06919332

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 14, NO. 2, FEBRUARY 2015 921

Group Partition and Dynamic Rate Adaptationfor Scalable Capacity-Region-AwareDevice-to-Device Communications

Yi-Shing Liou, Rung-Hung Gau, Senior Member, IEEE, and Chung-Ju Chang, Fellow, IEEE

Abstract—In this paper, we propose using group partition anddynamic rate adaptation for scalable throughput optimizationof capacity-region-aware device-to-device communications. Weadopt network information theory that allows a receiving deviceto simultaneously decode multiple packets from multiple trans-mitting devices, as long as the vector of transmitting rates isinside the capacity region. Based on graph theory, devices arefirst partitioned into subgroups. To optimize the throughput of asubgroup, instead of directly solving an integer-linear program-ming problem, we propose using a fast iterative algorithm to selectactive devices and using aggression levels for rate adaptation basedon channel state information. Simulation results show that theproposed algorithm is scalable and could significantly outperformthe greedy algorithm by more than 50%.

Index Terms—Device-to-device communications, capacity re-gion, graph coloring, rate adaptation, channel state information.

I. INTRODUCTION

TO deal with the exponential growth of mobile traffic,device-to-device (D2D) communications have been con-

sidered one of the key techniques in the Third GenerationPartnership Project (3GPP) Long Term Evolution Advanced(LTE-A) [1], [2]. With D2D communications capability, twoadjacent devices could directly send data to each other with-out using base stations as relays. D2D communications couldincrease the throughput of cellular networks and decrease theenergy consumption of mobile devices [1]–[3]. Unlike mobilead-hoc networks based on the IEEE 802.11 standards, D2Dcommunications in cellular networks could use licensed bandsto avoid interference from devices that use unlicensed bands.An overview of D2D standardization activities in 3GPP can befound in [1].

In this paper, we study the case in which the network consistsof cellular user equipments (UEs) and devices. A cellular userequipment sends data through the base station, while a sourcedevice directly sends data to the corresponding destination

Manuscript received April 15, 2014; revised July 28, 2014; acceptedOctober 2, 2014. Date of publication October 9, 2014; date of current versionFebruary 6, 2015. This work was supported in part by the Ministry of Sci-ence and Technology, Taiwan under Grant NSC 101-2221-E-009-005-MY3.This paper was presented in part at the IEEE Wireless Communications andNetworking Conference (WCNC), Istanbul, Turkey, April 2014. The associateeditor coordinating the review of this paper and approving it for publication wasS. Cui.

The authors are with the Department of Electrical Engineering, NationalChiao Tung University, Hsinchu 300, Taiwan (e-mail: [email protected];[email protected]; [email protected]).

Digital Object Identifier 10.1109/TWC.2014.2362523

device. We take into consideration the case in which a numberof source devices could have the same destination device.We adopt network information theory [4] for multiple accesschannels. In particular, a receiving device could concurrentlyreceive/decode distinct packets from multiple transmitting de-vices as long as the transmission rate vector is inside thecapacity region. When the proposed approach is used, devicesnever severely interfere with cellular user equipments.

For D2D communications, Liou et al. [5] proposed using agraph coloring algorithm to partition devices into subgroupsand solving an integer-linear programming problem to maxi-mize the throughput of a subgroup. In this paper, we focus onscalable algorithms for large-scale networks in which solvinginteger-linear programming problems is infeasible. Our majortechnical contributions of the paper include the following. First,for each subgroup, we propose using a fast approximationalgorithm to choose active devices and dynamically adjust-ing the data transmission rates based on aggression levels.Unlike most of the previous work on D2D communications,the approximation algorithm exploits the information-theoreticcapacity region of a multiple access channel to improve thenetwork throughput. In addition, the approximation algorithmremains applicable when only partial channel state informationis available. Furthermore, we show how to extend the proposedapproach to the case in which there are both short-lived D2Dcommunication sessions and long-lived D2D communicationsessions. Our simulation results show that the proposed ap-proach could significantly outperform the greedy algorithm andcould be near-optimal.

The rest of the paper is organized as follows. In Section II, webriefly introduce related work. In Section III, we present systemmodels for D2D communications. In Section IV, for scalableD2D communications, we show how to create a conflict graphin order to partition devices into subgroups. In Section V, wefirst formulate an integer-linear programming problem for max-imizing the throughput of a subgroup. In addition, we proposea fast approximation algorithm to decide the set of active de-vices and adjust data transmission rates based on channel stateinformation. In Section VI, we show mathematical propertiesof the proposed algorithm. In Section VII, we show simulationresults that justify the usage of the proposed approach. InSection VIII, we outline an implementation and an extensionof the proposed approach in cellular networks. Our conclusionsare included in Section IX. Mathematical proofs are included inthe Appendix.

1536-1276 © 2014 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

922 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 14, NO. 2, FEBRUARY 2015

II. RELATED WORK

Resource allocation had been extensively studied for enhanc-ing the network performance. Yu et al. [6] studied the problemof optimal resource allocation with power control betweencellular connections and D2D connections. Le [7] proposed amax–min fair resource allocation algorithm for cellular usersand a resource allocation algorithm with the rate protection forcellular users and D2D users. Feng et al. [8] studied a resourceallocation problem to maximize the overall network throughputwhile guaranteeing the quality of service for both D2D usersand regular cellular users. In [9], stochastic Petri Nets were usedfor performance analysis of D2D communications with randompacket arrivals. Chai et al. [10] proposed a resource assignmentalgorithm which allows D2D users to utilize the resource ofmultiple cellular users to reduce the interference to cellularusers. Lee et al. [11] proposed a semi-distributed resourceallocation algorithm based on the multiple set covering problemto maximize the spatial reuse of radio resources. Zhang et al.proposed [12] a graph-theoretic resource allocation scheme forD2D communications in OFDM-based cellular networks. Intheir scheme, to reflect that channel responses for distinct re-source blocks could be different, an edge weight in the conflictgraph is a matrix rather than a real number.

Power control and interference management had been widelyused to guarantee the quality of service for cellular users andimprove the network throughput. Kaufman et al. [13] proposeda distributed spectrum sharing mechanism which includespower control and route determination for the D2D users.Wang et al. [14] proposed resource sharing through powercontrol for D2D users. In particular, their design goal was tomaximize the throughput of the D2D network while guaran-teeing the quality of service of cellular users. Fodor et al.[15] investigated the performance of D2D communicationswith various LTE-based power control methods. Min et al.[16] proposed an interference management scheme to improvethe reliability of D2D communications without decreasing thetransmission power of cellular user equipments. Min et al. [17]proposed an interference limited area control scheme to managethe interference from cellular connections to D2D connections.Bao and Yu [18] used the partial location information to developan interference limited area in which the resources allocated tocellular users cannot be reused by D2D users. In [19], under theassumption that power control is ideal, the transmitting powerand SINR distributions of D2D networks in which cellular usersand D2D users use orthogonal resources were derived. In thispaper, we focus on binary power control and all active sourcedevices in a time slot have the same transmission power.

Recently, game theory has been applied to networks withcellular users and D2D users. Xu et al. [20] proposed anauction-game-based resource allocation algorithm for D2Dusers to mitigate the interference between cellular users andD2D users. Zhang et al. [21] proposed a distributed coalition-game-based resource assignment for both cellular users andD2D users assuming that both types of users have the samepriority. Wang et al. [22] modeled the power control on the D2Dusers as an auction game to extend the battery life. Wang et al.[23] developed a Stackelberg-game-based resource allocation

scheme to adjust the transmitting power of D2D users andguarantee the fairness. In this paper, we study the case in whichall devices are cooperative rather than selfish. Game theory isbeyond the scope of this paper.

Various medium access control algorithms were proposed toincrease the network performance. Hakola et al. [24] proposedmeans for selecting the optimal communication mode, whenboth cellular mode and D2D mode are available. Seppala et al.[25] introduced a concept of reliable multicast for D2D com-munications underlying cellular networks. Han et al. [26] de-signed an optimal channel reusing algorithm for the singlecell scenario based on the Hungarian algorithm. Zhou et al.[27] proposed an intra-cluster D2D retransmission scheme withoptimal resource utilization. Pei and Liang [28] proposed anovel spectrum sharing protocol, which allows D2D users toassist the two-way communications between the base stationand cellular users.

Many previous works on D2D communications do not takesuccessive interference cancellation [29] into consideration. Inthis paper, we study the case in which successive interferencecancellation is used. Recently, cooperative interference cancel-lation in the context of 3GPP D2D communications has beenproposed and discussed [30]. To the best of our knowledge,our previous work [5] is the first in the literature that exploitsnetwork information theory for D2D communications. Whileit is possible to extend other previous works on D2D com-munications to benefit from network information theory, suchextensions are beyond the scope of the paper.

