Uplink Rate Distribution in Heterogeneous Cellular Networks with Power Control and Load Balancing

Transcription

1 Uplink Rate Distribution in Heterogeneous Cellular Networks with Power Control and Load Balancing Sarabjot Singh, Xinchen Zhang, and Jeffrey G. Andrews Abstract Load balancing through proactive offloading users onto small cells is critical for tapping the potential of dense heterogeneous cellular networks HCNs). However, the impact of such offloading on the uplink performance is not well understood, primarily due to the lack of tractable models. Uplink power control and spatial interference correlation further complicate the mathematical analysis as compared to the downlink. In this paper, we propose a tractable and general model to characterize the uplink SIR and rate distribution in a K-tier HCN as a function of the association and power control parameters. Using the developed analysis, it is shown that the optimal degree of channel inversion for uplink power control) increases with load imbalance in the network. Minimum path loss association is shown to maximize uplink rate coverage. Moreover, with minimum path loss association and full channel inversion, uplink coverage is shown to be invariant of infrastructure density. I. INTRODUCTION Complementing existing cellular networks with low power access points APs), or small cells, leads to wireless networks comprising access points heterogeneous in tranmit powers and deployment density 2]. Although the mathematical modeling and performance analysis of HCNs particularly for downlink has gained significant attention in recent years 3] 6], attempts to model the uplink have been limited. In uplink intensive services like cloud storage and video chat, uplink performance is as important if not more) as that of the downlink. The insights for downlink design cannot be directly extrapolated to the uplink setting in HCNs, as the latter is fundamentally different due to i) the homogeneity of transmitters or user equipments UEs), ii) the use of uplink transmission power control to the desired AP, and iii) the correlation of the interference power from a UE with its path loss to its own serving AP. A. Background and related work Due to the significant AP transmission power disparity across different tiers in HCNs, the UE load under downlink max power association) is considerably imbalanced with macrocells being significantly more congested than small cells. It is now well established both empirically and theoretically) S. Singh is now with Nokia Inc., Berkeley, CA, X. Zhang is now with Qualcomm Inc., San Diego, CA, and J. G. Andrews is with WNCG, The University of Texas at Austin sarabjot.singh@nokia.com, xzhang7@alumni.nd.edu, jandrews@ece.utexas.edu). The work was done while first two authors were with WNCG, UT Austin. Date modified: March 5, 205. An extended version of this paper with full proofs and additional material has been submitted to IEEE Transactions on Wireless Communications ]. that biasing UEs towards small cells leads to significant improvement in downlink UE throughput see 2], 7], 8] and references therein). However, a complete analytical characterization of the uplink SIR and rate with such biasing is not available. Furthermore, power control is employed in uplink to conserve energy and to reduce other cell interference. Since the association strategy influences the statistics of path loss in HCNs, the aggressiveness of power control should be correlated with the UE-AP association strategy. Therefore, it is important to develop an analytical model to capture the interplay between load balancing and power control on the uplink performance. The use of spatial point processes particularly the Poisson point process PPP), for modeling HCNs and derivation of the corresponding downlink coverage and rate has been extensively explored as of late see 9] and references therein). However the analysis of the uplink in such a setting is highly non-trivial. Owing to orthogonal multiple access schemes, like OFDMA, there is one UE per AP located randomly within its coverage area that transmits on a given resource block, and hence an exact interference characterization is not available. Moreover, due to the uplink power control, the transmit power of an interfering UE is correlated with its path loss to the AP under consideration. Consequently, various generative models 0] 2] have been proposed to approximate uplink performance in OFDMA Poisson cellular networks. These models, however, only apply to certain special cases such as macro-only) for single tier networks 0] or full channel inversion with truncation and nearest AP association ]. They do not extend naturally to HCNs with flexible power control and association. The work in 2] adopts a similar approach to the one proposed in this paper for approximating the interfering UE process to derive the uplink SIR distribution in a two tier network with a simpler) linear power control and biased association. All these generative models, however, ignore the aforementioned conditioning, which may yield unreliable performance estimates. B. Contributions In this paper, we propose a generative model to analyze the uplink SIR and rate distribution, where the BSs of each tier are assumed to be distributed per an independent PPP and UEs employ a weighted path loss based association strategy. The interfering UE locations are modeled as an inhomogeneous PPP with intensity dependent on the association parameters. Further, the correlation between the uplink transmit power of each UE and its path loss to the AP under consideration

