 Research
 Open Access
CellularD2D resource reuse algorithms based on proportional fairness
 Wu Zheng^{1}Email authorView ORCID ID profile,
 Jing Hu^{2},
 Chen Liu^{2} and
 Youhua Fu^{2}
https://doi.org/10.1186/s1363801709539
© The Author(s). 2017
 Received: 10 April 2017
 Accepted: 28 September 2017
 Published: 24 October 2017
Abstract
Devicetodevice (D2D) communication can improve coverage, spectrum efficiency, and energy efficiency based on the current cellular network architecture. The fair scheduling for D2D communication in the orthogonal frequency division multiple accessbased cellular network is studied in this paper. Based on the proportional fairness criteria, the optimization objective is formulated to maximize the sum of the relative achievable rate of both cellular users (CUs) and D2D pairs. It is a hybrid optimization problem since it involves in both the resource reuse and the transmit power control. A practical proportional fairness scheduling (PFS) algorithm is proposed. Firstly, the admissible area is decided according to the transmit power limitation of terminals and the quality of service (QoS) requirement. Next, we seek the optimal transmit power combination for the CU and D2D pair in the admissible area and it is demonstrated that the optimal value can be selected from several points on the border of the admissible area. Lastly, when multiple subchannels can be employed for scheduling, based on the generated bipartite graph, Hungarian algorithm is adopted to realize maximum matching, i.e., select reusable CUs and D2D pairs to maximize the sum of the relative achievable rates. The simulation results show that PFS algorithm can guarantee the excellent performance of both throughput and fairness.
Keywords
 D2D
 Resource reuse
 Proportional fairness
 Relative achievable rate
1 Introduction
Devicetodevice (D2D) communication refers to the direct communication between proximity terminals without the involvement of the base station. Due to the frequency reuse employed by shortdistance transmission, the network capacity is greatly increased and the traffic load across the base station and core network is reduced, which leads to lower deployment costs and operating costs for mobile network operators [1, 2]. As one of the key techniques of 4th and 5th generation mobile network architecture evolution, D2D communication can improve system throughput and energy efficiency, especially for emergency communication and the typical mobile Internet applications, e.g., social networks. When the D2D user reuses the resources allocated to the cellular user (CU), the interference yields and thus the resource scheduling scheme needs to be implemented to protect CUs and improve overall performance.
Recently, several interference control and resource scheduling schemes are developed for D2D communication. Although interference control for D2D communication appears to the natural extension of that for traditional cellular communication, the key difference lies in the predefined quality of service (QoS) guarantee required for CUs. Without enforcing the QoS constraints in the throughput maximization, the D2D user is allocated to more resources for its shortdistance transmission and better transmission conditions, which causes the unfairness for CUs [3]. Therefore, the target of interference control is to maximize the throughput on the condition of satisfying the signal to interference and noise ratio (SINR) requirement [4, 5]. Based on QoS guarantee and maximum transmit power limitation, the maximum sum rate criterion is employed to set the optimal power and further decide whether the D2D users and the CUs can reuse the same resources or not [5]. Three D2D working modes, classified as cellular, orthogonal, and nonorthogonal resource reuse, are defined, and the resource allocation, power control, and mode selection are utilized to maximize throughput [6, 7]. In detail, given the conditions of no imperfect channel state information, power, and outage probability limitation, the maximization of ergodic sum rate is obtained [8]. Both centralized and distributed power control algorithms are investigated to limit the interference introduced by D2D communication and thus to guarantee the coverage probability of CUs, and also in the same recourse, D2D links are reused as much as possible [9]. The aforementioned papers just devote to maximize the throughput as the target of resource scheduling, which usually bring the results that some users occupy the resources for a long time and others lose the opportunities to be served, and thus the fairness among the users is a key performance index in order to improve the degree of satisfaction of each user. For the general multiuser communication, there are various fairness metrics, e.g., proportional fairness [10, 11], maxmin fairness [12], and channel access fairness [13]. The concept of proportional fairness is introduced and applied to schedule D2D users in the multicell scenario in [14], where a stepwise resource allocation mode is employed, i.e., a CU is first selected based on the proportional fairness criterion, and then a D2D pair is randomly selected to reuse the resource. In [15], fractional frequency reuse is adopted to depress the interference between CUs and D2D users, and the proportional fairnessbased resource allocation is implemented by optimizing the transmit power of D2D pair under the premise of the fixed transmit power of the CU. In [16], only D2D communications are considered and there is no resource reuse between the CUs and D2D pairs. The resources are divided into the dedicated resource area and the contention resource area. A twostep method is employed for scheduling D2D pairs: classify the D2D pairs into the dedicated resource area and contention resource area based on the computed SINR threshold and use the proportional fairness criterion to perform resource assignment in the dedicated resource area and in the contention resource area implement the resource reuse among multiple D2D pairs to achieve reuse gain. The theoretical analysis has demonstrated that the proportional fairness is the optimal tradeoff between throughput and fairness, which has the explanation from game theory [17]. The current research on the proportional fairness scheduling of the CUs and D2D pairs is usually simplified by some additional limitations, e.g., just the proportional fairness for CUs or the fixed transmit power for the CU. Inspired by these works, we present a framework of resource allocation for D2D communications underlaying cellular networks to maximize the sum of relative achievable rate of the CUs and D2D pairs while guaranteeing the QoS requirement of both CUs and D2D pairs. The framework includes three parts. First, the proportional fairness criterion is applied to achieve the sum of relative achievable rate of the CU and D2D pair. Then, the optimal power control scheme is investigated for each D2D pair and its possible CU partner to maximize the sum of relative achievable rate. Lastly, a maximum weight bipartite matchingbased scheme is designed to determine a specific CU partner for each D2D pair when multiple subchannels are employed.
The remainder of this paper is organized into the following sections. The scenario and the system model are introduced in Section 2. The optimization problem formulation of resource reuse for one CU vs. one D2D user based on the proportional fairness criterion is proposed in Section 3, and also the procedures of the optimal solution, with the name of proportional fairness scheduling (PFS) algorithm, are derived. In Section 4, the scheme for the resource reuse between multiple CUs and multiple D2D users is described when there are multiple subchannels. The system simulation for PFS algorithm is realized and the results are analyzed in Section 5. And the conclusion remarks are summarized in Section 6.
2 The scenario and system model

■ Mode 1: The MS requests transmission resources from the BS. The BS schedules transmission resources for transmission of D2D control information and data

■ Mode 2: A MS on its own selects resources from resource pools, which are denoted by the base station beforehand, and performs transport format selection to transmit D2D control information and data.
Mode 1, employed in our paper, is the fully centralized method, where the BS should know all the channel state information, and the MS needs to report the channel state information to the BS. In Mode 1, some extra overhead on control information, including the channel state information reporting and resource request message, has to be paid; the fully centralized scheduling method implemented by the BS can better match CUs and D2D pairs; and the appropriate resource allocation and power control can improve the transmit efficiency for the traffic data. When the mobile terminals have the feature of the static or low mobility, the channel state information can keep unchanged within a long time interval and thus the overhead on control information can be alleviated. In addition, we assume that there are K orthogonal subchannels which are scheduled for CUs and D2D pairs based on the channel state information and QoS requirement.
3 Fair resource reuse for CU and D2D pair
3.1 Fairness index and optimization problem
 (1)
According to constraints (6a)–(6d), compute the admissible area for CU i and D2D pair j when resource reusing occurs. If there exists such p _{ i } and p _{ j } to satisfy the aforementioned constraints, step into (2); else CU i and D2D pair j cannot be reused.
 (2)
Seek the optimal power combination \( \left({p}_i^{\ast },{p}_j^{\ast}\right) \) to make \( R\left({p}_i^{\ast },{p}_j^{\ast}\right)=\max R\left({p}_i,{p}_j\right) \).
