Average Throughput Analysis of Downlink Cellular Networks with Multi-Antenna Base Stations Rui Wang, Jun Zhang, S.H. Song and K. B. Letaief, Fellow, IEEE Dept. of ECE, The Hong Kong University of Science and Technology Email: {rwangae, eejzhang, eeshsong, eekhaled}@ust.hk Abstract Random spatial network models have been recently utilized in the performance analysis and system design for multicell networks. Such an approach has been mainly adopted to investigate the outage based system performance, such as the outage probability and outage throughput. However, these performance metrics are defined with a fixed-rate transmission, and cannot characterize the performance of data traffic, which normally adopts rate adaptation. In this paper, we will evaluate the average throughput of a space division multiple access SDMA) based cellular network by considering stochastically distributed base stations BSs) and mobile terminals MTs). The major difficulty for the performance analysis is the complicated distribution of the interference links. We shall provide an analytical framework for evaluating the average throughput by using the Moment Generating Function MGF) based method. Simulations will show that the proposed method is very accurate. In particular, the analytical result can be utilized to determine the optimum number of MTs to be served in SDMA networks that can maximize the network throughput. Index Terms Cellular networks, Poisson point process, average throughput, SDMA. I. INTRODUCTION Global mobile data traffic is going through an exponential growth, and this trend will continue. To accommodate the increasing data demand, cellular networks become more and more dense and irregular. As a result, regular network models, e.g., the hexagonal network model, are no longer appropriate and also become intractable as the network size grows. Compared to the regular models, the random spatial network model proposed by Baccelli et. al. ], where the locations of base stations BSs) are modeled as a homogeneous Poisson Point Process PPP), is more suitable for dense and irregular networks. With this stochastic model, the spatial variability of the network is taken into account and the analysis becomes tractable. The random spatial network model has been widely adopted to investigate the network performance, mainly with outage based metrics, such as the outage probability and outage throughput ], 3], 4]. However, such performance metrics are defined with a fixed-rate transmission, and thus are not suitable for data traffic that normally adopts rate adaptation. Under this circumstance, the average throughput is a better candidate. To analyze the average throughput, most previous papers directly integrate the coverage probability over the positive real axis, which, however, requires a triple integral at This work is supported by the Hong Kong Research Grant Council under Grant No. 63. least. Recently, Di Renzo et. al. 5] proposed a new analytical framework which significantly reduced the complexity in the average throughput analysis. However, the distribution and the density of mobile terminals MTs) were ignored in 5], i.e., it was assumed that there will always be MTs in each cell, but such assumption is not valid as the network gets denser. Thus the spatial distribution of both BSs and MTs should be taken into consideration. In this paper, we will analytically evaluate the network average throughput of cellular networks, where the BSs and MTs are modeled as two independent homogenous PPPs. As multi-antenna transmission is an effective technique for high data rate wireless systems, we consider the space division multiple access SDMA) scheme with each BS equipped with M antennas. We adopt a particular type of SDMA transmission. With T MTs in a cell, the BS will serve K mint,u) MTs, where U is defined as the maximum number of MTs that can be served in one cell. We treat U as a network design parameter which cannot vary among different cells. Such a transmission strategy is called as U-SDMA when U> 6], while the scenario with U =is the single-mt beamforming strategy. We call the scheme with U = M as full-sdma. Thus the value of U is critical for such system. The major difficulty in the performance evaluation is the non-identical channel gains of different interference links, given that the number of MTs in different cells are different. To overcome this difficulty, we define a normalized channel gain for each link, which will make the interference links identically distributed. Furthermore, to make the evaluation tractable, we will assume that the numbers of MTs in different cells are independent, which then makes different interference links independent. By using the Moment Generating Function MGF) based method, we shall get a simple expression of the network average throughput. A key finding is that there is an optimal value of U, denoted as U, that can maximize the network average throughput. Simulations show that when the BS density is much lower than the MT density, the network average throughput with U -SDMA grows linearly with the number of BS antennas, outperforming both full-sdma and single-mt beamforming. Simulation results also indicate that the value of U is always close to 3 5 M. Besides, the comparison between outage throughput and average throughput is given, which reveals the importance of selecting a suitable performance metric when considering different application scenarios.
