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networks by reusing cellular resources. The main drawback of underlay in-band D2D lies in the interference caused by D2D users to cellular communications.
IEEE ICC 2015 - Mobile and Wireless Networking Symposium 1

Joint Channel and Power Allocation in Underlay Multicast Device-to-Device Communications Hadi Meshgi1 , Dongmei Zhao1 and Rong Zheng2 1

Department of Electrical and Computer Engineering, McMaster University 2 Department of Computing and software, McMaster University

Abstract: In this paper, we present a framework of resource allocations for multicast device-to-device (D2D) communications underlaying a cellular network. The objective is to maximize the sum throughput of active cellular users (CUs) and feasible D2D groups in a cell, while guaranteeing a certain level of the signal-to-interference-plus-noise ratio (SINR) for both the CUs and D2D groups. We formulate the problem of power and channel allocations as a mixed integer nonlinear programming (MINLP) problem where each D2D group can reuse the channel of at most one CU and each CU can share their resources with at most one D2D group. A maximum weight bipartite matching based scheme is developed to assign the optimal channel for each feasible D2D group to reuse. A heuristic algorithm is then proposed which has less complexity compared to the matching algorithm. The performance of both schemes is evaluated through simulations. Numerical results demonstrate that the proposed heuristic scheme outperforms other heuristic schemes in the literature and can achieve closeto-optimal performance. I. I NTRODUCTION Device-to-Device (D2D) communication is a technology component for Long Term Evolution-Advanced (LTE-A) of the Third Generation Partnership Project (3GPP) [2]. In D2D communication, cellular users (CUs) in close proximity can exchange information over a direct link rather than transmitting and receiving signals through a cellular base station (BS). D2D users communicate directly while remaining controlled under the BS. From a technical perspective, using D2D communication may provide several benefits [3], [4]. First, D2D user equipments (UEs) may enjoy high data rate and low end-to-end delay due to the short-range direct communication. Second, proximate UEs can save energy and resources when communicating directly with each other. Third, offloading cellular traffic from the BS and other network components to a direct path between UEs will reduce the network load and increase its effective capacity. The majority of the literature in D2D communications uses the cellular spectrum for both D2D and cellular communications (i.e. in-band D2D) [5]. In-band D2D can be categorized into underlay and overlay [6] . Underlay inband D2D can improve the spectrum efficiency of cellular networks by reusing cellular resources. The main drawback of underlay in-band D2D lies in the interference caused by D2D

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users to cellular communications. Thus, efficient interference management and resource allocation is required to provide a target level of performance [7], [8]. In order to avoid this interference issue, it has also been proposed to dedicate part of the cellular resources to D2D communications in overlay inband D2D. In this case, designing a resource allocation scheme is very important to avoid wastage of any dedicated cellular resources [9]. Other works consider out-of-band instead of inband D2D communications so that the cellular spectrum would not be affected by D2D communications [10]. Most of the work in D2D resource allocation targets unicast scenario where multiple D2D pairs reuse the resources of CUs. They consider either throughput maximization [5] or quality-of-service (QoS) requirements [4]. The works in [11], [1], [12] consider both throughput and QoS simultaneously. In [11], throughput is maximized for a network with a single D2D pair and a single CU subject to spectral efficiency restrictions and energy constraints. There are few works for scenarios with multiple D2D users and CUs. For example, the QoS requirements for both CUs and D2D users have been investigated in [1] and [12]. In [1], a greedy heuristic algorithm has been proposed to solve the MINLP resource allocation problem that can decrease interference to the cellular network and maximize the total throughput. The authors in [12] present a framework of resource allocation for D2D communications underlaying cellular networks to maximize the overall network throughput of existing CUs and admissible D2D pairs while guaranteeing the QoS requirements for both CUs and D2D pairs. A scheme based on maximum weight bipartite matching was proposed to determine a specific CU partner for each admissible D2D pair. To the best of our knowledge, no work has been done to address resource management for D2D multicast communication. Multicast D2D transmissions, where the same packets for a UE are sent to multiple receivers, are important for scenarios such as multimedia streaming, device discovery, and public safety. Specially, D2D multicast communications are required features in public safety services like police, fire and ambulance [2]. Compared to communicating with each receiver separately in unicast D2D, multicast D2D transmission reduces overhead and saves resources. However, unlike the more commonly studied unicast D2D (see e.g. [11][12]), multicast D2D has its own challenges. When all users within a multicast group receive the same packets, the data rates of different receivers are different because of diverse link

