Wireless Networks Without Edges : Dynamic Radio Resource Clustering and User Scheduling

Transcription

1 Wireless Networks Withot Edges : Dynamic Radio Resorce Clstering and User Schedling Yhan D and Gstavo de Veciana Department of Electrical and Compter Engineering, The University of Texas at Astin dyhan123@gmail.com, gstavo@ece.texas.ed Abstract Celllar systems sing Coordinated Mlti-Point (CoMP) transmissions leveraging clsters of spatially distribted radio antennas as Virtal Base Stations (VBSs) have the potential to realize overall throghpt gains and, perhaps more importantly, can deliver sbstantial enhancement to poor performing edge sers. In this paper we propose a novel framework aimed at flly exploiting the potential of sch systems throgh dynamic radio resorce clstering and ser schedling which maximize system tility. The dynamic clstering problem is modeled as a maximm weight clstering problem which is NPhard, however, we show that by strctring the set of possible VBSs to be 2-decomposable it can be efficiently compted. We also propose to optimize over a class of power allocation policies to radio resorces, and ths VBSs, which allow dynamic ser schedling and flexible power allocations depending on instantaneos channel realizations. We se simlation to compare or approach with a state-of-the-art baseline which exploits dynamic freqency rese and opportnistic ser schedling, bt no clstering, and show edge sers throghpt gains are as high as 80% withot degrading the performance of others. I. INTRODUCTION The forth generation celllar systems based on OFDMA techniqes crrently being deployed achieve good coverage and high system throghpt. Still there is interest in realizing even higher per ser throghpt, particlarly for sers at the edge. Coordinated Mlti-Point (CoMP) techniqes provide a path to address this problem. They can be particlarly advantageos when leveraging spatially distribted radio resorces - we refer to these as Remote Radio Heads (RRHs). By clstering neighboring RRHs into Virtal Base Stations (VBSs) and encoraging cooperation among RRHs to redce or eliminate mtal interference, edge sers which were traditionally poorly served can become central to one or more VBSs, i.e., are well sitated relative to their serving VBSs. Ideally, if there is enogh freedom in choosing VBSs each ser cold be central - we refer to this as a no-edge wireless network. Given the benefits of CoMP and assming no comptational constraints, one cold in principle coordinate across all RRHs leveraging spatial diversity (in antennas) and removing all interference. Unfortnately, previos work (see e.g., [1]) has shown that coordination across the entire system does not bring as mch benefit as expected becase of measrement and signaling overheads. In practice cooperation is only possible, or desirable, within VBSs of limited size. This research was spported by Hawei Technologies Co. Ltd. If we clster RRHs into static sets of VBSs there may still be sers at the edge of neighboring VBSs which sffer from interference. To realize no-edge wireless networks it is ths necessary to se different VBSs in different sb-bands (or different times) garanteeing that each ser can be central to a VBS. In the extreme case, on each time slot and sb-band, the system has the freedom to dynamically clster RRHs into VBSs and to schedle sers that are central to the VBSs. We call this dynamic clstering. Advanced power control and ser selection strategies are also essential to reap the benefits of CoMP to deliver good performance to every ser. While this concept is attractive, there has not been mch work on how to achieve good clstering and ser schedling in this setting. A system employing CoMP and dynamic clstering cold also leverage the crrent trend towards a Clod-based Radio Access Network (C-RAN) compte infrastrctre [2]. In addition to the flexibility to dynamically adjsting VBSs, a C-RAN based system cold facilitate compte workload balancing among VBSs, is potentially more reliable and energy efficient by switching on/off RRHs and comptational resorces according to traffic loads. In this paper, we focs on delay-tolerant best-effort traffic and propose a framework that enables dynamic clstering and advanced ser schedling (power control and ser selection) algorithms to realize the promise of CoMP towards wireless networks withot edges. Related Work. Let s briefly introdce CoMP techniqes, see e.g., [3]. CoMP exploits cooperation among RRHs, or antennas co-located at a RRH, to provide better throghpt to a single ser or more aggressively to mltiple sers simltaneosly. Dirty Paper Coding (DPC) [4] is known to be the optimal (capacity achieving) theoretical soltion bt is difficlt to implement de to high complexity. In this paper, we consider a sboptimal CoMP techniqe called zeroforcing beamforming (ZFBF) that can be easily implemented. In ZFBF, a precoding vector is compted for each schedled ser by inverting the channel matrix of all schedled sers theoretically avoiding intra-vbs interference. In OFDMA-based celllar systems with Single-Inpt and Single-Otpt (SISO) transmissions, cell edge sers often see interference from adjacent cells, and/or sers therein, sharing the same freqency band inter cell interference. The dynamic Fractional Freqency Rese (FFR) scheme described in [3], [5] offers perhaps the most efficient and flexible means to mitigate

