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1 Optimal Deployment of mall Cell for Maximizing Average m Rate in Ultra-dense Networks Yang Yang Member IEEE Linglong Dai enior Member IEEE Jianjun Li Richard MacKenzie and Mo Hao Abstract In future 5G communication system demands of high data rate can be effectively met by ultra-dense networks UDN) where a lot of small cells are deployed within the conventional macro cells. In this paper we derive the optimal density of small cells in UDN to maximize the average sum rate AR). pecifically based on the stochastic geometry we use the homogeneous Poisson point process PPP) to characterize the random distribution of both user equipments UEs) and small cells. Then the closed-form successful transmission probabilities in both uplink and downlink transmissions are derived by Laplace transformation. After that we obtain AR of small cells and the maximization problem of AR is analyzed under the constraints of outage probabilies. Based on convex optimization the optimal density of small cell B for maximizing the AR of small cells is evaluated in a closed form. We also investigate the impact of system parameters on the optimal small cell B density. imulation results demonstrate that different maximized ARs can be reached under different constraints from cellular networks and the small cell performance is also influenced by the interference from macro cell UEs and small cells. Index Terms Ultra-dense networks UDN) small cells average sum rate stochastic geometry convex optimization. I. INTRODUCTION THE rapid growth of mobile traffic in recent years drives the fast development of wireless communication and future 5G system is expected to provide times higher network throughput and Gb/s mobile access rate ]. ch an explosion of mobile traffic causes the shortage of spectrum resources. In order to solve this problem ultra-dense networks UDN) has been proposed to effectively alleviate the traffic demand of macro base stations Bs) by densely deploying a large number of small cells in the networks ]. UDN also introduces other benefits including efficient reuse of spectrum improved data rate and so on which make it becomes one of the promising key technologies for future 5G system. However due to the aggravated interference generated by dense small cells the performance of macro cell networks will Y. Yang and L. Dai are with the Tsinghua National Laboratory for Information cience and Technology TNList) Department of Electronic Engineering Tsinghua University Beijing 84 China s: yy553 daill}@tsinghua.edu.cn). J. Li is with chool of Electric and Information Engineering Zhongyuan University of Technology jianjun.li@tsinghua.org.cn). R. MacKenzie is BT Innovate and Design Adastral Park Ipswich UK richard.mackenzie@bt.com). M. Hao is with the Tsinghua EM Advanced ICT LAB Tsinghua University Beijing 84 China haom@sem.tsinghua.edu.cn). This work was supported by the International cience and Technology Cooperation Program of China Grant No. 5DFG76) the National Natural cience Foundation of China Grant No. 6577) the British Telecom and Tsinghua EM Advanced ICT LAB and the China Postdoctoral cience Foundation Grant No. 6M5977). be degradated which results in a big challenge to the optimal deployment of small cells in UDN 3]. In order to reduce such interference some solutions have been proposed to enhance the network performance 4] 8] where the transmission rate is a key performance metric of UDN. pecifically based on the interference statistics between macro and small cells the transmission rate of UDN is evaluated for both dedicated and shared spectrum access under fixed grid-like macro cells 4]. The transmission rate is further optimized by using subchannel and power allocation 5] downlink hierarchical competition 6] and joint B association with power control 7] etc. where all optimization works are self-organized and noncooperative. When cooperative resource allocation is considered the optimization of transmission rate for small cells in a single macro B is investigated in 8]. However it is insufficient to just consider one or several macro cells with fixed locations in UDN. For practical UDN it is necessary to analyze the transmission rate of large-scale networks with the random deployment of macro cell users and small cells. In this paper the random distribution of macro cell users and small cell Bs is considered to derive the optimal deployment of small cells in UDN. pecifically based on the stochastic geometry we use the homogeneous Poisson point process PPP) to characterize the UDN with small cells sharing uplink frequency