High Throughput Opportunistic Cooperative Device-to-Device Communications With Caching
11 High Throughput Opportunistic CooperativeDevice-to-Device Communications With Caching
Binqiang Chen, Chenyang Yang and Gang Wang
Abstract —To achieve the potential in providing high through-put for cellular networks by device-to-device (D2D) communi-cations, the interference among D2D links should be carefullymanaged. In this paper, we propose an opportunistic cooper-ation strategy for D2D transmission by exploiting the cachingcapability at the users to control the interference among D2Dlinks. We consider overlay inband D2D, divide the D2D usersinto clusters, and assign different frequency bands to cooperativeand non-cooperative D2D links. To provide high opportunityfor cooperative transmission, we introduce a caching policy. Tomaximize the network throughput, we jointly optimize the clustersize and bandwidth allocation, where the closed-form expressionof the bandwidth allocation factor is obtained. Simulation resultsdemonstrate that the proposed strategy can provide ∼ throughput gain over traditional D2D communicationswhen the content popularity distribution is skewed, and canprovide ∼ gain even when the content popularitydistribution is uniform. Index Terms —Caching, D2D, Cooperative transmission, Inter-ference, High Throughput
I. I
NTRODUCTION
Device-to-device (D2D) communications enables directcommunications between two user devices without traversingthe base station (BS) or core network, and is a promisingway to achieve the high throughput goal of 5th generation(5G) cellular networks [1–4]. The typical use-cases of D2Dcommunications include cellular offloading, content distribu-tion, and relaying, etc. [5], where content delivery service hasattracted considerable attention recently, since it accounts forthe majority of the explosive increasing traffic load.Motivated by the observation that a large amount of contentdelivery requests are asynchronous but redundant, i.e., thesame content is requested repeatedly at different times, cachinghas long been studied as a technique to improve performanceof wired networks. Due to the rapid reduction in cost of storagedevice, caching at the wireless edge is also recognized asa promising way for delivering popular contents nowadays,which can improve the network throughput, energy efficiencyand the quality of user experience (QoE) [6–12]. However,different from wired networks, the performance of wirelessnetworks is fundamentally limited by the interference, whichinevitably limits the throughput gain from local caching.To take the advantage of the storage device at smart phones,cache-enabled D2D communications has been proposed re-cently, which can offload the content delivery traffic and
Binqiang Chen and Chenyang Yang are with the School of Electronicsand Information Engineering, Beihang University, Beijing, China, Emails: { chenbq,cyyang } @buaa.edu.cn. Gang Wang is with the NEC labs, China,Email: wang [email protected]. hence boost the network throughput significantly [13, 14].Since only the users in proximity communicate to each other,the interference in D2D networks is strong, which needs tobe carefully controlled. In an early work of studying cache-enabled D2D communications, the D2D users are divided intoclusters. Then, the intra-cluster interference among D2D linksis managed by using time division multiple access (TDMA),while the inter-cluster interference between D2D links issimply treated as noise [13]. In [14], only the D2D link fromone of the four adjacent clusters is allowed to be active atthe same time-frequency resource block, in order to avoidstrong inter-cluster interference among adjacent clusters. In[15], interference alignment was employed to mitigate theinterference among D2D links, but only three D2D links werecoordinated within each cluster, and the interference amongclusters was again treated as noise. In [16–18], cooperativerelay techniques were proposed to mitigate the interferencebetween cellular and D2D links, which however can notmanage the interference among the D2D links.It is well known that if several transmitters have the re-quired data for some users, they can jointly transmit to theusers without generating interference. In fact, if contents havebeen locally cached, cooperative transmission without dataexchange among transmitters becomes possible, which cantransform interference into spatial multiplexing gain. Based onsuch an interesting observation, a BS cooperative transmissionstrategy was proposed in [19] by exploiting the caches atBSs, where precoding and cache control were optimized toguarantee the QoE of users. Inspired by this work, a naturalquestion is: can we apply cooperative transmission in D2Dcommunications with caching?Fortunately, cooperative transmission is possible in practicedue to the following reasons. (i) In D2D communications, theD2D transmitter (DT) has been proposed to assist other usersin additional to transmitting data to its destined D2D receiver(DR), e.g., with cooperative relay [17]. The users have theincentive to do this if their own QoE can be improved ortheir costs can be compensated by some other rewards [20].(ii) To facilitate cooperative transmission, the global channelstate information (CSI) is required to compute the precodingmatrix. The CSI among D2D links can be obtained at DTsand the BS through channel probing and feedback [21]. Then,the precoding vectors can be computed at the BS and sent tothe cooperative DTs via multicast. (iii) The synchronizationamong cooperative DTs is more easier to be implemented thanthat in Ad-hoc networks, because it can be realized with theassist of the BS [2]. Besides, the synchronization can also berealized at users by using the methods proposed in [22]. a r X i v : . [ c s . I T ] J un In this paper, we propose an opportunistic cooperationstrategy for cache-enabled D2D communications to man-age the interference among D2D links. Different from theBS cooperative transmission strategy [19], the cooperationstrategy for D2D communications needs to be optimized ina different way. Considering that D2D communications isapplicable to users in proximity, we divide the D2D users intovirtual clusters. To maximize the opportunity of cooperativetransmission via D2D links, we take both redundant cachingand diversity caching into account in the users among theclusters, which differs from [19] where all BSs cache thesame files. When some users have cached the files requestedby other users called DRs, these users act as DTs to jointlytransmit the requested files to the