1 an incentive mechanism integrating joint power, channel...
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An Incentive Mechanism Integrating Joint Power,Channel and Link Management for Social-Aware
D2D Content Sharing and Proactive CachingChangyan Yi, Student Member, IEEE , Shiwei Huang and Jun Cai, Senior Member, IEEE
Abstract—In this paper, a downlink cellular traffic offloading framework with social-aware device-to-device (D2D) content sharing andproactive caching is studied. In the considered system model, each user equipment (UE) is intelligent to determine which content(s) torequest/cache and to share according to its own preference. As the central controller, the base station (BS) can establish cellulartransmissions and/or incentivize D2D communications for content dissemination (including proactive caching). By taking into accountwireless features, social characteristics and device intelligence, we formulate a welfare maximization problem integrating power control,channel allocation, link scheduling and reward design. To solve this complicated problem, we propose a novel mechanism whichconsists of a newly developed optimization approach, called basis transformation method, for the joint resource management, and aspecially devised pricing scheme for the reward determination. Theoretical and simulation results examine the desired properties of ourproposed mechanism, and demonstrate its superiority in improving social welfare, network capacity and utility of the BS.
Index Terms—Offloading, D2D, proactive caching, social-awareness, incentive mechanism, joint resource management.
F
1 INTRODUCTION
R ESULTED from the proliferation of smart mobile devicesand emergence of various Internet-based applications
(e.g., Youtube, Facebook and Twitter), wireless data traf-fic has been increasing tremendously and is expected tocontinue growing in coming years [1]. This ever-increasingdemand in wireless access is rapidly straining the capacityof existing cellular systems so that it becomes imperativeto develop new wireless technologies to ease this burdenby enhancing the utilization efficiency of limited radio re-sources. Device-to-device (D2D) communication [2], as anunderlay to traditional cellular networks, has been envi-sioned as a key paradigm in 5G wireless communicationsto offload cellular traffic by encouraging content sharingamong user equipments (UEs). With D2D communications,network capacity can be significantly improved throughproper spectrum reuse and load balancing while at the sametime decreasing power consumption.
Recent studies [3] also revealed that social multimedia,which contributed 70% of overall wireless traffic, becamea major contributor to downlink traffic overload at cellularbase stations (BSs). Thus, to further promote the offloadingcapacity of D2D communications, one promising solution isto proactively cache/seed popular contents at some UEs byexploring and exploiting social awareness so as to increasethe opportunities of future D2D content sharing [4]. To makefull use of this, it is required to determine based on contentpopularities and users’ social influences i) which contentsneed to be cached, and ii) which UEs should be seeded.
However, since UEs are commonly smart devices (e.g.,
• C. Yi, S. Huang and J. Cai are with the Department of Electrical and Com-puter Engineering, University of Manitoba, Winnipeg, MB, Canada R3T5V6. E-mail: [email protected], [email protected],[email protected].
smart phones) which are carried and controlled by humansthrough highly developed human-computer interactions,they are ordinarily intelligent and selfish [5], [6], and maynot be willing to participate in D2D content sharing andproactive caching unless they can obtain enough incentives.Thus, it is the duty of the BS to maintain an appropriatemechanism to encourage UEs by offering rewards (in theform of monetary renumeration, free data access or servicepayment reduction) for their resource consumptions, suchas power and storage, caused by D2D transmissions.
Unfortunately, designing such mechanisms integratingall previous features is challenging due to following aspects.
• D2D communications can cause inevitable interfer-ence to cellular networks as a result of spectrumreuse. Such interference may exist between cellularand D2D links, or among different D2D links ifthey share a same channel. The existence of mutualinterference results in the interaction among mul-tiple UEs, which makes the resource managementproblem much more complicated. Thus, to guaranteesatisfied communication qualities, the coordinationof interference becomes necessary.
• For multi-user D2D communications, it is essentialto carefully i) control transmission power, ii) allocateavailable channels, and iii) schedule communicationlinks (i.e., the pairing of transmitters and receivers).However, jointly optimizing power control, channelallocation and link scheduling falls in the areas ofcombinatorial optimization and mixed integer pro-gramming, which is extremely difficult to be solved.
• Because of the inherent relationships between con-tent sharing and proactive caching procedures,the formulation of a joint framework requires theconsideration of all constraints related to both
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interference-aware D2D communications and social-aware caching policies, which makes the resourcemanagement even more complex.
• In a D2D-enabled cellular network, all information,such as channel gains, power costs and content avail-abilities, needs to be collected at the BS for facilitatingthe network management. However, due to the infor-mation asymmetry [5], the BS may not be aware ofall information possessed by UEs so that intelligentUEs may strategically misreport their own privateinformation in order to maximize their obtained re-wards in serving as D2D transmitters. Hence, a well-designed mechanism should not only reward UEsbased on their contributions in traffic offloading, butalso prevent them from any untruthful behaviors.
To address all aforementioned issues, in this paper,we study a downlink cellular traffic offloading frameworkand propose an efficient mechanism for social-aware D2Dcontent sharing with proactive caching. In the consideredsystem model, each UE can individually determine whichcontent(s) to request/cache and to share according to itsown preference (e.g., willingness/interests, buffer size andbattery status), and reports the BS its unit-power cost inoffering D2D service. As the central controller, the BS in turnmanages the content dissemination for all UEs by enablingboth cellular and D2D communications. With the objec-tive of maximizing the social welfare (including benefitsfrom content sharing and proactive caching, total powercosts of all UEs and the BS, and penalties for potentialservice dissatisfactions) so as to incentivize both the BSand UEs to participate in the system, we formulate anoptimization problem integrating power control, channelallocation, link scheduling and reward design. To solvethis complicated problem, we first relax constraints relatedto rewards (i.e., conditions for incentive compatibility andindividual rationality) and develop a novel approach, calledbasis transformation method, for joint power, channel and linkmanagement. After that, we propose a new reward schemebased on the solution of the joint optimization problem, andprove that it can satisfy all design requirements.
Main contributions of this paper are as follows:
• A joint design of resource management and incentivemechanism for social-aware D2D content sharingwith proactive caching is proposed, which can ef-fectively offload cellular traffic and can significantlyimprove system’s energy and spectral efficiency.
• A basis transformation method is developed tojointly optimize power, channel and link manage-ment in two steps. Although this 2-step approachcannot theoretically provide the optimality or ap-proximation guarantees, it can solve the problem inpolynomial-time and ensure a good performance asshown in simulations.
• Based on the solution to the joint optimization, anovel reward scheme is devised, which satisfies bothincentive compatibility and individual rationality.
• Simulations are conducted to show that our pro-posed mechanism can improve social welfare, net-work capacity and utility of the BS compared tocounterparts.
The rest of this paper is organized as follows: Section2 gives a review of related works. Section 3 describes theconsidered system model and the problem formulation. InSection 4, a basis transformation method for solving the jointoptimization of power control, channel allocation and linkscheduling is proposed. The design of reward scheme ispresented in Section 5. Simulation results are provided inSection 6, followed by conclusions in Section 7.
2 RELATED WORKS
As one of the key technologies to boost wireless networkcapacity, D2D communication has attracted a lot of researchattentions recently. For instance, Xu et al. in [7] introduced areverse iterative combinatorial spectrum auction to optimizethe system sum rate of D2D communications. Gu et al.in [8] designed an optimal proportionally fair schedulingscheme for maximizing the logarithmic sum of data ratesin D2D-cellular networks. However, both of them assumedthat transmission powers of all UEs were pre-known.
To achieve network welfare maximization with highenergy efficiency, different power control schemes for D2Dcommunications have been proposed in the literature. Forexample, Zhang et al. in [9] formulated a power allocationproblem, where the objective was to maximize the systemthroughput of D2D-assisted wireless caching networks. In[10], Cheng et al. developed an optimal power control algo-rithm in D2D-enabled wireless networks, where the quality-of-service requirements were taken into account. Both [9]and [10] addressed single-channel network scenarios only.Besides these, power control for D2D networks with im-perfect channel state information was investigated in [11],and a two-stage game theoretic framework integrating jointresource block and power allocations for interference mit-igation in D2D underlay cellular networks was built andanalyzed in [12]. Both [11] and [12] focused on uplink D2Dcommunications, while link scheduling for achieving besttransmitter and receiver pairing was ignored.
With the rapid development of mobile social networks,social-aware proactive caching/seeding has become increas-ingly popular in traffic offloading. Taghizadeh et al. in[4] presented a distributed cooperative caching scheme forminimizing content provisioning cost in social wireless net-works. Wang et al. in [13] constructed a traffic offload-ing framework, in which a group of UEs were chosenas seeds based on their social influences. In [14], Gregoriet al. illustrated a jointly transmission and caching policyfor D2D-enabled small cell networks. In [15], Jiang et al.focused on maximizing cellular traffic offloading with D2Dcommunications via content caching and user pairing. How-ever, in these works, either the inherent interaction betweencaching and D2D systems was ignored [4], [13], or it wasassumed that the system models under consideration wereinterference-free [14], [15].
Incentive mechanism design has been widely discussedin various wireless networks for encouraging intelligent andselfish mobile users to participate in the system manage-ment. Yi et al. in [16] developed a recall-based spectrumsharing mechanism for heterogeneous secondary users incognitive radio networks. Zhang et al. in [5] proposed
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TABLE 1IMPORTANT NOTATIONS IN THIS PAPER
Symbol MeaningN set of active UEsM set of downlink channelsNR, NC set of service and content requesters(i, j) transmission link from ∀i ∈ N ∪BS to ∀j ∈ Nxi,j,m indicator of joint link scheduling and channel allocationPi,j,m transmission power assigned to (i, j) on channel mGi,j,m channel gain of link (i, j) on channel mγj SINR at receiver UE jqj content request from UE jδj penalty for service dissatisfaction to service requester jIj social impact of receiver UE jQi,share set of contents that UE i is willing to shareci unit-power cost of transmitter iπi reward to UE i for serving as a D2D transmitterUi utility function of D2D transmitter iS network social welfareNa set of admitted receivers in the system
a contract-theoretic approach for incentivizing D2D com-munications in cellular networks. However, most mecha-nisms in the literature [17]–[20] were designed for single-dimensional resource management problem so that theycannot be applied in our considered model, where a three-dimensional management with joint power control, channelallocation and link scheduling has to be addressed.
Different from existing works, this paper addresses fol-lowing issues specified to social-aware content sharing andproactive caching in D2D-enabled cellular networks.
• Downlink cellular traffic offloading problem withboth social-ware proactive caching and interference-aware D2D communications is investigated.
• Joint optimization of power control, channel alloca-tion and link scheduling is formulated and solvedwith the aim of maximizing the network welfare.
