Energy Efficiency Analysis of Multi-Antenna Cellular Networks with Strategic Sleep Relay

Yonghong Chen , Jie Yang, Xuehong Cao1,,＊ Shibing Zhang

1 Department of Communication and Information Engineering, Nanjing University of Posts and Telecommunications, Nanjing 210003, China

2 Xinglin College, Nantong University, Nantong 226019, China

3 Department of Communication Engineering, Nanjing Institute of Technology, Nanjing 211167, China

4 School of Electronics and Information, Nantong University, Nantong 226019, China

Abstract: Energy efficiency (EE) of cellular networks has attracted considerable attention recently. However, EE of relay-assisted cellular networks where the macro base stations(MBSs) are equipped with the multi-antenna has not been thoroughly addressed. This paper considered the downlink transmission of multi-antenna relay-assisted cellular networks,meanwhile, a strategic sleep scheme was used in relay stations (RSs), which dynamically adjusted the RS working mode according to whether the number of users serviced by the relay exceeds a given threshold. A geometric model was built to derive the coverage probability and mean achievable rate from the MBSs to user (UE), the MBS to RS, the RS to UE links and analyze the system EE. It is shown that the energy efficiency of cellular network with strategic sleep RS is slightly higher than that of cellular network with non-sleeping strategy. Furthermore, the MBS equipped with multi-antenna has better impact on energy efficiency and spectral efficiency than the MBS with single antenna.

Keywords: energy efficiency; coverage probability; strategic sleep; multi-antenna; mean achievable rate

I. INTRODUCTION

With the growth of mobile traffic, the energy consumption of mobile network becomes more and more serious. For the future mobile communication system, the research of network architecture and transmission mechanism to save energy consumption, and improve the energy efficiency, is a widespread concern of the academia and industry [1-2]. Small cell networks are emerging as an inevitable solution for fifth generation (5G) cellular networks[3]. The roles of computation and transmission power in 5G small cell networks had been investigated in [4].

In the cellular network, relay can be used to expand the coverage area, combat the fading and improve the spectrum efficiency and data rate. Energy efficient relay-assisted networks have been studied in some literatures [5-11].In 2010, G. Andrews Jeffrey first modeled the distribution of the cellular network with the homogeneous Poisson Point Processes (PPP)in stochastic geometry [5]. Stochastic geometry is a powerful mathematical tool for modeling, analysis, and design of wireless networks with random topologies, which has been prov-en to be a more tractable model for the situation where the locations of base stations (BSs),relay stations (RSs), and users (UEs) all can be modeled as a spatial PPP [6-7]. In [8], the author considered joint relay selection and link scheduling to maximize the network throughput in relay-assisted cellular networks. The joint macro-relay networks comprising a conventional cellular network plus embedded RS hotspots offered an economically viable solution to achieve high cellular user capacity and low energy consumption [9]. Reference [10]proposed an energy-efficient user scheduling and relay selection scheme that minimized the scheduling duration of each user and the total transmit power consumption under the constraints of individual minimum data rate requirements for relay-assisted cellular networks. In [11], a pricing-based approach was investigated to achieve energy-efficient power allocation in relay-assisted multiuser networks and a network price to the power consumption as a penalty for the achievable sum rate, and studied its impact on the tradeoff between the EE and the spectral efficiency (SE).

To improve the EE, a lot of methods have been proposed in almost all aspects of cellular networks, such as multiple input multiple output (MIMO) technology [12-13], Energy harvesting (EH) technology [14-15], which enabled devices to harvest energy from their surrounding environment. Reference [12]proposed a multiuser multiantenna random cellular network model with minimum distance constraint for adjacent BSs. In [13],the simulation results indicated that MIMO systems were more efficient than single input single output (SISO) systems. In addition, BS sleep scheduling scheme is an effective way to reduce the total energy consumption and improve the EE. Reference [16] developed a theoretical framework for BS energy saving that encompasses dynamic BS operation (i.e., BS switching on/off) and proposed simple greedyon and greedy-off algorithms. Reference [17]and [18] proposed static BS sleep schemes by dynamically switching BSs depending on the traffic load. In [19], a type of sleeping control and active BSs’ optimal transmitting time strategy was considered according to the users’ QoS. In [20-21], the authors investigated BS sleep operations by considering transmission delay, spectral efficiency, respectively.In [22], the authors investigated the network sleep schemes by applying the Markov Decision Processes to reduce energy consumption.Sleeping control and power matching for energy-delay trade-offs in one cell of a cellular network with burst traffic was studied in [23].The authors in [24] proposed the concept of cell zooming, which was indeed a distributed BS sleep scheduling algorithm through minimizing the number of active BSs. The work in [25] focused on random BS switch-off strategy in HetNets, where the optimal switchoff probability for MBS was derived based on minimizing BS energy consumption. The simplest BS switch-off strategy was proposed in [26] based on only traffic statistics over time, where a small base station (SBS) was switched off based on a fixed timer and this timer was manually con figured for a statistical cycle when the traffic load was very low.

