User Association in Dense mmWave Networks based on Rate Requirements
UUser Association in Dense mmWave Networksbased on Rate Requirements
Veljko Boljanovic ∗ , Forough Yaghoubi † , and Danijela Cabric ∗∗ Electrical and Computer Engineering Department, University of California, Los Angeles, CA, USA † Communication Systems Division, KTH Royal Institute of Technology, Stockholm, SwedenEmail: [email protected], [email protected], [email protected]
Abstract —Commonly considered user association frame-works in millimeter-wave communications are based on the sumrate maximization, and they essentially neglect user specificrate and service requirements. Furthermore, new features ofmillimeter-wave communications including spatial multiplexing,connectivity to multiple coordinated base stations and densebase station deployment are not considered. In this work, wepropose a two-step optimization framework for single-shot userassociation in dense millimeter-wave networks which takes intoaccount users’ rate requirements, multi-connectivity, and hybridtransceiver architecture for spatial multiplexing. The proposedframework considers multiple RF chains at each base stationand assigns them to different users that also have multiple RFchains for connectivity with more than one base station. In thefirst step, the objective of the user association is to maximizethe number of users with satisfied rate requirements while min-imizing network underutilization. In the second step, remainingRF chains are assigned to users, whose rate requirement hasnot been met, such that network sum rate is maximized. Thisis a novel problem formulation for user associate in millimeter-wave networks. We propose low complexity sub-optimal userassociation algorithms based on this formulation, numericallyevaluate the optimal and sub-optimal solutions, and comparethem to the conventional association approaches in terms of thenumber of associated users and network sum rate.
I. I
NTRODUCTION
Millimeter-wave (mmWave) communications will have thekey role in providing high data rates in the fifth generation(5G) of cellular systems [1]. Besides the abundant spectrumat mmWave frequencies, spatial multiplexing enabled byhybrid analog-digital transceiver architectures with multipleRF chains will be leveraged for rate improvements andconnectivity management. Hybrid antennas array architectureprovides flexibility for data rate increase using simultaneousbeamforming (BF) and spatial mutliplexing (SM) betweenthe base station (BS) and user equipment (UE) [2]. Thus,in dense mmWave networks with large number of the UEs,the RF chains at the BS become an important resource forconnectivity optimization. Due to its energy efficiency, it isexpected that hybrid architecture will be implemented at UEside as well. Having multiple RF chains at the UE will allowthe UE to connect to multiple BSs at the same time, which isoften referred to as multi-connectivity [3]. This new featureof multi-connectivity paired with high data rate requirementsand limited number of RF chains per UE and BS makes the
This work is supported by NSF under grant 1718742. problems of user association (UA) and resource allocation(RA) in mmWave networks very challenging and differentfrom microwave frequency approaches.The UA/RA at microwave frequencies were extensivelystudied for heterogeneous networks (HetNets), e.g., in [4]–[6] with the objective of maximizing network utility function.The utility function is often assumed to be logarithmic,which encourages load balancing in HetNets, while the setof constraints usually includes available resources at theBS or UE. In particular, the main Quality-of-Service (QoS)constraint in [4]–[6] is to maintain the minimum SNR orthe maximum interference level, which may not necessarilylead to an acceptable rate for the UEs. A UE with a veryhigh SNR associated with a heavily loaded BS may notget high effective rate due to potential sharing of the BSresources among many served UEs. More recently, UA/RAin mmWave networks were studied [7]–[10]. Similarly tothe microwave HetNets, the optimization frameworks in [7]–[10] are focused on the sum rate maximization, while thesets of constraints mainly addressed resource availabilityand maximum interference levels. In [8], UA/RA problem isconsidered for hybrid architecture with multiple RF chains,but it does not address multi-connectivity and users’ data raterequirements. On the other hand, work [10] considers users’data rate requirements, but it does not study architectures withmultiple RF chains and multi-connectivity. Finally, withoutconsidering users’ data rate requirements, work [9] studiesBSs and UEs with