III. SYSTEM MODELS AND PROBLEM FORMULATION

Consider a cell in a cellular network where there is a basestation (BS) in the cell. Let C be the total number of orthog-onal channels that could be used for D2D communicationsand W be the bandwidth of a channel. The C channels areindexed by 1, 2, 3, . . . , C, respectively. Let N ≥ 2 be the totalnumber of devices in the cell. The devices are indexed by1, 2, 3, . . . , N , respectively. A device is either a source deviceor a destination device. Let Ns be the total number of sourcedevices and Nd = N −Ns be the total number of destinationdevices. Without loss of essential generality, it is assumed thatthe source devices are indexed by 1, 2, 3, . . . , Ns, respectively.Let S = {1, 2, . . . , Ns} be the set composed of the indexesof source devices and D = {Ns + 1, Ns + 2, . . . , N} be theset composed of the indexes of destination devices. A sourcedevice always has data to send and is equipped with an omnidi-rectional antenna. We focus on unicast applications in which asource device has only one destination. Let Di be the index ofdestination device associated with source device i. Accordingto network information theory [4], two or more source devicescould successfully send different data to the same destinationdevice at the same time through the same frequency band aslong as the data rate vector is inside the capacity region. LetSj be the set composed of the indexes of source devices asso-ciated with destination device j. Define Sj = {i|Di = j, 1 ≤i ≤ Ns}. Namely, Sj is composed of the indexes of sourcedevices that want to transmit data to destination device j. Notethat Ns ≥ Nd. Regarding the total number of cellular user

LIOU et al.: GROUP PARTITION AND DYNAMIC RATE ADAPTATION FOR D2D COMMUNICATIONS 923

equipments (UEs) in the network, we study two cases. In thefirst case, there is no UE. It corresponds to the case when anUE and a source device always use orthogonal channels. In thesecond case, there are C UEs. In addition, it is assumed that UEk uses channel k, ∀ 1 ≤ k ≤ C. While a cellular UE has to usea BS to relay its data, a source device directly sends data to thecorresponding destination device.

The time domain is partitioned into time slots of equal length.The length of a time slot is smaller than the channel coherencetime of the wireless channel. Let P k be the maximum trans-mission power of UE k. Let P k(t) be the transmission powerof UE k in time slot t. Let Rk(t) be the data transmissionrate of UE k in time slot t. Let hk

0(t) be the gain of channelk from UE k to the BS in time slot t. Let hk

j (t) be the gainof channel k from UE k to destination device j in time slot t.Let σ2 be the power spectral density of the background white

Gaussian noise. Define Rk = 12 · E

[W log2

(1 +

Pkhk0 (t)

σ2

)].

If W log2

(1 +

Pk ·hk0 (t)

σ2

)≥ Rk, P k(t) = P k and Rk(t) =

Rk. Otherwise, P k(t) = 0 and Rk(t) = 0.Let gki,j(t) be the gain of channel k from device i to device j

in time slot t. Let gki,0(t) be the gain of channel k from device ito the BS in time slot t. Let Pi(t) be the transmission power ofdevice i in time slot t. Let Ri(t) be the data transmission rate ofdevice i in time slot t. Let Fi(t) be the index of the channel usedby source device i to transmit data in time slot t. Define S(t) ={i|i ∈ S, Pi(t) > 0, Ri(t) > 0}. Then, S(t) corresponds to theset of source devices that are selected by the base station totransmit data in time slot t. Define Sk(t) = {i|i ∈ S, Pi(t) >0, Ri(t) > 0, Fi(t) = k}, ∀ k. Namely, Sk(t) corresponds tosource devices that use channel k to transmit data in timeslot t. Define Sk

j (t) = Sk(t) ∩ Sj . Then, Skj (t) corresponds to

source devices that use channel k to transmit data to destinationdevice j in time slot t. Define Ikj (t) = Sk(t)− Sk

j (t). Ikj (t)

corresponds to source devices that use channel k in time slott but do not want to transmit data to destination device j intime slot t. Define D(t) = ∪i∈S(t)Di and Dk(t) = ∪i∈Sk(t)Di,∀ k. If |Sk

j (t)| ≥ 2, in time slot t, through a multiple accesschannel [4], source devices with indexes in Sk

j (t) use channel kto transmit data to destination device j with interference fromsource devices with indexes in Ikj (t) and UE k. According tonetwork information theory [4], for each source device withindex in S(t) to successfully transmit data to the correspondingdestination device in time slot t, the following constraints haveto be satisfied for each (k, j), where 1 ≤ k ≤ C and j ∈ Dk(t).

Ri(t)

≤W log2

(1+

Pi(t)·gki,j(t)σ2+P k(t)·hk

j (t)+∑

u:u∈Ikj(t)Pu(t)·gku,j(t)

),

∀ i ∈ Skj (t),∑

i:i∈TRi(t)

≤W log2

(1+

∑i:i∈TPi(t)·gki,j(t)

σ2+P k(t)·hkj (t)+

∑u:u∈Ik

j(t)Pu(t)·gku,j(t)

),

∀T ⊂ Skj (t). (1)

In addition, the BS should be able to receive/decode datatransmitted from UE k in time slot t. Therefore,

Rk(t)≤W log2

(1+

P k(t) · hk0(t)

σ2+∑

u:u∈Sk(t) Pu(t)·gku,0(t)

). (2)

Let P be the maximum transmission power of a device ina time slot. In this paper, we study the case in which Pi(t) ⊂{P, 0}, ∀ i, t. Transmission power control is beyond the scopeof this paper. Then, if i ∈ S(t), Pi(t) = P . Otherwise, Pi(t) =0. In addition, it is assumed that at the beginning of time slott, with cooperation from UEs and devices, the base stationcould know the values of all channel gains, which are obtainedthrough channel estimation.

At time slot t, given the values of channel gains, to optimizethe network throughput, the base station solves the followinginteger-linear programming problem.

max∑C

k=1

∑i:i∈Sk(t)

Ri(t)

subject to

Sk(t) ⊂ {1, 2, . . . , Ns}, ∀ 1 ≤ k ≤ C,

Sα(t) ∩ Sβ(t) = ∅, ∀ 1 ≤ α < β ≤ C,

Dk(t) = Δ(Sk(t)

), ∀ 1 ≤ k ≤ C,

Rk(t) ≤ W log2

(1 +

P k(t) · hk0(t)

σ2 +∑

u:u∈Sk(t) Pu(t) · gku,0(t)

),

∀ 1 ≤ k ≤ C,

Ri(t)

≤W log2

(1+


j (t)+∑


),

∀ j ∈ Dk(t), i ∈ Skj (t),∑

i:i∈TRi(t)

≤W log2

(1+



∑u:u∈Ik

j(t)Pu(t)·gku,j(t)

),

∀ j ∈ Dk(t), T ⊂ Skj (t),

Ri(t) ≥ 0, ∀ 1 ≤ i ≤ Ns,

S(t) = ∪Ck=1S

k(t),

Ri(t) = 0, ∀ i ∈ S(t). (3)

Note that we have to find the optimal values of Sk(t), ∀ 1 ≤k ≤ C, and Ri(t), ∀ i ∈ S(t). The second constraint reflectsthat a source device uses at most one channel in time slot t.The third constraint means that Dk(t) is a function of Sk(t).The last constraint means that the data rate of source device imust be zero if it does not use any channel.


We now show that the optimization problem in (3) is aninteger-linear programming problem [31]. Define binary vari-ables Bi,k,t’s as follows. If source device i uses channel k totransmit data in time slot t, Bi,k,t = 1. Otherwise, Bi,k,t = 0.Then, Bi,k,t = 1 if and only if i ∈ Sk(t). In addition, the firsttwo constraints in (3) are equivalent to the two constraints,Bi,k,t ∈ {0, 1}, ∀ 1 ≤ i ≤ Ns, 1 ≤ k ≤ C and

∑Ck=1 Bi,k,t ≤

1, ∀ 1 ≤ i ≤ Ns. Furthermore, for each fixed collection ofBi,k,t’s, the optimization problem in (3) becomes a linearprogramming problem [31]. Therefore, the above optimizationproblem is an integer-linear programming problem.

Since an integer-linear programming problem is NP-hard ingeneral, we propose using the following three-stage approachfor scalable D2D communications. First, we create a conflictgraph to partition devices into subgroups. In particular, sourcedevices in subgroup k are candidate devices that could usechannel k to transmit data, ∀ k ∈ {1, 2, . . . , C}. We elaborateon the first stage in Section IV. Second, for each channel,we use a fast iterative algorithm to select active devices thatactually use the channel. Last, we use aggression levels forfast rate adaptation based on channel state information. Weelaborate on the last two stages in Section V.

IV. PARTITIONING DEVICES BASED ON

COLORING A CONFLICT GRAPH

We [5] proposed an algorithm that partitions source devicesinto subgroups based on graph theory. When the algorithm isused, source devices in subgroup k become candidate devicesthat could use channel k, ∀ 1 ≤ k ≤ C. For the completenessof the paper and the convenience of the readers, we brieflyintroduce the algorithm in this section.