2 2 inter cell interference for sch systems. A virtal schedler and a power-control loop are proposed to adapt power across cells and sb-bands. However, Mlti-User Mltiple-Inpt Mltiple- Otpt (MU-MIMO) techniqes are generally employed to enable higher system capacity. In sch systems sers may also see interference associated with transmissions to sers within the same cell intra-cell interference. The work in [6], [7] extends the framework in [5] to MU-MIMO scenarios that adopt a fixed set of beams which limits the flexibility and is not considered to be efficient. Moreover, cooperation across RRHs is not considered in these works. This paper aims to address this challenge. The work in [5] [6] [7] makes se of the reslt in [8] which generalizes the well-known proportional fair schedler to an opportnistic gradient schedling algorithm that aims to maximize a concave system tility fnction. This paper proves the asymptotic optimality of sch an algorithm and also provides a theoretical fondation for or proposed framework. In [9] the athors propose the se of different VBSs on different sb-bands so that each ser can be central to its associated VBS, bt the paper only lists a fixed set of VBSs for each sb-band to garantee fll coverage. They also do not consider systems which make clstering decisions dynamically and do not take into accont opportnistic ser schedling. Additional motivating work incldes [10] which focses on a single cell with mltiple antennas that spports ZFBF. They propose an efficient semi-orthogonal ser selection (SUS) algorithm that selects semi-orthogonal sers for simltaneos transmission and prove that their ZFBF-SUS framework can achieve the same asymptotic sm rate as DPC when the nmber of sers is large. However, it does not extend to larger systems and does not distingish edge sers and central sers. Or Contribtions. To or knowledge, this is the first work to propose a framework leveraging CoMP, via dynamic clstering and opportnistic ser schedling aimed at maximizing the overall system tility. To address the large nmber of degrees of freedom associated with solving highly-copled problems of RRH clstering, ser selection and power control, along with transmission precoding, and meet real-time comptational constraints, we propose a decomposition of this complex problem, along with new strctral properties that lead to efficient comptation. We provide an efficient soltion to finding optimal clsters. In particlar dynamic clstering is redced to a Maximm Weight Clstering (MWC) problem which is proven to be NP-hard. However, in the wireless context where cooperation wold happen amongst RRHs that are close by, we propose to strctre the set of possible VBSs to satisfy 2- decomposability property nder which dynamic clstering in each sb-band can be solved efficiently with complexity O( R 1.5 ) where R is the nmber of RRHs in the system. In or approach, dynamic clstering comptation is centralized while the ser schedling for single VBSs can be decentralized and solved at possibly distribted comptational resorces associated with VBSs. To exploit opportnistic gains associated with ser selection and power control withot increasing comptational complexity, we propose a flexible power allocation policy which adapts power levels across radio resorces and sb-bands. With these in hand, or schedler can choose to assign power levels opportnistically to schedled sers. This enables the system to achieve good FFR patterns and at the same time allows the schedler more flexible se of power to serve sers and ths more freedom to achieve opportnistic benefits. Finally, we provide an initial performance evalation based on simlations. By comparing with an aggressive benchmark that does not exploit dynamic clstering bt does perform dynamic soft FFR (see e.g. [5]) and opportnistic ser schedling, we show that dynamic clstering improves the rate of edge sers by 80.4% withot degrading the performance of others. Moreover and interestingly, we show how dynamic clstering is prone to move edges to other locations. Paper Organization. The paper is organized as follows: Section II introdces or system model, the problem we aim to solve and the conceptal overview of or approach. Section III discsses or problem decomposition and strctral constraints to ensre efficient soltion of dynamic clstering and ser schedling while Section IV addresses the adaptation of the power allocation policy. Simlation reslts are exhibited in Section V and Section VI points to ftre work. II. SYSTEM MODEL AND PROBLEM FORMULATION In this paper, a vector x is by defalt a row vector. We se x to denote the conjgate, x T the transpose and x H the conjgate transpose of vector x. We consider the downlink of a mlti-ser mlti-rrh OFDMA-based celllar network. The operational freqency band is divided into eqal sb-bands of size W Hz; J denotes the set of sb-bands. σ 2 is the noise power in a sb-band. The system operates in discrete time, over time-slots t = 0, 1,. U and R represent finite sets of sers and RRHs, respectively. We let S denote the cardinality of the set S. The RRHs are indexed from 1 to R. At time t, each r R is associated with a set of sers U {r} (t) U. An example association policy 1 wold be U {r} (t) iff r is the closest RRH to at time t. In each sb-band an RRH r R cold work by itself (in which case {r} is a singleton VBS) or cooperate with neighboring RRHs to form a VBS. There may be constraints on forming VBSs, for example, the maximm nmber of RRHs in a VBS or the maximm distance between RRHs in a VBS. We denote by V the collection of all allowed VBSs. A VBS V V is a set of RRHs that can coordinate their transmissions. (Withot loss of generality, we assme that all antennas of an RRH belong to the same VBS at a certain time in a given sb-band.) For example, let V = {r 1, r 2 } V denote a VBS V containing RRH r 1 and r 2 and U V (t) = U {r} (t) denote the set of sers that r V cold be served by VBS V at time t. 1 Other association policies cold be considered, or in fact no sch policy needs to be specified. Yet this simplifies description of the setting for now.

