---
abstract: 'In this paper, we consider a two-dimensional heterogeneous cellular network scenario consisting of one base station (BS) and some mobile stations (MSs) whose locations follow a Poisson point process (PPP). The MSs are equipped with multiple radio access interfaces including a cellular access interface and at least one short-range communication interface. We propose a nearest-neighbor cooperation communication (NNCC) scheme by exploiting the short-range communication between a MS and its nearest neighbor to collaborate on their uplink transmissions. In the proposed cooperation scheme, a MS and its nearest neighbor first exchange data by the short-range communication. Upon successful decoding of the data from each other, they proceed to send their own data, as well as the data received from the other to the BS respectively in orthogonal time slots. The energy efficiency analysis for the proposed scheme is presented based on the characteristics of the PPP and the Rayleigh fading channel. Numerical results show that the NNCC scheme significantly improves the energy efficiency compared to the conventional non-cooperative uplink transmissions.[^1]'
author:
- 
bibliography:
- 'manu\_icc2015-01.bib'
title: 'On Energy Efficiency of the Nearest-Neighbor Cooperative Communication in Heterogeneous Networks'
---

Cooperative communication; Poisson point process; heterogeneous cellular network

Introduction
============

Nowadays many of mobile stations (MSs), e.g, smart cellular phones, tablets and PADs, are equipped with multiple radio access interfaces, e.g, cellular radio access, wireless local area network (WLAN), Bluetooth interfaces and etc.. As multi-mode MSs, they can constitute heterogeneous cellular networks (HCNs) and make it possible to improve the performance of cellular uplinks by serving as a relay to their neighboring MSs. By some of short-range communication methods provided by the multi-mode MSs, they can communicate with each other with significantly high efficiency and quite low cost. Then the MSs can exploit the short-range communication links among them along with the uplinks to the base station (BS) to form the cooperative communication, which can improve the performance of the HCNs with regard to rate, outage probability, coverage, energy efficiency and etc.. This paper focuses on the improvement of the energy efficiency of uplink cellular communications based on cooperation between neighboring MSs in HCNs.

Cooperative diversity has already emerged as a new and effective technique to combat fading and to decrease energy consumption in wireless networks. The nearest neighbor relay scheme that relay is chosen to be the nearest-neighbor to the user towards the BS (access-point) always has been applied in [@SAK2010Relay12; @ALT2012Balance14]. [@SAK2010Relay12] proposes and analyzes the performance of two schemes: a distributed nearest-neighbor relay assignment in which users can act as relays, and an infrastructure-based relay assignment in which fixed relay nodes are deployed in the network to help the users forward their data. [@ALT2012Balance14] explores the balance between cooperation through relay nodes and aggregated interference generation in large decentralized wireless networks using decode-and-forward by the nearest neighbor relay scheme. [@ESM2010EnergyEfficient13] proposes an energy-efficient cooperative multicasting scheme by properly selecting relay agents (RAs) based on their location, channel condition and coverage. [@NB2013Contract15] studies the relay selection schemes to reduce energy consumption, and the optimal number of cooperative is also given. Besides, [@YU2010Channel18] is based on coded cooperation, which combines cooperation and channel coding together. To save bandwidth and improve the information transmission rate, network coding [@Li2003tit4] is often used after the MSs receive each others information successfully. But based on some criteria, [@ka2014Unicast17] finds more scenarios where network coding has no gain on throughput or energy saving. Further more, many existing works concentrate on the resource allocation in cooperative networks. [@Adeane2006spawc6] presents both a centralized and a distributed power allocation schemes to optimize the BER performance of cooperative networks. To maximize the overall throughput, [@Zhang2013globecom7] proposes an optimal power allocation. An adaptive coded cooperative protocol based on incremental redundancy by a ACK/NACK feedback is proposed in [@Alazem2008WiMob9].

Many of the above works have presented valuable theories, methods and technologies of cooperative communication. But there are still some improvements left to perform. Zou *et al*. in [@zou2013tcom] investigate user terminals cooperating with each other in transmitting their data packets to the BS, by exploiting the multiple network access interfaces, which is called inter-network cooperation. Given a target outage probability and data rate requirements, they analyze the energy consumption of conventional schemes as compared to the proposed inter-network cooperation. The results show that the inter-network cooperation can significantly improve the energy efficiency of the uplink cellular communications. However in practical view there are some limits of this scheme work. In [@zou2013tcom], it is required that the cooperative MSs have the same distances to the BS and know the instantaneous fading coefficients of both the short-range communication channel and each cellular channel to form an orthogonal matrix used in cooperative. In contrast, our paper intends to propose a more general and yet efficient cooperative scheme for HCNs, in which the cooperative MSs do not need to locate at the same distance away from the BS and are able to perform the cooperative communication without knowing the channel status.