resources. Then we derive the successful transmission probabilities of both uplink and downlink transmissions in small cells. With the help of convex optimization we further analyze the average sum rate AR) maximization problem of small cells under the constraints of outage probabilities. Finally the optimal small cell B density for maximizing AR of samll cells are derived in closed form. imulation results show that different maximized ARs can be reached under different constraints from cellular networks. In addition we show that the small cell performance is not only constrained by the transmission of macro cell but also influenced by the interference from macro cell user equipments UEs) and small cells. The rest of this paper is organized as follows. ection II briefly describes the system model of UDN. ection III presents the successful transmission probabilities in UDN and the AR of small cells. ection IV shows the optimization analysis of small cell B density for the maximum AR of small cells. imulation results are provided in ection V. Finally our conclusions are summarized in ection VI. imulation codes are provided to reproduce the results presented in this paper:

2 UDN ]. Therefore the signal-to-interference ratio IR) at the typical macro cell B IR M is Fig.. ystem model. II. YTEM MODEL As shown in Fig. we consider a UDN that small cells are densely deployed within the conventional macro cells. The uplink frequency resources of the macro cells are shared by small cells. Macro cell UEs are modeled as an independent homogeneous Poisson point process PPP) Φ M with density on the two-dimensional plane R. imilarly the small cell Bs satisfy an independent homogeneous PPP Φ with density λ. The traffic in the UDN is assumed as full buffer. The powers of each macro cell UE small cell B and UE are defined as P and P respectively. By denoting the uplink transmission probability as Pr u Pr u ) the downlink transmission probability Pr d should be Pr d = Pr u Pr d ). With the help of stochastic geometry a typical receiver of small cells is assumed to be located at the origin on R 9]. If this typical receiver is receiving signals in downlink it is a small cell UE while this typical receiver is a small cell B when it is receiving signals in uplink. Then according to Palm theory ] this typical receiver does not influence statistics of the PPP. imilarly as the uplink frequency resources of macro cell networks are reused by small cells the analysis will be performed on a typical receiver which is a macro cell B in the UDN. The propagation channel in the UDN includes both path loss and Rayleigh fading with the form P r = P t δ tr Rtr where P t and P r denote the power of transmitter and receiver respectively. R tr is the distance between the transmitter and receiver and denotes the path loss exponent with. δ tr stands for Rayleigh fading coefficient which follows an exponential distribution with unit mean in UDN ]. III. AVERAGE UM RATE OF MALL CELL In this section the successful transmission probabilities in UDN are analyzed. Then based on the uplink and downlink transmissions in small cells the average sum rate AR) of small cells is also obtained. A. ccessful Transmission Probabilities in UDN First we consider the condition that the typical macro cell B suffers the interferences from both macro cell networks and small cells in the UDN. Compared with the distances from other interfering small cells to the typical macro cell B the distance between the small cell UE and its serving B can be neglected because the coverage of each small cell is small in IR M = δ M R M δ j R j + P δ k R k + j Φ M k Φ A) k Φ B) P δ k R k where δ M denotes the Rayleigh fading coefficient and R M denotes the distance from the desired macro cell UE to the typical macro cell B. imilarly δ j and R j represent the Rayleigh fading coefficient and the distance from node j to the origin in the macro cell while δ k and R k are corresponding parameters of the node k in small cells. We define the compact point sets A and B as A = m node m is the receiver in uplink transmission in small cells} and B = n node n is the receiver in downlink transmission in small cells}. Then the following Lemma presents the successful transmission probability of the typical macro cell B in UDN. Lemma. The successful transmission probability of the typical macro cell B in UDN satisfies: Pr IR M ξ M ) = exp η M λ η M Pr P u + Pr P d ) where Pr ) represents the probability ξ M is the IR threshold of uplink transmission of macro cell UE η M = πrm ξ M + ) Γ ) Γ ) denotes the gamma function with the form Γ z) = e t t z dt. Proof: imilar