2 2 is captured. Using the above approach, the complementary cumulative distribution function CCDF) of the the uplink SIR and rate is derived for a K-tier HCN as a function of the offloading and power control parameters. The general expression is simplified for certain plausible scenarios. Using the developed analysis, it is shown that, minimum path loss association maximizes uplink rate coverage. For such an association and full channel inversion, the SIR coverage is shown to be independent of infrastructure density. Further, the optimal degree of power control is shown to increase with increasing disparity in association weights and decrease with increase in the SIR threshold. II. SYSTEM MODEL A co-channel deployment of a K-tier HCN is considered, where the locations of the APs of the k th tier are modeled as a 2-D homogeneous PPP Φ k R 2 of density λ k. Further, the UEs in the network are assumed to be distributed according to an independent homogeneous PPP Φ u with density λ u. The signals are assumed to experience path loss with a path loss exponent PLE) α and the power received from a node UE/AP) at X R 2 transmitting with power P at Y R 2 is PH X,Y LX, Y ), where H R + is the fast fading power gain and L is the path loss. The random channel gains are assumed to be Rayleigh distributed with unit average power, i.e., H exp), and LX, Y ) S X,Y X Y α, where S R + denotes the large scale fading i.e. shadowing). S is assumed i.i.d across all UE-AP pairs. WLOG, the analysis in this paper is done for a typical UE located at the origin O. The AP serving this typical UE is referred to as the tagged BS. The thermal noise is ignored in the analysis. A. Uplink power control Let B X Φ denote the AP serving the UE at X R 2 and define L X LX, B X ) to be the path loss between the UE and its serving base station. A fractional pathloss inversion based power control is assumed for the uplink transmission, where a UE at X transmits with power P X = L ɛ X, where 0 ɛ is the power control fraction PCF). Thus, with ɛ = 0, each UE transmits with constant power some preset target), and with ɛ =, the path loss is fully compensated. Orthogonal access is assumed in the uplink and hence at any given resource block, there is at most one UE transmitting in each cell. Let Φ b u be the point process denoting the location of UEs transmitting on the same resource as the typical UE. The uplink SIR of the typical UE at O) on a given resource block is SIR = H 0,B0 L ɛ O X Φ bu Lɛ X H. ) X,B O LX, B O ) Henceforth channel power gain between interfering UEs and the tagged BS {H X,BO } are simply denoted by {H X }, and are assumed i.i.d. The index O is dropped wherever implicitly clear. B. Weighted path loss association Every UE is assumed to be using weighted path loss for association in which a UE at X associates to an AP of tier K X where ) K X = arg max min LX, Y ), 2) Y Φ k k {,...,K} T k where T k is the association weight for k th tier same for all APs of the corresponding tier). Note that if all the association weights are identical, it results in minimum path loss association. For ease of notation, we define ˆT k T k T K k =... K, as the ratio of the association weight of an arbitrary tier to that of the serving tier of the typical user defined in 2)) under association weights {T k }. As a result of the above association model, the uplink association cell of an AP of tier k located at X is ) C X = {Y R 2 : T k LX, Y ) T j min LZ, Y ), Z Φ j j =... K. Note that the described association strategy is stationary 3] and hence the resulting association cells are also stationary. It is assumed that each AP has at least one user in its association region with data to transmit in uplink. Assuming an equal partitioning of the total uplink resources among the associated uplink users as accomplished by proportional fair or round robin scheduling), the rate of the typical user is Rate = W log + SIR), 4) N where W is the bandwidth, N denotes the total number of uplink users. III. UPLINK SIR AND RATE DISTRIBUTION A. Uplink SIR coverage The uplink SIR CCDF of the typical UE is given by Pτ) PSIR > τ) = PK = k)p k τ), 5) k= where P k τ) PSIR > τ K = k) ) HL ɛ = P X Φ bu Lɛ X H > τ K = k XLX, B) = E exp L ɛ τi) K = k ] = E L I K=k L ɛ τ) ], where I = X Φ b Lɛ u X H XLX, B) is the uplink interference and L I K=k is the Laplace transform of interference conditional on k th tier being the serving tier. The following lemma characterizes the path loss distribution of a typical UE in the given system model. Lemma. Path loss distribution at the desired link. The probability distribution function PDF) of the path loss of a typical UE to its serving BS is f L l) = l K a j exp G j l ), l 0, } 3)