active MTs in the i-th cell is an M matrix, denoted as H i =h i,h i...h iki ], where h ij j ) CN, I) is an M vector. The precoding matrix at the i-th BS is denoted as W i, the columns of which are equal to the. normalized columns of H i H i i) H Based on such network model and channel model, the received signal for the typical MT, denoted as the -th MT, is given as y = d α h W Pt K s + i d α i h i W i Pt s i +n, ) BS MT Fig.. A sample network with U =. II. NETWORK MODEL AND PERFORMANCE METRICS In this section, we will first introduce the network model and present a particular SDMA scheme for such networks. Then the main performance metric, i.e., the average throughput, will be defined, with a comparison with outage throughput. A. The Network Model We will consider a downlink cellular network, where the M- antenna BSs and single-antenna MTs are distributed according to two independent homogeneous PPPs, denoted as Φ b and Φ m. The density of BSs and MTs are denoted as λ b and λ m, respectively. We assume that each MT is served by the nearest BS and perfect channel state information CSI) of the served MTs is available at each BS. SDMA is applied, and each BS will serve multiple MTs with zero-forcing precoding. Given the limited number of transmit antennas M), it may happen that not all the MTs in the same cell can be served simultaneously. Accordingly, we define U as the maximum number of MTs that can be served in one cell, where U M. Specifically, if there are T MTs in a cell, the BS will serve K mint,u) MTs. For the case that T>U, the BS will randomly choose U MTs to serve. For the case that T =, the BS will not transmit any signal and it is called as an inactive BS. Correspondingly, BSs which need to transmit signals are called as active BSs and the set of active BSs is denoted as Φ a. An MT is called active if it is picked to be served. Such a transmission strategy is called as U-SDMA when U >, while the scenario with U =is the single-mt beamforming strategy. Fig. shows an example of the network. We consider the Rayleigh fading channel with additive white Gaussian noise AWGN). The variance of the Gaussian noise and the pathloss exponent are denoted as σn and α, respectively. The transmit power at each BS is. We denote the BS located at x i Φ b as the i-th BS and denote the cell served by the i-th BS as the i-th cell. The number of active MTs in the i-th cell is denoted as. We assume that the power of the i-th BS is allocated equally to the active MTs. The channel matrix between the i-th BS and the where s i is a vector denoting the transmit signals in the i-th cell, d i denotes the distance between the i-th BS and -th MT, and n denotes the AWGN at the typical MT. As zero-forcing precoding is applied, the equivalent channel gain from the i-th BS to the typical MT is g = h W GammaM K +,) and g i = h i W i Gamma,) 7]. From ), the received signal-to-interference plus noise ratio SINR) for the typical MT is given as SINR = i:x i Φ a\x K g d α g i d α i + σn where Φ a \x denotes the set of interfering BSs., ) B. Performance Metrics To characterize the performance of wireless data services, we will use the network average throughput as the performance metric. To obtain the network throughput, we first obtain the average throughput for the typical MT. With k active MTs in the typical cell, the average throughput of the typical MT can be written as R u k ) = E ln + SINR) K = k ] ) ] K g = E ln + d α K = k. i:x i Φa\x K g i id α i +σn 3) Then, the network average throughput is defined as the sum of the average throughput for all MTs being served in a unit area. It can be written as R a = λ b E K K R u K )]. 4) For comparison, the outage throughput is determined by the outage probability p o, which is defined as the probability that the received SINR is smaller than a given threshold ˆγ, i.e., p o P SINR ˆγ]. 5) The outage throughput of the network is defined as the average number of successfully transmitted nats per sec Hz unit-area. Given a cell with k active MTs, the outage throughput of this cell is k p o k )] ln + ˆγ). Therefore, the outage throughput of the network is given by 6]: R o = λ b E K {K p o K )] ln + ˆγ)}. 6) From the definition, we can see that the outage throughput is evaluated with a fixed threshold ˆγ. Thus the rate in the system is fixed as ln + ˆγ). When SINR > ˆγ, the data can