3 3 We let P t,j denote a partition of R based on sets in V for se at time t in sb-band j, in other words, for any V 1, V 2 P t,j, and if V 1 V 2 then V 1 V 2 =, V P t,j V = R. The set of all possible partitions of R indced by V is denoted P(R, V). Note that each ser can be associated with more than one VBS bt it can only be served by one at a given time per sb-band. Sppose each RRH is eqipped with n t transmit antennas and each ser has n r receive antennas. In this paper we assme n r = 1. ZFBF is employed to enable simltaneos transmissions to mltiple sers. In particlar, k (< n t ) sers per RRH can be served at the same time, ths, p to k V sers can be served simltaneosly by VBS V in a given sbband. For simplicity we assme flat and fast fading, i.e., the same fading is experienced in all sb-bands bt can vary across slots - the framework of this paper can be extended to freqency-selective fading scenarios. Let h {r} (t) = (t),..., h r,nt (t)) represent the complex channel vector from RRH r to ser at time t and h R (t) = (h {1} (t),..., h { R } (t)) to be the channels from all RRHs to. We also define h V (t) = (h {r} (t) r V ) to be the channel vector from VBS V to ser at time t 2. Note that h V (t) is the signal channel if U V (t) and an interference channel otherwise. We focs on best-effort traffic, so each ser has an associated concave, strictly increasing tility fnction U ( X ) of its long-term time average rate X. The choice of tility fnctions U may take into accont fairness, qality of service and priorities among different sers. Additional reqirements sch as meeting a minimm throghpt per ser cold also be inclded in or setting. Or goal is to find a dynamic clstering and ser schedling strategy that maximizes the system tility given by (h r,1 U = U U ( X ) and more generally tracks changes in the system while attempting to meet this goal. A. Gradient Schedler Inspired by proportional-fair schedler, [5], [8] introdced a gradient algorithm to tackle general tility maximization problems sch as the above. For or setting, let d j denote a clstering and ser schedling decision in sb-band j and D denote a finite set 3 of possible decisions. Also let { x (t) U} denote the long term average rate estimates p 2 For consistency, the complex nmbers in h V (t) have the same orders as they appear in h R (t). 3 Althogh power is a continos vale, we can consider power as a discrete variable that has a large nmber of possible vales. to time t and R j (d j, t) denote the rate ser wold achieve at time t in sb-band j nder decision d j. The algorithm and the main theoretical reslt developed in [8] as applied to or problem is stated below. Theorem 1: Sppose the channels from the RRHs to sers have stationary distribtions and U ( ) are concave increasing tility fnctions. In the convex set of all feasible long-term achieved rate vectors, let x opt = ( x opt opt 1,..., x U ) be the optimal vector maximizing system tility U. If at each time t and in sb-band j, the system chooses a decision d opt j sch that d opt U j arg max d j D X x(t)r j (d j, t), U and for all the average rate estimate x (t) is pdated as: x (t + 1) = (1 β) x (t) + βjr j (d opt j, t), with arbitrary initial vale x (0). Then, as β 0 and t, both the estimate rate x (t) and the achieved average rate X converge to x opt for every ser. Underlying each decision d at each time t and sb-band j, we need to: Select a partition P t,j P(R, V) corresponding to a collection of VBSs. Select p to k V sers for each VBS V in P t,j. Assign power to each schedled ser. Determine precoding vectors for all schedled sers associated with a VBS. Althogh we listed these for items seqentially, they correspond to copled decisions and the challenge is to find comptationally efficient approach that can be sed in practice. B. Parameterizing Power Allocation Policy To decople the dynamic clstering and ser schedling decisions across sb-bands and VBSs we will fix a power allocation policy bt adapt its parameters. Definition 1: A feasible power allocation policy is a pair (P, Φ). The RRH sb-band power allocation matrix P is P = (p j,r j J, r R), where p j,r denotes the power allocated to RRH r on sb-band j and mst satisfy p j,r p where p is the power constraint j J per RRH. The power available to a VBS V on sb-band j is then given by p V j = p j,r. Recall that V may serve r V p to k V sers simltaneosly, for which it will need to assign transmit power levels. The VBS sb-band power level allocation matrix Φ is given by Φ = (φ V j,l j J, V V, l = 1,..., k V ), where φ V j,l is the ratio of the lth highest power level for VBS V in sb-band j to the total available power p V j and mst k V satisfy φ V j,l 1 and φv j,l 1 φ V j,l 2 for l 1 < l 2. l=1