DRs. Because only someD2D links can employ cooperative transmission, we assigndifferent frequency bands to cooperative and non-cooperativelinks to avoid mutual interference. To maximize the averagenetwork throughput without compromising the experience ofnon-cooperative users, we jointly optimize the cluster size andbandwidth allocation under the minimal average user data rateconstraint.The contributions of this paper are summarized as follows: • We propose an opportunistic cooperation strategy to man-age the interference among D2D links, which improve thenetwork throughput remakably. • We jointly optimize the cluster size and bandwidth allo-cation and obtain the closed-form expression of optimalbandwidth allocation factor.The rest of the paper is organized as follows. Section IIpresents the system model. Section III introduces the co-operation strategy, derives the average network throughputand average user data rate, and jointly optimizes bandwidthallocation and cluster size. Section IV provides numerical andsimulation results. Section V concludes the paper.II. S
YSTEM M ODEL
Consider a cellular network, where M single-antenna usersare uniformly located in a square hotspot within a macro cell,where the area is with side length of D c as shown in Fig. 1.Each user is willing to store N files in its local cache and canact as a helper to share files. When a helper conveys a filein local cache via D2D link to a DR requesting the file, thehelper becomes a DT. The BS is aware of the cached files ateach user and coordinates the D2D communications. A. Content Popularity
We consider a static content catalog including N f filesthat the users may request, where the files are indexed in adescending order of popularity, e.g., the 1st file is the mostpopular file. The probability that the i th file is requested by auser is assumed to follow a Zipf distribution, P N f ( i ) = i − β / N f (cid:88) j =1 j − β , (1)where (cid:80) N f i =1 P N f ( i ) = 1 , and the parameter β reflects skew-ness of the popularity distribution, with large β meaning thata few files are requested by the majority of users [23]. B. Communication Protocol
D2D links can be established among users in proximity.A widely used communication protocol for D2D communi-cations is that two user equipments (UEs) can communicateif their distance is smaller than a given distance [13, 24]. Torestrict the D2D link distance and make the analysis tractable,the square hotspot area is divided into B smaller squareareas called clusters, where the side length of each clusteris D = D c / √ B . Only the users within the same cluster canestablish D2D link. For mathematical simplicity, we assumethat the number of users per cluster is K = M/B and eachuser is assumed to transmit with the same power P as in [14].We consider overlay inband D2D [5], and assume that afixed bandwidth of W is assigned to the D2D links.III. O PPORTUNISTIC C OOPERATION S TRATEGY
In this section, we first introduce a caching policy toprovide high opportunity for cache-enabled cooperative D2Dtransmission. Then, we propose an opportunistic cooperativetransmission policy. Finally, we optimize two key parametersin the strategy to maximize the network throughput.
A. Caching Policy
To maximize the probability that a user can fetch filesthrough D2D links, the users within a cluster should cachedifferent files. To maximize the probability of cooperativetransmission among DTs in different clusters, the files cachedat the users of each cluster should be the same. This suggestthat the caching policy needs to balance the diversity of contentwith the redundancy of the replicas of popular contents. To thisend, we consider the following caching policy.According to the user cache size N , all files are dividedinto K = N f /N groups. The k th file group G k consists ofthe ( k − N + 1 th to the kN th files where ≤ k ≤ K , e.g.,the 1st file group G contains the most popular N files. Then,the probability that a user requests a file within the k th filegroup G k can be obtained as P k = kN (cid:88) i =( k − N +1 P N f ( i ) = (cid:80) kNi =( k − N +1 i − β (cid:80) N f j =1 j − β . (2)In every cluster, the k th user caches the k th file group G k . Then, every user in each cluster caches different files, DT ij r clusrter i ii r clusrter j DR D Macro Cell
Hot SpotBS c D Wait UEScheduled UE:
Fig. 1. Cluster division model, “UE” means user equipment. i.e., diversity caching is achieved within each cluster, andthe most popular KN files are cached in every cluster, i.e.,redundant caching is achieved among clusters. Because eachcluster contains K users, the file groups with indices exceeding K (i.e., G k , k > K ) are not cached at users.When K = K , all the N f files can be cached at the usersin each cluster and all user requests can be served via D2Dlinks, therefore it is not necessary to assign more than K users to each cluster. For this reason, we assume K ≤ K .In practice, the files can be proactively downloaded by theoperator from the BS to the cache at each user via multicastduring off-peak times according to the user demand statistics. B. Opportunistic Cooperative Transmission Policy
According to whether a user can find the requested file inits local cluster, we can classify the users into two types.
D2D users:
If the file requested by a user is cached atany UE in the cluster it belongs to (called local cluster ofthe user ), then the user can directly obtain the file with D2Dcommunication. Such a user is referred to as a D2D user.Besides, if the file requested by a user is in its local cache,it can retrieve the file immediately with zero delay, but weignore this case for analysis simplicity as in [14].
Cellular users:
If the file requested by a user is not cachedin the UEs within its local cluster, the user fetches the filefrom the BS and becomes a regular cellular user. The numberof cellular users is denoted as N b . F9 G2
Requets the 9 th file in the 2 rd file group N-Coop user Cellular user
Cluster 1 Cluster 2
Cluster 3 Cluster 4
Coop userThe 3 rd user cached with the 3 rd file group. F1 G1 F5 G1 F2 G1 F16 G4 F11 G3 F18 G4 F4 G1 F10 G2 F9 G2 F8 G2 F15 G3
Hit {1,2,4}Hit {1,2} Hit {1,3,4}Hit {1,3}Hit {1,2,4} The cluster hit the file group {1,2,4}
F13 G3 Fig. 2. Illustration of the opportunistic cooperaton strategy. Catalog size N f = 20 , B = 4 clusters in the hotspot, K = 3 users in each cluster andeach user caches N = 5 files. All files are divided into groups accordingto content popularity, e.g., G = { , , , , } . For easy understanding, we introduce the strategy with thehelp of an example.