3.2 Admission control for CU vs. D2D pair
Observing Fig. 2a, point A is outside of the square area, which represents CU i and D2D pair j cannot share the same resource based on constraints (6a)–(6d). In Fig. 2b–d, point A is within the square area and, any point in the shaded area, denoted by the transmit power for CU i and D2D pair j, satisfies constraints (6a)–(6d). Therefore, the shaded area is called the admissible area.
3.3 Optimal transmit power control for CU vs. D2D pair
The admissible area is denoted by Ω, corresponding to the shadow region of the three possible scenarios represented in Fig. 2b–d. Here, we investigate how to allocate power for the CU and for its reuse partner, D2D transmitter, to maximize the relative achievable rate, i.e., seek the optimal power combination \( \left({p}_i^{\ast },{p}_j^{\ast}\right) \) within Ω to make \( R\left({p}_i^{\ast },{p}_j^{\ast}\right)=\max R\left({p}_i,{p}_j\right) \). It is demonstrated in this paper that only several power combinations need to be computed and compared, and thus the optimal solution can be achieved, which avoids the complex optimization algorithms.
Proposition 1. When CU i and D2D pair j reuse the resource, the optimal transmit power satisfies \( {p}_i^{\ast }={p}_{i,\max } \) or \( {p}_j^{\ast }={p}_{j,\max } \) [21].
Hence, the solution of (6) will have \( {p}_i^{\ast }={p}_{i,\max } \) or \( {p}_j^{\ast }={p}_{j,\max } \), i.e., the optimal power combination occurs in the border of Ω.
Proposition 2. Under the premise of p _{ j } = p _{ j, max} or p _{ i } = p _{ i, max}, i.e., one variable is constant, the function R(p _{ i }, p _{ j }) is convex.
Since E ≤ 0, we have
Now, when CU reuses with D2D pair, the optimal power combination \( \left({p}_i^{\ast },{p}_j^{\ast}\right) \) satisfied with constraints (5a)–(5d) can be achieved.
4 Resource matching for multiple CUs and D2D pairs
In the mathematical field of graph theory, a bipartite graph is a graph whose vertices can be divided into two disjoint sets \( \mathbb{U} \) and \( \mathbb{V} \) (that is, \( \mathbb{U} \) and \( \mathbb{V} \) are each independent sets) such that every edge connects a vertex in \( \mathbb{U} \) to one in \( \mathbb{V} \). A bipartite graph is often denoted by \( \mathbb{G}=\left(\mathbb{U},\mathbb{V},\mathbb{E}\right) \), whose partition has the parts \( \mathbb{U} \) and \( \mathbb{V} \), with \( \mathbb{E} \) denoting the edges of the graph. Compared with our proposed problem of reusing resources between CU and D2D pair, the set of active CUs, denoted by ℂ, and the set of D2D pairs, denoted by \( \mathbb{D} \), are disjoint sets, we have \( \mathbb{C}\cap \mathbb{D}==\varnothing \). If CU i and D2D pair j reuse the resource, there exists an edge to connect the vertex C _{ i } and D _{ j }, which belong to ℂ and \( \mathbb{D} \), respectively; else there is no edge between C _{ i } and D _{ j }. Therefore, the problem can be expressed by a bipartite graph.
The classic KuhnMunkres (KM) algorithm [22] can be employed to solve the aforementioned bipartite graph maximum weight matching.