4 4 Under sch a policy the schedler can flexibly decide which sers to serve and what power levels they will se allowing it to opportnistically exploit channel variations. It may be desirable to trn off an RRH if there are few or no sers associated with it. In this case the power allocation policy cold set the associated allocation for that RRH to zero. C. Conceptal Overview of Or Approach Or approach contains two main components as shown in Fig.1.!!!!!!!!!!!!!!!1)!!Dynamic!Clstering!&!!!!!!!!!!!!!!!!!!!!!!!User!Schedling! 8 Instantaneos!channels! Average!channels! characteris>cs!!!!!!!!!!!!2)!!adap>ng!power!alloca>on!policy! 8 System!model!based!on!average! channels! Fig. 1. Overview of Approach Power!! Alloca>on!! Policy! The first component takes as inpts a power allocation policy for the system and performs dynamic clstering and ser schedling. To exploit channel variations and achieve opportnistic gains, it makes se of the instantaneos channels that are fed back from sers to the RRHs. The second component adapts the power allocation policy across radio resorces and sb-bands. To do so, it approximates the gradients of system tility to the power allocation parameters and performs a gradient ascent algorithm. III. DYNAMIC CLUSTERING AND USER SCHEDULING In this section, we assme we are given a power allocation policy (P, Φ), the crrent long term average rate estimates ( x (t) U) for sers p to time t and instantaneos channel measrements (h R (t) U). Or goal is to make a clstering and ser schedling decision for each sb-band j. To simplify notation, we will sppress the time index. Definition 2: A clstering and ser schedling decision d j for sb-band j consists of a partition P(d j ) P(R, V) for sb-band j and a ser schedling decision s V (d j ) for each VBS V P(d j ). A ser schedling decision s V for VBS V is a power allocation vector s V = (l 1, l 2,..., l U ) {0, 1,..., k V } U where { l, if U V and is assigned power level l in s V, l = 0, otherwise. We reqire that no two sers be assigned the same power level and ths s V has at most k V non-zero elements. Let D denote the finite set of possible clstering and ser schedling decisions and D(V ) denote the finite set of ser schedling decisions for VBS V. Following the reslt in Theorem 1, or goal is to find a decision satisfying d opt U j arg max d j D X R j (d j ). (1) x U A key isse here is to compte the achieved rate R j (d j ) for ser V nder a given decision d j. In particlar, given s V (d j ) for VBS V and sing ZFBF, we can compte the precoding vector w (s V (d j )) (see e.g., [10]) for each schedled ser in V sing 0 for that are not schedled. For ser along with V (where it is possible that = ), we define g V, (s V (d j )) = h V w (s V (d j )) T 2 to be the effective gain from VBS V to ser nder the precoding vector of ser associated with decision s V (d j ). Let p j (s V (d j )) denote the power assigned to ser in sbband j nder decision s V (d j ). Ths, p j (s V (d j )) eqals to p V j φv j,l if l is the power level assigned to ser and eqals to 0 if is not schedled nder decision s V (d j ). Since mltiple precoding vectors are sed simltaneosly, there may be some intra-vbs interference I,j intra from sers in the same VBS and inter-vbs interference I,j inter from other VBSs. However, I,j intra shold close to 0 if channel measrements are accrate and perfect ZFBF is sed. So we have R j (d j ) = W log 2 (1 + gv, (s V (d j ))p j (s V (d j )) σ 2 + I intra ), (2) where I inter,j = I intra,j = U V, V P(d j),v V U V,j + Iinter,j g V, (s V (d j ))p j(s V (d j )), g V, (s V (d j ))p j(s V (d j )). Problem (1) is difficlt to solve becase the nmber of decisions D is large and the ser schedling decisions in different VBSs are copled. Instead, we will explore a decomposition of this problem. A. Decopling Dynamic Clstering and User Schedling Note that the achieved rate R j (d j ) for ser V not only depends on s V (d j ) bt also depends on the clstering decision P(d j ) and the ser schedling decisions in other VBSs throgh I,j inter. Or approach is to find an approximation for I,j inter and ths for R j (d j ) sch that problem (1) can be decomposed into sb-problems which in trn can be efficiently solved. The optimal decision determined nder d opt j or approximation shold still generate a large vale for (1). Sppose each RRH r / V transmits independently, i.e., withot mtal cooperation, then an achievable pper bond for I inter,j is given by Ĩ inter,j r / V h {r} 2 p j,r. (3) We arge that Ĩinter,j is a good estimate, becase or algorithm is geared at choosing appropriate clstering and ser schedling decisions to make I,j inter small in the first place which means a small error in I,j inter shold not lead to a large bias in R j (d j ).

5 5 By replacing I inter,j with Ĩinter,j, we get the approximation for instantaneos achieved rate for ser served by VBS V that depends only on s V (d j ), i.e., decisions local to VBS V rather than all decisions across the network, i.e., R j (s V (d j )) = W log 2 (1 + gv, (s V (d j ))p j (s V (d j )) σ 2 + I,j intra + ). Ĩinter,j (4) Now the problem to be solved can be approximated and written as d opt U j arg max d j D X Rj (s V (d j )), (5) x V P(d j) U V i.e., a sm over VBSs in partition P(d j ) of the marginal tilities nder ser schedling decision s V (d j ). Let s define U w j (V ) = max s V D(V ) X Rj (s V ) x U V to be the weight of VBS V in sb-band j which captres the maximm marginal tility for VBS V, then solving (5) is eqivalent to finding a weight for each VBS and finding a partition that gives maximm sm weight. In Sbsection III-B we tackle the problem of picking a partition assming weight for each VBS is known and in Sbsection III-C we propose a sboptimal scheme to compte approximations of the VBSs weights. B. Maximm Weight Clstering Problem