1) Cooperative D2D Users:
If there exists at least one userin a cluster requesting the files in G k , then we say that thecluster hits the k th file group . In Fig. 2, the users in the firstcluster respectively request the files in G , G and G , andhence the first cluster hits the { , , } th file groups.If every cluster hits the same file group G k , the k th user ineach cluster who caches the file group G k can act as a DT,and all DTs in these clusters cooperatively transmit files tothe DRs requesting the files in G k . Those DRs are referredto as cooperative D2D users ( Coop users for short), whosenumber is denoted as N c .In Fig. 2, every cluster hits the st file group. Hence, the stusers in all the four clusters who cache the files in G can actas DTs to cooperatively transmit files with indices { , , , } respectively to the rd user in cluster , the rd user in cluster , the nd user in cluster , and the rd user in cluster , asshown in Fig. 3. F4 F2 F1 F5 Cluster 1 Cluster 2 Cluster 3 Cluster 4
Fig. 3. Illustration for cooperative transmission from multiple DTs to DRs.
The remaining users except the cellular and Coop users are non-cooperative D2D users ( N-Coop users for short), whosenumber is N n = M − N b − N c .
2) Interference Control:
Due to the random locations of theDTs in proximity, the interference in the network needs to becarefully controlled even with the cooperative transmission.
Inter-type interference:
To avoid the mutual interferencebetween
Coop users and
N-Coop users , we assign ηW forCoop users and the remaining bandwidth (1 − η ) W for N-Coop users, where η is the bandwidth allocation factor and ≤ η < . A large value of η means that more bandwidth isallocated to Coop users. Intra-cluster interference:
Considering that the userswithin each cluster can not cooperate due to caching differentfiles, we randomly select one Coop D2D link and one N-CoopD2D link respectively in each cluster to transmit at the sametime to avoid intra-cluster interference as in [13, 14].
Inter-cluster interference:
There is no inter-cluster inter-ference among Coop users owing to the joint transmissionfrom multiple DTs, but there exists inter-cluster interferencebetween N-Coop users, which is regarded as noise. Though further improvement is possible by allowing cooperation amongless than B clusters (called partial cooperation), we only consider the fullcooperation among all clusters for mathematical tractability. The impact ofpartial cooperation is shown via simulation in Section IV.
3) Operation Modes:
Due to the opportunistic nature ofestablishing the cooperative D2D links, the network mayoperate in the following two modes. • In Mode , there does not exist any file group hit by everycluster, i.e., all D2D users are N-Coop users. Then, allthe DTs transmit independently, and the bandwidth W isassigned to the N-Coop users, i.e., η = 0 . • In Mode , there exist file groups hit by every cluster,i.e., there exist Coop users. Then, < η < .To become the Coop users, the users are not necessary torequest the same file , but to request the files in the same group .Hence the cooperative probability, i.e., the probability that thenetwork operates in Mode 1 , is high, which increases with thenumber of users in each cluster K . To see this, we derive thecooperative probability P c as follows.A cluster hits the k th file group if at least one of the K users in the cluster requests a file in G k , whose probability isdenoted as P hk . It is the complement of the probability that nouser requests any file in the k th file group, which is (1 − P k ) K .Then, from (2), P hk can be obtained as P hk = 1 − (1 − P k ) K (3)which increases with K , because ≤ P k ≤ .Cooperative probability is the probability that there existsat least one file group hit by all the B clusters. It is thecomplement of the probability that there is no file group hitby all the B clusters, and hence can be derived as P c = 1 − K (cid:89) k =1 (1 − ( P hk ) B ) , (4)where − ( P hk ) B is the probability that the number of clustershitting the k th file group is less than B , which decreases withthe growth of K since B = M/K . Therefore, for a givenvalue of M , P c is an increasing function of K .
4) Key Parameters:
Since only one Coop D2D link percluster is allowed to be active each time, B users out ofall Coop users can be scheduled simultaneously in Mode active Coop users is N a = B in Mode , and is N a = 0 in Mode With the cooperativeprobability, the average number of active Coop users can beobtained as ¯ N a = BP c + 0(1 − P c ) = BP c . (5) ¯ N a characterizes how many interference-free D2D links can transmit at the same time-frequency resources in average,which can reflect the multiplexing gain. In general, the numberof interference-free D2D links demonstrates the same trendwith the network throughput, as to be verified in SectionIV. Hence, a large value of ¯ N a implies a high networkthroughput. When the number of users per cluster K is large,the cooperative probability is high, but the number of activeCoop users is small since N a = B and B = M/K . Thissuggests that there is a tradeoff between two counter-runningeffects: a small value of K leads to more active Coop users if For the considered strategy, the number of active Coop users is less thanthe number of Coop users, i.e., N a ≤ N c . the system operates in Mode 1; a large value of K yieldshigh cooperative probability. In other words, to maximizethe network throughput, the cluster size should be optimized,which is reflected by the number of users per cluster K sincethe number of users in the hotspot M is given.Due to the multiplexing gain and interference-free transmis-sion, the average data rate of Coop users usually exceeds thatof N-Coop users. As a result, the overall network throughputwill be reduced if we simply assign identical bandwidth tothese two types of D2D users. Owing to the same reason, sim-ply allocating all the bandwidth to Coop users can maximizethe network throughput, but no N-Coop users can be served.This indicates that the bandwidth allocation factor η should beoptimized to maximize the throughput of the network underthe constraint on the data rate of each user to avoid unfairness. C. Optimization of Cluster Size and Bandwidth Allocation
In this subsection, we jointly optimize the bandwidth allo-cation factor η and cluster size K to maximize the averagenetwork throughput under a constraint that the average userdata rate is larger than a given value, µ (Mbps). Because weassume overlay D2D communications, only D2D users areconsidered in the network throughput.In the sequel, we first derive the average network throughputachieved by all D2D users. Then, we derive the average datarate for each of Coop and N-Coop users. Finally, we find theoptimal cluster size K and bandwidth allocation factor η .