The main contributions of this paper are summarized as follows. At first, we present a nearest-neighbor cooperative communication (NNCC) scheme in a HCN consisting of different radio access networks, i.e., a short-range communication network and a cellular network. Then we compare the proposed NNCC scheme to conventional schemes without user cooperation under target outage probability and data rate requirements. Secondly, we derive the energy efficiency of NNCC scheme in a Rayleigh fading environment. Further more, given a target outage probability, data rate requirements and distances between MSs and the BS, we derive the cumulative distribution function (CDF) and the probability density function (PDF) of energy consumption by considering the MSs that follow a Poisson point process (PPP).

The remainder of this paper is organized as follows. Section \[sec:Cellular-Uplink-Transmission\] presents the network model and the NNCC scheme. In Section \[sec:Energy-Consumption-Analysis\], we present the desired power consumption analysis, then we derive the CDF and the PDF of the system desired power consumption by considering stochastic spatial distribution of cooperative MSs. Section \[sec:Numerical-Results\] gives the numerical results. Finally, Section \[sec:Conclusions\] concludes the paper.

System Model\[sec:Cellular-Uplink-Transmission\]
================================================

In this section, we first present a two-dimensional network model of a HCN environment. Then, we propose a NNCC scheme by exploiting the short-range network to assist cellular uplink transmissions.

![\[fig:System Model\]System Model](huitu2){width="0.75\columnwidth"}

Network Model
-------------

Consider a HCN consisting of a BS and some MSs whose locations follow a homogeneous Poisson point process (PPP) with density $\rho$. The MSs are assumed to equip with multiple radio access interfaces including at least a short-range communication interface and a cellular access interface. The MSs can communicate with their neighboring MSs by the short-range communication. The packet size of data is assumed to be the same across cellular communication link and all short-range communication links between MSs. Without loss of generality, we consider the BS is located at the origin of coordinate, and a specific MS U1 located at coordinate $\left(r_{0},0\right)$ intends to communicate with the BS. Our model is shown in Fig. \[fig:System Model\], U1 will choose its nearest neighboring MS, denoted by U2, to cooperatively communicate with the BS. According to the properties of the homogeneous PPP, the distance $r$ between U1 and U2 satisfies the following PDF [@stoyan1995stochastic] $$f_{r}(r)=2\pi\rho r\exp\left(-\pi\rho r^{2}\right),\label{eq:fr}$$ then the coordinate of U2 can be obtained as $\left(r_{0}+r\cos\theta,r\sin\theta\right)$, where $\theta$ follows a uniform distribution between $-\frac{\pi}{2}$ and $\frac{3\pi}{2}$. Denote the distance between U1 and the BS by $r_{1}$. The distance $r_{\mathrm{2}}$ between the MS U2 and BS can be obtained as

$$r_{\mathrm{2}}^{2}=r{}^{2}+r_{1}^{2}+2r_{1}r\text{\ensuremath{\cos}}\left(-\theta\right)=r{}^{2}+r_{1}^{2}+2r_{1}r\cos\theta.\label{eq:d2b}$$

We consider a general channel model that incorporates the radio frequency, path loss and fading effects in characterizing wireless transmissions, i.e.,

$$\mathcal{P}_{{\rm \mathrm{R}}}=\mathcal{P}_{{\rm \mathrm{T}}}\left(\frac{\lambda}{4\pi d}\right)^{2}G_{{\rm \mathrm{T}}}G_{{\rm \mathrm{R}}}\left|h\right|^{2},\label{eq:pt-pr}$$

where $\mathcal{P}_{\mathrm{R}}$ is the received power, $\mathcal{P}_{\mathrm{T}}$ is the transmitted power, $\lambda$ is the carrier wavelength, $d$ is the transmission distance, $G_{\mathrm{T}}$ is the transmit antenna gain, $G_{\mathrm{R}}$ is the receive antenna gain, and $h$ is the channel fading coefficient. In this paper, we consider a Rayleigh fading model to characterize the channel fading, i.e., $|h|^{2}$ is modeled as an exponential random variable.