In this paper, we investigate the energy efficiency of the multi-antenna cellular network with strategic sleep relay.

To the best of our knowledge, researches on RS sleep are relatively scarce in relay-assisted cellular networks, and the analysis of multi-antenna cellular networks with strategic sleep relay has not been reported. In this paper, we consider a multi-antenna relay-assisted cellular network, meanwhile, strategic sleep scheme is adopted in RS. We analyze the energy efficiency of multi-antenna cellular networks with strategic sleep relay by using stochastic geometry. The expression of energy efficiency for the proposed algorithm is obtained.

The rest of this paper is organized as follows. Section II describes the system model.In Section III, we propose a RS sleep strategy and give the distribution of users under the sleep strategy. Section IV presents the coverage probability and mean achievable rate of different links and the expression of system energy efficiency. Some simulation results are discussed in Section V. Conclusions are stated in Section VI.

II. SYSTEM MODEL

Suppose there is a downlink of multi-antenna relay aided cellular network where the Macro base stations (MBSs) and relay stations (RSs)are deployed in the Euclidean plane according to independent homogeneous Poisson Point Processes (HPPPs) ΦMand ΦRwith densities λMand λR, respectively. Users (UEs) are distributed according to an independent HPPP with density λU. In this network, the MBSs are equipped with multiple antennas, each RS communicates to the closest MBS and has a circular coverage area with radius R. UEs are classified into two groups according to their locations, users who are located in the nearest RSs’ coverage area (R-UE), which communicate to the MBS with the help of RS, the other users communicate directly to the closest MBS (M-UE). The RSs adopt the strategic sleep scheme, and each RS has two working modes, active mode and sleeping mode. When the number of users served by the RS exceeds a certain threshold, the RS works in active mode, otherwise it works in sleeping mode.

Fig. 1. The cellular network model with strategic sleep relay.

Denote the distance between a M-UE and the MBS is r1, the distance between a RS and the MBS is r2, the distance between a R-UE and the RS is r3. Since the RSs or M-UEs are connected to the nearest MBS,and the distance between M-UEs or RSs and interfering MBSs is farther than r1or r2, the probability density functions (PDF)of r1and r2areand[27]. Because a R-UE locates in the RS’s coverage, the distance r3follows the distributionwhere R is the circular radius the RS covered.And more, the transmit power from MBS to M-UE and RS are denoted as PMUand PMR,respectively, the transmit power for all RSs is denoted as PRU. The cellular network model with strategic sleep relay is shown in figure 1.

The system is evenly divided into NTsub-channels. Each user transmits its data with one sub-channel. The data transmission is divided into two phases. In the first phase,NM1sub-channels are used to serve the M-UEs by MBS, NMRsub-channels are used to serve the RSs by MBS, NM1+ NMR= NT. In the second phase, each relay decodes the data and forwards those to its R-UEs with NRUsub-channels, the remainder NM2sub-channels are used for the direct link between UEs and MBS. Because each RS has no overlap coverage area we need only NRUsub-channels for all RSs. Therefore, we have NM2+ NRU= NT,and the number of RSs which work simultaneously in the cell is L = NMR/ NRU. Due to the single user beamforming in the multi-antenna technique used in MBS, the channel power distributions of both the direct and the interfering links under Rayleigh fading follow the Gamma distribution [28].

III. SLEEP STRATEGY FOR RS

In order to improve the energy efficiency of the network, we propose a relay sleep strategy,in which the working mode of RS is dynamically changed according to the number of users served by the relay whether exceeds a given threshold. Each RS has two working modes, active mode and sleep mode. When the number exceeds a given threshold Uth, the relay works in active mode with probability Pa,otherwise the relay works in sleep mode with probability (1-Pa). Suppose nRUis the number of R-UEs, nRU～ Poisson (λUπR2) in its serving area, we obtain the active mode working probability as follows

Denote the cell size of the Voronoi cell which the MBS covers as S. The accurate probability density function (PDF) of S is hard to derive. However, it has been proved that a simple approximation PDF is enough for practical purposes [29]. The approximate PDF is given by