The proposed algorithm is composed of two phases. Inthe first phase, the algorithm creates a conflict graph anddetermines the number of required subgroups based on graphcoloring. In the second phase, the algorithm partitions sourcedevices into subgroups.

In the first phase, to determine the total number of requiredsubgroups, a conflict graph G = (V,E), where V is the vertexset and E is the edge set, is created. In particular, vertex i ∈ Vcorresponds to source device i in the network. In addition,(i, j) ∈ E if and only if source device i and source device jshould not use the same channel to transmit data at the sametime. Let mi,j be the mean channel gain from device i todevice j. Recall that Di is the index of the destination deviceassociated with source device i. Consider source device i andsource device j. Let T̃o(t) be the throughput per channel in timeslot t when the two source devices use two orthogonal channels.Then, based on information theory [4],

T̃o(t) =1

2

[W log2

(1 +

P · gi,Di(t)

σ2

)

+W log2

(1 +

P · gj,Dj(t)

σ2

)]. (4)

Let T̃s(t) be the throughput per channel in time slot t whenthe two source devices share the same channel. If Di = Dj , the

two source devices transmit data to the same destination devicethrough a multiple access channel. Otherwise, source devicei interferes with the reception at destination device Dj , whilesource device j interferes with the reception at destinationdevice Di. Recall that log2(x) + log2(y) = log2(xy). Then,based on network information theory [4],

T̃s(t) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩W log2

(1 +

P ·gi,Di(t)

σ2+P ·gj,Di(t)

)(1 +

P ·gj,Dj(t)

σ2+P ·gi,Dj(t)

),

if Di = Dj ,

W log2

(1 +

P(gi,Di(t)+gj,Dj

(t))σ2

), if Di = Dj .

(5)

Since log2(1 + x) is a concave function of x, based onJensen’s inequality [4], for an arbitrary random variable X ,E[log2(1 +X)] ≤ log2(1 + E[X]). Therefore,

E

[1

2

[W log2

(1 +

P · gi,Di(t)

σ2

)+ W log2

(1 +

P · gj,Dj(t)

σ2

)]]≤ 1

2

[W log2

(1+

P ·mi,Di

σ2

)+W log2

(1+

P ·mj,Dj

σ2

)].

(6)

Define To the approximated average throughput per channelin a time slot when the two source devices use two orthogonalchannels as follows.

To=1

2

[W log2

(1+

P ·mi,Di

σ2

)+W log2

(1+

P ·mj,Dj

σ2

)]. (7)

Similarly, define Ts the approximated average throughput perchannel in a time slot when the two source devices share thesame channel as follows.

Ts=

⎧⎪⎪⎪⎨⎪⎪⎪⎩W log2

(1 +

P ·mi,Di

σ2+P ·mj,Di

)(1 +

P ·mj,Dj

σ2+P ·mi,Dj

),

if Di = Dj ,

W log2

(1 +

P(mi,Di+mj,Dj )

σ2

), if Di = Dj .

(8)

To reduce the computational complexity, we propose parti-tioning devices at time zero based on the value of (To, Ts) ratherthan partitioning devices every time slot based on the value of(T̃o(t), T̃s(t)).

If To > Ts, it is better for source device i and source devicej to transmit data via orthogonal channels. In this case, sourcedevice i and source device j are seen as two conflicting devices.Therefore, i, j ∈ V and (i, j) ∈ E. Otherwise, (i, j) ∈ E (andi, j ∈ V ). In [5], it has been proved that if Di = Dj , Ts > To.Thus, it is more efficient for two source devices with the samedestination to share a channel rather than use two orthogonalchannels. In this paper, all sources to the same destination usethe same channel.

Let Nc be the total number of colors required for coloringthe vertices of G so that adjacent vertices are colored withdifferent colors. Given the conflict graph, we could use thesequential coloring algorithm [32] to find out Nc. Fig. 1 shows


Fig. 1. A conflict graph [5].

a conflict graph in which a circle represents a source device anda cross represents a destination device. In addition, a dashedline represents the association between a source device andthe corresponding destination device. The number adjacentto a circle is the index of the corresponding source device.A solid line represents an edge of the conflict graph. Forexample, source device 6 and source device 7 cannot use thesame channel, since (6, 7) ∈ E. In the conflict graph, there are7 vertices and 6 edges.

We now elaborate on the second phase. Pseudo codes for thesecond phase can be found in [5] and are omitted due to limit ofspace. The second phase is used to assign each source device asubgroup. Recall that C is the total number of channels. Sourcedevices are assigned priorities based on the average through-put up to date. In particular, to improve fairness, the sourcedevices with smaller throughput up to date are assigned higherpriorities. Define Δ(H) = ∪i∈HDi, ∀H ⊂ {1, 2, 3, . . . , Ns}.If Nc ≤ C, we use an iterative algorithm to sequentially assignsource devices to subgroups. Let Vk be the set composed ofthe indexes of the source devices that currently belong tosubgroup k, 1 ≤ k ≤ C. Define V i

k = Vk ∪ {i}. According tothe iterative algorithm, source device i is assigned to subgroupk∗1(i), where

k∗1(i)=arg maxk:1≤k≤C

W

×[∑

j:j∈Δ(Vk∪{i})log2

(1+

P∑

l:l∈V ik∩Sj

ml,j

σ2+P∑

l:l∈V ik−(V i

k∩Sj)ml,j

)

−∑

j:j∈Δ(Vk)log2

(1+

P∑

l:l∈Vk∩Sjml,j

σ2+P∑

l:l∈Vk−(Vk∩Sj)ml,j

)].

(9)

Note that in each step, the iterative algorithm assigns a sourcedevice to the optimal subgroup that maximizes the throughputincrement.

If Nc > C, the following procedure is used. First, the se-quential coloring algorithm is used to color vertices in V from

high-priority vertices to low-priority vertices. Let θ(i) be theindex of the color to which source device i is assigned. LetV0 be the set composed of the indexes of source devices thatcannot not transmit data. Let Q be the set composed of theindexes of source devices that transmit data. If θ(i) > C, i ∈V0. Otherwise, i ∈ Q. We propose using the above iterativealgorithm with Equation (9) to determine the index of thesubgroup to which source device i belongs, ∀ i ∈ Q.

We now take into consideration the source devices that donot correspond to vertices in the conflict graph. Define V c ={i|1 ≤ i ≤ Ns, i ∈ V }. The source devices with indexes in V c

are first sorted in descending order according to their priorities.Next, an iterative algorithm is used as follows. Consider sourcedevice i. Let ei,k be the interference level from source device ito destination devices in subgroup k. In particular, if |Vk| ≥ 1,ei,k is set to maxj:j∈Vk

mi,Dj. Otherwise, ei,k is set to 0. To

minimize the maximum interference from source device i todestination devices, source device i is assigned to subgroupk∗0(i), where

k∗0(i) = arg mink:1≤k≤C

ei,k

= arg mink:1≤k≤C

maxj:j∈Vk

mi,Dj. (10)

When the partition algorithm terminates, Vk consists of theindexes of the candidate source devices that might use channelk to transmit data. A source device with index i belongs tosubgroup k if and only if i ∈ Vk.

V. THROUGHPUT OPTIMIZATION FOR A SUBGROUP

In this section, we focus on the problem of throughputoptimization for each subgroup. Recall that a source deviceis associated with a single destination device. Consider a sub-group of source devices in a time slot. In order to optimize thethroughput for the subgroup in the time slot, it is necessaryto find the optimal set of active source devices and the corre-sponding optimal data rates. We first formulate the optimizationproblem as an integer-linear programming problem and thenpropose a fast approximation algorithm for solving it.

A. An Integer-Linear Programming Approach

Consider subgroup k in time slot t. Recall that Vk is the set ofcandidate source devices that might use channel k to transmitdata. Recall that Sk(t) is the set of source devices that usechannel k to transmit data in time slot t. Then, Sk(t) ⊂ Vk. Inorder to maximize the throughput of subgroup k in time slot t,we have to solve the following optimization problem.

max∑

i:i∈Sk(t)Ri(t)

subject to

Sk(t) ⊂ Vk,

Dk(t) = Δ(Sk(t)

),

Rk(t) ≤ W log2

(1 +

P k(t) · hk0(t)

σ2 +∑

u:u∈Sk(t) Pu(t) · gku,0(t)

),


Ri(t)

≤ W log2

(1+


j (t)+∑


),

∀ j ∈ Dk(t), i ∈ Skj (t),∑

i:i∈TRi(t)

≤W log2

(1+



∑u:u∈Ik

j(t)Pu(t)·gku,j(t)

),

∀ j ∈ Dk(t), T ⊂ Skj (t),

Ri(t) ≥ 0, ∀ i ∈ Sk(t). (11)

The third constraint is used to ensure that UE k could success-fully use channel k to transmit data to the BS in time slot t.The above problem is an integer-linear programming problem,which is NP-hard in general. In the worst case, there are 2|Vk |

feasible solutions for Sk(t). For each fixed value of Sk(t),the above optimization problem becomes a linear programmingproblem. In principle, a linear programming problem can besolved by the simplex algorithm or the interior-point algorithms[31]. However, for each fixed value of Sk(t), the correspondinglinear programming problem has

∑j:j∈Dk(t) 2

|Skj (t)| − 1 con-

straints. Therefore, when the total number of source devices islarge, it is impractical to find an optimal solution for the integer-linear programming problem in (11).

