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Abstract

In this work, we propose a clustering-based multihop relaying with the partial relay selection scheme for an energy harvesting relaying network and analyze the performance in the framework of the decode-and-forward relaying and adaptive power splitting protocol over symmetric and asymmetric fading channel models. In particular, we analyze the system performance in terms of the outage probability, effective transmission rate, and throughput. Through extensive numerical analysis, we show that the proposed scheme can substantially outperform the conventional multihop relaying without clustering as well as direct transmission, which suggests that the proposed scheme can be used to extend the network coverage without any extra energy from the network. We also demonstrate that the proposed scheme can compensate for the performance loss due to poor radio frequency to DC conversion efficiency as well as path loss by exploiting the gain associated with multihop relaying as well as the diversity gain achieved through the partial relay selection scheme. Moreover, we investigate the relationship between the total number of relay nodes in the network and the number of hops and show that there is an optimal number of hops that can maximize the throughput for a given transmission power of the source. The effect of the asymmetric channels in our clustering-based multihop relaying is also investigated and it is revealed that the existence of Rician fading will help improve the throughput at the destination side, rather than the source side, as opposed to the conventional multihop relaying scenarios without clustering.

Keywords

Energy harvesting power splitting outage probability effective transmission rate throughput partial relay selection decode and forward

Introduction

For severely energy-constrained wireless networks such as mobile ad hoc wireless networks and wireless sensor networks, multihop transmission is a viable option to overcome wireless impairments and efficiently enhance network coverage. In the multihop relaying, the communication between the source and destination is established with the help of intermediate nodes that relay the information of the source to the destination.¹ For the energy-constrained relay nodes, relaying the information of the source could be subject to severe energy limitation, which may considerably affect the lifetime of their own network. A radio-frequency (RF) energy harvesting (EH) technique can be a viable option to prolong the lifetime of such energy-constrained wireless networks;² the relay can harvest energy from the information source to assist the source transmission as discussed by Zhang and colleagues.^3,4 Such a relaying scheme, where the intermediate nodes take part in relaying while exploiting the energy harvested from the signal transmitted by the same source, is analyzed in the works by Krikidis et al.⁵ and Nasir et al.^6,7

In the work by Krikidis et al.,⁵ a classical three-node cooperative network with an EH relay node is studied and analyzed in terms of the outage probability for a battery model with discretized levels and finite capacity. The outage probability and the ergodic capacity for the time switching (TS) EH and power splitting (PS) EH protocols in the framework of amplify-and-forward (AF) and decode-and-forward (DF) protocols are derived by Nasir et al.,^6,7 respectively. In the work by Ding et al.,⁸ the application of wireless information and power transfer (WIPT) to the cooperative networks with randomly located relays is investigated for the DF framework. In the work by Ju et al.,⁹ the outage probability expressions for the PS relaying and TS relaying protocols with given PS and TS ratios are derived in the framework of DF relaying, and the optimal ratios that maximize the transmission rate are determined. In the work by Atapattu et al.,¹⁰ closed-form expressions for the average signal-to-noise ratio (SNR), outage probability, and throughput are derived for the TS EH protocol in the framework of DF relaying, and an expression for the optimal TS ratio is obtained. The results are also extended to an opportunistic relay selection (ORS) scheme. In the work by Haghifam et al.,¹¹ closed-form expressions for the outage probability and the network throughput are derived in the framework of adaptive and non-adaptive power allocation schemes. Furthermore, a closed-form expression for the optimal relay position which can achieve the minimum outage probability is derived. Readers interested in the conventional two-hop EH network can find further details in the works by Krikidis et al.,¹² Gu and Aissa,¹³ and Zhong et al.¹⁴ and references therein.

All the above-mentioned studies, however, deal with the dual-hop relaying, which may not be necessarily practical for low-power wireless sensor networks where the communication link between the source and destination may not be supported by a single relay due to severe path loss, shadowing, and fading. For the conventional multihop relaying networks without EH (i.e. the networks with fully powered relays), extensive work has been done, which can be found in the works by Boyer et al.,¹⁵ Ribeiro et al.,¹⁶ Zhao and Valenti,¹⁷ Nguyen et al.,¹⁸ Li and Li,¹⁹ and Karagiannidis²⁰ and references therein. However, the studies on multihop relaying with WIPT are rather scarce; the performance of multihop relaying with EH relays is evaluated by Mao et al.²¹ under the framework of TS and PS EH protocols. Through numerical simulations, it is shown that in order to extend the network coverage, the TS along with AF relaying is a better combination. However, it does not provide any analytical expression for the outage probability or the achievable throughput. Apart from that, Mao et al.²¹ deal with multihop relaying as a wireless network of a cascaded point-to-point links and, as a consequence, the average throughput diminishes as the number of nodes increases toward infinity.²²

Motivated by the above observation, in this work, we propose a clustering-based multihop relaying with the partial relay selection (PRS) scheme for the EH relay networks, which can improve the performance of the multihop relaying with WIPT, by combining the benefits of the multihop relaying (i.e. gain achieved by the path loss reduction) and relay selection (i.e. diversity gain).^17,18,23 The main contributions of this work are summarized as follows:

We derive the closed-form expressions for the outage probability, effective transmission rate, and throughput for the proposed scheme in the framework of the DF relaying and adaptive power splitting (APS) protocol, considering both symmetric and asymmetric channel models. Here, the symmetric channel model refers to the case where all the wireless links are subject to the same fading statistics (e.g. all the links are Rayleigh fading), whereas the asymmetric channel model refers to the case where the different wireless links are subject to different fading (e.g. mixed Rician–Rayleigh or Rayleigh–Rician fading).

