Energy efficiency analysis of relayassisted cellular networks
 Huan Yu^{1, 2},
 Yunzhou Li^{2}Email author,
 Marios Kountouris^{3},
 Xibin Xu^{2} and
 Jing Wang^{2}
https://doi.org/10.1186/16876180201432
© Yu et al.; licensee Springer. 2014
Received: 21 September 2013
Accepted: 7 March 2014
Published: 14 March 2014
Abstract
To meet the demand for higher throughput, improved coverage and enhanced reliability, future wireless cellular networks face significant technical challenges. One promising solution is to place relay stations between transmitters and receivers in the cellular network. Meanwhile, as energy consumption reduction has been an important concern for the wireless industry, energyefficient communications is of prime interest for future networks. In this paper, we study whether and how relays can improve the energy efficiency of cellular networks. Specifically, the energy efficiency of relayassisted cellular networks is analyzed using tools of stochastic geometry. We first derive the coverage probability for the macro base station (MBS) to user (UE), the MBS to relay station (RS), and the RS to UE links, and then we model the power consumption at the MBS and RS. Based on the analytical model and expressions, the energy efficiency of relayassisted cellular networks is then evaluated and is shown to be strictly quasiconcave on the transmit power for MBS to UE link or the RS to UE link. Numerical results show that the energy efficiency first improves while it hits a ceiling as the MBS density increases.
Keywords
1 Introduction
With the exponential growth of wireless traffic, future cellular networks face huge challenges for catering higher data rate and transmission reliability. One effective and promising way to meet the traffic demands is to deploy relay stations (RSs) in the traditional cellular networks, as a means to increase coverage, throughput, and reliability [1]. Meanwhile, as the energy consumption reduction has become a global trend, the research for energyefficient wireless networks has increasingly attracted attention recently [2].
There have been some works in the literature about energyefficient relayassisted networks from the link layer to the network layer [3–8]. For the link layer, the energy efficiency problem for multicarrier amplifyandforward (AF) relay links is considered in [3]. The optimization problem cast in the paper is nonconvex and is converted into a quasiconcave problem by two approximate methods, thus providing a suboptimal solution. In [4–6], energyefficient twoway relay links under different forwarding schemes are studied. The energy efficiency for oneway relay transmission and twoway relay transmission is compared when the AF strategy is adopted in [4], showing that when the bidirectional data amounts are equal, twoway relay transmission performs better; otherwise oneway relay transmission may offer higher energy efficiency. In [5], hybrid relay transmission transmitting partial messages with oneway relay transmission and the remaining messages with twoway relay transmission is proposed. Simulation results show that the performance of hybrid relay transmission is better than that of the twoway relay transmission and oneway relay transmission. Energy efficiency of the twoway relay link using decodeandforward (DF) strategy is investigated in [6] and an improved bisection algorithm is proposed to find the optimal energy efficiency. For the network layer, in [7, 8], energyefficient multirelay networks are studied, in which the source node sends massage to the destination node via the relay nodes and the best one or several relay nodes by a certain criterion are selected to forward the message from source node in order to achieve the highest energy efficiency.
In previous studies on the performance of traditional cellular networks, the cell shape is often supposed to be hexagonal or square, which may not be the case in realistic scenarios (e.g., due to shadowing) or in heterogeneous and uncoordinated networks. Besides, complex timeconsuming systemlevel simulations must be used. In [9], an approach to derive the coverage and rate in cellular networks based on stochastic geometry is proposed, which is more tractable compared to the traditional grid models. Under this approach, in [10], the design of energyefficient heterogeneous cellular networks through the employment of base station sleeping mode strategies as well as small cells are investigated, in which some methods can be considered in the relayassisted cellular networks. The relationship among the energy efficiency and the intensity of noncooperative users (UEs) and cooperative UEs is studied in [11] based on the stochastic geometry approach. In [12], the effect of base station density on the energy efficiency of relayassisted cellular networks is investigated using the stochastic geometry model. However, the paper only considers the special case where the path loss exponent α=4; while in our work, we derive expressions that are valid for any α>2.