proof can be found in ] with the help of Laplace transformation so the detail is omitted here due to space limit. imilarly by denoting δ and R as the Rayleigh fading coefficient and the distance from the typical small cell UE to its serving small cell B we have the following lemma. Lemma. The successful transmission probability of the typical small cell UE in uplink transmission satisfies: Pr IR ξ ) = exp η P λ η u + Pr P d P 3) where η = πr ξ + ) Γ ) and ξ is the IR thresholds of uplink transmission in small cell. In addition by denoting δ and R as the Rayleigh fading coefficient and the distance from the typical small cell B to its desire small cell UE we have the following lemma. Lemma 3. The successful transmission probability of the typical small cell B in downlink transmission satisfies: Pr IR ξ ) = exp η P λ η P u P + Pr d 4) where η = πr ξ + ) Γ ) and ξ is the IR thresholds of downlink transmission in small cell. )

3 3 B. Average m Rate in mall Cells In UDN the AR in small cells is defined as ]: f n λ ) = Pr n λ R n n u d} 5) where R n with subscript n u d} denotes the average rates of the uplink n is u in this case) and downlink n is d in this case) transmission respectively. Particularly R n satisfies the following form 3]: R n = sup ξ n W log + ξ n ) Pr IR n ξ n ) n u d} 6) where W is the bandwidth of macro cell uplink transmission and this bandwidth is also reused by small cells. Then we have the following definition: Definition. The AR of uplink transmission in small cells is f λ ) = W Pr u λ log + ξ ) exp η P λ η u + Pr P d P while the AR of downlink transmission in small cells is f λ ) = W Pr d λ log + ξ ) exp η P λ η P u P + Pr d. Thus the AR of small cells in UDN f λ ) is 7) 8) f λ ) = f λ ) + f λ ). 9) IV. OPTIMIZATION OF AR FOR MALL CELL When small cells reuse the frequency resources of macro cell networks the reliable transmission of macro cells cannot be disturbed i.e. the transmission of small cells must guarantee the outage probabilities at the macro cell B in UDN. Considering the transmission in small cells is divided as uplink and downlink transmissions we have the following four constraints: λ λ max ) ] e λ P M η M λ η M Pr P u P +Pr d M θ M ) ] e λ M η P P λ η Pr u+pr d P θ ) ] e λ M η P P λ η Pr u P +Pr d θ 3) where λ max is the maximum density of small cell B in UDN θ M θ and θ θ M θ θ ]) are the outage probability thresholds of macro cell B small cell B and small cell UE respectively. o we get the following AR optimization problem of small cells: max f λ ) s.t. ) ) ) 3) From inequality ) we have λ P M P M 4) η M ln θ M ) Pr u P + Pr d P sup. 5) From inequality ) we have λ P M P η ln θ ) Pr u P + Pr d P sup. 6) From inequality 3) we have λ P M P η ln θ ) Pr u P + Pr d P sup 3. 7) Combine 5)-7) with ) the upper bound of small cell density satisfies λ sup max λ sup λ sup λ sup 3 }. = η u + Pr P d P P ) Let A = W Pr u log + ξ ) A = η ] B = W Pr d log + ξ ) B = η P ) We have the following Proposition. and = η P u P + Pr d ]. Proposition. The maximum value of function f λ ) locates } at the interval λ l λ h ] where λ l = min and λ h = max }. Proof: According to 7) and 8) we know f λ ) = A λ e A λ f λ ) = B λ e B λ. Take the first derivative of both f λ ) and f λ ) and make both of them equal to we get the maximum values of function f λ ) and f λ ) as and respectively. In addition we know that the function f λ ) increases monotonically when λ ) and f λ ) decreases monotonically when λ +. imilarly the function f λ ) increases monotonically when λ ) and it decreases monotonically with λ +. Denote λ l = min } λ h = max }. Without loss of generality we suppose λ l = and λ h = ). Then according to the results above for λ ) it can be known that f λ ) < f and f λ ) < f because λ < <. o f λ ) < f when λ ). imilarly f λ ) < f when λ +. The same results can be obtained if we define λ l = and λ h =. Therefore the maximum value of function f λ ) locates at the interval λ l λ h ]. Based on Proposition we can further derive the following Theorem. for maxi- Theorem. The optimal small cell density λ max mizing AR of small cells satisfies: λ max = λsup λ sup < λ λ λ λ sup 8)