Picking an optimal partition is called a Maximm Weight Clstering Problem which we define as follows: Maximm Weight Clstering (MWC) Problem: The Maximm Weight Clstering (MWC) Problem (V, w j ), where V is the collection of allowed VBSs and w j = (w j (V ) V V) represents the associated non-negative weights for VBSs, is to determine a partition P opt j sch that P opt j arg max P P(R,V) V P w j (V ). Fig.2 exhibits an MWC Problem. The sets R, V and two possible partitions P1, P2 are shown while the weights are not given in the figre. For this simple example, P(R, V) = {P1, P2}. The goal is to find which partition has maximm weight. Theorem 2: MWC is an NP-hard problem. Proof: MWC can be transformed from the problem of three-dimensional matching [11] which is NP-hard. In general, a brte-force algorithm which goes over all possible partitions in P(R, V) wold have a time complexity of O(2 V ). Knth s Algorithm X [12] provides a backtracking algorithm that finds all possible partitions. However, the time complexity is still exponential in V which is not sfficiently efficient to be applied each time slot on each sb-band. To simplify the soltion to dynamic clstering, we consider constraining the set of possible VBSs, V, so that MWC can be Fig User RRH An example MWC Problem. solved efficiently. Some definitions are needed to nderstand or approach. Definition 3: Two RRHs r 1 and r 2 are eqivalent nder V, written as r 1 V r2, iff V V, if r 1 V and V > 1, then r 2 V. In other words, for any possible partition in P(R, V), r 1 and r 2 are either singletons or in the same VBS. For the example in Fig.2, RRH 2 and RRH 5 are eqivalent. In P1, they belong to VBS {2, 5, 6} and in P2 they are in {1, 2, 5}. RRH 3 and RRH 7 are also eqivalent. Collorary 1: The above eqivalence relation is reflexive, symmetric and transitive, so it partitions R into a set of eqivalence classes E V = {E 1, E 2,... } where E V is a partition of R, and i and r 1, r 2 E i, r 1 V r2. Note that althogh E V is a partition of R it need not be in P(R, V). For the example in Fig.2, there are five eqivalence classes, i.e. E 1 = {1}, E 2 = {2, 5}, E 3 = {6}, E 4 = {3, 7}, E 5 = {4}. Definition 4: A collection of VBSs V is said to be 2- decomposable if for any V V, there exists E i, E j E V sch that V E i E j. Theorem 3 below shows the benefit of satisfying this property. It is proven in the Appendix. Theorem 3: Given a MWC Problem (V, w j ), if V is 2- decomposable, then it is eqivalent to a maximm weight matching problem which is solvable in polynomial-time. For the example in Fig.2, V is 2-decomposable. Constrcting the set of VBSs V that satisfy 2- decomposability still allows flexibility towards achieving a no-edge network and ths realizing the benefits of dynamic clstering. First, inclding all singleton VBSs to a 2- decomposable V wold maintain this property bt provide freedom to the schedler. Second, this property allows VBSs inclding more than two RRHs, sch as VBS {1, 2, 5} in Fig.2. As mentioned in Section I, VBSs of interest generally have limited size making it more likely they satisfy 2- decomposability. Taking Fig.2 as an example, by switching between P1 and P2, all for sers can be schedled and served by VBSs with respect to which they are central. By inclding more

6 6 VBSs in V withot losing 2-decomposability, for example, all singleton VBSs and VBS {2, 5, 3, 7}, more areas can be garanteed good coverage. Ths, no-edge wireless network can increasingly be achieved nder this choice of V in this example. The maximm weight matching problem is well stdied in graph theory and can be solved efficiently sing Edmond s matching algorithm (see e.g., [13]). The analysis in the Appendix shows the time complexity wold be O( R 1.5 ). C. User Schedling for Each VBS We refer to the comptation of the weight w j (V ) for VBS V in sb-band j as a User Schedling Problem. Recall that the weights for all V V serve as inpts to the MWC Problem considered in the previos sbsection. User Schedling (US) Problem: The User Schedling (US) Problem (p j, φ V j, x, h, U V ), where p j = (p j,r r R), φ V j = (φ V j,l l = 1,..., k V ), x = ( x U V ) and h = (h R U V ), is to determine an optimal decision s opt V consisting of an optimal ser grop S opt and power level assignment that gives a maximm weight U w j (V ) = max s V D(V ) X Rj (s V ). x U V Solving this reqires an exhastive search over all possible ser grops. The time complexity for this task wold be k V ) C(i, V ) where C(i, V ) 4 captres the complexity i=1 ( UV i of evalating a ser grop of size i in a VBS of size V, i.e., determining precoding vectors and power level assignments. Ths, a sboptimal algorithm that is easy to implement is of interest. We propose a two phase approach. In Phase 1 we se a greedy algorithm to select ser grop S 0 and in Phase 2 we find the optimal power level assignment for S 0. As a reslt, we compte the marginal tility for this decision which is only an approximation for w j (V ). Phase 1 - Sboptimal User Selection: Inspired by [10], we claim that in dense networks where there are many sers in each VBS, the optimal ser grop S opt consists of sers that have nearly orthogonal channels, i.e., the precoding vectors for sch sers have similar direction as their channels, reslting