j)$ is ${\omega }_{\mathrm{ij}}(t_0)$ , the weight of link $(l_i,l_j)$ is ${\omega }_{\mathrm{ij}}(t_0+\mathrm{\Delta }t)$ over $\mathrm{\Delta }t$ , and the change in the weight of link $(l_i,l_j)$ is described as follows

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{\Delta }{w}_{\mathrm{ij}}=w_{\mathrm{ij}}(t_0+\mathrm{\Delta }t)-w_{\mathrm{ij}}\left(t_0\right)}\\ \multicolumn{1}{c}{=\sqrt{{\lambda }_1{\lambda }_2{N}_0^2(N_0+Q_i(t_0+\mathrm{\Delta }t))\frac{P_{\mathrm{ij}}(t_0+\mathrm{\Delta }t)}{R_{\mathrm{ij}}(t_0+\mathrm{\Delta }t)}(\frac{1}{R_{\mathrm{ij}}(t_0+\mathrm{\Delta }t)}+\frac{1}{R_{\mathrm{proc}}})}}\\ \multicolumn{1}{c}{-\sqrt{{\lambda }_1{\lambda }_2{N}_0^2(N_0+Q_i\left(t_0\right))\frac{P_{\mathrm{ij}}\left(t_0\right)}{R_{\mathrm{ij}}\left(t_0\right)}(\frac{1}{R_{\mathrm{ij}}\left(t_0\right)}+\frac{1}{R_{\mathrm{proc}}})}}\end{array}$$

(35)

Assume that $Q_i(t_0)=Q_i(t_0+\mathrm{\Delta }t)$ and $R_i(t_0)=R_i(t_0+\mathrm{\Delta }t)$ , that is, no new node $c_k$ is set as the next hop of node $c_i$ to send data, and the influence of the change in the spectrum environment on the transmission rate of the link can be disregarded, then

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{\Delta }{w}_{\mathrm{ij}}=\sqrt{{\lambda }_1{\lambda }_2{N}_0^2(N_0+Q_i)\frac{1}{R_{\mathrm{ij}}}(\frac{1}{R_{\mathrm{ij}}}+\frac{1}{R_{\mathrm{p}}})}}\\ \multicolumn{1}{c}{\times (\sqrt{P_{\mathrm{ij}}(t_0+\mathrm{\Delta }t)}-\sqrt{P_{\mathrm{ij}}\left(t_0\right)})}\\ \multicolumn{1}{c}{=\sqrt{{\lambda }_1{\lambda }_2{N}_0^2(N_0+Q_i)\frac{1}{R_{\mathrm{ij}}}(\frac{1}{R_{\mathrm{ij}}}+\frac{1}{R_{\mathrm{p}}})}}\\ \multicolumn{1}{c}{\times (\sqrt{\frac{{\gamma }_{\mathrm{LU}}N{I}_j}{A_{\mathrm{ij}}(t_0+\mathrm{\Delta }t)G}}-\sqrt{\frac{{\gamma }_{\mathrm{ULU}}N{I}_j}{A_{\mathrm{ij}}\left(t_0\right)G}})}\\ \multicolumn{1}{c}{=\sqrt{{\lambda }_1{\lambda }_2{N}_0^2(N_0+Q_i)\frac{1}{R_{\mathrm{ij}}}(\frac{1}{R_{\mathrm{ij}}}+\frac{1}{R_{\mathrm{p}}})\frac{{\gamma }_{\mathrm{LU}}N{I}_j}{G}}}\\ \multicolumn{1}{c}{\times (\frac{1}{\sqrt{A_{\mathrm{ij}}(t_0+\mathrm{\Delta }t)}}-\frac{1}{\sqrt{A_{\mathrm{ij}}\left(t_0\right)}})}\\ \multicolumn{1}{c}{=\sqrt{{\lambda }_1{\lambda }_2{N}_0^2(N_0+Q_i)\frac{1}{R_{\mathrm{ij}}}(\frac{1}{R_{\mathrm{ij}}}+\frac{1}{R_{\mathrm{p}}})\frac{{\gamma }_{\mathrm{LU}}N{I}_j}{G·{10}^{\frac{A_0}{10}}}}}\\ \multicolumn{1}{c}{\times (\frac{1}{d_{\mathrm{ij}}{(t_0+\mathrm{\Delta }t)}^{\frac{\alpha }{2}}}-\frac{1}{d_{\mathrm{ij}}{\left(t_0\right)}^{\frac{\alpha }{2}}})}\\ \multicolumn{1}{c}{=U(d_{\mathrm{ij}}{(t_0+\mathrm{\Delta }t)}^{-\frac{\alpha }{2}}-d_{\mathrm{ij}}{\left(t_0\right)}^{-\frac{\alpha }{2}})}\end{array}$$

(36)

where $N{I}_j$ denotes the noise and interference received by the node of $j$ .