3 3 where 2 α, a k = λ k πe S ], G k = K a ˆT j j, and the corresponding PDF, conditioned on the serving the tier being k, is f L K=k l) f L l K = k) = G k l exp G k l ), l 0, where A k PK = k) = a k G k is the probability of the typical UE associating with tier k. Proof. The proof follows by generalizing the results in 4], 5]. The above distribution is not, however, identical to the distribution of the path loss between an interfering UE and its serving BS, since the latter is the conditional distribution given that the interfering UE does not associate with the tagged BS. This correlation is formalized in the corollary below. Corollary. Path loss distribution at an interfering UE. The PDF of the path loss of a UE at X associated with tier j conditioned on it not lying in the association cell C B ) of the tagged BS at B of tier k and the corresponding path loss LX, B) = y, is f LX l K X = j, K = k, X / C B, LX, B) = y) G j = exp G k y ) l exp G j l ), 0 l T j y. T k Proof. Conditioned on the fact that the UE does not belong to the association cell of the tagged BS of tier k, the corresponding path loss is bounded as L X Tj T k LX, B). Noting that G j Tj T k ) = Gk results in the constrained distribution. Due to uplink orthogonal access within each AP, only one UE per AP transmits on the typical resource block and hence contributes to interference at the tagged AP. Therefore Φ b u is not a PPP but a Poisson-Voronoi perturbed lattice as per 6]) and hence the functional form of the interference or the Laplace functional of Φ b u) is not tractable. However, based on the following remark, we propose an approximation to characterize the corresponding process as an inhomogeneous PPP. Remark. Thinning probability Conditioned on a BS of tier k being located at V R 2, a UE at U R 2 associates with V with probability PB U = V ) = exp G k LV, U) ). Assumption. Proposed interfering UE point process. Conditioned on the tagged BS being located at B and of tier k, the propagation process of interfering UEs from tier j to B, N u,j {LX, B)} X Φ b u,j is assumed to be Poisson with intensity measure function Λ u,j dx) = a j x exp G k x ))dx). The basis of the above assumption is the fact that only one UE per AP can potentially interfere with the typical UE in the uplink. Assuming the potential interfering UEs from tier j to be a PPP with density λ j, the propagation process of these UEs to the tagged BS has intensity measure derivative a j x. However, conditioned on the fact that these UEs do not associate with the tagged BS, the intensity measure is thinned as per the probability of Remark. Assumption 2. Tier-wise independence. The point process of interfering UEs from each tier are assumed to be independent, i.e., the intensity measure of the interfering UEs propagation process N u is Λ u dx) K Λ u,jdx). Assumption 3. Independent path loss. The path losses {L X } are assumed to follow the Gamma distribution given by Corollary, assumed independent but not identically distributed) for all X Φ b u. Lemma 2. The Laplace transform of interference L Ik s) L I K=k s) at the tagged BS of tier k under the proposed model is exp K )]) s ˆT j a je L K=j L ɛ) sˆt j C, L ɛ 6) where C x) 2 F,, 2, x), where 2 F is the Gauss-Hypergeometric function. Proof. Let L Ikj s) denote the Laplace of interference from tier j UEs, then L Ik = K L I kj from Assumption 2). Now, L Ikj s) = E exp s X Φ b u,j L ɛ XH XLX, B) a) = E + sl ɛ X Φ b XLX, B) u,j =E ] E LX + sl ɛ X N X X u,j ]) ) b) = exp E Lx Λ u,jdx) x>0 + sl ɛ xx ]) ) c) = exp E L x>0 + sl ɛ x L < ˆT jx, K x = j Λ u,jdx) ] ) = exp E L x>0 + sl ɛ ) x L < ˆT jx, K x = j Λ u,jdx) ]) = exp E L K=j a jl dt ˆT j, + sˆt j) L ɛ t / where a) follows from the i.i.d. nature of {H X }, b) follows from the Laplace functional also known as probability generating functional) of the assumed PPP N u,j, c) follows from Corollary, and the last equality follows with change of variables t = xˆt j /L). The final result is then obtained by using the definition of Gauss-Hypergeometric function. Using the above Lemma and 5), the uplink SIR coverage is given in the following Theorem. Theorem. The coverage probability for the proposed uplink generative model is given in 7) at the top of next page) The coverage expression for the most general case involves two folds of integral and a lookup table for the Hypergeometric function. The expression is, however, further simplified for the special cases in the following corollaries. Useful bounds can hence be obtained in closed-form Corollary 2) as below.