Based on our analytical results, we compare the effective transmission rate of the proposed scheme and the conventional multihop relaying approach without clustering²¹ and show that the proposed scheme can substantially outperform the conventional approach.

Furthermore, we demonstrate that the proposed scheme can compensate for the performance loss due to poor RF-to-DC conversion efficiency and could be used to extend the network coverage without any extra energy from the network.

We study the relationship between the number of the total relay nodes in the network and the number of hops and reveal that there is an optimal number of hops that can maximize the throughput for a given transmission power of the source.

We compare the throughput performance of the proposed scheme over various asymmetric fading channels and reveal the conventional observation that the multihop relaying networks over asymmetric fading channels can achieve maximum throughput in Rician–Rayleigh fading environment^24,25 (i.e. the links closer to the source are subject to Rician fading and those closer to the destination are subject to Rayleigh fading), which does not necessarily hold true in the case of the proposed scheme with relay node clustering.

To the best of our knowledge, an analysis in terms of the outage probability, effective transmission rate, and the throughput of the clustering-based multihop relaying with EH relays under the scenario of WIPT has not been reported in the literature. A closely related system model is studied in Nguyen et al.,²³ where the throughput of a wireless sensor network with EH nodes in the framework of clustering-based DF relaying is analyzed under the assumption that the EH nodes can harvest energy from the surrounding environment, whereas in this work, EH nodes can harvest energy only through the signal transmitted by the information source.

The remainder of the article is organized as follows. In section “System and channel models,” we describe the system and channel models considered throughout the article. In section “Performance analysis,” the closed-form expressions for the outage probability, effective transmission rate, and the throughput for the clustering-based multihop relaying network over symmetric and asymmetric fading environment are derived. In section “Numerical results,” the numerical results under various scenarios of multihop relaying are presented and the insights gained from them are discussed. Finally, conclusions are given in section “Conclusion.”

System and channel models

Multihop wireless sensor networks

As illustrated in Figure 1, we consider a clustering-based multihop wireless EH sensor network where a single source node $S_{0}$ and a single destination node $S_{K}$ are connected through $K - 1$ clusters of relays (i.e. K-hop relay channel). In each cluster $k \in {1, 2, \dots, K - 1}$ , the relay nodesare denoted by $S_{k, m}$ , where $m \in {1, 2, \dots, M_{k}}$ represents the node index for each cluster and $M_{k}$ is the number of the relay nodes in the $k th$ cluster. Therefore, the total number of relay nodes is given by

$\begin{matrix} N & = \sum_{k = 1}^{K - 1} M_{k} \end{matrix}$ (1)

Figure 1.

Clustering-based multihop relaying network.

It is assumed that there is no direct link between the source and destination. Therefore, the source communicates with the destination through K-hop relaying channel. We assume that the relays are grouped into clusters on the basis of their geographical proximity (or equivalently average SNR) as discussed by Zhao and Valenti,¹⁷ Nguyen et al.,¹⁸ and Belding-Royer.^26,27 By this assumption, the nodes in the $k th$ hop and those in the $k th$ cluster become equivalent. We thus use the terms the $k th$ hop and the $k th$ cluster interchangeably in what follows. However, it should be noted that if the number of hops is K, then that of the clusters is $K - 1$ .

It is considered that there is no cluster head, and thus all the relays in a cluster are treated equally.^17,18 Note that when $M_{1} = \dots = M_{K - 1} = 1$ , the proposed model is equivalent to the conventional multihop relaying without clustering where the message is sent over a predetermined route.^1,21

Channel and access models

We consider that each hop takes T seconds to transmit the signal as in the work by Mao et al.²¹ and all the physical links between the nodes are frequency flat block fading, where the fading coefficient remains constant during one block time, that is, for T seconds, but changes independently to the next. If a link experiences Rayleigh fading, then the channel gain $h_{k, m} \in R$ experienced by the $m th$ relay in the $k th$ cluster follows a random variable (RV) with exponential distribution.²⁸ The corresponding probability density function (pdf) of $h = h_{k, m}$ is expressed as

$p_{h} (x) = \frac{1}{λ_{k, m}} \exp (- \frac{x}{λ_{k, m}})$ (2)

where $λ_{k, m} = E (h_{k, m})$ , with $E (\cdot)$ representing an expectation operation. Since the clustering algorithm based on the geographical proximity of nodes is employed,^26,27 this system model ensures that all the links in one hop have the same average channel power. As a result, we have $λ_{k, m} = λ_{k}$ for $\forall m \in {1, \dots, M_{k}}$ . If a link experiences Rician fading, then the pdf of $h = h_{k, m}$ can be expressed as

$p_{h} (x) = \frac{(V + 1) e^{- V}}{λ_{k, m}} e^{- \frac{(V + 1) x}{λ_{k, m}}} I_{0} (\sqrt{\frac{4 V (V + 1) x}{λ_{k, m}}})$ (3)

where V is the Rician factor defined as the ratio of the powers of the line-of-sight (LoS) component to the scattered components, and $I_{0} (\cdot)$ is the zeroth-order modified Bessel function of the first kind.²⁴

Furthermore, we assume that the noise level of the additive white Gaussian noise (AWGN) observed by all the relay nodes in the $k th$ cluster is identical with its average power $σ_{k}^{2}$ .