• Considering the intelligence and selfishness of mo-bile users, a reward scheme is designed to motivatethe participation of the BS and UEs in a truthful way.
3 SYSTEM MODEL AND PROBLEM DESCRIPTION
In this section, the system model for social-aware D2D con-tent sharing and proactive caching is first described. Then,strategies of UEs and the utility of the BS are defined. Afterthat, the mechanism design problem with joint resourcemanagement is formulated. For convenience, Table 1 listssome important notations used in this paper.
3.1 D2D-Enabled Communication Model
Consider a downlink cellular network, which consists ofone BS, a set N of active UEs and a set M of orthogonalchannels. The time is partitioned into frames1. At the begin-ning of any time frame t, each UE independently declares arequest for one piece of content to the BS for either fulfilling
1. Here, a time frame defines a period of content sharing andproactive caching. Such frame-by-frame content sharing and cachingframework has been widely discussed in the literature [14], [21], [22].
BS
UE4UE5
UE1
Service Requester
Caching Requester
Content Sharing
Proactive Caching
Interference
UE2
UE3
Fig. 1. An illustration of D2D content sharing and proactive caching.
its own service requirement or proactive caching purpose2.We categorize all UEs at time frame t into two sets N t
R andN tC , which stand for service and caching requesters, respec-
tively. After receiving the service/caching request from aUE, the BS decides whether to serve it or not, and choosesto either establish a cellular transmission link (i.e., BS →UE) by adopting OFDMA or incentivize another UE whoowns the same content to share through a D2D link (i.e., UE→ UE) in an underlay access manner. Mutual interferenceexists among links assigned on a same channel. We furtherassume that packet lengths of contents are heterogeneous,and each UE has a finite buffer for content storage.
An example of the considered D2D-enabled communica-tion model is illustrated in Fig. 1. In this figure, {UE1, UE3,UE5} and {UE2, UE4} are service and caching requesters,respectively. Suppose that a cellular link (i.e., BS → UE1)and two D2D links (i.e., UE3 → UE2 and UE4 → UE5)share a same channel for transmissions. Then, receiversUE1, UE2 and UE5 are interfered by transmitters {UE3,UE4}, {BS, UE4} and {BS, UE3}, respectively. In addition,since the spectrum resource is limited, it is also possiblethat the service/caching requests from UEs may not beserved. However, the dissatisfaction due to unserved servicerequests may result in penalty to the BS (which will befurther discussed in Section 3.4).
Similar to [7], [23], in this paper, each downlink channelcan be accessed by one cellular link and multiple D2Dlinks simultaneously. Furthermore, all UEs are consideredto have a full-duplex communication mode [24]. Due tothe hardware limitations, we follow the convention in theliterature [9], [15] that there are at most one outgoing linkand one incoming link at each UE, where each link workson a single channel.
Let xti,j,m ∈ {0, 1} be an indicator of the joint linkscheduling and channel allocation for the data transmissionfrom transmitter i ∈ N ∪ BS to receiver j ∈ N on channelm ∈ M in time frame t. xti,j,m = 1 means that transmitter iand receiver j are paired to set up a communication link,denoted by (i, j), occupying channel m during the timeframe t, and xti,j,m = 0, otherwise. Then, in any time frame
2. From the view of the BS, serving caching requesters can proactivelyseed most popular contents to a certain number of socially popular UEsso as to increase the opportunities of future traffic offloading. From theview of the UE, proactively caching the popular content can increaseits chance of being a future D2D transmitter so as to obtain rewards tobenefit itself.
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t, we must have∑i∈N∪BS
∑m∈M
xti,j,m ≤ 1, ∀j ∈ N , (1)∑j∈N
∑m∈M
xti,j,m ≤ 1, ∀i ∈ N , (2)∑j∈N
xtBS,j,m ≤ 1, ∀m ∈M, (3)
where (1) and (2) denote that there are at most one incominglink and one outgoing link for any receiving UE j andtransmitting UE i, respectively. Inequality (3) implies thatthe BS can establish multiple cellular links, but each of themhas to be scheduled on a distinct channel.
Besides, the BS or each UE has a total power budgetfor transmission in each time frame t, namely P t,maxi ,∀i ∈N ∪BS. Thus, we have
0 ≤∑j∈N
∑m∈M
xti,j,mPti,j,m ≤ P
t,maxi , ∀i ∈ N ∪BS, (4)
where P ti,j,m ≥ 0 denotes the transmission power assignedto the transmitter of link (i, j) on channel m in frame t.
The BS is responsible to jointly determine link, chan-nel and power management, i.e., xti,j,m and P ti,j,m,∀i ∈N ∪ BS, ∀j ∈ N ,∀m ∈ M,∀t. Similar to [7], [25], weassume that the BS has a full acquisition of the instantaneouschannel state information (CSI) of all links, which can beestimated by each receiver and then fed back to the BS viadedicated control channels. Note that compared to regu-lar data transmissions (for content sharing), the overheadcaused by control signalling (e.g., the report of CSI) isgenerally negligible [26]. Moreover, by applying techniques,such as CSI feedback compression and signal flooding [7],this overhead can be further reduced. Thus, in this paper,the cost in collecting CSI is ignored.
Consider independent block fading on each channel sothat channel coefficients remain invariant during one timeframe (but may vary from one frame to another). Then, theinstantaneous channel gain Gti,j,m between transmitter i ∈N ∪BS and receiver j ∈ N on channel m ∈M in any timeframe t can be formulated as
Gti,j,m = |hti,j,m|2(dti,j)−η
, (5)
where |hti,j,m|2 captures the Rayleigh fading effect and iscommonly modeled as an exponentially distributed randomvariable with a unit mean; dti,j is the distance betweentransmitter i and receiver j in time frame t, so that (dti,j)
−η
signifies the path loss effect, where the path loss exponentη ≥ 2. Since all transmission links are interfered witheach other if they share a same channel, the signal-to-interference-plus-noise ratio (SINR) at receiving UE j intime frame t can be expressed as
γtj =P tdes,j
P tint,j + σ2, ∀j ∈ N , (6)
where P tdes,j and P tint,j denote desired signal power andinterference at UE j in time frame t, respectively; σ2 rep-resents the additive white Gaussian noise variance. Let qtjdenote the content that is requested by UE j,∀j ∈ N , intime frame t, and Nqtj be the set of all UEs which own and
would like to share content qtj . Then, we can calculate P tdes,jand P tint,j as
P tdes,j=∑
i∈Nqtj∪BS
∑m∈M
xti,j,mPti,j,mG
ti,j,m, (7)
P tint,j=∑
i∈Nqtj∪BS
∑i′∈N∪BS\{i}
∑j′∈N
∑m∈M
xti,j,mxti′,j′,mP
ti′,j′,mG
ti′,j,m,(8)
where i represents the transmitter who is paired with re-ceiver UE j (and thus i is chosen from Nqtj ∪ BS), and i′
represents the transmitter that interferes receiver UE j.According to the Shannon capacity formula, the receiv-
ing data rate of UE j in frame t is
Rtj = W log(1 + γtj), ∀j ∈ N , (9)
where W is the bandwidth of each channel. Traditionally,wireless content sharing networks impose a requirementthat a piece of content cannot be split or partially transmit-ted [14], [15]. Hence, the achievable transmission rate fromUE i to UE j has to satisfy a condition that
Rtj ≥( ∑i∈N∪BS
∑m∈M
xti,j,m
)· Lqtj/T, ∀j ∈ N ,∀t, (10)
where Lqtj and T denote the size of content qtj and the lengthof time frame, respectively. Inequality (10) indicates that ifUE j is scheduled as a receiver for content qtj during timeframe t, its requested content has to be completely receivedbefore the end of the time frame no matter whether thecellular or D2D link is used. With a simple manipulation,condition (10) can be rewritten as
γtj ≥ γtj,th ·( ∑i∈N∪BS
∑m∈M
xti,j,m
), ∀j ∈ N ,∀t, (11)
where γtj,th can be derived from (9) and (10) as
γtj,th = 2Lqtj/TW
− 1. (12)
3.2 Social AwarenessIn this subsection, social awareness is investigated. Specifi-cally, content popularities and user impacts are defined.
3.2.1 Content PopularityLet Q = {q1, q2, . . . , qK} be the set of all available contents,where subscript k ∈ {1, 2, . . . ,K} represents the particularrank of content popularity. Here, we define that a contentqk ∈ Q is more popular (or likely to be requested morefrequently) if it has a higher ranking, i.e., a smaller value ofk. Then, the popularity of content qk ∈ Q can be modeledby the Zipf distribution [4] as
fk =1/kβ∑K
k′=1 1/k′β, (13)
where fk describes the probability of qk being interested byUEs, and β ≥ 0 is a predefined Zipf exponent reflecting theskew of the popularity distribution.
3.2.2 User ImpactsFor each UE i ∈ N , its social impact results from twoaspects: i) spreading impact [13], denoted by Φi, which eval-uates the potential of UE i in propagating contents to others
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in social networks; and ii) mobility impact [6], denotedby Ψi, which measures the capability of UE i in sharingcontents via physical encounters. Both Φi and Ψi can bemathematically expressed according to the statistic infor-mation of users’s behaviors observed from history. Sincethe focus of this paper is on the joint resource managementand incentive mechanism design, we omit the discussions ofderiving Φi and Ψi and suggest interested readers to referto [6], [13] for detailed steps.
3.3 Strategies of UEs
Since UEs are ordinarily intelligent, their individual strate-gies have to be carefully taken into account in the re-source management for D2D content sharing and proactivecaching. Though each UE can act as a receiver (in eithercellular or D2D link) and a transmitter (in D2D link) simul-taneously, its content receiving and transmitting processescan actually be treated separately within any time frame.Therefore, in this subsection, we explore the strategies ofeach UE in different roles (i.e., receiver and transmitter).
3.3.1 As a receiverSince there is at most one incoming link at each UE for onecontent reception, each UE j ∈ N can request at most onecontent from the set of all available contents Q in each timeframe t. If UE j is a service requester, i.e., j ∈ N t
R, thenits content request qtj is directly determined by its servicerequirement. While, for UE j as a caching requester, i.e.,j ∈ N t
C , it will select its content request qtj based on a localoptimization by considering heterogeneities in content sizesand its own buffer limit as
qtj = arg maxqk∈{Q\Qtj∪∅}
f tj,total(Qtj ∪ qk), (14)
where Qtj is the set of existing contents cached by UE j atthe beginning of time frame t, and thus Q\Qtj ∪ ∅ includesall contents that may be interested to UE j in frame t. Notethat qtj = ∅ if UE j has no further caching demand. Here,f tj,total(Qtj ∪ qk) represents the total popularity of contentsin UE j’s buffer after time frame t, and is obtained from
f tj,total(Qtj ∪ qk) = maxθk′
∑qk′∈Qtj∪qk
fk′ · θk′ (15)
s.t.,∑
qk′∈Qtj∪qkLk′ · θk′ ≤ Cj , (16)
θk′ ∈ {0, 1}, (17)
where fk′ and Lk′ are popularity and size of content qk′ ,respectively; Cj stands for the buffer limit of UE j; θk′ is abinary decision which indicates whether content qk′ shouldbe stored or not, and hence (16) means that the total sizeof all contents stored by UE j should not exceed its bufferlimit. Obviously, (15) is a single knapsack problem whichcan be solved easily by using dynamic programming [27].