in large effective gains and ths large rates. If sers were perfectly orthogonal, the precoding vectors wold be hv hv h V h V. By sing as precoding vectors in (4) and assming total power p V j eqally split among M = min[k V, U V ] sers, we get an estimate R j (S) for the rate of ser S when ser grop S is selected, R j(s) h V 2 pv j M = W log 2 (1 + σ 2 + I intra,j + ), Ĩinter,j 4 A reasonable estimate wold be that compting precoding vectors takes O((i n t V ) 3 ) and power assignment takes O(i 4 ). where with I intra,j = S, EI(h V, h V ) = hv EI(h V, h V )pv j M, h V H h V 2, (6) representing the Effective Intra-VBS interference (EI) to generated by and Ĩinter,j given in (3). In this comptation we may generate non-zero I intra,j by sing hv as precoders. h V However, once ser grop S 0 is determined, we can afford to compte actal precoding vectors which gives zero intra-vbs interference in the power assignment process. Frther, we define w j(v, S) = U X R j(s), S x to be the estimate of marginal tility if ser grop S is selected and power is eqally split among M sers. The sboptimal ser grop S 0 is constrcted as below: We start with an empty S 0. At each step we pick a ser opt sch that opt arg max U V \S 0 w j(v, S 0 {}) and let S 0 S 0 {}. The algorithm terminates when S 0 = M. Phase 2 - Power Level Assignment: In Phase 2, the ser selection S 0 is determined and we can compte the precoding vector for each S 0. Since I,j intra is 0, Rj (s V ) in (4) depends on s V only throgh the power level assigned to ser nder schedling decision s V. In this phase we se R jl to denote R j (s V ) for s V that assigns power level l to ser. We constrct a bipartite graph [14] G = (S 0, Y, L) where S 0 is the sboptimal ser grop, Y is the set of power levels and there is an edge in L connecting each ser and each power level l with associated weight U X x Rjl. The power assignment problem in Phase 2 now becomes that of finding the maximm weight matching (definition can be fond in Appendix) in the bipartite graph G which can be efficiently solved sing Hngarian algorithm (see e.g., [15]). Finally we compte the marginal tility of this sboptimal decision and pass it to MWC. Now we analyze the complexity of or approach. In Phase 1, there are at most k V iterations and in each iteration it needs O( U V ) operations. So time complexity is O(k V U V ). In Phase 2, the comptation for edge weights takes O(k 2 V 2 ) and solving maximm weight matching takes O(k 4 V 4 ). Ths, time complexity for ser schedling for VBS V in each sb-band is O(k V U V + k 4 V 4 ). IV. ADAPTATION OF POWER ALLOCATION POLICY Or goal in this section is to adapt the power allocation policy (P, Φ) so as to increase system tility. Doing so reqires compting the gradients of system tility to the power

7 7 allocation policy s parameters. This is hard becase tility and gradients depend on stochastic channel variations. To address this we introdce a virtal system based on average channel measrements and estimate the gradients for the virtal system. This is still difficlt becase power affects system tility implicitly throgh average rates which in trn depend on dynamic clstering, opportnistic ser selection and power assignment. Let D j,r and Dj,l V be the gradient estimates of virtal system tility to p j,r and φ V j,l, respectively. Given a clstering and ser schedling decision, it is easy to express achievable rates as a fnction of power allocation parameters and ths easy to compte gradients nder this decision. However, the virtal system tility is maximized by taking each decision with some fraction of time and ths estimating D j,r and Dj,l V reqires averaging the gradients for each decision by the time fraction it will be taken. The work in [5] provides a soltion to this problem. The virtal system rns virtal schedler which implicitly captres the time fractions associated with decisions and comptes fraction-weighted gradients. Specifically, the virtal schedler rns a fixed nmber n v virtal slots. On each virtal slot in sb-band j, it ses average channels to make a clstering and ser schedling decision ˆd opt j and comptes virtal rates ˆR opt j ( ˆd j ) according to (2). It then ses small averaging parameters β 1 and β 2 to pdate virtal average rate ˆx for each, and D j,r for each r R, ˆx = (1 β 1 )ˆx + β 1 J ˆR opt j ( ˆd j ), U D j,r = (1 β 2 )D j,r + β 2 X ˆR opt j ( ˆd j ), p ˆx j,r U and Dj,l V for each V V and l = 1,..., k V, Dj,l V = (1 β 2 )Dj,l V U + β 2 X ˆR j ( ˆx U φ V j,l opt ˆd j ) However, nlike the SISO scenario in [5], in the ZFBF context we select sers (in III-C) to exploit instantaneos orthogonality by sing normalized channels as precoding vectors while in virtal schedler the orthogonality between average channels may not be representative for real world. The challenge is to captre the degree of instantaneos orthogonality based on average channels. For example, sers with nearly orthogonal average channels (e.g., in VBS {r 1, r 2 } a ser close to r 1 and a ser close to r 2 ) are very likely orthogonal while sers with non-orthogonal average channels are less likely to remain so. Let s divide [0, π 2 ] into m angle ranges. Let (U 1, U 2, V 1,2 ) be a random triplet representing a typical pair of sers concrrently schedled on the same