The NNCC Scheme
---------------

![\[fig:NNCC\]In NNCC scheme, transmissions happen in short range network channel in time slot 1, and in cellular network channel in time slot 2 (and time slot 3).](schemes22){width="0.7\columnwidth"}

In the NNCC scheme, when MS U1 and its nearest neighbor MS U2 intend to send data $D_{1}$ and $D_{2}$ to the BS, respectively, they cooperate with each other according to the following steps:

1.  U1 and U2 exchange their data over the short-range communication network in time slot 1.

2.  If both U1 and U2 succeed in decoding the data from each other, defined as the case $\delta=0$, both of them will send their own data as well as the data received from the other side to the BS in time slot 2 and time slot 3, respectively, i.e., they send both $D_{1}$ and $D_{2}$ to the BS over two orthogonal cellular uplink channels. Otherwise, defined as the case $\delta=1$, U1 and U2 will send only their own data to the BS separately in time slot 2, just like a conventional non-cooperation communication.

Assuming that the short-range channels among MSs and the cellular channels to the BS are orthotropic and there is no interference at the BS among the MSs’ signals, we only consider the channel noise when analyzing the performance of the scheme. Considering that Ui transmits $D_{{\rm i}}$ to Uj with the signal power $\mathcal{P}_{ij}^{\mathrm{NC}}$, we can obtain the received signal-to-noise-ratio (SNR) between MSs by NNCC scheme as

$$\gamma_{\mathrm{ij}}^{\mathrm{NC}}=\frac{\mathcal{P}_{\mathrm{ij}}^{\mathrm{NC}}}{N_{0}B_{\mathrm{s}}}\left(\frac{\lambda_{\mathrm{s}}}{4\pi r}\right)^{2}G_{\mathrm{U1}}G_{\mathrm{U2}}\left|h_{\mathrm{ij}}\right|^{2},\label{eq:r12nc}$$

where $\mathrm{i}=1\,\mathrm{or}\,2$, $\mathrm{j}=2\,\mathrm{or}\,1$, ${\rm i}\neq{\rm j},$ $\lambda_{s}$ is the carrier wavelength of the short-range communication, $G_{{\rm U}1}$ is the antenna gain at U1, $G_{\mathrm{U2}}$ is the antenna gain at U2, and $h_{{\rm ij}}$ is the fading coefficient of the channel from Ui to Uj. The noise is modeled as $N_{0}B_{s}$, where $N_{0}$ is the noise power spectral density and $B_{s}$ is the channel bandwidth.

In step 2, MSs will transmit the data to the BS, and the received SNR at the BS from MS U1 or U2 over the cellular channel can be obtained as

$$\gamma_{\mathrm{ib}}=\frac{\mathcal{P}_{\mathrm{ib}}}{N_{0}B_{\mathrm{c}}}\left(\frac{\lambda_{\mathrm{c}}}{4\pi r_{i}}\right)^{2}G_{\mathrm{Ui}}G_{\mathrm{BS}}\left|h_{\mathrm{ib}}\right|^{2},\label{eq:r1bnc1}$$

where $\mathrm{i}=1\,\mathrm{or}\,2$, $\lambda_{c}$ is the cellular carrier wavelength, $B_{c}$ is the cellular spectrum bandwidth, $G_{\mathrm{BS}}$ is the receive antenna gain at BS, and $h_{\mathrm{ib}}$ is the fading coefficient of channel from Ui to BS.

Energy Efficiency Analysis of NNCC scheme\[sec:Energy-Consumption-Analysis\]
============================================================================

In this section, we analyze the energy efficiency of the proposed NNCC scheme compared to the conventional scheme without cooperation, under the requirements of target outage probability $P_{\mathrm{out}}$ and data rate $R$.

Energy Consumption in the NNCC scheme
-------------------------------------

\[thm:NNCC\_energy\_consumpution\]Under the situation that the BS succeeds in receiving the complete data from both MSs, the power consumption for the NNCC scheme can be obtained as $$\mathcal{P}^{\mathrm{NC}}=\mathcal{P}_{\mathrm{12}}^{\mathrm{NC}}+\mathcal{P}_{\mathrm{21}}^{\mathrm{NC}}+(1+(1-P_{{\rm out}})^{2})(\mathcal{P}_{\mathrm{1b}}^{\mathrm{NC}}+\mathcal{P}_{\mathrm{2b}}^{\mathrm{NC}}),\label{eq:pnc1}$$ where is the target outage probability,and to meet the target outage probability, where $\mathcal{P}_{12}^{\mathrm{NC}}$, $\mathcal{P}_{21}^{\mathrm{NC}}$, $\mathcal{P}_{1\mathrm{b}}^{\mathrm{NC}}$ and $\mathcal{P}_{2\mathrm{b}}^{\mathrm{NC}}$ are the desired transmission power from U1 to U2, U2 to U1, U1 to the BS and U2 to the BS, respectively, which are given by and .