multiple RF chains and multi-connectivity,but it puts them in the context of time scheduling andhandover in mmWave networks.In this work, we propose an optimization framework forUA/RA in dense mmWave networks. By abstracting eachRF chain at BS and UE as assignable system resource, theUA/RA can be considered as the RF-chain-wise associationbetween multiple BSs and UEs. Unlike previous works, ourproposed framework jointly considers users’ data rate re-quirements, and multi-connectivity. Moreover, our objectivefunction is primarily to maximize the number of UEs withsatisfied rate requirements, and then to assign the remainingBS RF chains to UEs such that network underutilizationis avoided. We show that our optimization problem is NP-hard, and then develop a sub-optimal algorithm to solve theproblem in polynomial time.The rest of the paper is organized as follows. In Section II, a r X i v : . [ c s . I T ] M a y e introduce the system and channel models. In Section III,we introduce the proposed UA/RA optimization framework.Section IV describes the proposed sub-optimal algorithm.In Section V, we compare our framework with existingUA/RA approaches. Finally, conclusions are summarized inSection VI.II. S YSTEM AND C HANNEL M ODELS
We consider downlink (DL) of a standalone mmWavenetwork with a set of BSs B and a set of UEs U , operatingat frequency f . There are N BS BSs in B , and N UE UEs in U . We assume sub-array hybrid architecture with N RFBS
RFchains at each BS and N RFUE
RF chains at each UE. Each RFchain at each BS controls a uniform linear array (ULA) with N aBS antenna elements, and each RF chain at each UE has aULA with N aUE antenna elements. We consider each RF chainat BS as a virtual BS and define a set of all RF chains at allBSs B v , whose cardinality is N BS N RFBS . Similarly, we define aset of all RF chains at all UEs U v with cardinality N UE N RFUE .There are N BS N RFBS N UE N RFUE multiple input multiple output(MIMO) channels between RF chains from B v and U v .Let i and j be arbitrary RF chains from U v and B v ,respectively. Using the bandwidth B , i and j communicateover a single-path mmWave MIMO channel represented bymatrix H ij ∈ C N aUE × N aBS . We consider dense urban microenvironment and we model the channel path loss accordingto [11]. Under assumption that capacity achieving code isused, achievable data rate between i and j can be approxi-mated with the link capacity. Since highly directional trans-mission in mmWave networks is noise-limited rather thaninterference-limited [2], the link capacity c ij is calculated asfollows c ij = B log | √ P t N RFBS a HUE (ˆ θ i ) H ij a BS ( ˆ φ j ) | BN , (1)where P t and N represent the transmit power and noisespectral density, respectively. The beamforming vectors areequal to the spatial response vectors a UE (ˆ θ i ) and a BS ( ˆ φ j ) defined as follows a UE (ˆ θ i ) = [1 , e − jπ sin(ˆ θ i ) , ..., e − j ( N aUE − π sin(ˆ θ i ) ] T (cid:112) N aUE , (2) a BS ( ˆ φ j ) = [1 , e − jπ sin( ˆ φ j ) , ..., e − j ( N aBS − π sin( ˆ φ j ) ] T (cid:112) N aBS . (3)The imperfect angular estimates ˆ θ i and ˆ φ j are assumed to beobtained through practical beam training. They are modeledas Gaussian random variables ˆ θ ij ∼ N ( θ ij , σ AoA ) and ˆ φ ij ∼ N ( φ ij , σ AoD ) , where θ ij and φ ij represent true angleof arrival (AoA) and angle of departure (AoD), respectively.The imperfect estimates ˆ θ i and ˆ φ j can negatively affect thecapacity (achievable rate) between i and j due to decreasein beamforming gain. All RF chain pairs ( i, j ) that correspond to the same BS-UE pairexperience the same path loss because of spatial consistency. III. O
PTIMIZATION F RAMEWORK
In an urban dense environment, 3GPP specifices thateach UE should get data rate of at least
Mbps [12].Note that UEs data requirements can be significantly higherand reach the order of several Gbps. Conventional UA/RAschemes often do not consider heterogeneous UEs datarequirements, and resources are often allocated to UEs withhigh capacity links. In this work, we take data requirementsinto account, and we design a new two-step optimizationframework for UA/RA, assuming that RF chains at BSsrepresent the resources. In the first step, we maximize thenumber of associated UEs with satisfied data requirements.This maximization is done with minimal number of BS RFchains to avoid network underutilization. Simultaneously, RFchains are allocated such that the sum rate among associatedUEs is maximal. Since minimal amount of resources is usedin the first step, in the second step the remaining RF chainsare used to serve non-associated users and maximize networksum rate.