To clearly understand the equation, we convert it into polar coordinate. Let $c_i$ be located at the original point. The coordinates of the node of $c_j$ is $({\rho }_j,0)$ at time $t_0$ , $V_{\mathrm{ij}}$ is the relative rate between nodes $c_i$ and $c_j$ , and ${\theta }_{\mathrm{ij}}$ is the relative movement direction of nodes $c_i$ and $c_j$ as shown in Figure 3. The distance of nodes $c_i$ and $c_j$ may be defined as follows

$$d_{\mathrm{ij}}{\left(t_0\right)}^{-\frac{\alpha }{2}}={\rho }_j^{-\frac{\alpha }{2}}$$

(37)

$$\begin{array}{c}\multicolumn{1}{c}{d_{\mathrm{ij}}{(t_0+\mathrm{\Delta }t)}^{-\frac{\alpha }{2}}}\\ \multicolumn{1}{c}{={({\rho }_j^2+{\left(v_{\mathrm{ij}}\mathrm{\Delta }t\right)}^2-2{\rho }_j{v}_{\mathrm{ij}}\mathrm{\Delta }t\cos (\pi -{\theta }_{\mathrm{ij}}))}^{-\frac{\alpha }{4}}}\\ \multicolumn{1}{c}{={({\rho }_j^2+{\left(v_{\mathrm{ij}}\mathrm{\Delta }t\right)}^2+2{\rho }_j{v}_{\mathrm{ij}}\mathrm{\Delta }t\cos {\theta }_{\mathrm{ij}})}^{-\frac{\alpha }{4}}}\end{array}$$

(38)

Figure 3.

The sketch of two nodes in the polar coordinate.

In equation (38), ${\theta }_{\mathrm{ij}}$ and $V_{\mathrm{ij}}$ can be separately described as

$${\theta }_{\mathrm{ij}}=\arccos \left(\cos ({\theta }_j-{\theta }_i)\right)$$

(39)

$$v_{\mathrm{ij}}=\sqrt{v_i^2+v_j^2-2v_i{v}_j\cos {\theta }_{\mathrm{ij}}}$$

(40)

In equation (40), the variables $V_i$ , $V_j$ , and ${\theta }_{\mathrm{ij}}$ are toxicity independence. The probability density function of $V_{\mathrm{ij}}$ can be obtained by transforming multiple variables with the uniform distribution of $V_i$ , $V_j$ , and ${\theta }_{\mathrm{ij}}$ between $[0,\pi ]$ . We have excluded unnecessary details in this article. Therefore, the distribution of $V_{\mathrm{ij}}$ can be obtained by converting the three metavariables, which are described as follows

$$f_{V_{\mathrm{ij}}}\left(v_{\mathrm{ij}}\right)={\int }_0^{v_{max}}{\int }_0^{v_{max}}{f}_{V_i{V}_j{V}_{\mathrm{ij}}}(v_i,v_j,v_{\mathrm{ij}})d{v}_{i}d{v}_j$$

(41)

where $f_{V_i{V}_j{V}_{\mathrm{ij}}}(v_i,v_j,v_{\mathrm{ij}})$ can be represented as

$$\begin{array}{c}\multicolumn{1}{c}{f_{V_i{V}_j{V}_{\mathrm{ij}}}(v_i,v_j,v_{\mathrm{ij}})=\frac{f_V\left(v_i\right)f_V\left(v_j\right)}{\pi }\left|\frac{\partial {\theta }_{\mathrm{ij}}}{\partial v_{\mathrm{ij}}}\right|}\\ =\frac{1}{\pi v_{max}^2}\frac{1}{\sqrt{1-\left(\frac{v_i^2+v_j^2-v_{\mathrm{ij}}^2}{2v_i{v}_j}\right)}}·\frac{v_{\mathrm{ij}}}{v_i{v}_j}\hfill \end{array}$$

(42)

