Cross-Layer Network Codes for Content Delivery in Cache-Enabled D2D Networks
Mohammed S. Al-Abiad, Md. Zoheb Hassan, Md. Jahangir Hossain
CCross-Layer Network Codes for Content Delivery inCache-Enabled D2D Networks
Mohammed S. Al-Abiad, Student Member, IEEE, Md. Zoheb Hassan, Student Member, IEEE, and Md. JahangirHossain, Senior Member, IEEE
Abstract —In this paper, we consider the use of cross-layernetwork coding (CLNC), caching, and device-to-device (D2D)communications to jointly optimize the delivery of a set ofpopular contents to a set of user devices (UDs). In the consideredD2D network, a group of near-by UDs cooperate with eachother and use NC to combine their cached files, so as thecompletion time required for delivering all requested contentsto all UDs is minimized. Unlike the previous work that considersonly one transmitting UD at a time, our work allows multipleUDs to transmit simultaneously given the interference amongthe active links is small. Such configuration brings a new trade-off among scheduling UDs to transmitting UDs, selecting thecoding decisions and the transmission rate/power. Therefore,we consider the completion time minimization problem thatinvolves scheduling multiple transmitting UDs, determining theirtransmission rates/powers and file combinations. The problemis shown to be intractable because it involves all future codingdecisions. To tackle the problem at each transmission slot, we firstdesign a graph called herein the D2D Rate-Aware IDNC graphwhere its vertices have weights that judiciously balance betweenthe rates/powers of the transmitting UDs and the number of theirscheduled UDs. Then, we propose an innovative and efficientCLNC solution that iteratively selects a set of transmitting UDsonly if the interference caused by the transmissions of the newlyselected UDs does not significantly impact the overall completiontime. Simulation results show that the proposed solution offerssignificant completion time reduction compared with the existingalgorithms.
Index Terms —Cross layer network coding, content delivery,device-to-device communications, power optimization, real-timeapplications.
I. I
NTRODUCTION
A. Overview T HE exploding amount of mobile traffic, e.g., streamingapplications, YouTube videos, video-on demand, con-sume large bandwidth and high transmission energy of thesource limited cellular networks. Moreover, if the cloud base-stations (CBSs) are fully loaded, it is not possible for the CBSsto schedule all user devices (UDs). To circumvent these chal-lenges, D2D communications have widely been consideredas a promising technology [1]–[3]. The performance of D2Dcommunications can be further improved by pushing somepopular contents to the UDs near to the CBSs. This integratedsystem is referred to as cache-enabled D2D system. Cache-enabled D2D system draws remarkable benefits for alleviatingthe traffic congestion of the cellular network and reducing boththe CBS involvement and end-to-end latency. In this work,we consider cache-enabled D2D system, where multiple UDs
Mohammed S. Al-Abiad, Md. Zoheb Hassan, and Md. Jahangir Hossainare with the School of Engineering, University of British Columbia, Kelowna,BC V1V 1V7, Canada (e-mail: [email protected], [email protected],[email protected]). cache some popular contents and cooperate among them todeliver their cached contents that are requested by other UDs.As such, all the requested contents are delivered to all UDswithin the lowest possible network delay.Network coding (NC) has been shown to be promisingfor improving throughput and minimizing decoding delayand completion time for numerous applications in wirelessnetworks [2]–[5]. Specifically, random linear NC (RLNC)can achieve the optimal throughput of wireless broadcastnetworks [5]. However, this throughput achievement comes atthe expense of complex encoding (i.e., mixing contents usingcoefficients from a large Galois field), high decoding delay,and prohibitive computational complexity. This is suitable fordelay-tolerant contents and UDs with high capabilities andbuffer sizes. The report by CISCO [6] shows that a significantportion of network traffic is popular contents (popular videosand photos) that are frequently requested by UDs in shorttime. Therefore, it is crucial to deliver these delay-sensitivecontents with minimum possible delay. For this purpose,instantly decodable NC (IDNC) is adopted [7]. IDNC performssimple encoding XOR operation at the transmitter and simpledecoding XOR operation at the receiver, and thus instantuse of the received contents. Accordingly, it is suitable forimplementation in small and low cost UDs [8]. Therefore,D2D communication and IDNC technique can be exploitedto deliver popular contents to UDs with the lowest possibledelay while offloading the CBSs. For instance, consider that acontent consists of set of files f , f , and f is wanted by setof UDs u , u , and u . Suppose that the CBS transmitted thewanted contents to the UDs and due to channel impairmentsUD u i did not obtain file f i for ≤ i ≤ . These missedfiles can be traditionally re-transmitted from the CBS to eachUD until all UDs obtain them correctly. As a result, the CBSrequires at least uncoded transmissions for delivering thesefiles, which degrade system performance [9]. However, UDscan be either file cachers that can deliver their cached files toother UDs or file requesters that can receive the wanted filesfrom other UDs. In our considered example, UD u holds files f and f , and accordingly, it can transmit the binary XORcombination f ⊕ f to UDs u and u . Then, UD u holds f and can provide it to UD u . As a result, transmissionsare required for delivering all files to all UDs. Therefore, thecooperation among UDs can be utilized with IDNC to combinefiles and transmit them to interested UDs via D2D links. Assuch, the requested files can be delivered to the requestingUDs quickly while offloading the CBS’s resources. B. Related Works and Motivation
The content delivery problem, known as completion timeminimization problem , in IDNC-enabled networks is consid- a r X i v : . [ c s . I T ] J a n red based on layer functionalities as follows. From onlynetwork-layer perspective, IDNC schedule is adopted to solvethe problem in real-time applications in terms of minimizingthe number of transmissions [8], [12]–[15]. In particular, theserelated works modeled the status of physical channels by fileerasure probabilities and integrated such erasures in the codingdecisions, e.g., see for example [12], [15]. This improvesthe system’s performance from network-layer perspective byscheduling many UDs to the same resource block, but itdegrades the performance from physical-layer perspectivethrough selecting the minimum rates of all the scheduledUDs. This results in prolonged transmission duration and thus,consumes the time resources of network. Unlike network-layerIDNC that depends solely on file combinations for aiding thecoding decisions, rate-aware IDNC (RA-IDNC) also dependson the channel capacities of different UDs. This allows anew degree-of-freedom, such as, choosing the transmittingUDs, their transmission rates, and IDNC file combinations, tooptimize the content delivery problem. The authors of [16]–[24] used RA-IDNC in centralized and decentralized networksfor optimizing different system parameters. For example, theauthors of [19] used RA-IDNC scheme in cloud radio accessnetworks (C-RANs) for completion time minimization. Theauthors of [20], [22] developed cross-layer IDNC to optimizethroughput and Quality-of-Service (QoS) of UDs in central-ized C-RANs and Fog-RANs, respectively.For D2D systems, the authors of [24] considered a vanilla-version of the completion time minimization problem. In-deed, the problem was considered by simply selecting thetransmitting UD and its NC combination. However, the maindrawback of the work in [24] is that only one UD is allowedto transmit coded file in each transmission slot. Thus, theyignored the interference caused by different transmitting UDsto the scheduled UDs. Actually, in D2D networks, UDs arespatially distributed in a region which creates an opportunityto judiciously select multiple transmitting UDs that schedulea significant set of other UDs. Such configuration brings anew trade-off between scheduling UDs to transmitting UDsand choosing the coded files and the transmission rate/power.However, solving the completion time minimization problemwhile jointly considering the previously