Performance Analysis for Multihop Cognitive Radio Networks With Energy Harvesting by Using Stochastic Geometry

Cognitive multihop relaying has been widely considered for device-to-device (D2D) communications for applications in the physical layer of the Internet of Things. In this article, we construct a multihop cellular D2D communications system model with energy harvesting (EH) in underlay cognitive radio networks. The locations of primary user equipments (PUEs) and cellular base stations are considered as a Poisson point process in this model. The transmit power of secondary devices is collected from the power beacon with time-switching EH policy. Two charging policies for different applications are considered in this article. Then, the end-to-end outage probability analysis expressions of these two scenarios for the transmission scheme subject to interferences from PUEs are derived. The optimal harvesting time ratio is obtained to get the maximum capacity for end-to-end D2D communications. The analytical results are validated by performing the Monte Carlo simulation of the end-to-end outage probability, which is based on the half-duplex transmission scheme. The results of this article provide a potential pathway to reduce reliance on grid or battery energy supplies and, hence, further strengthen the benefits for the environment and deployment of future smart devices.


I. INTRODUCTION
The Internet of Things (IoT) and the Tactile Internet have become essential research directions to accelerate the development of fifth generation (5G) mobile networks and beyond.IoT is a promising technology which aims to revolutionize and connect the global world via heterogeneous smart devices through seamless connectivity [1].Massive IoT applications require an enormous number of connected smart devices, such as deployments in shipping environments, smart-homes (buildings), smart-cities, smart energy systems, and agricultural monitoring environments, etc., which need to update regularly to the cloud with a low end-to-end cost [2]- [5].
Cognitive radio (CR) technology is available to provide the opportunity for unlicensed users (secondary users) to share the wireless spectrum without the need for expensive spectrum licenses as long as the secondary users can protect the data of licensed users (primary users) [6].Existing wireless networks such as wireless sensor networks also benefit from CR technology by integrating it into their existing infrastructure.The CR network has three paradigms: underlay, overlay and interweave [7].In the underlay mode, if the interference caused by the transmission frequency of the secondary device to the primary device is strictly controlled, the secondary device can transmit with the same frequency as the primary one [8]- [10].The overlay pattern allows the secondary devices to transmit by adopting the same frequency spectrum as the primary devices.However, the premise is that the channel state information is known between the primary and secondary devices, and secondary devices use part of transmit power to communicate with each other; in the meanwhile, the remaining transmit power of the secondary devices is used to support the transmission of primary devices.The authors of [11] investigated simultaneous wireless information and power transfer in a cooperative overlay spectrum sharing system, and exact expressions for user outage probability for primary users was also provided.In addition, in interleaved communication systems, CR can also transmit signals using spectral holes so as not to interfere with other communications [12].
Device-to-device (D2D) communications has been considered as one of the key technologies in a 5G cellular network, and relates to direct transmission between devices [13]- [16].D2D communications can improve spectrum efficiency, reduce power consumption, and efficiently offload traffic from the base station/access points [17].In [18], a cross-cell fractional reuse-based frequency resource multiplexing scheme was proposed for multi-cell D2D communications to reduce interference between adjacent cells.In order to obtain a better access probability, secondary users are treated as relays to enhance the primary users in [19].In [20], the authors optimized the transmission rate of the D2D users when modelling D2D users as cognitive secondary users.A full-duplex relay-assisted D2D communications system was proposed in [21], and the exact closed-form expression for the outage probability was obtained.In [22] the energy efficiency of the D2D users has been maximized according to the minimum rate requirement of the D2D users and the cellular users.In addition, cooperative D2D communications in an uplink cellular network was investigated in [23]; and the optimal spectrum and power allocation were obtained to maximize the total average achievable rate.From the perspective of IoT network architecture, the exchange of information between two IoT devices usually requires relay assistance [24].Therefore, one of the common evaluations for the performance of D2D communication systems is to undertake end-to-end performance analysis.In [25], the authors have quantified the system throughput and energy efficiency with the average transmission time to investigate the end-toend outage performance.In [26], the authors stated that the connectivity of a path can be used for the determination of the maximum end-to-end outage probability in the context of the route selection.