Building on [9], a downlink relayassisted cellular network model is proposed based on the stochastic geometry theory.

The coverage probabilities for the MBSRS, the MBSUE, and the RSUE links are derived, and the power consumption for MBSs as well as RSs are modeled.

The energy efficiency expression for relayassisted cellular networks is analytically deduced under specific power consumption models.

The energy efficiency is proved to be strictly quasiconcave on the transmit power for MBS to UE link or RS to UE link, and an algorithm based on bisection method is proposed to find the optimal value.
The rest of this paper is organized as follows: in Section 2, the system model is described. In Section 3, the energy efficiency for the downlink relayassisted cellular networks is derived. Simulation results are shown in Section 4, and conclusions are given in Section 5.
2 System model
The distance between MUE (or RS) and the target MBS is denoted by r. As Θ_{ M } is a HPPP with density λ_{ M }, the probability density function (pdf) of r is f_{ r }(r)=2π λ_{ M }r exp(−π λ_{ M }r^{2}) [9]. Furthermore, since the distribution of RSs follows a HPPP and RUEs are distributed according to some independent stationary point process within each relay’s circular area, the distance between the RUE and its corresponding RS r follows a distribution with pdf f_{ R }(r)=2r/R^{2}.
3 Energy efficiency analysis
In this section, the coverage probabilities are first derived, and then, the energy efficiency expression is given for certain power consumption models.
3.1 Coverage probability
where P is the transmit power for node s, h is the channel power gain due to the smallscale fading, r is the distance between s and d, α is the path loss exponent, I is the aggregate interference from all the other active transmitters operating in the same frequency band, and σ^{2} is the variance of the additive white Gaussian background noise. All channels are assumed to be subject to Rayleigh fading, i.e., h∼ exp(1).
3.1.1 Coverage probability for the MBSUE link
where ${\text{SINR}}_{\text{MU}}=\frac{{P}_{\text{MU}}{h}_{0}{r}^{\alpha}}{{I}_{\text{MU}}+{\sigma}^{2}}$ and ${I}_{\text{MU}}=\phantom{\rule{1pt}{0ex}}\sum _{i\in {\Theta}_{M}\setminus \left\{{m}_{0}\right\}}\phantom{\rule{0.3em}{0ex}}{P}_{\text{MU}}{h}_{i}{r}_{i}^{\alpha}$ and f_{ r } (r)=2π λ_{ M }r exp(−π λ_{ M }r^{2}).
Theorem 1
where $\rho \left({\Gamma}_{\text{MU}},\alpha \right)\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}{\Gamma}_{\text{MU}}^{2/\alpha}{\int}_{{\Gamma}_{\text{MU}}^{2/\alpha}}^{\infty}\frac{1}{1+{v}^{\alpha /2}}\mathrm{d}v$.
In the interferencelimited regime where the background thermal noise is ignored (σ^{2}→0), the coverage probability can be simplified as ${p}_{c\text{\_MU}}=\frac{1}{1+\rho \left({\Gamma}_{\text{MU}},\alpha \right)}$.
Proof
where ${L}_{{I}_{\text{MU}}}\left(s\right)$ is the Laplace transform of I_{MU}.
The result of Equation 5 is deduced from the property of the probability generating functional (PGFL) for PPP, which for some function g(x), it satisfies $\mathbb{E}\left[{\prod}_{x\in \Theta}g\left(x\right)\right]=exp\left(\lambda {\int}_{{R}^{2}}\left(1g\left(x\right)\right)\mathrm{d}x\right)$[13], and λ is the density of the interference nodes.
Combining the results of Equations 5 and 6, Equation 4 is obtained.
When σ^{2}→0, it can be easily derived for Equation 4 that ${p}_{c\text{\_MU}}=\frac{1}{1+\rho \left({\Gamma}_{\text{MU}},\alpha \right)}$.