cached files at users,their transmission rates/powers, NC, and D2D communicationshas not explored yet. Furthermore, developing a joint cross-layer IDNC for the completion time minimization in D2Dnetworks is new to the area of NC. Therefore, our setting inthis work is much more realistic than the one used in [24] asit enables for both selection of multiple transmitting UDs andoptimization of the employed transmission rates using powercontrol on each transmitting UD.The completion time minimization problem is motivatedby real-time applications, i.e., video streaming. In these ap-plications, UDs need to obtain a set of popular files fromother transmitting UDs with the minimum possible completiontime, given the required minimum rates for QoS. Unlike pre-loads that can be done at much lower rates or at off-peaktimes, our work delivers popular contents to UDs with theminimum possible completion time. Consider that a popularvideo representing a frame of files is requested by a set of UDs located in a playground. Many UDs in the playground areinterested in receiving this frame. At any given time, considerthat UDs have already cached some files and requested someother files from that frame. In order to deliver the requestedfiles in that frame without any interruption, UDs should receivetheir requested files with a minimum possible delay. For such acase, users can re-XOR the transmitted files from transmittingUDs to progressively and immediately use the decoded filesat the application layer. Such progressive file decoding at theUDs meets the delay requirements and streaming quality. C. Contributions
Unlike previously-discussed existing works that consideredthe optimization factors (e.g., NC, users’ cached and requestedfiles, UD scheduling, QoS guarantee requirements, and poweroptimization) and their corresponding problems separately,our work develops a framework that jointly considers allthe aforementioned factors. To this end, we develop a novel cross-layer network coding (CLNC) optimization frameworktaking NC and rate/power optimization into account. The maincontributions of our work are summarized as follows. • The completion time minimization problem is shown tobe computationally intractable due to the interdependenceamong variables such as the UDs’ cached and requestedfiles, power optimization, channel qualities, and codingdecisions. Using the lower bound on the completion timeused in the literature, we tackle the problem and solve itonline at each transmission slot. • We design a D2D-RA-IDNC graph to efficiently trans-form the completion time minimization problem to amaximal weight independent set (MWIS) problem. Thedesigned D2D-RA-IDNC graph represents all the feasiblerates and NC decisions for all potential transmitting UDs.The problem is then reformulated as an MWIS problemthat can be efficiently solved using low complexity graphtheoretical solution. The designed weights of the verticesin this graph balances between the transmission rate andnumber of scheduled UDs in each transmission. • We develop a CLNC solution that efficiently iteratesbetween finding the MWIS in the designed D2D-RA-IDNC graph and optimizing the power of the transmittingUDs using a function evaluation (FE) method. In eachiteration, a new transmitting UD is selected only if theresultant interference does not significantly degrade thecompletion time performance. The complexity of ourdeveloped CLNC solution is analyzed. • We compare our proposed scheme with existing baselineschemes. Simulation results demonstrate that our pro-posed CLNC solution significantly minimizes the com-pletion time compared with existing algorithms.The rest of this paper is organized as follows. Section IIoverviews the system model. The completion time approxi-mation and problem formulation are illustrated in Section III.In Section IV, we present the graph construction and prob-lem transformation and propose cross layer network codingsolution in Section V. Finally, we present selected simulationresults in Section VI and conclude the work in Section VII.I. S
YSTEM M ODEL
A. System Overview
We consider a cache-enabled D2D system with one cloudbase-station (CBS) and N user-devices (UDs), denoted bythe set N = { u , u , · · · , u N } . We adopt a fully connectedD2D model where D2D links are usually implemented withlow-range transmission technologies, such as Bluetooth andWiFi. Therefore, we assume that each UD is connected to allother UDs. Each UD is assumed to be equipped with singleantenna and used half-duplex channel. Accordingly, each UDcan either transmit or receive at a given transmission slot.Unlike the work in [24] that ideally considered interference-free setup, our realistic work considers that UDs use thesame frequency band and can cooperate by utilizing D2Dlinks and transmit simultaneously. With such a cooperationamong UDs for content delivery, the CBS dose not need totransmit requested contents to UDs. Therefore, the CBS isresponsible for selecting a set of transmitting UDs and theirNC combinations and power allocations, that deliver requestedcontents to requesting UDs. Accordingly, the whole processof coding decisions in this work is executed at the CBS anddepends on selecting the transmission rate and transmit powerallocation of each transmitting UD.Let F denote a frame of F files, F = { f , f , · · · , f F } ,each of size B bits. This data frame represents a popularcontent, i.e., YouTube video, and constitutes the set of mostfrequent requested files by the UDs for any given time period.We assume that UDs proactively cached some files from F and stored them in their local caches, i.e., C u k represents theset of the files locally cached at UD u k . We assume that UD u k requests a set of files, from the frame F , and denoted by thedemand set W u k = F\C u k . Following the caching model in[24], each file from F is cached by at least one UD in N whichleads to the fact that ∪ u k ∈N C u k = F . The set of UDs thathaving non-empty demand sets at the t -th transmission slot isdenoted by N w,t , i.e., N w,t = { u k ∈ N |W u k ,t (cid:54) = ∅ } . Withoutloss of generality, we assume that all UDs have non-empty demand sets. Otherwise, they can simply be ignored from theset N without affecting the system performance. When an UDreceives its requested files, it acts as a transmitting UD thatprovides its received files to the interested UDs. The goal isto deliver the requested files to the UDs within the lowestpossible completion time by leveraging NC and D2D links.Let γ u k ,u i denote the channel gain between UD u k andUD u i and Q max denote the maximum transmit power forD2D link. We consider slow fading channels, and accordingly, γ u k ,u i is considered to be fixed during a single transmissionbut may change independently from one file transmissionto another file transmission. Then, the achievable capacityof a D2D pair ( u k , u i ) is given by C u k ,u i = log (1 + SINR u k ,u i ( Q )) , ∀ u k ∈ A , where SINR u k ,u i ( Q )) is the corre-sponding signal-to-interference plus noise-ratio experienced byUD u i when it is scheduled to UD u k . This SINR is given bySINR u k ,u i ( Q )) = Q u k | γ u k ,u i | N + (cid:80) u m (cid:54) = u k Q u m | γ u m ,u i | , ∀ u k , u m ∈A , where A is the set of transmitting UDs, N is the noisepower, Q u k , Q u m are the transmit powers of UDs u k and u m which both are bounded by Q max , and Q = [ Q u k ] is a rowvector containing the power levels of the transmitting UDs.The channel capacities of all pairs of D2D links can be storedin an N × N capacity status matrix (CSM) C = [ C u k ,u i ] , ∀ ( u k , u i ) . Since UD u k cannot transmit to itself, C u k ,u k = 0 . B. Rate-Aware NC and Expression of the Completion TimeMetric
