Distributed Resource Allocation for Relay-Aided Device-to-Device Communication Under Channel Uncertainties: A Stable Matching Approach

Distributed Resource Aocation for Reay-Aided Device-to-Device Communication Under Channe Uncertainties: A Stabe Matching Approach Monowar Hasan, Student Member, IEEE, and Ekram Hossain, Feow, IEEE Abstract Wireess device-to-device D2D communication underaying ceuar network is a promising concept to improve user experience and resource utiization. Unike traditiona D2D communication where two mobie devices in the proximity estabish a direct oca ink bypassing the base station, in this work we focus on reay-aided D2D communication. Reay-aided transmission coud enhance the performance of D2D communication when D2D user equipments UEs are far apart from each other and/or the quaity of D2D ink is not good enough for direct communication. Considering the uncertainties in wireess inks, we mode and anayze the performance of a reay-aided D2D communication network, where the reay nodes serve both the ceuar and D2D users. In particuar, we formuate the radio resource aocation probem in a two-hop network to guarantee the data rate of the UEs whie protecting other receiving nodes from interference. Utiizing time sharing strategy, we provide a centraized soution under bounded channe uncertainty. With a view to reducing the computationa burden at reay nodes, we propose a distributed soution approach using stabe matching to aocate radio resources in an efficient and computationay inexpensive way. Numerica resuts show that the performance of the proposed method is cose to the centraized optima soution and there is a distance margin beyond which reaying of D2D traffic improves network performance. Index Terms Device-to-device D2D communication, LTE- A L3 reay, uncertain channe state information, distributed resource aocation, stabe matching. I. INTRODUCTION We consider reay-assisted device-to-device D2D communication underaying LTE-A ceuar networks where D2D user equipments UEs are served by the reay nodes [], [2]. When the ink condition between two D2D UEs is too poor for direct communication, the D2D traffic can be transmitted via a reay node, which performs scheduing and resource aocation for the D2D UEs. We refer to this as reay-aided D2D communication which can be an efficient approach to provide better quaity of service QoS for communication between distant D2D UEs. For this, we utiize the LTE-A Layer-3 L3 reay [3]. We consider scenarios in which the potentia D2D UEs are ocated in the same macroce; however, the proximity and ink condition may not be favorabe for direct communication. Therefore, they communicate via reays. The M. Hasan is with the Department of Computer Science, University of Iinois Urbana-Champaign, Iinois, USA; E. Hossain is with the Department of Eectrica and Computer Engineering, University of Manitoba, Winnipeg, Canada emais: mhasan@iinois.edu, Ekram.Hossain@umanitoba.ca. This work was supported by the Natura Sciences and Engineering Research Counci of Canada NSERC through a Strategic Project Grant STPGP 430285. radio resources e.g., resource bocks [RBs] and transmission power at the reays are shared among the D2D communication inks and the two-hop ceuar inks. We formuate an optimization probem to aocate radio resources at a reay node in a muti-reay muti-user orthogona frequency-division mutipe access OFDMA ceuar network e.g., LTE-A network. Due to the NP-hardness of the resource aocation probem, we utiize time sharing strategy and provide an asymptoticay optima centraized soution. Considering the random nature of wireess channes, we reformuate the resource aocation probem using the worst-case robust optimization theory. The uncertainties in ink gains are modeed using eipsoida uncertainty sets. Each reay node can centray sove the probem taking channe uncertainty into consideration. However, considering the high e.g., cubic to the number of UEs and RBs computationa overhead at the reay nodes, we provide a distributed soution based on stabe matching theory which is computationay inexpensive e.g., inear with the number of UEs and RBs. We aso anayze the stabiity, uniqueness, and optimaity of the proposed soution. Considering the computationa and signaing overheads and ack of scaabiity of the centraized soutions, game theoretica modes have been widey used for wireess resource aocation probems. However, the anaytica tractabiity of equiibrium in such game-theoretica modes requires specia properties for the objective functions, such as convexity, which may not be satisfied for many practica cases [4]. In this context, resource aocation using matching theory has severa beneficia properties [4], [5]. For exampe, the stabe matching agorithm terminates for every given preference profie. The outcome of matching provides suitabe soutions in terms of stabiity and optimaity, which can accuratey refect different system objectives. Besides, with suitabe data structures, a Pareto optima stabe matching e.g., aocation of resources to the UEs can be obtained quicky for onine impementation. The goa of this work is to design a practica radio resource scheme for reay-aided D2D communication in a muti-user muti-reay OFDMA network. As opposed to most of the work in the iterature where channe gain information is assumed to be perfect, we capture the dynamics of random and timevarying nature of wireess channes. To this end, we deveop a ow-compexity distributed soution based on the theory of stabe matching and demonstrate how this scheme can be impemented in a practica LTE-A system. The major contributions of this work can be summarized as foows: We mode and anayze the radio resource aocation

2 probem for reay-aided D2D communication underaying an OFDMA ceuar network considering uncertainties in channe gains. We formuate an optimization probem to maximize system capacity in a two-hop network whie satisfying the minimum data rate requirement for each UE and imiting the interference to other receiving nodes. We show that the convexity of the optimization probem is conserved under bounded channe uncertainty in both the usefu and interference inks. Using the theory of stabe matching, we deveop a distributed iterative soution, which considers bounded channe uncertainty. The stabiity, uniqueness, optimaity, and compexity of the proposed soution are anayzed. We aso present a possibe impementation approach of our proposed scheme in an LTE-A system. Numerica resuts show that the proposed distributed soution performs cose to the upper bound of the optima soution obtained in a centraized manner; however, it incurs a ower e.g., inear compared to cubic computationa compexity. Through simuations, we aso compare the performance of the proposed approach with a traditiona underay D2D communication scheme and observe that after a distant margin, reaying of D2D traffic improves network performance. We organize the rest of the paper as foows. We briefy review the reated work in Section II. Section III presents the system mode and reated assumptions. Foowed by the formuation of the nomina resource aocation probem in Section IV, we reformuate the resource aocation probem considering wireess channe gain uncertainties in Section V. We deveop the stabe matching-based distributed resource aocation agorithm in Section VI. Theoretica anaysis of the proposed soution is presented in Section VII. In Section VIII, we present the performance evauation resuts before we concude the paper in Section IX. II. RELATED WORK Despite the fact that the resource aocation probems in ceuar D2D communication have been intensivey studied in the recent iterature, ony a very few work consider reays for D2D communication and incorporate wireess ink uncertainties in the formuation of the resource aocation probem. In [6], a greedy heuristic-based resource aocation scheme is proposed for both upink and downink scenarios where a D2D pair shares the same resources with ceuar UE CUE ony if the achieved signa-to-interference-pusnoise ratio SINR is greater than a given SINR requirement. A resource aocation scheme based on coumn generation method is proposed in [7] to maximize