1) Average Network Throughput:
Recall that only one CoopD2D link (if any) and one N-Coop D2D link are scheduledper cluster in each time. Then, the average throughput of thenetwork operating in
Mode ¯ R = E { W B (cid:88) i =1 R ni } ( a ) = W B ¯ R ni , (6)where the expectation is taken over small scale channel fadingand user location, ( a ) comes from the fact that all users arerandomly located and transmit with equal power, and R ni and ¯ R ni are the instantaneous and average data rate per unitbandwidth per second of the N-Coop link in the i th cluster,respectively.Analogically, the average throughput of the network oper-ating in Mode ¯ R = E { ηW B (cid:88) i =1 R ci + (1 − η ) W B (cid:88) i =1 R ni } = W B ( η ¯ R ci + (1 − η ) ¯ R ni ) , (7)where R ci and ¯ R ci are the instantaneous and average data rateper unit bandwidth per second of the Coop link in the i thcluster, respectively.Further considering the cooperative probability P c in (4),the average throughput of the network is ¯ R = P c ¯ R + (1 − P c ) ¯ R = W B ( P c η ¯ R ci + (1 − P c η ) ¯ R ni ) . (8) Proposition 1:
The average data rate per unit bandwidth persecond of the N-Coop link in the i th cluster is ¯ R ni = log ( Q ( α )) − log ( Q ( α )) − , (9)where Q ( α ) (cid:44) (cid:82) √ r − α g ( r ) dr +8 (cid:82) √ r − α f ( r ) dr , Q ( α ) (cid:44) (cid:82) √ r − α f ( r ) dr , and f ( r ) and g ( r ) are in closed-form expres-sion defined in Appendix A. Proof:
See Appendix A.In Proposition 1, Q ( α ) and Q ( α ) are easy to be computednumerically. We can see that ¯ R ni only depends on the path lossexponent α . Proposition 2:
The average data rate per unit bandwidth persecond of the Coop link in the i th cluster is ¯ R ci = log (1 + P D − α Bσ Q ( α )) . (10) Proof:
See Appendix B.By substituting (9) and (10) into (8), the average networkthroughput can be obtained as ¯ R = W BP c η (log ( Q ( α )) − log ( Q ( α )) − W B (1 − P c η ) log (1 + P D − α Bσ Q ( α ) . (11)
2) Average User Data Rate:
Since only one N-Coop userand one Coop user (if exists) are active in a cluster each time,with round robin scheduling, the average data rates of N-Coopand Coop users can be respectively obtained from (9) and (10)as follows ¯ R nu = W (1 − η ) E { BR ni N n } ( a ) ≈ W B (1 − η ) ¯ R ni ¯ N n = W B (1 − η )¯ N n log (1 + P D − α Bσ Q ( α )) , ¯ R cu = W η E { BR ci N c } ( b ) ≈ W Bη ¯ R ci ¯ N c = W Bη (log ( Q ( α )) − log ( Q ( α )) − N c , (12)where (a) and (b) come from the fact that R ni and N n areindependent random variables and the same to R ci and N c ,thus E { R ni /N n } = E { R ni } E { /N n } ≈ E { R ni } / E { N n } =¯ R ni / ¯ N n according to (A.4) and E { R ci /N c } ≈ ¯ R ci / ¯ N c ana-logically. ¯ N c = E { N c } and ¯ N n = E { N n } are the averagenumbers of Coop users and N-Coop users, respectively. Proposition 3:
The average number of Coop users is ¯ N c = (cid:88) Φ N B (cid:89) i =1 K ! (cid:81) K k =1 ( P k ) n ik (cid:81) K j =1 n ij ! K (cid:88) k =1 B (cid:88) i =1 ζ ( k ) n ik , (13)which can be approximated as ¯ N c ≈ ¯ N c + ¯ N c , (14)where ζ ( k ) , n ik , ¯ N c and ¯ N c are defined in Appendix C. Proof:
See Appendix C.Though we can use similar way to derive the average num-ber of N-Coop users ¯ N n as for ¯ N c , the resulting expression is complicated. Considering that N n + N c + N b = M , we canobtain ¯ N n by deriving the average number of cellular users ¯ N b = E { N b } . Since all requests follow a Zipf distributionindependently, the number of users that can not fetch files viaD2D is a random variable following a Binomial distributionand N b ∼ B ( M, − (cid:80) Kk =1 P k ) . Therefore, ¯ N b = M (1 − (cid:80) Kk =1 P k ) . Then, the average number of N-Coop users is ¯ N n = M − ¯ N c − ¯ N b . (15)With the average number of Coop and N-Coop users ¯ N c and ¯ N n , we can obtain the corresponding average user datarate ¯ R cu and ¯ R nu using (12).