Let f u k ,t denote the XOR file combination to be sent byUD u k to the set of scheduled UDs u ( f u k ,t ) at the t -th trans-mission. For simplicity, we use time index t to represent the t -th transmission slot, i.e., t = 1 refers to the first transmissionslot. The file combination f u k is an element of the power set P ( C u k ) of the cached files at UD u k . At every transmission t , each scheduled UD in u ( f u k ,t ) can re-XOR f u k ,t withits previously cached files to decode a new requested file.To ensure successful reception at the UDs, the maximumtransmission rate of a particular transmitting UD is equalto the minimum achievable capacity of its scheduled UDs.Therefore, the set of targeted users by UD u k is expressed as u ( f u k ) = (cid:8) u i ∈ N w (cid:12)(cid:12) | f u k ∩ W u i | = 1 and R u k ≤ C u k ,u i (cid:9) .Without loss of generality, the set of all targeted UDs, when |A| transmitting UDs transmit the set of combinations f ( A ) , isrepresented by u ( f ( A )) , where u k , f u k , u ( f u k ) are elementsin A , f ( A ) , and u ( f ( A )) , respectively .Let T u k denote the duration of the transmission from the u -th UD. The duration for transmitting f u k from UD u k withrate R u k to u ( f u k ) is T u k = BR uk seconds. For transmissionsynchronization, all transmitting UDs in the set A adopt acommon transmission rate, denoted as R . Otherwise, differenttransmitting UDs will have different transmission rates, andthus, they will have different transmission durations. So, UDswho finish the transmission first must wait for those whotransmit with the slowest-rate to start a new transmission at thesame time. Therefore, we adopt one transmission rate for alltransmitting UDs, and accordingly, the transmission durationfor sending any coded/uncoded file from any transmittingUD is denoted by T t and expressed by T t = BR seconds.Consequently, UDs that are not targeted at transmission slot t , experience T t seconds of delay has a cumulative delay asdefined below. Definition 1:
Any UD with non-empty demand set experiences T t seconds of time delay if it does not receive any requestedfile at t -th transmission. The accumulated time delay of UD u i is the sum of T t seconds at each transmission until t -thtransmission, denoted by T u i ( t ) , and expressed as T u i ( t ) = T u i ( t −
1) + (cid:40) T t if u i / ∈ u ( f ( A )) T t if u i ∈ A . (1)Let T u i denote the completion time of UD u i until itreceives the requested files. The completion time for UD u i includes two parts, its accumulated time delay T u i due to re-ceiving a non-instantly decodable file and the time duration ofsending all instantly decodable transmissions. In other words, The term “targeted users” is given for the scheduled UDs who receive aninstantly-decodable transmission. The symbol |X | represents the cardinality of the set X . ABLE I: Variables and parameters of the systemVariable Definition N Set of N UDs N w Set of N UDs that want files F Set of F popular files B File size f u k The encoded file of of UD u k u ( f u k ) Set of targeted UDs by UD u k A Set of A transmitting UDs C Set of all achievable capacities W u i Set of wanted files by UD u i C u i Set of locally cached files by UD u i R u k Transmission rate of UD u k T u k The transmission duration of UD u k T u i The completion time of UD u i R u k Set of all achievable rates of UD u k such completion time is divided into consecutive instantlyand non instantly decodable transmission for each UD in N w until it obtains all requested files. Subsequently, the overallcompletion time T = max u i ∈N { T u i } is the time required untilall UDs recover all files. The used notations and variables aresummarized in Table I.To minimize the overall completion time, we need tofind the optimal schedule from the beginning of the D2Dtransmission phase at t = 1 until all UDs obtained allrequested files at t = |S| . Here, S is defined as a col-lection of transmitting UDs, file combinations and trans-mission rates/powers until all UDs in N w receive all F files, i.e., S = {A ( t ) , P ( C u i ( t )) , R ( t ) } , ∀ t ∈ { , ..., |S|} .Thus, the optimal schedule S ∗ that minimizes the overallcompletion time of all UDs is S ∗ = arg min S∈ S { T ( S ) } =arg min S∈ S { max u i ∈N w { T u i ( S ) }} , where S is the set of allpossible D2D transmission schedules, i.e., S ∈ S . This optimalschedule can be formulated as follows Theorem 1.
The minimum overall completion time problem ina D2D multihop network can be formulated as a transmissionschedule selection problem such that: S ∗ = arg min S∈ S (cid:26) max u i ∈N w (cid:26) B. | W u i (0) | ˜ R u i ( S ) + T u i ( S ) (cid:27)(cid:27) , (2) where | W u i (0) | is the initial demand set size of UD u k , T u i ( S ) is the accumulative time delay of UD u i in schedule S and ˜ R u i ( S ) is the harmonic mean of the transmission rates of timeindices that are instantly decodable for UD u i in schedule S .Proof. The proof of Theorem 1 is omitted in this paperbecasue we can use the same steps that was used in [18]for C-RAN networks. Therefore, a sketch of the proof isgiven as follows. We first show that the completion timecan be expressed as the sum of instantly and non-instantlydecodable transmission times from |A| transmitters via D2Dlinks. Afterward, we need to proof that the number of instantlydecodable transmissions to UD u l is equal to the number ofits requested files |W u l , | and the number of non-instantlydecodable transmissions matches the time delay in definition1. Finally, we extend the results of the optimal schedule inTheorem 1 in [18] that was used in C-RAN system to thecoordinated D2D setting with multiple transmitters. (cid:4) u S u . .
53 30512 . u : u : Ist time-slot 2nd time-slot f u : u u u u u f f f f Wants = { φ } f f Wants = { f } f f f Wants = { f } f f f Wants = { f } f f Wants = { f ,f } f f Wants = { f ,f } C u ,u i u C u ,u i C u ,u i C u ,u i C u ,u i C u ,u i u . .
53 35512 u u u . . f f f ⊕ .
33 sec. f f ⊕ Fig. 1: D2D system containing UDs and their correspondingrequested/received files and rates. For example, u receives f , f and requests f , f . The sets of files that locally cached atUD u is: C u = { f , f , f } .The optimal NC transmission schedule that reduces theoverall completion time in a D2D network is the solution of theoptimization problem in Theorem 1. Such schedule requires toexploit the heterogeneity of UDs’ channel capacities and theinterdependence of UDs’ file reception. Actually, the decisionat the current transmission slot is dependent on the futurecoding situations, which makes the optimization problem anti-causal. Therefore, it can be inferred that finding the optimalschedule S ∗ is intractable [19], [24]. C. Example of RA-IDNC Transmissions in D2D System
This example illustrates the aforementioned definitions andconcepts to ease the analysis of the completion time minimiza-tion problem reformulation in next section. Consider a simpleD2D network that shown in Fig. 1 which consists of users,users’ received and requested files and their rates. For example, u receives f , f and requests f , f . Each file is assumedto have a size of bits. To minimize the completion time forthis example, one possible schedule is given as follows. First time slot:
The u -th and u -th UDs can use their cachedfiles to transmit f u = f ⊕ f and f u = f with rates R u = 2 . and R u = 2 . bits/s, respectively, to the sets u ( f u ) = { u , u } and u ( f u ) = { u , u } . Given this, wehave the following transmission durations of u -th UD and u -th UD, respectively: T u = . = 4 , T u = . = 4 seconds.The decoding process at UDs side can be explained as follows. • The u -th UD already has f , so it can XOR the combi-nation ( f ⊕ f ) with f (i.e., ( f ⊕ f ) ⊕ f ) to retrieve f . Thus, the transmission is instantly decodable for UD u . • The u -th UD already has f , so it can XOR the combi-nation ( f ⊕ f ) with f (i.e., ( f ⊕ f ) ⊕ f ) to retrieve f . Thus, the transmission is instantly decodable for UD u . • The u -th and u -th UDs can receive f from u -th UD.Thus, the transmission is instantly decodable for UDs u and u .Therefore, the updated demand sets after the first time slotare: W u = { f } , W u = ∅ , W u = ∅ , W u = { f } , W u = { f } . Note that T t, = 4 seconds. Second time slot:
The u -th UD can use its cached files totransmit f u = f ⊕ f with rate R u = 3 bits/s to the set u ( f e ) = { u , u , u } which requires transmission time T u = t, = = 3 . seconds. The decoding process at UDs sidecan be explained as follows. • The u -th UD already has f , so it can XOR the combi-nation ( f ⊕ f ) with f (i.e., ( f ⊕ f ) ⊕ f ) to retrieve f . Thus, the transmission is instantly decodable for UD u . • The u -th UD already has f , so it can XOR the combi-nation ( f ⊕ f ) with f (i.e., ( f ⊕ f ) ⊕ f ) to retrieve f . Thus, the transmission is instantly decodable for UD u . • The u -th UD already has f , so it can XOR the combi-nation ( f ⊕ f ) with f (i.e., ( f ⊕ f ) ⊕ f ) to retrieve f . Thus, the transmission is instantly decodable for UD u .By the end of second time slot, all UDs will have theirrequested files. Therefore, the total transmission time is T t, + T t, = 4 + 3 .