the spectrum utiization by finding the minimum transmission ength i.e., time sots for D2D inks whie protecting the ceuar users from interference and guaranteeing QoS. A new spectrum sharing protoco for D2D communication overaying a ceuar network is proposed in [8], which aows the D2D users to communicate bi-directionay whie assisting the two-way communications between the enb and the CUE. A graphbased resource aocation method for ceuar networks with underay D2D communication is proposed in [9]. A twophase resource aocation scheme for ceuar network with underaying D2D communication is proposed in [0]. In [], the mode seection and resource aocation probem for D2D communication underaying ceuar networks is investigated and the soution is obtained by partice swarm optimization. The works above do not consider reays for D2D communication. A distributed reay seection method for reay-assisted D2D communication system is proposed in [2]. In [3], [4], authors investigate the maximum ergodic capacity and outage probabiity of cooperative reaying in reay-assisted D2D communication considering power constraints at the enb. Taking the advantage of L3 reays supported by the 3GPP standard, in [], a centraized resource aocation approach is proposed for reay-assisted D2D communication assuming that perfect channe information is avaiabe. A gradient-based distributed resource aocation scheme is proposed in [2] for reay-aided D2D communication in a muti-reay network under uncertain channe information. In this muti-reay network, the interference ink gain between a UE and other reays to which the UE is not associated with is modeed with eipsoida uncertainty sets. However, the uncertainty in direct channe gain between reay and the UE is not considered. In this paper, we remode the previous formuation and extend the work in [2] by incorporating uncertainties in both the usefu and interference inks. In particuar, we present a distributed resource aocation agorithm using stabe matching considering the uncertainties in wireess channe gains e.g., channe quaity indicator [CQI] parameters according to the LTE-A terminoogy. Athough not in the context of D2D communication, matching theory has been used in the iterature to address the radio resource aocation probems in wireess networks. A spectrum aocation agorithm using matching theory is proposed in [5] for a cognitive radio network CRN under perfect channe assumption. In [6], a two-sided stabe matching agorithm is appied for adaptive muti-user scheduing in an LTE-A network. A distributed matching agorithm is proposed in [7] for cooperative spectrum sharing among mutipe primary and secondary users with incompete information in a CRN. In [8], a distributed agorithm is proposed to sove the user association probem in the downink of sma ce networks SCNs. A matching-based subcarrier aocation approach is proposed in [9] for services with couped upink and downink QoS requirements. The radio resource e.g., subcarrier and power aocation probem for a fu-dupex OFDMA network is modeed as a transmitter-receiver-subcarrier matching probem in [20]. The matching-based soutions proposed in [5] [20] do not consider D2D-enabed networks. In the context of D2D communication, most of the works e.g., [], [6], [7], [9] [], [3], [4] provide centraized soutions. Aso, note that in [6] [], the effect of reaying on D2D communication is not investigated. Moreover, the wireess ink uncertainty is not considered in [], [6] [6], [8] [20]. Different from the above works, we propose a stabe matching-based distributed radio resource aocation approach considering the channe gain uncertainties in a muti-reay and muti-user reay-aided

3 Work on D2D communication Work utiizes matching theory TABLE I SUMMARY OF RELATED WORK AND COMPARISON WITH PROPOSED SCHEME Reference Probem focus Reay-aided Channe information Soution approach Soution type Optimaity [6] Resource aocation No Perfect Proposed greedy heuristic Centraized Suboptima [7] Resource aocation No Perfect Coumn generation based Centraized Suboptima greedy heuristic [8] Resource aocation No * Perfect Numerica optimization Semi-distributed Pareto optima [9] Resource aocation No Perfect Interference graph cooring Centraized Suboptima [0] Resource aocation No Perfect Two-phase heuristic Centraized Suboptima [] Resource aocation, No Perfect Partice swarm optimization Centraized Suboptima mode seection [2] Resource aocation, Yes Perfect Proposed heuristic Distributed Suboptima mode seection [3] Theoretica anaysis, Yes Perfect Statistica anaysis Centraized Optima performance evauation [4] Performance evauation Yes Perfect Heuristic, simuation Centraized N/A [] Resource aocation Yes Perfect Numerica optimization Centraized Asymptoticay optima [2] Resource aocation Yes Uncertain # Gradient-based iterative update Distributed Suboptima [5] Resource aocation N/A Perfect One-to-one matching Centraized Optima [6] Cross ayer scheduing N/A Perfect Many-to-one matching Centraized N/A [7] Spectrum sharing N/A Compete, One-to-one matching Distributed Pareto optima incompete [8] Ce association N/A Perfect Many-to-one matching Distributed N/A [9] Resource aocation N/A Perfect Many-to-one matching Distributed N/A [20] Resource aocation N/A Perfect One-to-one matching Centraized N/A Proposed Resource aocation Yes Uncertain Matching theory many-to-one Distributed Weak Pareto scheme matching optima * D2D UEs serve as reays to assist CUE-eNB communications. No information is avaiabe. # Uncertainty in channe gain in the direct ink between UEs reays and reays enb is not considered. Not appicabe for the considered system mode. D2D communication scenario. A summary of the reated work and comparison with our proposed approach is presented in Tabe I. A. Network Mode III. SYSTEM MODEL Let L = {, 2,..., L} denote the set of fixed-ocation L3 reays in the network Fig. in [2]. The system bandwidth is divided into N orthogona RBs denoted by N = {, 2,..., N} which are used by a the reays in a spectrum underay fashion. The set of CUEs and D2D pairs are denoted by C = {, 2,..., C} and D = {, 2,..., D}, respectivey. We assume that association of the UEs both ceuar and D2D to the corresponding reays are performed before resource aocation. Prior to resource aocation, D2D pairs are aso discovered and the D2D session is setup by transmitting known synchronization or reference signas [2]. We assume that the CUEs are outside the coverage region of the enb and/or having bad channe condition, and therefore, the CUE-eNB communications need to be supported by the reays. Communication between two D2D UEs requires the assistance of a reay node due to poor propagation condition. The UEs assisted by reay are denoted by. The set of UEs assisted by reay is U = {, 2,..., U } such that U {C D}, L, U = {C D}, and U =. In the second hop, there coud be mutipe reays transmitting to their associated D2D UEs. We assume that mutipe reays transmit to the enb in order to forward CUEs traffic using orthogona channes and this scheduing of reays is done by the enb. In our system mode, taking advantage of the capabiities of L3 reays, scheduing and resource aocation for the UEs is performed in the reay nodes to reduce the computationa oad at the enb. B. Achievabe Data Rate Let γ n,, denote the unit power SINR for the ink between UE U and reay using RB n in the first hop and γ n,,2 be the unit power SINR for the second hop. Note that, in the second hop, when the reays transmit CUEs traffic i.e., {C U }, γ n,,2 denotes the unit power SINR for the ink between reay and the enb. On the other hand, when a reay transmits to a D2D UE i.e., {D U }, γ n,,2 refers to the unit power SINR for the ink between reay and the receiving D2D UE for the D2D-pair. Let i,j 0 denote the transmit power in the ink between i and j over RB n and B RB is the bandwidth of an RB. The achievabe data rate 2 for in the first hop can be expressed as r n, = B RB og 2 +., γn,, Simiary, the achievabe data rate in the second hop is r n,2 = B RB og 2 +. Since we consider a two-, γ n,,2 Scheduing of reay nodes by the enb is not within the scope of this work. 2 We wi present the rate expressions in Section IV-A.