In the Voronoi cell with sleep strategy, the HPPPs density of the RSs number is PaRλ.Then, the numbers of RSs and M-UEs, nRand nMU, follow the Poisson distributions as follows

The distributions of average numbers of RSs and M-UEs are given by

Since the number of the R-UEs are distributed according to HPPP with intensity λUin the circular coverage area with size πR2, the distribution of the average number of R-UEs nRUis given by

When the relays work with active mode with probability Pa, the remainder direct. It is obviously that. For any transmission between MBS and UEs in the first phase, there are NM1available sub-channels. Then, the probability with which a sub-channel is occupied can be obtained by [31]

Similarly, the occupied probability of sub-channel between MBS and RSs in the first phase is given by

The occupied probability of sub-channel between MBS and UEs (including M-UEs and R-UEs which the RS is sleeping) in the second phase is given by

The occupied probability of sub-channel between RSs and R-UEs in the second phase is given by

The spatial process of MBS that use a given sub-channel (MBS to the UEs and RSs in the first phase, and UEs in the second phase)are the independent thinning of original HPPP MSB by the probability pM1,pMRand pM2with density pM1Mλ, pMRMλ and pM2Mλ, respectively.

IV. ENERGY EFFICIENCY ANALYSIS

4.1 Coverage probability

Coverage probability PCis defined as the probability with which the received signal power to interference plus noise power ratio (SINR)is lager or equal to a given threshold T, i.e.,PC= Pr(S INR ≥T ). The SINR can be given by

Where P is the transmitted signal power, h is the channel power gain, r is the distance between the receiver and the tagged transmitter,α is the path loss exponent, I is the cumulative interference power from all the other transmitters, σ2is the power of the additive white Gaussian background noise. The distri-bution of SINR depends on h.

4.1.1 Coverage probability of MBS-UE link Suppose the MBS has k antennas and the single user beamforming technique is adopted.Then, the channel power distribution of MBS,h0, follows the Gamma distribution Γ(k,1)under Rayleigh fading channel.

Coverage probability for the MBS-UE link in the first phase is obtained as (13)shown in the bottom at this page, whereis the aggregate interference from all the other transmitters. Generally, the interference is much larger than additive white Gaussian noise in the cell networks.Then, we have

Where

Similarly, we obtain the coverage probability for the MBS-UE link in the second phase as(17) shown in the bottom at this page.

4.1.2 Coverage probability of MBS-RS link The coverage probability of MBS-RS is obtained as follows

where

4.1.3 Coverage probability of RS-UE link

Now, we have L RSs, the distribution densities of MBS to L RSs is LpMRMλ. RSs which re-ceive SINR larger than a threshold can decode the messages successfully and can be denoted by coverage probability PC_MR. Considering the working probability of the RSs with sleep strategy Pa, the coverage probability PC_MRof the MBS-RS link and the occupied probability of sub-channel between RSs and R-UEs pRU, we have the density of RSs to R-UEs as follows

The channel power distribution of RS,l0, follows the exponential distribution l0～exp(1) under Rayleigh fading channel.Then, we get the coverage probability of RSUE link as follows

where

4.2 Mean achievable rate

For a positive random variable X,the average ergodic achievable rate for a given link is defined as τ=E [ln(1 +SINR)]. The mean achievable rate for the first phase of MBS-UE link can be given by

where

Similarly, we obtain the mean achievable rate for the MBS-UE link in the second phaseas follows

where

The mean achievable rate of MBS-RS τMRis obtained as follows

where

The mean achievable rate of RS-UE τRUis given by

and

4.3 Power consumption

For the downlink, the power consumption at each MBS and RS can be given by

where PMand PRare the transmit power of MBS and RS, 1/ΔMand 1/ΔRare the adjustment coefficient of the power amplifier of MBS and RS, PM0and PR0are the static power consumption for MBS and RS, respectively.

The total transmit power consumption for each MBS consists of three parts, the trans-mit power over MBS-UE links in the first phase, the transmit power over MBS-UE links in the second phaseand the transmit power over MBS-RS linksThen, we can obtain the average total power consumption of MBS as follows

On one hand, the number of active RSs locate in a Voronoi cell is PaλR/λM, the average number of active RSs which decode data successfully is PC_MRpMRL , the average power consumption of the RSs is ΔRPRUpRUNRU+PR0. On the other hand,the average number of the other active RSs which decode date unsuccessfully is PaλR/λM− PC_MRpMRL, and the RSs only consume static power PR0. If we denote the consume power of RSs, which are sleeping, as PR,sleep, we can obtain the average total power consumption of RSs as follows