Due to the limited error correction capability in practical communication systems, both the short-range and cellular communications cannot achieve the Shannon capacity. Therefore, let $\Delta_{\mathrm{s}}>1$ and $\Delta_{\mathrm{c}}>1$ denote the performance gaps for the short-range communication and the cellular communication from their respective capacity limits, respectively. Using and considering the performance gap $\Delta_{\mathrm{s}}$ away from Shannon capacity, we obtain the maximum achievable rate from U1 to U2 of the short-range communication of the NNCC scheme as

$$\begin{aligned}
C_{\mathrm{12}}^{\mathrm{NC}} & =B_{\mathrm{s}}\log_{2}(1+\frac{\gamma_{\mathrm{12}}^{\mathrm{NC}}}{\Delta_{\mathrm{s}}})\nonumber \\
 & =B_{\mathrm{s}}\log_{2}\left(1+\frac{\mathcal{P}_{\mathrm{12}}^{\mathrm{NC}}G_{\mathrm{U1}}G_{\mathrm{U2}}\left|h_{12}\right|^{2}}{\Delta_{\mathrm{s}}N_{0}B_{\mathrm{s}}}\left(\frac{\lambda_{\mathrm{s}}}{4\pi r}\right)^{2}\right).\label{eq:c12NC}\end{aligned}$$

In a Rayleigh fading channel, all random variables $\left|h_{12}\right|^{2}$, $\left|h_{21}\right|^{2}$, $\left|h_{\mathrm{1b}}\right|^{2}$ and $\left|h_{\mathrm{2b}}\right|^{2}$ follow independent exponential distributions with means $\sigma_{12}^{2}$, $\sigma_{21}^{2}$, $\sigma_{\mathrm{1b}}^{2}$ and $\sigma_{\mathrm{2b}}^{2}$, respectively. As we know, an outage event occurs when the channel capacity falls below the required data rate. Using and considering the performance gap $\Delta_{\mathrm{s}}$ away from Shannon capacity, we can obtain the outage probability of the short-range transmission from U1 to U2 as

$$\begin{aligned}
P_{\mathrm{out12}}^{\mathrm{NC}} & =\Pr\left(C_{\mathrm{12}}^{\mathrm{NC}}<R\right)\nonumber \\
 & =1-\exp\left(-\frac{16\pi^{2}\triangle_{\mathrm{s}}N_{0}B_{\mathrm{s}}r^{2}\left(2^{\frac{R}{B_{\mathrm{s}}}}-1\right)}{\mathcal{P}_{\mathrm{12}}^{\mathrm{NC}}\sigma_{\mathrm{12}}^{2}G_{\mathrm{U1}}G_{\mathrm{U2}}\lambda_{\mathrm{s}}^{2}}\right).\label{eq:pout12nc}\end{aligned}$$

Assuming $P_{\mathrm{out12}}^{\mathrm{NC}}=P_{\mathrm{out}}$, we can obtain the desired power consumption of MSs for short-range communication $\mathcal{P}_{\mathrm{ij}}^{\mathrm{NC}}$ from as $$\mathcal{P}_{\mathrm{ij}}^{\mathrm{NC}}=-\frac{16\pi^{2}\triangle_{\mathrm{s}}N_{0}B_{\mathrm{s}}\left(2^{\frac{R}{B_{\mathrm{s}}}}-1\right)}{\sigma_{\mathrm{ij}}^{2}G_{\mathrm{U1}}G_{\mathrm{U2}}\lambda_{\mathrm{s}}^{2}\ln\left(1-P_{\mathrm{out}}\right)}r^{2}.\label{eq:p12nc}$$

Given $\sigma_{\mathrm{21}}^{2}=\sigma_{\mathrm{12}}^{2}$, we can obtain $\mathcal{P}_{\mathrm{12}}^{\mathrm{NC}}=\mathcal{P}_{\mathrm{21}}^{\mathrm{NC}}=\zeta r^{2}$, where $\zeta=-\frac{16\pi^{2}\triangle_{\mathrm{s}}N_{0}B_{\mathrm{s}}\left(2^{\frac{R}{B_{\mathrm{s}}}}-1\right)}{\sigma_{\mathrm{ij}}^{2}G_{\mathrm{U1}}G_{\mathrm{U2}}\lambda_{\mathrm{s}}^{2}\ln\left(1-P_{\mathrm{out}}\right)}$.