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IEEE ICC 2015 - Mobile and Wireless Networking Symposium 2

conditions between each receiver and the transmitter. If the transmitter transmits at a rate higher than the maximum rate that a user can handle, the user cannot decode any of the transmitted data at all. Therefore, a common approach is to transmit at the lowest rate of all users within a group determined by the user with the worst channel condition. This assures that multicast services can be provided to all the subscribed users. On the one hand, as all multicast users within a group receive the same data rate, the total sum rate grows with the number of active users of the group. On the other hand, the lowest transmission rate typically decreases as the number of users increases since it is based on the user with the Least Channel Gain (LCG) [13]. In this paper, we consider multicast D2D communications underlaying cellular networks and present a joint power and channel allocation scheme to maximize the total throughput of all CUs and D2D groups within a cell. We formulate the problem of power and channel allocation as an MINLP problem where one D2D group can reuse the channel of one CU and the channel of each CU can be reused only by one D2D group. Also, a certain level of SINR is considered to guarantee the QoS requirements for both CUs and D2D groups. Inspired by the work in [12], we use the maximum weight bipartite matching algorithm for the case where each D2D group can reuse the channel of at most one CU and each CU can share their resources with at most one D2D group. Then, we propose a heuristic algorithm for resource allocation with less complexity compared to the matching-based algorithm. The proposed heuristic algorithm has close-to-optimal performance and outperforms other heuristic schemes in the literature [1]. The remainder of the paper is organized as follows. In Section II the system model is described and the problem of power and channel allocation is formulated. The matchingbased optimal resource allocation is presented in Section III, and the greedy heuristic algorithm is presented in Section IV. Numerical results are demonstrated in Section V, and Section VI concludes the paper.

Define a set of binary variables ykm with ykm = 1 if the kth D2D group reuses channel m and ykm = 0 otherwise. We consider one D2D group uses at most one channel, and each channel can be reused by at most one D2D group. That is, M X

yk,m ≤ 1, ∀k ∈ K,

(1)

yk,m ≤ 1, ∀m ∈ M.

(2)

m=1 K X k=1

The channel quality of receiver d in the kth D2D group at channel m is given by D2D βk,m,d =

GD2D k,m,d Cell GC2D Pnoise + Pm k,m,d

,

(3)

where Pnoise is the aggregate power of background noise, GD2D k,m,d is the link gain to D2D receiver d from its desired D2D transmitter at channel m, GC2D k,m,d is the link gain from CU m Cell is the transmission to D2D receiver d in group k, and Pm power of CU m. For the kth D2D group, its transmission condition at channel m is determined by the receiver with the worst condition. Define D2D D2D βk,m = min βk,m,d .

(4)

d∈Dk

Then, the normalized transmission rate (bits/s/Hz) of the kth D2D group is given by rkD2D =

M X

D2D D2D yk,m log2 (1 + Pk,m βk,m ),

(5)

m=1 D2D where Pk,m is the transmission power of the kth D2D group transmitter at channel m. The aggregate transmission rate of the kth D2D group is given by

RkD2D = |Dk |rkD2D .

(6)

For CU m, its channel quality is given by

II. S YSTEM M ODEL AND P ROBLEM F ORMULATION We study resource allocation for multicast D2D communcations underlaying uplink (UL) transmissions in LTE networks. UL resource sharing is considered since reusing downlink resources is more difficult and less effective than reusing uplink resources in a fully loaded cellular network, as demonstrated in [14]. Consider K groups of multicast D2D users coexisting with M CUs. We assume a fully loaded cellular network scenario. That is, there are M channels, each occupied by one CU. We use m ∈ M = {1, 2, . . . , M } to index both the mth CU and the channel it occupies, and k ∈ K = {1, 2, . . . , K} to the D2D groups. We consider a single cell scenario and assume that advanced intercell interference mitigation works on top of our scheme. Each D2D transmitter multicasts messages to a group of receivers, and each D2D receiver belongs to only one multicast group. We use Dk to represent the set of D2D receivers in the kth multicast group, and |Dk | is the total number of receivers in the group. As a special case, when |Dk | = 1, the scenario becomes unicast.

Cell βm =

GCell m Pnoise +

PK

k=1

D2D GD2C yk,m Pk,m k,m

,

(7)

where GCell is the link gain of CU m to the cellular base m station and GD2C k,m is the link gain from the kth D2D transmitter to the cellular base station at channel m. Therefore, the normalized transmission rate for CU m is Cell Cell Cell Rm = log2 (1 + Pm βm ).