be successfully transmitted; otherwise the transmission will fail. On the other hand, the average throughput is evaluated with rate adaptation. Different conclusions will be drawn when using different performance metrics, as will be shown in Section IV. III. AVERAGE THROUGHPUT ANALYSIS FOR SDMA NETWORKS In this section, we will first derive the average throughput for a typical MT using the MGF-based method, and then derive the network average throughput for the SDMA network. As the number of active MTs in one cell depends on the spatial distribution of both BSs and MTs, there are two main difficulties for evaluating the average throughput. One is that as the inactive BS will not transmit any signal, we should only consider the interference from active BSs. The other one is that the equivalent transmit power gains / ) are not constant and the equivalent channel gains g i ) are not identically distributed, as both of them are related to the number of active MTs in the cell ), which is different among different cells. To the best of our knowledge, there has been no analytical result of the average throughput with the SINR given in ). To resolve these issues, we define normalized channel gains as V = g, K 7) { gi V i = if if =. 8) The expression of SINR in ) can thus be rewritten as V d α SINR = i:x i Φ b \x V i d α. 9) i + σn In this way, we can still work with the original BS set Φ b, while the randomness of the number of MTs has been absorbed into the normalized channel gain. However, the evaluation of the interference is still intractable, as the normalized interference channel gains V i ) are correlated with each other due to the correlation among the numbers of active MTs in different cells. To simplify the evaluation, we assume that the numbers of active MTs in different cells are independent of each other, which was also used in previous works 4], 6]. Thus the normalized interference channel gains V i ) are independent and identically distributed i.i.d.). In Section IV, we will demonstrate the accuracy of such approximation through simulation. The SINR expression in 9) is of the same form as that for the single-mt beamforming scenario with the assumption that all the BSs are active. Following the results in 5], we can get the average throughput for the typical MT with k active MTs in the typical cell in an interference limited scenario as R u k )=Eln + SINR) K = k ] M V K = z) dz M Vi z)+az) z, ) where A z) =Γ ) α z j+ ) j+ M j+) V i z) Γ )] α + j, j= ) M V K s) = E expsv ) K = k ] is the MGF of V conditioned on K = k, and M Vi s) =EexpsV i )] is the MGF of V i. To evaluate M Vi s), we need to first obtain the distribution for the number of active MTs in one cell K), which is given by 6]: μ μ Γk+μ)ρ k if k U Γμ)k! p K k) = ρ +μ)k+μ, μ μ Γi+μ)ρ i i=u if k = U Γμ)i! ρ +μ)i+μ ) where ρ λ b λ m is the BS-MT density ratio and μ is a constant that can be obtained through data fitting. As shown in 8], μ is equal to 3.6. Based on the distributions of g i and, we use the law of total expectation to obtain the MGF of V i, based on which we can then evaluate Az). As the result of Az) is complicated, an identity in 9]5..) is utilized to simplify the result. Then, the expressions of the MGFs and Az) are given in the following lemma. Lemma : M V K z), M Vi z), and Az) in ) can be expressed as M Vi z) = Az) = M V K z) = α) U z + zk ) M k+), 3) p K k i ) + z ) ki + p K ), 4) k i p K k i )k ki+ i z + k i ) ki+) F k i +,, ) ] α, z. z + k i 5) Proof: The proof is omitted due to space limitation. By substituting ), 5) and ) into 4), we can obtain the expression of network average throughput in the following theorem. Theorem : The average throughput in the interference limited scenario over the Rayleigh fading channel is E K E KM V dz R a = λ b 6) M Vi z)+az) z where E K = p K k )k, 7) E KM V = k = p K k )k + z ) M k+), 8) k k =
Average Throughput nats/s/m /Hz).5 x 3.5.5 ρ=. ρ=. ρ=.5 Numerical results Simulation results Average Throughput nats/s/m /Hz).9.8.7.6.5.4.3.. x 3 U = U = M U = U Average Throughput nats/s/m /Hz) 5 x 3 4.5 4 3.5 3.5.5 5 5 U 5 5 M with ρ=.5 5 5 M with ρ= Fig.. Average throughput vs. U over a Rayleigh fading channel, with M =, λ m = 3 m, α =4, =38dBm ], and the noise power considered in simulation is 97.5dBm ]. M Vi z) = Az) = p K k i ) + z ) ki + p K ), 9) k i α) U z p K k i )k ki+ i z + k i ) ki+) F k i +,, ) ] α, z. z + k i ) Based on the above expressions, we can identify some properties of the network average throughput. In particular, when U, M and ρ are fixed, the integral in 6) is unrelated to the BS density. As a result, the network average throughput increases linearly with respect to the BS density. Moreover, this expression is for a particular value of U, and thus it can help to pick an optimal value of U to maximize the average throughput, which will be demonstrated in the next section. By substituting U = into Theorem, we can get the expression of the network average throughput for the special case of single-mt beamforming. IV. NUMERICAL AND SIMULATION RESULTS In this section, we will present simulation results to verify our analytical expressions. Important insights will be obtained. In particular, the advantage of SDMA and the key difference between the two throughput metrics, i.e., outage throughput and average throughput, will be demonstrated. A. Performance Gains of SDMA In Fig., we show the average throughput of the SDMA system with different values of U, where the simulation results are also included for comparison. It can be observed that our numerical results fit the simulation curves very well, which validates the interference-limited assumption and also shows the accuracy of our approximation in the evaluation. Furthermore, the maximum throughput is achieved when the number of active MTs in one cell is less than the total number Fig. 3. Average throughput vs. M over a Rayleigh fading channel, with λ m = 3 m, α =4. ρ M TABLE I OPTIMAL U, WITH λ m = 3 m. α=3 α=4 α=5 3 5 M.5.5.5.5.5.5 4 3 3 3 3 3 3 3 6 4 4 4 4 4 4 4 8 5 5 5 5 5 5 5 6 6 6 7 6 7 6 7 7 8 8 8 8 8 4 8 9 9 9 9 9 9 6 9 8 3 3 3 of BS antennas. Thus, it is not optimal to fully utilize the M antennas to serve M MTs, which is because serving too many MTs will reduce the beamforming gain for each of them. In Fig. 3, we compare the average throughput of three schemes: the single-mt scheme with U =, the U -SDMA where U is equal to the optimal value U that maximizes the network average throughput, and the full-sdma scheme with U = M 6]. It can be observed that the U -SDMA scheme always outperforms the single-mt scheme. Furthermore, when λ b λ m, U -SDMA also provides a significant average throughput gain compared to full-sdma, and the network average throughput with U -SDMA increases almost linearly with respect to the number of BS antennas. However, when λ b λ m, U -SDMA and full-sdma provide similar performance. The optimal U with different values of α, ρ and M is shown in Tab. I. We can find that the optimal U is always close to 3 5 M, and further investigation is needed to show it analytically.