A. Step 1
Let z ∈ { , } |U| be a vector of binary associationvariables, where z u is if user u is served and otherwise.Let r be a vector of data rate requirements r u for all u ∈ U .Let x ∈ { , } |U v ||B v | be a vectorized matrix of binaryassociation variable for all RF chain pairs, where x ij is if user RF chain i is connected to BS RF chain j and otherwise. Let c be a vector whose elements are capacities c ij from (1), associated with corresponding x ij . We definethe function F as the number of associated UEs, and F asthe number of allocated BS RF chains with negative sign.Mathematically, F and F can be expressed as follows F = (cid:88) u ∈U z u , F = − (cid:88) i ∈U v (cid:88) j ∈B v x ij . (4)The goal in Step 1 is to maximize F with maximal F .To achieve this, a multi-criterion optimization problem needsto be solved. Since the problem is constrained on users’data requirements and available resources at BSs, it can beformulated as follows max z , x λ F + λ F (5a) s . t . (cid:88) i ∈U v x ij ≤ , ∀ j ∈ B v , (5b) (cid:88) j ∈B v x ij ≤ , ∀ i ∈ U v , (5c) (cid:88) i ∈U v (cid:88) j ∈B v j → b x ij ≤ N RFBS , ∀ b ∈ B , (5d) (cid:88) i ∈U v i → u (cid:88) j ∈B v x ij ≤ z u N RFUE , ∀ u ∈ U , (5e) (cid:88) i ∈U v i → u (cid:88) j ∈B v x ij c ij ≥ z u r u , ∀ u ∈ U , (5f) x ij ∈ { , } , ∀ i ∈ U v , j ∈ B v , (5g) z u ∈ { , } , ∀ u ∈ U , (5h)here j → b means that the RF chain j ∈ B v belongs to theBS b ∈ B . Similarly, i → u means that the RF chain i ∈ U v belongs to the UE u ∈ U . Constraints (5b) and (5c) guaranteethat each RF chain can be connected to up to 1 RF chain.Constraints (5d) and (5e) relate to the maximum number ofRF chains at the BS b and UE u , respectively. The variable z u in (5e) ensures that RF chains of UE u are not used if u is notassociated. The data requirement constraint in (5f) guaranteesthat all associated UEs have their data rate requirementssatisfied. The constants λ and λ in (5a) represent weightswhich can be obtained through scalarization. The set of allpossible values for F and F includes the optimal trade-offcurve that is a piece-wise linear (PWL) function consistingof discrete points. It is possible to find a hyperplane definedby [ λ , λ ] which touches the optimal trade-off curve at thepoint where F is maximized and F is maximal. Commonly,the weight λ is fixed, and then the optimal values for λ are found. Based on capacities of its links, a UE could needfrom to N RFUE
RF chains to satisfy its data rate requirement.This range defines a set of slopes of the optimal trade-offcurve. They could take values from the set { , , ..., N RFUE , } .The optimal values for λ directly depend on the slopes. Anexample for finding λ when λ = 1 and N RFUE = 2 is depictedin Fig. 1. If λ = 1 , the number of associated users, i.e.,objective F , is not maximized and there are multiple optimalpoints. If λ = N RFUE , there are again multiple optimal pointsand F is not necessarily maximized. Further, if λ = 0 , theobjective F is maximized, but the number of used RF chainsis not minimal. It can be observed that F is maximized withminimal number of RF chains (the red point is certainlyachieved) if λ ∈ (cid:16) , N RFUE (cid:17) . Note that this result holds forany N RFUE and any optimal trade-off curve. In this work, wechoose λ = N RFUE +1 , and then ( a) gets the following form max z , x (cid:88) u ∈U z u − (cid:88) i ∈U v (cid:88) j ∈B v N RFUE + 1 x ij . (6)Note that with [ λ , λ ] = (cid:104) , N RFUE +1 (cid:105) , the sum rateamong associated UEs is not necessarily maximal since theyare associated using arbitrary links that satisfy their datarequirements and maximize (6). We further extend λ byadding the term c ij r u to its denominator and reformulate (6)as follows max z , x (cid:88) u ∈U z u − (cid:88) u ∈U (cid:88) i ∈U v i → u (cid:88) j ∈B v N RFUE + 1 + c ij r u x ij . (7)The last expression can be considered as ( a) with a new pairof functions F (cid:48) and F (cid:48) and [ λ (cid:48) , λ (cid:48) ] = [1 , . The formulationin (7) ensures that the sum rate of associated UEs is maximal.The objective function increases more if UEs are associatedwith links that have higher capacity c ij . If capacity c ij wasnot divided by corresponding r u , minimal number of usedRF chains would not be guaranteed. To see this, assume u and u are two UEs from U , where u experience only lowcapacity links with all BSs, but it can satisfy its low datarate requirement with one RF chain, and u that has high [λ ,λ ] F F -1-2 12 λ =1 λ =1/N RF λ =0 ( ( Fig. 1. PWL trade-off curve with discrete points. We fix λ = 1 and findthe values of λ for which the red optimal point is certainly achieved. capacity links with all BSs, but it needs two RF chains tosatisfy its extremely high rate requirement. If capacity c ij wasnot divided by corresponding r u , high capacity user u wouldbe favored even though it requires more RF chains. In otherwords, the objective could see higher reward in associating u than u . To solve this issue, we introduce the relativecapacity term c ij r u which ensures that the sum rate is maximalamong UEs associated with minimal number of RF chains.With the relative capacity term, the objective function seeshigher reward if UEs are associated with smaller number ofRF chains. Finally, the optimization problem in Step 1 canbe restated as follows max z , x (cid:88) u ∈U z u − (cid:88) u ∈U (cid:88) i ∈U v i → u (cid:88) j ∈B v N RFUE + 1 + c ij r u x ij s . t . (5b)-(5h) (8)In summary, Step 1 in the optimization framework maxi-mizes the number of associated UEs with satisfied data raterequirements using minimal number of RF chains at BSs. Inaddition, it ensures maximal sum rate among associated UEswhen minimal resources are used. B. Step 2
After Step 1, some number of RF chains at BSs mightstill be available. Since no more UEs can satisfy their datarequirements, the reminder of resources at BSs is fullyexploited and distributed among non-associated UEs usingmax-sum-rate scheme. Note that the proposed frameworkreduces to the sum rate maximization among all users if nousers are associated in Step 1.Let U = U A ∪ U NA , where U A and U NA are sets ofassociated and non-associated UEs in Step 1, respectively.Now let U NA v be a set of all RF chains at non-associated UEs.Let B v = B A v ∪ B NA v , where B A v and B NA v are sets of assignedand non-assigned RF chains from BSs in Step 1, respectively.Let N RFBS ( b ) be the number of remaining RF chains at theBS b ∈ B after Step 1. Similarly as in Step 1, let x s be avector of binary association variables x ij for all RF chainpairs { i, j } , i ∈ U NA v , j ∈ B NA v . Note that the vector x s consists of the subset of variables from the vector x . Let c ij e the capacity associated with corresponding variable x ij from x s . The sum rate can be maximized as follows max x s (cid:88) i ∈U NA v (cid:88) j ∈B NA v x ij c ij (9a) s . t . (cid:88) i ∈U NA v x ij ≤ , ∀ j ∈ B NA v , (9b) (cid:88) j ∈B NA v x ij ≤ , ∀ i ∈ U NA v , (9c) (cid:88) i ∈U NA v (cid:88) j ∈B NA v j → b x ij ≤ N RFBS ( b ) , ∀ b ∈ B , (9d) (cid:88) i ∈U NA v i → u (cid:88) j ∈B NA v x ij ≤ N RFUE , ∀ u ∈ U NA , (9e) x ij ∈ { , } , ∀ i ∈ U NA v , j ∈ B NA v . (9f)As in Step 1, constraints (9b) and (9c) guarantee that eachRF chain can be connected to up to 1 RF chain. Constraints(9d) and (9e) ensure that the number of used RF chainsdoes not exceed the amount of available resources at theBS and non-associated UE, respectively. The formulation in(9) maximizes the sum rate in Step 2 regardless of user datarate requirements, but other approaches are also possible. Forexample, the association scheme in Step 2 can be designedto distribute remaining resources among non-associated UEsaccording to their data rate requirements. With this approach,UEs with higher data requirements would get more resources.We leave these alternative designs for future work.IV. P ROPOSED A LGORITHM