As discussed before, case $\delta=0$ implies that both U1 and U2 succeed in decoding each others signals through short range communications, and $\delta=1$ means that either U1 or U2 (or both) fails to decode in the short-range transmissions. We can describe $\delta=0$ and $\delta=1$ as follows.

$\delta=0$: $$B_{\mathrm{s}}\log_{2}\left(1+\frac{\gamma_{12}^{\mathrm{NC}}}{\Delta_{\mathrm{s}}}\right)\geqslant R\,{\rm and}\,B_{\mathrm{s}}\log_{2}\left(1+\frac{\gamma_{21}^{\mathrm{NC}}}{\Delta_{\mathrm{s}}}\right)\geqslant R.\label{eq:xita0}$$

$\delta=1:$ $$B_{\mathrm{s}}\log_{2}\left(1+\frac{\gamma_{12}^{\mathrm{NC}}}{\Delta_{\mathrm{s}}}\right)<R\,{\rm or}\,B_{\mathrm{s}}\log_{2}\left(1+\frac{\gamma_{21}^{\mathrm{NC}}}{\Delta_{\mathrm{s}}}\right)<R.\label{eq:xita1}$$

Denote the target outage probability for short-range communication between U1 and U2 by $P_{{\rm out}}$, given $P_{\mathrm{out}1\mathrm{2}}^{\mathrm{NC}}=P_{\mathrm{out21}}^{\mathrm{NC}}=P_{\mathrm{out}}$, we have $$\Pr\left(\delta=0\right)=\left(1-P_{{\rm out}}\right)^{2},$$ and $$\Pr\left(\delta=1\right)=1-\left(1-P_{{\rm out}}\right)^{2}.$$

Moreover, denote the target outage probability for cellular communication from U1, U2 to the BS by $P_{\mathrm{out}}^{{\rm NC}}$, and given $P_{\mathrm{out}1\mathrm{b}}^{\mathrm{NC}}=P_{\mathrm{out2b}}^{\mathrm{NC}}=P_{\mathrm{out}}^{{\rm NC}}$, we have $\Pr\left(C_{\mathrm{1b}}<R\right)=\Pr\left(C_{\mathrm{2b}}<R\right)$, then we obtain the outage probability of the NNCC scheme by from and as

$P_{\mathrm{out}}=\left(1-P_{{\rm out}}\right)^{2}*\left(P_{\mathrm{out}}^{{\rm NC}}\right)^{2}+\left(1-\left(1-P_{{\rm out}}\right)^{2}\right)*\left(1-\left(1-P_{\mathrm{out}}^{{\rm NC}}\right)^{2}\right)$.

Then, we obtain $$P_{\mathrm{out}}^{{\rm NC}}=\frac{\sqrt{\left(1-\epsilon\right)^{2}+P_{\mathrm{out}}\left(2\epsilon-1\right)}-\left(1-\epsilon\right)}{2\epsilon-1}\text{,}\label{eq:p11}$$ where $\epsilon=\left(1-P_{{\rm out}}\right)^{2}$.

We can obtain the power consumption of Ui for cellular communication from as $$\mathcal{P}_{\mathrm{ib}}^{\mathrm{NC}}=-\frac{16\pi^{2}\triangle_{\mathrm{c}}N_{0}B_{\mathrm{c}}\left(2^{\frac{R}{B_{\mathrm{c}}}}-1\right)}{\sigma_{\mathrm{ib}}^{2}G_{\mathrm{Ui}}G_{\mathrm{BS}}\lambda_{\mathrm{c}}^{2}\ln\left(1-P_{\mathrm{out}}^{{\rm NC}}\right)}r_{i}^{2}.\label{eq:p1bnc1}$$

Given $\sigma_{\mathrm{1b}}^{2}=\sigma_{\mathrm{2b}}^{2}$, we can obtain $\mathcal{P}_{\mathrm{1b}}^{\mathrm{NC}}=\eta r_{1}^{2}$ and $\mathcal{P}_{\mathrm{2b}}^{\mathrm{NC}}=\eta r_{2}^{2}$, where $\eta=-\frac{16\pi^{2}\triangle_{\mathrm{c}}N_{0}B_{\mathrm{c}}\left(2^{\frac{R}{B_{\mathrm{c}}}}-1\right)}{\sigma_{\mathrm{ib}}^{2}G_{\mathrm{U2}}G_{\mathrm{BS}}\lambda_{\mathrm{c}}^{2}\ln\left(1-P_{\mathrm{out}}^{{\rm NC}}\right)}$.