Using the Theorem 1, the following corollaries can be easily obtained.
Corollary 1
In the interferencelimited regime, the coverage probability p_{c _MU} increases with increasing P_{MU}, and when P_{MU} goes to infinity, $\underset{{{P}_{\text{MU}}\to \infty}_{}}{\text{lim}}{p}_{c\text{\_}\text{MU}}=\frac{1}{1+\rho \left({\Gamma}_{\text{MU}},\alpha \right)}$.
Corollary 2.
With σ^{2}>0, the coverage probability p_{c _MU} decreases with increasing Γ_{MU}.
3.1.2 Coverage probability for the MBSRS link
where ${\text{SINR}}_{\text{MR}}=\frac{{P}_{\text{MR}}{g}_{0}{r}^{\alpha}}{{I}_{\text{MR}}+{\sigma}^{2}}$, ${I}_{\text{MR}}=\sum _{i}{P}_{\text{MR}}{g}_{i}{r}_{i}^{\alpha}$, for i in all other active MRSs in the MBSRS frequency band except the target MBS m_{0}.
Theorem 2.
where λ = min{λ_{ M },λ_{ R }} and $\rho \left({T}_{\text{MR}},\alpha \right)\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}{\Gamma}_{\text{MR}}^{2/\alpha}{\int}_{{\Gamma}_{\text{MR}}^{2/\alpha}}^{\infty}\frac{1}{1+{v}^{\alpha /2}}\mathrm{d}v$.
In the interferencelimited regime, i.e., σ^{2}→0, the coverage probability can be simplified as ${p}_{c\text{\_MR}}=\frac{{\lambda}_{M}}{{\lambda}_{M}+\mathrm{\lambda \rho}\left({\Gamma}_{\text{MR}},\alpha \right)}$.
The derivation of the coverage probability ${p}_{c\text{\_MR}}$ is almost the same as that of ${p}_{c\text{\_MU}}$ in the previous subsection except that the density used for the derivation of the Laplace transform of I_{MR} is λ instead of λ_{ M }. As we assume that each MBS serves at most one RS at any time in each cell, the density of interference nodes is the smaller of λ_{ M } and λ_{ R }. Thus, the density used for the derivation of the Laplace transform of I_{MR} is λ= min{λ_{ M },λ_{ R }}.
Similar corollaries can be obtained as follows.
Corollary 3.
With σ^{2}>0, the coverage probability p_{c _MR} increases with increasing P_{MR}, and when P_{MR} goes to infinity, $\underset{{{P}_{\text{MR}}\to \infty}_{}}{\text{lim}}{p}_{c\text{\_MR}}=\frac{{\lambda}_{M}}{{\lambda}_{M}+\mathrm{\lambda \rho}\left({\Gamma}_{\text{MR}},\alpha \right)}$.
Corollary 4.
With σ^{2}>0, the coverage probability p_{c _MR} decreases with increasing Γ_{MR}.
3.1.3 Coverage probability for the RSUE link
where ${\text{SINR}}_{\text{RU}}=\frac{{P}_{\mathrm{R}}{l}_{0}{r}^{\alpha}}{{I}_{\text{RU}}+{\sigma}^{2}}$ and ${I}_{\text{RU}}=\sum _{i}{P}_{\mathrm{R}}{l}_{i}{r}_{i}^{\alpha}$, for i denoting all other active RSs except r_{0}.
As the distance between RS and UE follows a different distribution, the form of ${p}_{c\text{\_RU}}$ is not the same as for ${p}_{c\text{\_MR}}$ and ${p}_{c\text{\_MU}}$.
Theorem 3.
In the interferencelimited regime, the coverage probability can be expressed as${p}_{c\text{\_RU}}\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}\frac{1+\rho \left({\Gamma}_{\text{MR}},\alpha \right)}{\mathrm{\lambda \pi}{R}^{2}\mu \left({\Gamma}_{\text{RU}},\alpha \right)}\left(1exp\right.\left.\left(\frac{\mathrm{\lambda \pi}{R}^{2}\mu \left({\Gamma}_{\text{RU}},\alpha \right)}{1+\rho \left({\Gamma}_{\text{MR}},\alpha \right)}\right)\right)$.