(8)

A threshold is set for the SINR of each D2D group and CU transmission. For the kth D2D group, D2D D2D D2D Pk,m βk,m ≥ yk,m γth ,

(9)

and for CU m, Cell Cell Cell Pm βm ≥ γth .

(10)

Given these SINR threshold constraints, we can approximate the capacity in higher SINR cases by removing the term “1”

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IEEE ICC 2015 - Mobile and Wireless Networking Symposium 3

from the logarithm functions in both (5) and (8). The maximum power constraints for CUs and D2D groups, respectively, are given by Cell Cell Pm ≤ Pmax , ∀m ∈ M,

(11)

and M X

D2D D2D Pk,m ≤ Pmax , ∀k ∈ K.

(12)

1) Feasibility check and power allocateion: In order to determine whether D2D group k can reuse channel m and to find the transmission power of the feasible D2D group and CU, we need to solve problem P2. P2 is a reduced version of Problem P1 by considering only one D2D group and one CU and the objective is to maximize the total transmission throughput.  Cell P2. max RkD2D + Rm

m=1

s.t.

Our objective is to maximize the aggregate data transmission rate of all the D2D groups and CUs. Combining (1) – (12), we formulate the joint power control and channel allocation problem to maximize the sum throughput of multicast D2D groups and cellular users as follows, K X

P1 max

RkD2D

RkD2D =

PM

Cell Rm =

m=1

K X

,

Cell Cell Pm ≤ Pmax , M X D2D D2D Pk,m ≤ Pmax .

D2D D2D yk,m |Dk | log2 (Pk,m βk,m ), ∀k ∈ K, m ∈ M

m=1

 Cell Cell log2 Pm βm , ∀m ∈ M

D2D Gk,m,d Cell GC2D , Pnoise +Pm k,m,d

∈M

∀m ∈ M

∀k ∈ K, m ∈ M, d ∈ Dk

yk,m ≤ 1, ∀m ∈ M

PM

yk,m ≤ 1, ∀k ∈ K

m=1

!

Cell Cell Cell Pm βm ≥ γth , D2D G k,m,d D2D , ∀d ∈ Dk βk,m ≤ Cell GC2D Pnoise + Pm k,m,d

!

PK

k=1

Cell Cell Pm Gm D2D GD2C Pnoise + Pk,m k,m

D2D D2D PkD2D βk,m ≥ γth ,

Cell Rm

k=1 D2D D2D PkD2D βk,m ≥ yk,m γth , ∀k ∈ K, m Cell Cell Cell Pm βm ≥ γth , ∀m ∈ M GCell Cell PK m =P βm D2D D2C , noise + k=1 yk,m Pk,m Gk,m D2D βk,m ≤

Cell Rm = log2

m=1

k=1

s.t.

+

M X

D2D D2D RkD2D = |Dk | log2 (Pk,m βk,m ),

P2 is a geometric programming problem and can be converted to a convex optimization problem using geometric programming techniques [15]. We solve problem P2 for all k and m combinations. Define a candidate channel set Ck for D2D group k. If the problem is feasible, D2D group k is admissible to channel m (i.e., eligible to use channel m), and m is added to Ck . For m ∈ Ck , denote the optimum transmission power for the kth D2D transmitter and the mth D2D Cell , respectively, and the optimum sum CU as Pk,opt and Pm,opt sum D2D throughput as Rk,m . For m ∈ / Ck, we define Pk,opt = 0, P Cell GCell

yk,m ∈ {0, 1}, ∀k ∈ K, m ∈ M Cell Cell Pm ≤ Pmax , ∀m ∈ M PM D2D D2D m=1 Pk,m ≤ Pmax , ∀k ∈ K

The above optimization problem is an MINLP problem. In general, MINLP problems are known to be NP-hard and no efficient polynomial-time solutions exist, as the complexity may increase exponentially with the problem size. However, we find out that P1 has a special structure. Next, we devise a solution to problem P1 using a matching-based algorithm.

sum Cell Cell m = log2 max , and Rk,m = Pmax Pm,opt . Pnoise 2) Maximizing total throughput: Given the achievable throughput for each D2D group to reuse each cellular channel, the following problem is used to find the optimum channel allocation in order to maximize the total throughput,

P3.

s.t.