3.3.2 As a transmitterTo join the network as a potential transmitter of D2D com-munications in time frame t, each UE i ∈ N is required toreport a set of contents that it is willing to share, denoted byQti,share, and its unit-power cost cti. Intuitively, Qti,share isdetermined by the buffer status of UE i and its willingness
of sharing at the beginning of time frame t. In other words,UE i can determine Qti,share as any subset of contents thatare stored in its buffer for sharing based its own interest. ctiis related to a variety of factors, such as the battery status,charging cost and application purposes, and it is also aprivate information of each UE i. Unlike Qti,share, cti hasto be truthfully elicited by the BS to facilitate the resourcemanagement and the reward design. However, it is possiblethat UE i, as an intelligent entity, may strategically reportcit 6= cti if and only if it can benefit from such behavior. By
considering the total power cost and reward in serving as aD2D transmitter, the net utility of each UE i,∀i ∈ N , in timeframe t, with actual unit-power cost cti but reporting cit canbe expressed as
U ti (cit|cti) =
∑j∈N
∑m∈M
xti,j,m(πti − P ti,j,mcti), (18)
where xti,j,m, P ti,j,m and πti are the joint link and channelscheduling, power assignment and reward from the BS forincentivizing D2D communications, respectively. Note thatthe BS is unaware of the truthful cti, and thus all its decisions(i.e., xti,j,m, P ti,j,m and πti ) are based on the reported ci
t.Clearly, the first term
∑j∈N
∑m∈M xti,j,mπ
ti indicates the
reward, and the second term∑j∈N
∑m∈M xti,j,mP
ti,j,mc
ti
shows the total power cost.As essential requirements for guaranteeing robustness of
a mechanism, the following conditions have to be met.
• Individual rationality: UEs can always obtain non-negative utilities in the system, i.e.,∑j∈N
∑m∈M
xti,j,m(πti−P ti,j,mcti) ≥ 0, ∀i ∈ N ,∀t. (19)
• Incentive compatibility: No UE can improve its util-ity by misreporting its unit-power cost, i.e., cti =arg maxcit U
ti (ci
t|cti),∀i ∈ N , or equivalently
U ti (cit|cti) ≥ U ti (cit|cti), ∀i ∈ N ,∀t. (20)
3.4 Utility of the BS
According to previous discussions, the BS may suffer cost ineach time frame t from
1) the power consumption for cellular transmissions:∑j∈N
∑m∈M xtBS,j,mP
tBS,j,mc
tBS , where ctBS is the
unit-power cost of the BS;2) the total reward paid to UEs for traffic offloading:∑
i∈N∑j∈N
∑m∈M xti,j,mπ
ti ;
3) the penalty for possible service dissatisfactions:∑j∈N tR
(1 −∑i∈N∪BS
∑m∈M xti,j,m)δtj , where δtj is a
pre-determined penalty for breaking the service con-tract with UE j ∈ N t
R.
Besides, the BS can also gain benefits through contentsharing and proactive caching, and such gain mainly resultsfrom seeding contents to UEs for future D2D offloadingopportunities. Clearly, the potential benefit introduced byeach UE j in each time frame t, denoted by Itj , dependson UE j’s spreading impact Φj and mobility impact Ψj , its
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requested content popularity fqtj , and unit-power cost ctj .Thus, we define
Itj = F(
ΦjΨjfqtjctj
), ∀j ∈ N ,∀t, (21)
where F(·) is a pre-known general increasing functionwhich matches the intuition that the potential of each UEin D2D offloading increases with its social influence, butdecreasing with its power cost. Some special examples ofF(·) can be found in [28], [29].
Therefore, the net utility of the BS in time frame t is
U tBS =∑
i∈N∪BS
∑j∈N
∑m∈M
xti,j,mItj−( ∑j∈N
∑m∈M
xtBS,j,mPtBS,j,mc
tBS
+∑i∈N
∑j∈N
∑m∈M
xti,j,mπti +∑j∈N tR
(1−∑
i∈N∪BS
∑m∈M
xti,j,m)δtj
).
Note that in order to encourage the BS to run the mecha-nism, we should also guarantee
U tBS ≥ 0, ∀t. (22)
3.5 Optimization Problem Formulation
Define the social welfare of the considered D2D contentsharing and proactive caching system in each frame t as
St = U tBS +∑i∈N
U ti
=∑
i∈N∪BS
∑j∈N
∑m∈M
xti,j,mItj−∑
i∈N∪BS
∑j∈N
∑m∈M
xti,j,mPti,j,mc
ti
−∑j∈N tR
(1−∑
i∈N∪BS
∑m∈M
xti,j,m)δtj . (23)
Our objective is to maximize the social welfare by de-signing efficient joint resource management and rewardscheme. Thus, the overall optimization problem can beformulated as
maxXt,P t,Πt
St (24)
s.t., (1)− (4), (6)− (8), (11), (12), (19), (20), (22),
xti,j,m ∈ {0, 1}, P ti,j,m ≥ 0,∀i ∈ N ∪BS, ∀j ∈ N ,∀m, (25)
πti ≥ 0, ∀i ∈ N , (26)
where decisions are Xt = {xti,j,m}i∈N∪BS,j∈N ,m∈M, P t ={P ti,j,m}i∈N∪BS,j∈N ,m∈M and Πt = {πti}i∈N . However,solving this problem is extremely challenging due to thefollowing facts: i) Because of constraints (1) – (3), the prob-lem become an independent set problem in graph theory,and unfortunately the considered graph does not have anyspecial form such as bipartite, tree or planar graphs; ii)Even given the link scheduling (i.e., independent set), de-termining the optimal channel allocation is still intractablebecause of the complicated interference constraint in (8)and abundant user-channel combinations, which falls in thearea of combinatorial optimization [26]; and iii) Even aftera relaxation of removing all constraints except (1)–(3), (25)and (26), our problem turns to find the maximum weightindependent sets (MWIS), which is well-known as NP-hardness and is also difficult in implementing any approxi-mation [30]. To address all these challenges, in the following
sections, novel solution procedures are proposed. Specifi-cally, we first develop a basis transformation method withlow-computational complexity to solve the joint resourcemanagement problem, and then design a correspondingreward scheme which not only maintains the same socialwelfare, but also satisfies incentive compatibility and in-dividual rationality. Since this optimization is carried outindependently for each time frame, we omit the subscript tfrom all notations in the following context for clarity.
4 JOINT POWER CONTROL, CHANNEL ALLOCA-TION AND LINK SCHEDULING
In this section, we study the joint optimization of powercontrol, channel allocation and link scheduling, i.e., X andP , by ignoring the reward design. Note that since rewardsare not included in the objective function (23) but onlyincluded in constraints (19), (20) and (22), decoupling theresource management and the reward design will not affectthe resulted social welfare, if the determination of Π dis-cussed in the next section is based on given X and P .
4.1 Reformulation and Decomposition of the ProblemIt can be observed that the first and third terms of theobjective function (23) only depend on the set of receiversadmitted in the system regardless of any other decisions.This means that these two terms will become constants ifthe set of admitted receivers (denoted by Na) are given. Forthe second term, the integer variable xi,j,mj can be removedbecause it can be determined by the value of Pi,j,mj . Thatis, if Pi,j,mj > 0, then xi,j,mj = 1, i.e., receiver j is paired totransmitter i using channel mj , and xi,j,mj = 0, otherwise.The constraints on xi,j,mj (i.e., (1) – (3)) can be convertedto impose constraints on Pi,j,mj . In this way, the objectivefunction becomes linear in Pi,j,mj , which makes the prob-lem more tractable. To capitalize on this, we reformulate ourproblem by considering Na as a new decision variable andrelaxing reward-related constraints (19), (20) and (22) as
Main Problem:
minma,Pa,Na
∑j∈N
∑i∈Nqj∪BS
ciPi,j,mj+∑j∈Na
Ij −∑
j∈NR\{Na∩NR}
δj (27)
s.t., (1)− (3),
γj =
∑i∈Nqj∪BS
Gi,j,mjPi,j,mj∑k∈N\{j}
∑i∈Nqk∪BS
1(mj = mk)Gi,j,mkPi,k,mk + σ2,
γj ≥ γj,th, ∀j ∈ Na, (28)∑j∈N
Pi,j,mj ≤ Pmaxi , ∀i ∈ Nq,a ∪ {BS}, (29)
ma ≥ 0,Pa ≥ 0, (30)
where Nq,a = ∪j∈NaNqj denotes the set of all potentialUEs being able to and willing to transmit their contents toother UEs. The variable mj stands for the channel allocatedto receiver j ∈ Na, and ma = [m1,m2, · · · ,m|Na|]T is thevector form of channel allocation variables. 1(mj = mk)is a binary indicator variable: 1(mj = mk) = 1 ifmj = mk, and 1(mj = mk) = 0, otherwise. Thepower variables are grouped into the vector form Pa =[Ps11,1,m1
, Ps21,1,m1, · · · , Ps1|Na|,|Na|,m|Na| , Ps2|Na|,|Na|,m|Na| , · · · ]
T ,
7
where sτj ∈ Nqj ∪ BS denotes one of potential transmittersto receiver j ∈ Na and τ ∈ {1, 2, · · · , |Nqj | + 1} isthe dummy index enumerating all potential transmitters.Constraints (28) and (29) are the updated SINR requirementsand power budgets from (11) and (4), respectively.Furthermore, we can rewrite constraints (1) – (3) as
• Condition 1: For any receiver j ∈ Na, at most one of{Pi,j,mj |∀i ∈ Nqj ,∀mj ∈ M} is greater than zero,i.e., there is at most one incoming link;
• Condition 2: For any transmitter i ∈ Nq,a, at most oneof {Pi,j,mj |∀j ∈ Na,∀mj ∈ M} is greater than zero,i.e., there is at most one outgoing link;
• Condition 3: For the BS , at most one of{PBS,j,mj |∀j ∈ Na} is greater than zero, i.e., thereis at most one cellular link on each given channelmj .