B. A Fast Approximation Algorithm

Instead of finding an optimal solution for the above integer-linear programming problem, to reduce the computationalcomplexity, we propose a fast approximation algorithm. Theproposed approximation algorithm is composed of two phases.In the first phase, the algorithm selects active source devicesthat transmit data in the current time slot. There are twovariants for the first phase. The first variant is source-based,while the second variant is destination-based. When the source-based approximation algorithm is used, in each round, thealgorithm selects an additional active source device. Whenthe destination-based approximation algorithm is used, in eachround, the algorithm chooses a destination device and all as-sociated source devices become active source devices. In thesecond phase, the algorithm determines the data rates of activesource devices. Pseudo codes for the main function of theapproximation algorithm are shown in Algorithm 1. Pseudocodes for the first phase of the approximation algorithm areshown in Algorithm 2 and pseudo codes for the second phaseof the approximation algorithm are shown in Algorithm 3.

Algorithm 1: The approximation algorithm (for subgroup k)

Input: W , σ2, P , P k, Rk, hk0(t), h

kj (t), gi,j(t).

Output: P k(t), Rk(t), Sk(t), Ri(t), ∀ i ∈ Sk(t).1: Determine P k(t), Rk(t), Sk(t) by Algorithm 2.2: Calculate Ri(t), ∀ i ∈ Sk(t) by Algorithm 3.

Algorithm 2: Selecting the active source devices that usechannel k to transmit data in time slot t

Input: W , σ2, P , P k, Rk, hk0(t), gi,j(t).

Output: P k(t), Rk(t), Sk(t).1: Ψk(t) ← ∅2: if W log2

(1 +

Pk·hk0 (t)

σ2

)≥ Rk then

3: // UE k transmits data in time slot t.4: Set P k(t) = P k and Rk(t) = Rk.5: else6: Set P k(t) = 0 and Rk(t) = 0.7: end if8: repeat9: Obtain (v∗, j∗) by Equation (13) or Equation (14).

10: if v∗ > Λk,t(Ψk(t)) then

11: Add Θj∗ to Ψk(t).12: end if13: until v∗ ≤ Λk,t(Ψ

k(t))14: Set Sk(t) = Ψk(t).15: if Rk(t) == Rk then

16: Rk(t) ← W log2

(1 +

Pk ·hk0 (t)

σ2+∑

β:β∈Sk(t)P ·gk

β,0(t)

).

17: end if

Algorithm 3: Rate adaptation based on channel state infor-mation for subgroup k

Input: W , σ2, P , hkj (t), gi,j(t), P

k(t), Sk(t).Output: Ri(t), ∀ i ∈ Sk(t).

1: // Calculate date transmission rate for source device i,∀ i ∈ Sk(t).

2: for j ∈ Δ(Sk(t)) do3: Calculate αk

j (t) by Equation (15).4: for i ∈ Sk

j (t) do5: Calculate Ri(t) by Equation (16).6: end for7: end for

Consider channel k in time slot t. We now elaborate onthe first phase, which is an iterative algorithm. We first setthe temporary values of P k(t) and Rk(t) as follows. If

W log2

(1 +

Pk ·hk0 (t)

σ2

)≥ Rk, P k(t) = P k and Rk(t) = Rk.

Otherwise, P k(t) = 0 and Rk(t) = 0. Recall that Sk(t) is theset composed of the indexes of the source devices that actuallyuse channel k to transmit data in time slot t. Let Ψk(t) be a setthat is used to determine the value of Sk(t). Initially, Ψk(t) =∅. If the source-based approximation algorithm is used, defineΘj = {j}, ∀ j ∈ S. On the other hand, if the destination-basedapproximation algorithm is used, define Θj = Sj , ∀ j ∈ D. LetΛk,t(X) be the throughput of channel k in time slot t, when the


set of active source devices that use channel k to transmit datain time slot t is equal to X . In particular,

Λk,t(X)

= W∑

i:i∈Δ(X)

log2

(1 +

∑α:α∈X∩Si

P × gkα,i(t)

σ2 +∑

β:β∈X−SiP × gkβ,i(t)

).

(12)

When the source-based approximation algorithm is used,the iterative algorithm first solves the following optimizationproblem.

maxΛk,t

(Ψk(t) ∪ {j}

)subject to

j ∈ Vk −Ψk(t),

Rk(t) ≤ W log2

(1 +

P k(t) · hk0(t)

σ2 +∑

β:β∈Ψk(t)∪{j} P · gkβ,0(t)

).

(13)

Note that the last constraint is used to assure that UE k couldsuccessfully transmit data to the base station with rate Rk(t).

Let v∗ be the optimal value and j∗ be an optimal solution forthe above optimization problem. When the above optimizationproblem is infeasible, v∗ = −∞. If v∗ ≤ Λk,t(Ψ

k(t)), the firstphase terminates. Otherwise, the integer j∗ is added into the setΨk(t) and the iterative algorithm continues.

Similarly, when the destination-based approximation algo-rithm is used, the iterative algorithm first solves the followingoptimization problem.

maxΛk,t

(Ψk(t) ∪Θj

)subject to

j ∈ Δ(Vk −Ψk(t)

),

Rk(t) ≤ W log2

(1 +

P k(t) · hk0(t)

σ2 +∑

β:β∈Ψk(t)∪ΘjP · gkβ,0(t)

).

(14)

Let v∗ be the optimal value and j∗ be an optimal solution forthe above optimization problem. If v∗ ≤ Λk,t(Ψ

k(t)), the firstphase terminates. Otherwise, all elements of the set Θj∗ areadded into the set Ψk(t) and the iterative algorithm continues.

When the first phase terminates, Sk(t) is set to bethe current value of Ψk(t). In addition, if the temporaryvalue of Rk(t) is Rk, the final value of Rk(t) is set to

W log2

(1 +

Pk·hk0 (t)

σ2+∑

β:β∈Sk(t)P ·gk

β,0(t)

)in order to increase the

throughput of UE k. Note that the same algorithm is used foreach channel in each time slot.

We now introduce the second phase. Consider destinationdevice j, where j ∈ Δ(Sk(t)). Recall that Sk

j (t) is composedof the indexes of source devices that use channel k to transmit

data to destination device j in time slot t. The proposedalgorithm obtains the value of Ri(t), ∀ i ∈ Sk

j (t), as follows.Recall that Ikj (t) constitutes of the indexes of the source devicesthat use channel k to transmit data to destination devices otherthan destination device j in time slot t. Define ηkj (t) = P k(t) ·hkj (t) +

∑β:β∈Ik

j(t) P · gkβ,j(t). Then, ηkj (t) equals the power

of the interference at destination device j through channel k intime slot t.

Let αkj (t) be the aggression level of the source devices that

use channel k to transmit data to destination device j in timeslot t. In particular,

αkj (t) =

log2

(1 +

∑i∈Sk

j(t)

P ·gki,j(t)

σ2+ηkj(t)

)∑

i∈Skj(t) log2

(1 +

P ·gki,j

(t)

σ2+ηkj(t)

) . (15)

In time slot t, for each i ∈ Skj (t), the proposed algorithm

chooses the value of Ri(t) as follows.

Ri(t) = αkj (t)W log2

(1 +

P · gki,j(t)σ2 + ηkj (t)

). (16)

Then,

∑i:i∈Sk

j(t)

Ri(t) =αkj (t)W

∑i∈Sk

j(t)

log2

(1 +


)

=W log2

(1 +

∑i∈Sk

j(t) P · gki,j(t)

σ2 + ηkj (t)

). (17)

Namely, the sum rate for source devices that transmit data todestination device j through channel k in time slot t reaches theupper bound of the sum rate for a multiple access channel [4].However, (17) does not imply (1) in general.

VI. PROPERTIES OF THE PROPOSED ALGORITHM

We show analytical results in this section. We first prove thatwhen two source devices simultaneously use the same channelto transmit data to the same destination device according to (15)and (16), both source devices succeed for sure.

Theorem 1: If |Skj (t)| = 2 and the values of Ri(t)’s, ∀ i ∈

Skj (t), are determined by (15) and (16), both source devices

with indexes in Skj (t) successfully use channel k to transmit

data to destination device j in time slot t.Proof: See Appendix.

The above theorem means that if a destination device has twosources in a time slot, the proposed aggression-level-based ap-proach maximizes the sum rate with a very low computationalcomplexity.