The optimization problems in (8) and (9) are BinaryInteger Programs (BIP), which are known to be NP-hard.This means that even small-size problems with a few BSsand moderate number of UEs have prohibitive computa-tional complexity. To solve these optimization problems, wepropose a low complexity solution based on relaxation androunding. First, we relax the BIP in (8) and formulate thefollowing low complexity Linear Program (LP) max z , x (cid:88) u ∈U z u − (cid:88) u ∈U (cid:88) i ∈U v i → u (cid:88) j ∈B v N RFUE + 1 + c ij r u x ij s . t . (5b)-(5f) , ≤ x ij ≤ , ∀ i ∈ U v , j ∈ B v , ≤ z u ≤ , ∀ u ∈ U , (10)The solution x ∗ to (10) is fractional, meaning that itselements are not necessarily integer values or . Since weconsider a single-shot UA and not the long-term average,elements in x ∗ should be rounded either to or . Usingsimple rounding technique where values x ∗ ij ≥ . arerounded to , and zero otherwise, is not a good way to findthe rounded solution x ro . This technique can round too manyelements to , and thus x ro can violate multiple constraintsand become infeasible. Similarly, it can round too many Algorithm 1
Proposed rounding algorithm Inputs: C S , r Outputs: z ro , x ro Initialization: z ro = , x ro = X ro = vec − ( x ro ) Define matrix of values V and matrix of indices I for n = 1 : N RFUE do while true do d = [ V [: , u ] , I [: , u ]] = sort (cid:0) C S [: , u ] , (cid:48) descend (cid:48) (cid:1) d u = argmin k (cid:16)(cid:80) km =1 V [ m, u ] ≥ r u (cid:17) , ∀ u if d u (cid:54) = n, ∀ u then break while end if z ro [ u ∗ ] = 1 , for [ ∼ , u ∗ ] = max ( (cid:80) nm =1 V [ m, :]) X ro [ I [1 : n, u ∗ ] , u ∗ ] = 1 C S [: , u ∗ ] = C S [ k, :] = T , ∀ k of assigned BS RF chains end while end for x ro = vec ( X ro ) elements to , and then x ro becomes a poor solution withfew associated UEs. We propose a sub-optimal polynomialtime rounding algorithm used to obtain x ro from x ∗ .Let S be the support of the vector x ∗ . Let vector c S be equal to the vector c for indices found in S , and zerofor all other indices. Let X ∗ = vec − ( x ∗ ) , where vec − () reshapes a vector of size R N BS N RFBS N UE N RFUE into a matrix ofsize R N BS N RFBS N RFUE × N UE . Similarly, let C S = vec − ( c S ) bea matrix of capacities on used links. Columns in X ∗ and C S correspond to different UEs, and rows correspond to allpossible links that UEs can have.We first compare elements C S to the corresponding datarate requirements r u , u = 1 , ..., N UE , to see how many RFchains each associated UE needs. The needed number of RFchains is stored in the demand vector d . We identify all UEsthat can be associated using 1 RF chain, i.e., we find allpositions in d where d u = 1 . Among these UEs, we find theone that has link with the highest capacity and associate theUE using this link. To exclude the associated UE from furtherconsideration, its corresponding column in C S is set to .Similarly, we exclude the used BS RF chain from furtherconsideration by setting all corresponding rows in C S to T . In the next iteration, the demand vector d is determinedagain, and the whole procedure is repeated for d u = 1 ifthere are still UEs that require 1 RF chain. If that is not thecase, the procedure first repeats for d u = 2 , then for d u = 3 ,and so on until d u = N RFUE . Note that when d u > , we pickthe UE with the highest aggregated capacity on d u best links.The algorithm