Finally, all the nodes are assumed to be half duplex and equipped with only a single antenna as in the works by Krikidis et al.,⁵ Nasir et al.,^6,7 Ding et al.,⁸ Ju et al.,⁹ Atapattu et al.,¹⁰ and Haghifam et al.¹¹ In order to avoid the inter-relay interference and to ensure the orthogonality among transmitting nodes, a time division multiple access (TDMA) scheme with K time slots, each consisting of T seconds, is considered for simplicity.

Relay selection scheme

For the relay selection within each cluster k, we consider the PRS scheme where a relay that provides the best channel gain between the transmitter and the receiving cluster is selected from a set of $M_{k}$ relays.²⁹ With the distributed relay selection algorithm proposed by Bletsas et al.,³⁰ selection can be done locally, that is, in a distributive fashion. The channel gain of the relay selected at the $k th$ cluster can be found as

$h_{k}^{*} \overset{Δ}{=} max_{m \in {1, \dots, M_{k}}} h_{k, m}$ (4)

Throughout this work, we assume that only the receiver side has full channel state information (CSI), which can be obtained through the pilot symbols.³¹

Battery model

We assume that the relay node is equipped with a discrete-level energy battery of size $E_{B}$ , which is capable of holding energy for an immediate use only, due to leakage. Thus, all the harvested energy should be used in the same information block, and such a system model is also known as a harvest-and-use (HU) architecture.^{6,7,9,10,12–14,21,32,33} We also assume that the battery is discretized into L energy levels as $ϵ^{(ℓ)} \overset{Δ}{=} ℓ E_{B} / L$ with $ℓ \in {1, \dots, L}$ , and the battery is said to be in the state $s_{ℓ}$ when its stored energy is equal to $ϵ^{(ℓ)}$ . Apparently, for sufficiently large energy levels, the discrete battery model can closely approximate a continuous battery model.^5,34,35

APS protocol

For the information relaying with EH, the APS protocol proposed by Ding and colleagues^8,36 is employed, where the selected EH relay in the $k th$ cluster will dynamically split the received signal into the two streams with PS ratio $α_{k}$ and $(1 - α_{k})$ for information decoding and EH, respectively. For a given $α_{k}$ , the energy harvested by the selected relay in the $k th$ cluster can be expressed as

$E_{k} = \frac{ζ (1 - α_{k}) P_{k - 1} h_{k}^{*}}{d_{k}^{η}}$ (5)

where $P_{k - 1}$ is the transmission power of the transmitting node in the previous cluster, $d_{k}$ is the distance between the $(k - 1) th$ and $k th$ clusters, $η$ is the path loss exponent, $ζ \in (0, 1]$ is the RF-to-DC conversion efficiency, and $h_{k}^{*}$ is the channel gain of the selected relay in the $k th$ cluster defined in equation (4). Note that $P_{0}$ corresponds to the transmission power of the source.

By definition, the battery energy harvested by each relay is discrete and finite. Therefore, the energy of the selected relay can be expressed as

$ϵ_{k} = max_{ℓ \in {1, \dots, L}} ϵ^{(ℓ)} where ϵ^{(ℓ)} \leq E_{k}$ (6)

The transmission power of the relay selected in the $k th$ cluster can be expressed as

$P_{k} = max (0, \frac{ϵ_{k} - E_{C}}{T})$ (7)

where $E_{C}$ is the required energy that should be consumed by the receiver circuit of the selected relay for information decoding and re-encoding. For the APS scheme, the channel capacity corresponding to the $k th$ hop, that is, the link from the $(k - 1) th$ cluster to the $k th$ cluster, can be expressed as²¹

$C_{k} = \log_{2} (1 + α_{k} \frac{P_{k - 1} h_{k}^{*}}{σ_{k}^{2} d_{k}^{η}}) 1_{ϵ_{k} > E_{C}}$ (8)

where $σ_{k}^{2}$ is the average power of the corresponding AWGN and $1_{E}$ is the indicator function that returns 1 when the event E occurs and returns 0 otherwise.³⁷ The indicator function in equation (8) represents the circuit power constraint on the $k th$ relay for its information decoding. Note that $C_{K}$ is the capacity observed by the destination node $S_{K}$ . Since we consider the DF relaying scheme for information relaying, the relay needs to successfully decode the received signal before further processing. Hence, the PS ratio $α_{k}$ for a given transmission rate R can be calculated as

$α_{k} = min (1, \frac{2^{R} - 1}{h_{k}^{*} γ_{k}})$ (9)

where

$γ_{k} = \frac{P_{k - 1}}{σ_{k}^{2} d_{k}^{η}}$ (10)

Outage probability

From equations (5) and (9), it can be observed that when the selected relay cannot decode the received signal successfully (i.e. $α_{k} \geq 1$ ), it will not harvest any energy due to the assumption of the HU architecture that it cannot retain the harvested energy for the later use. When $α_{k} < 1$ , the relay first feeds the $(1 - α_{k})$ portion of the received signal to the EH unit, and if the harvested energy satisfies $ϵ_{k} > E_{C}$ (i.e. the harvested energy is higher than that required for the information decoding and re-encoding), the information decoding unit utilizes the harvested energy to decode the received signal. Therefore, the outage at the destination will be caused by either of the following events: (1) any of the previous hop is in outage, that is, $ϵ_{k} < E_{C}$ with $\forall k \in {1, \dots, K - 1}$ , or (2) none of the previous hops is in outage but the destination is in outage. Therefore, the outage event can be expressed as