Proof.
where step (a) follows from the fact that f_{ R }(r)=2r/R^{2} and l_{0}∼ exp(1).
where λ= min{λ_{ M },λ_{ R }} and $\mu \left({\Gamma}_{\text{RU}},\alpha \right)={\Gamma}_{\text{RU}}^{2/\alpha}{\int}_{0}^{\infty}\frac{1}{1+{v}^{\alpha /2}}\mathrm{d}v$.
One difference is the density used in the expression. At any time, there is at most only one RS communicating with the target MBS, and only the RSs with received SINR larger than the target SINR Γ_{MR} forward messages to the RUEs, so the density used in the above formula is $\lambda {p}_{c\text{\_MR}}$.
Besides, the distance between RUE and active RSs can be any value, so the integration limits start form 0 (not r) to ∞ in μ(Γ_{RU},α).
Combining Equations 11 and 13, the expression of ${p}_{c\text{\_RU}}$ is obtained.
Using Theorems 2 and 3, the following corollaries can be acquired.
Corollary 5.
With σ^{2}>0, the coverage probability p_{c _RU} increases with decreasing P_{MR} or increasing P_{RU}.
Corollary 6.
With σ^{2}>0, the coverage probability p_{c _RU} increases with increasing Γ_{MR} or decreasing Γ_{RU}.
Corollary 7.
When λ_{ R }<λ_{ M }, the coverage probability p_{c _RU} decreases with increasing λ_{ R }. When λ_{ R }≥λ_{ M }, the coverage probability p_{c _RU} remains the same with increasing λ_{ R }.
Proof.
When λ_{ R }<λ_{ M }, λ=λ_{ R }, increasing λ_{ R } can increase λ and decrease p_{c _RU}. However, when λ_{ R }≥λ_{ M }, λ=λ_{ M } as λ_{ R } rises, as a result, p_{c _RU} remains the same with increasing λ_{ R }.
3.2 Power consumption model
where 1/β_{ M }, 1/β_{ R } denote the efficiency of the power amplifier for MBS and RS, P_{ M } and P_{ R } account for the total transmit power for MBS and RS, and P_{M 0} and P_{R 0} are the static power consumption for MBS and RS, respectively.
3.3 Performance metrics
where τ_{ M } and τ_{ R } are the area spectrum efficiency over all the MBSUE links and MBSRSUE links, respectively. ${P}_{M\text{\_ave}}$ and ${P}_{R\text{\_ave}}$ denote the average network area power consumption for MBSs and RSs, respectively.
As at most λ= min(λ_{ M },λ_{ R }) MBSs per square meter transmit signals to the RSs, the density used for P_{MR} is λ.
Note that the 1/2 factor is due to halfduplex operation of the relay nodes, and $\frac{{w}_{R}}{{w}_{M}+{w}_{R}}$ is the rate of occupancy of the frequency bands for the MBSRSUE link.
Combining Equations 17 to 21, the expression of energy efficiency for the relayassisted cellular networks is obtained.
The following theorem demonstrates the quasiconcavity of energy efficiency function.
Theorem 4.
The energy efficiency function η_{EE} is strictly quasiconcave on P_{MU} or P_{RU}.
Proof.