Notice that in case of $\delta=0$, cooperation communication is employed and there are energy consumption at both time slot 2 and time slot 3, resulting in that a total power consumption of $2(\mathcal{P}_{\mathrm{1b}}^{\mathrm{NC}}+\mathcal{P}_{\mathrm{2b}}^{\mathrm{NC}})$ is consumed by U1 and U2 in transmitting to BS. In case of $\delta=1$, U1 and U2 consume a total power consumption of $(\mathcal{P}_{\mathrm{1b}}^{\mathrm{NC}}+\mathcal{P}_{\mathrm{2b}}^{\mathrm{NC}})$ for transmitting to BS. Therefore, considering both the short-range communication and cellular transmissions, the total power consumption by the NNCC scheme is given by $$\begin{aligned}
 & \mathcal{P}^{\mathrm{NC}}\\
 & =\mathcal{P}_{\mathrm{12}}^{\mathrm{NC}}+\mathcal{P}_{\mathrm{21}}^{\mathrm{NC}}+\left(2Pr\left(\delta=0\right)+Pr\left(\delta=1\right)\right)(\mathcal{P}_{\mathrm{1b}}^{\mathrm{NC}}+\mathcal{P}_{\mathrm{2b}}^{\mathrm{NC}})\\
 & =\mathcal{P}_{\mathrm{12}}^{\mathrm{NC}}+\mathcal{P}_{\mathrm{21}}^{\mathrm{NC}}+(1+(1-P_{{\rm out}})^{2})(\mathcal{P}_{\mathrm{1b}}^{\mathrm{NC}}+\mathcal{P}_{\mathrm{2b}}^{\mathrm{NC}}).\end{aligned}$$

In order to compare our method and the traditional method, we give the same target interrupt probability definition about two schemes, and then we will derive the power consumption for the conventional scheme without user cooperation, in which both U1 and U2 succeed in transmitting their data to the BS separately. Similarly to the power consumption analysis of the NNCC scheme, assuming that $P_{\mathrm{out1b}}^{\mathrm{C}}=P_{\mathrm{out2b}}^{\mathrm{C}}=P_{{\rm out}}^{{\rm C}}$, the total power consumption of U1 and U2 under conventional non-cooperative communication can be obtained by $$\mathcal{P}^{\mathrm{C}}=\sum_{i=1}^{2}\mathcal{P}_{\mathrm{ib}}^{\mathrm{C}},\label{eq:pt}$$ where $$\mathcal{P}_{\mathrm{ib}}^{\mathrm{C}}=-\frac{16\pi^{2}\triangle_{\mathrm{c}}N_{0}B_{\mathrm{c}}r_{\mathrm{i}}^{2}\left(2^{\frac{R}{B_{\mathrm{c}}}}-1\right)}{\sigma_{\mathrm{ib}}^{2}G_{\mathrm{Ui}}G_{\mathrm{BS}}\lambda_{\mathrm{c}}^{2}\ln\left(1-P_{{\rm out2}}^{{\rm C}}\right)},\label{eq:p1bt}$$ and

$$P_{{\rm out}}^{{\rm C}}=1-\sqrt{1-P_{\mathrm{out}}}.\label{eq:pOUT12C}$$

As and show, the NNCC scheme can save more energy than the conventional scheme because it can work under larger target outage probability. and give the relations between the desired transmission powers and the target outage probabilities as well as other impact factors, e.g., path loss, fading, and thermal noise, under the NNCC scheme and the conventional non-cooperative scheme, respectively. Based on them, some further performance analysis, such as the energy efficiency analysis in Section , can be presented.