33 = 7 . seconds.The above example demonstrates the benefit of NC andD2D communications in minimizing the completion time. Wecan further improve this result by allocating the power levelsefficiently to the transmitting UDs.III. C OMPLETION T IME A PPROXIMATION AND P ROBLEM R EFORMULATION
Following [18], [24], we approximate the completion timein Theorem 1 to select a set of transmitting UDs, file combina-tions, and transmission rates/powers at each transmission slot t without going through all future possible coding decisions. Toachieve this, at each transmission slot t , a lower bound on thecompletion times of all UDs is computed. This lower boundis computed separately for each UD and does not requireto exploit the interdependence of UDs’ file reception andchannel capacities. In fact, this lower bound metric facilitatesthe mapping of the transmission schedule selection problem in(2) into an online maximal independent set selection problem. Corollary 1.
A lower bound on completion time ¯ T i ( t ) of UD u i ∈ N w in a given time index t can be approximated as ¯ T u i ( t ) ≈ B. | W u i (0) | ˜ R u i + T u i ( t ) , (3) where T u i ( t ) is the accumulative time delay experienced byUD u i until time index t and ˜ R u i is the harmonic mean ofthe channel capacities from all UDs.Proof. The expression in (3) matches the expression in Theo-rem 1, except T u i ( S ) and ˜ R u i ( S ) of Theorem 1 are replacedby T u i ( t ) and ˜ R u i , respectively. The best case scenario isthat all transmissions starting from time slot t are instantlydecodable for UD u i . Thus, it experiences no further timedelay, i.e., T u i ( S ) = T u i ( t ) . In addition, since a fullyconnected D2D model is adopted, UD u i can receive a missingfile from any other UD until it receives all F files. Therefore, ˜ R u i ( S ) is replaced by ˜ R u i , where ˜ R u i is the harmonic meanof the channel capacities from all other UDs to UD u i . Thisis an approximation as ˜ R u i is exactly equal to ˜ R u i ( S ) if UD u i receives an equal number of files from other UDs with therates of channel capacities. (cid:4) Using the approximated completion time (3) at each trans-mission slot t , we now ready to reformulate the completiontime minimization problem in Theorem 1 with the aim todevelop a cross-layer network coding framework that decidesthe set of transmitting UDs A for sending f u k to the UDs u ( f u k ) , and their transmission rate/power { R u k , Q u k } , ∀ u k ∈A . As such, all files are delivered to all UDs with minimumcompletion time. Therefore, the completion time minimizationproblem in fully D2D connected system can be formulated as P1 : min f uk ,r uk ,Q uk A∈P ( N ) (cid:26) max u i ∈N w ¯ T u i (t) (cid:27) (4a) subject to (C1): u ( f u k ) ∩ u ( f u m ) = ∅ , ∀ u k (cid:54) = u m ∈ A , (C2): f u k ⊆ P ( H u k ) , ∀ u k ∈ A , (C3): ≤ Q u k ≤ Q max , ∀ u k ∈ A , (C4): R u k ≥ R th , ∀ u k ∈ A , where (C1) states that the sets of targeted UDs from alltransmitting UDs are disjoint, i.e., each UD must be scheduledto only one transmitting UD; (C2) ensures that all filesto be combined using XOR operation at each transmittingUD u k are stored in its cache ; (C3) bounds the maximumtransmit power of transmitting UDs, and (C4) guarantees theminimum transmission rate R th required to meet the QoS raterequirements.The optimization variables in (P1) contain the NC schedul-ing parameters u ( f u k ) , potential set of transmitting UDs A ,and their adopted power allocations. It can be seen that prob-lem (P1) is intractable. However, by analyzing the problem,next section successfully transforms it into MWIS problemusing graph theory technique.IV. G RAPH C ONSTRUCTION AND P ROBLEM T RANSFORMATION
The formulated problem in ( P1 ) is similar to MWIS prob-lems in several aspects. In MWIS, two vertices should benon-adjacent in the graph, and similarly, in problem ( P1 ),same UD cannot be scheduled to two different UDs (i.e., C1).Moreover, the objective of problem ( P1 ) is to minimize themaximum completion time, and similarly, the goal of MWISis to maximize the number of vertices that have high weights.Therefore, the feasible NC schedules can be considered to bethe MWISs. Consequently, we focus on graph-based methods,and in what follows, we will construct a graph that allows usto transform problem ( P1 ) into MWIS-based problem. A. D2D Rate-Aware IDNC Graph
In this sub-section, we construct a weighted undirectedgraph, referred to D2D-RA-IDNC graph, that considers allpossible conflicts for scheduling UDs, such as transmission,network coding, and transmission rate. Let G ( V , E ) representthe D2D-RA-IDNC graph where V , E stand for the set of allthe vertices and the edges, respectively. In order to construct G , we need first to generate the vertices and connect them.Let N w ⊂ N denote the set of UDs that still wantssome files. Hence, the D2D-RA-IDNC graph is designed byenerating all vertices for the u k -th possible transmitting UD, ∀ u k ∈ N . The vertex set V of the entire graph is the union ofvertices of all UDs. Consider, for now, generating the verticesof UD u k . Note that transmitting UD u k can encode its IDNCfile f u k using its previously received files C u k . Therefore, eachvertex is generated for each single file f h ∈ W u i ∩ C u k that isrequested by each UD u i ∈ N w and for each achievable rate r of UD u k that is defined below. Definition 2:
The set of achievable rates R u k ,u i from UD u k to UD u i is a subset of achievable rates R u k that are lessthan or equal to channel capacity r u k ,u i . It can be expressedby R u k ,u i = { r ∈ R u k | r ≤ C u k ,u i and u i ∈ N w } . The above definition emphasizes that u i -th UD can receivea file from transmitting UD u k if the adopted transmissionrate r is in the achievable set R u k ,u i . Therefore, we generate |R u k ,u i | vertices for a requesting file f h ∈ C u k ∩ W u i , ∀ u i ∈N w . In summery, a vertex v kr,i,f is generated for each associa-tion of transmitting UD u k , a rate r ∈ R u k ,u i , and a requestingfile f h ∈ C u k ∩ W u i of user u i ∈ N w . Similarly, we generateall vertices for all UDs in N .Given the above generated vertices, in what follows, weconnect them to construct the D2D-RA-IDNC graph. Allpossible conflict connections between vertices (conflict edgesbetween circles) in the D2D-RA-IDNC graph are provided asfollows. Two vertices v kr,i,h and v kr (cid:48) ,i (cid:48) ,h (cid:48) representing the sametransmitting UD u k are linked with a coding-conflict edgeif the resulting combination violate the instant decodabilityconstraint. This event occurs if one of the following holds. • The combination is not-instantly decodable, i.e., f h (cid:54) = f h (cid:48) and ( f h , f h (cid:48) ) / ∈ C u k (cid:48) × C u k . • The transmission rate is different, i.e., r (cid:54) = r (cid:48) .Similarly, two vertices v kr,i,h and v k (cid:48) r (cid:48) ,i (cid:48) ,h (cid:48) representing dif-ferent transmitting UDs u k (cid:54) = u k (cid:48) are conflicting if • The transmission rate is different, i.e., r (cid:54) = r (cid:48) . • The same UD is scheduled, i.e., u i = u i (cid:48) .Therefore, two vertices v kr,i,h and v k (cid:48) r (cid:48) ,i (cid:48) ,h (cid:48) are adjacent bya conflict edge in E if they satisfy one of the followingconnectivity conditions (CC). • CC1: u k = u k (cid:48) and ( f h (cid:54) = f h (cid:48) ) and ( f h , f h (cid:48) ) / ∈ C u k (cid:48) ×C u k . • CC2: r (cid:54) = r (cid:48) . • CC3: u k (cid:54) = u k (cid:48) and u i = u i (cid:48) . B. Problem Transformation
In this sub-section, we transform the network-coded userscheduling and power optimization problem ( P1 ) into MWISproblem, and consequently, we start by the following defini-tions. Definition 3:
Any independent set (IS) I in graph G mustsatisfies: i) I i ⊆ G ; ii) ∀ v, v (cid:48) ∈ I i , we have ( v, v (cid:48) ) / ∈ E . Definition 4:
A maximal IS in an undirected graph cannot beexpanded to add one more vertex without affecting the pairwisenon-adjacent vertices.