We now study the case in which not all channel gains areknown to the base station. Consider source devices that transmitdata to destination device j through channel k in time slot t.Let i be an integer in Sk

j (t). Let mki,j be the mean channel

gain of channel k from source device i to destination device j.


Suppose that the base station knows the values of gkα,j(t)’s,∀α ∈ Sk

j (t)− {i} but does not know the value of gki,j . In thiscase, based on (15), αk

j (t) is approximated by α̂ki,j(t), which is

defined as follows.

α̂ki,j(t)

= log2

(1 +

P ·mki,j +

∑α:α∈Sk

j(t)−{i} P · gkα,j(t)

σ2 + ηkj (t)

)

×

⎡⎢⎣log2(1 +

P ·mki,j

σ2 + ηkj (t)

)

+∑

α∈Skj(t)−{i}

log2

(1 +

P · gkα,j(t)σ2 + ηkj (t)

)⎤⎥⎦−1

. (18)

Note that the above equation is very similar to (15) except thatgki,j(t) is replaced by mk

i,j .Define f(gki,j(t)) = αk

j (t)− α̂ki,j(t). Note that αk

j (t) de-pends on gki,j(t) but α̂k

i,j(t) is independent of gki,j(t). Iff(gki,j(t)) < 0, destination device j fails to receive any datafrom source devices with indexes in Sk

j (t). If f(gki,j(t)) ≥ 0

and |Skj (t)| = 2, based on Theorem 1, two source devices

with indexes in Skj (t) successfully transmit data to destination

device j through channel k in time slot t.Consider the case in which Sk

j (t) = {i+, i−}, the value ofgki+,j(t) is known, and the value of gki−,j(t) is unknown. Toanalyze the function f in this case, for each fixed positive realnumber a, define the function φa(b) as follows.

φa(b) =ln(1 + a+ b)

ln(1 + a+ b+ ab), ∀ b ∈ [0,∞). (19)

Define α+ =Pgk

i+,j(t)

σ2+ηkj(t)

. Define a function s such that

s(gki−,j(t)) =Pgk

i−,j(t)

σ2+ηkj(t)

. Then, s is an increasing function of

gki−,j(t). Based on (15),

αkj (t) = φα+

(s(gki−,j(t)

)). (20)

In addition, based on (18),

α̂ki−,j(t) = φα+

(s(mk

i−,j

)). (21)

Thus,

f(gki−,j(t)

)=αk

j (t)− α̂ki−,j(t)

=φα+

(s(gki−,j(t)

))− φα+

(s(mk

i−,j

)).

(22)

It was conjectured that if gki−,j(t) > mki−,j , f(gki−,j(t)) > 0,

since the unknown channel gain is underestimated. It is clearthat f(mk

i−,j) = 0. Then, it is conjectured that the function f

has only one positive root. However, the conjectures are incor-rect. It is possible that the function f have two positive roots andf(gki−,j(t)) < 0 even when gki−,j(t) > mk

i−,j . The followingtheorem contains a sufficient condition for the function f tohave two positive roots.

Theorem 2: If Skj (t) = {i+, i−}, the value of gki+,j(t) is

known, the value of gki−,j(t) is unknown, and φα+(mk

i−,j) >

minx:x≥0 φα+(x), then there exists a positive real number b̂ ∈

(0,∞) such that μ(̂b) = 0 and the function f has two or morepositive roots.

Proof: See Appendix.When the condition of Theorem 2 is satisfied, f(x) is not a

monotonic function of x.We now analyze the overall computational complexity of the

proposed approach. Consider a graph G = (V,E). Let deg(v)be the degree of vertex v, ∀ v ∈ V . For each vertex v, it takesdeg(v) steps to collect the colors that have been used by itsneighbors and at most |V | more steps to find an appropriatecolor. According to graph theory [32],

∑v∈V deg(v) = 2|E| ≤

|V |(|V |−1)2 < |V |2. Recall that N is the total number of devices

in the network. Thus, there are at most N vertices in theconflict graph and the computational complexity of the grouppartition based on the sequential coloring algorithm is at mostO(N2 +N ·N) = O(N2). The computational complexity ofthe first phase of the fast approximation algorithm is upperbounded by O(N3). The computational complexity of the sec-ond phase of the fast approximation algorithm is upper boundedby O(N2). Therefore, the overall computational complexity isupper bounded by O(N3).

VII. SIMULATION SETUP AND RESULTS

In this section, we show simulation setup and results. Toobtain discrete event simulation [33] results, we wrote C++programs that could call MATLAB for solving linear program-ming problems. The service region is a 20 by 20 square. Werandomly create 10 networks. In a network, the BS is deployedat the center of the service region. Regarding the total numberof UEs in the network, we study two cases. In the first case,there are no UEs in the network. The case corresponds to thecase in which an UE and a device use orthogonal channels. Inthe second case, there are C UEs in the network and differentUEs use distinct channels. Each UE is randomly deployed 3units away from the BS. Source devices are randomly deployedin the service region. A source device is associated with adestination device. There are two classes of destination devices.A destination device in the first class is associated with asource device, while a destination device in the second classis associated with two or more source devices. The distancebetween a source device and the corresponding destinationdevice is a random variable that is uniformly distributed in[0, 3]. In this section, P = 1, σ2 = 0.01, and W = 10 (MHz).Unless explicitly stated, a communication link between a sourcedevice and a destination device is a Rayleigh fading channel.In addition, for each link, channel gains at distinct time slotsare independent and identically distributed random variables.The mean channel gain of each link is the reciprocal of the


Fig. 2. The performance of the global optimal algorithm and the proposedapproximation algorithm in small networks.

squared distance between the source device and the destinationdevice. In addition, C = 5. To obtain the network throughput,a simulation instance contains 10 000 time slots. In this section,we show normalized throughput in units of bps/Hz.

We compare the performance of five algorithms: the globaloptimal algorithm, the local optimal algorithm [5], the greedyalgorithm, the proposed source-based approximation algorithm,and the proposed destination-based approximation algorithm.The global optimal algorithm solves the integer-linear program-ming problem in (3) to maximize the overall throughput ofthe network. The local optimal algorithm uses the proposedgraph coloring approach to partition devices into subgroups andsolves the integer-linear programming problem in (11) to maxi-mize the throughput of each subgroup. Both the global optimalalgorithm and the local optimal algorithm are feasible only invery small networks, since a general integer programming prob-lem is NP-hard. When the greedy algorithm is used, in a timeslot, only the D2D source devices with the largest C channelgains might transmit data. In a time slot, the active set containsall active source devices in the time slot. When the source-based approximation algorithm is used, in each round, a sourcedevice is added into the active set. When the destination-basedapproximation algorithm is used, in each round, a destinationdevice is selected and the corresponding source device is addedinto the active set. None of the five algorithms are aware of thelocations of devices.

In Fig. 2, we show the performance of the proposed approxi-mation algorithm and the global optimal algorithm, when thereare two orthogonal channels, 4 source devices and 4 destinationdevices. In this case, the source-based approximation algorithmis identical to the destination-based approximation algorithm.When there is no user equipment in the network, the networkthroughput of the proposed approximation algorithm is at least93% of that of the global optimal algorithm. When there areuser equipments in the network, the network throughput of theproposed approximation algorithm is at least 82% of that of theglobal optimal algorithm.

Fig. 3. The performance of the local optimal algorithm and the proposedapproximation algorithm in small networks.

In Fig. 3, we show the network throughput of the lo-cal optimal algorithm, the greedy algorithm, and the source-based approximation algorithm. The network throughput of thedestination-based approximation algorithm is almost identicalto that of the source-based approximation algorithm and isnot shown due to limit of space. We study the case when thenetwork contains 13 source devices and 10 destination devices.Seven destination devices belong to the first class, while threedestination devices belong to the second class. A destinationdevice in the first class is associated with one source device,while a destination device in the second class is associatedwith two source devices. The network throughput of the source-based approximation algorithm is very close to that of the localoptimal algorithm. When there are UEs in the network, on av-erage, the network throughput of the approximation algorithmcould be 20% larger than that of the greedy algorithm.

We also study the case when the total number of source de-vices in the network is either 25 or 50. Five destination devicesbelong to the second class. In addition, a destination device insecond class is associated with three source devices. In Fig. 4,we show the network throughput for the two approximationalgorithms and the greedy algorithm, when there is no UE.Regardless of the total number of source devices in the net-work, the approximation algorithms are superior to the greedyalgorithm. The throughput of the source-based approximationalgorithm is almost identical to that of the destination-based ap-proximation algorithm. Furthermore, when there are 25 sourcedevices, the throughput of the source-based approximationalgorithm could be 1.77 times larger than that of the greedyalgorithm. When there are 50 source devices, the throughput ofthe source-based approximation algorithm could be 2.42 timeslarger than that of the greedy algorithm.