Energy harvesting (EH) is an emerging technology for enabling green, sustainable, and autonomous wireless networks.In particular, radio frequency (RF) energy harvesting provides key benefits in wireless transmission.However, one of the barriers to connect massive smart devices is supplying sufficient energy to operate the network in a self-sufficient manner whilst maintaining the quality of service [27].Recent research has shown that a combination of different energy harvesting schemes, such as time splitting, power splitting, and antenna switching, can be utilized to collect the energy for the devices [28].The time-splitting protocol means the receiver at the device switches over time between harvesting energy device and decoding information, whereas the power splitting protocol means that part of the received signal is used for EH and the rest is used for information processing in [29], [30].In this paper, a time-splitting scheme is employed to collect energy of the D2D secondary devices since such time splitting is known to require less complex circuitry [31].Two charging strategies are proposed based on the time-splitting protocol, one for identical charging channels and the other for varying charging channels.Both of these charging strategies can effectively charge the device during the cognitive massive D2D connections based on different deployment scenarios.Furthermore, these two charging strategies are able to act as the baselines, which enable the development of a high energy-efficiency network in the near future, such as one having the ability to support battery-free IoT devices [32].In [33], the authors proposed combined energy management for queuing control to deal with the delay requirements of tactile communication in the presence of energy constraints on IoT devices.
Currently, stochastic geometry is an important mathematical tool for analysing large scale ad hoc, cognitive and cellular networks [34].In [35], a model for cognitive D2D communications was presented using RF energy harvesting from the ambient interference in a multi-channel cellular network, and stochastic geometry was used to analyse and evaluate the outage probabilities of the system.The authors of [36] investigated a power transfer model and an information signal model to realize energy harvesting and secure information transmission.Energy harvesting applications with multi-hop communications and optimization of energy harvesting time have also not been considered.
In this paper, we model a multi-hop D2D cognitive radio network with a time-splitting energy harvesting scheme.The Distance operation transmit power of secondary devices is determined by jointly considering the mutual constraint of the preset maximum transmit power, peak interference power and the amount of energy harvesting.And then, the end-to-end outage probability of the proposed multi-hop D2D cognitive system is analyzed.
The main contributions are listed as below: • Building a multi-hop cognitive D2D communications system model with energy harvesting, where the locations of the PUEs and primary base stations (PBSs) are represented by a homogeneous Poisson point process (PPP).• Proposing two charging policies, one is for identical charging channels and the other is for varying charging channels.After that, this article analyzes the end-to-end outage probability for the multi-hop CR D2D communications by considering the interference from the PUEs and PBSs.• Deriving the optimal charging time ratio to maximize the end-to-end transmission rate of the proposed system.• Performing Monte Carlo simulations to validate the proposed system, and confirming that the simulation results match with the theoretical expressions.
The structure of the remaining part of the paper is organised as follows: Section II introduces the proposed cellular D2D communications with energy harvesting in cognitive radio networks.Section III presents two energy harvesting strategies regarding different scenarios.Section IV provides the theoretical analysis of end-to-end outage probability.Section V shows the optimisation of energy harvesting time ratio.Section VI presents all numerical simulation results along with discussions to verify the rationality of the proposed system.All outcomes of this study are summarized in Section VII, and abbreviations and notifications for the paper are listed in Table I.