3) Joint Optimization of η and K : The bandwidth alloca-tion factor and cluster size that maximize the average networkthroughput under the constraint of average user data rate canbe optimized from the following problem max η,K ¯ Rs.t. ¯ R cu ≥ µ, ¯ R nu ≥ µ, < η ≤ , KB = M. (16)Since the number of users per cluster K is an integer, wecan find the joint optimal solution by first finding optimal η for any given K and then enumerating K until the value of ¯ R computed by (11) achieves the maximum under the twoconstraints.By taking the derivative of ¯ R in (8) with respect to η , wehave ∂ ¯ R∂ ¯ η = W BP c ( ¯ R ci − ¯ R ni ) . The constraints ¯ R cu ≥ µ and ¯ R nu ≥ µ can be respectively rewritten as η ≤ W B ¯ R ni − µ ¯ N n W B ¯ R ni and η ≥ ¯ N c µW B ¯ R ci according to (12). Therefore, if ¯ R ci ≥ ¯ R ni , ¯ R is anincreasing function of η , then the optimal solution of problem(16) for any given K is η ∗ K = W B ¯ R ni − µ ¯ N n W B ¯ R ni ; otherwise, ¯ R isa decreasing function of η , and η ∗ K = ¯ N c µW B ¯ R ci . Proposition 4: If ¯ I i ≥ Bσ , then ¯ R ci ≥ ¯ R ni . Proof:
See Appendix D.Since the power of interference among N-Coop users ¯ I i defined in (A.1) is much larger than noise variance σ in D2Dcommunications and B is finite, the condition ¯ I i ≥ Bσ inProposition 4 is easy to be satisfied. Consequently, the optimalvalue of η for a given K is η ∗ K = 1 − µ ¯ N n W B ¯ R ni , (17)which only depends on the average data rate of N-Coop link ¯ R ni and the number of N-Coop users ¯ N n .The joint optimal solution K and η can be found by one-dimensional searching and η ∗ is with closed-form expression,and is hence with low complexity. Because K ∗ and η ∗ dependon W , µ , B , M , N f , β and N , which are usually fixed for along time, they are unnecessary to be updated frequently.IV. S IMULATION AND N UMERICAL R ESULTS
In this section, we validate the analysis and evaluate theperformance of the proposed opportunistic cooperation strat-egy by simulation and numerical results.
In the simulation, we consider a square hotspot area with theside length D c = 100 m, where M = 180 users are randomlylocated. Such a setting reflects relatively high user density asin [25], where more than one user is located within an areaof × m . The path-loss model is . . ( r ) [14]. Each user is with transmit power P = 23 dBm. W = 20 MHz, and σ = − dBm. The file catalog size N f = 300 ,and each user is willing to cache N = 10 files. The parameterof Zipf distribution is β = 0 ∼ . The user data rate constraintis µ = 1 ∼ Mbps. This setup is used in the sequel unlessotherwise specified.
A. Impact of Cluster Size and Number of Total Users
In Fig. 4, we provide numerical results of the averagenumber of active Coop users ¯ N a obtained from (5) andsimulation results for the average sum rate of the active Coopusers versus the number of users per cluster K . It is shownfrom Fig. 4(a) that ¯ N a increases with β . This is becausewith large β , a few popular files are requested by majority ofusers, which leads to high cooperative probability P c . Withthe increase of K , ¯ N a first increases and then decreases. It isshown from Fig. 4(b) that with the growth of K , the averagesum rate of active Coop users exhibits the same trends with ¯ N a , which agrees to the analysis in section III.A. K(b) S u m R a t e o f A c t i v e C oop U s e r s ( bp s / H z ) - =1 - =0.8 - =0.6 - =0.4 - =0.2 - =0 K(a) A v e r age N u m be r o f A c t i v e C oop U s e r s - =1 - =0.8 - =0.6 - =0.4 - =0.2 - =0 Fig. 4. ¯ N a and average sum rate of active Coop users versus K . In Fig. 5, we simulate the cooperative probability P c . Itis shown from Fig. 5(a) that P c increases with K , whichagrees with the analysis after (4). Moreover, the cooperativeprobability is high although the full cooperation is allowedonly when all clusters hit the same file group, especiallywhen β is not small, say β > . . In Fig. 5(b), we showthe impact of partial cooperation by changing the minimalnumber of clusters allowed to cooperate. As expected, P c can be improved if we allow partial cooperation amongclusters, but not significant. Therefore, the throughput gainfrom partial cooperation over full cooperation is marginal asto be illustrated later. We set such a minimal number because when we allow fewer clusters tocooperate, the multiplexing gain will reduce despite that P c will be higher,and then the resulting throughput will reduce. When this minimal number ofclusters is six, it becomes the full cooperation strategy. In Fig. 6, we provide numerical results of the optimal clustersize K ∗ and the resulting average number of active Coop users ¯ N a for different number of total users in the hotspot area M .When M = 1000 , there are users in an area of × m , corresponding to a very high traffic load [25]. It is shownfrom Fig. 6(a) that K ∗ decreases with β as expected. Withthe growth of M , K ∗ first increases and then approaches aconstant that equals to K = N f /N . This is because when K = K , all files in the catalog can be cached at the users ineach cluster. Assigning more than K users to each cluster cannot increase the number of Coop users. It is shown from Fig.6(b) that with the growth of M , the average number of activeCoop users monotonously increases when β is large but firstincreases and then decreases when β is small. This impliesthat if the user density is large but β is small, the proposedopportunistic cooperation strategy may be not useful. B. Accuracy of the Approximations