VBS V 1,2. We se h V 1,2 U, h V 1,2 1,2 1,2 V V 1 U and h 2 U, h 1 U for their instantaneos and 2 average channels, respectively. Let θ U 1,U 2 denote the angle of their average channels compted by θ U 1,U 2 =. arccos( h V 1,2 U 1 hv 1,2 H U 2 h V 1,2 U 1 h V 1,2 U 2 ). We define [ EI(h V 1,2 U, h V 1,2 γ i = E 1 U ) [ 2 (i 1)π EI( h V 1,2 U, h V 1,2 1 U ) θ U 1,U 2 2m, iπ ) ], 2m 2 [ ) where the fnction EI is given in (6) and (i 1)π 2m, iπ 2m is the ith angle range. The vector γ = (γ 1,..., γ m ) represents a set of averaged qantities that the system keeps track of for se in the virtal system. Given a pair of sers and, we let γ(θ, ) = γ i where θ, falls into the ith angle range. Now in virtal schedler opt to find ˆd j, when we select sboptimal ser grop S 0 in VBS V, we replace EI(h V, h V ) with ÊI( h V, h V ) = h V h V H h V 2 γ(θ, ). Adaptation of Power Allocation Policy Given the gradient estimates, we pdate (P, Φ) as below. 1) Update p j,r : for each RRH, we increase the power allocated to the sb-band with largest positive gradient and decrease the power allocated to the sb-band with smallest negative gradient. (see [5] for details). 2) Update φ V j,l : for each VBS V and sb-band j, we find the level l small with smallest gradient and level l large with largest gradient and exchange a small amont between them while maintaining the ranking of the power levels. V. SIMULATIONS We consider a grid of 7 RRHs as shown in Fig.3. It is easy to check that the set of VBSs V exhibited in Fig.3 is 2- decomposable and there are for eqivalence classes, i.e. E 1 = {1}, E 2 = {2, 3}, E 3 = {4, 5}, E 4 = {6, 7}. The propagation and transmission parameters are listed in Table I. 5 Fig RRH setp and set of VBSs sed in simlations. Sppose the operational bandwidth is divided into 3 sbbands. We randomly and niformly generate the locations of 100 sers which are stationary. The fll bffer model where all sers always have traffic to send is sed in all simlations. We assme flat and fast Rayleigh fading conditions and se the log tility fnction. An averaging parameter β = 0.01 is sed in the actal schedler. All simlations are rn for 3000 time slots. The power allocation policy is adapted every 10 time slots and n v = 30 virtal slots are performed in virtal schedler. The two averaging parameters are chosen to be β 1 = 0.005, β 2 = 0.01.

8 8 Parameters Vales Inter-RRH Distance 1 Km Path Loss Model L = log 10 (d) Bandwidth 1 MHz Noise Power Density -174 dbm/hz Tx Power per RRH 40 dbm Tx Antenna Gain 15 db Rx Antenna Gain 76 db Rx Noise Figre 7 db Nm of Antennas per RRH (n t) 2 k 2 TABLE I PROPAGATION AND TRANSMISSION PARAMETERS IN SIMULATIONS. Users Y Coordinate (m) PC/DC edges "Edge" Users Positions NO PC/NO DC PC/NO DC PC/DC NO PC/NO DC & PC/NO DC edges RRHs 800 We se PC to signify a system that adapts the power allocation policy to achieve good FFR patterns and DC to represent a system sing dynamic clstering. Or approach is ths denoted PC/DC. To evalate the benefits of these we consider three baseline systems. PC/NO-DC: This system ses PC bt no DC. This represents an aggressive state-of-the-art approach which spports ZFBF, explores dynamic soft FFR and opportnistic ser schedling, similar to e.g., [5]. NO-PC/DC: This system ses DC bt no PC. NO-PC/NO-DC: This system ses neither PC nor DC. For NO-DC systems we assme no cooperation among RRHs and for NO-PC systems we set the power allocation policy sch that power p for a RRH is split eqally among sb-bands and then frther eqally split among power levels. Note the channel realizations are the same in all systems for comparison. Moving Edges: Sppose the 10 sers experiencing the worst throghpt nder each system roghly represent the location of the edges. Fig.4 shows the edges for three simlated systems. The edge sers in NO-PC/NO-DC can be seen to be at traditional edges lying at the borders between RRH cells. PC/NO-DC aims to mitigate interference by exploiting soft FFR patterns bt does not change the locations of edge sers by mch. Under or PC/DC, most of the traditional edge sers are well covered, ths dynamic clstering moves the edges to other locations. These new edges depend on the selected set of VBSs V. By adding more VBSs to V withot breaking 2-decomposability, we can provide good coverage to any location sch that even edge sers cold get high throghpt. Throghpt Improvement for Traditional Edge Users: To qantitatively evalate the throghpt gains achieved by dynamic clstering for traditional edge sers, we focs on the worst 10 sers nder PC/NO-DC and compare those same sers rates nder PC/DC. By ranking the sers according to their average rates nder PC/NO-DC, Fig.5 exhibits the average rates for all sers in PC/NO-DC and PC/DC and zooms in the comparison for worst 10 sers. We can see that dynamic clstering sbstantially improves the throghpt of edge sers in PC/NO-DC withot degrading the performance of others. More precisely, the mean throghpt of 10 edge sers is increased by 80.4% and 81 sers get better rates in Fig. 4. X Coordinate (m) Edge sers positions in three systems. PC/DC while there are only 3 sers whose rates decrease by more than 5%. The Jain s fairness of average ser rates also increases from 0.65 to Average Rate (bps) x Average Rate Comparison Fig x