Obviously, the reformulated problem (called the “MainProblem” hereinafter) is still NP-hard. Thus, to solve thisproblem, we introduce the following 2-step approach: i) thefirst step is to determine an optimal set of admitted receiversNa by considering the impact of users’ potential benefits Ijand penalties δj , and establishing an initial feasible resourcemanagement scheme; and ii) the second step is to furtheroptimize the resource management given the set Na withthe objective of power cost minimization.Remark 1. This 2-step approach may not guarantee optimality
because admitted receivers with larger Ij and δj in the firststep may require transmitters in higher power consumptions,leading to the increase of power cost in the second step.However, as proved later on, our proposed solution can sig-nificantly reduce the the complexity of the original problem.Moreover, since each step jointly optimizes link, channel andpower allocations, this approach results in a relatively goodperformance as shown in simulations.
For the first step, in order to determine whether each UEj ∈ N should be included in the set Na as a receiver inthe system, we have to take into account both its potentialoffloading benefit Ij if it is admitted in the system and thepossible penalty δj if not. Besides, we have to guarantee thatthe set of receivers admitted in the system is feasible underthe constraints on power and spectrum resources. Hence,our objective is to allow the system accommodating as manyrequesters with larger Ij and δj as possible while ensuringthe feasibility of the main problem by jointly adjustingpower control, channel allocation and links scheduling.Therefore, we can formulate our first subproblem as
Subproblem 1: minm,P ,y,a,z
∑j∈N
(Ij + δj)aj (31)
s.t.,∑
i∈Nqj∪BS
Gi,j,mjγj,th
Pi,j,mj − yj + aj (32)
−∑
k∈N\{j}
∑i∈Nqk∪BS
1(mj=mk)Gi,j,mkPi,k,mk =σ2,∀j ∈ N ,
∑j∈N
Pi,j,mj + zi = Pmaxi , ∀i ∈ Nq ∪BS, (33)
m,P ,y,a, z ≥ 0, Conditions 1, 2 and 3, (34)
where m = [m1,m2, · · · ,m|N |]T and P = [Ps11,1,m1,
Ps21,1,m1, · · · , Ps1|N|,|N |,m|N| , Ps2|N|,|N |,m|N| , · · · ]
T ; (32) and
(33) are the equality forms of SINR constraint (28) andpower constraint (29), respectively. yj is the surplus vari-able, aj is the artificial variable and zi is the slack vari-able. y = [y1, y2, · · · , y|N |]T , a = [a1, a2, · · · , a|N |]Tand z = [zs1 , zs2 , · · · , zs|Nq| , zBS ]T are the vectors ofsurplus, artificial and slack variables, respectively, wheresτ ∈ Nq = ∪j∈NNqj is one of all potential transmitterand τ = 1, 2, · · · , |Nq| is the dummy index. Note that insubproblem 1, the surplus variable yj can be set as zero inthe optimal solution, and thus it can be removed withoutloss of generality. This is because if yj > 0 and aj > 0,we can always find a smaller aj making yj = 0 to satisfythe constraint (32) and at the same time achieve a smallerobjective function value. On the other hand, if yj > 0,Pi,j,mj > 0 and aj = 0, we can also find a smaller Pi,j,mjallowing yj = 0 to meet the SINR requirement (32) andkeep the objective function value unchanged. Finally, Na isobtained from the optimal artificial variables a∗, where UE jis chosen as a receiver only if a∗j = 0, and a∗j > 0, otherwise.
Given the set Na obtained from subproblem 1, we canformulate the second subproblem which aims to minimizethe total power cost with joint resource management as
Subproblem 2: minma,Pa,za
∑j∈Na
∑i∈Nqj∪BS
Pi,j,mjci (35)
s.t.,∑
i∈Nqj∪BS
Gi,j,mjγj,th
Pi,j,mj (36)
−∑
k∈N\{j}
∑i∈Nqk∪BS
1(mj=mk)Gi,j,mkPi,k,mk =σ2,∀j ∈ Na,∑j∈Na
Pi,j,mj + zi = Pmaxi , ∀i ∈ Nq,a ∪BS, (37)
ma,Pa, za ≥ 0, Conditions 1, 2 and 3, (38)
where za = [zs1 , zs2 , · · · , zs|Nq,a| , zBS ]T . Subproblem 2 issimilar to the main problem except thatNa is already knownand all constraints are written in the equality forms. Notethat the artificial variable aj that was included in constraint(36) has been removed because we must have a∗j = 0 in theoptimal solution. Otherwise, we can always find a smallerPi,j,mj allowing aj = 0 to improve the objective function.
It can observed that subproblems 1 and 2 have a similarstructure and both of them become linear programming ifconditions 1, 2 and 3 are relaxed and the channel alloca-tion is given. One initial feasible solution to subproblem1 can be directly obtained by letting aj = σ2,∀j ∈ N ,zi = Pmaxi ,∀i ∈ Nq ∪ BS, P = 0, and m randomly overM. While an initial feasible solution to subproblem 2 isgiven by the final solution to subproblem 1 with aj = 0,i.e., the initial ma, Pa and za of subproblem 2 are part ofm∗, P ∗ and z∗ of subproblem 1, respectively. Starting fromthe initial feasible solutions, both subproblems 1 and 2 canbe solved by following the same algorithm. Thus, in the nextsubsection, we will only take the solution to subproblem 1as an example to describe our developed method.
4.2 Basis Transformation Method
The basic idea of the proposed method is to introduce thebases of subproblem 1 where each basis determines onefeasible solution. It starts from an initial feasible basis and
8
Initialize the basis B
Compute the solution under the basis B,
i.e., (42)
Update B by replacing the column of leaving variable
by the entering one
Optimality conditions (43) are satisfied?
Start
End
Find a pair of entering and leaving variables,
i.e., (45)
Yes
No
Fig. 2. The flow chart of the simplex method.
gradually transforms the basis to improve the objectivefunction until convergence. To this end, we first introducean auxiliary problem by relaxing conditions 1-3 and givingan arbitrary channel allocation m as follows:
Auxiliary Problem: minχ
wTχ (39)
s.t., Aχ = b,χ ≥ 0. (40)
Here, χ = [P ;a; z] is the vector of joint power, ar-tificial and slack variables, where the notation “;” de-notes the column extension. w = [0Ns ; I + δ;0|Nq|+1]is the coefficient vector of the objective function, whereI = [I1, I2, · · · , I|N |]T , δ = [δ1, δ2, · · · , δ|N |]T , Ns =∑j∈N (|Nqj |+ 1), and 0k (1k) denotes the all-zero (all-one)
column vector with dimension k. b = [σ21|N |;Pmax]T is
the right hand side (RHS) constant vector of constraints,where Pmax = [Pmax1 , P2,
max , · · · , Pmax|Nq| , PmaxBS ]T . A
is the coefficient matrix associated with constraints (32)and (33), where the coefficient column associated withvariable Pi,j,mj is A[Pi,j,mj ] = [[−Gi,1,mj1(mj =m1), · · · , −Gi,j−1,mj1(mj = mj−1), Gi,j,mj/γj,th,−Gi,j+1,mj1(mj = mj+1), · · · , −Gi,N,mj1(mj = mN )]T ;0i−1; 1; 0|Nq|−i+1]. Similarly, the coefficient column associ-ated with variable aj is A[aj ] = [0j−1; 1;0N−j ;0|Nq|+1],the column with respective to zsτ is A[zsτ ] =[0N ;0τ−1; 1;0|Nq|−τ+1], sτ ∈ Nq , and the column associ-ated with zBS is A[zBS ] = [0N ;0|Nq|; 1].
The auxiliary problem can be solved by the traditionalsimplex method [31], and the flow chart of which is shownin Fig. 2. Starting from the initial feasible basis B =[A(a1), · · · , A(aN ), A(zs1), · · · , A(zs|Nq|), A(zBS)], we cancompute the associated solution under the basis B as
χB = B−1b ≥ 0, χN = 0, (41)
where χB and χN are basic and non-basic variable vectors,respectively. Then, we need to check whether this solutionsatisfies the optimality conditions [31]
uN = wN −NTλ ≥ 0, (42)
where N is the constraint coefficient matrix associatedwith non-basic variables, i.e, A = [B,N ], and wN is thecorresponding coefficient vector of the objective function. λis the dual solution of the auxiliary problem, where
λ = (BT )−1wB. (43)
If the optimality conditions are violated, it is required toupdate the basis B by replacing one of its columns withthat of N such that the objective is improved. This processis known as pivot operation and is described as follows.
Pivot Operation for the auxiliary problem: The leavingcolumn from the basis B and the new entering columnare determined by the associated leaving and entering vari-ables. The potential entering variable could be χe such thatue < 0, e ∈ IN , where IN denotes the index set of non-basicvariables χN . Based on (40) with χN = 0 except the newentering variable χe, we have Bχ′B +Aeχe = b, where Ae
is the coefficient column associated with variable χe. Whenthe entering variable χe is allowed to increase from zero,the original basic variables are decreased accordingly fromχB = B−1b toχ′B = B−1b−B−1Aeχe = χB−B−1Aeχe.Let ρ = B−1Ae. We have χ′B = χB − ρχe. The in-crease of the entering variable should satisfy χ′B ≥ 0, i.e.,χB − ρχe ≥ 0. This is equivalent to require χe ≤ χi/ρifor any ρi > 0 (note that if ρi ≤ 0, the corresponding basicvariable is always no less than zero no matter how much χeincreases from zero, i.e., χ′i = χi − ρiχe ≥ 0). Therefore, theleaving variable index can be determined by
l = arg mini{χi/ρi|ρi > 0, i ∈ IB}, (44)
where IB is the index set of basic variables χB .Modified Pivot Operation for subproblem 1: To solve
subproblem 1, the pivot operation has to be modified tosatisfy conditions 1-3 as well as allowing the change ofchannel allocation m.
a) Guarantees of Conditions 1-3: The conditions 1-3 poseadditional limitations on the pivot operations of the powerentering variables Pi,j,mj , i ∈ Nq ∪ BS, j ∈ N . To easethe explanation, we first define the power variable setassociated with receiver j,∀j ∈ N as Pr,j = {Pi,j,mj |i ∈Nqj ∪ {BS}}. Similarly, define the power variable set asso-ciated with transmitter i,∀i ∈ Nq as Pt,i = {Pi,j,mj |j ∈ N}.For the transmitter BS and channel m, define the powervariable set associated with the BS and channel m asPt,BS,m = {PBS,j′,mj′ |mj′ = m, j′ ∈ N}. Then, thefeasibility of any potential entering variable Pi,j,mj can bedetermined by the following 4 cases:
• Case I: There is already one basic variable in bothPr,j and Pt,i (Pt,BS,mj , if i = BS), denoted byPi′,j,mj , i′ 6= i and Pi,j′,mj′ , j
′ 6= j (mj′ = mj ifi = BS), respectively. The variable Pi,j,mj cannot bean entering variable and should be removed fromthe candidate entering variable set. This is becausethere is at most one leaving variable at each pivotoperation. If the variable Pi,j,mj joins in the basicvariable set, there will exist two basic variables inPr,j or Pt,i (Pt,BS,mj if i = BS), which violatesCondition 1 or Condition 2 (Condition 3, if i = BS).