And, the average network power consumption is given by

4.4 Energy efficiency

For the MBS-UE link in the first phase, there are NM1available sub-channels, each of which has the occupied probability pM1and the coverage probability. The area average achievable rate is obtained by

Similarly, the area average achievable rate in the second phase is given by

The area average achievable rate of the MBS-UE links over two phases is given by

For the MBS-RS-UE link, the link is divided into two sub-links. Only when the received SINRs of two sub-links are both larger than the threshold, the link is considered successful one. Then, the achievable rate of the link is depend on the smaller rate between τMRand τRU. The average achievable rate of the MBSRS-UE links over the area,τR, is given by

Based on the analysis above, the spectral efficiency τ is given by

The energy efficiency of the network with single antenna can be obtained by setting k=1 in formula (14), (17), (20), (27), (30) and (32).The energy efficiency of the network that the RS without sleep strategy can be obtained by setting Pa=1.

V. SIMULATION RESULTS AND ANALYSIS

In this section, we provide some simulation results of the energy efficiency of multi-antenna strategic sleep relay-assisted cellular networks proposed in this paper. In order to simplify the simulation, we suppose that each MBS is equipped with two antennas, each RS and user has only one antenna. We use the default simulation parameters of the system model in table 1. In order to better compare the performance of the multi-antenna strategic sleep relay-assisted cellular network, the simulation curve of reference [15] are also given in figure 2 and figure 5.

Figure 2 describes the relationship between the energy efficiency of the cellular network and the density of MBSs under different situations. With the increase of the density of MBSs, the energy efficiency grows fast at the very beginning, and then slightly decreases. And more, the energy efficiency of the network with two antennas is higher than that with single antenna, a gain of energy efficiency about 23% is achieved when λM=2× 10−5m−2. Compared with the EH algorithm in reference [15] , the proposed algorithm improves the energy efficiency by 16%.The energy efficiency of the two antennas network with sleep strategy is slightly higher than that without sleep strategy, a gain of energy efficiency about 5.2% is achieved. The optimal energy efficiency can be achieved according to the quasi-convex curve. When the density exceeds the optimal value, the overlapped coverage areas of MBSs would take place. It decreases the energy efficiency.

Figure 3 shows the relationship between the energy efficiency and sleep threshold, Figure 4 shows the relationship between the spectral efficiency and sleep threshold. When the number of the user serviced in a RS nRUis larger than the given sleep threshold Uth, the RS works in the active mode, otherwise the RS works in the sleep mode. The larger the Uthis, the smaller the relay activation probability,and the smaller the spectral efficiency. Although the RS sleeping reduces the spectrum efficiency, it saves the transmission power of the RS. Therefore, the energy efficiency will be improved accordingly.

Figure 5 gives the relationship between the energy efficiency and the transmit power of MBSs. It can be seen from figure 5 that the energy efficiency increase first and then decrease. There exists a proper transmit power to achieve the optimum value of energy efficiency. When MBS’s transmit power continues to increase, although the area average achievable rate will increase, the average total power consumption of MBSs will increase more, so the energy efficiency will decrease. As same as figure 2, the energy efficiency with two antennas is higher than that with single antenna, the energy efficiency with sleep strategy is slightly higher than that without sleep strategy.

Table I. System parameters.

Fig. 2. Energy efficiency ηEE versus the MBSs densities λM.

Fig. 3. Energy efficiency ηEE versus the sleep threshold Uth.

Fig. 4. Spectral efficiency ηEE versus the threshold Uth.

Fig. 5. Energy efficiency ηEE versus the MBSs transmit power PMU.

VI. CONCLUSION

In this paper, we investigate the energy efficiency of the multi-antenna cellular network with strategic sleep relay. We derive the coverage probabilities and the mean achievable rates of MBS-UE links, MBS-RS links and RS-UE links. According to the average network power consumption and the area spectral efficiency, the energy efficiency of the cellular network is derived out. Through the simulation we obtain that the proposed multi-antenna cellular network with strategic sleep relay has the better performance than that of the single antenna relay-assisted cellular network and energy efficiency can be improved by setting an appropriate threshold and transmit power of MBSs.

ACKNOWLEDGEMENTS

This work is partly supported by the National Natural Science Foundation of China(Grant No. 61371112, No.61701221 ), the Jiangsu Natural Science Foundation (No.BK20160781), Jiangsu Higher Education Institutions Natural Science Foundation (No.16KJB510013, 16KJB510038), the Research Innovation Project for College Graduates of Jiangsu Province (No. KYLX16_0662), and the Natural Science Foundation of Nantong University Xinglin College (No. 2016K116).