pseudo-code is provided in Algorithm 1.The sub-optimality of the proposed algorithm comes fromthe fact that it does not maximize the sum rate of associatedUEs. In fact, this is a greedy approach which tries to max-imize the number of associated UEs with minimal numberf RF chains by choosing the UE with best links in eachiteration. The greedy selection of best links often comes atprice of lower number of associated UEs. Consequently, theproposed solution leaves more RF chains for the second step,where the sum rate is maximized.Once vectors z ro and x ro are obtained, the optimal x ∗ s inStep 2 can be obtained by relaxing and solving (9). Theorem 1.
The relaxation of (9) has an integral optimalsolution x ∗ s , whose elements are 0 or 1.Proof. See Appendix A.V. N
UMERICAL E VALUATION
In this section, we evaluate the proposed low complexityalgorithm for new UA/RA framework, and compare it toexisting association schemes, including the max-sum-rate andmax-SNR. Neiher the max-sum-rate nor max-SNR considerusers’ data requirements and they both allocate BS RF chainsto the UEs with high capacity links. The max-sum-rate jointlyconsiders all RF chains at all BSs to maximize the networksum rate, while the max-SNR considers resources at one BSat a time and thus allocates them sequentially.We consider a scenario with N BS = 5 BSs and N UE = 30 UEs randomly placed within the area among BSs. Thedistance between neighboring BSs is m. We assumehybrid sub-array architecture at both BSs and UEs, with N RFBS = 5
RF chains at each BS and N RFUE = 2
RF chainsat each UE. Each BS RF chain controls N aBS = 32 antennas,while each UE RF chain controls N UE = 8 antennas. Thedata rate requirements are drawn from uniform distribution r u ∼ U ( R min , R max ) , ∀ u , where R min = 0 . Gbps and R max can vary. We consider the downlink communication with BStransmit power P t = 30 dBm, and noise spectral density of N = − dBmHz . All RF chain pairs { i, j } , i ∈ U v , j ∈ B v ,operate over the same bandwidth of B = 200 MHz. True an-gles θ i and φ j are drawn from uniform distribution U ( − π , π ) for each RF chain pair ( i, j ) , and standard deviations ofestimation errors are σ AoD = 1 ◦ and σ AoA = 3 ◦ for AoDand AoA, respectively.The UA/RA in the proposed two-step optimization frame-work is presented in Fig. 2 for one scenario realization, with R max = 2 Gbps. The two subfigures compare the associationsbased on the optimal and proposed solutions to (8) and (9).Association based on the optimal solutions maximizes thenumber of UEs with satisfied data rate requirement in Step1, and it leaves small number of RF chains for Step 2. On theother hand, the UA based on the proposed sub-optimal so-lutions associates less users with satisfied data requirementsin Step 1 due to its greedy nature. This means that moreRF chains are available in Step 2, where they are givento non-associated UEs through the max-sum-rate approachformulated in (9). When hybrid transceiver architecture isconsidered, a BS tends to allocate multiple RF chains for asingle UE with good links in Step 2. This explains why manyUEs are provided with two RF chains in Step 2 in Fig. 2(b). -200 -150 -100 -50 0 50 100 150 200
Distance [m] -200-150-100-50050100150200250 D i s t an c e [ m ] Base stationsAssociated (Step 1)Associated (Step 2)Non-associated (a) User association based on optimal solutions to (8) and (9). -200 -150 -100 -50 0 50 100 150 200
Distance [m] -200-150-100-50050100150200250 D i s t an c e [ m ] Base stationsAssociated (Step 1)Associated (Step 2)Non-associated (b) User association based on proposed solutions to (8) and (9).Fig. 2. User association for one scenario realization. Thin and thick linesillustrate the use of 1 and 2 RF chains for communication, respectively.