4 hop communication, the end-to-end data rate 3 for on RB n is haf of the minimum achievabe data rate over two hops [22], i.e., R u n = { } 2 min r n,, rn,2. IV. RESOURCE ALLOCATION: FORMULATION OF THE NOMINAL PROBLEM In the foowing, we present the formuation of the resource aocation probem assuming that perfect channe gain information is avaiabe. This formuation is referred to as the nomina probem since the uncertainties in channe gains are not considered. For each reay, the objective of radio resource i.e., RB and transmit power aocation is to obtain the assignment of RB and power eve to the UEs that maximizes the system capacity, which is defined as the minimum achievabe data rate over two hops. Let the maximum aowabe transmit power for UE reay is Pu max P max and et the QoS i.e., data rate requirement for UE be denoted by Q u. The RB aocation indicator is a binary decision variabe {0, }, where {, if RB n is assigned to UE = 2 0, otherwise. A. Objective Function N Let R u = R u n denote the achievabe sum-rate over aocated RBs. We consider that the same RBs wi be used by the reay in both the hops i.e., for communication between reay and enb and between reay and D2D UEs. The objective of resource aocation probem is to maximize the end-to-end rate for each reay L as foows: max, n,,p, N U where the rate of UE over RB n R u n = B RB og 2 + 2 min B RB og 2 + R u n 3, γn,,, γ n,,2 In 3, the unit power SINR for the first hop, γ n,, = j,j L h n,,. u j u j,j gn u + 4 j,, σ2 where h n i,j,k denotes the direct ink gain between node i and j over RB n for hop k {, 2}, σ 2 = N 0 B RB in which 3 In a conventiona D2D communication approach where two D2D UEs communicate directy without a reay, the achievabe data rate for D2D UE n u D over RB n can be expressed as R u = B RB og 2 + P u n γ u n, where γ n u = h n u,u j Ûu j, g n hn u,j +σ2 u,u is the channe gain in the ink between the D2D UEs and Ûu denotes the set of UEs transmitting using the same RBs as u. N 0 denotes therma noise. The interference ink gain between reay UE i and UE reay j over RB n in hop k is denoted by g n i,j,k, where UE reay j is not associated with reay UE i. Simiary, the unit power SINR for the second hop 4, h n,,2, u x u n j j,u g n {C U } j j,enb,2 +σ2 u j {D U j}, γ n,,2 = j,j L j,j L h n,,2 u j j,u j g n j,,2 +σ2, {D U } 5 where h,u,2 denotes the channe gain between reay-enb ink for CUEs e.g., {C U } or the channe gain between reay and receiving D2D UEs e.g., {D U }. From, the maximum data rate for UE over RB n is achieved when, γn,, =, γ n,,2. Therefore, in the second hop, the power P,u aocated for UE, can be expressed as a function of power aocated for transmission in the first hop, P u, as foows:, = γn,, γ n, hn,,,,2 h n,,2,. Hence, the data rate for over RB n can be expressed as R u n = 2 B RB og 2 +, γn,,. Considering the above, the objective function in 3 can be rewritten as max,, N U 2 xn B RB og 2 +, γn,,. 6 For each reay L in the network, the objective of resource aocation probem is to obtain the RB [ and power aocation vectors, ] i.e., T x = x,..., xn,..., x U,..., x N U and [ ] T, P = P,,..., P N,,..., P U,,..., P N U, respectivey, which maximize the data rate. B. Constraint Sets In order to ensure the required data rate for the UEs whie protecting a receiver nodes from harmfu interference, we define the foowing set of constraints. The constraint in 7 ensures that each RB is assigned to ony one UE, i.e., U, n N. 7 The foowing constraints imit the transmit power in each 4 According to LTE-A standard, the L3 reays are abe to peform simiar operation as an enb. Besides, the reays in the network are interconnected through X2 interface for better interference management [23]. Since the reays can estimate the CQI vaues and hence the interference eve using X2 interface, it is straightforward to account for interference in 4 and 5. Consequenty, interference from other transmitter nodes e.g., UEs associated to other reays in the first hop or other reays in the second hop wi appear as a constant term in 4 and 5.