$\begin{matrix} P_{out} & = 1 - \Pr {(\cap_{k = 1}^{K - 1} (ϵ_{k} > E_{C})) \cap (C_{K} \geq R)} \end{matrix}$ (11)

From equation (11), it can be observed that the outage at the destination depends on all the previous hops and this generally leads to difficulty in precise mathematical analysis. However, the discrete-level battery model introduced in section “Battery model” would make the subsequent analysis mathematically tractable. For the APS protocol, the channel gain required to harvest $ϵ^{(ℓ)} = ℓ E_{B} / L$ units of energy for the information relaying at the relay in the $k th$ cluster, or equivalently, the channel gain to reach the state $s_{ℓ}$ in the $k th$ hop, can be given by

$Γ_{k} (ℓ) = (2^{R} - 1) \frac{σ_{k}^{2} d_{k}^{η}}{P_{k - 1}} + \frac{(ϵ^{(ℓ)} + E_{C}) d_{k}^{η}}{ζ P_{k - 1}}$ (12)

where the first term represents the amount of channel gain required to ensure the successful decoding of the received signal at the relay in the $k th$ hop, and the second term denotes the amount of channel gain required to harvest $ϵ^{(ℓ)} + E_{C}$ units of energy by the relay, with $ϵ^{(ℓ)}$ representing the amount of the harvested energy left after information decoding and re-encoding, that is, the amount of energy that will be used for the information relaying to the $(k + 1) th$ cluster.

Effective transmission rate and throughput

For a fixed transmission rate R and outage probability $P_{out}$ , the effective transmission rate can be defined as³²

$Ψ = R {1 - P_{out}}$ (13)

The network throughput indicates the bandwidth efficiency of the network which is one of the important performance metrics of the multihop relaying networks,¹⁸ which can be defined in the TDMA framework as³⁸

$τ = \frac{Ψ}{K}$ (14)

Remark

Based on the above system, network, and channel models, we are particularly interested in the following aspects:

The performance of the network (in terms of effective transmission rate and throughput) should depend on the number of relays as well as the number of clusters. Then, for a given number of total relay nodes N in the network, how we should allocate them among different clusters?

Increasing the number of hops K should help reducing path loss, but it also reduces the bandwidth efficiency due to the assumption of TDMA. Thus, is it reasonable to increase K from the viewpoint of network throughput?

These issues will be investigated in section “Numerical results” through numerical analysis.

Performance analysis

We analyze the performance of the proposed system over various fading models. We start with the most general scenario in our framework, that is, all the links follow Rician fading. The results for other channel settings can be obtained directly from this result.

We consider that all the relay nodes are EH nodes and take part in the relaying with harvested energy. The corresponding channel gains are modeled as RVs as described in section “Channel and access models.” For simplicity of our subsequent analysis, we assume that ${h_{k, m}}$ is independent and identically distributed (iid) with its pdf given by equation (3). Consequently, we have $λ_{k, m} = E (h_{k, m}) = λ$ for all $k \in {1, \dots, K}$ and $m \in {1, 2, \dots, M_{k}}$ .

Probability expressions for harvested energy

We are interested in the probability that the battery of the relay selected in the $k th$ cluster reaches the energy level of $ℓ \in {1, 2, \dots, L}$ after information decoding and re-encoding (i.e. the amount of energy left for information transmission to the next cluster) and we denote this by $q_{k} (ℓ)$ . In our system model, the only node that always transmits with constant power $P_{0}$ is the source, that is, the first link in the multihop relay channel. The transmission power of the relay nodes in the subsequent clusters is modeled as a RV by nature. Upon derivation of a theoretical expression for $q_{k} (ℓ)$ , these two cases, that is, whether the information is transmitted by the source ( $k = 1$ ) or relay ( $1 < k < K$ ), will be separately dealt with in what follows.

Energy harvested at the first hop

Based on the PRS scheme where the relay with the best channel condition from the source is chosen according to equation (4), the probability of harvesting $ϵ^{(ℓ)}$ units of energy for information relaying (i.e. the amount of harvested energy left after the information decoding) with $ℓ \in {1, \dots, L}$ can be calculated by the following two steps.

Case 1: battery being partially charged

When the energy harvested by the relay is greater than or equal to $ϵ^{(1)}$ and less than $ϵ^{(L)}$ , the battery is said to be partially charged. The corresponding probability $q_{k} (ℓ)$ in the first hop ( $k = 1$ ) can be expressed as

$\begin{matrix} q_{1} (ℓ) = \Pr {Γ_{1} (ℓ) \leq h_{1}^{*} < Γ_{1} (ℓ + 1)} \\ = Π_{m = 1}^{M_{1}} F_{h_{1, m}} (Γ_{1} (ℓ + 1)) - Π_{m = 1}^{M_{1}} F_{h_{1, m}} (Γ_{1} (ℓ)) \\ = F_{h}^{M_{1}} (Γ_{1} (ℓ + 1)) - F_{h}^{M_{1}} (Γ_{1} (ℓ)) \end{matrix}$ (15)

where $M_{1}$ is the number of rely nodes in the first cluster, $Γ_{1} (ℓ)$ is the required channel gain derived in equation (12), and the expression $F_{X} (x_{0}) = \Pr (X < x_{0})$ is the cumulative distribution function (cdf) of the RV X. Since ${h_{k, m}}$ is iid with its pdf given by equation (3), for $h = h_{k, m}$ , we have