4 4 Pτ) = a k k= l>0 l exp G k l τl ɛ T j /T k ) a j E L K=j L ɛ) τtj l ɛ )] C T k L ɛ dl. 7) Corollary 2. The uplink coverage is lower bounded by Corollary 5. ɛ = ) With full channel inversion, the coverage P l = exp τ π 2 K ) ɛ ɛ) K )) is a k a k sinπ) sinπɛ) G 2 ɛ G ɛ a k k= k k= k Pτ) = exp G k τ K ) ) ) Tj a j C τ Tj. T k G j T k k= Proof. Using the proof of Lemma 2, we have Corollary 6. ɛ = 0, T j = T k j, k) Without power control ] and with minimum path loss association, the coverage is L Ik s) exp E L K=j x>0 + sl ɛ ) Λ u,j dx) x ] Pτ) = a l exp al ) l>0 exp E L K=j x>0 + sl ɛ ) x a jx dx exp a )]) τlel L τl C dl. L Corollary 7. ɛ =, T j = T k j, k) With full channel inversion and with minimum path loss association, the coverage a) = exp s π a j E L K=j L ɛ ], sinπ) is where a) follows by the change of variables t = x sl ɛ ) 2/α and noting that Pτ) = exp τ ) C τ). dt 2π = 0 +t α/2 α sin2π/α). Now using the Remark 2. Density invariance. Corollary 7 highlights the coverage expression independence of uplink coverage on infrastructure density in Pτ) E exp π sinπ) τ L ɛ) a j E L K=j L ɛ ] HCNs with minimum path loss association and full channel inversion. exp π sinπ) τ E L ɛ)] K a j E L K=j L ɛ ] C. Validation, The proposed model and the corresponding analysis are validated in a two tier setting with λ = 5, α = 3.5, and where the last inequality follows from Jensen s inequality. Noting that E ] L K=j L ɛ = Γ+ɛ) G and E S assumed Lognormal with 8 db variance. Same parameter L ɛ)] = values are used in the numerical results of the next section ɛ j K a j Γ2 ɛ) and Γ + ɛ)γ2 ɛ) = πɛ ɛ) unless otherwise specified. Fig. shows the SIR distribution G 2 ɛ sinπɛ) leads to the j comparison between the simulations and analysis Theorem final result. ) for different association weights and small cell density. B. Special cases For the following special cases, the coverage expression is further simplified. Corollary 3. K = ) The coverage in a single tier network with density λ is Pτ) = a l exp a l ) l>0 exp )]) τl τl ɛ a E L L ɛ) ɛ C L ɛ dl, where a = λ πe S ]. The above expression differs from the one in 0] due to the interference characterization. In 0], the distribution of path loss of each UE to its serving BS was assumed i.i.d. Corollary 4. T j = T k j, k) The uplink coverage in a K-tier network with minimum path loss BS association is same as the coverage of single tier network with density λ = K k= λ k. A value of = 20 db corresponds to a typical power difference between macrocells and small cells. As can be observed, the proposed analysis accurately matches the simulation results. D. Rate distribution The rate of a user depends on both the SIR and load at the tagged AP as per 4)), which in turn depends on the corresponding association area. The weighted path loss association and PPP placement of APs leads to complex association cells whose area distribution is not known. However, the association policy is stationary 3] and hence the mean association area of a typical AP of tier k is A k λ k. The mean load approximation proposed in 7] can be used to quantify the uplink rate distribution as Rρ) PRate > ρ) = A k P 2ˆρ N k k ). k= where the load at each AP is approximated by its respective mean, Nk E N K = k] = +.28 A kλ u λ k.