5 of the hops to the maximum power budget: N N U, P u max, U 8, P max. 9 Simiar to [24], we assume that there is a maximum toerabe interference threshod imit for each aocated RB. The constraints in 0 and imit the amount of interference introduced to the other reays and the receiving D2D UEs in the first and second hop, respectivey, to be ess than some threshod, i.e., U U, gn u,, In th,, g n,u, n N 0,2 In th,2, n N. The minimum data rate requirements for the CUE and D2D UEs is ensured by the foowing constraint: R u Q u, U. 2 The binary decision variabe on RB aocation and nonnegativity condition of transmission power is defined by {0, },, 0, U, n N. 3 Note that in constraint 0 and, we adopt the concept of reference user. For exampe, to aocate the power eve considering the interference threshod in the first hop, each UE associated with reay node obtains the reference user u associated with the other reays and the corresponding channe gain g n u,, for n according to the foowing equation: u = argmax j g n,j,, U, j, j L. 4 Simiary, in the second hop, for each reay, the transmit power wi be adjusted accordingy considering interference introduced to the receiving D2D UEs associated with other reays considering the corresponding channe gain g n,u,2 for n, where the reference user is obtained by u = argmax u j C. Centraized Soution g n,u j,2, j, j L, u j {D U j }. 5 Coroary. The objective function in 6 and the set of constraints in 7-3 turn the optimization probem to a mixed-integer non-inear program MINLP with non-convex feasibe set. Therefore, the formuation described in Section IV is NP-hard. A we-known approach to sove the above probem is to reax the constraint that an RB is used by ony one UE by using the time-sharing factor [25]. In particuar, we reax the optimization probem by repacing the non-convex constraint. Thus represents the sharing factor where each denotes the portion of time that RB n is assigned to UE and satisfies {0, } with the convex constraint 0 < the constraint U, n. Besides, we introduce a new variabe, = xn, 0, which denotes the actua transmit power of UE on RB n [26]. Then the reaxed probem can be stated as foows: P2 max R u 6,, U subject to 7, 2 and N Pu max, 7 N, H n, Sn, U, gn u,, U U H n, Sn, gn,u,2 0 < where γ n,, = R u = N j,j L P max 8 I n th,, n 9 I n th,2, n 20,, 0, n, 2 h n, 2 xn B RB og 2, Hn u j,j gn u j,, +σ2 + Sn, γn,,, = hn,, h n,,2. and Coroary 2. The objective function in 6 is concave, the constraint in 2 is convex, and the remaining constraints in 7, 7-2 are affine. Therefore, the optimization probem P2 is convex. Since P2 is a non-inear convex probem, each reay can sove the optimization probem using standard agorithms such as interior point method [27, Chapter ]. The centraized optimization-based soution is summarized in Agorithm. Each reay ocay soves the optimization probem P2 and informs the other reays the aocation vectors using X2 interface. The process is repeated unti the data rate is maximized, e.g., R t R t < ɛ for, where R = R u U is the sum data rate for reay obtained by soving the optimization probem at iteration and ɛ is a sma vaue. The duaity gap of any optimization probem satisfying the time-sharing condition becomes negigibe as the number of RBs becomes significanty arge. The optimization probem P2 satisfies the time-sharing condition, and therefore, the soution of the reaxed probem is asymptoticay optima [28]. Given the parameters of other reays e.g., x j, P j j, j L, at each iteration of Agorithm the aocation vectors e.g., x, P obtained at each reay provide ocay optima soution for. In addition, Agorithm aows the reays to perform aocation repeatedy with a view to finding the best possibe aocation. If the data rate at the t + -th iteration is not improved compared to that in the previous iteration t, the agorithm terminates, and the aocation at iteration t wi be the resutant soution. Each of the iteration of Agorithm outputs the soution of reaxed version of

6 Agorithm Optimization-based resource aocation : Each reay L estimates the CQI vaues from previous time sot and determines reference gains g n u,, and gn,u,2,, n. 2: Initiaize t := 0. 3: repeat 4: Update t := t +. 5: Each reay L: soves the optimization probem P2 and muticasts the aocation variabes x, P to a reay j, j L over X2 interface. cacuates the achievabe data rate based on current aocation as R t := R u t. U 6: unti data rate not maximized and t < T max. 7: Aocate resources i.e., RB and transmit power to associated UEs for each reay. the origina NP-hard optimization probem. Since the soution of reaxed probem gives us the upper bound, and at the termination of Agorithm, we obtain an upper bound of the achievabe sum rate. V. RESOURCE ALLOCATION UNDER CHANNEL UNCERTAINTY For worst-case robust resource optimization probems, the channe gain is assumed to have a bounded uncertainty of unknown distribution. An eipsoid is often used e.g., [29] [3] to approximate such an uncertainty region. A. Uncertainty Sets Let the variabe F n,u j, denote the normaized channe gain which is defined as foows: F n gn u,u j, j,, h n,,, u j U j, j, j L. 22 In addition, et F n, denote the uncertainty set that describes the perturbation of ink gains for over RB n. The normaized gain is then denoted by n F F n n n,u j, = F,u j, + F,u j, 23 where,u j, is the nomina vaue and F n,u j, is the perturbation part. The uncertainty in the CQI vaues is modeed under an eipsoida approximation as foows: F n, = F n n,u j, + F,u j, : F n,u j, 2 ξ n u ;, n j,j L 24 where ξ n u 0 is the uncertainty bound in each RB. Using 22, we rewrite the rate expression for over RB n as R n = 2 B RB og 2 + where σ n σ2 h n,, j,j L, F n,u j, u j,j + σn and F n,u j, is given by 23. 25 B. Reformuation of the Optimization Probem Considering Channe Uncertainty Utiizing uncertainty sets simiar to 24 in the constraints 8-20, the optimization probem P2 can be equivaenty represented under channe uncertainty as foows: U P3 U max,, subject to N Hn, + Hn, ḡ n u,, + gn min F n,u j,, gn u,,, H n,, Hn, gn,u,2 7, 2, 7, 2 and, u,,, U Hn,ḡn,u,2 + Hn, gn,u,2, j,j L U F n N U R u 26 P max 27 I n th,, n 28 I n th,2, n 29,u j, 2 ξ n 2, u u, n 30 H n, 2 ξ 2 2 3 U g n u,, 2 U H n, gn,u,2 2 ξ n 3 u 2, n 32 ξ n 4 u 2, n 33 where for any parameter y, ȳ denotes the nomina vaue and y represents the corresponding deviation part; ξ 2, ξ n 3 u, and ξ n 4 u are the maximum deviations e.g., uncertainty bounds of corresponding entries in CQI vaues. In P3, R u is given by R u = N 2 xn B RB og 2 + j,j L n F,u j,, + F n,u j, u j,j + σn. 34 The above optimization probem is subject to an infinite number of constraints with respect to the uncertainty sets and hence becomes a semi-infinite programming SIP probem [32]. In order to sove the SIP probem it is required to transform P3 into an equivaent probem with finite number of constraints. Simiar to [29], [30], we appy the Cauchy- Schwarz inequaity [33] and transform the SIP probem. More specificay, utiizing Cauchy-Schwarz inequaity, we obtain the foowing: F n,u j, Sn j,j L u j,j j,j L ξ n u F n,u j, 2 j,j L j,j L u j,j 2 u j,j 2. 35