Definition 5:
The independent set I is referred to as an MWISof G if it satisfies: i) I is an IS in graph G ; ii) the sum weightsof the vertices in I offers the maximum among all ISs of G .Therefore, the MWIS will be denoted as I . Based on the aforementioned designed D2D-RA-IDNC graphand definition of MWIS, we have the following proposition.
Proposition 1.
The problem of minimizing the approximatedcompletion time in ( P1 ) at the t -th transmission is equivalentlyrepresented by the MWIS selection among all the ISs in the G graph, where the original weight ω o ( v ) of each vertex v isgiven by ω o ( v ) = 2 N w − d ui +1 ¯ T u i ( t ) (cid:16) rB (cid:17) , (5) where d u i is the order of UD u k in the group that arrangesall UDs in N w ( t ) in decreasing order of lower bound oncompletion times [24].Proof. The proof of the proposition follows similar stepsof [18], and consequently, the detailed steps are omitted.Accordingly, a sketch of proof is provided as follows. First,we need to sufficiently show that there is a mapping betweenthe set of maximal ISs in the D2D-RA-IDNC graph and theset of feasible transmissions. Then, the weight of each IS isthe objective function to P1. The authors in [18] showed thatthere exists a one-to-one mapping between the set of feasibletransmissions and the set of ISs in the RA-IDNC graph. Here,we extend the results of [18] to the D2D-RA-IDNC graphby showing that the feasible transmissions between differenttransmissions are non-adjacent, i.e., the constraint CC3. Sinceeach feasible transmission by a transmitting UD is an IS andthey are non-adjacent, then the union of both sets is also anIS. From CC3, the same UD cannot be targeted by distincttransmitting UDs. Therefore, all vertices in the sub-graphrepresenting transmitting UD u k are non-adjacent to verticesin the sub-graph of transmitting UD u k (cid:48) as long as the targetedUDs are distinct. Therefore, each feasible association betweentargeted UDs-transmitting UDs, file combinations, and thetransmission rate is represented by a maximal IS. Conversely,it can readily be seen that each IS represents a feasiblecondition as it does not violate the connectivity conditionsCC1, CC2, and CC3. Indeed, for I , the transmission of thecombination u u k = ⊕ v kr,i,h ∈I f by transmitting UD u k at rate r is instantly for all UDs u ( f u k ) = ∪ v kr,i,h ∈I u .To finish the proof, we show that the weight of the IS is theobjective function of ( P1 ). Let the weight of vertex v kr,i,h bedefined as in (5) and I be the set of maximal ISs in the D2D-RA-IDNC graph G . Consider I ∈ I is the MWIS that has themaximum vertex weights. By the designed graph G , all thefeasible decisions of transmitting UDs, transmitted files andtransmission rates/powers are mapped to the set of all maximalISs. Consequently, the completion time reduction problem canbe reformulated as a maximal IS selection problem in graph G such as arg max A∈P ( N ) f uk ∈P ( C uk ) Q uk ∈{ , ··· ,Q max } r ∈R uk (cid:88) u i ∈X N w − d ui +1 ¯ T u i ( t ) (cid:16) rB (cid:17) = max I ∈I (cid:88) v ∈ I N w − d ui +1 ¯ T u i ( t ) (cid:16) rB (cid:17) = max I ∈I (cid:88) v ∈ I ω o ( v ) . (6)Consequently, the problem of choosing transmitting UDs, filecombinations, and transmission rates/powers that results inminimizing the completion time is equivalent to the MWISelection problem over the D2D-RA-IDNC graph. (cid:4) It is readily known that finding the MWIS is NP-completeproblem [25]. Consequently, solving Proposition 1 is NP-hard.In the next section, we greedily select a maximal IS using thevertices’ weights defined in (5).V. P
ROPOSED S OLUTION
In this section, we develop an efficient cross-layer networkcoding solution that judiciously selects multiple transmittingUDs simultaneously and their coding decisions and transmit-ting rates/powers. As shown in SINR expression, the increasein the number of transmitting UDs also increases interferenceof a transmission channel caused by multiple transmittingUDs and therefore, reduces the channel capacity. To controlthe deleterious impact of interference on channel capacities,a power allocation mechanism is employed that efficientlyselects the set of transmitting UDs and allocates the transmit-ting power to the transmitting UDs such that: (1) a potentialnumber of UDs can be targeted with an IDNC combination,and (2) the channel capacities of the transmitting UDs to thetargeted UDs still improves the objective function. The overallsteps of our proposed solution are as follows. We first presenta power allocation algorithm for the given set of transmittingUDs and the scheduled/targeted UDs to these transmittingUDs. Next, we provide a greedy algorithm that selects a set oftransmitting and targeted UDs considering known/predefinedpower allocations. Finally, by combining the aforementionedalgorithms, we present an innovative cross-layer NC solution.
A. Transmit Power Allocation Algorithm
In this sub-section, we derive optimal power allocationsto maximize sum-throughput for a given set of transmittingUDs. We assume that the system has A transmitting UDs,i.e., A = { , , · · · , A } and the UDs receiving data fromthe u k -th transmitting UD is denoted by the set u ( f u k ) . Thepower optimization problem to maximize the sum-capacity of A transmitting UDs is formulated as max { Q k } A (cid:88) k =1 N k s.t. ≤ Q k ≤ Q max , ∀ k (7)where N k = (cid:80) u i ∈ u ( f uk ) log (1 + SINR u k ,u i ) . The near-optimal power allocation for the u k -th transmitting UD isobtained in the following proposition. Proposition 2.
Let (cid:98) Q k be the given transmit power of the k -th transmitting UD at the t -th iteration. A converged powerallocation is obtained by updating power at the ( t + 1) -th iteration, ∀ t , according to the following power updateequation Q k = (cid:80) u i ∈ u ( f uk ) SINR uk,ui SINR uk,ui (cid:80) m =1 m (cid:54) = k (cid:80) u j ∈ u ( f um ) (cid:16) SINR um,uj SINR um,uj (cid:17) γ uk,ui (cid:98) Q m γ um,uj Q max (8)where SINR u m ,u j , ∀ m, j , is obtained by applying the value (cid:98) Q m in the expression of end-to-end SINR. Algorithm 1
Transmit Power Allocations for A Given Set ofTransmitting UDs Input:
Set of transmitting UDs, the file combinations, andthe associated UDs with each transmitting UDs. Initialize: (cid:98) Q u k = Q o , ∀ k = 1 , , · · · A , t = 1 . repeat Update the power allocation of the u k -th transmittingUD, ∀ u k , by applying (8). Set (cid:98) Q u k = Q u k , ∀ k = 1 , , · · · A , and t = t + 1 until Objective function of (7) converges or t > t max . Output:
Final transmission power for all the transmittingUDs.
Proof.
The proof follows similar steps of [26, Lemma 2].Particularly, although (7) is a non-convex power allocationproblem, a local optimal solution to (7) can be obtained byobtaining the stationary point of the objective function. Toobtain a stationary power allocation for the u k -th transmittingUD, we need to solve ∂ N k ∂Q k = 0 . In particular, we obtain ∂ N k ∂Q k = 1 Q k (cid:88) u i ∈ u ( f uk ) SINR u k ,u i SINR u k ,u i (9) − (cid:88) m =1 ,m (cid:54) = k (cid:88) u j ∈ u ( f um ) (cid:18) SINR u m ,u j SINR u m ,u j (cid:19) γ u k ,u i (cid:98) Q m γ u m ,u j . Therefore, by solving ∂ N k ∂Q k = 0 , we obtain Q k = (cid:80) u i ∈ u ( f uk ) SINR uk,ui SINR uk,ui (cid:80) m =1 ,m (cid:54) = k (cid:80) u j ∈ u ( f um ) (cid:16) SINR um,uj SINR um,uj (cid:17) γ uk,ui Q m γ um,uj (10)By solving (10), one can obtain the stationary point forthe objective function of the u k -th transmitting UD, ∀ u k .However, a closed-form power allocation by solving (10) isintractable. Accordingly, we adopt an iterative approach toobtain a near-optimal stationary power allocation. To this end,we denote (cid:98) Q k as the given power allocation for the u k -thtransmitting UD, ∀ u k , and evaluate the R.H.S. of (10) for thegiven power allocations. Finally, by projecting R.H.S of (10) tothe feasible region of the power allocations, we obtain (8). (cid:4) Based on
Proposition 2 , an iterative algorithm to obtaintransmit power allocations for a given set of transmitting UDsis provided as Algorithm 1. The convergence of Algorithm 1is justified as follows.