In Fig. 7, we evaluate the accuracy of all the approximationsused in deriving the throughput and the number of users bysimulation and numerical results. In Fig. 7(a), the numericalresults of ¯ R ci and ¯ R ni are respectively obtained from (9) and(10) by changing the path loss exponent α from to . We K(a) C oope r a t i v e P r obab ili t y - =1 - =0.8 - =0.6 - =0.4 - =0.2 - =0 Minimal Number of Coop Cluster(b) C oope r a t i v e P r obab ili t y - =1 - =0.8 - =0.6 - =0.4 - =0.2 - =0 K=30,B=6K=20,B=9
Fig. 5. P c of (a) full cooperative, and (b) partial cooperation. M (a) O p t i m a l C l u s t e r S i z e - =0 - =0.2 - =0.4 - =0.6 - =0.8 - =1 M, with K * (b) A v e r age N u m be r o f A c t i v e C oop U s e r s - =1 - =0.8 - =0.6 - =0.4 - =0.2 - =0 Fig. 6. K ∗ and ¯ N a versus the number of total users in the hotpot M . can see that with the growth of α , the average data rateper unit bandwidth per second of the Coop link decreasesbut that for the N-Coop link increases. This is because bothsignal and interference power decrease when α increases, butthe interference power decreases more rapidly due to largerdistance of interference link than that of signal link. In Fig.7(b), the numerical results of the number of Coop users ¯ N c and the number of N-Coop users ¯ N n are respectively obtainedfrom (14) and (15). As expected, when K increases, thenumber of Coop users increases due to the decrease of thenumber of clusters. However, the number of N-Coop users firstdecreases and then increases slowly with K . This is becausewith the growth of K , more files can be cached in each clusterand thus the total number of D2D users increases, and thenumber of N-Coop users changes as in (15). We can see thatthe approximations are accurate, except the average data rateper unit bandwidth per second of the Coop link when α islarge, which comes from the first order approximation in (A.5). Pathloss Exponent - , (a) A v e r age D a t a R a t e ( bp s / H z ) Coop-Num.Coop-Sim.N-Coop-Num.N-Coop-Sim. K, - =1(b) A v e r age N u m be r o f U s e r s Coop-Users-Num.Coop-Users-Sim.N-Coop-Users-Num.N-Coop-Users-Sim.
Fig. 7. Average data rate per unit bandwidth per second and number ofusers. “Num.”: Numerical results, “Sim.”: Simulation results.
C. Optimal Bandwidth Allocation and Network Throughput
In Fig. 8(a), we present the optimal solution of problem (16) η ∗ versus the Zipf parameter β . We can see that η ∗ increaseswith β . This is because the number of interference-free links, ¯ N a , increases with β , and then allocating more bandwidth toCoop users can increase the network throughput. η ∗ decreasesas µ increases, because more bandwidth is needed for N-Coopusers to support higher user data rate.In Fig. 8(b), we provide simulation results for the maxi-mal network throughput. In the legends, “ η = 0 ” refers tothe strategy in [13] (without interference coordination) and“TDMA” is the strategy in [14] (without strong inter-clusterinterference), which serve as the baselines for comparison.“ η = 0 . , K = 30 ” refers a cooperation strategy withoutoptimizing K and η , which allocates equal bandwidth to Coopusers and N-Coop users. “ η ∗ , K = 30 ” refers to a cooperationstrategy without optimizing K but only optimizing η . Wecan see that optimizing the bandwidth allocation becomesnecessary when β > . , while optimizing the cluster size isalways necessary but the gain from optimization grows with β . With K ∗ and η ∗ , the throughput gain over the baselinefor β = 1 is ∼ , which demonstrates thatthe proposed opportunistic cooperation strategy can boost thenetwork throughput remarkably. Even when β = 0 , where filepopularity follows a uniform distribution, the throughput gainis still ∼ . We also demonstrate the performance of“Partial-Coop”, i.e., allowing cooperation among less than B clusters. It is shown that the performance can be improved byallowing partial cooperation, but the gain is marginal. - , with K * (a) O p t i m a l A ll o c a t i on F a c t o r * =1 =1.5 =2 - , =1(b) N e t w o r k T h r oughpu t ( bp s ) K * , * K * , * * ,K=30 =0.5,K=30 =0 TDMA
Full-CoopPartial-Coop
Fig. 8. η ∗ and maximal average network throughput versus β . V. C
ONCLUSIONS
In this paper, we proposed an opportunistic cooperationstrategy for cache-enabled D2D communications. We jointlyoptimized the cluster size and the bandwidth allocated to Coopand N-Coop users to maximize the network throughput withminimal user data rate constraint. Simulation results showedthat the proposed strategy can boost the throughput even whenthe content popularity follows a uniform distribution, and thegain over existing strategies is remarkable when the popularitydistribution is more skewed.A