7 Simiary, N H n, Sn U, ξ 2 N U g n u,,sn ξn 3 u U H n, gn,u,2sn, ξn 4 u U U U, 2 36, 2 37, 2. 38 Note that, as presented in Section V-A, to tacke the uncertainty in channe gains, we have considered the worst-case approach, e.g., the estimation error is assumed to be bounded by a cosed set uncertainty set. Hence, from 35-38, under the worst-case channe uncertainties, the optimization probem P3 can be rewritten as P4, where R u is given by 43. The transformed probem is a second-order cone program SOCP [27, Chapter 4] and the convexity of P4 is conserved as shown in the foowing proposition. Proposition. P4 is a convex optimization probem. Proof: Using an argument simiar to that in footnote 4, the objective function of P4 in 39 is concave. The constraints in 7, 7, 2 are affine and the constraint in 2 is convex. In addition, the additiona square root term in the eft hand side of the constraints in 40, 4, and 42 is the inear norm of the vector of power variabes, with order 2, which is convex [27, Section 3.2.4]. Therefore, the optimization probem P4 is convex. P4 is sovabe using standard centraized agorithms such as interior point method. The joint RB and power aocation can be performed simiar to Agorithm and an upper bound for the soution can be obtained under channe uncertainty. It is worth noting that soving the above SOCP using interior 3 point method incurs a compexity of O x + P at each reay node where y denotes the ength of vector y. Besides, the size of the optimization probem increases with the number of network nodes. Despite the fact that the soution from Agorithm outputs the optima data rate, considering very short scheduing period e.g., miisecond in LTE- A network, it may not be feasibe to sove the resource aocation probem centray in practica networks. Therefore, in the foowing, we provide a ow-compexity distributed soution based on matching theory. That is, without soving the resource aocation probem in a centraized manner using any reaxation technique e.g., time-sharing strategy as described in the preceding section, we appy the method of two-sided stabe many-to-one matching [34]. VI. DISTRIBUTED SOLUTION APPROACH UNDER CHANNEL UNCERTAINTY The resource aocation approach using stabe matching invoves mutipe decision-making agents, i.e., the avaiabe RBs and the UEs; and the soutions i.e., matching between UE and RB are produced by individua actions of the agents. The actions, i.e., matching requests and confirmations or rejections are determined by the given preference profies. That is, for both the RBs and the UEs, the ists of preferred matches over the opposite set are maintained. For each RB, the reay hods its preference ist for the UEs. The matching outcome yieds mutuay beneficia assignments between RBs and UEs. Stabiity in matching impies that, with regard to their initia preferences, neither RBs nor UEs have an incentive to ater the aocation. A. Concept of Matching A matching i.e., aocation is given as an assignment of RBs to UEs forming the set of pairs, n U N. Note that a UE can be aocated more than one RB to satisfy its data rate requirement; however, according to the constraint in 7, one RB can be assigned to ony one UE. This scheme corresponds to a many-to-one matching in the theory of stabe matching. More formay, we define the matching as foows [35]. Definition. A matching µ for L is defined as a function, i.e., µ : U N U N such that i µ n U { } and µ n {0, } ii µ N and µ {, 2,..., κ u } where the integer κ u N, µ = n µn = for n N, U and µ j denotes the cardinaity of matching outcome µ j. The above definition impies that µ is a one-to-one matching if the input to the function is an RB. On the other hand, µ is a one-to-many function, i.e., µ is not unique if the input to the function is a UE. In order to satisfy the data rate requirement for each UE, we introduce the parameter κ u denoting the number RBs which are sufficient to satisfy the minimum rate requirement Q u. Consequenty, the constraint in 2 is rewritten as N = κ u,. Generay this parameter is referred to as quota in the theory of matching [5]. Each user wi be subject to an acceptance quota κ u over RBs within the range κ u N and aowed for matching to at most κ u RBs. The outcome of the matching determines the RB aocation vector at each reay, e.g., µ x. B. Utiity Matrix and Preference Profie Let us consider the utiity matrix U under the worst-case uncertainty, which denotes the achievabe data rate for the UEs in different RBs, defined as foows: U = R R N. R U.... R N U 44 where U [i, j] denotes the entry of i-th row and j-th coumn in U, and R u n is given by 45. Each of the UEs and RBs hods a ist of preferred matches where a preference reation can be defined as foows [36, Chapter 2]. Definition 2. Let be a binary reation on any arbitrary set Ξ. The binary reation is compete if for i, j Ξ, either

8 P4 N U H n, Sn max R u 39 U subject to 7, 2, 7, 2 and, + ξ N 2 2, P max 40,, U ḡ n u,,sn + ξn 3 u U U Hn,ḡn,u,2Sn, + ξn 4 u U U,, 2 n I th,, n 4 2 n I th,2, n. 42 R u = = N N 2 xn B RB og 2 2 xn B RB og 2 + + j,j L, F n,u j, Sn u + j,j ξn ḡ n u j,, Sn u j,j j,j L u j,j L, hn x u n,, S n u j,j n + h,, ξn u j,j L 2 + σ n u j,j n 2 43 S + σ 2 = 2 B RB og 2 + R n j,j L u j, h n,, ḡ n u n j,, u j,j + h,, ξn u j,j L x n u j u j,j 2 45 + σ 2 i j or j i or both. A binary reation is transitive if i j and j k impies that i k for k Ξ. The binary reation is a weak preference reation if it is compete and transitive. The preference profie of a UE U over the set of avaiabe RBs N is defined as a vector of inear order P u N = U [, i] i N. The UE prefers RB n to n 2 if n n 2, and consequenty, U [, n ] > U [, n 2 ]. Likewise, the preference profie of an RB n N is given by P n U = U [j, n] j U. C. Agorithm for Resource Aocation Based on the discussions in the previous section, we utiize an improved version of matching agorithm adapted from [37, Chapter.2] to aocate the RBs. The aocation subroutine, as iustrated in Agorithm 2, executes as foows. Whie an RB n is unmatched i.e., unaocated and has a non-empty preference ist, the RB is temporariy assigned to its first preference over UEs, i.e.,. If the aocation does not exceed κ u, the aocation wi persist. Otherwise, the worst preferred RB from s matching wi be removed even though it was previousy aocated. The iterations are repeated unti there are unaocated pairs of RB and UE. The iterative process dynamicay updates the preference ists and hence eads to a stabe matching. Once the optima RB aocation is obtained, the transmit power of the UEs on assigned RBs is obtained as foows. We coupe the cassica generaized distributed constrained power contro scheme GDCPC [38] with an autonomous power contro method [39] which considers the data rate requirements of UEs whie protecting other receiving nodes from interference. More specificay, at each iteration t, the transmission power for each aocated RB is updated as foows: where, t = ˆmax Λt, ˆP nmax if Λt 46 ˆ,, otherwise Λt = = min P max N 2Q 2 R t,, x u n Hn, +ξ2 t 47 P max N U 48