Energy Efficiency Analysis based on PPP\[sub:Energy-Efficiency-Analysis\] 
--------------------------------------------------------------------------

In order to get more performance analysis about the NNCC scheme, we put the NNCC scheme on a more general environment, we will consider MSs satisfy Poisson point process, which meet actual situation. The result is more practical and performance analysis is more accurate. Moreover, considering the stochastic spatial distribution of the MSs, we can derive the CDF and PDF of the desired transmission power which meet the target outage probability and rate requirement.

\[thm:pdf\_cdf\_ee\] When the spatial distribution of the MSs follows a homogeneous PPP with density $\rho$, the PDF of is and the CDF of is where $$\varepsilon=1+(1-P_{{\rm out}})^{2}\text{,}\label{eq:I}$$ $$R_{2}=\frac{-\varepsilon\eta r_{1}\cos\theta+\varDelta_{R}}{2\zeta+\varepsilon\eta},\label{eq:r2}$$ and $$R_{1}=\frac{-\varepsilon\eta r_{1}\cos\theta-\varDelta_{R}}{2\zeta+\varepsilon\eta}.\label{eq:r1}$$ $Q_{1}$ stands for and $Q_{2}$ stands for

Due to independence of random variables $r$ and $\theta$, considering $\theta$ follows the uniform distribution between $-\frac{\pi}{2}$ and $\frac{3\pi}{2}$, $r$ satisfies , the PDF of $r$ and $\theta$ can be obtained as $$f(r,\theta)=\rho r\exp\left(-\pi\rho r^{2}\right).\label{eq:frjiao-1-1-1}$$ Substituting into , we obtain $$\mathrm{\mathcal{P}}^{\mathrm{NC}}=\left(2\zeta+\varepsilon\eta\right)r^{2}+2\varepsilon\eta r_{1}\cos\theta r+2\varepsilon\eta r_{1}^{2}.\label{eq:grjiao}$$ So the the CDF of $\mathrm{\mathcal{P}}^{\mathrm{NC}}$ is

$$\begin{aligned}
 & F_{\mathcal{P}^{\mathrm{NC}}}\left(p^{{\rm NC}}\right)=\Pr\left(P^{{\rm NC}}\leqslant p^{{\rm NC}}\right)\nonumber \\
 & =\Pr\left(\left(2\zeta+\varepsilon\eta\right)r^{2}+2\varepsilon\eta r_{1}\cos\theta r+2\varepsilon\eta r_{1}^{2}\leqslant p^{{\rm NC}}\right).\label{eq:cdf_pnc}\end{aligned}$$

Based on and , we can acquire the CDF and PDF of desired power for NNCC scheme as expresses.

From , we will know the PDF and CDF of $\mathrm{\mathcal{P}}^{\mathrm{NC}}$ and can easily obtain its expectation. It can help us to find the energy efficiency. Based on , it is easy to derive the relations between desired transmission power of the NNCC scheme and the target outage probability as well as other impact factors. Considering a successful transmission delivering both data of U1 and U2 by the total power consumption $\mathrm{\mathcal{P}}^{\mathrm{NC}}$, the energy efficiency of the NNCC scheme can be derived by $$EE^{\mathrm{NC}}=\frac{2R}{E\left(\mathrm{\mathrm{\mathcal{P}}^{\mathrm{NC}}}\right)}.\label{eq:ENERGY EFFICIENCY}$$

In , some numerical results about the desired transmission power and the energy efficiency of the NNCC scheme are given.

Numerical Results\[sec:Numerical-Results\]
==========================================

In this section, we present the analytical results of the proposed NNCC scheme and compare it to the existed schemes. We compare different schemes under the same definition of target outage probability. In our simulation, the frequency and bandwidth of the short-range communication are given as $f_{{\rm s}}=2.4{\rm \,GHz}$ and ${\rm B_{s}}=2{\rm \,MHz}$, respectively. In the cellular communication they are given as $f_{{\rm c}}=2100{\rm \,MHz}$ and ${\rm B}_{{\rm c}}=5\,{\rm MHz}$, respectively. The antenna gains of U1 and U2 are set as $G_{{\rm U1}}=G_{{\rm U2}}=0\,{\rm dB}$ and the BSs antenna gain is set as $G_{{\rm BS}}=5\,{\rm dB}$. The performance gaps $\Delta_{{\rm s}}$ and $\Delta_{{\rm c}}$ of short-range and cellular communication are given by $\Delta_{{\rm s}}=4\,{\rm dB}$ and $\Delta_{{\rm c}}=2\,{\rm dB}$.