yk,m

K X

K X M X

sum yk,m Rk,m

k=1 m=1

yk,m ≤ 1, ∀m ∈ M

k=1 M X

III. M ATCHING - BASED O PTIMAL R ESOURCE A LLOCATION In this section, we will solve the MINLP problem in P1 by dividing it into two subproblems. In the first problem, for each D2D group k and each CU m we find their transmission power so that the sum throughput of the D2D group and the CU is maximized. If this problem is feasible D2D group k is allowed to reuse the channel of CU m and we choose them as a candidate partner for the second problem; Otherwise we remove them from the list of feasible partners. The second problem is then to find the best CU partner for each D2D group among all feasible candidates so that the total throughput of all D2D groups and CUs is maximized.

max

yk,m ≤ 1, ∀k ∈ K

m=1

yk,m ∈ {0, 1}, ∀k ∈ K, m ∈ M. This problem can be cast as a maximum weight bipartite matching problem, where the D2D groups and the cellular channels each corresponds to one group of vertices in the bipartite graph, and the edge connecting D2D group k and sum channel m has a weight Rk,m . The Hungarian algorithm [17] can be used to solve the bipartite matching problem in polynomial time. To calculate the computational complexity, consider M ≥ K and the complexity of solving P2 is a function

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IEEE ICC 2015 - Mobile and Wireless Networking Symposium 4

of the size of each D2D group, f (|DK |). Therefore, the complexity of the matching-based optimal resource allocation is O(M 3 ) + O(M × K × f (|DK |)). In the next section, we propose a heuristic channel allocation algorithm with less complexity compared to the matching-based algorithm. IV. H EURISTIC C HANNEL ALLOCATION ALGORITHM In this section we propose a heuristic scheme corresponding to the MINLP problem. The pseudo code is given in Algorithm 1. To increase cellular throughput and D2D throughput, higher values of SINR is desirable. From (3) and (7), it can be deduced that having smaller GC2D k,m,d helps to reduce interference D2D from CU m to D2D group k, which will result in higher βk,m and D2D throughput. Also, higher values of GCell leads to m higher cellular throughput. Algorithm 1 tries to pair up the CU with higher GCell with the D2D group with lower GC2D m k,m,d . This is the intuition behind sorting the CUs in line 1 and finding C2D Gm,k in line 3. Here, we assume that each CU sends the channel information between itself and D2D receivers through control channels to the BS. After finding the D2D group k ∗ for CU m in line 6, we will calculate the minimum transmission D2D power for each m and its corresponding k ∗ to satisfy γth Cell and γth , respectively. Using (11) and (12) and substituting D2D Cell from (4) and (7), respectively, we can find βk,m and βm Cell D2D Pk∗ ,m and Pm as follows,  Cell Cell D2D γth Pnoise GCell +γth γth Pnoise GC2D m k∗ ,m,d  PkD2D ∗ ,m = maxd∈Dk∗ D2D Cell D2D C2D D2C Cell Gk∗ ,m,d Gm



Cell Pm

=

−γth

γth

Gk∗ ,m,d Gk∗ ,m

Cell Cell D2C γth Pnoise +PkD2D ∗ ,m γth Gk∗ ,m GCell m

(13) If the calculated transmission powers are less than maximum transmission powers, we choose k ∗ as a D2D group partner for cellular user m and solve the power allocation problem in P2. Then, k ∗ is removed from the D2D group list as in line 11, otherwise we try the next cellular user.

V. SIMULATION RESULTS We consider a single cell network where cellular users are randomly distributed in the cell. We assume that all the CUs satisfy the QoS requirement before adding D2D groups to the cell. The distance-based path loss and slow Rayleigh fading are considered for the transmission channel. The proposed algorithms have been implemented in Matlab together with the CVX convex optimization package. Default parameters used in the simulations are given in Table I. TABLE I: Default Simulation Parameters

Parameter Cell Radius (R) Number of D2D goups (K) Number of D2D receivers in each group Number of CUs (M ) Pnoise Pathloss exponent (α) D2D Pmax Cell Pmax Cell γth D2D γth D2D cluster size (r)

Value 1 km 4 3 20 -114dBm 3 20 dBm 20 dBm 10 dB 10 dB 50m

We follow the clustered distribution model in [16], where clusters of radius r are randomly located in a cell and the D2D users in each group are randomly distributed in the corresponding cluster. Four metrics are used to evaluate the performance: The sum throughput, Rsum , the D2D throughput, RD2D , the success rate, and fairness index. Rsum and RD2D are defined as follows, Rsum = RCell + RD2D , (14) X RD2D = RkD2D , (15) k∈A