Throughput nats/s/m /Hz)..9.8.7.6.5.4.3. Average throughput numerical results) Average throughput simulation results) Outage throughput simulation results) M=,3,5 throughput with different numbers of BS antennas in single- MT beamforming scenario. It is observed that the average throughput increases linearly with respect to logm and the slope increases with ρ. This phenomenon does not exist for the outage throughput case, where the curves converge to a constant when increasing M. The above comparisons clearly demonstrate the difference between the two performance metrics. Different metrics may give different conclusions, and thus it is critical to pick a metric suitable for the considered application scenario.. M=,3,5 BS user Density Ratio ρ Fig. 4. Average throughput and Outage throughput vs. BS density with U = U over Rayleigh fading channel, with λ m = 3 m, α =4, = 38dBm ], and the noise power considered in simulation is 97.5dBm ]. The threshold of outage throughput is set as ˆγ =. Throughput nats/s/m /Hz) 6 x 3 5 4 3 Average throughput numerical results) Average throughput simulation results) Outage throughput simulation results) Number of BS antennas M ρ= ρ=. Fig. 5. Average throughput and Outage throughput vs. M with U = over Rayleigh fading channel, with λ m = 3 m, α =4, =38dBm ], and the noise power considered in simulation is 97.5dBm ]. The threshold of outage throughput is set as ˆγ =. B. Outage Throughput vs Average Throughput In this subsection, we will demonstrate the difference between outage throughput and average throughput. Fig. 4 compares the outage throughput and average throughput with U -SDMA. Simulation results for average throughput are also included to validate our analytical results. It can be observed that, with rate adaptation, we can achieve a much higher throughput. Furthermore, when the BS density λ b is larger than the MT density λ m, the average throughput increases logarithmically with the BS density. However, for the outage throughput, increasing λ b brings very limited gain when λ b is large compared to λ m. Specifically, the outage throughput tends to converge to a constant when increasing the BS density λ b, which is because a fixed rate threshold is adopted. Note that the outage throughput curves will change with different threshold values, but the trend remains the same. In Fig. 5, we plot the average throughput and outage ρ= V. CONCLUSION In this paper, we evaluated the network average throughput for downlink SDMA systems, where we took the spatial distributions of the BSs and MTs into consideration by modeling them as two independent homogeneous PPPs. In particular, we obtained an analytical expression for the network average throughput. It was observed that the U -SDMA system outperforms both the full-sdma and the single-mt beamforming systems in terms of the network average throughput, and the gain is more significant when λ b λ m. Furthermore, the analytical result can be utilized to determine the optimal value of the maximum number of active MTs in one cell, and it was shown that the optimum value is close to 3 5 M.We also demonstrated the differences between outage throughput and average throughput, which indicates the importance of selecting a suitable performance metric. REFERENCES ] F. Baccelli, M. Klein, M. Lebourges, and S. Zuyev, Stochastic geometry and architecture of communication networks, J. Telecommun. Syst., vol. 7, no. -3, pp. 9 7, Jun. 997. ] J. G. Andrews, F. Baccelli, and R. K. Ganti, A tractable approach to coverage and rate in cellular networks, IEEE Trans. Commun., vol. 59, no., pp. 3 334, Nov.. 3] H. S. Dhillon, R. K. Ganti, F. Baccelli, and J. G. Andrews, Modeling and analysis of K-tier downlink heterogeneous cellular networks, IEEE J. Sel. Areas Commun., vol. 3, no. 3, pp. 55 56, Apr.. 4] C. Li, J. Zhang, and K. B. Letaief, Throughput and energy efficiency analysis of small cell networks with multi-antenna base stations, IEEE Trans. Wireless Commun., vol. 3, no. 5, pp. 55 57, May 4. 5] M. Di Renzo, A. Guidotti, and G. Corazza, Average rate of downlink heterogeneous cellular networks over generalized fading channels: a stochastic geometry approach, IEEE Trans. Commun., vol. 6, no. 7, pp. 35 37, Jul. 3. 6] C. Li, J. Zhang, and K. Letaief, Performance analysis of SDMA in multicell wireless networks, in Proc. of IEEE Globecom, Atlanta, GA, Dec. 3. 7] J. Zhang and J. G. Andrews, Adaptive spatial intercell interference cancellation in multicell wireless networks, IEEE J. Sel. Areas Commun., vol. 8, no. 9, pp. 455 468, Dec.. 8] D. Weaire, J. Kermode, and J. Wejchert, On the distribution of cell areas in a Voronoi network, Philosophical Magazine B, vol. 53, no. 5, pp. L L5, 986. 9] M. Abramowitz and I. A. Stegun, Handbook of mathematical functions: with formulas, graphs, and mathematical tables. Courier Dover Publications,. ] E. Lang, S. Redana, and B. Raaf, Business impact of relay deployment for coverage extension in 3GPP LTE-Advanced, in Proc. of IEEE Int. Conf. on Commun. ICC) Workshops, Dresden, Jun. 9. ] H. Holma and A. Toskala, LTE for UMTS-OFDMA and SC-FDMA based radio access. John Wiley & Sons, 9.