![\[fig:power changes with distance\]Energy consumption by various transmission schemes with target outage probability $P_{\mathrm{out}}=10^{-3}$, effective rate $R=100000\,\mathrm{bits/s}$, and inter-user distance $r=20\,\mathrm{m}$. ](power_3_kinds){width="0.94\columnwidth"}

Energy consumption is influenced by some parameters, such as distances between the MSs and the BS, distance between MSs, the density of MS and the target outage probability. In Fig. \[fig:power changes with distance\], we discuss the impact of distances between the MSs and the BS on energy consumption. We compare the desired transmission power of the NNCC scheme to the conventional non-cooperative scheme and the inter-network cooperative communication scheme by orthogonal matrix proposed in [@zou2013tcom]. The result shows that the power consumption of the NNCC scheme is significantly lower than the non-operative scheme and slightly higher than the inter-network cooperative scheme. It demonstrates the effectiveness of the NNCC scheme, while considering the simplicity and feasibility of the NNCC scheme, which does not need the knowledge of the instantaneous status of the channel and can be easily applied in the scenario of the MSs with stochastic spatial distribution. Similarly to Fig. \[fig:power changes with distance\], we present the comparison of the energy efficiency among the schemes in Fig. \[fig:Energy efficiency two way\].

![\[fig:Energy efficiency two way\]Energy efficiency by various transmission schemes with target outage probability $P_{\mathrm{out}}=10^{-3}$, effective rate $R=100000\mathrm{\,bits/s}$, and inter-user distance $r=20\mathrm{\,m}$. ](ee_comp_3_kinds){width="0.94\columnwidth"}

In Fig. \[fig:power and density \], we present the relations between the desired transmission power consumption and the density of the MSs under different distances between the MS U1 and the BS. It is shown that the power consumption decreases as the density of MSs increases, because the distance between the cooperative MSs tends to be smaller when the density of the MSs increases, and thus less power consumption is required in step 1. Fig. \[fig:power and outage probability\] demonstrates that the energy consumptions of the NNCC scheme decreases as the target outage probability increases with various densities of the MSs.

![\[fig:power and density \]Relation between the desired the transmission power and the density of MSs, with target outage probability $P_{\mathrm{out}}=10^{-3}$, effective rate $R=10000000\mathrm{\,bits/s}$, and U1-BS distance $r_{1}=2000\mathrm{\,m}$. ](rel_density_power){width="0.94\columnwidth"}

![\[fig:power and outage probability\]Desired transmission power with regard to the target outage probability with effective rate $R=1000000\,\mathrm{bits/s}$, and U1-BS distance $r_{1}=150\mathrm{\,m}$.](power_in_diff_out_prob){width="0.94\columnwidth"}

Conclusions\[sec:Conclusions\]
==============================

In this paper, we propose a cooperative communication scheme, namely the NNCC scheme, in which a MS in HCN and its nearest neighbor MS exploit a short-range communication network to assist the cellular transmissions. The energy efficiency of the NNCC scheme is analyzed and derived in closed-form expressions, which provide insights into the relations between energy efficiency and many important factors, e.g., MS density, the distance between the MSs and the BS, the target outage probability and etc.. Numerical results show that the NNCC scheme is simple, yet efficient compared to the existing schemes. Although this article studies collaboration between two MSs only, it’s obvious that there are vast energy saving comparing with the traditional scheme, no matter from the aspects of the distance between BS and MSs or expectation. In this paper, we have only considered the scenario of a single BS and allow only two MSs cooperate with each other. In the future, we will extend the work to the multi-cell scenario with multi-MS cooperative communications.

[^1]: The corresponding author is Lijun Wang. The authors would like to acknowledge the support from the International Science and Technology Cooperation Program of China (Grant No. 2014DFA11640, 2012DFG12250 and 0903), the National Natural Science Foundation of China (NSFC) (Grant No. 61471180, 61271224 and 61301128), the NSFC Major International Joint Research Project (Grant No. 61210002), the Ministry of Science and Technology 863 program (Grant No. 2014AA01A707), the Hubei Provincial Science and Technology Department (Grant No. 2013CFB188), the Fundamental Research Funds for the Central Universities (Grant No. 2013ZZGH009 2013QN136, and 2014QN155), and Special Research Fund for the Doctoral Program of Higher Education (Grant No. 20130142120044), and EU FP7-PEOPLE-IRSES (Contract/Grant No. 247083, 318992 and 610524).
