 Research
 Open access
 Published:
Energy efficiency analysis of relayassisted cellular networks
EURASIP Journal on Advances in Signal Processing volume 2014, Article number: 32 (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.
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.
In our work, we analyze the energy efficiency for relayassisted cellular networks using models based on stochastic geometry [9], where the macro base stations (MBSs) and relay stations(RSs) locations are distributed according to independent homogeneous Poisson point process (HPPP). The mobile users (UEs) in the network are divided into two types, i.e., UEs that communicate directly with MBSs (MUEs) and UEs that communicate with MBSs via the help of RSs (RUEs). The main contributions of this paper are summarized as follows:

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
We consider a relayassisted cellular network where the MBSs and RSs are located according to independent HPPPs Θ_{ M } and Θ_{ R } with densities λ_{ M } and λ_{ R }, respectively, in the Euclidean plane. Halfduplex relay nodes using DF strategy are considered, where each RS connects to the geographically closest MBS and has coverage of a disk of radius R, i.e., the RSs deployed in the Voronoi cell of an MBS are connected with it (cf. Figure 1). UEs that communicate directly with the MBS (MUE) are distributed according to some independent stationary point process and connect with the closest MBS. Meanwhile, UEs that communicate with the MBS via the RS (RUE) are arranged with reference to some independent stationary point process within each RS’s circular area. Note that these two different types of UEs are not distinguished in Figure 1.
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}.
As shown in Figure 2, in each cell, the MBSUE link and the MBSRSUE link work in nonoverlapping frequency bands with bandwidth ω_{ M } and ω_{ R }, respectively. For the MBSRSUE link with halfduplex relays, the transmission is divided into two phrases. In the first phrase, the MBS sends messages to the RS, and in the second phrase, the RS decodes the messages and forwards them to the user. The time duration of both phases is equal. We assume that each MBS serves at most one MUE and RS at any time. If more than one MUE (RS) are located in the MBS’s cell, orthogonal resource sharing such as time division multiple access (TDMA) is performed. Similarly, each RS serves one RUE at any time and TDMA may be used if multiple RUEs are distributed in the RS’s serving area. Once TDMA is selected, the MBS can schedule the resource and select the RS to work with, the RSs only need to know which one is selected.
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
The signaltointerferenceplusnoise ratio (SINR) in the downlink from the transmitter (MBS or RS) s to the receiver (RS or UE) d is given by
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).
The coverage probability is defined as the probability that the receive SINR is above a certain threshold Γ and can be written as
3.1.1 Coverage probability for the MBSUE link
First, we assume that MUEs that are directly connected to MBS are distributed according to an independent stationary point process. Second, the transmit power from MBS to MUE is P_{MU}, the receive threshold is denoted as Γ_{MU}, and I_{MU} is the aggregate interference from all the other active MBSs except the target MBS m_{0}. Then, the coverage probability for a typical MBSUE link is defined as
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
The coverage probability for the MBSUE link {p}_{c\text{\_MU}} is given by
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
The details of the derivation can be found in [9], and we just provide here the key points. From Equation 3, the definition of SINR and the assumption that h_{0}∼ exp(1), the coverage probability can be rewritten as
where {L}_{{I}_{\text{MU}}}\left(s\right) is the Laplace transform of I_{MU}.
Considering the definition of Laplace transform and the assumption that h_{ i }∼ exp(1), {L}_{{I}_{\text{MU}}}\left(s\right) is given by
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
Suppose the relay can successfully decode the signals from the MBS if the received SINR is larger than or equal to Γ_{MR}, then the coverage probability for the MBSRS link can be written as
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.
The coverage probability for the MBSRS link {p}_{c\text{\_MR}} is given by
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
We assume that RUEs receiving messages by the aid of RSs are located according to some stationary point process in a circular area around the RS and of radius R. We denote P_{R} as the transmit power of the RS, Γ_{RU} as the receive threshold, and I_{RU} as the sum of interference of all other active RSs except the target RS r_{0}. The coverage probability for a typical RSUE link is defined as
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.