9 Agorithm 2 RB aocation using stabe matching Input: The preference profies P u N, P nu ; U, n N. Output: The RB aocation vector x. : Initiaize x := 0. 2: whie with N x u n < κ u or n with x u n = 0, U and P nu do 3: u mp := most preferred UE from the profie P nu. 4: Set u mp :=. /* Temporariy aocate the RB */ N 5: if x j u mp > κ ump then j= 6: n p := east preferred resource aocated to u mp. 7: Set x n p u mp := 0. /* Revoke aocation due to quota vioation */ 8: end if N 9: if x j u mp = κ ump then j= 0: n p := east preferred resource aocated to u mp. : /* Update preference profies */ 2: for each successor ˆn p of n p on profie P ump N do 3: remove ˆn p from P ump N. 4: remove u mp from Pˆnp U. 5: end for 6: end if 7: end whie and n ˆP, is obtained as ˆ, = min P n,, min ˆP n max, ϖ n,. 49 n In 49, the parameter P, is chosen arbitrariy within the n nmax range of 0 P, ˆP and ϖ n, is given by ϖ n, = min I n th,, ḡ n u,,+ξn 3 I n th,2 H n,ḡn,u,2+ξn 4. 50 Based on the RB aocation, the reay informs the parameter ˆP u nmax and each UE updates its transmit power in a distributed manner using 46. Each reay independenty performs resource aocation and aocates resources to corresponding associated UEs. The joint RB and power aocation agorithm is given in Agorithm 3. VII. ANALYSIS OF THE PROPOSED SOLUTION In the foowing, we anayze the performance of our proposed distributed resource aocation approach under bounded channe uncertainty. More specificay, we anayze the stabiity, optimaity, and uniqueness of the soution, and its computationa compexity. A. Stabiity Definition 3. a The pair of UE and RB, n in U N is acceptabe if and n prefer each other to be matched to being remain unmatched. b A matching µ is caed individuay rationa if no agent i.e., UE or RB j prefers to remain unmatched to µ j. Definition 4. A matching µ is bocked by a pair of agents i, j if they each prefer each other to the matching they obtain by µ, i.e., i µ j and j µ i. Agorithm 3 Joint RB and power aocation agorithm Phase I: Initiaization : Each reay L estimates the nomina CQI vaues from previous time sot and determines reference gains ḡ n u,, and ḡ n,u,2,, n. max Pu := N 2: Initiaize t := 0,,, n and U based on CQI estimates. Phase II: Update 3: for each reay L do 4: repeat 5: Update t := t +. 6: Buid the preference profie P nu for each RB n N based on utiity matrix and inform corresponding entries of U to UEs. 7: Each UE U buids the preference profie P u N. 8: Obtain RB aocation vector using Agorithm 2. 9: Update the transmission power using 46 for, n and update the utiity matrix U. 0: Inform the aocation variabes x, P to each reay j, j L and cacuate the achievabe data rate based on current aocation as R t := R u t. U : unti data rate not maximized and t < T max. 2: end for Phase III: Aocation 3: For each reay, aocate resources i.e., RB and transmit power to the associated UEs. From Definition 3, 4, the matching µ is bocked by RB n and UE if n prefers to µ n and either i prefers n to some ˆn µ, or ii µ < κ u and n is acceptabe to. Using the above definitions, the stabiity of matching can be defined as foows [4, Chapter 5]. Definition 5. A matching µ is stabe if it is individuay rationa and there is no pair, n in the set of acceptabe pairs such that prefers n to µ and n prefers to µ n, i.e., not bocked by any pair of agents. Proposition 2. The assignment performed in Agorithm 2 abides by the preferences of the UEs and RBs and it eads to a stabe aocation. Proof: See Appendix A. Note that the aocation of RBs is stabe at each iteration of Agorithm 3. Since after evauation of the utiity, the preference profie of UEs and RBs are updated and the routine for RB aocation is repeated, a stabe aocation is obtained. B. Uniqueness Proposition 3. If there are sufficient number of RBs i.e., N U, and the preference ists of a UEs and RBs are determined by the U N utiity matrix U whose entries are a different and obtained from given uncertainty bound, then there is a unique stabe matching. Proof: See Appendix B. C. Optimaity and Performance Bound Definition 6. A matching µ is weak Pareto optima if there is no other matching µ that can achieve a better sum-rate,

0 i.e., µ µ, where the inequaity is component-wise and strict for one user. Proposition 4. The proposed resource aocation agorithm is weak Pareto optima under bounded channe uncertainty. Proof: See Appendix C. Coroary 3. Since x satisfies the binary constraint in 2, and the optima aocation x, P satisfies a the constraints in the optimization probem P4, for a sufficient number of avaiabe RBs, the data rate obtained by Agorithm 3 gives a ower bound of the soution under channe uncertainty. D. Convergence and Computationa Compexity Proposition 5. The subroutine for RB aocation terminates after some finite number of steps T. Proof: Let the finite set X represent a possibe combinations of UE-RB matching where each eement x j i X denotes that RB j is aocated to UE i. Since no UE rejects the same RB more than once see ine 7 in Agorithm 2, the finiteness of the set X ensures the termination of RB aocation subroutine in finite number of steps. Note that the distributed approach repaces the optimization routine in the centraized approach with the matching agorithm Agorithm 2. Since by Proposition 5 we show matching agorithm terminates after finite number of iterations, utimatey Agorithm 3 wi end up with a oca Pareto optima soution after some finite number of iterations. In ine 6-7 of Agorithm 3, the compexity to output the ordered set of preference profies for the RBs using any standard sorting agorithm is O NU og U and for each UE, the compexity to buid the preference profie is O N og N. Let U N β = P u N + P n U = 2NU be the tota ength = of input preferences in Agorithm 2, where P j denotes the ength of the profie vector P