RCell =

M X

Cell Rm

(16)

m=1

Algorithm 1 Resource allocation algorithm 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16:

M: List of cellular users in decreasing order of GCell m K : List of D2D groups C2D GC2D m,k = mind∈Dk Gk,m,d m=1 while K 6= ∅ or m ≤ M do k ∗ = arg mink∈K GC2D m,k , Cell Find PkD2D from (13) ∗ ,m and Pm D2D Cell Cell if PkD2D ≤ P and Pm ≤ Pmax then ∗ ,m max ∗ D2D k transmits on channel m, Cell Solve P2 to find RkD2D and Rm , ∗ ∗ K = K \ {k }, m = m + 1. else m = m + 1, end if end while

In this algorithm, problem P2 is solved M times in the worst case, and thus the complexity of the heuristic algorithm is O(M 2 ) + O(M × f (|DK |)).. This is considerably less than the complexity of matching-based algorithm.

where A is a set of D2D groups that found their CU partner. The success rate is defined as the ratio of the number of D2D groups that found their CU partner, |A|, and the total number of D2D groups. Moreover, we adapt the heuristic scheme in [1] for multicast D2D and compare it against our scheme. The process of selecting the CU candidate and D2D group in [1] is the same as our proposed algorithm. However, the uplink resource allocation algorithm in [1] assumes maximum power for the CU and the D2D group. Figs. 1 – 3 compare the performance of the three algorithms for different D2D cluster sizes (r) and different cell radii (R). From these figures, we observe that both the sum and the D2D throughput as well as the success rate decrease with the D2D cluster size. Since the channel gain of D2D link decreases when the cluster radius increases, more transmission power is required for the D2D groups to satisfy the SINR threshold constraint causing more interference to the reused CU partner. Also, it is seen that the D2D throughput and the success rate of the matching, the proposed heuristic, and the heuristic in [1] increase with the cell radius. This is because increasing the cell radius will increase the distance between the CUs and D2D receivers in groups and the distance to the BS. Hence,

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Fig. 2: D2D rate versus D2D cluster radius.

Fig. 1: Sum rate versus D2D cluster radius.

1

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the interference from CUs to the D2D receivers and the interference from D2D transmitters at the BS will be decreased. Recall that the D2D rate is the maximum throughput achieved by the admitted D2D groups. Interestingly, it is also shown that the sum throughput of the three algorithms increases with the reduced cell size. Although decreasing the cell size leads to lower D2D throughput, it increases the cellular throughput because of the increased link gain between the CUs and the base station. Therefore, by decreasing the cell size, RCell increases and RD2D decreases. Since in this set of simulations, the number of CUs is more than D2D users in the cell, RCell is moredominant and as a result Rsum increases.

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In Figs. 4 – 6 the performance of the three algorithms D2D Cell for different SINR thresholds (γth = γth = γth ) with different numbers of CUs (M ) are illustrated. It is seen that increasing the SINR threshold leads to decreasing sum rate, D2D rate, and success rate since it tightens the SINR threshold constraint and limits the chances for a D2D group to find a CU partner. It can be seen that the success rate and the D2D throughput improves slightly with increasing number of CUs since there are more potential candidates for D2D groups to reuse.

Fig. 3: Success rate versus D2D cluster radius.

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500

In all these figures, the matching-based algorithm is the optimal solution for the channel allocation and power control problem, and hence has the best performance. The proposed heuristic algorithm is suboptimal but is close to optimal. The heuristic algorithm in [1] has the worst performance among three algorithms. However, it incurs the lowest computation complexity (O(M 2 ) i.e., quadratic with respect to the number of cellular users). For example, on a desktop PC with the processor speed of 2.93 GHz and RAM of 4 GB, the total computation time to execute the matching-based algorithm, our proposed heuristic, and the adapted algorithm in [1] for 20 cellular users and 4 D2D groups with 3 receivers each are 18.41, 1.96, and 0.0035 seconds, respectively.