In Fig. 3, the proposed UA/RA framework is comparedwith existing UA schemes, including max-sum-rate and max-SNR, for the same scenario realization as in Fig. 2. Asexpected, the UA/RA based on the optimal solutions to (8)and (9) results in the highest number of associated UEswith satisfied data rate requirements. The proposed sub-optimal UA/RA performs better than conventional associationschemes in terms of the number of associated users. Whenconventional schemes are used for association, only UEs withgood links are provided with the opportunity to satisfy theirdata rate requirements regardless of how low or high theirrequirements are. Unsurprisingly, the max-sum-rate schemeachieves the highest network sum rate. The sum rate in theproposed framework is higher with the sub-optimal UA/RAthan with the optimal. The main reasons for this are sub-optimal maximization with the proposed solutions in Step 1and consequent use of more RF chains in Step 2 where sumrate is maximized among non-associated UEs.In Fig. 4, the average number of associated UEs withsatisfied data rate requirements and the average achieved sumrate are presented as functions of the maximum requirement R max . To obtain results in both subfigures, we perform 30Monte Carlo runs with different positions of UEs and users’data requirements r u , ∀ u , to find the averages for different R max . By jointly considering both subfigures, we see that theUA/RA based on the proposed solutions represents a trade- Max-SNR Max-sum-rate Optimal Proposed05101520253035 N u m be r o f U E s Associated (req. satisfied) Associated (req. not satisfied) Non-associated
Fig. 3. Comparison in terms of the number of associated UEs and achievednetwork sum rate for one scenario realization. The associated UEs can havetheir data requirements satisfied or not. off between max-sum-rate scheme and UA/RA based on theoptimal solutions. When the proposed solutions are used, thenumber of associated UEs with satisfied data requirementsis higher than with the max-sum-rate scheme, and lowerthan with optimal solutions. On the other hand, the use ofthe proposed solutions results in higher sum rate than withthe optimal solutions, but lower than with the max-sum-ratescheme. As the number of associated UEs in Step 1 of theproposed framework decreases in Fig. 4(a), the sum rateincreases in Fig. 4(b) since more BS RF chains are availablefor rate maximization in Step 2. Note that the sum ratesfor the max-sum-rate and max-SNR schemes in Fig. 4(b) donot increase because these schemes associate UEs with goodlinks in a single step, regardless of UEs’ data requirements.VI. C
ONCLUSIONS
We proposed a new UA/RA framework which, unlikeexisting UA/RA approaches, considers features of mmWavenetworks and diverse user data rate requirements. The firststep in the framework maximizes the number of associatedUEs with satisfied data rate requirements using minimalamount of resources, while the second step uses the re-maining resources to maximize the sum rate among non-associated UEs. We proposed sub-optimal solutions to theNP-hard problems in both steps, and our numerical resultsshowed that the proposed solutions represent a good trade-offbetween the optimal solutions to the NP-hard problems andexisting max-sum-rate approach.A
PPENDIX
A. Integer solution to relaxed problem
The relaxation of ( ) can be transformed into a min-cost network flow problem with integer edge capacities andinteger vertex supplies/demands.Let G = ( V, E ) be a directed graph (network flow), withthe source s and the sink t . The set of vertices V includesthe source s , base stations b from B , BS RF chains j from B NA v , UE RF chains i from U NA v , UEs u from U NA , and sink t . The set E includes edges between: • the source s and the BS b, ∀ b ∈ B , with edge capacity c ( s, b ) = N RFBS ( b ) and edge cost ω ( s, b ) = 0 , Maximum data requirement R max [Gbps] A ss o c i a t ed U E s ( Q o s ) Max-SNR Max-sum-rate Optimal Proposed (a) Average number of associated UEs with satisfied QoS requirements.