Proposition 3.
Algorithm 1 provides a stable and localoptimal solution to (7) .Proof.
We can proof
Proposition 2 by resorting to the gametheory. In fact, the proposed power allocation update can beconsidered as a non-cooperative power control game (NCPCG)where each transmitting UDs act as a rational and selfishplayer, and wants to maximize its utility by choosing thebest possible power allocation strategy. To this end, the utilityfunction of the u k -th transmitting UD is given at the top ofthe next page, where Q − k denotes the power allocation forthe transmitting UDs other than the u k -th UD. k ( Q k , Q − k ) = (cid:88) u i ∈ u ( f uk ) log (cid:32) Q k | γ u k ,u i | N + (cid:80) Km =1 ,m (cid:54) = k Q m | γ u m ,u i | (cid:33) + K (cid:88) m =1 ,m (cid:54) = K (cid:88) u j ∈ u ( f um ) log (cid:32) Q m | γ u m ,u j | N + (cid:80) Kn =1 ,n (cid:54) = k,m Q n | γ u n ,u i | + Q k | γ u k ,u i | (cid:33) The utility function has two parts where the first part isthe payoff in terms of the achievable throughput and thesecond term is the payoff for creating less interference to theother players in the system. Obviously, the first and secondterms monotonically increase and decrease with the increaseof transmission power, Q k , respectively. We denote the R.H.Sof (8) as F k (cid:16) { (cid:98) Q k } (cid:17) . We can readily demonstrate that if Q k < F k (cid:16) { (cid:98) Q k } (cid:17) , U k ( Q k , Q − k ) monotonically increases,and if Q k > F k (cid:16) { (cid:98) Q k } (cid:17) , U k ( Q k , Q − k ) monotonically de-creases. Therefore, U k ( Q k , Q − k ) is a quasi-concave utilityfunction. From [27, Theorem 3.2], for a non-cooperative gamewith quasi-concave utility functions, a Nash-equilibrium (NE)point must exists and it is obtained as the best responsestrategy of the players in the game. Note that, in an NEpoint, no player can improve its utility by taking an alternativestrategy, and consequently, the overall solution must converge.We can easily justify that (8) is same as the best responsestrategy of the k -th transmitting UD, ∀ k . Consequently, theiterative power allocation procedure, given in Algorithm 1,must converge to a stable point. We also emphasize that(8) is derived by satisfying the Karush-khun-Tucker (KKT)conditions for (7). Hence, a stable power allocation that isobtained by iteratively solving (8) must converge to a localoptimal solution to (7). Accordingly, Algorithm 1 provides astable and local optimal solution to (7) . (cid:4) B. Greedy Maximal Independent Set Selection Algorithm
In this sub-section, we describe a maximal IS selection algo-rithm based on a greedy vertex search in the D2D-RA-IDNCgraph and the priority of vertices defined in Proposition 1.Such a greedy vertex search approach was adopted in [10],[11] without adopting the physical-layer rate, but demonstratedits efficiency for completion time minimization. For simplicity,we use v and v (cid:48) instead of v kr,i,h and v kr,i (cid:48) ,h (cid:48) , respectively. Let E v,v (cid:48) be the adjacency connector of vertices v and v (cid:48) in graph G such that E v,v (cid:48) = (cid:40) if v is not adjacent to v (cid:48) in G , otherwise . (11)Further, let g v denote the weighted degree of vertex v , whichcan be expressed as g v = (cid:80) v (cid:48) ∈G E v,v (cid:48) ω o ( v (cid:48) v ) , where ω o ( v (cid:48) ) isthe priority of vertex v (cid:48) defined in (5). Finally, the modifiedweight of vertex v is defined as ω m ( v ) = ω o ( v ) g v = 2 N w − d ui +1 ¯ T u i ( t ) (cid:16) rB (cid:17) n v . (12)To this end, at each step, the vertex search method addsa new vertex based on the maximum weight. Essentially, avertex v ∗ that has the maximum weight ω m ( v ∗ ) is selected The convergence of transmission power update equation, given by (8), isjustified for asymptotically high signal-to-noise (SNR) ratio regime in [26].However, using
Proposition
2, we justify that the considered power allocationconverges without the assumption of asymptotic high SNR. and added to the maximal independent set I (i.e., I = { v ∗ } ).Then, the subgraph G ( I ) , which consists of vertices in graph G that are not connected to vertex v ∗ is extracted and consideredfor the next step. Next, a new maximum weight vertex v (cid:48) ∗ isselected from subgraph G ( I ) . We repeat this process until nomore vertices that are not connected to all the vertices in themaximal independent set I . The steps of the greedy vertexsearch selection are summarized in Algorithm 2.The transmitting UD in I generates a coded file by XORingall the files identified by the vertices in I . It also adopts thetransmission rate corresponding to the vertices of I . It is worthmentioning that the MWIS I and its corresponding modifiedweights in (12) provide the following potential benefits: • The modified weight of each vertex in I shows thefollowing. The first term (cid:0) rB (cid:1) provides a balance be-tween the transmission rate/power and the number ofscheduled UDs to transmitting UD u k . The second term N w − d ui +1 ¯ T u i ( t ) classifies the UDs based on their com-pletion time lower bounds. As such, we give them priorityfor scheduling. More importantly, through the weighteddegree g , the modified weight of a vertex v has a largeoriginal weight and it is not connected to a large numberof vertices that have high original weights. • Each UD is scheduled only to a transmitting UD thatcached one of its missed files. • The transmitting UD delivers an IDNC file with anadopted transmission rate/power that provides a lowercompletion time to a set of UDs. This adopted rateensures the QoS rate guarantee and no larger than thechannel capacities of all scheduled UDs.
C. Cross-layer NC Solution
The iterative proposed CLNC solution maximizes theweighted sum rate subject to completion time reduction con-straints, i.e., the problem of determining transmitting UDs andtheir transmission rates/powers and transmitted file combina-tions in a coordinated fashion. Particularly, we iterate betweensolving the completion time reduction problem for fixedtransmit power and optimizing the power level for a givenschedule of completion time reduction. The main philosophyof this heuristic is to iteratively include more transmittingUDs and allocate transmission powers subject to the reductionin the completion time. At each iteration, it first determinesthe scheduled UDs by the set of chosen transmitting UDs asdescribed in Algorithm 2. Then, given the resulting network-coded user scheduling, it executes a power allocation algorithmto determine the power level of the transmitting UDs thatmaximizes the sum-rate and minimizes the completion time asdescribed in Algorithm 1. The steps of the proposed iterativesolution is described as in Algorithm 3. lgorithm 2
Greedy Maximal Weight Independent Set(MWIS) Selection Generate D2D-RA-IDNC graph G . Initialize I = ∅ . Set G ( I ) ← G . while G ( I ) (cid:54) = ∅ do ∀ v ∈ G ( I ) : compute ω o ( v ) and ω m ( v ) using (5) and(12), respectively. Select v ∗ = arg max v ∈G ( I ) { ω m ( v ∗ ) } . Set I ← I ∪ v ∗ . Obtain G ( I ) . end while In Algorithm 3, A is the set of selected transmitting UDs, M w is the set of UDs having non-empty Wants set, and X isthe set of all the targeted UDs. Proposition 4.