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R EFERENCES 160

[1] M. Zulhasnine, C. Huang, and A. Srinivasan “Efficient resource allocation for device-to-device communication underlaying LTE network”, in Proc. IEEE Int. Conf. on Wireless and Mobile Computing, Networking and Commun, 2010, pp. 368-375. [2] “3rd generation partnership project; technical specification group SA; study on architecture enhancements to support proximity services (ProSe) (Release 12)”, TR 23.703 V0.4.1, June 2013. [3] X. Lin, J. G. Andrews, and A. Ghosh, “A comprehensive framework for device to device communications in cellular networks”, submitted to IEEE Journal on Selected Areas in Communications, May 2013. Available at arXiv preprint arXiv:1305.419. [4] G. Fodor, E. Dahlman, G. Mildh, S. Parkvall, N. Reider, et al.,“Design aspects of network assisted device-to-device communications”, IEEE Commun. Mag., vol. 50, no. 3, pp. 170-177, 2012. [5] K. Doppler, M. Rinne, C. Wijting, C. Ribeiro, and K. Hugl,“Deviceto- device communication as an underlay to LTE-Advanced networks”, IEEE Commun. Mag., vol. 47, no. 12, pp. 42-49, 2009. [6] A. Asadi, Q. Wang, V. Mancuso, “A survey on device-to-device communication in cellular networks”, IEEE Communications Surveys and Tutorials, vol, no. 4, pp. 1801-1819, 2014. [7] T. Peng, Q. Lu, H. Wang, S. Xu, and W. Wang,“Interference avoidance mechanisms in the hybrid cellular and device-to-device systems”, in Proc. of IEEE PIMRC, 2009, pp. 617-621. [8] P. Hanis et al., “Interference-aware resource allocation for device-todevice radio underlaying cellular networks”, IEEE 69th Vehic. Tech. Conf. Spring, 2009. [9] Y. Pei and Y.-C. Liang, “Resource allocation for device-to-device communication overlaying two-way cellular networks”, IEEE Transactions on Wireless Communications, vol. 12, no. 7, pp. 3346?3351, Jul. 2013. [10] A. Asadi, V. Mancuso, “WiFi Direct and LTE D2D in action”, Wireless Days (WD), 2013 IFIP , pp. 1-8, 13-15 Nov. 2013 [11] C. Yu, K. Doppler, C. Ribeiro, and O. Tirkkonen,“Resource sharing optimization for device-to-device communication underlaying cellular networks”, IEEE Trans. Wireless Commun., vol. 10, no. 8, pp. 27522763, 2011. [12] D. Feng, L. Lu, Y. Yuan-Wu, G. Y. Li, G. Feng, and S. Li, “Device-todevice communications underlaying cellular networks”, IEEE Transactions on Communications, vol. 61, no. 8, pp. 3541-3551, August 2013. [13] A. Richard, A. Dadlani, and K. Kim,“Multicast Scheduling and Resource Allocation Algorithms for OFDMA-Based Systems: A Survey”, IEEE Comm. Surveys and Tutorials, vol. 15, no. 1, pp. 240?254, 2013. [14] K. Doppler, M.P. Rinne, P. Janis, C.B. Ribeiro, and K. Hugl, “Deviceto- device communications; functional prospects for LTE-Advanced networks”, IEEE ICC, pp. 1-6, June 2009. [15] M. Chiang,“Geometric programming for communication systems”, Foundations and Trends of Communications and Information Theory, vol. 2, no. 1-2, Aug. 2005. [16] B. Kaufman and B. Aazhang,“Cellular networks with an overlaid device to device network”, in Proc. 2008 IEEE Asilomar Conf. on Signals, Syst. and Comput., pp. 1537-1541. [17] H. Khun “The Hungarian method for the assignment problems”, Naval Research Logistics Quarterly 2, pp.83-97 1955.

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Fig. 6: Success rate versus γth .

VI. C ONCLUSIONS In this paper, we have considered joint power and channel allocation for multicast D2D communications sharing uplink resource in a fully loaded cellular network. To maximize the overall throughput while guaranteeing the QoS requirements of both CUs and D2D groups, we formulate the optimization problem and find the optimum solution using maximum weight bipartite matching. A low-complexity heuristic algorithm is also proposed and is shown via simulations to have close to optimal performance. Both schemes outperform an adapted heuristic algorithm proposed in the literature for unicast communication. In this work, each channel can be allocated to only one D2D link. Consequently, the radio resources in the cellular network may not be efficiently utilized. To increase the spectrum utilization, as our future work, we will investigate scenarios where multiple D2D links are allowed to share the same resource with the cellular users as long as D2D transmissions are not harmful to the cellular transmissions.

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