II. SYSTEM MODEL
A cognitive radio multi-hop D2D communications system with energy harvesting is shown in Fig. 1, where the signal is sent from the secondary source device D 1 to the secondary destination device D N +1 with N hops by adopting the decodeand-forward transmission scheme.The signal is transmitted from secondary devices to PBSs and PUEs and the signal received at the secondary devices from PBSs and PUEs are treated as interference.The locations of the PBSs and the PUEs are modelled as homogeneous PPPs Φ B and Φ C with densities λ B and λ C , respectively.All secondary devices are equipped with a half-duplex antenna, so that each secondary device cannot simultaneously transmit and receive the signal.We assume that all secondary devices are powered by a fixed power beacon (PB).
We assume independent Rayleigh fading channel models with path loss.The channel coefficients can be expressed as , where d m,n and α represent the distance and the pathloss exponent between two devices, m and n, respectively.β m,n is a complex Gaussian random variable with unit variance.Therefore, |h m,n | 2 is the channel gain and m,n is defined as the average channel power.Assuming that before the signal is passed to the next secondary device, the current secondary device and all remaining secondary devices receive energy from the PB.Moreover, since the time splitting scheme is used, the EH phase takes place during time ratio τ i (0 < τ i < 1), where i is the index of the secondary devices.In this paper, we assume The harvested power at the ith secondary device is given by where η ∈ (0, 1) denotes the energy harvesting efficiency dependent on the design of an efficient rectifier [37] and j represents the number of times of charging at the ith secondary device.P T is the transmit power at the PB.d p,i is the distance between the PB and ith secondary device.
In actual deployment, the transmitted signal power cannot exceed the rated power P max , and if it exceeds P max , it can only be transmitted at P max .For the underlay CR networks, the secondary device uses the same spectrum band as the primary device to transmit signals, therefore, the transmit power of the secondary device must be less than the peak interference power at the PBSs.Thus, it is necessary to consider comprehensively the constraints of the peak interference power I th at the PBSs, the rated power P max and the total harvested transmit energy P Di .The transmit power of the ith secondary device should satisfy the following constraint where d i,b and h i,b denote the distance and the channel coefficient between the ith secondary device and the PBSs, respectively.The received signal at D i from D i−1 can be expressed as where h i−1,i and d i−1,i represent the coefficient of the channel and the distance between two secondary devices, respectively; h c,i and d c,i denote the channel coefficient and the distance between PUEs and the ith secondary device.x i−1 is the signal sent from D i−1 to D i , and x c is the interference signal from the PUEs.P c denotes the transmit power of the PUEs.n i is the additive white Gaussian noise (AWGN) at D i with variance σ 2 n , which is normalized to unity.As a result, the SINR at the ith secondary device can be expressed as The proposed EH strategies will be presented in the next section.

III. ENERGY HARVESTING STRATEGIES
In this section, two charging scenarios are proposed.The effect of charging link on the energy term P Di are considered.
• Case1: identical charging channels For this case, the channels between PB and the ith secondary device are assumed to be unchanged regardless of the hops, h p,i,j = h p,i , ∀j ∈ {1, 2, • • •, N+1}, which can be considered as block fading channels [38].The cumulative distribution function (CDF) of P Di is The probability density function (PDF) of P Di is obtained as • Case2: varying charging channels The channels between PB and the ith secondary device are changed with hops in this case.This application scenario can be used in the real-time geometry-based channel emulator in [39].The CDF of P Di can be expressed as follows where according to the definition of the CDF of the regularized Gamma distribution Γ(i) , and γ(i, z K ) is the lower incomplete Gamma function.The term K = ητ Pt and Γ(i) = (i − 1)! is the Gamma function.Then, the PDF of P Di can be written as Based on the proposed system model with different EH strategies, the theoretical analysis of the end-to-end outage probability will be given in the next section.
IV. OUTAGE PROBABILITY ANALYSIS Based on the above charging scenarios proposed in the last section, we derive the end-to-end outage probability for these two EH scenarios.The definition of outage event between the (i − 1)th secondary device and ith secondary device is given by where R = 2 R th − 1, and R th is the target rate.

+1
, the CDF of A can be first obtained by using the probability generating functional lemma as where ψ = 2πρ B α Γ( 2 α ), and Γ(•) is the gamma function.Therefore, the PDF of A can be given as The CDF of D can be expressed as . Therefore, the PDF of D can be obtained as From the above mathematical derivation of SINR γ Di , the outage probability in (9) can be obtained as The outage probability between i − 1th secondary device and ith secondary device for EH case 1 and case 2 can be obtained as The derivations of W Ψ 1 (x, y), W Ψ 2 (x, y), W Ψ 3 (x, y), Ψ ∈ {C 1 , C 2 }, can be found in Appendix I.Although the general closed-form expressions for case 1 and case 2 cannot be given, the numerical results can be implemented by MATLAB [40].By using the DF scheme, the end-to-end outage probability of the proposed system can be obtained as The discussion of the optimization of EH time for two EH strategies will be given in the next section.