We prove η_{EE} is strictly quasiconcave on P_{MU} here, and the process of the proof that η_{EE} is strictly quasiconcave on P_{RU} is similar. As defined in [17], a function f:R^{ n }→R is said to be strictly quasiconcave if its sublevel set S_{ α }={xx∈d o m f,f(x)≥α} is strictly convex for every α. When α≤0, S_{ α } is obviously convex on P_{MU}. When α>0, S_{ α } is equivalent to ${S}_{\alpha}=\left\{{P}_{\text{MU}}\alpha \left({P}_{M\text{\_ave}}+{P}_{R\text{\_ave}}\right){\tau}_{M}{\tau}_{R}\le 0\right\}$. It can be easily proved that p_{c _MU} is strictly concave on P_{MU}, so −τ_{ M } is strictly concave on P_{MU}. Besides, ${P}_{M\text{\_ave}}$ linearly increases with P_{MU}, ${P}_{R\text{\_ave}}$ and τ_{ R } are not relevant with P_{MU}. As a result, S_{ α } is also strictly convex on P_{MU} and Theorem 4 follows.
Due to the strict quasiconcavity of η_{EE} on P_{MU} or P_{RU}, the optimal energy efficiency exists for various values of P_{MU} or P_{RU}.
The above problem aims at finding the optimal energy efficiency under the constraint that the coverage probability for the MBSUE link is larger than or equal to a certain threshold θ. As η_{EE} is strictly quasiconcave on P_{MU}, the bisection method can be used to find the optimal η_{EE} and the corresponding P_{MU} (denoted P_{MU_o p t}). If p_{c _MU}(P_{MU_o p t})≥θ, then η_{EE}(P_{MU_o p t}) is the optimal value. Else, as p_{c _MU} strictly increases monotonically on P_{MU}, P_{MU_o p t} is less than the power ${\stackrel{\u0304}{P}}_{\text{MU}\text{\_}\mathit{\text{opt}}}$ which makes ${p}_{c\text{\_}\text{MU}}\left({\stackrel{\u0304}{P}}_{\text{MU}\text{\_}\mathit{\text{opt}}}\right)=\theta $ and ${\eta}_{\text{EE}}\left({\stackrel{\u0304}{P}}_{\text{MU}\text{\_}\mathit{\text{opt}}}\right)$ is the optimal value according to the quasiconcavity of η_{EE}. The detailed algorithm to find the optimal η_{EE} based on bisection method is described in Algorithm 1.
Algorithm 1 The algorithm to find optimal η_{ EE }based on the bisection method
4 Simulation results
Parameter values used in the simulations
Parameter  Value 

λ_{ M }, λ_{ R }  10^{−5} m^{−2}, 10^{−4} m^{−2}, 
P_{MU}, P_{RU}, P_{MR}  43 dBm, 30 dBm, 33 dBm 
R  40 m 
α  4 
w_{ M }, w_{ R }  40 MHz, 4 MHz 
Γ_{MR}, Γ_{RU}, Γ_{MR}  10 dB, 10 dB, 10 dB 
σ ^{2}  −60 dBm 
β_{ M }, β_{ R }  5.32, 4.8 
P_{M0}, P_{R0}  118.7 W, 7.5 W 
4.1 Simulation results for coverage probability
4.2 Simulation results for energy efficiency
5 Conclusions
In this paper, we study the energy efficiency of downlink relayassisted cellular networks where MBSs and RSs are distributed according to independent HPPPs. First, we build on a tractable stochastic geometrybased model to derive the coverage probabilities for MBSUE, MBSRS, and RSUE links. Second, using power consumption models for both macrocell and relay networks, we obtain expressions for the network energy efficiency. Our results reveal that there is a fundamental tradeoff between energy efficiency and transmit MBS power, and the main takeaway is that deploying more MBSs generally triggers higher energy efficiency; however this gain hits a ceiling as the density of MBSs increases.
Declarations
Acknowledgements
This work is supported by the National Basic Research Program of China (No. 2012CB316000), the National Natural Science Foundation of China (No. 61201192), the National S&T Major Project (No. 2013ZX03004007), International Science and Technology Cooperation Program (No. 2012DFG12010), China’s 863 Project (No. 2012AA011402), Keygrant Project of Chinese Ministry of Education (No. 313005), the Open Research Fund of National Mobile Communications Research Laboratory, Southeast University (No. 2012D02), and TsinghuaQualcomm Joint Research Program.
Authors’ Affiliations
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