• Case II: One of Pr,j is a basic variable, denoted byPi′,j,mj , and none of Pt,i (Pt,BS,mj if i = BS)is a basic variable. The entering of Pi,j,mj shouldguarantee the leaving of the basic variable Pi′,j,mj ,i.e., χl = Pi′,j,mj where the leaving variable index lis given by (44).
• Case III: None of Pr,j is a basic variable and one ofPt,i (Pt,BS,mj if i = BS) is a basic variable, denoted
9
by Pi,j′,mj′ . The entering of Pi,j,mj should guaranteethe leaving of Pi,j′,mj′ , i.e., χl = Pi,j′,mj′ .
• Case IV: There is no basic variables in both Pr,j andPt,i (Pt,BS,mj if i = BS). In this case, there is noadditional limitations on the entering variable.
b) Change of channel allocation: When the channel that re-ceiver j occupies is changed from mj to m′j , the power vari-able Pi,j,mj becomes Pi,j,m′j , and the associated coefficientcolumn becomes A[Pi,j,m′j ] = [[−Gi,1,m′j1(m′j = m1), · · · ,−Gi,j−1,m′j
1(m′j = mj−1), Gi,j,m′j/γj,th,-Gi,j+1,m′j1(m′j =
mj+1), · · · ,−Gi,N,m′j1(m′j = mN )]T ;0|Nq|+1]. Note thatfor this potential entering variable Pi,j,m′j in the pivotoperation, the coefficient row of the basis B associatedwith the SINR constraint (32) of receiver j may also changeaccordingly, which is different from the entering variablePi,j,mj which maintains the same channel allocation. Denoteby Bj and B′j the coefficient rows associated with the SINRconstraint (32) of receiver j with respect to the old and newbases B and B′, respectively. The new coefficient row B′jcan be obtained based on Bj as well as the associated leav-ing variable as follows: Replacing the leaving variable by theentering variable Pi,j,m′j , the corresponding element of thecoefficient row becomes the coefficient of variable Pi,j,m′j ,i.e, Gi,j,m′j/γj,th. Besides, in B′j , the elements associatedwith other basic power variables Pi′′,j′,mj′ ,∀j
′ ∈ N\{j}should be changed to −Gi′′,j,mj′1(mj′ = m′j), and theelements associated with any basic artificial (slackness) vari-ables aj′ ,∀j′ ∈ N (zi′ ,∀i′ ∈ Nq ∪ {BS}) should keep thesame as those of Bj .
Since the change of channel allocation may alter boththe coefficient column and row, the pivot operation by onlyexchanging two columns between the basis B and non-basis N may be not sufficient for improving the objectivefunction. Thus, to ensure that the entering variable Pi,j,m′jis feasible and is effective to improve the objective functionwith a higher probability, we verify this entering variable bytwo steps:
• Step 1) Column Test: Under the condition that thecoefficient row keeps invariant, we check the en-tering variable Pi,j,m′j by changing the coefficientcolumn only. This process is the same as that ofthe entering variable Pi,j,mj except that the coeffi-cient column associated with the entering variablebecomes A[Pi,j,m′j ]. If the column test passes andfinds the leaving variable χl, then continue to Step2). Otherwise, stop and remove this variable Pi,j,m′jfrom the candidate entering variable set.
• Step 2) Row Test: Under the condition that the coef-ficient column keeps invariant, we verify if this en-tering variable Pi,j,m′j can still improve the objectivefunction value by replacing the coefficient row Bj
by B′j , where B′j is determined under the leavingvariable χl given by Step 1). If the row test alsosucceeds, then this variable Pi,j,m′j can be selected asan entering variable. Otherwise, it has to be removedfrom the candidate entering variable set. The processof the row test will be described as follows.
The row test aims to check if the change of coefficientrow can improve the objective value of subproblem 1. To
this end, we need to compare the following two problems.
PB-1: minχB
wTBχB (45)
s.t., BχB = b,χB ≥ 0, (46)PB-2: min
χB′wTB′χB′ (47)
s.t., B′χB′ = b,χB′ ≥ 0, (48)
where PB-1 is the subproblem of the auxiliary problem un-der the basis B while PB-2 is based on PB-1 by introducingthe entering variable Pi,j,m′j and changing the associatedcoefficient row of the basis B. Specifically, B′ is obtainedby replacing the coefficient row Bj of the basis B by B′j ,and the new coefficient vector of the objective functionwB′ is obtained by replacing the coefficient of the leavingvariable χl in wB by that of the entering variable Pi,j,m′j .Obviously, if PB-2 has a smaller optimal objective valuethan that of PB-1, i.e., wT
B′χ∗B′ < w
TBχ∗B , then the row test
succeeds. Otherwise, it fails. Although the solution to PB-1is already known as shown in (41), a direct comparison ofPB-1 and PB-2 still requires to resolve PB-2. By consideringthe fact that we may need to repeatedly check multiplepotential entering variables in the pivot operation, directcomparison becomes time consuming. Therefore, instead ofdirectly comparing PB-1 and PB-2, we resort to their dualproblems where the coefficient row becomes the column onesuch that some principles from column test can be applied.The dual problems of PB-1 and PB-2 can be respectivelyformulated as
DPB-1: maxλB
bTλB (49)
s.t., BTλB ≤ wB, (50)DPB-2: max
λB′bTλB′ (51)
s.t., B′TλB′ ≤ wB′ . (52)
According to the strong duality theory [32] that the optimalobjective value of the primal problem PB-1 (PB-2) equalsthat of the dual problem DPB-1 (DPB-2), the row test isequivalent to comparing dual problems DPB-1 and DPB-2.
It is known that the objective function coefficient of theentering variable Pi,j,m′j in subproblem 1 always equals zerowhile the coefficient of the leaving variable (Pi,j,mj , aj , orzi) could be 0 or 1. This means that any element of wB′ isalways smaller than or equal to the corresponding elementof wB , i.e., wB′ ≤ wB . Thus, the feasible region of DPB-2is smaller than or equal to that of the following problem.
DPB-3: maxλB′
bTλB′ (53)
s.t., B′TλB′ ≤ wB . (54)
Obviously, DPB-3 is obtained by changing the RHS constantvector wB′ of constraints of DPB-2 to wB . Let V ∗DPB−1,V ∗DPB−2 and V ∗DPB−3 be the optimal objective values ofthe problems DPB-1, DPB-2 and DPB-3, respectively. Then,based on the feasible regions of DPB-2 and DPB-3, it isknown that the optimal objective value of DPB-2 is smallerthan or equal to that of DPB-3, i.e., V ∗DPB−2 ≤ V ∗DPB−3.Therefore, if V ∗DPB−3 < V ∗DPB−1, we must have V ∗DPB−2 ≤V ∗DPB−3 < V ∗DPB−1. In other words, in order to guaranteethe success of the row test (i.e., V ∗DPB−2 < V ∗DPB−1), it is
10
sufficient to ensure that the problem DPB-3 has a smallerobjective value than DPB-1 (i.e., V ∗DPB−3 < V ∗DPB−1).
To check if V ∗DPB−3 < V ∗DPB−1 holds, we can examinethe following extension problem.
PE: maxλB ,λB′
j
bTλB + σ2λB′j (55)
s.t.,[BT (B′j)
T] [λBλB′j
]≤ wB, (56)
where λB′j denotes the variable corresponding to the coef-ficient column (B′j)
T of DPB-3 while the coefficient σ2 isthe objective function coefficient of λB′j in DPB-3, whichcorresponds to the RHS constant σ2 of the SINR constraintof receiver j. In fact, PE is obtained by adding the coefficientcolumn (B′j)
T into DBP-1. It can be seen that DBP-1 andDBP-3 are subproblems of PE under bases BT and B′T ,respectively. This means that if the objective function of PEunder basis B′T has a smaller value than that under basisBT , then we have V ∗DPB−3 < V ∗DPB−1.
To achieve our goal, we formulate the Karush-Kuhn-Tucker (KKT) conditions [31] [32] for the optimality of PEas follows: [
BT (B′j)T] [λBλB′j
]+ µ = wB, (57)
µ ≥ 0, (58)v ≥ 0, (59)Bκ = b, Bjκ = σ2, κ− v = 0, (60)
µTv = 0. (61)
Equalities (57) and (58) are the primal feasibility conditions,where we have transformed the inequality constraint (56)into the equality (57) by adding the slackness variablesµ; (59) is the dual feasibility condition, where v is thedual variable vector associated with the slackness variableconstraints µ ≥ 0; (60) is the stationary condition, where κis the dual variable vector associated with constraint (57);and (61) is the complementary slackness condition.