j. From Proposition 5 and [37, Chapter ] it can be shown that, if impemented with suitabe data structures, the time compexity of RB aocation subroutine is inear in the size of input preference profies, i.e., Oβ O NU. Since Phase II of Agorithm 3 runs at most fixed T max iterations, at each reay node, the compexity of the proposed soution is inear in N and U. E. Signaing Over Contro Channes Assuming that the reays obtain the CQI prior to resource aocation, the centraized approach does not require any exchange of information between a reay node and the associated UEs to perform resource aocation. However, in the distributed approach, the reay node and the UEs need to exchange information to update the preference profies and transmit power. In both the approaches, the reay nodes need to exchange the aocation variabes among themseves e.g., over X2 interface in order to cacuate the interference eves at the receiving nodes. In the distributed approach, the exchange of information between a UE and the reay node during execution of the resource aocation agorithm can be mapped onto the standard LTE-A scheduing contro messages. For scheduing in LTE- A networks, the exchanges of messages over contro channes are as foows [40]. The UEs wi periodicay sense the physica upink contro channe PUCCH by transmitting known sequences as sounding reference signas SRS. When data is avaiabe for upink transmission, the UE sends the scheduing request SR over PUCCH. The reay, in turn, uses the scheduing grant SG over physica downink contro channe PDCCH to aocate the appropriate RBs to the UE. Once the aocation of RBs is received, the UE reguary sends buffer status report BSR using PUCCH in order to update the resource requirement, and in response, the reay sends the acknowedgment ACK over the physica hybrid-arq indicator channe PHICH. Given the above scenario, the UEs may provide the preference profie P u N with the SR and BSR messages. The reays may provide the corresponding vaues in the utiity matrix, e.g., û u, = U [, j] j=,,n and ˆP nmax inform the parameter using SG and ACK messages. Once the RB and power aocation is performed, the reays muticast the aocation information over X2 interface. In what foows, we anayze signaing overhead for our proposed soution. For a sufficient number of avaiabe RBs, we consider two cases: a number of avaiabe RBs at each reay is equa to the number of UEs e.g., N = U and κ u =, ; and b the number of RBs is greater than the number of UEs e.g., N > U. For the first case, once Agorithm 2 terminates, a RBs are aocated to the UEs. This is because, by the definition of individua rationaity see Definition 3, none of the agents i.e., UE or RB wants to remain unaocated. Hence, at the end of any iteration ˆt of Agorithm 2, there are N ˆt unaocated RBs at each reay. Therefore, the maximum number of iterations, say T max are required when a the RBs are aocated, e.g., N T max = 0, and therefore, Tmax = N. Since at each iteration ˆt, N ˆt + messages are exchanged, the tota number of messages exchanged in Agorithm 2 can be quantified as Ω = T max i= N i + = NN +. 5 2 In Agorithm 3, each reay exchanges the aocation parameters e.g., x, P over X2 interface. If Agorithm 3 executes T < T max iterations, the overa signaing overhead e.g., number of messages exchanged is given by Ω max = T Ω + = T N 2 + N + 2. 52 2 Likewise, for the second case e.g., when N > U, Agorithm 2 terminates when there are no unaocated UEs with ess N than their quota requirement e.g., < κ u. Hence the maximum number of iterations T max = U, and the numnber of messages exchanged in Agorithm 2 for the second case is

given by Ω = T max i= U N i + = N i + i= = N + U U U +. 53 2 Therefore, the overa signaing overhead for the second case can be expressed as foows: Ω max = T 2N + U U U + + 2. 54 2 As can be seen from 52 and 54, the overhead of signaing increases with the number of UEs and the avaiabe RBs. However, the proposed distributed approach has significanty ower computationa compexity than the centraized approach inear compared to cubic and it offers performance improvement over the existing soutions see Section VIII-B. A. Simuation Setup VIII. PERFORMANCE EVALUATION We deveop a discrete-time simuator in MATLAB and evauate the performance of our proposed soution. We simuate a singe three-sectored ce in a rectanguar area of 700 m 700 m, where the enb is ocated in the center of the ce and three reays are depoyed, i.e., one reay in each sector. The CUEs are uniformy distributed within the reay ce. The D2D UEs are ocated according to the custered distribution mode [42]. In particuar, the D2D transmitters are uniformy distributed over a radius D r,d ; and the D2D receivers are distributed uniformy in the perimeter of the circe with radius D d,d centered at the corresponding D2D transmitter Fig. 2 in [2]. Both D r,d and D d,d are varied as simuation parameters and the vaues are specified in the corresponding figures. The simuation resuts are averaged over 200 network reaizations of user ocations and channe gains. We consider a snapshot mode and a the network parameters are assumed to remain unchanged during a simuation run. For propagation modeing, we consider distance-dependent path-oss, shadow fading, and muti-path Rayeigh fading see Section VII-A in [2]. We measure the uncertainty in channe gains as percentages and assume simiar uncertainty bounds in the CQI parameters for a the UEs. For exampe, uncertainty bound ξ = ξ n u = ξ 2 = ξ n 3 u = ξ n 4 u = 0.25 refers that uncertainty e.g., estimation error in the CQI parameters for, n, is not more than 25% of their nomina vaues. The simuation parameters are simiar to those in Tabe II in [2]. B. Resuts A summary of the observations from the performance evauation resuts is provided in Tabe II. Convergence and goodness of the soution: In Fig., we show the convergence behavior of our proposed distributed agorithm. In particuar, we pot the average achievabe data rate for the UEs in different network reaizations versus the number of iterations. The agorithm starts with uniform power aocation over RBs, which provides a higher data rate at the TABLE II SUMMERY OF PERFORMANCE RESULTS Observations The proposed distributed soution converges to a stabe data rate within a few iterations and performs cose to the optima data rate with significanty ess computationa compexity. As the distance between D2D peers increases, the data rate for direct communication decreases. In such cases reaying of D2D traffic can improve the end-to-end data rate between D2D peers. After a distance threshod, reay-aided D2D communication provides considerabe gain in terms of the