4 4 where λ satisfies λ l f λ ) < when λ λ l λ h ) λ f λ ) = and λ λ l λ h ) λ h f λ ) > when λ λ l λ h ) 9) where f λ ) means the first derivative of f λ ) with respect to λ. λ is the solution of f λ ) = when f λ ) has zero point on λ l λ h ) where λ = arg max λ A λ 3 } f λ ) Proof: The first derivative of f λ ) is ) f λ ) = A e A λ λ ) + B e B λ λ ). ) First if f λ ) < then according to Proposition we can know that f λ ) can get the maximum value as λ l since f λ ) monotonically decreases in λ l λ h ). econd if f λ ) > f λ h ) is the maximum value as f λ ) monotonically increases in λ l λ h ). Third consider f λ ) is a continuously bounded function in the close set λ l λ h ] and if λ λ l λ h ) which leads to f λ ) = f λ ) must be the local maximum or minimum value in λ l λ h ]. o f λ ) is the maximum value when λ = arg max f λ ). Last combine those three λ λ } points above we get λ then consider the constraints of power and the outage probabilities of both macro and small cell transmissions the optimal small cell density λ max for maximizing AR of small cells is obtained as equation 8). V. IMULATION REULT In this section we evaluate AR of small cells with the main simulation parameters provided in Table I. TABLE I MAIN IMULATION PARAMETER. Parameter Physical Meaning Default Value Path loss 4 Macro cell UE density 6 macro cell UE/m λ mall cell B density 5 small cell B/m The power of macro cell UE 35 dbm P The power of small cell UE 5 dbm P The power of small cell B 3 dbm Pr u /Pr d The probability of uplink/downlink communication in a small cell.6/.4 ξ M IR threshold of the macro cell uplink transmission -5 db ξ /ξ IR threshold of the small cell uplink/downlink transmission db R M The distance from macro cell B to the typical cellular user m R /R The uplink/downlink distance 5m/6m Fig. a) and b) show the AR of small cells with two different feasible regions of small cell B density. We can see the maximum AR of small cells locates in a interval of small AR of small cells bit/s/hz/m ) Fig...5 x 5 a).5 uplink AR downlink AR total AR maximum uplink AR maximum downlink AR maximum total AR feasible region boundary x 4 b) x mall cell B density mall Cell B/m ) AR of small cell vs. small cell B density. x 4 cell B density where the two boundary values of that interval correspond to the maximum uplink AR and downlink AR. This phenomenon verifies Proposition before. When the small cell B density is not very high the AR of small cells is increasing as the small cell B density becomes high which means more small cell Bs will bring larger performance gain. However as the small cell B density continues to increase the interference caused by small cells in the UDN cannot be neglected so the AR becomes to fall down. In Fig. a) the powers of small cell UE and B are set as 4 dbm and dbm we can see the downlink AR of small cells is lower than the uplink AR since the receivers in downlink transmission of small cells cannot endure much interference than uplink receivers. In Fig. b) the powers of small cell UE and B are set as 5 dbm and 3 dbm respectively and we can see the downlink AR is larger than the uplink AR but the growth speed of downlink AR is slower than that of the uplink AR when small cell B density is low. This is caused by the fact that the distance of uplink transmission is shorter than that of downlink transmission which makes the uplink signals suffer less propagation loss than the downlink signals. In addition by comparing these two figures we can find that because the power of small cell UE and B in Fig. a) is low macro cells can allow more small cells to reuse their frequency resources then we can see the feasible region of small cell B density is wide and the maximum AR of small cells can be obtained. Fig. 3 shows the maximum AR of small cells against the macro cell UE density and power of macro cell UE. From Fig. 3 a) we can see the maximum AR of small cells is decreasing as the macro cell UE density becomes high. This is because high macro cell UE density will cause more interference to the small cells and the constraints to small cells become more strict. When the macro cell UE density is high enough the receivers in small cells are severely affected by the interference from macro cell UEs which leads AR come to zero. It can also be known that high-power macro cell UE will cause more interference to small cells so the maximum AR of small cells is lower while the power of macro cell UE is higher. In Fig. 3 b) it can be seen that the maximum AR of small cell is increasing as the power of macro cell UE