$F_{h} (x) = 1 - Q (\sqrt{2 V}, \sqrt{(V + 1) \frac{2 x}{λ}})$ (16)

where V is the Rician factor and $Q (\cdot, \cdot)$ is the generalized first-order Marcum Q-function.²⁸

Case 2: battery being fully charged

When the energy harvested by the relay node is greater than or equal to $ϵ^{(L)}$ , the battery is considered to be fully charged. The corresponding probability can be expressed as

$\begin{matrix} q_{1} (L) = \Pr {h_{1}^{*} \geq Γ_{1} (L)} \\ = 1 - F_{h}^{M_{1}} (Γ_{1} (L)) \end{matrix}$ (17)

Energy harvested at the $k th$ hop ( $1 < k < K$ )

Except for the relay in the first hop, all the other relays harvest energy through the signal transmitted by the preceding EH relay with their own discrete-level batteries. Hence, the channel gain required to harvest $ϵ^{(ℓ)}$ units of energy for the information relaying of the selected node in the $k th$ cluster, provided that the transmitting node in the previous hop (i.e. $(k - 1) th$ hop) has the harvested energy of $ϵ^{(j)}$ units with $j \in {1, \dots, L}$ , can be calculated as

$Γ_{k} (ℓ | j) = (2^{R} - 1) \frac{σ_{k}^{2} d_{k}^{η} T}{ϵ^{(j)}} + \frac{(ϵ^{(ℓ)} + E_{C}) d_{k}^{η} T}{ζ ϵ^{(j)}}$ (18)

Similar to the case of $k = 1$ , we derive $q_{k} (ℓ)$ , that is, the probability that the energy $ϵ^{(ℓ)}$ is left for the selected relay in the $k th$ cluster, in the following two steps.

Case 1: battery being partially charged

When the energy harvested by the relay is greater than or equal to $ϵ^{(1)}$ and less than $ϵ^{(L)}$ (partially charged), the corresponding probability $q_{k} (ℓ)$ can be written as

$\begin{matrix} q_{k} (ℓ) = \sum_{j = 1}^{L} q_{k - 1} (j) \\ \times [F_{h}^{M_{k}} (Γ_{k} (ℓ + 1 | j)) - F_{h}^{M_{k}} (Γ_{k} (ℓ | j))] \end{matrix}$ (19)

Case 2: battery being fully charged

When the energy harvested by the relay is greater than or equal to $ϵ^{(L)}$ (fully charged), the corresponding probability can be expressed as

$\begin{matrix} q_{k} (L) & = \sum_{j = 1}^{L} q_{k - 1} (j) [1 - F_{h}^{M_{k}} (Γ_{k} (L | j))] \end{matrix}$ (20)

Outage probability expressions

Whenever the destination fails to receive or decode the signal transmitted by the source, the system is said to be in outage.³⁹ The channel gain required to successfully decode the signal at the $k th$ hop, provided that the power of the $(k - 1) th$ hop is given by $P_{k - 1} = ϵ^{(ℓ)} / T$ with $ℓ \in {1, \dots, L}$ , is

$Γ_{K} (ℓ) = (2^{R} - 1) \frac{σ_{K}^{2} d_{K}^{η} T}{ϵ^{(ℓ)}}$ (21)

With the help of equation (11), the outage probability for the proposed scheme with transmission rate R can be expressed as

$\begin{matrix} P_{out} = 1 - \sum_{ℓ = 1}^{L} q_{K - 1} (ℓ) {1 - F_{h} (Γ_{K} (ℓ))} \\ = 1 - \sum_{ℓ = 1}^{L} q_{K - 1} (ℓ) Q (\sqrt{2 V}, \sqrt{(V + 1) \frac{2}{λ} Γ_{K} (ℓ)}) \end{matrix}$ (22)

where $q_{k} (ℓ)$ is derived in equations (15) and (17) for $k = 1$ and in equations (19) and (20) for $k > 1$ .

The effective transmission rate and throughput can be obtained from equations (13) and (14), respectively, based on the outage probability expression developed above.

Other fading scenarios

Rayleigh fading

If all the wireless links experience Rayleigh fading, from equations (2) and (3), it can be readily observed that the results are obtained by substituting $V = 0$ for the Rician cases, that is, with $F_{h} (x)$ of equation (16) replaced by

$F_{h} (x) = 1 - \exp (- \frac{x}{λ})$ (23)

The corresponding outage probability is given by

$\begin{matrix} P_{out} & = 1 - \sum_{ℓ = 1}^{L} q_{K - 1} (ℓ) \exp (- \frac{1}{λ} Γ_{K} (ℓ)) \end{matrix}$ (24)

Note that $q_{k} (ℓ)$ can be found in the same manner as that of the Rician case again with $F_{h} (x)$ of equation (16) replaced by equation (23).