5 = -20 db Simulation Analysis = 0 db Simulation Analysis = 6λ = 4λ SIR coverage 0.5 ε = 0 SIR coverage 0.5 ε = ε = ε = SIR threshold db) SIR threshold db) a) Fig. : Comparison of uplink SIR distribution obtained from analysis and simulation. b) Optimal PCF = 0 db = -0 db = -20 db SIR threshold db) Fig. 2: Variation of optimal PCF with SIR threshold in a two tier setting with = 5λ. IV. OPTIMAL POWER CONTROL AND ASSOCIATION The coverage probability expression in Theorem can be used to numerically find the optimal power control and association weights. To get more direct insights, we focus on the coverage lower bound P l and obtain the following proposition. Proposition. Minimum path loss association maximizes P l ɛ 0, ]. Further, ɛ = 0.5 maximizes P l with minimum path loss association. Proof. Using Corollary 2, P l is maximized with {T j } given by {T j } K = arg min = + arg min a k a k G 2 ɛ G ɛ k= k k= k i j a ia j ɛ i T ɛ j T it j ) K a, jt j ) 2 where the last equation is minimized with T j = T k j, k. Moreover, for such a case ) P l τ) = exp τ 2/α π2 ɛ ɛ), sinπ) sinπɛ) which is maximized for ɛ = 0.5. Since the lower bound overestimates the interference by neglecting the correlation with the path loss and hence treating it as if originating from an ad-hoc network), the result of optimal PCF of 0.5 is inline with the results for ad-hoc wireless networks 8], 9] derived under quite different modeling assumptions, though). The SIR threshold plays a vital role in determining the optimal PCF. Power control is more beneficial for cell edge UEs as they suffer from higher path loss and as a result the optimal PCF obtained using Theorem ) decreases with SIR threshold, as shown in Fig. 2. Further, as can be observed a higher association weight imbalance leads to a uniform increase in the optimal ɛ, as the path losses in the network increase statistically). The trend of uplink edge fifth percentile) and median rate with association weights is shown in Fig. 3. As can be seen, irrespective of the PCF and density, minimum path loss association maximizes both these metrics, since minimum path loss association leads to identical load distribution across all APs and therefore balances the load. Note that these results and insights for uplink are in contrast with the corresponding result This is consistent with the result in 0] for single-tier networks.