For equation (41), it is difficult to directly obtain the numerical results. Therefore, the probability density function of $v_{\mathrm{ij}}$ can be simulated to obtain the change range of $v_{\mathrm{ij}}$ . Existing research shows that the random variable of $v_{\mathrm{ij}}$ is between 0 and 80 m/s when $v_{max}=40\mathrm{m}/\mathrm{s}$ , in which the interval is set to $[0,2\times v_{max}]$ . From the preceding analysis, we can obtain the following formula

$$\begin{array}{c}\multicolumn{1}{c}{E\left\{d_{\mathrm{ij}}{(t_0+\mathrm{\Delta }t)}^{-\frac{\alpha }{2}}\right\}}\\ \multicolumn{1}{c}{=\frac{1}{\pi }{\int }_0^{2v_{max}}{\int }_0^{\pi }\begin{array}{c}{({\rho }_j^2+{\left(v_{\mathrm{ij}}\mathrm{\Delta }t\right)}^2+2{\rho }_j{v}_{\mathrm{ij}}\mathrm{\Delta }t\cos {\theta }_{\mathrm{ij}})}^{-\frac{\alpha }{4}}\hfill \\ \multicolumn{1}{c}{f_{V_{\mathrm{ij}}}\left(v_{\mathrm{ij}}\right)d{\theta }_{\mathrm{ij}}d{v}_{\mathrm{ij}}}\end{array}}\\ \multicolumn{1}{c}{=-\frac{1}{\pi }{\int }_0^{2v_{max}}{\int }_0^{\pi }\begin{array}{c}{({\rho }_j^2+{\left(v_{\mathrm{ij}}\mathrm{\Delta }t\right)}^2+2{\rho }_j{v}_{\mathrm{ij}}\mathrm{\Delta }t\cos {\theta }_{\mathrm{ij}})}^{-\frac{\alpha }{4}}\hfill \\ \multicolumn{1}{c}{f_{V_{\mathrm{ij}}}\left(v_{\mathrm{ij}}\right)\stackrel{-1}{\sin }{\theta }_{\mathrm{ij}}d\cos {\theta }_{\mathrm{ij}}d{v}_{\mathrm{ij}}}\end{array}}\\ \multicolumn{1}{c}{=\frac{1}{\pi }{\int }_0^{2v_{max}}{\int }_{-1}^1\begin{array}{c}{(1-x^2)}^{-\frac{1}{2}}{({\rho }_j^2+{\left(v_{\mathrm{ij}}\mathrm{\Delta }t\right)}^2+2{\rho }_j{v}_{\mathrm{ij}}\mathrm{\Delta }\mathrm{tx})}^{-\frac{\alpha }{4}}\hfill \\ \multicolumn{1}{c}{f_{V_{\mathrm{ij}}}\left(v_{\mathrm{ij}}\right)\mathrm{dxd}{v}_{\mathrm{ij}}}\end{array}}\end{array}$$

(43)

Evidently, the factor of $({\rho }_j^2+(v_{\mathrm{ij}}\mathrm{\Delta }t{)}^2+2{\rho }_j{v}_{\mathrm{ij}}\mathrm{\Delta }\mathrm{tx})$ is monotonically deceasing from −1 to 1, then

$$\begin{array}{c}\multicolumn{1}{c}{E\left\{d_{\mathrm{ij}}{(t_0+\mathrm{\Delta }t)}^{-\frac{\alpha }{2}}\right\}}\\ \multicolumn{1}{c}{\le \frac{1}{\pi }{\int }_0^{2v_{max}}{\int }_{-1}^1{(1-x^2)}^{-\frac{1}{2}}{({\rho }_j-v_{\mathrm{ij}}\mathrm{\Delta }t)}^{-\frac{\alpha }{2}}{f}_{V_{\mathrm{ij}}}\left(v_{\mathrm{ij}}\right)\mathrm{dxd}{v}_{\mathrm{ij}}}\\ \multicolumn{1}{c}{<\frac{1}{\pi }{\int }_0^{2v_{max}}{\int }_{-1}^1{(1-x^2)}^{-\frac{1}{2}}{({\rho }_j-v_{\mathrm{ij}}\mathrm{\Delta }t)}^{-\frac{\alpha }{2}}\mathrm{dxd}{v}_{\mathrm{ij}}}\end{array}$$

(44)