Maximum data requirement R max [Gbps] S u m r a t e [ G bp s ] Max-SNR Max-sum-rate Optimal Proposed (b) Average sum rate in the network.Fig. 4. Comparison in terms of the number of associated UEs with satisfieddata requirements and average network sum rate. • the BS b, ∀ b ∈ B , and BS RF chain j, ∀ j ∈ B NA v , withedge capacity c ( b, j ) = 1 , if j → b , and c ( b, j ) = 0 otherwise, and edge cost ω ( b, j ) = 0 , • the BS RF chain j, ∀ j ∈ B NA v , and UE RF chain i, ∀ i ∈ U NA v , with edge capacity c ( j, i ) = 1 , and edgecost ω ( j, i ) = c ij , where c ij are capacities from (1), • the UE RF chain i, ∀ i ∈ U NA v , and UE u, ∀ u ∈ U , withedge capacity c ( i, u ) = 1 , if i → u , and c ( i, u ) = 0 otherwise, and edge cost ω ( i, u ) = 0 , • the UE u, ∀ u ∈ U , and sink t , with edge capacity c ( u, t ) = N RFUE and edge cost ω ( u, t ) = 0 .The graph G is depicted in Fig. 5.Let ( m, n ) be an edge between vertices m, n ∈ V . Let x ( m, n ) be the flow over the edge ( m, n ) . Let E (cid:48) be the setof all vertices excluding s and t , i.e., V (cid:48) = V \ { s, t } . Themin-cost network flow problem, which is equivalent to therelaxation of (9), can be stated as follows max( m, n ) ∈ E (cid:88) ( m,n ) ∈ E x ( m, n ) ω ( m, n ) (11a) s . t . (cid:88) n :( m,n ) ∈ E x ( m, n ) − (cid:88) n :( n,m ) ∈ E x ( n, m ) = 0 , ∀ m ∈ V (cid:48) , (11b) (cid:88) n :( s,n ) ∈ E x ( s, n ) − (cid:88) n :( n,s ) ∈ E x ( n, s ) = (cid:88) b ∈B N BSRF ( b ) , (11c) (cid:88) n :( t,n ) ∈ E x ( t, n ) − (cid:88) n :( n,t ) ∈ E x ( n, t ) = − (cid:88) b ∈B N BSRF ( b ) , (11d) ≤ x ( m, n ) ≤ c ( m, n ) , ∀ ( m, n ) ∈ E (11e)The right-hand sides in (11b)-(11d) represent supply/demand Ss BS RF chains UE RF chains UEs Sink t Source s CapacityCost
Fig. 5. Directed graph G for the min-cost network flow problem. Zerocapacity links are not included in the figure. of each vertex in V . Both supplies/demands and capacities c ( m, n ) , ∀ ( m, n ) ∈ E , are integers. It was proved thatthe min-cost network flow problem always have an integersolution when edge capacities and vertex supplies/demandsare integer [13]. Thus, there is an integer optimal solution x ∗ s to the relaxation of (9).R EFERENCES[1] J. G. Andrews, S. Buzzi, W. Choi, S. V. Hanly, A. Lozano, A. C. K.Soong, and J. C. Zhang, “What will 5G be?”
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