The coverage probability for the MBSRS link {p}_{c\text{\_RU}} is given by
where λ= min{λ_{ M },λ_{ R }} and
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(\right)separators="">\n \n \n \n \u2212\n \n \n \lambda \pi \n \n \n R\n \n \n 2\n \n \n \mu \n \n \n \n \n \Gamma \n \n \n RU\n \n \n ,\n \alpha \n \n \n \n \n 1\n +\n \rho \n \n \n \n \n \Gamma \n \n \n MR\n \n \n ,\n \alpha \n \n \n \n \n \n \n \n.
Proof.
From the definition of {p}_{c\text{\_RU}}, SINR_{RU}, and I_{RU}, we can rewrite {p}_{c\text{\_RU}} as
For analytical convenience, we assume that {\u0128}_{\text{RU}}={I}_{\text{RU}}+{P}_{\mathrm{R}}{l}_{0}{r}^{\alpha}, which encompasses the signals from all the transmitting RSs including r_{0}. We substitute I_{RU} with ({\u0128}_{\text{RU}}{P}_{\mathrm{R}}{l}_{0}{r}^{\alpha}) and the coverage probability {p}_{c\text{\_RU}} is given by
where step (a) follows from the fact that f_{ R }(r)=2r/R^{2} and l_{0}∼ exp(1).
The derivation of {L}_{{\u0128}_{\text{MR}}} is similar to that of {L}_{{I}_{\text{MU}}} except some differences as shown below.
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.
When σ^{2}→0, {p}_{c\text{\_MR}}=\frac{1}{1+\rho \left({\Gamma}_{\text{MR}},\alpha \right)} and then {p}_{c\text{\_RU}} can be written as
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
In this section, we model the power consumption of MBSs and RSs in downlink relayassisted cellular networks. The relation between total power consumption P_{tot} and transmit radiated power P_{ T } is modeled as [14–16]P_{tot}=β P_{ T }+P_{0}, where 1/β is the efficiency of the power amplifier, and P_{0} is the static power consumption, which includes signal processing overhead, battery backup, cooling power consumption, etc. As a result, for MBS and RS, the total power consumption P_{M _tot}, P_{R _tot} can be given by
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
We define the energy efficiency for the relayassisted cellular networks as
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.
The throughput attained at the MBSUE link is given by {p}_{c\text{\_MU}}\frac{{w}_{M}}{{w}_{M}+{w}_{R}}\underset{2}{log}\left(1+{\Gamma}_{\text{MU}}\right) where \frac{{w}_{M}}{{w}_{M}+{w}_{R}} is the rate of occupancy of the frequency bands for the MBSUE link. Thus, the area spectrum efficiency over all MBSUE links is
The MBS sends messages to MBSUE with transmit power P_{MU}, and to the RS with transmit power P_{MR} only in the one of the two phrases, so the average macrocell network area power consumption is
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.
As a DF strategy is adopted by the RS, then the MBSRSUE link operates only if both the relay and the user decode the messages received successfully, namely, the SINR attained at the RS and the RUE must be larger than Γ_{MR} and Γ_{RU}, respectively. Thus, the area spectrum efficiency for all the MBSRSUE links is given by
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.
At any time, the number of RSs taking part in the communication is at most the same as the number of MBSs (λ relays per square meters), and only the relays decoding signals for MBSs successfully forward signals to corresponding RUEs (\lambda {p}_{c\text{\_MR}} relays per square meter), the other relays (λ_{ R }−λ p_{c _MR} relays per square meter) do not send signals but only consume the static power part. Thus, the average network area power consumption for RSs is as follows:
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}.
We now study the following optimization problem:
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
In this section, we evaluate the derived analytical results through simulation. We use the default values of the system model in Table 1 unless otherwise stated. The simulations are divided into two parts: coverage probability and energy efficiency.