5 Numerical results
Simulation parameters
Parameter  Value 

Cell radius  500 m 
Maximum distance for the D2D pair  200 m 
Carrier frequency (f _{c})  2 GHz 
Uplink bandwidth  10 MHz 
Number of subchannels  10 
Maximum transmit (Tx) power  23 dBm 
Path loss model  \( {\displaystyle \begin{array}{l}{PL}_{\mathrm{LOS}}=40{\mathrm{log}}_{10}(d)+7.5617.3{\mathrm{log}}_{10}\left({h}_1^{\prime}\right)17.3{\mathrm{log}}_{10}\left({h}_2^{\prime}\right)+2.7{\mathrm{log}}_{10}\left({f}_c\right)\\ {}{PL}_{\mathrm{NLOS}}=\left(44.96.55{\mathrm{log}}_{10}\left({h}_1\right)\right)\cdot {\mathrm{log}}_{10}(d)+5.83{\mathrm{log}}_{10}\left({h}_1\right)+9.78+34.97{\mathrm{log}}_{10}\left({f}_{\mathrm{c}}\right)\end{array}} \) where the probability of LOS can be expressed by \( {P}_{\mathrm{LOS}}=\min \left(\frac{18}{d},1\right)\cdot \left(1{e}^{\frac{d}{36}}\right)+{e}^{\frac{d}{36}} \) (1) Cellular communication: The distance between CU and BS is denoted by d; h _{1} = 10m, \( {h}_1^{\prime }=9\mathrm{m} \), and \( {h}_2^{\prime }=0.5\mathrm{m} \) denote the antenna height of BS, the effective height of BS, and the effective height of CU, respectively; (2) D2D communication: The distance between D2D pair is denoted by d; h _{1} = 1.5m, \( {h}_1^{\prime }=0.5\mathrm{m} \) and \( {h}_2^{\prime }=0.5\mathrm{m} \) denote the antenna height of D2D receiver, the effective height of D2D receiver, and the effective height of D2D transmitter, respectively. 
Shadowing  Lognormal distribution with standardization deviation 7 dB 
Multiplepath fading  Rayleigh distribution 
In Fig. 5, it is indicated that more than 10% CUs and D2D pairs are not served when max C/I algorithm is applied. As for PFS algorithm, the ratio is close to 0 for both CUs and D2D pairs. More users can get the serving opportunities, and thus, the fairness is improved.
Numerical analysis for PFS and max C/I
PFS algorithm  Max C/I algorithm  

Average throughput mean  Average throughput variance  Average Tx power (dBm)  Average throughput mean  Average throughput variance  Average Tx power (dBm)  
CUs  3.18  2.69  22.78  4.20  5.51  22.98 
D2D pairs  6.29  4.47  21.43  8.17  7.05  22.41 
The mean indicates the average throughput for all the CUs or D2D pairs within the 2000 scheduling slots for 1000 times of user distribution, and the variance denotes the deviation level from the mean value, where the smaller variance implies better fairness. PFS algorithm achieves the gain of fairness at the expenditure of some throughput. As for CUs, the improvement of fairness is about 51% while the loss of throughput is about 24%; and for D2D pairs, 37 vs. 23%. The average transmit power is also listed in Table 2, which indicates the similar power consumption for both algorithms.
Numerical analysis for PFS with different power control methods
PFS optimal power control  PFS maximum power control  

Average throughput mean  Average throughput variance  Average Tx power (dBm)  Average throughput mean  Average throughput variance  Average Tx power (dBm)  
CUs  3.18  2.69  22.78  3.35  3.44  23.00 
D2D pairs  6.29  4.47  21.43  7.61  6.09  23.00 
Compared with the maximum power transmission, for CUs the PFS algorithm with the optimal power control can achieve 22% fairness gain while yield 5.1% throughput loss; For D2D pairs, it can achieve both 27% fairness gain and 17% throughput loss. It can be predicted that in the multicell environments the power control is much more important because of the extra interference from the adjacent cells.
The simulation results can be explained as follows. Because the distance between the D2D pair is shorter and LOS path existed with greater possibility, the channel gain between the D2D pair is usually better than that between the CU and the base station, and generally, the data rate of D2D pair is larger than that of CU, which can be observed in Fig. 6. For PFS algorithm, the objective is to maximize the sum of relative achievable rate, denoted by max(α _{ i } R _{ i } + β _{ j } R _{ j }), where the relative achievable rate of the D2D pair, β _{ j } R _{ j }, is the main contribution. Therefore, the transmit opportunities of CUs are blocked. In this sense, when more users, especially more D2D pairs, the proportional fairness becomes the proportional fairness dedicated for D2D pairs.