Considering $2<\alpha <4$ and ${\rho }_j>>v_{\max }\mathrm{\Delta }t$ , then

$$\begin{array}{c}\multicolumn{1}{c}{E\left\{d_{\mathrm{ij}}{(t_0+\mathrm{\Delta }t)}^{-\frac{\alpha }{2}}\right\}}\\ \multicolumn{1}{c}{<\frac{-2}{(2-\alpha )\mathrm{\Delta }t}[{({\rho }_j-2v_{max}\mathrm{\Delta }t)}^{\frac{2-\alpha }{2}}-{\rho }_j^{\frac{2-\alpha }{2}}]}\\ \multicolumn{1}{c}{=\frac{-2{\rho }_j^{\frac{2-\alpha }{2}}}{(2-\alpha )\mathrm{\Delta }t}[{(1-\frac{2v_{max}\mathrm{\Delta }t}{{\rho }_j})}^{\frac{2-\alpha }{2}}-1]}\\ \multicolumn{1}{c}{\approx 2v_{max}{\rho }_j^{-\frac{\alpha }{2}}}\end{array}$$

(45)

which can yield the following formula

$$\begin{array}{c}\multicolumn{1}{c}{E\left\{\mathrm{\Delta }{w}_{\mathrm{ij}}\right\}=U·E\{d_{\mathrm{ij}}{(t_0+\mathrm{\Delta }t)}^{-\frac{\alpha }{2}}-d_{\mathrm{ij}}{\left(t_0\right)}^{-\frac{\alpha }{2}}\}}\\ \multicolumn{1}{c}{<U{\rho }^{-\frac{\alpha }{2}}(2v_{max}-1)}\end{array}$$

(46)

The preceding analysis shows that data are transmitted through the route that satisfies routing capacity. Moreover, the impact of node mobility on link weight is not particularly evident and can be disregarded. Therefore, the CROMC protocol can be preferably adapted to wireless mobile sensor networks.

Results and discussion

Simulation parameter

In this section, we describe the simulation parameters used in our experiment as shown in Table 2.

Table 2.

The simulation parameters in this study.

Parameter	Value
Network size	$5000\times 5000{\mathrm{m}}^2$
Source node region	$[0,1000]\times [0,1000]$
PU	25
SU	50
Spectrum	3.5–3.75 GHz
Bandwidth of carrier	0.5 MHz
Bandwidth of common channel	2 MHz
Noise spectrum density	–180 dBm/Hz
Subcarrier of PU	10
Subcarrier of SU	$<5$
SNR of PU	10 dB
SNR of SU	5 dB
Maximum power of PU	0.1 W
Maximum power of SU	0.05 W
Data	500 kbit
Arrival rate of PU	6/h–50/h
Service time of PU	120 s

PU: primary user; SU: secondary user; SNR: signal-to-noise ratio.

In addition, the parameters $R_p,E_p,{\lambda }_1,{\lambda }_2,k_1,\mathrm{and}{k}_2$ are set to 100 Mbps, 10–5 J/bit, 0.5, 0.5, 0.5, and 0.5, respectively.

The Cauchy–Schwarz power of $\eta$ should be analyzed prior to accessing the CROMC protocol. Equations (29) and (30) show that the Cauchy–Schwarz power of $\eta$ is related to the transmission success rate and transmission power of the spectrum after negotiation. Figure 4 presents the detailed experimental results of the average transmission success rate with different $\eta$ values after 50 times.

Figure 4.

Values of success rate for different values of $\eta$ .

As shown in Figure 4, the success rate value increases gradually with an increase in $\eta$ and exhibits a steady change trend when $\eta >6000$ . Therefore, to obtain a high transmission success rate, the value of $\eta$ is set between 6000 and 7000 in this study.

Simulation results

Figure 5 illustrates the detailed results with different $\eta$ values. As shown in Figure 5, the total energy consumption of $\eta$ decreases sharply with a gradual increase in average throughput. The possible reason for this result is that a low throughput leads to a poor spectrum condition of communication, which consumes more power. In addition, a large power of $\eta$ requires more power with the same throughput.

Figure 5.

Vaules of total energy comsumption for different values of average throughput with a threshold of $\eta$ between 6000 and 7000.

To study the routing capability of CROMC, different hops of CROMC are presented in Figure 6. Notably, the average distance between the source node and the base station in the stimulation scenario is approximately 3000 m, and the maximum distance of one hop is less than 1000 m. Four hops are executed from the source station to the base station under this scenario. Therefore, our proposed method has fewer hops.

Figure 6.

Percentage of different hops.