4.1 Simulation results for coverage probability
In this part, we evaluate the relationship between coverage probability and transmit power of MBS or MBS density. Figure 3 shows the coverage probability for the MBSUE link p_{c _MU} versus the transmit power of MBS P_{MU} for different target SINRs Γ_{MU}. It can be seen from the figure that the coverage probability increases with the transmit power, and the growth rate is high when the transmit power is relatively small, whereas it approaches zero when the transmit power is relative large. Besides, as the target SINR increases, the coverage probability decreases, and higher target SINR results in larger changes of coverage probability with the variation of transmit power. Note that for Γ_{MU} = 20 dB, the curve is almost flat, while for Γ_{MU} up to 0 dB, the curve becomes the most curved.
Figure 4 plots the coverage probability for the MBSRS link p_{c _MR} versus the MBS density λ_{ M } for different values of pathloss exponent α. The curves can be divided into two parts due to the value of λ. When λ_{ M } is smaller than that of RS λ_{ R }, λ equals to λ_{ M }, and for pathloss exponent increasing, the curves become more and more curved. When the pathloss exponent is 3, the coverage probability remains almost the same as the density changes in this range. However, when the pathloss exponent becomes 4 or 5, the coverage probability increases as the density grows. Moreover, when λ_{ M }≥λ_{ R }, λ equals to the constant value λ_{ R }, then all the curves increase slowly with λ_{ M }.
The relationship between coverage probability for RSUE link p_{c _RU} and density of MBS λ_{ M } for different pathloss exponents α is shown in Figure 5. Compared to p_{c _MR} or p_{c _RU}, the expression of p_{c _RU}, which is relevant to p_{c _MR} and the density of both MBSs and RSs, is quite different and causes different curve shapes. When λ_{ M } is smaller than λ_{ R }, the value of λ is λ_{ M }, and the coverage probabilities almost linearly decrease as λ_{ M } increases. However, when λ_{ M } becomes larger than λ_{ R }, the value of λ is the constant value λ_{ R }, and then the coverage probabilities just slightly decrease as the density of MBS increases. This is because when λ_{ M } becomes larger than λ_{ R }, p_{c _RU} only varies with p_{c _MR}. From Figure 4, we can see that p_{c _MR} slightly increases when λ_{ M } is larger than λ_{ R }, which results in the shape of p_{c _RU}.
4.2 Simulation results for energy efficiency
The energy efficiency for downlink relayassisted cellular networks is simulated in this part. Figure 6 shows how the energy efficiency varies with the density of MBSs for different densities of RSs. It can be seen that the energy efficiency increases with the density of MBSs and saturates when the density goes to infinity. That is to say, the growth of area spectral efficiency is faster than the growth of average network power consumption as the density of MBSs increases. But when the density of MBSs is relatively high, continuously increasing MBSs cannot further increase the energy efficiency. In addition, higher density of RSs brings about lower energy efficiency in the case of all other parameters constant. This is due to the access assumption in our model. We assume that each MBS serves at most one MUE and RS at any time, and also, each RS serves one RUE at any time. As a result, if the density of RSs is larger than that of MBSs, on average only λ_{ M } RSs per square meter can be served at a time, the area spectral efficiency remains the same but the average area power consumption, mainly the average area static power consumption of RSs increases with the growth of density of RSs.
Figure 7 displays the energy efficiency with respect to transmit power of MBS P_{MU} for different values of RS densities. As predicted in Theorem 4, the energy efficiency first increases and then decreases, showing a quasiconvex trend as the transmit power varies. The optimal energy efficiency can be found using the algorithm shown in Table 1. If the threshold θ is set to zero and the tolerance ε is 10^{−5}, the optimal energy efficiency is 6.40×10^{−4}bps/Hz/W, 5.26×10^{−4}bps/Hz/W and 4.48×10^{−4}bps/Hz/W corresponding to the three curves when λ_{ R }=5×10^{−5}m^{−2},1×10^{−4}m^{−2},1.5×10^{−4}m^{−2}, respectively. Besides, the figure also shows that while the density of RSs grows, the energy efficiency drops, which means that the growth of area spectrum efficiency by increasing the density of RSs cannot compensate for the growth of average network power consumption.
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.