Based on the KKT conditions, the solution to PE underthe basis BT can be obtained by letting the non-basicvariable λB′j and the slackness variables µ be zero. Then, wehave the solution λB = (BT )−1wB which is the dual solu-tion (43) to the auxiliary problem under the basisB. Besides,let κ = χB ≥ 0 which is the primal solution (41) to theauxiliary problem under the basis B. It can be verified thatλB and κ are also the primal and dual optimal solutions tothe problem DBP-1 by checking its KKT conditions whichare very similar to the KKT conditions of the extensionproblem PE. The verification process is omitted here forsimplicity. Based on the principle of simplex method, it canbe shown that when adding the coefficient column (B′j)
T
into the basis BT and allowing the entering variable λB′j toincrease from zero, every unit increase in λB′j will result inan increase by ∆ = σ2−B′jκ in the objective function of PE.Note that ∆ < 0 means the negative increase (i.e., decrease).Similar to (44), when the entering variable λB′j joins thebasis BT , the leaving variable index can be determined as
l′ = arg mini{λi/ρ′i|ρ′i > 0, i ∈ I ′B}, (62)
Algorithm 1: Basis Transformation Method1 Initialize the basic variable set as
XB = {aj , zi, ∀j ∈ N , ∀i ∈ Nq ∪ {BS}} and the initial basis isan identity matrix B = I ;
2 Initialize channel allocation: Let each receiver j randomly selecta channel from the channel setM with equal probabilities(mj ∈M);
3 while true do4 Compute primal and dual solutions χB and λ based on (41)
(43);5 Let flag = 1;6 for χi ∈ XN ∪ Pc (Pivot Operation) do7 Compute the Lagrangian multiplier ui and the leaving
variable index l based on (42) (44);8 if χi ∈ Pm ∪ Pc then9 Decide if χi is feasible based on Cases I-IV;
10 if χi ∈ Pm && ui < 0 && χi is feasible (ColumnTest) then
11 Replace χl by χi and update the basis B;12 Let flag = 0 and break;13 else if χi ∈ Pc && ui < 0 && χi is feasible then14 Compute ∆ = σ2 −B′
jχB and the leavingvariable index l′ for the problem PE based on(62);
15 if ∆ < 0 && λl′ = λBj (Row Test) then16 Replace χl by χi and update the basis B;17 Let flag = 0 and break;
18 else if χi /∈ Pm ∪ Pc && ui < 0 then19 Replace χp by χi and update the basis B;20 Let flag = 0 and break;
21 if flag = 1 then22 Output the solution of current basic variables χB and
let χN = 0; Break and Stop;
23 Denote by XN the set of non-basic variables;24 Denote the set of power variables changing channel allocation
from mj to m′j as
Pc = {Pi,j,m′j|∀j ∈ N , i ∈ Nqj ,m
′j ∈M\mj};
25 Denote the set of power variables maintaining channelallocation mj as Pm = {Pi,j,mj |∀j ∈ N , i ∈ Nqj }.
where i denotes the index of vectors λB and ρ′ =(BT )−1B′j . I ′B is the index set of all elements of λB . If theentering of variable λB′j ensures the leaving of variable λBj ,i.e., λl′ = λBj , then the new basis is obtained by replacingthe coefficient column (Bj)
T with (B′j)T . This new basis
is exactly the coefficient matrix B′T of DPB-3. Similarly, itcan be verified by the KKT conditions that the solution toPE under the new basis B′T is also the optimal solutionto DPB-3. Therefore, the decrease in the objective functionof PE from the basis BT to B′T implies that DPB-3 has asmaller optimal objective value than DPB-1. To sum up, if∆ = σ2 −B′jκ = σ2 −B′jχB < 0 and λl′ = λBj where l′
is given by (62), then the row test is successful.Algorithm 1 summarizes all detailed steps of our pro-
posed basis transformation method. In addition, a toy ex-ample is presented in Appendix A for illustration purpose.Lemma 1. The complexity of Algorithm 1 is O(|N |4|M|).
Proof: Please see Appendix B.
5 AN INCENTIVE-COMPATIBLE AND INDIVIDUAL-RATIONAL REWARD SCHEME
Denote the solution of the joint power control, channel allo-cation and link scheduling obtained in the previous section
11
as (X∗, P ∗). In this section, we propose a pricing schemeΠ∗ based on (X∗, P ∗) to determine the reward for each UEin D2D communications such that constraints (19), (20) and(22) can be met. Note that the well-known Vickrey-Clarke-Groves (VCG) mechanism [33] is inapplicable to our casebecause the corresponding resource management problemis NP-hard. In addition, unlike existing pricing schemes [16],[34], [35] for approximate allocation algorithms where userswere assumed to be single-minded (i.e., win or lose) or allo-cation decisions were single-dimensional, in this paper, themanagement outcomes are three-dimensional (i.e., powercontrol, channel allocation and link scheduling) so that theinteractions and relationships among UEs in the competitionbecome much more complex.
5.1 Design of the Pricing rule
The idea of the proposed pricing rule is under the frame-work of [36], where the reward (payment) for each useris determined based on the welfare improvement (impair-ment) it offers. Therefore, we define the reward for eachUE as a function of the power costs of other potentialtransmitters (e.g., the BS or other UEs) that this UE blocksin (X∗, P ∗).
Definition 1 (Blocks). Given (X∗, P ∗) and the resultedsocial welfare S(X∗, P ∗), suppose that UE i∗ is chosenas a D2D transmitter with
∑m∈M x∗i∗,j∗,m = 1 and∑
m∈M x∗i′,j∗,m = 0,∀i′ ∈ Ki∗ . We say that UE i∗
blocks the potential transmitter i′,∀i′ ∈ Ki∗ , if afterremoving UE i∗ from the solution (i.e., X∗\{xi∗,j∗,m},P ∗\{Pi∗,j∗,m}), i′ can be feasibly added as a newtransmitter for receiver j∗ with
∑m∈M xi′,j∗,m =
1,∑m∈M Pi′,j∗,m ≥ 0 and the difference be-
tween S(X∗\{xi∗,j∗,m} ∪ {xi′,j∗,m}, P ∗\{Pi∗,j∗,m} ∪{Pi′,j∗,m}) and S(X∗\{xi∗,j∗,m}, P ∗\{Pi∗,j∗,m}) islarger than zero (which indicates that the welfare canbe improved compared to that of leaving j∗ unserved).
Note that we can easily find the set of potential trans-mitters blocked by any UE i∗ ∈ N , i.e., Ki∗ , by checking alltransmitters that are able to feasibly replace i∗.
Based on Definition 1, the reward for each UE i∗ ∈ Ncan be calculated as follows:
• If UE i∗ is not chosen as a D2D transmitter in(X∗, P ∗), its reward πi∗ = 0.
• If UE i∗ is chosen as a D2D transmitter in (X∗, P ∗)with
∑m∈M x∗i∗,j∗,m = 1, and blocks some other
potential transmitters in a set Ki∗ 6= ∅, then
πi∗ = mini′∈Ki∗
∑m∈M
Pi′,j∗,mci′ . (63)
• If UE i∗ is chosen as a D2D transmitter in (X∗, P ∗)with
∑m∈M x∗i∗,j∗,m = 1, but does not block any
other potential transmitter, i.e., Ki∗ = ∅, then
πi∗ = Ij∗ + 1NR(j∗) · δj∗ , (64)
where 1NR(j∗) = 1 if j∗ ∈ NR, and 1NR(j∗) = 0otherwise. Note that (64) represents the total advan-tage (including the benefit in future D2D offloadingand the save in penalizing from potential servicedissatisfaction) that UE i∗ brings to the BS.
TABLE 2MAIN SIMULATION PARAMETERS.
Parameter ValueCell radius 500 mChannel bandwidth 180 KHzNoise spectral density −174 dBm/HzNoise figure at UE 9 dBAntenna gain BS: 14 dBi; UE: 0 dBiMaximum transmit power BS: 46 dBm; UE: 23 dBmUnit power cost BS: 1; UE: randomly over [0, 1]Penalty for dissatisfaction randomly over [0, 1]Content size randomly from 100 Mb to 1 GbTime frame length 1024 secondsTotal amount of contents 200Zipf distribution exponent 0.8
Number of downlink channels5 10 15 20 25 30 35
Socia
l w
elfare
0
5
10
15
20
25
Proposed mechanism (basis transformation)
Iterative auction with fixed power allocation
Coloring-based scheduling algorithm
Fig. 3. Performance comparison with existing D2D algorithms.
5.2 Property AnalysisIn this subsection, we first prove that the designed rewardscheme can certainly guarantee the satisfaction of individualrationality and incentive compatibility (i.e., constraints (19),(20) and (22)), and then illustrate the total complexity of theoverall mechanism.Theorem 1. Given (X∗, P ∗), the designed reward scheme
satisfies individual rationality constraints (19) and (22).
Proof: Please see Appendix C.Theorem 2. Given (X∗, P ∗), the designed reward scheme
satisfies incentive compatibility constraint (20).
Proof: Please see Appendix D.Theorem 3. The overall mechanism, which consists of the
developed basis transformation method for joint re-source management and the designed pricing rule forreward determination, results in a total complexity ofO(2|N |4|M|+ |N |2/4).
Proof: Please see Appendix E.
6 SIMULATION RESULTS
In this section, simulations are conducted to examine theperformance of our proposed mechanism. Table 2 lists thevalues of main simulation parameters. Similar settings havealso been employed in [7], [10], [21]. The social impacts ofUEs are modeled following the same way as in [6], [13].
In Fig. 3, we first evaluate the system performanceof our proposed mechanism based on the designed basistransformation method by comparing it with two existing
12
algorithms, i.e., the iterative auction with fixed power al-location [7] and the coloring-based scheduling algorithm[9]. In [7], D2D links with fixed power allocations wereiteratively assigned to available channels in an underlaymanner, while in [9], D2D links were grouped by a coloring-based algorithm and then assigned to different channelswith power adjustments. From the figure, it is shown thatthe social welfare increases with the number of downlinkchannels for all three algorithms. This is because with morespectrum resources, i) more service requesters can be servedso that less penalty for service dissatisfaction is induced;and ii) mutual interference becomes lower so that higherpower efficiency is achieved. Besides, we can clearly see thatour proposed algorithm outperforms the other two. This isbecause the iterative auction algorithm with fixed powerallocation does not enable power control so that it maysuffer from a large power cost, while the coloring-basedscheduling algorithm decouples the channel and powerallocations, which may degrade the total amount of admit-ted D2D links (especially when the number of downlinkchannels is small). However, when the number of channelsis relatively large, nearly all D2D links can be admitted, sothat the performance gap between our proposed mechanismand the coloring-based algorithm becomes less obvious.
Since none of existing works can deal with our consid-ered social-aware joint content sharing and caching prob-lem, for comparison purpose, the following two intuitivecaching mechanisms are simulated as benchmarks.
• Mechanism based on random proactive caching (RC mech-anism): UEs randomly determine the contents forproactive caching. In addition, the BS manages thetransmission services for service requesters and ran-domly seed contents to caching requesters.
• Mechanism without proactive caching (NC mechanism):UEs only demand contents related to their servicerequirements, so that no proactive caching is applied.
In both RC and NC mechanisms, underlaying D2Dcommunications are enabled. However, neither RC nor NCmechanism utilize the social information to build up a jointframework of content sharing and proactive caching.
Fig. 4 shows the performance of different caching mecha-nisms on social welfare with different numbers of downlinkchannels. It can be observed that, because of the benefit fromproactive caching, our proposed mechanism and RC mech-anism result in higher social welfare than NC mechanism.Besides, with the consideration of social awareness and wellcoordination of interference, our proposed mechanism canfurther lead to a better resource management (i.e., selectinga better subset of UEs for seeding and consuming lesspower in transmissions) than that in RC mechanism, andthus achieves the best performance. However, when thenumber of downlink channels is relatively large (e.g., 30channels in our simulation), the performance gap betweenRC mechanism and our proposed mechanism turns to bevery small. This is because in networks with excessiveamount of channels, nearly all UEs can be served evenunder RC mechanism, so that the advantage of seeding UEsby their social influences becomes negligible.