■ The D2D pairs can only be scheduled in certain subchannels, and some subchannels are dedicated for the scheduling of CUs;

■ Limit the maximum transit power of D2D pairs, it may be realized by system broadcast information to indicate the D2D pairs in the cell. And the permitted transmit power of D2D pairs can be adjusted according to the traffic load in the cell.
As for the complexity of the algorithm, for the scenario of N active CUs and M pairs of D2D users, the complexity to compute the optimal power combination is O(N ⋅ M) and that of bipartite graph maximum weight matching employing KM algorithm is O(N ^{3}).
6 Conclusions and discussions
In this paper, we investigated on the scheduling for CU and D2D pair that share the uplink subchannel resource in the cellular network. In order to achieve the excellent tradeoff between throughput and fairness, the PFS algorithm is proposed based on the proportional fairness criteria, which divides into the admissible area decision, solving the optimal power combination, and utilizing bipartite graph for resource reuse in multiple subchannels. Several factors, e.g., sleeping feature and hybrid automatic repeat request process limitation, restrict the number of active users in a scheduling slot, and the complexity of PFS algorithm can be acceptable. The simulation results also verify that the PFS algorithm can guarantee the good throughput and fairness for both CUs and D2D pairs.
Declarations
Acknowledgements
This work is supported by the Natural Science Foundation of Jiangsu Province of China under Grant BK21030874, the Natural Science Foundation of China under Grants 61372126 and 61302101, and Jinling Institute of Technology Project No. 2015b29.
Authors’ contributions
WZ carried out the research on D2D communication, participated in the proposed algorithm design and the simulation, and drafted the manuscript. JH participated in the simulation. CL conceived of the study, participated in the algorithm design, and helped in drafting the manuscript. YF helped in drafting the manuscript. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
 P Schulz, M Matthe, H Klessig, et al., Latency critical IoT applications in 5G: perspective on the design of radio interface and network architecture. IEEE Commun. Mag. 55(2), 70–78 (2017)View ArticleGoogle Scholar
 G Fodor, E Dahlman, G Mildh, et al., Design aspects of network assisted devicetodevice communications. IEEE Commun. Mag. 50(3), 170–177 (2012)View ArticleGoogle Scholar
 CH Yu, O Tirkkonen, K Doppler, C Ribeiro, in Proc. IEEE Inter. Conf. Commun. (ICC), Power Optimization of DevicetoDevice Communication Underlaying Cellular Communication (IEEE, Dresden, 2009), pp. 1–5Google Scholar
 C Xu, L Song, Z Han, Q Zhao, X Wang, X Cheng, B Jiao, Efficiency resource allocation for devicetodevice underlay communication systems: a reverse iterative combinatorial auction based approach. IEEE J. Sel. Areas. Commun. 31(9), 348–358 (2013)View ArticleGoogle Scholar
 D Feng, L Lu, Y Wu, GY Li, G Feng, S Li, Devicetodevice communications underlaying cellular networks. IEEE Trans. Commun. 61(8), 3541–3551 (2013)View ArticleGoogle Scholar
 CH Yu, K Doppler, CB Ribeiro, O Tirkkonen, Resource sharing optimization for devicetodevice communication underlaying cellular networks. IEEE Trans. Wirel. Commun. 10(8), 2752–2763 (2011)View ArticleGoogle Scholar