Asymmetric fading channels

The previous results can be extended to the cases where the channel is composed of mixture of Rayleigh and Rician fading with slight modification by adopting $F_{h} (x)$ of equation (16) when the channel is Rician and that of equation (23) when the channel is Rayleigh.

For example, if the first K’ hops (with $K' < K$ ) are subject to Rayleigh fading and the remaining $K - K'$ hops are subject to the Rician fading, then since the last hop to destination is Rician, the outage probability can be expressed as equation (22) with $Γ_{K} (ℓ)$ given in equation (21), but $q_{k} (ℓ)$ should be calculated according to the fading statistic of each link $F_{h} (x)$

$F_{h} (x) = {\begin{matrix} 1 - \exp (\frac{- x}{λ}), \\ for k \in {1, 2, \dots, K'} \\ 1 - Q (\sqrt{2 V}, \sqrt{(V + 1) \frac{2 x}{λ}}), \\ for k \in {K' + 1, 2, \dots, K} \end{matrix}$ (25)

Numerical results

In this section, based on the theoretical analysis carried out in the previous section, we present several numerical results and discuss the insights gained from them. The outage probability, effective transmission rate, and throughput are used as our performance measures. The accuracy of the developed analytical results are also verified by Monte Carlo simulations.

We initially consider a homogeneous linear network case consisting of $(K - 1)$ clusters, each of which has the same number of nodes denoted by M, that is, $M = M_{1} = M_{2} = \dots = M_{K - 1} = N / (K - 1)$ . Also, unless otherwise stated, the following parameters are chosen as our baseline example scenario: the distance between source to distance is $d_{SD} = 5 m$ .⁸ and the clusters are located with equal distance, that is, $d_{k} = d_{SD} / K$ for $\forall k \in {1, 2, \dots, K}$ . Furthermore, we normalize the fading coefficient such that $λ = 1$ , the path loss exponent is $η = 3$ , and in the case of Rician fading, Rician factor is chosen as $V = 10$ . The transmission power of the source is $P_{0} = 10$ dBm, the circuit power is $P_{C} = P_{0} / 100$ (i.e. $1 %$ of the transmission power of the source), the noise power is $σ_{k}^{2} = - 10$ dBm for $\forall k \in {1, 2, \dots, K}$ , and the RF-to-DC conversion efficiency is $ζ = 0.8$ . The transmission rate is $R = 1$ and the number of the battery levels is $L = 1000$ .

We also consider the following four models for fading:

Model I: All the wireless links are subject to Rayleigh fading.

Model II: Only the first link, that is, $S_{0} \to S_{1}$ , is subject to Rician fading and the remaining links are subject to Rayleigh fading.

Model III: The first two links, that is, $S_{i} \to S_{j}$ with $(i, j) \in {(0, 1), (1, 2)}$ are subject to Rician fading and the remaining links are subject to Rayleigh fading.

Model IV: Only the last link, that is, $S_{K - 1} \to S_{K}$ is subject to the Rician fading and the rest of the links are subject to Rayleigh fading.

In most part of this section, we assume that all the channels are subject to Rayleigh fading (i.e. model I) as an example of the severe environment. The mixed fading cases are explored only in section “Effect of asymmetric channel” (except for comparison of theoretical and simulation results in the next subsection).

Comparison of theoretical and simulation results

We first compare our theoretical derivation with the simulation results. Figure 2 compares the outage probability based on both the theoretical analysis (indicated by the solid lines) and the corresponding simulation (discrete points) over channel models I and II, with two relay node cases ( $M = 2$ ) and $K = 5$ hops. The transmission power of the source $P_{0}$ is chosen as our parameter. From the results, it can be observed that the theoretically derived outage probabilities are in excellent agreement with the simulation results, irrespective of the source power $P_{0}$ and channel model, validating our theoretical analysis. For the rest of this section, we only plot the numerical results based on the theoretical analysis.

Figure 2

Outage probability versus power transmitted by the source $P_{0}$ (solid lines: theoretical analysis, discrete points: simulation). Parameters: $R = 1$ , $ζ = 0.8$ , $P_{C} = P_{0} / 100$ , $σ_{k}^{2} = - 10$ dBm, $λ_{k} = 1$ , $d_{SD} = 5$ m, $d_{k} = d_{SD} / K$ , $K = 5$ , $M = 2$ , $V = 10$ , and $L = 1000$ .

Conventional versus proposed schemes

Figure 3 shows the relationship between the effective transmission rate and the number of hops K for various cases in terms of the number of relay nodes per cluster M. These results are obtained over Rayleigh fading environment (model I). The case with $M = 1$ represents the effective transmission rate of the conventional approach (i.e. without clustering). From the results, it can be observed that the proposed approach outperforms the conventional one irrespective of the number of hops K. We can also observe that due to the attenuation of channel associated with path loss, the system performance of multihop relaying improves as the number of hops increases (i.e. gain in terms of path loss). Furthermore, we can also observe that as the number of relay nodes in each cluster increases, the performance of the proposed scheme also improves since the proposed clustering scheme can exploit the spatial diversity gain as well with the help of PRS scheme.

Figure 3.

Effective transmission rate versus the number of hops with several cases of relay nodes per cluster M. Parameters: $R = 1$ , $ζ = 0.8$ , $P_{0} = 10$ dBm, $P_{C} = P_{0} / 100$ , $σ_{k}^{2} = - 10$ dBm, $λ_{k} = 1$ , $d_{SD} = 5$ m, $d_{k} = d_{SD} / K$ , and $L = 1000$ .