Fig. 5 illustrates the social welfare obtained by differentcaching mechanisms under different densities of UEs. In this
figure, we can see that the social welfare decreases when thedensity of UEs gets higher. Clearly, the mutual interferencein the network increases with the density of UEs, whichleads to a much larger power cost for content dissemination.Note that even though the opportunities of D2D communi-cations may increase for higher density of UEs, this benefitmay be offset by the traffic burden from service requesters.Similar to Fig. 4, our proposed mechanism outperforms bothRC and NC mechanisms because of the employment ofsocial-aware proactive caching and the joint optimization onresource management. Moreover, it also demonstrates thatthe superiority of our proposed mechanism is more obviousin scenarios with higher densities of UEs.
Fig. 6 compares three different caching mechanisms interms of network capacity (i.e., sum of data rates on alltransmission links) and D2D capacity (i.e., sum of data rateson all D2D links) with respect to different numbers of down-link channels. It is intuitive that the overall network capacityincreases with the number of downlink channels. Besides, itis shown that our proposed mechanism and RC mechanismcan reach higher network capacities than NC mechanism.This results from more D2D communications enabled dueto proactive caching. In addition, our proposed mechanismshows an even better performance than RC mechanism innetwork capacity because social-awareness has been takeninto account in caching decisions so that more popularcontents have been seeded to more social influential UEs.This figure also shows that D2D capacity is not growingapparently with the growing number of downlink channels.This is because with the increase of downlink channels, mostof UEs are served by cellular links instead of D2D links. Theexplanation is twofold: 1) the considered network (with 30UEs distributed in a cell with 500 m radius) is relativelysparse so that transmission qualities of cellular links may becomparable with or even better than most D2D links; 2) unitpower costs of UEs (D2D transmitters) are set to be similarto that of the BS so that D2D links are not preferable if theirtransmission qualities cannot be better than cellular links.
Fig. 7 presents the comparison among different mecha-nisms on the transmission capacity by varying the densityof UEs. It can be seen that the network capacity decreaseswith the density of UEs regardless of which mechanismis adopted. This is because in order to guarantee moretransmissions for more service requesters (so as to avoid alarger penalty), the network will be overwhelmed by inter-ferences so that the total number of transmissions is actuallydecreased. More interestingly, Fig. 7 also shows that theD2D capacity first increases with the density of UEs. This isbecause more D2D opportunities emerge in dense networks.However, after a certain point, since the interference issuebecomes dominant, the D2D capacity decreases. Note thatour proposed mechanism still outperforms the other two,and explanations follow the same as that for Fig. 6.
To further demonstrate the advantage of our proposedmechanism, in Figs. 8 and 9, we evaluate the performanceof different caching mechanisms in terms of the utility of theBS with respect to different numbers of downlink channelsand different densities of UEs, respectively. Fig. 8 illustratesthat utilities of the BS under all mechanisms increase withthe number of downlink channels. The reason is that if morechannels are available, more service and caching requests
13
Number of downlink channels10 15 20 25 30 35 40 45 50
So
cia
l w
elfa
re
0
5
10
15
20
25
Proposed mechanismRC mechanismNC mechanism
Fig. 4. Social welfare vs. number of channels.
Density of UEs (Number of UEs/Km2)
50 100 150
Socia
l w
elfare
0
5
10
15
20
25
Proposed mechanism
RC mechanism
NC mechanism
Fig. 5. Social welfare vs. density of UEs.
Number of downlink channels5 10 15 20 25
Tra
nsm
issio
n c
apacity (
Mbps)
0
2
4
6
8
10
12
14
16
18
Network capacity, Proposed mechanism
Netwwork capacity, RC mechanism
Network capacity, NC mechanism
D2D capacity, Proposed mechanism
D2D capacity, RC mechanism
D2D capacity, NC mechanism
Fig. 6. Capacity vs. number of channels.
Density of UEs (Number of UEs/Km2)
50 100 150
Tra
nsm
issio
n c
apacity (
Mbps)
0
2
4
6
8
10
12
14
16
18
Network capacity, Proposed mechanism
Netwwork capacity, RC mechanism
Netwwork capacity, NC mechanism
D2D capacity, Proposed mechanism
D2D capacity, RC mechanism
D2D capacity, NC mechanism
Fig. 7. Capacity vs. density of UEs.
Number of downlink channels5 10 15 20 25
Utilit
y o
f th
e B
S
0
5
10
15
Proposed mechanism
RC mechanism
NC mechanism
Fig. 8. BS’s utility vs. number of channels.
Density of UEs (Number of UEs/Km2)
50 100 150
Utilit
y o
f th
e B
S
0
2
4
6
8
10
12
14
16
18Proposed mechanism
RC mechanism
NC mechanism
Fig. 9. BS’s utility vs. density of UEs.
can be fulfilled with less power consumption, and thus theBS can obtain a higher benefit and suffer a lower cost. Incontrast, Fig. 9 reveals the trend that utilities of BS decreasewith the increase of density of UEs. This is because ahigher density of UEs can result in more transmission powerconsumption and more service dissatisfactions due to thestrong interference, so that the power cost and the penaltyon the BS will certainly increase. More importantly, we canobserve from Figs. 8 and 9 that the utility of the BS under ourproposed mechanism is much higher than those under bothRC and NC mechanisms. This is because with social-awareproactive caching and joint optimization of resource man-agement, our proposed mechanism can create more low-costand highly influential D2D offloading opportunities, so thatthe power consumption of the BS and the total rewards paidto UEs can be in turn decreased. In addition, both figuresverify that our proposed mechanism can always guaranteenon-negative utility for the BS.
7 CONCLUSION
In this paper, an incentive mechanism for downlink cellulartraffic offloading with social-aware D2D content sharing andproactive caching has been studied. By considering wirelessfeatures, social characteristics and device intelligence, awelfare maximization problem is formulated. After that, abasis transformation method and a novel pricing schemeare developed to solve the joint resource management (i.e.,power control, channel allocation and link scheduling) andreward design. Both theoretical and simulation results showthat our proposed mechanism can satisfy all desired proper-ties, and can improve social welfare, network capacity andutility of the BS compared to counterparts.
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Changyan Yi (S’16) received the B.Sc. degreefrom Guilin University of Electronic Technology,China, in 2012, and M.Sc. degree from Uni-versity of Manitoba, Winnipeg, MB, Canada, in2014. He is currently working toward the Ph.D.degree in Electrical and Computer Engineering,University of Manitoba. He was awarded EdwardR. Toporeck Graduate Fellowship in Engineeringin 2014, 2015, 2016 (three times), Universityof Manitoba Graduate Fellowship (UMGF) for2015-2018, and IEEE ComSoc Student Travel
Grant for IEEE Globecom 2016. His research interests include algo-rithmic game theory, queueing theory and their applications in radioresource allocation, prioritized scheduling and network economics.
Shiwei Huang received the B.E. and M.S. de-grees in telecommunication engineering fromGuilin University of Electronic Technology,Guilin, China, in 2010 and 2012, respectively.He is currently working toward the Ph.D. degreein telecommunications with the Department ofElectrical and Computer Engineering, Univer-sity of Manitoba, Winnipeg, MB, Canada. Hiscurrent research interests include mobile edgecomputing, D2D communication, wireless relaynetworks, cooperative spectrum sensing, and
energy-efficient communications.
Jun Cai (M’04-SM’14) received the B.Sc. andM.Sc. degrees from Xi’an Jiaotong University,Xi’an, China, in 1996 and 1999, respectively, andthe Ph.D. degree from the University of Waterloo,ON, Canada, in 2004, all in electrical engineer-ing. From June 2004 to April 2006, he was withMcMaster University, Hamilton, ON, as a NaturalSciences and Engineering Research Council ofCanada Postdoctoral Fellow. Since July 2006, hehas been with the Department of Electrical andComputer Engineering, University of Manitoba,
Winnipeg, MB, Canada, where he is currently an Associate Profes-sor. His current research interests include energy-efficient and greencommunications, dynamic spectrum management and cognitive radio,radio resource management in wireless communications networks, andperformance analysis. Dr. Cai served as the TPC Co-Chair for IEEEVTC-Fall 2012 Wireless Applications and Services Track, IEEE Globe-com 2010 Wireless Communications Symposium, and IWCMC 2008General Symposium; the Publicity Co-Chair for IWCMC in 2010, 2011,2013, and 2014; and the Registration Chair for QShine in 2005. Healso served on the editorial board of the Journal of Computer Systems,Networks, and Communications and as a Guest Editor of the specialissue of the Association for Computing Machinery Mobile Networksand Applications. He received the Best Paper Award from Chinacom in2013, the Rh Award for outstanding contributions to research in appliedsciences in 2012 from the University of Manitoba, and the OutstandingService Award from IEEE Globecom in 2010.
15
APPENDIX AA TOY EXAMPLE FOR ALGORITHM 1For clarity, we consider a very simple case with a singlechannel and two UEs to explain the proposed basis transfor-mation method. In this single channel scenario, the channelindex is omitted for notation simplicity. All parametersare set as follows: GBS,1 = d−3
BS,1 = (100)−3 = 10−6,GBS,2 = d−3
BS,2 = (200)−3 = 1.25 × 10−7, G2,1 = G1,2 =
d−31,2 = (100)−3 = 10−6, G2,2 = G1,1 = (0.1)−3 = 103,γ1,th = 2, γ2,th = 5, σ2 = −90 dBm = 10−12W , PmaxBS =46 dBm = 40W , Pmax1 = Pmax2 = 23 dBm = 0.2W ,I1 + δ1 = 0.1, I2 + δ2 = 0.2.
With these settings, subproblem 1 can be rewritten as
min (I1 + δ1)a1 + (I2 + δ2)a2
s.t.GBS,1γ1,th
PBS,1 +G2,1
γ1,thP2,1 −GBS,1PBS,2
−G1,1P1,2 + a1 = σ2,
−GBS,2PBS,1 −G2,2P2,1 +GBS,2γ2,th
PBS,2
+G1,2
γ2,thP1,2 + a2 = σ2,
PBS,1 + PBS,2 + zBS = PmaxBS ,
P1,2 + z1 = Pmax1 ,
P2,1 + z2 = Pmax2 ,
Conditions 1, 2 and 3,
where conditions 1,2 and 3 are stated as follows:
• At most one element of set {PBS,1, P2,1} isgreater than zero, and at most one element of set{PBS,2, P1,2} is greater than zero, i.e., there is at mostone incoming link for each receiver.
• At most one element of set {P1,2} is greater thanzero, and at most one element of set {P2,1} is greaterthan zero, i.e., there is at most one outgoing link foreach transmitter.