In Fig. 5, we show the network throughput for the approxi-mation algorithms and the greedy algorithm, when there are fiveUEs and the total number of source devices is either 25 or 50.The throughput difference between the two approximation al-gorithms is negligible. When there are 25 source devices, the


Fig. 4. Throughput comparison, when there is no UE.

Fig. 5. Throughput comparison, when there are UEs.

throughput of the approximation algorithms is about 1.53 timeslarger than that of the greedy algorithm. When there are 50source devices, the throughput of the approximation algorithmsis about 2.03 times larger than that of the greedy algorithm.When there are UEs, a source device should not interfere withthe UE that uses the same channel. Thus, a source device maynot be able to transmit data even if its instantaneous channelgain is quite large. Therefore, the throughput when there areUEs is smaller than that when there is no UE.

We also evaluate the performance of the approximationalgorithm when the base station is unaware of some channelgains. Recall that mk

i,j is the mean channel gain of channel kfrom source device i to destination device j. If the base stationdoes not know the value of gki,j(t), where i ∈ Sk

j (t), it usesmk

i,j instead to determine the aggression level and the value ofRi(t). We study the case in which there are 15 source devicesand 1 destination device. In Fig. 6, the variable in the x-axisis the total number of known channel gains at the base station.

Fig. 6. The network throughput, when some of the channel gains areunknown.

Fig. 7. Performance comparison of the graph coloring approach and themodulo approach.

Note that in the simulation, the expected value of a Rayleighrandom variable is the same as that of a Rician random variable.When all communication links are Rayleigh fading channels,as the total number of known channel gains increases from2 to 8, the network throughput increases as expected. Similarly,when all communication links are Rician fading channels, asthe number of known channel gains increases from 2 to 8, thenetwork throughput increases. On the other hand, as the numberof known channel gains increases from 10 to 14, the networkthroughput decreases irrespective of the fading channel model.

In Fig. 7, we compare the performance of two group par-tition approaches: the proposed graph coloring approach andthe modulo approach. According to the modulo approach, thesource device with identification number x can only use channely = x mod C. We study the case in which C = 5, Ns = Nd =25, and devices form 5 clusters. In particular, in the serviceregion, there are 5 circles with radius 3 centered at (0, 0), (7, 7),


(−7, 7), (−7, −7), and (7, −7), respectively. Destination devicei is randomly deployed in circle i mod C. In addition, eachsource device associated with destination device i is randomlydeployed in circle i mod C. When each destination device isassociated with a unique source device, on average, the networkthroughput of the proposed graph coloring approach is 58.79%larger than that of the modulo approach. When each of the first5 destination devices has 3 sources and each of the remainingdestination devices has only one source, on average, the net-work throughput of the proposed graph coloring approach is10.87% larger than that of the modulo approach. We also studythe performance of the two group partition when source devicesare randomly deployed according to the first paragraph of thesection. When there are 50 source devices in the network, onaverage, the network throughput of the graph coloring approachis 7.9% larger than that of the modulo approach. Due to the limitof space, the corresponding figure is not included.

In terms of network throughput, partitioning based on bothnetwork topology and instantaneous channel gains is expectedto outperform the graph coloring approach in which the lo-cations of devices are not known/used. On the other hand,the proposed graph coloring approach does not require anylocalization techniques and does not have to collect informationon the locations of devices.

VIII. DISCUSSIONS

We now outline an implementation of the proposed approachin cellular networks. When Orthogonal Frequency DivisionMultiplexing (OFDM) is used at the physical layer, a channelis composed of adjacent subcarriers. In addition, a subcarrierbelongs to only one channel. A time slot is composed of twoparts: the control sub-slot and the data sub-slot. The controlsub-slot consists of four phases. Consider time slot t. In thefirst phase of the control sub-slot, there are Ns time intervalsfor channel estimation. At the ith channel estimation timeinterval, source device i broadcasts pilot signals for channelestimation through C orthogonal channels and each destinationdevice j estimates the value of (g1i,j(t), g

2i,j(t), . . . , g

Ci,j(t)). It is

assumed that channel estimation is perfect in the paper. The im-pacts of channel estimation errors on the network performanceare beyond the scope of this paper. More details on channelestimation in MIMO-OFDM wireless communication systemscan be found in [34]. In the second phase of the control sub-slot,there are Nd time intervals for feedback. At the jth feedbacktime interval, destination device j sends the values of gki,j(t)’sto the base station. Although it is possible to use an appropriaterandom multiple access scheme instead in the second phase,random multiple access is beyond the scope of this paper. Thebase station could collect the values of hk

j (t)’s through a similarprocedure. In the third phase of the control sub-slot, to optimizethe network throughput (at time slot t), the base station usesthe proposed approach to approximately solve the optimizationproblem in (3). In the fourth phase of the control sub-slot,the base station broadcasts its decision. In particular, the basestation sends the value of (Pi(t), Ri(t)) to source device i. Inthe data sub-slot, each source device i transmits data with powerPi(t) and rate Ri(t).

Fig. 8. Network throughput when there are long-lived and short-lived D2Dsessions.

We now discuss how to extend the proposed approach tothe case in which there are short-lived D2D communicationsessions in addition to Ns long-lived D2D communicationsessions. A long-lived D2D communication session always hasdata to send from the source device to the destination de-vice. The arrivals of short-lived D2D communication sessionsare modeled as a Poisson process with rate λ. The lifetimesof short-lived D2D communication sessions are modeled asindependent and identically distributed exponential randomvariables with mean 1

μ . Among the C channels, Cs channelsare reserved for short-lived D2D sessions, while C − Cs chan-nels are used by long-lived D2D sessions. When a short-livedsession arrives/starts, the associated source device randomlychooses one of the Cs channels to use till the end of the session.According to M/M/∞ queueing theory [35], the expectedvalue of the total number of short-lived sessions is equal to λ

μ .

Let x be a real number such that λ/μNs

= xC−x . We propose using

a loading-aware algorithm in which Cs = �x�. In Fig. 8, weshow the performance of the proposed loading-aware algorithmand the baseline algorithm in which Cs = �C

2 �, when C = 6.When �x� = 1, the proposed loading-aware algorithm alwaysoutperforms the baseline algorithm. The average performanceimprovement is 13%. When �x� = 2, on average, the networkthroughput of the proposed loading-aware algorithm is 3.6%larger than that of the baseline algorithm. When �x� = 3, thenetwork throughput of the loading-aware algorithm is identicalto that of the baseline algorithm.

IX. CONCLUSION

In this paper, we have proposed using group partition anddynamic rate adaptation for scalable throughput optimization ofcapacity-region-aware device-to-device communications. Wehave adopted network information theory that allows a receiv-ing device to simultaneously decode distinct packets from mul-tiple transmitting devices, as long as the vector of transmitting


rates is inside the capacity region. We have proposed creatinga conflict graph to partition devices into subgroups based ongraph theory. To optimize the throughput of a subgroup, insteadof directly solving an integer-linear programming problem,we have proposed an iterative algorithm to quickly selectthe set of active devices and using aggression levels for fastrate adaptation based on channel state information. Simulationresults show that the proposed approach is scalable and couldsignificantly outperform the greedy algorithm when it is in-feasible to obtain an optimal solution. Future work includesfurther improving the network throughput when the arrivals ofD2D communication sessions form a renewal process and thelifetime of a D2D communication session is a general randomvariable based on queueing theory.

APPENDIX

For the convenience of readers, we list key variables asfollows.

hk0(t) the channel gain from cellular UE k to the base

station in time slot t.gki,0(t) the gain of channel k from source device i to the

base station in time slot t.hkj (t) the channel gain from cellular UE k to destina-

tion device j in time slot t.gki,j(t) the gain of channel k from source device i to

destination device j in time slot t.P k(t) the transmission power of cellular user equip-

ment k in time slot t.Pi(t) the transmission power of source device i in

time slot t.Rk(t) the data transmission rate of cellular user equip-

ment k in time slot t.Ri(t) the data transmission rate of source device i in

time slot t.Sk(t)/Sk

j (t) the set composed of the indexes of source de-vices that use channel k to transmit data (todestination device j) in time slot t.

Dk(t) the set composed of the indexes of destinationdevices that use channel k to receive data in timeslot t.

Ikj (t) the set composed of the indexes of source de-vices that use channel k but do not want totransmit data to destination device j in timeslot t.

Theorem 1: If |Skj (t)| = 2 and the values of Ri(t)’s, ∀ i ∈

Skj (t), are determined by (15) and (16), both source devices

with indexes in Skj (t) successfully use channel k to transmit

data to destination device j in time slot t.Proof:

1) Since |Skj (t)| = 2, without loss of essential generality, it

is assumed that Skj (t) = {m,n}. Then, according to (15),

αkj (t) =

log2

(1 +

P ·gkm,j(t)+P ·gk

n,j(t)

σ2+ηkj(t)

)log2

((1 +

P ·gkm,j

(t)

σ2+ηkj(t)

)(1 +

P ·gkn,j

(t)

σ2+ηkj(t)

)) .