Effect of RF-to-DC conversion efficiency

The relationship between the effective transmission rate and the RF-to-DC conversion efficiency $ζ$ is shown in Figure 4 for various cases in terms of the number of relay nodes per cluster, where the number of hops is chosen as $K = 5$ . These results are obtained over Rayleigh fading environment (model I). It can be observed that while the effective transmission rate of the conventional network is very sensitive to the value of $ζ$ , the proposed scheme can cope with the effects of poor $ζ$ by the diversity gain, which makes the proposed scheme very attractive for practical multihop sensor network scenarios operated with low energy conversion efficiency.

Figure 4.

Effective transmission rate versus RF-to-DC conversion efficiency. Parameters: $R = 1$ , $P_{0} = 10$ dBm, $P_{C} = P_{0} / 100$ , $ζ = 0.8$ , $σ_{k}^{2} = - 10$ dBm, $λ_{k} = 1$ , $d_{SD} = 5$ m, $K = 5$ , $d_{k} = d_{SD} / K$ , and $L = 1000$ .

Resource allocation

From the above analysis, we can conclude that the performance of the proposed scheme is dominated by the number of relays present in each cluster, which gives rise to the following issue stated in section “Remark”: for a given number of total relay nodes N in the network, what is the best strategy to distribute them among different clusters?

In order to investigate this issue, in Figure 5, the effective transmission rate of the network is compared for the four different kinds of topologies listed in Table 1, where $N = 8$ relays are distributed among four different clusters with distinctive patterns. These results are obtained over Rayleigh fading environment (model I).

Figure 5.

Effective transmission rate versus power transmitted by the source with the four different network topologies. Parameters: $R = 1$ , $ζ = 0.8$ , $P_{0} = 10$ dBm, $P_{C} = P_{0} / 100$ , $K = 5$ , $σ_{k}^{2} = - 10$ dBm, $λ_{k} = 1$ , $d_{SD} = 5$ m, $d_{k} = d_{SD} / K$ , and $L = 1000$ .

Table 1

Network topologies investigated ( $K = 5$ ).

Network ID	Network topologies [M₁M₂M₃M₄]
1	[2 2 2 2]
2	[3 2 2 1]
3	[1 2 2 3]
4	[3 3 1 1]

We observe from Figure 5 that the effective transmission rate achieved with topology 1 is maximum among the four topologies compared, followed by 2, 4, and 3. From these results, it can be concluded that in the EH multihop networks, the best performance can be achieved when all the clusters enjoy the same level of diversity (i.e. the result of network 1, where each cluster has the equal number of relay nodes).

Furthermore, if it would not be possible to assign the relay nodes equally to each cluster, the cluster closer to the source should be provided with a larger number of relay nodes instead of those closer to the destination. This stems from the fact that in the proposed EH network, the receiver performance strongly depends on the transmission power, and the transmit power of the source node is only the energy source that can be used for forwarding the information toward the destination through multiple relays.

Throughput improvement by path loss reduction

From the numerical results analyzed so far, it is obvious that by increasing the number of hops, gain due to the reduction of path loss in each hop can be achieved, but at the same time, the bandwidth efficiency is reduced by a factor of $1 / K$ as indicated by equation (14). Thus, as stated in section “Remark,” it may be interesting to investigate whether the use of multihop relaying can be justified from the perspective of bandwidth efficiency.

Figure 6 shows the throughput as a function of the transmission power of the source $P_{0}$ under the assumption of the homogeneous network of $M = 10$ but with various number of hops K. These results are obtained over Rayleigh fading environment (model I). From the results, it can be observed that the number of hops that can achieve the maximum throughput differs depending on the transmission power, and in the case of low $P_{0}$ , larger number of K results in better throughput. However, since the achievable throughput in this case is reduced by a factor of $1 / K$ , the direct transmission (i.e. $K = 1$ ) eventually outperforms the others as $P_{0}$ increases under the considered parameter setting. Nevertheless, it can be concluded that when the transmission power is limited, which is the case with typical sensor networks, larger gain due to the path loss reduction achieved by multihop can effectively increase the throughput.

Figure 6.

Throughput versus power transmitted by the source. Parameters: $R = 1$ , $ζ = 0.8$ , $σ_{k}^{2} = - 10$ dBm, $λ_{k} = 1$ , $M = 10$ , $d_{SD} = 5$ m, $d_{k} = d_{SD} / K$ , $P_{C} = P_{0} / 100$ , $η = 3$ , and $L = 1000$ .

Throughput improvement versus total number of relay nodes

In section “Resource allocation,” we have observed that for a given number of the total relay nodes N, the effective transmission rate can be improved if the nodes are distributed equally for a given number of clusters $K - 1$ . In what follows, we investigate how the throughput can be improved by increasing N for a different number of hops K. To this end, for a given pair of N and K, we equally allocate the relay nodes among all the clusters (i.e. the homogeneous case) whenever possible. For a fair comparison, if any relay nodes are left after this equal allocation process, the remaining relay nodes are assigned to the first cluster following the observation in section “Resource allocation.”

Figure 7 shows the relationship between the throughput and the number of relays in the cluster for various number of hops. These results are obtained over Rayleigh fading environment (model I). It can be observed that for a given number of total relay nodes N, the number of the hops K that maximizes the throughput may be different. When N is small, increasing K would result in better throughput, but as N increases, the resulting throughput soon reaches its upper limit, and thus reducing K will eventually improve the throughput.