6 6 0 x 04.8 x 06 9 ε =.6 ε = Fifth percentile rate bps) = 0λ = 5λ Median rate bps) = 0λ = 5λ Normalized association weight ) db) a) Edge rate, ɛ = Fig. 3: Variation of edge and median rate with association weights Normalized association weight ) db) b) Median rate, ɛ = 0 for downlink, where maximum SIR association equivalent to maximum downlink received power association) is optimal for downlink SIR coverage 6], and hence a conservative association bias was shown to be optimal for rate coverage 7], 20]. V. CONCLUSION A tractable and general framework is proposed for characterizing uplink SIR and rate coverage in K-tier HCNs. To the best of the authors knowledge, this is the first work to derive and validate the uplink SIR distribution for HCNs incorporating load balancing and power control. This work provides tools to analyze the trade off between uplink and downlink performance with regards to load-balancing and further help quantify the potential gains of decoupled where the BS serving the uplink and that serving downlink need not be same) association strategies 2]. REFERENCES ] S. Singh, X. Zhang, and J. G. Andrews, Joint rate and SINR coverage analysis for decoupled uplink-downlink biased cell associations in Het- Nets, IEEE Trans. Wireless Commun., Dec Submitted, available at: 2] A. Ghosh et al., Heterogeneous cellular networks: From theory to practice, IEEE Commun. Mag., vol. 50, pp , June ] J. G. Andrews, F. Baccelli, and R. K. Ganti, A tractable approach to coverage and rate in cellular networks, IEEE Trans. Commun., vol. 59, pp , Nov ] 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. 30, pp , Apr ] S. Mukherjee, Distribution of downlink SINR in heterogeneous cellular networks, IEEE J. Sel. Areas Commun., vol. 30, pp , Apr ] H.-S. Jo, Y. J. Sang, P. Xia, and J. G. Andrews, Heterogeneous cellular networks with flexible cell association: A comprehensive downlink SINR analysis, IEEE Trans. Wireless Commun., vol., pp , Oct ] J. G. Andrews, S. Singh, Q. Ye, X. Lin, and H. S. Dhillon, An overview of load balancing in HetNets: Old myths and open problems, IEEE Wireless Commun. Mag., vol. 2, pp. 8 25, Apr ] A. Damnjanovic et al., A survey on 3GPP heterogeneous networks, IEEE Wireless Commun. Mag., vol. 8, pp. 0 2, June 20. 9] H. ElSawy, E. Hossain, and M. Haenggi, Stochastic Geometry for Modeling, Analysis, and Design of Multi-tier and Cognitive Cellular Wireless Networks: A Survey, IEEE Communications Surveys & Tutorials, vol. 5, pp , July ] T. Novlan, H. Dhillon, and J. Andrews, Analytical modeling of uplink cellular networks, IEEE Trans. Wireless Commun., vol. 2, pp , June 203. ] H. ElSawy and E. Hossain, On stochastic geometry modeling of cellular uplink transmission with truncated channel inversion power control, IEEE Trans. Wireless Commun., vol. 3, pp , Aug ] H. Lee, Y. Sang, and K. Kim, On the uplink SIR distributions in heterogeneous cellular networks, IEEE Commun. Let., vol. to appear, pp , Dec ] S. Singh, F. Baccelli, and J. G. Andrews, On association cells in random heterogeneous networks, IEEE Wireless Commun. Lett., vol. 3, pp , Feb ] B. Blaszczyszyn, M. K. Karray, and H.-P. Keeler, Using Poisson processes to model lattice cellular networks, in Proc. IEEE Intl. Conf. on Comp. Comm. INFOCOM), pp , Apr ] P. Madhusudhanan, J. Restrepo, Y. Liu, and T. Brown, Downlink coverage analysis in a heterogeneous cellular network, in IEEE Global Commun. Conf. GLOBECOM), pp , Dec ] B. Błaszczyszyn and D. Yogeshwaran, Clustering comparison of point processes, with applications to random geometric models, in Stochastic Geometry, Spatial Statistics and Random Fields, vol. 220 of Lecture Notes in Mathematics, pp. 3 7, Springer International Publishing, ] S. Singh and J. G. Andrews, Joint resource partitioning and offloading in heterogeneous cellular networks, IEEE Trans. Wireless Commun., vol. 3, pp , Feb ] N. Jindal, S. Weber, and J. Andrews, Fractional power control for decentralized wireless networks, IEEE Trans. Wireless Commun., vol. 7, no. 2, pp , ] F. Baccelli, J. Li, T. Richardson, S. Shakkottai, S. Subramanian, and X. Wu, On optimizing CSMA for wide area ad-hoc networks, in Intl. Symp. on Modeling and Optimization in Mobile, Ad Hoc and Wireless Networks WiOpt), pp , May ] Q. Ye, B. Rong, Y. Chen, M. Al-Shalash, C. Caramanis, and J. G. Andrews, User association for load balancing in heterogeneous cellular networks, IEEE Trans. Wireless Commun., vol. 2, pp , June ] H. Elshaer, F. Boccardi, M. Dohler, and R. Irmer, Downlink and uplink decoupling: a disruptive architectural design for 5G networks, in IEEE Global Commun. Conf. GLOBECOM), Dec. 204.