 HJ Chou, RY Chang, Joint mode selection and interference management in devicetodevice communications underlaid MIMO cellular networks. IEEE Trans. Wirel. Commun. 16(2), 1120–1134 (2017)View ArticleGoogle Scholar
 L Wang, H Tang, H Wu, GL Stuber, Resource allocation for D2D communications underlay in Rayleigh fading channels. IEEE Trans. Veh. Technol. 66(2), 1159–1170 (2017)View ArticleGoogle Scholar
 N Lee, X Lin, JG Andrews, RW Heath, Power control for D2D underlaid cellular networks: modeling, algorithms and analysis. IEEE J. Sel. Areas. Commun. 33(1), 1–13 (2015)View ArticleGoogle Scholar
 D Calabuig, JF Monserrat, N Cardona, Proportionally fair scheduler for heterogeneous wireless systems. Trans. Emerg. Tel. Technol. 23(1), 1–5 (2011)Google Scholar
 M Ayhan, Y Zhao, HA Choi, in Proc. IEEE Global Commun. Conf. (GLOBECOM), Utilizing Geometric Mean in Proportional Fair Scheduling: Enhanced Throughput and Fairness in LTE DL (IEEE, Washington, 2016), pp. 1–6Google Scholar
 QD Vu, KG Nguyen, M Juntti, in Proc. IEEE Global Commun. Conf. (GLOBECOM), MaxMin Fairness for Multicast Multigroup Multicell Transmission under Backhaul Constraints (IEEE, Washington, 2016), pp. 1–6Google Scholar
 N Torabi, BS Ghahfarokhi, in Proc. 2014 4th Int. Conf. Comput. Knowl. Eng. (ICCKE), A TDMABased Channel Access Scheme for Achieving Fairness in InterVehicle Communications (IEEE, Mashhad, 2014), pp. 747–752Google Scholar
 RL Batista, CF Silva, JM Silva, TF Maciel, FR Cavalcanti, in Proc. IEEE Wireless Commun. Netw. Conf. Workshops (WCNCW), What Happens with a Proportional Fair Cellular Scheduling when D2D Communications Underlay a Cellular Network (IEEE, Istanbul, 2014), pp. 260–265Google Scholar
 ST Shah, J Gu, SF Hasan, MY Chung, SCFDMAbased resource allocation and power control scheme for D2D communication using LTEA uplink resource. EURASIP J. Wireless Commun. Netw. 2015, 137 (2015)View ArticleGoogle Scholar
 HH Wang, JC Chen, ZN Liu, in Proc. IEEE Global Commun. Conf. (GLOBECOM), Resource Allocation in CentralControlled DevicetoDevice Communications Networks (IEEE, Atlanta, 2013), pp. 4871–4876Google Scholar
 F Kelly, Charging and rate control for elastic traffic. Euro. Trans. Telecommun. 8(1), 33–37 (1997)View ArticleGoogle Scholar
 3rd Generation Partnership Project (3GPP) TS 36.300 V14.3.0, Technical Specification Group Radio Access Network; EUTRA and EUTRAN; Overall Description (2017)Google Scholar
 DM Blough, G Resta, P Santi, in Proc. IEEE Conf. Comput. Commun. (INFOCOM), InterferenceAware Proportional Fairness for MultiRate Wireless Networks (IEEE, Toronto, 2014), pp. 2733–2741Google Scholar
 TD Nguyen, Y Han, A proportional fairness algorithm with QoS provision in downlink OFDMA systems. IEEE Commun. Letters. 10(11), 760–762 (2006)View ArticleGoogle Scholar
 A Gjendemsjo, D Gesbert, GE Oien, SG Kiani, in Proc. International. Symposium on Modeling and Optimization in Mobile, Ad Hoc and Wireless Networks (WiOPT), Optimal Power Allocation and Scheduling for twoCell Capacity Maximization (IEEE, Boston, 2006), pp. 1–6Google Scholar
 A Gibbons, Algorithmic Graph Theory (Cambridge University Press, Cambridge, 1985), pp. 136–147Google Scholar
 3rd Generation Partnership Project (3GPP) TR 36.843 V12.0.1, Technical Specification Group Radio Access Network; Study on LTE Device to Device Proximity Services; Radio Aspects (2014)Google Scholar