Since all variables in the right-hand side of the aboveequation are positive, αk

j (t) > 0.

2) Note thatP ·gk

m,j(t)

σ2+ηkj(t)

> 0 andP ·gk

n,j(t)

σ2+ηkj(t)

> 0. In addition,

1 + a+ b < (1 + a)(1 + b), ∀ a, b > 0. Furthermore,since log2(x) is an increasing function ofx, log2(1 + a+ b) < log2((1 + a)(1 + b)),

∀ a, b > 0, Thus, log2

(1 +

P ·gkm,j(t)+P ·gk

n,j(t)

σ2+ηkj(t)

)<

log2

((1 +

P ·gkm,j(t)

σ2+ηkj(t)

)(1 +

P ·gkn,j(t)

σ2+ηkj(t)

))and therefore

αkj (t) < 1.

3) Since 0 < αkj (t) < 1, ∀ i ∈ Sk

j (t), Ri(t) =

αkj (t) log2

(1 +

P ·gki,j(t)

σ2+ηkj(t)

)< log2

(1 +

P ·gki,j(t)

σ2+ηkj(t)

).

4) Based on the definition of αkj (t),

∑i:i∈Sk

j(t)

Ri(t) =αkj (t)

∑i:i∈Sk

j(t)

log2

(1 +


)

= log2

(1 +

∑i:i∈Sk

j(t) P · gki,j(t)

σ2 + ηkj (t)

).

5) Based on 3 and 4, for channel k in time slot t,(Rm(t), Rn(t)) is inside the capacity region of the mul-tiple access channel from source device m and sourcedevice n to destination device j. Therefore, accordingto network information theory, all source devices withindexes in Sk

j (t) successfully use channel k to transmitdata to destination device j in time slot t.

Theorem 2: If Skj (t) = {i+, i−}, the value of gki+,j(t) is

known, the value of gki−,j(t) is unknown, and φα+(mk

i−,j) >

minx:x≥0 φα+(x), then there exists a positive real number b̂ ∈

(0,∞) such that μ(̂b) = 0 and the function f has two or morepositive roots.

Proof:

1) Based on the proof of Theorem 1, φa(b)∈(0, 1], ∀ a,b>0. In addition, φa(0) = 1, ∀ a > 0.

2) Consider a fixed value of a.

limb→∞

φa(b)

= limb→∞

ln(1 + a+ b)

ln(1 + a+ b+ ab)

= limb→∞

ln(1 + a+ b)

ln(1 + a) + ln(1 + b)

=

[limb→∞

ln(1 + a)

ln(1 + a+ b)+ lim

b→∞

ln(1 + b)

ln(1 + a+ b)

]−1

= 1.

3) φ′a(b)={(1+b) ln(1 + a)−[(1 + a+ b) ln(1 + a+ b)−

(1 + b) ln(1 + b)]} × {(1 + b)(1 + a+ b)× ln2((1 +a)(1 + b))}−1.


4) Define c = 1(1+b)(1+a+b) ln2((1+a)(1+b))

. Then, c > 0.

Define μ1(b) = (1 + b) ln(1 + a). In addition, defineμ2(b) = (1 + a+ b) ln(1 + a+ b)− (1 + b) ln(1 + b).Furthermore, define μ(b)=μ1(b)− μ2(b). Then, φ′

a(b)=c · [μ1(b)−μ2(b)]=c·μ(b). In addition, μ1(0) = ln(1 +a) and μ2(0) = (1 + a) ln(1 + a). Thus, μ1(0)−μ2(0)<0 and φ′

a(0) < 0.5) Define ε =

√ln(1 + a)− 1. Then, ε > 0. Since

limb→∞1+a+b1+b = 1, there exists b1 ∈ (0,∞) such that

1+a+b1+b < 1 + ε, ∀ b ≥ b1. Since limb→∞

ln(1+a+b)ln(1+b) = 1,

there exists b2 ∈ (0,∞) such that ln(1+a+b)ln(1+b) < 1 + ε,

∀ b ≥ b2. Define b+ = max(b1, b2). Then, 0 < b+ < ∞.In addition,

μ1(b+)− μ2(b+)

= (1 + b+) ln(1 + a)− (1 + a+ b+) ln(1 + a+ b+)

+ (1 + b+) ln(1 + b+)

= (1 + b+) ln(1 + a+ b+ + ab+)

− (1 + a+ b+) ln(1 + a+ b+).

Furthermore,

(1 + b+) ln(1 + a+ b+ + ab+)

(1 + a+ b+) ln(1 + a+ b+)

=1 + b+

1 + a+ b+× ln(1 + b+)

ln(1 + a+ b+)× ln(1 + a)

> (1 + ε)−1 × (1 + ε)−1 × (1 + ε)2

= 1.

Thus, μ(b+) = μ1(b+)− μ2(b+) > 0. Since μ(0) <0, μ(b+) > 0, and μ(b) is a continuous function of b,based on the intermediate value theorem for continuousfunctions, there exists a positive real number b̂ ∈ (0, b+)

such that μ(̂b) = 0.6) Recall that f(gki−,j(t)) = φα+

(s(gki−,j(t)))−φα+

(s(mki−,j)). Since φα+

(b) is a continuous functionof b, φα+

(b) ∈ (0, 1], ∀ b ∈ [0,∞), φα+(0) = 1 =

limb→∞ φα+(b) and there exists b− ∈ (0,∞) such

that φα+(b−) < 1, minx:x≥0 φα+

(x) exists. Defineb∗ = argminx:x≥0 φα+

(x). Note that b∗ ∈ (0,∞).If mk

i−,j < b∗, since φα+(b∗) < φα+

(mki−,j) <

limb→∞ φα+(b), based on the intermediate value theorem

for continuous functions, there exists a positive realnumber r ∈ (b∗,∞) such that φα+

(r) = φα+(mk

i−,j)

and therefore f(r) = f(mki−,j) = 0. On the other hand,

if mki−,j > b∗, since φα+

(b∗) < φα+(mk

i−,j) < φα+(0),

there exists a positive real number r ∈ (0, b∗) such thatφα+

(r) = φα+(mk

i−,j) and therefore f(r) = 0. Thus,

both mki−,j and r are roots of the function f . Namely, the

function f has two or more positive roots.7) Based on 5 and 6, we have completed the proof.

REFERENCES

[1] X. Lin, J. G. Andrews, A. Ghosh, and R. Ratasuk, “An overview of 3GPPdevice-to-device proximity services,” IEEE Commun. Mag., vol. 52, no. 4,pp. 40–48, May 2014.

[2] D. Feng et al., “Device-to-device communications in cellular networks,”IEEE Commun. Mag., vol. 52, no. 4, pp. 49–55, May 2014.

[3] M. N. Tehrani, M. Uysal, and H. Yanikomeroglu, “Device-to-device com-munication in 5G cellular networks: Challenges, solutions, future direc-tions,” IEEE Commun. Mag., vol. 52, no. 5, pp. 86–92, May 2014.

[4] T. M. Cover and J. A. Thomas, Elements of Information Theory, 2nd ed.Hoboken, NJ, USA: Wiley, 2006.

[5] Y.-S. Liou, R.-H. Gau, and C.-J. Chang, “Group partition for capacity-region-aware device-to-device communications,” in Proc. IEEE WCNC,Apr. 2014, pp. 1708–1713.

[6] C.-H. Yu, K. Doppler, C. B. Ribeiro, and O. Tirkkonen, “Resource shar-ing optimization for device-to-device communication underlaying cellularnetworks,” IEEE Trans. Wireless Commun., vol. 10, no. 8, pp. 2752–2763,Aug. 2011.

[7] L. B. Le, “Fair resource allocation for device-to-device communicationsin wireless cellular networks,” in Proc. IEEE GLOBECOM, Dec. 2012,pp. 5451–5456.

[8] D. Feng et al., “Device-to-device communications underlaying cellu-lar networks,” IEEE Trans. Commun., vol. 61, no. 8, pp. 3541–3551,Aug. 2013.

[9] L. Lei, Y. Zhang, X. Shen, C. Lin, and Z. Zhong, “Performance analy-sis of device-to-device communications with dynamic interference usingstochastic Petri nets,” IEEE Trans. Wireless Commun., vol. 12, no. 12,pp. 6121–6141, Dec. 2013.

[10] Y. Chai, Q. Du, and P. Ren, “Partial time–frequency resource allocationfor device-to-device communications underlaying,” in Proc. IEEE ICC,Jun. 2013, pp. 6055–6059.

[11] D. H. Lee, K. W. Choi, W. S. Jeon, and D. G. Jeong, “Resource alloca-tion scheme for device-to-device communication for maximizing spatialreuse,” in Proc. IEEE WCNC, Apr. 2013, pp. 112–117.