References
Laneman JN, Tse DNC, Wornell GW: Cooperative diversity in wireless networks: efficient protocols and outage behavior. IEEE Trans. Inf. Theory 2004, 50(12):30623080. 10.1109/TIT.2004.838089
Li GY, Xu Z, Xiong C, Yang C, Zhang S, Chen Y, Xu S: Energyefficient wireless communications: tutorial, survey, and open issues. IEEE Wireless Commun. Mag 2011, 18(6):2835.
Yu H, Qin H, Li Y, Zhao Y, Xu X, Wang J: Energyefficient power allocation for nonregenerative OFDM relay links. Sci. China Information Sci 2013, 56(2):185192.
Sun C, Yang C: Is twoway relay more energy efficient? in Proceedings of the Global Telecommunications Conference (GLOBECOM 2011), 2011 IEEE . Houston, Texas, USA: IEEE; 2011:16.
Sun C, Yang C: Energy efficient hybrid oneway and twoway relay transmission strategy. In Proceedings of the Acoustics, Speech and Signal Processing (ICASSP), 2012 IEEE International Conference on. Kyoto, Japan: IEEE; 2012:24972500.
Yu H, Li Y, Zhong X, Wang L, Wang J: The analysis of the energy efficiency for the decodeandforward twoway relay networks. 2013.
Madan R, Mehta N, Molisch A, Zhang J: Energyefficient cooperative relaying over fading channels with simple relay selection. IEEE Trans. Wireless Commun 2008, 7(8):30133025.
Huang R, Feng C, Zhang T, Wang W: Energyefficient relay selection and power allocation scheme in AF relay networks with bidirectional asymmetric traffic. In Proceedings of the Wireless Personal Multimedia Communications (WPMC), 2011 14th International Symposium on. Brest, France: IEEE; 2011:15.
Andrews JG, Baccelli F, Ganti RK: A tractable approach to coverage and rate in cellular networks. IEEE Trans. Commun 2011, 59(11):31223134.
Soh YS, Quek TQS, Kountouris M, Shin H: Energy efficient heterogeneous cellular networks. IEEE J. Selec. Areas Commun 2013, 31(5):840850.
Deng N, Zhang S, Zhou W, Zhu J: A stochastic geometry approach to energy efficiency in relayassisted cellular networks. In Proceedings of the Global Telecommunications Conference (GLOBECOM 2012), 2012 IEEE. Anaheim, California, USA: IEEE; 2012:34843489.
Wei H, Deng N, Zhao M, Zhou W, Dong P: Station density effect on energy efficiency of relayassisted cellular networks. In Proceedings of the 1st IEEE International Conference on Communications in China (ICCC). Beijing, China: IEEE; 2012:411415.
Stoyan D, Kendall WS, Mecke J, Ruschendorf L: Stochastic Geometry and Its Applications, Volume 2, Chichester, UK. Chichester: Wiley; 1995.
Auer G, Giannini V, Desset C, Godor I, Skillermark P, Olsson M, Imran MA, Sabella D, Gonzalez MJ, Blume O, Fehske A: How much energy is needed to run a wireless network? IEEE Wireless Commun. Mag 2011, 18(5):4049.
Fehske AJ, Richter F, Fettweis GP: Energy efficiency improvements through micro sites in cellular mobile radio networks. In GLOBECOM Workshops, 2009 IEEE. Hawaii, USA: IEEE; 2009:15.
Arnold O, Richter F, Fettweis G, Blume O: Power consumption modeling of different base station types in heterogeneous cellular networks. In Future Network and Mobile Summit, 2010. Florence, Italy: IEEE; 2010:18.
Boyd SP, Vandenberghe L: Convex Optimization. Cambridge, UK: Cambridge University Press; 2004.
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ original submitted files for images
Below are the links to the authors’ original submitted files for images.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Yu, H., Li, Y., Kountouris, M. et al. Energy efficiency analysis of relayassisted cellular networks. EURASIP J. Adv. Signal Process. 2014, 32 (2014). https://doi.org/10.1186/16876180201432
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/16876180201432