User Ranking in PC/NO DC Rate in PC/NO DC Rate in PC/DC 10 edge sers nder PC/NO DC Average rates for sers in PC/NO-DC and PC/DC. Benefits of DC vs. Benefits of PC: Both DC and PC aim at improving the edge throghpt which in trn reslts in higher fairness. We se 10-percentile mean throghpt (i.e. the mean throghpt of worst 10% sers) as a measre of the edge throghpt for a system. To evalate and compare the benefits of DC and PC separately, we compte the 10-percentile mean throghpt and Jain s fairness of all sers average rates in for systems and compare the gains of adding DC or PC in Fig.6. It trns ot that both techniqes bring significant benefits to edge sers bt DC provides the higher gain for or simlation setp. We note that the above reslts are representative of different realizations of fading channels for niformly distribted sers. However, it will of interest to investigate the reslts for nonniform ser distribtions. VI. FUTURE WORK There are several interesting topics for ftre work. Since we focsed on downlink and delay-tolerant best-effort traffic,

9 9 Throghpt gain: 54% Fairness gain: 0.06 NO-PC/NO-DC Edge throghpt: 0.13 Fairness: 0.65 Throghpt gain: 31% Fairness gain: 0.00 NO-PC/DC Edge throghpt: 0.20 Fairness: 0.71 PC/NO-DC Edge throghpt: 0.17 Fairness: 0.65 Throghpt gain: 35% Fairness gain: 0.02 PC/DC Edge throghpt: 0.27 Fairness: 0.73 Throghpt gain: 59% Fairness gain: 0.08 Fig percentile mean throghpt and Jain s fairness in for systems. The nit for edge throghpt is 10 6 bps. it wold be of interest to integrate CoMP and network layer schedling for plink and other types of traffic. Evalating the framework by taking into accont ser mobility and nonniform ser distribtion are also goals for ftre work. APPENDIX PROOF OF THEOREM 3 We begin by introdcing maximm weight matching problem. Definition 5: In an edge-weighted graph G = (N, L) where N is the set of vertices and L represents the set of edges, a matching is a collection of edges withot common vertices. And a Maximm Weight Matching (MWM) is defined as the matching that has the maximm sm weights of the edges in the matching. As shown in the left figre of Fig.7, the edges in dashed lines form a matching. a(e 1 ) a(e 2 ) a(e 3 ) a(e 4 ) w j ({1}) w j ({1, 2, 5}) w j ({2, 5, 6}) w j ({3, 6, 7}) w j ({3, 4, 7}) b(e 1 ) a(e 5 ) w j ({4}) b(e 5 ) Fig. 7. A matching in a general graph and the converted graph for the MWC Problem in Fig.2. Sppose we are given Maximm Weight Clstering (MWC) Problem (V, w j ) where V is 2-decomposable. A set E E V is said to be singleton-decomposable if r E, {r} V, i.e., each RRH in E is a candidate VBS in V. We define a new set of weights ω(e) for each E E V as below, If E V and E is singleton-decomposable, max[ w j ({r}), w j (E)]. r E If E V and E is not singleton-decomposable, w j (E). ω(e) = If E / V and E is singleton-decomposable, w j ({r}). r E If E / V and E is not singleton-decomposable, 0. Next, we constrct a graph G = (N, L) for this MWC Problem as follows. The right figre in Fig.7 is the graph for the example problem in Fig.2. For each element E E V that has an positive weight ω(e), we add two vertices a(e) and b(e) to N. For each element E E V whose ω(e) is 0, we add only one vertex a(e) to N. For each element E E V that has an positive weight ω(e), we pt an edge between a(e) and b(e) and associate weight ω(e) to that edge. For example, the edge between a(e 1 ) and b(e 1 ) in Fig.7. For each VBS V V, if V is the nion of two eqivalence classes, say V = E i E j, we add an edge to connect a(e i ) and a(e j ) with weight w j (V ). For example, the edge between a(e 1 ) and a(e 2 ) in Fig.7. After constrcting the graph, we claim that solving MWM in graph G gives s the answer for the MWC problem. It is not difficlt to prove this claim and we omit it to save space. Let m denote the nmber of vertices and n denote the nmber of edges in a graph, the time complexity for solving MWM is O(n m). In or scenario, m 2 E V 2 R = O( R ), n E V + V R + V. In practical systems of interest, an RRH cooperates with neighboring RRHs rather than RRHs that are far away which means each RRH belongs to a small nmber of VBSs. Ths, V = O( R ) which reslts in time complexity in or scenario becoming O( R 1.5 ). ACKNOWLEDGEMENT The athors wold like to thank Andrés García Saavedra, Alan Gatherer, Robert W. Heath Jr. and Namyoon Lee for their comments and feedbacks on this work. REFERENCES [1] A. Lozano, R. W. H. Jr., and J. G. Andrews, Fndamental limits of cooperation, IEEE Trans. Inf. Theory, vol. PP, p. 1, March [2] China Mobile, C-RAN The Road Towards Green RAN, Oct [3] E. Pateromichelakis, M. Shariat, A. l Qdds, and R. Tafazolli, On the evoltion of mlti-cell schedling in 3gpp lte / lte-a, IEEE Commn. Srv. Ttor., vol. 15, pp , May [4] M. Costa, Writing on dirty paper, IEEE Trans. Inf. Theory, vol. 29, pp , May [5] A. L. Stolyar and H. Viswanathan, Self-organizing dynamic fractional freqency rese for best-effort traffic throgh distribted inter-cell coordination, in Proc. INFOCOM 09, April 2009, pp [6] G. Wnder, M. Kasparick, A. Stolyar, and H. Viswanathan, Selforganizing distribted inter-cell beam coordination in celllar networks with best effort traffic, WiOpt 2010, pp

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