5 5 Maximum AR of small cells bit/s/hz/m ) x 5.5 x 5 a) Macro cell UE density Macro Cell UE/m ) b) =x 6 Macro Cell UE/m ) =x 6 Macro Cell UE/m ) =3dBm =35dBm =4dBm x Power of macro cell UE dbm) Fig. 3. The maximum AR of small cells vs. Density and power of macro cell UE. increases. This can be explained by the fact that high power of macro cell UE can endure more interference from small cells more small cell Bs can be deployed into the networks. As the power continues to increase the interference generated by macro cells becomes larger then the maximum AR of small cells begins to decrease. In addition lower macro cell UE density can lead maximum AR of small cells to be larger. This is because the interference from macro cell UEs to the small cells becomes smaller. Fig. 4. The maximum AR of small cells vs. small cell uplink and downlink power. Finally Fig. 4 illustrates the relationship among the maximum AR uplink power and downlink power of small cells. We can see the performance gain in the uplink transmission is larger than that in the downlink transmission of small cells. This can be explained as follows: ) The uplink transmission distance is shorter than the downlink transmission distance which makes the signal experienced less propagation loss in the uplink; ) The probabilities of uplink and downlink transmission are set as.6 and.4 respectively which means the number of links in uplink transmission is more than that in the downlink transmission so we can see that the uplink transmission dominates the whole performance gain. An extreme example in Fig. 4 is when uplink and downlink transmission are 5 dbm and 35 dbm the maximum AR of small cells is very low because not only the uplink transmission cannot bring much performance gain to the small cells but also the downlink transmission will generate serious interference to the whole networks. VI. CONCLUION In this paper we have optimized the small cell B density in UDN for maximizing AR of small cells. By modeling the whole networks as the homogeneous PPPs we have investigated the successful transmission probabilities of both macro cell networks and small cells. The AR expression of small cells was evaluated and an optimization problem of small cell B density was formulated by considering outage probabilities and density constraints. Then we proved that the maximum AR of small cells locates in a fixed interval of small cell B density. The boundaries of the density were also shown. Finally the optimal small cell density was derived in closed form for maximizing AR of the small cells in UDN. The impact of the parameters such as optimal small cell B density and macro cell UE power were discussed through simulation results which verify that the maximum AR of small cells is influenced by both the constraints from macro cell networks and the interference in UDN. REFERENCE ] D. Lopez-Perez M. Ding H. Claussen and A. H. Jafari Towards Gbps/UE in cellular systems: Understanding ultra-dense small cell deployments IEEE Commun. rveys & Tutorials vol. 7 no. 4 pp. 78 Jun. 4. ] P. Agyapong M. Iwamura D. taehle W. Kiess and A. Benjebbour Design considerations for a 5G network architecture IEEE Commun. Mag. vol. 5 no. pp Nov. 4. 3] U. iddique H. Tabassum E. Hossain and D. I. Kim Wireless backhauling of 5G small cells: Challenges and solution approaches IEEE Wireless Commun. Mag. vol. no. 5 pp. 3 Oct. 5. 4] H. Tabassum Z. Dawy E. Hossain and M.-. Alouini Interference statistics and capacity analysis for uplink transmission in two-tier small cell networks: A geometric probability approach IEEE Trans. Wireless Commun. vol. 3 no. 7 pp Jul. 4. 5] H. Zhang C. Jiang N. C. Beaulieu. He and X. Chu Cooperative bargaining resource allocation for cognitive small cell networks in Proc. IEEE Global Communications Conference IEEE GLOBECOM 4) Dec. 4 pp ]. Guruacharya D. Niyato D. I. Kim and E. Hossain Hierarchical competition for downlink power allocation in OFDMA femtocell networks IEEE Trans. Wireless Commun. vol. no. 4 pp Apr. 3. 7] V. N. Ha and L. B. Le Distributed base station association and power control for heterogeneous cellular networks IEEE Trans. Veh. Technol. vol. 63 no. pp Jan. 4. 8] H. Zhang C. Jiang N. Beaulieu X. Chu X. Wang and T. Quek Resource allocation for cognitive small cell networks: A cooperative bargaining game theoretic approach IEEE Trans. Wireless Commun. vol. 4 no. 6 pp Jun. 5. 9] F. Baccelli and B. Blaszczyszyn tochastic Geometry and Wireless Networks: Theory. Now Publishers Inc. 9. ] M. Haenggi tochastic Geometry for Wireless Networks. Cambridge University Press. ] R. Q. Hu and Y. Qian Heterogeneous cellular networks. John Wiley & ons 3. ] Q. Ye M. Al-halash C. Caramanis and J. G. Andrews A tractable model for optimizing device-to-device communications in downlink cellular networks in Proc. IEEE International Conference on Communications IEEE ICC 4) Jun. 4 pp ]. halmashi E. Björnson M. Kountouris K. W. ng and M. Debbah Energy efficiency and sum rate when massive MIMO meets deviceto-device communication in Proc. IEEE International Conference on Communication Workshop IEEE ICC 5 Workshop) Jun. 5 pp

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