The CLNC solution achieves improved sum-rate compared to the interference free solution of [24].Proof.
At each iteration of the proposed Algorithm 3, the setof transmitting UDs is updated. Let denote A ( i ) be the set oftransmitting UDs at the i -th iteration of Algorithm 3. Recall,only a finite number of UDs can be the transmitting UDs ina given TS. Thus, the set of transmitting UDs is evolved as A (1) → A (2) → · · · → A ( final ) . (13)Note that A (1) contains only one transmitting UD. Particularly,the proposed scheme initially selects a transmitting UD withmaximum number of potential receiving UDs, and such atransmitting UD is included in A (1) . Subsequently, at eachiteration Algorithm 3 adds one more transmitting UDs withthe existing set of transmitting UDs given that the total sum-rate is improved. To this end, at each iteration, Algorithm3 updates the power allocations of all the transmitting UDsto maximize the overall sum-capacity. Essentially, at eachiteration of Algorithm 3, the sum-rate is non-decreasinglyimproved. Obviously, the sum-rate of the A ( final ) set mustoutperform the sum-rate achieved by the A (1) set. Recallthat the interference free solution of [24] selects only onetransmitting UD each TS. Thus, the achievable sum-rate ofthe interference free solution of [24] becomes same to thesum-rate achieved by the A (1) set. Hence, the CLNC solutionachieves improved sum-rate compared to the interference freesolution of [24]. (cid:4) Remark 1: By exploiting power allocation and time-varyingchannel of the UDs, the proposed CLNC solution activatesmultiple transmitting UDs at each TS. However, for severelystrong inter-device interference channel, the power allocationmay not improve the sum-capacity. In this case, the proposedsolution activates only one transmitting UD. Consequently,the the interference free solution of [24] is a special caseof the proposed CLNC solution. Hence, the proposed CLNCsolution always achieves a lower completion time comparedto the interference free solution.Remark 2: When we schedule many UDs to the transmittingUDs, the number of targeted UDs is increased from the sideinformation optimization, however, the sum-capacity may notbe maximized. Consequently, we optimize the power/rate of
Algorithm 3
Cross-layer Network-Coded (CLNC) ResourceScheduling Algorithm Initialize: A = ∅ , M w = N w , and X = ∅ . Initialize:
Transmission power level Q o ∈ Q feasible foreach potential transmitting UD. Compute SINR(s) setting transmission power Q o andconsidering no interference. repeat Construct the D2D-RA-IDNC graph using Section IV-Aby considering using M w as the set of potential trans-mitting or targeted UDs. Select maximal independent set I using Algorithm 2.Let the transmitting UD be u k , the file combinationbe f in the maximal independent set I , and the set ofpotential targeted UDs by u k be u ( f u k ) . Compute the lower bound on the individual completiontime of targeted UDs in X set using (3) and increasedelay of the non-targeted UDs in M w \X by Br ( t ) . Set A = {A , u k } . Consider the UDs in M w \ ( { u k } ∪ X ( f u k )) as futuretransmitting UDs or more targeted UDs. Compute the SINRs by setting the transmission poweras Q o and considering interference from the UDs in theset A . Optimize the transmission power of the transmittingUDs A using Algorithm 1 to maximize the sum-capacity. If the sum-capacity is improved,
A ← A ∪ u i and M w ← M w \ u i . if |A| > then For each receiving UD, u i , compute R u i . If R u i ≥ R th and u i / ∈ X , update X ← X ∪ u i and M w ←M w \ u i . On the other hand, if R u i < R th and u i ∈X , update X ← X \ u i and M w ← M w ∪ u i . Repeatthis step ∀ u i ∈ u ( f u k ) and ∀ u k ∈ A . end if ∃ u k ∈ A such that the none of UDs in u ( f u k ) setsatisfies the rate constraint, M w ← M w ∪ u k and A ←A \ u k . Recompute ¯ T and store the solution that achieves theminimum completion time. until No UDs can be added to the set A . Output:
Overall completion time ¯ T . the transmitting UDs to maximize the sum-capacity. Thus, ourCLNC solution not only increases the number of targeted UDs,but also maximizes the sum-capacity.D. Complexity Analysis For any arbitrary D2D network setting and at any iterationof the proposed algorithm, we need to construct the D2D-RA-IDNC graph, calculate the power allocation of the transmittingUDs, and find the MWIS.Since each UD caches only a set of files, the total numberof vertices in D2D-RA-IDNC graph corresponding to thatUD is V = |C| × N . Therefore, we construct the D2D-RA-IDNC graph for all UDs by generating O ( V N ) vertices.uilding the adjacency matrix needs a computational com-plexity of O ( V N ) . For the vertex search algorithm, weneed first to calculate the weights of all vertices, and thenfinding the MWIS. It is easily to note that all UDs havingvertices in the independent set have the same transmissionrate as they initially corresponding to the same transmittingUD. Thus, the algorithm needs |R| maximal ISs. Note thateach maximal IS has at most V vertices as each UD canbe targeted by at most one file (i.e., one vertex for eachtargeted UD) per transmission. Each iteration with a givenrate needs a complexity of O ( V N ) for weight calculationsof the MWIS. It also needs searching for at most N − vertices. Then, the complexity of the algorithm for findingthe maximal ISs for all rates and their sum weights, at most,is O ( V N |R| + ( N − |R| ) = O ( |R| ( V N + N − . Thecomputational complexity of constructing the D2D-RA-IDNCgraph, building the adjacent matrix, and finding the MWIS is O ( |R| ( V N + N − O ( V N ) = O ( V N ) .On the other hand, calculating the power allocation for anyfixed D2D schedule needs C p = O ( | u u | × | u u | × · · · | u u K | ) .Finally, Algorithm 3 iterates between constructing the D2D-RA-IDNC graph and finding its corresponding MWIS andoptimizing power levels of the transmitting UDs, thus leadingto an overall computational complexity of O ( T ( V N + C p )) ,where T is the number of iterations.VI. N UMERICAL R ESULTS
In this section, we present some numerical results thatcompare the completion time performance of our proposedCLNC scheme with existing coded and uncoded schemes.We consider a D2D network where UDs are distributedrandomly within a hexagonal cell of radius m. We assumethe channel gains between UDs follow the standard path-loss model, which consists of three components: 1) path-loss of
148 + 37 . ( d u k ,u i ) dB, where d u k ,u i representsthe distance between u k -th UD and u i -th UD in km; 2)log-normal shadowing with dB standard deviation and 3)Rayleigh channel fading with zero-mean and unit variance.We consider that the channels are perfectly estimated. Thenoise power and maximum’ UD power are assumed to be − dBm/Hz and Q max = − . dBm/Hz, respectively, andthe bandwidth is MHz. Unless otherwise stated, we initiallyconsider that each UD already has about and of F files for the considered schemes. To evaluate the performanceof our proposed scheme with different thresholds ( R th = 0 . ,and R th = 5 ), we simulate various scenarios with differentnumber of UDs, number of files, file sizes, and demand ratioof UDs. These thresholds represent the minimum transmissionrates required for QoS. The performances of our joint solutionfor R th = 0 . and R th = 5 are shown in solid and dash redlines, respectively.For the sake of comparison, our proposed schemes arecompared with the following existing schemes. • Uncoded Broadcast:
This scheme picks a random UDthat broadcasts an uncoded file from its cache set that ismissing at the largest number of other UDs. Moreover,this scheme uses the minimum channel capacity from thetransmitting UD to all other UDs as the transmission rate.