V. OPTIMAL ENERGY HARVESTING TIME
The EH time plays an essential role in the time-switching scheme because of its influence on system capacity.From the perspective of providing the potential pathway to reduce battery energy supplies, the EH time for each independent hop should be optimized when the system achieves maximum capacity.The nonlinear cost function for optimal time ratio can be mathematically expressed as follows, where, the capacity of the system is given as A. The optimization of EH time ratio for case 1 In case 1, we assume that the charging links remain unchanged for each charging process.The second order derivative of Cap can be obtained as where .

B. The optimization of EH time ratio for case 2
In case 2, the charging links are changed for each charging process, therefore, we can obtain the second order derivative of Cap as follows where .
Through mathematical operations for both two cases above, it is worth noting that the term τ − 1 is a negative and the other term is positive.Therefore, the above second order derivatives of the objective function is less than zero so the unique concave point can be found for both cases.The optimal harvesting time ratio is determined by the CVX function for different numbers of hops as shown in Table II and Table III.Target rate [bits/sec/Hz] VI.SIMULATION RESULTS In this section, the simulation and theoretical results are given to verify the accuracy of our analysis.In excess of 10 5 independent Monte Carlo simulations introduce the following results.First, the noise power is assumed to be unity, and the path loss exponent of α equals to 4. The maximum rated transmit power P max to noise ratio of each secondary device is set to 50 dB 1 .Additionally, the interference power cannot exceed I th and is set to equal to 0.05.The energy harvesting efficiency is assumed as 0.5.The location of all secondary devices are fixed from (−4, 0) to (2, 0), respectively.All theoretical results are represented by a curve with denoted T , and simulation results are represented by various hollow shapes denoted as S.

A. Results for case 1
Fig. 2 demonstrates the variation of end-to-end outage probabilities with respect to varying system target rates.Both 1 In reality, this is a tuning variable which can be adjusted, either higher or lower, according to real-world applications requirement.simulations and theoretical results match.Furthermore, the power beacon supplies 30 dB power to the whole system continuously.The density of PUEs Φ C and the density of cellular base stations Φ B are both set to 1 × 10 −5 m −2 .The end-to-end outage probability is increasing while the target rate increases.As the number of hops increases, the end-to-end outage probability of the system decreases.When the target rate achieves 2 bits/sec/Hz, the end-to-end probability of 2-hop system is 0.2383, the end-to-end probability of 3-hop system is 0.0903, and the end-to-end probability of 5-hop system drops to 0.0285.In order to demonstrate the effect of the density of PBSs on the end-to-end outage probability, the simulation and theoretical results for 5 hops at Φ B = 1 × 10 −4 m −2 are also presented in this figure.Due to the density of PBSs increase, the interference power increases.As a result, the end-to-end probability increases.In order to assess the influences of energy harvesting time on the system performance of case 1, Fig. 3 indicates the variation of end-to-end outage probability in terms of the EH time ratio.The optimal EH time ratio and the corresponding end-to-end outage probability are marked in Fig. 3.This optimal EH time ratio was obtained in Section V.It shows that the end-to-end probability is reduced when the number of hops is increasing.For example, when τ = 0.6, the end-toend outage probability of 2-hop system is 0.0603, similarly, the end-to-end outage probability of 3-hop system and 5-hop system are 0.0244 and 0.0088, respectively.Beyond this point, end-to-end outage probability is increasing dramatically as the time ratio increases.When the harvesting time is increasing, the secondary devices can collect more energy.However, due to the constraints of peak interference power and rated power, only a proportion of the harvested energy is effectively used.Additionally, the charging time ratio gradually increases, the term 1 − τ will gradually decrease in (9), resulting in an increase in the target SINR and an increase in the probability of the final interruption.In Fig. 4, the simulation and theoretical results of the end-toend outage probability for each hop scenario are indicated with respect to the power variation of the power beacon.In general, when the power is more than 40 dB, the system end-to-end outage probability tends to be stable.It is because the transmit power of the secondary devices is limited by the interference power and the rated transmit power.  of cellular base stations Φ B are both set to 1 × 10 −5 m −2 .It is shown that the end-to-end outage probability is increasing, when the target rate is increasing.For instance, the end-toend probability of 2-hop system is 0.0973 when the target rate is 2 bits/sec/Hz, similarly, when the number of hops is increased to 3, the end-to-end probability is 0.0391 and when the number of hops is increased to 5, the end-to-end probability is reduced to 0.011.Moreover, a larger number of hops leads to a lower end-to-end outage probability trend.