Figure 7.

Throughput versus the number of total relay nodes with different number of hops. Parameters: $R = 1$ , $ζ = 0.8$ , $P_{0} = 10$ dBm, $P_{C} = P_{0} / 100$ , $σ_{k}^{2} = - 10$ dBm, $λ_{k} = 1$ , $d_{SD} = 5$ m, $d_{k} = d_{SD} / K$ , $η = 3$ , and $L = 1000$ .

The above result also suggests that when there are no sufficient relay nodes in terms of achievable diversity in the network, the system performance can be improved with the help of the gain associated with path loss reduction achieved by increasing the number of hops. If there are a number of the relay nodes in the network that can guarantee sufficient diversity, it would be better to reduce the number of hops K such that the throughput limit (which is a factor of $1 / K$ ) can be enhanced.

Effect of asymmetric channel

Finally, we investigate the effect of asymmetric channel model where different hops may experience different fading phenomena. The four models described at the beginning of this section are evaluated in what follows.

Figure 8 shows the relationship between the throughput and the number of relay nodes per cluster in various fading environments with transmission rate $R = 2$ . When the number of relays per cluster is small, the system achieves the maximum throughput with channel model III, due to the fact that the end-to-end performance of the DF relaying with EH relay node is dominated by the channel gains of the hops closer to the source, which is in line with the results observed in the work by Suraweera et al.²⁴ and Ding et al.²⁵. However, as the number of the relay nodes in each cluster increases, the throughput achieved with model IV becomes maximum, whereas the throughput values achieved with the other three channel models become identical.

Figure 8.

Throughput versus the number of relay nodes in each cluster with the four different fading models. Parameters: $R = 2$ , $ζ = 0.8$ , $P_{0} = 10$ dBm, $σ_{k}^{2} = - 10$ dBm, $λ_{k} = 1$ , $d_{SD} = 5$ m, $K = 5$ , $d_{k} = d_{SD} / K$ , $V = 10$ , $P_{C} = P_{0} / 100$ , and $L = 1000$ .

The reason for this contrast is that in the proposed scheme, all the hops enjoy the diversity gain due to the PRS scheme except for the last hop that consists of only one node (i.e. destination). As a consequence, the relays in the intermediate hops can compensate for the absence of the LoS channel by exploiting the spatial diversity. In the case of model IV, each cluster can enjoy the diversity gain in the intermediate hops due to the PRS scheme, and the last hop can also enjoy the LoS channel, which makes it the best channel model. This observation could be useful for engineers to design the routing protocol of the EH sensor networks that enables relay node clustering.

Conclusion

We have proposed a clustering-based multihop relaying with the PRS scheme for an EH relaying network, which can improve the system performance by exploiting the gain due to the path loss reduction as a result of multihop relaying and the gain due to the spatial diversity based on the PRS scheme. The results suggest that with the help of gain in path loss reduction as well as diversity gain, the proposed scheme can compensate for possible performance degradation associated with poor RF-to-DC conversion efficiency, which makes the proposed scheme attractive for the practical multihop sensor network scenarios operated with low energy conversion efficiency and low power.

We have also compared the performance of the proposed scheme with the conventional multihop relaying scheme without clustering as well as the direct transmission and numerically outlined the benefit of the clustering approach; it may enhance the network coverage without consuming any extra power, suitable for lifetime improvement of energy-constrained sensor networks. The performance has been also investigated under the constraint of the total number of relay nodes with different number of hops, and it has been shown that there is an optimal number of hops that can maximize the throughput for a given transmission power of the source. The effect of the asymmetric channels has been also investigated. It has been revealed that in the case of the proposed clustering, compared to the environment where all the links are subject to Rayleigh fading, the existence of Rician fading will help improve the throughput more effectively at the destination side, rather than the source side, as opposed to the conventional multihop relaying scenarios without clustering.

Footnotes

Academic Editor: Antonio Lazaro

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research,authorship,and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research,authorship,and/or publication of this article: This work was supported in part by the Strategic Information and Communications R&D Promotion Programme (No. 0159-0053) and the Ministry of Internal Affairs and Communications,Japan.

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Performance analysis of the clustering-based multihop wireless energy harvesting sensor networks over symmetric and asymmetric fading channels

Abstract

Keywords

Introduction

System and channel models

Multihop wireless sensor networks

Channel and access models

Relay selection scheme

Battery model

APS protocol

Outage probability

Effective transmission rate and throughput

Remark

Performance analysis

Probability expressions for harvested energy

Energy harvested at the first hop

Case 1: battery being partially charged

Case 2: battery being fully charged

Energy harvested at the k th hop ( 1 < k < K )

Case 1: battery being partially charged

Case 2: battery being fully charged

Outage probability expressions

Other fading scenarios

Rayleigh fading

Asymmetric fading channels

Numerical results

Comparison of theoretical and simulation results

Conventional versus proposed schemes

Effect of RF-to-DC conversion efficiency

Resource allocation

Throughput improvement by path loss reduction

Throughput improvement versus total number of relay nodes

Effect of asymmetric channel

Conclusion

Footnotes

Declaration of conflicting interests

Funding

References

Energy harvested at the $k th$ hop ( $1 < k < K$ )