[12] R. Zhang, X. Cheng, L. Yang, and B. Jiao, “Interference-aware graphbased resource sharing for device-to-device communications underlayingcellular networks,” in Proc. IEEE WCNC, Apr. 2013, pp. 140–145.

[13] B. Kaufman, J. Lilleberg, and B. Aazhang, “Spectrum sharing schemebetween cellular users and ad-hoc device-to-device users,” IEEE Trans.Wireless Commun., vol. 12, no. 3, pp. 1038–1049, Mar. 2013.

[14] J. Wang, D. Zhu, C. Zhao, J. C. F. Li, and M. Lei, “Resource sharing ofunderlaying device-to-device and uplink Cellular communications,” IEEECommun. Lett., vol. 17, no. 6, pp. 1148–1151, Jun. 2013.

[15] G. Fodor, D. D. Penda, M. Belleschi, M. Johansson, and A. Abrardo,“A comparative study of power control approaches for device-to-devicecommunications,” in Proc. IEEE ICC, Jun. 2013, pp. 6008–6013.

[16] H. Min, W. Seo, J. Lee, S. Park, and D. Hong, “Reliability improvementusing receive mode selection in the device-to-device uplink period under-laying cellular networks,” IEEE Trans. Wireless Commun., vol. 10, no. 2,pp. 413–418, Feb. 2011.

[17] H. Min, J. Lee, S. Park, and D. Hong, “Capacity enhancement using aninterference limited area for device-to-device uplink underlaying cellularnetworks,” IEEE Trans. Wireless Commun., vol. 10, no. 12, pp. 3995–4000, Dec. 2011.

[18] P. Bao and G. Yu, “An interference management strategy for device-to-device underlaying cellular networks with partial location information,”in Proc. IEEE PIMRC, Sep. 2012, pp. 465–470.

[19] M. C. Erturk, S. Mukherjee, H. Ishii, and H. Arslan, “Distributions oftransmit power and SINR in device-to-device networks,” IEEE Commun.Lett., vol. 17, no. 2, pp. 273–276, Feb. 2013.

[20] C. Xu et al., “Efficiency resource allocation for device-to-device underlaycommunication systems: A reverse iterative combinatorial auction basedapproach,” IEEE J. Sel. Areas Commun., vol. 31, no. 9, pp. 348–358,Sep. 2013.

[21] R. Zhang, L. Song, Z. Han, X. Cheng, and B. Jiao, “Distributed resourceallocation for device-to-device communications underlaying cellular net-works,” in Proc. IEEE ICC, Jun. 2013, pp. 1889–1893.

[22] F. Wang et al., “Energy-aware resource allocation for device-to-deviceunderlay communication,” in Proc. IEEE ICC, Jun. 2013, pp. 6076–6080.

[23] F. Wang, L. Song, Z. Han, Q. Zhao, and X. Wang, “Joint schedulingand resource allocation for device-to-device underlay communication,”in Proc. IEEE WCNC, Apr. 2013, pp. 134–139.

[24] S. Hakola, T. Chen, J. Lehtomaki, and T. Koskela, “Device-to-device(D2D) communication in cellular network—Performance analysis of op-timum and practical communication mode selection,” in Proc. IEEEWCNC, Apr. 2010, pp. 1–6.


[25] J. Seppala, T. Koskela, T. Chen, and S. Hakola, “Network controlleddevice-to-device (D2D) and cluster multicast concept for LTE and LTE-Anetworks,” in Proc. IEEE WCNC, Mar. 2011, pp. 986–991.

[26] T. Han, R. Yin, Y. Xu, and G. Yu, “Uplink channel reusing selectionoptimization for device-to-device communication underlaying cellularnetworks,” in Proc. IEEE PIMRC, Sep. 2012, pp. 559–564.

[27] B. Zhou, H. Hu, S.-Q. Huang, and H.-H. Chen, “Intracluster device-to-device relay algorithm with optimal resource utilization,” IEEE Trans.Veh. Technol., vol. 62, no. 5, pp. 2315–2326, Jun. 2013.

[28] Y. Pei and Y.-C. Liang, “Resource allocation for device-to-device commu-nications overlaying two-way cellular networks,” IEEE Trans. WirelessCommun., vol. 12, no. 7, pp. 3611–3621, Jul. 2013.

[29] D. Tse and P. Viswanath, Fundamentals of Wireless Communications.Cambridge, U.K.: Cambridge Univ. Press, 2005.

[30] R. Tanbourgi, H. Jakel, and F. K. Jondral, “Cooperative interference can-cellation using device-to-device communications,” IEEE Commun. Mag.,vol. 52, no. 6, pp. 118–124, Jun. 2014.

[31] D. G. Luenberger and Y. Ye, Linear and Nonlinear Programming, 3rd ed.New York, NY, USA: Springer-Verlag, 2008.

[32] G. Chartrand and O. R. Oellermann, Applied and Algorithmic GraphTheory. New York, NY, USA: McGraw-Hill, 1993.

[33] S. M. Ross, Simulation, 5th ed. New York, NY, USA: Academic Press,2012.

[34] Y. S. Cho, J. Kim, W. Y. Yang, and C. G. Kang, MIMO-OFDM Wire-less Communications With MATLAB. Hoboken, NJ, USA: Wiley-IEEEPress, 2010.

[35] D. Gross, J. F. Shortle, J. M. Thompson, and C. M. Harris, Fundamentalsof Queueing Theory, 4th ed. Hoboken, NJ, USA: Wiley, 2008.

Yi-Shing Liou received the B.S. degree in indus-trial technology education from National KaohsiungNormal University, Kaohsiung, Taiwan, in 2006, theM.S. degree in applied electronics technology fromNational Taiwan Normal University, Taipei, Taiwan,in 2009, and the Ph.D. degree in communicationsengineering from National Chiao Tung University,Hsinchu, Taiwan, in 2014. He is currently withthe Department of Electrical Engineering, NationalChiao Tung University. His research interests in-clude cross-layer medium access control in wireless

networks, device-to-device communications in cellular networks, and accessnetwork selection in heterogeneous networks.

Rung-Hung Gau received the B.S. degree in elec-trical engineering from National Taiwan University,Taipei, Taiwan, in 1994, the M.S. degree in electri-cal engineering from the University of California atLos Angeles, Los Angeles, CA, USA, in 1997, andthe Ph.D. degree in electrical and computer engi-neering from Cornell University, Ithaca, NY, USA,in 2001. He is currently a Professor with the Depart-ment of Electrical Engineering, National Chiao TungUniversity, Hsinchu, Taiwan. His research interestsinclude cross-layer design for medium access control

in wireless networks, device-to-device communications in cellular networks,machine-to-machine communications/Internet of Things, mobility manage-ment, multicast flow control, and stochastic processes and queueing theory withapplications to communications networks.

Chung-Ju Chang (F’06) was born in Taiwan in1950. He received the B.E. and M.E. degrees inelectronics engineering from National Chiao TungUniversity, Hsinchu, Taiwan, in 1972 and 1976, re-spectively, and the Ph.D. degree in electrical en-gineering from National Taiwan University, Taipei,Taiwan, in 1985. From 1976 to 1988, he was with theTelecommunication Laboratories, Directorate Gen-eral of Telecommunications, Ministry of Communi-cations, Taiwan, as a Design Engineer, Supervisor,Project Manager, and then Managing Director. He

also acted as a Science and Technology Advisor for the Minister of theMinistry of Communications during 1987 and 1989. In 1988, he joined theFaculty of the Department of Electrical and Computer Engineering, College ofElectrical Engineering and Computer Science, National Chiao Tung University,as an Associate Professor. He has been a Professor since 1993 and a ChairProfessor since 2009. He was the Director of the Institute of CommunicationEngineering from August 1993 to July 1995, Chairman of the Departmentof Communication Engineering from August 1999 to July 2001, Dean of theResearch and Development Office from August 2002 to July 2004, and Directorof the Center for Information and Communications Research, Aim for TopUniversity Plan, sponsored by the Ministry of Education from 2006 to 2010.Also, he was an Advisor for the Ministry of Education to promote the educationof communication science and technologies for colleges and universities inTaiwan during 1995 and 1999. He was acting as a Committee Member ofthe Telecommunication Deliberate Body, Taiwan. His research interests includeperformance evaluation, radio resources management for cellular mobile com-munication systems, and traffic control for broadband networks. He is a memberof the Chinese Institute of Engineers and the Chinese Institute of ElectricalEngineers. He serves as the Editor of the IEEE Communications Magazineand an Associate Editor of the IEEE TRANSACTIONS ON VEHICULAR TECH-NOLOGY. He was a recipient of the Outstanding Research Award in 2003 and2009 as well as the Outstanding Scholar Research Project in 2008 from theNational Science Council, Taiwan; the TECO Award from TECO TechnologyFoundation in 2006; and the Science and Technology Chair from Far EasternY.Z. Hsu Science and Technology Memorial Foundation in 2013.

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