• For the BS, at most one element of set {PBS,1, PBS,2}is greater than zero, i.e., there is at most one cellularlink on the channel.
Relaxing conditions 1, 2 and 3, subproblem 1 is trans-formed into an auxiliary problem as
minχ
wTχ
s.t. Aχ = b,χ ≥ 0,
where χ = [PBS,1, P2,1, PBS,2, P1,2, a1, a2, zBS , z1, z2]T ,w = [0, 0, 0, 0, I1 + δ1, I2 + δ2, 0, 0, 0]T , andb = [σ2, σ2, PmaxBS , Pmax1 , Pmax2 ]T . The coefficient matrixA = [A(PBS,1),A(P2,1),A(PBS,2),A(P1,2),A(a1),A(a2),A(zBS),A(z1),A(z2)], which can be further calculated as
A =
GBS,1γ1,th
G2,1
γ1,th−GBS,1 −G1,1 1 0 0 0 0
−GBS,2 −G2,2GBS,2γ2,th
G1,2
γ2,th0 1 0 0 0
1 0 1 0 0 0 1 0 00 0 0 1 0 0 0 1 00 1 0 0 0 0 0 0 1
,whereA(χi) denotes the coefficient column associated withvariable χi.
The initial basic variable set is chosen as{a1, a2, zBS , z1, z2} and the non-basic variable set is{PBS,1, P2,1, PBS,2, P1,2}. The corresponding initial basis is
B = [A(a1),A(a2),A(zBS),A(z1),A(z2)].
Since χB = B−1b, the associated basic variable vector canbe calculated as
χB =
a1
a2
zBSz1
z2
=
σ2
σ2
PmaxBS
Pmax1
Pmax2
=
10−12
10−12
400.20.2
.The non-basic variables are all set to be zero, i.e.,
χN =
PBS,1P2,1
PBS,2P1,2
=
0000
.Since the objective function coefficient vector associatedwith basic variables can be written as wB = [I1 + δ1, I2 +δ2, 0, 0, 0]T = [0.1, 0.2, 0, 0, 0]T , the dual solution is
λ = (BT )−1wB = wB = [I1 + δ1, I2 + δ2, 0, 0, 0]T
= [0.1, 0.2, 0, 0, 0]T .
We now check if the optimality condition uN = wN −NTλ ≥ 0 is satisfied.
Here N = [A(PBS,1),A(P2,1),A(PBS,2),A(P1,2)], andwN = [0, 0, 0, 0]T . Thus,
uN = wN −NTλ
= −
GBS,1γ1,th
−GBS,2 1 0 0G2,1
γ1,th−G2,2 0 0 1
−GBS,1 GBS,2γ2,th
1 0 0
−G1,1G1,2
γ2,th0 1 0
I1 + δ1I2 + δ2
000
=
−GBS,1γ1,th
(I1 + δ1) +GBS,2(I2 + δ2)
− G2,1
γ1,th(I1 + δ1) +G2,2(I2 + δ2)
GBS,1(I1 + δ1)− GBS,2γ2,th
(I2 + δ2)
G1,1(I1 + δ1)− G1,2
γ2,th(I2 + δ2)
=
−2.5× 10−8
2009.5× 10−8
200
.Since uN (1) = −2.5 × 10−8 is less than zero, which doestnot satisfy the optimality condition, the corresponding non-basic variable PBS,1 is chosen as the entering variable,which will become one of basic variables for the nextround calculation and the corresponding coefficient columnA(PBS,1) will be added in the basis B.
Meanwhile, one of existing basic variables must beremoved to maintain the total number of basic variablesunchanged. To determine the leaving variable, we firstcalculate the intermediate vector ρ as
ρ = B−1Ae = B−1A(PBS,1)
= [5× 10−7,−1.25× 10−7, 1, 0, 0]T ,
whereAe = A(PBS,1). Then, the leaving variable index can
16
be determined as
l = arg mini{χi/ρi|ρi > 0, i ∈ IB}
= arg min1, 3
{χB(1)
ρ(1),χB(3)
ρ(3)
}= arg min
1, 3
{a1
ρ(1),zBSρ(3)
}= arg min
{10−12
5× 10−7,
40
1
}= arg min
{2× 10−6, 40
}= 1.
Hence, the leaving variable is a1, which will become a non-basic variable with value of 0 in the next round.
To sum up, the entering variable is PBS,1 and theleaving variable is a1. Thus, in the next round, thebasic variable set is updated from {a1, a2, zBS , z1, z2}to {PBS,1, a2, zBS , z1, z2}, and the non-basic variableset is updated from {PBS,1, P2,1, PBS,2, P1,2} to{a1, P2,1, PBS,2, P1,2}. The basis B is updated by replacingthe leaving variable coefficient column A(a1) with theentering variable coefficient column A(PBS,1) as
B = [A(PBS,1),A(a2),A(zBS),A(z1),A(z2)].
A similar calculation process of finding the new enteringvariable and leaving variable is repeated for updating Buntil either the optimality condition uN ≥ 0 is satisfied orno more entering variable can be found to guarantee condi-tions 1, 2 and 3 (as explained in Section IV-B-a). Eventually,the final solution for this toy example is obtained as
χB =
PBS,1a2
P1,2
z1
z2
=
40
5× 10−6
2× 10−8
0.20.2
, χN =
a1
P2,1
PBS,2zBS
=
0000
.
APPENDIX BPROOF OF LEMMA 1
Proof: The proposed basis transformation method isbased on the simplex method which is well known to beremarkably efficient in practice even though it has exponen-tial worst-case complexity in some pathological instances[37]. However, it can achieve polynomial average-case com-plexity under various probability distributions of randomcoefficient matrices [38], [39]. The average number of pivotoperations is bounded by O([min(nc, nv)]
2) [39], where ncand nv denote the numbers of constraints and unknownvariables, respectively. When applying the basis transforma-tion method, we are required to check not only the powervariables Pi,j,mj maintaining the channel allocation, but alsothe power variables Pi,j,m′j changing the channel allocationwhich can be treated as hidden unknown variables in thelinear auxiliary problem. These indicate that there are totally(∑j∈N |Nqj |)|M| power variablesP , |N | artificial variables
a and |Nq| + 1 surplus variables z. Let Nqj be the averagenumber of transmitting UEs over receivers j ∈ N . Then,there are Nqj |N ||M| power variables in average. Thus, wehave nv = Nqj |N ||M|+|N |+|Nq|+1 ≤ |N |2|M|+2|N |+1
in average and nc = |N | + |Nq| + 1 ≤ 2|N | + 1 becauseNqj ≤ |N | and |Nq| ≤ |N |. Hence, the average numberof pivot operations is bounded by O(|N |2). At each pivotoperation, since at most nv variables have to be checked,the complexity of Algorithm 1 is O(|N |4|M|).
APPENDIX CPROOF OF THEOREM 1
Proof: Obviously, the utility of each UE is zero if itdoes not serve as a D2D transmitter. Otherwise, by substi-tuting the expression of πi∗ (i.e., (63) or (64)), the utility ofUE i∗ ∈ N is given by
Ui∗ = πi∗ −∑m∈M
Pi∗,j∗,mci∗
=
mini′∈Ki∗
∑m∈M
Pi′,j∗,mci′ −∑
m∈MPi∗,j∗,mci∗
Ij∗ + 1NR(j∗) · δj∗ −∑
m∈MPi∗,j∗,mci∗
≥ 0.
The above inequality holds since if UE i∗ is selected as atransmitter in (X∗, P ∗), we must have
∑m∈M Pi∗,j∗,mci∗ ≤
mini′∈Ki∗
∑m∈M Pi′,j∗,mci′ and
∑m∈M Pi∗,j∗,mci∗ ≤
Ij∗ + 1NR(j∗) · δj∗ .Furthermore, in a special case that the BS rewards all UEs
according to (64), we have UBS = 0. Since the reward maybe alternatively calculated by (63), which produces a smalleror equal value than that of (64), implied from definition 1,we can also conclude that UBS ≥ 0.
APPENDIX DPROOF OF THEOREM 2
Proof: We consider two different cases to prove thatno UE i∗ can improve its utility by misreporting.
• Case I: UE i∗ is chosen as the D2D transmitter forreceiver j∗ in (X∗, P ∗), and obtains utility Ui∗ ≥ 0when reporting ci∗ truthfully. If UE i∗ misreports ci∗by ci∗ 6= ci∗ , there could be two possible outcomes:i) UE i∗ is no longer a D2D transmitter, and getsUi∗ = 0; or ii) UE i∗ is still the D2D transmitterfor receiver j∗. Since Na remains unchanged basedon our proposed basis transformation method inSection 4, UE i∗ blocks the same set of other potentialtransmitters, i.e., Ki∗ = Ki∗ . Thus, Ui∗(ci∗ |ci∗) =Ui∗(ci∗ |ci∗).
• Case II: UE i∗ is not chosen as a D2D transmitter in(X∗, P ∗) when reporting ci∗ truthfully, and obtainsutility Ui∗ = 0 . Intuitively, the value of Ui∗ may bechanged only if UE i∗ becomes a D2D transmitter bymisreporting a lower unit-power cost, i.e., ci∗ < ci∗ .In this case, we must have
∑m∈M Pi∗,j∗,mci∗,j∗,m ≤
mini′∈Ki∗
∑m∈M Pi′,j∗,mci′ (or Ij∗+1NR(j∗)·δj∗ ≤∑
m∈M Pi∗,j∗,mci∗ ), and thus
Ui∗(ci∗ |ci∗) = πi∗ −∑m∈M
Pi∗,j∗,mci∗
=
mini′∈Ki∗
∑m∈M
Pi′,j∗,mci′−∑
m∈MPi∗,j∗,mci∗ ≤ 0;
Ij∗ + 1NR(j∗) · δj∗ −∑
m∈MPi∗,j∗,mci∗ ≤ 0.
17
Therefore, in both Cases I and II, Ui∗ = Ui∗(ci∗ |ci∗) ≥Ui∗(ci∗ |ci∗),∀ci∗ 6= ci∗ ,∀i∗ ∈ N .
APPENDIX EPROOF OF THEOREM 3
Proof: Since the developed basis transformationmethod has to be conducted for both subproblems 1 and2, the total complexity of the joint power control, chan-nel allocation and link scheduling will be O(2|N |4|M|)according to Lemma 1. Besides, given (X∗, P ∗), we candirectly observe that the designed reward scheme requiresto check at most |W|(|N | − |W|) possibilities in transmitterreplacement for determining Π∗, where |W| ∈ [0, |N |] de-notes the number of granted D2D transmitters in (X∗, P ∗),and thus this leads to an additional complexity of at mostO(|W|(|N | − |W|)) ≤ O(|N |2/4).