B. Results for case 2
The increased PBSs density leads to an increased interference power, and hence, increases the end-to-end outage probability.
For instance, the end-to-end probability is higher while the density of PBSs is 1 × 10 −4 m −2 , compare to the density of PBSs is 1 × 10 −5 m −2 To assess the impact of the energy harvesting time ratio of system performance, Fig. 6 represents the variation of the end-to-end outage probability in terms of harvesting time ratio.The simulation and theoretical results are perfectly matched.When the harvesting time ratio is from 0.3 to 1, although the secondary devices can harvest more energy, the effective usable energy is limited due to the defined constraints in this paper.Moreover, the transmission time ratio is reduced so that the end-to-end outage probability increases dramatically due to increasing the target SINR in (9).In details, when the EH time ratio achieves 0.8, for the 2-hop, 3-hop and 5-hop, the end-to-end outage probability are 0.0409, 0.0168, and 0.0068, respectively.
In Fig. 7, the simulation and theoretical results of end-toend outage probability for each hop scenario are given as a function of the power variation of the power beacon.It clearly shows that by having a fixed power of the power beacon, the end-to-end probabilities of the proposed system are reduced with increasing the number of hops.The numerical integration results are correspond exactly with the simulation results.When the power is more than 40 dB, the system endto-end outage probability tends to be stable, which means that the system always maintains the best performance while other parameters are unchanged.Besides, the variation of the hop number has no effect upon this stabilized point.

VII. CONCLUSION
In this paper, a multi-hop underlying CR D2D communications system with energy harvesting was constructed.This proposed model took into account the influences of randomly distributed primary cellular user equipments and cellular base stations by applying a Poisson point process.The network architecture consisted of multi-hop secondary devices that collect transmit energy from the fixed power beacon with a time-switching energy harvesting policy.The mutual constraints of the peak interference power, the rated power and the harvested energy were considered in this study.Two energy harvesting scenarios were performed based on the developed system model to demonstrate the effects of variable charging links.Furthermore, the end-to-end outage probability analysis expression for the half-duplex transmission scheme subject to interference from PUEs and PBSs was derived in this paper.The analytical results were validated by performing Monte Carlo simulations of the end-to-end outage probability.Moreover, the optimal charging time ratio was obtained while the maximum end-to-end transmission rate was maintained for two cases.By analyzing these two cases, it can be concluded that the charging time ratio has the most significant impact on the system performance.

Fig. 1 .
Fig. 1.System model of the buffer-aided link selection in multi-relay cooperative networks.
where setting k = |h i−1,i | 2 at (a), so the PDF of k should be f k (k) = e −k and for (b), it holds for the probability generating functional; and then )fP D i (x)fA(y)dtdxdy+ )fP D i (x)fA(y)dtdxdy + )fA(y)fP D i (x)dtdydx + fP D i (x)fA(y)dtdxdy.

4 Fig. 2 .
Fig. 2. Comparison of theoretical and numerical end-to-end outage probability with target rate for case 1.

1 Fig. 3 .
Fig. 3.The comparison of theoretical and numerical the end-to-end outage probability with EH time ratio for case 1.

Fig. 4 . 4 Fig. 5 .
Fig.4.The comparison of theoretical and numerical the end-to-end outage probability with power of power beacon for case 1.

Fig. 5 2 Fig. 6 .
Fig.5shows the comparison of end-to-end outage probability for different densities of PUEs and PBSs versus the target rate.We assumed that the density of PUEs Φ C and the density

Fig. 7 .
Fig. 7. Comparison of theoretical and numerical end-to-end outage probability with power of the power beacon for case 2.