10 15 20 25 30
Number of UDs N A v e r age C o m p l e t i on T i m e ( s e c ond s ) Classical IDNC-10 UncodedRLNCRA-IDNCProposed CLNC R th1
Proposed CLNC R th2
Fig. 2: Average completion time in sec. vs the number of UDs N .
10 15 20 25 30
Number of file F A v e r age C o m p l e t i on T i m e ( s e c ond s ) Classical IDNC-10 UncodedRLNCRA-IDNCProposed CLNC R th1
Proposed CLNC R th2
Fig. 3: Average completion time in sec. vs the number of files F . • Cooperative RLNC : This RLNC algorithm picks the UDwith the highest side information rank as the transmittingUD in a D2D transmission [28]. The picked UD encodesall files using random coefficient from a large Galoisfield. However, this algorithm discards the dynamic trans-mission rates and for the transmission to be successfullyreceived by all other UDs, the minimum channel capacityfrom the transmitting UD to all other UDs is adopted asthe transmission rate. • Cooperative IDNC : This IDNC algorithm considers co-operation among UDs and jointly selects a set of transmit-ting UDs and their XOR file combinations [12]. However,this algorithm focuses on serving a large number ofUDs with a new file in each time index to reduce theoverall completion time. Due to ignoring the dynamicrate adaptation, the minimum channel capacity from thetransmitting UDs to all targeted UDs is adopted as thetransmission rate.For completeness of this work, we also compare our pro-posed scheme with the recent RA-IDNC work in [24]. In thisscheme, RA-IDNC scheme is employed for D2D network thatallows only one transmitting UD to transmit at a time.In Fig. 2, we depict the average completion time versus theumber of UDs N . We consider a D2D model with a frameof files and a file size of Mbits. From this figure, we canobserve that the proposed CLNC scheme offers an improvedperformance in terms of completion time minimization ascompared to the other schemes for all considered numberof UDs. This improved performance is due the fact thatour proposed scheme judiciously selects potential UDs fortransmitting coded files to a set of schedule UDs, adopts thetransmission rate, and optimizes the transmission power ofeach transmitting UD. This in turn aides the file combinationselection process. The uncoded broadcast scheme sacrificesthe rate optimality by scheduling the maximum number ofUDs. Although uncoded scheme needs a fewer number oftransmissions, at least F transmissions, it requires longertransmission durations for frame delivery completion. Thisleads to a high completion time. On the other hand, theRA-IDNC scheme improves the selection of file process byadapting the transmission rate, but it suffers from activatingonly one transmitting UD at each transmission slot. This is aclear limitation of the RA-IDNC scheme as it does not fullyexploit the simultaneous transmissions from multiple UDs.The proposed CLNC scheme strikes a balance between theaforementioned aspects by jointly selecting the number oftargeted UDs and the transmission rate of each transmittingUD such that the overall completion time is minimized. Thisresults in a full utilization of simultaneous transmissions frommultiple transmitting UDs. Consequently, an improved per-formance of our proposed scheme compared to the RA-IDNCscheme is achieved. Moreover, our proposed scheme improvesthe used rates using power control on each transmitting UD.In Fig. 3, we show the average completion time versusthe number of files F . Fig. 3 considers different sizes offrames. The simulated D2D system composed of UDs andfile size of Mbits. For the same reason as mentioned forFig. 2, our proposed scheme outperforms other schemes. Itcan be observed from the figure that increasing the framesize leads to an increased completion time of all schemes.This is because for few files, the opportunities of mixingfiles using IDNC in the proposed and other NC schemes arelimited. As a result, all NC schemes have roughly similarperformances. As the number of files increases, the increasein the completion time with our proposed scheme is low.This is due to the fact that our proposed scheme judiciouslyallows each transmitting UD to decide on a set of files tobe XORed. As such, they are beneficial to a significant setof UDs that have relatively good channel qualities. Notethat uncoded broadcast and RLNC schemes complete frametransmissions in fewer transmissions ( F transmissions) thanour developed scheme. However, each of their transmissiondurations is longer than a single transmission of the proposedschemes since they are adopting the transmission rates to theminimum of all achievable capacities.In Fig. 4, we illustrate the impact of increasing the filesize B on the average completion time. Fig. 8 shows the sizeof such popular files and how long it takes for the proposedsolution to deliver a complete frame to UDs. In this figure,we simulate the D2D system composed of UDs and files. We observe that the completion time performances of File size B -2 A v e r age C o m p l e t i on T i m e ( s e c ond s ) Classical IDNC-10 UncodedRLNCRA-IDNCProposed CLNC R th1
Proposed CLNC R th2
Fig. 4: Average completion time in sec. vs file size B . Demand Ratio A v e r age C o m p l e t i on T i m e ( s e c ond s ) Classical IDNC-10 UncodedRLNCRA-IDNCProposed CLNC R th1
Proposed CLNC R th2
Fig. 5: Average completion time in sec. vs demand ratio µ .all schemes increase linearly with the file size. This agreeswith the completion time expression in Corollary 1, where itwas shown that T increases linearly with B . From physical-layer consideration, as B increases, more bits are neededfor delivering files. Thus, time delay is increased to receivefiles from transmitting UDs. It can be seen that the proposedscheme in all above figures outperforms all other schemesfor different rate thresholds as shown with red lines. As therate threshold increases, the completion time improvementincreases. This is because as the rate threshold increases, acertain number of UDs is scheduled and the transmission rateof the transmitting UDs becomes high. Thus, the role of ourproposed scheme for completion time minimization and QoSoptimization technique becomes more noticeable.In Fig. 5, we illustrate the impact of changing the demandratio µ on the average completion time. This ratio representsthe demand portion of the requested files of UDs. In thisfigure, we simulate the D2D system composed of UDsand files each with a size of Mbits. We can observethat the completion time performance of our proposed schemeoutperforms the performances of other schemes for the wholerange of µ . It can be seen from the figure that increasingthe demand ratio leads to an increased completion time of allschemes. This is because for high demand ratio, the numberof transmissions for delivering all the files to all the UDsf all schemes increases. As a result, the completion timeperformance of the considered schemes increases.Finally, we provide some observations from our presentedsimulation results as follows. First, it is always beneficial fromnetwork-layer perspective to schedule many UDs with IDNCfiles as in the classical IDNC scheme. However, selecting theminimum transmission rate of all the scheduled UDs degradesthe completion time performance of the classical scheme.Second, although the uncoded brodcast and RLNC schemesschedule almost all the UDs, they adopt the transmission ratesto the minimum of all the scheduled UDs. Thus, its completiontime performance is degraded, especially for large networksizes since selecting the minimum transmission rate of anincreasing set is always minimum. Third, RA-IDNC schemeovercomes the limitations of the aforementioned schemesbut suffers from selecting only one transmitting UD. Thislimitation further degrades the completion time performanceof the RA-IDNC scheme in large network sizes. This due tothe fact that RA-IDNC scheme always selects one transmitterregardless of the size of the network. Conversely, our trans-mission framework is more practically relevant as it considersdifferent transmitting UDs and optimizes the employed ratesusing power control on each transmitting UD.VII. C ONCLUSION
In this paper, we have studied the joint optimization ofCLNC and D2D communications for the file delivery phasewith the goal of minimizing the completion time while guar-anteeing UD’s QoS, subject to the UD’s cache files, therequired minimum rate, power allocation, and NC constraints.The completion time minimization problem in interference-allowed setup is solved over a set of transmitting UDs, theirpower allocation, dynamic rate selection and transmitted filecombinations. By using a graph theory technique, we proposeda novel and efficient approach that uses cross-layer NC forpower optimization and UDs coordinated scheduling in D2Dnetworks. Specifically, our proposed solution judiciously iter-ates between finding the MWIS in the D2D-RA-IDNC graphand optimizing the power allocation, subject to the resultantinterference of the newly added transmitting UDs. Simulationresults show that the proposed interference-allowed solutionreduces the completion time compared to the interference-freesolution as well as conventional network coding algorithms.R
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