To evaluate the performance of CROMC, we investigated the average throughput and the total power consumption of the user with an increment of 50 between 100 and 500. Figures 7 and 8 present the different values of $\eta$ for the effects of average throughput and total power consumption, respectively. As shown in Figures 7 and 8, the average throughput and total consumption change slightly with an increase in $\eta$ . In Figure 7, the average throughput is 200 Kbps because the multi-hop is a continuous transmission system. The average throughput can be calculated based on transmission rate per hop and hops. Compared with the transmission route with an average of 4–5 hops, the transmission rate of each hop is approximately 1 Mbps.

Figure 7.

Vaules of average throughput for different values of $\eta$ with a threshold between 6000 and 7000.

Figure 8.

Vaules of total energy consumption for different values of $\eta$ with a threshold between 6000 and 7000.

The maximum number of subcarriers in each hop cannot exceed 5 according to the simulation parameters. The maximum transmission rate modulated by M = 1/2 is 1 Mbps. Hence, the transmission rate of each hop can basically reach the maximum number of subcarriers in the simulation scenario, that is, the proposed protocol can achieve the maximum average throughput.

As shown in equations (31) and (32), the corresponding link weights present energy consumption balance and unbalance, which indicates the corresponding simulation results in Figures 7–9. From these figures, we can clearly see that the average throughput of the system will be reduced and the total power consumption will be increased. The primary reason is that the routing with the better spectrum is skipped when energy consumption balance is considered. Moreover, the standard deviation of energy (as shown in Figure 9) is relatively small after transmission under this circumstance, that is, the residual energy of all SUs in the network is basically equal.

Figure 9.

Vaules of the energy standard deviation for different values of $\eta$ with a threshold between 6000 and 7000.

To evaluate the performance of CROMC, we further compare the results of CROMC with the two other latest methods on multichannel sensor networks. The first one is EFP, ¹² which is based on a hop-by-hop transport scheme and seeks the minimum power consumption, the other one is HMCA, ¹⁴ which is based on hybrid TDMA techniques and seeks the maximum throughput under the premise of delay control. Similar to the above analysis, $\eta =6500$ is used. The detailed descriptions are presented in Figures 10 and 11, respectively.

Figure 10.

Results comparison of CROMC, EFP, and HMCA using hops.

Figure 11.

Results comparison of CROMC, EFP, and HMCA using average throughput.

From Figure 10, we can clearly see that our proposed method achieved a high proportion of fewer hops, when the number of hops is 3 and 4. Especially, the highest percentage of the number of hops is obtained when the number of hops is 4, which is 54.5%. And, the highest percentage of the number of hops of EFP and HMCA is obtained when the number of hops is 5, which is 36% and 42.2%, respectively. It shows that CROMC is superior to the two methods on the number of hops. This is because one of the goals of CROMC is to minimize the number of transmission hops.

In Figure 11, we study how the method behaves in terms of total energy consumption and let average throughput from $1.15\times {10}^5$ to $2.0\times {10}^5\mathrm{bps}$ with a $0.1\times {10}^5$ increment. With the gradual increase of average throughput, the total energy consumption decreases. With the gradual increase of average throughput, the total energy consumption decreases. HMCA achieves the larger energy consumption value, because HMCA is mainly used to solve the problem of maximum throughput within the allowable range of delay, and the percentage of the number of large hops is more than that of CROMC, as shown in Figure 10; therefore, the higher value of power consumption is achieved. EFP obtains the higher power consumptions value than that of CROMC, the reason is that the mobility of nodes increases the power consumption. Notably, the ratio of the number of large hops of EFP is higher than that of CROMC, as shown in Figure 10. CROMC considers the minimum number of hops and the minimum power consumption from the perspective of global network, so it achieves the minimum energy consumption with the increase of throughput, which is as shown in Figure 11.

Conclusion

Spectrum selection plays an important role in the research on routing protocols. Realizing on-demand information transmission with limited spectrum resources in large-scale wireless mobile sensor networks has always been a popular topic among researchers.

To maximize the throughput and lifetime of a network for on-demand data transmission under spectrum resource and hardware constraints, CROMC, which is based on the cross-layer design, was proposed. Our proposed protocol can effectively select the communication spectrum between links via spectrum negotiations among cluster heads, determine data transmission rate and transmission power between links, and transmit data by selecting minimum weight routing from the base station. The experimental result shows that our proposed method can effectively maximize network throughput and lifetime with lower computational complexity, which is also widely applied to large-scale wireless mobile sensor networks.