achievabe data rate for the D2D UEs. Even for a reativey arge distance between the reay node and a D2D UE, reaying can provide a better data rate compared to direct communication for distant D2D peers. There is aso a trade-off between achievabe data rate and robustness against channe uncertainty. End to end data rate bps 2.5.5 0.5 3 x 06 2 x 05 0 9 8 Data rate Ceuar UE Data rate D2D UE 7 4 5 6 7 8 9 Ref. figures Figs. -2 Fig. 3 2 4 6 8 0 2 Number of iterations Figs. 4-5 Fig.. Convergence of the proposed soution where the number of CUEs and D2D UEs served by each reay node is 5 and 3, receptivey e.g., U = 8. D r,d and D d,d are set to 50 m, and uncertainty in CQI parameters is assumed to be not more than 25%. first iteration; however, it may cause severe interference to other receiving nodes. As the agorithm executes, the aocations of RB and power are updated considering the interference threshod and data rate constraints. From this figure it can be observed that the soution converges to a stabe data rate very quicky e.g., in ess than 0 iterations. We compare the performance of our proposed scheme with a dua-decomposition based suboptima resource aocation scheme proposed in [43]. We refer to this scheme as existing agorithm. In this scheme, the reay node aocates RBs considering the data rate requirement and the transmit power is updated in an iterative manner by updating the Lagrange dua variabes. For detais refer to [43, Agorithm 2]. The compexity of this agorithm is of O NU og N + N og U +, where denotes the number of iterations it takes for the power aocation vector to converge [43]. In Fig. 2a, we show the performances of the proposed distributed scheme and the existing agorithm, and the upper bound of the optima soution which can be obtained in a centraized manner using Agorithm. We use the MATLAB optimization toobox to obtain this upper bound. We pot the average achievabe data rate for the UEs versus the

2 Average achievabe data rate bps Efficiency %.8.6.4.2 0.8 0.6 0.4 0.8 0.6 0.4 0.2 0 x 0 6 Optima upper bound Proposed scheme Existing agorithm 5 8 2 24 27 30 33 Tota number of UEs a Proposed scheme Existing agorithm 5 8 2 24 27 30 33 Tota number of UEs b Fig. 2. a Average achievabe data rate for optima upper bound, distributed stabe matching and existing agorithm. b Efficiency of the proposed soution and the existing agorithm. Tota number of UEs i.e., C +D are varied from 9 + 6 = 5 to 5 + 8 = 33. D r,d and D d,d are assumed to be 50 m. tota number of UEs. The average data rate is given by u {C D} C+D R ach u R avg =, where Ru ach is the achievabe data rate for UE u. Note that, for a given number of RBs, increasing the number of UEs decreases the data rate. Reca that, the compexity of both the proposed and reference schemes is inear with the number of RBs and UEs; and for the optima soution, the compexity is cubic to the number of RBs and UEs. As can be seen from this figure, the proposed approach outperforms the existing agorithm and performs cose to the optima soution. In order to obtain more insights into the performance, in Fig 2b, we pot the efficiency of the proposed scheme and existing agorithm for different number of UEs. Simiar to [44, Chapter 3], we measure the efficiency as η = R R optm, where R optm is the network sum-rate for optima soution. The parameters R prop and R exst denote the data rate for the proposed and existing schemes, respectivey, which are used to cacuate the corresponding efficiency metric η prop and η exst. The coser the vaue of η to, the nearer the soution is to the optima soution. Ceary, the efficiency of the existing agorithm is ower compared to the proposed scheme. From the figure we observe that even in a dense network scenario i.e., C + D = 5 + 8 = 33 the proposed scheme performs 80% cose to the optima soution compared to 60% for the existing agorithm; however, with much ess computationa compexity. 2 Impact of reaying: We compare the performance of the proposed method for reay-aided D2D communication with a conventiona underay D2D communication scheme. In this scheme [6], an RB aocated to CUE can be shared with at most one D2D ink. The D2D UEs share the same RBs aocated to a CUE using Agorithm 3 and communicate directy with their peers without a reay if the data rate requirements for both the CUEs and D2D UEs are satisfied; otherwise, the D2D UEs refrain from transmitting. We refer to this underay D2D communication scheme [6] as the reference scheme. Notice that this conventiona e.g., direct D2D communication approach can save haf of the RBs. Therefore, as mentioned in footnote 3, data rate of in the reference scheme is given by Ru ref = N B RB og 2 + P u n γ u n. On the contrary, data rate of in the proposed reay-aided approach is given by Ru prop = N 2 xn B RB og 2 +, γn,,. i Average achievabe data rate vs. distance between D2D UEs: The average achievabe data rates of D2D UEs for both the proposed and reference schemes are iustrated in Fig. 3. Athough the reference scheme outperforms when the distance between the D2D UEs is sma i.e., d < 40 m, our proposed approach, which uses reays for D2D traffic, can greaty improve the data rate especiay when the distance increases. This is due to the fact that when the distance increases, the performance of direct communication deteriorates due to increased signa attenuation. Besides, when the D2D UEs share resources with ony one CUE, the spectrum may not be utiized efficienty, and therefore, the achievabe rate decreases. As a resut, the gap between the achievabe rate with our proposed agorithm and that with the reference scheme widens when the distance increases. ii Gain in aggregate achievabe data rate vs. varying distance between D2D UEs: The gain in terms of aggregate achievabe data rate under both uncertain and perfect CQI is shown in Fig. 4. We cacuate the rate gain as foows: R gain = Rprop R ref R ref 00%, where R prop and R ref denote the aggregate data rate for the D2D UEs in the proposed scheme and the reference scheme, respectivey. The figure shows that, compared to direct communication, with the increasing distance between D2D UEs, reaying provides considerabe gain in terms of achievabe data rate and hence spectrum utiization. As expected, the gain reduces under channe uncertainty since the agorithm becomes cautious against channe fuctuations and aocates RBs and power accordingy to protect the receiving nodes in the network. Note that there is a trade-off between performance gain and robustness against channe uncertainty. For exampe, when the distance D r,d = 50 m, the performance gain of reaying under perfect CQI is 30%. In the case of uncertain CQI,