PPENDIX AP ROOF OF P ROPOSITION i th cluster can be expressed as γ ni = P | h ii | r − αii I i + σ , (A.1)where P is the transmit power, σ is the variance of whiteGaussian noise, I i = P (cid:80) Bj =1 ,j (cid:54) = i r − αij | h ij | is the total powerof inter-cluster interference, h ij and r ij are respectively thechannel coefficient and distance between the DT and the DR, α is the path loss exponent, and both the interference channelcoefficient h ij ( i (cid:54) = j ) and the desired channel coefficient h ii follow a complex Gaussian distribution with zero mean andunit variance. Due to the short distance between D2D links, the D2Dnetwork is interference-limited and hence the noise can beignored, i.e., I i (cid:29) σ .Then, from (A.1) the data rate per unit bandwidth persecond for the N-Coop link in the i th cluster is R ni =log (1 + P | h ii | r − αii I i ) .Considering that | h ij | follows an Exponential distribution,which is a special case of the Gamma distribution, the in-terference power I i (which is a sum of random variablesfollowing a Gamma distribution) can be approximated as aGamma distribution [26]. Further consider that for a Gammadistributed random variable X with parameters k and θ , E { ln( X ) } = ψ ( k ) + ln( θ ) , where ψ ( k ) is the Digammafunction [27]. Then, the average data rate per unit bandwidthper second of the N-Coop link can be obtained according toProposition 9 in [26] as E h { R ni } ≈ log (1 + P r − αii ¯ I i ) , (A.2)where E h {·} represents the expectation taken over small scalechannel fading, ¯ I i = P (cid:80) Bj =1 ,j (cid:54) = i r − αij is the average totalpower of the inter-cluster interference.Since channel fading and user location are distributed inde-pendently, the average data rate per unit bandwidth per secondof the N-Coop link taken over both channel fading and userlocation can be obtained as ¯ R ni = E p { log (1 + P r − αii ¯ I i ) } , (A.3)where E p {·} denotes the expectation taken over user location.Because the joint probability density function (pdf) of thedistances among D2D users is hard to obtain, we introducethe first order approximation to derive the expression of ¯ R ni .Specifically, for a random variable X , the expectation of afunction of X , ϕ ( X ) , can be approximated as [28] E { ϕ ( X ) } = E { ϕ ( µ x + X − µ x ) }≈ E { ϕ ( µ x ) + ϕ (cid:48) ( µ x )( X − µ x ) } = ϕ ( µ x ) , (A.4)where µ x = E { X } and the approximation is accurate whenthe variance of X is small. With this approximation, ¯ R ni in(A.3) can be approximated as ¯ R ni ≈ log ( E p { P r − αii + ¯ I i } ) − log ( E p { ¯ I i } ) . (A.5)The pdf of the signal link distance r ii can be obtained from[29] by variable substitution r = r ii /D as g ( r ) = 1 D r ( r − r + π ) , ≤ r < r(cid:15) − r ( r + 2)+4 r (arcsin( r ) − arccos( r )) , ≤ r < √ . (A.6)To simplify the analysis, the interference link distance r ij isassumed to have the same distribution f ( r ) , where r = r ij /D .We can derive the pdf of the interference link distance r ij asfollows.Denote the position of the DR in the i th cluster as ( x i , y i ) and the position of the DT in the j th cluster as ( x j , y j ) , / ij ij d r D D ( , ) i i x y ( , ) j j x y Fig. 9. The distance of users in two adjacent cluster as illustrated in Fig. 9. The link distance between themnormalized by the cluster side D can be expressed as d ij = √ ( x i − x j ) +( y i − y j ) D = (cid:112) (∆ x ) + (∆ y ) , where ∆ x = x i − x j D and ∆ y = y i − y j D . The pdf of | ∆ y | can be obtained accordingto [29] as p ∆ y ( v ) = (cid:40) − v ) , ≤ v ≤ , otherwise . (A.7)Analogically, the pdf of | ∆ x | can be obtained as p ∆ x ( u ) = (cid:40) − | − u | , ≤ u ≤ , otherwise . (A.8)Then, the cumulative probability distribution function (cdf)of d ij is F d ( r ) = P { d ij ≤ r } = P { (cid:112) | ∆ x | + | ∆ y | ≤ r } = (cid:90) (cid:90) dudvp ∆ x, ∆ y ( u, v ) ( a ) = (cid:90) (cid:90) dudvp ∆ x ( u ) p ∆ y ( v ) , (A.9)where (a) comes from the fact that | ∆ x | and | ∆ y | areindependent random variables.When ≤ r ≤ , from (A.8) and (A.7), we can obtain p ∆ x, ∆ y ( u, v ) = 2 u (1 − v ) , where ≤ u, v ≤ . Then, the cdfin (A.9) can be derived as F d ( r ) = (cid:90) (cid:90) dudvp ∆ x, ∆ y ( u, v )= (cid:90) r dv (cid:90) √ r − v u (1 − v ) du = 23 r − r , (A.10)When ≤ r ≤ √ , we have p ∆ x, ∆ y ( u, v ) = u (1 − v ) , ≤ u ≤ , ≤ v ≤ − u )(1 − v ) , ≤ u ≤ , ≤ v ≤ , otherwise , and the corresponding cdf can be derived as F d ( r ) = (cid:90) √ r − dv (cid:32)(cid:90) u (1 − v ) du + (cid:90) √ r − v − u )(1 − v ) du (cid:33) + (cid:90) √ r − dv (cid:90) √ r − v u (1 − v ) du = 54 + (cid:15) (cid:18) − r + (1 + r ) (cid:15) + 23 (cid:15) − (cid:15) (cid:19) + r (cid:18) − ( 3 r ) + 12 − r (cid:19) , (A.11)where (cid:15) (cid:44) √ r − .Analogically, when √ ≤ r ≤ , the cdf can be derived as F d ( r ) = (cid:90) dv (cid:90) u (1 − v ) du + (cid:90) dv (cid:90) √ r − v − u )(1 − v ) du = − (cid:15) − r r sin − ( 1 r ) + 43 ( (cid:15) − r ) . (A.12)When ≤ r ≤ √ , the cdf can be derived as F d ( r ) = 1 − (cid:90) √ r − dv (cid:90) √ r − v u (1 − v ) du = − − r − r − r (cid:18) sin − ( ξr ) − sin − ( 1 r ) (cid:19) + r ξ + 2 (cid:15) − ξ + 43 (cid:15) + 14 ξ, (A.13)where ξ (cid:44) √ r − .Finally, by combining (A.10) (A.11) (A.12) and (A.13), andconsidering the pdf f ( r ) = dF ( r ) dr , the pdf of the interferencelink distance r ij is f ( r ) = 1 D r − r , ≤ r < r − r + 2 r − r(cid:15) + r(cid:15) − r (cid:15) + 4 r arcsin( (cid:15)r ) , ≤ r < √ r(cid:15) + 4 r arcsin( r ) − r − r , √ ≤ r < − r − r + 4 r(cid:15) − r arcsin( ξr ) − arcsin( r ) − rξ + rξ + r ξ , ≤ r < √ (A.14)where r = r ij /D , (cid:15) (cid:44) √ r − , and ξ (cid:44) √ r − .Since the interference generated by DTs far away from theDR can be ignored due to pathloss, we only consider dominantinterference generated from the nearest eight clusters aroundthe i th cluster as shown in Fig. 1. Then, by substituting thepdf of interference and signal link distance into (A.5), we can obtain the average data rate per unit bandwidth per second ofthe N-Coop link in the i th cluster as ¯ R ni ≈ log (cid:32)(cid:90) √ D P r − αii g ( r ii D ) dr ii +8 (cid:90) √ D P r − αij f ( r ij D ) dr ij (cid:33) − log (cid:32) (cid:90) √ D P r − αij f ( r ij D ) dr ij (cid:33) = log (cid:32)(cid:90) √ P r − α g ( r ) dr + 8 (cid:90) √ P r − α f ( r ) dr (cid:33) − log (cid:32) (cid:90) √ P r − α f ( r ) dr (cid:33) = log ( Q ( α )) − log ( Q ( α )) − , (A.15)This proves Proposition 1.A PPENDIX BP ROOF OF P ROPOSITION
Mode 1 , the cooperative DTs jointly transmit the re-quested files to the Coop users with zero-forcing beamforming,which is of low complexity and hence practical. Then, theSINR at the DR of the active Coop link in the i th cluster canbe expressed as γ ci = P (cid:107) h i (cid:107) δ i σ ≈ P (cid:80) Bj =1 r − αij | h ij | Bσ , (B.1)where h i = [ (cid:113) r − αi h i , (cid:113) r − αi h i , ..., (cid:113) r − αiB h iB ] is the com-posite channel vector between all DTs and the DR, ≤ δ i ≤ ,and a larger value of δ i indicates a better orthogonalitybetween h i and h j for i (cid:54) = j . The approximation comes fromthe fact δ i ≈ ( BN t − B + 1) /B = 1 /B [30], where N t is thenumber of antennas per DT and N t = 1 in this paper. Thisapproximation is accurate when the variance of δ i is small.Using the same approximation as deriving (A.2), the averagedata rate per unit bandwidth per second of the Coop link isobtained as ¯ R ci ≈ E p { log (1 + P (cid:80) Bj =1 r − αij Bσ ) } . (B.2)By applying the first-order approximation in (A.4), using(A.6) and (A.14), and only considering dominant signal, wecan obtain the average data rate per unit bandwidth per secondfor the Coop link as ¯ R ci ≈ log (cid:32) Bσ + 8 (cid:90) √ D P r − αij f ( r ij D ) dr ij + (cid:90) √ D P r − αii g ( r ii D ) dr ii (cid:33) − log (cid:0) Bσ (cid:1) = log (1 + P D − α Bσ Q ( α )) . (B.3)This proves Proposition 2. A PPENDIX CP ROOF OF P ROPOSITION i th cluster who requestfiles in G k as n ik , ≤ k ≤ K , ≤ i ≤ B .Since the users request files independently, the probabilitythat the combination of the numbers of users in the i th clusterwho request files in each file group is { n i , n i , ..., n iK } (cid:44) N i can be derived as p N i = K (cid:89) m =1 C n im K − (cid:80) m − j =1 n ij K (cid:89) k =1 ( P k ) n ik ( a ) = K ! (cid:81) K k =1 ( P k ) n ik (cid:81) K j =1 n ij ! , (C.1)where ( a ) comes from C mn C kn − m = n ! m ! k !( n − m − k )! .Only when all B clusters hit a file group G k and k ≤ K (i.e., n ik > is satisfied for k ≤ K and any i , ≤ i ≤ B ),the users requesting the files within G k are Coop users, andwe call G k a hit file group .The number of Coop users for all hit file groups can beobtained as N c = K (cid:88) k =1 B (cid:88) i =1 ζ ( k ) n ik , (C.2)where ζ ( k ) = (cid:100) (cid:80) Bi =1 u ( n ik ) − B + 1) (cid:101) + ( k ≤ K ) indicatesthat whether G k is a hit file group , the function u ( x ) = 1 when x > , otherwise u ( x ) = 0 , and (cid:100) Λ (cid:101) + = max(Λ , .Denote N = {N , N , ..., N B } , and Φ N = {N | n ik ≥ , (cid:80) K k =1 n ik = K } represents a set of all possible combi-nations of N . Then, by taking average of N c in (C.2) over Φ N , we can derive the average number of Coop users as ¯ N c = (cid:88) Φ N B (cid:89) i =1 p N i N c = (cid:88) Φ N B (cid:89) i =1 K ! (cid:81) K k =1 ( P k ) n ik (cid:81) K j =1 n ij ! K (cid:88) k =1 B (cid:88) i =1 ζ ( k ) n ik . (C.3)The number of Coop users can be computed accurately using(C.3). However, the cardinality of Φ N is K ( K − B , and hencethe computational complexity exponentially increases with K .For example, when K = K = 15 and B = 9 , we obtain | Φ N | = 1 . × .In the sequel, we seek an alternative solution.Noticing that the probability that multiple hit file groups exist simultaneously decreases with the growth of the numberof hit file groups, we only consider the case where only one ortwo hit file groups exist to approximate the average number ofCoop users as follows. This approximation is accurate whenthe number of clusters is large.The probability that a hit file group only contains G k canbe obtained from (3) as (cid:81) Kj =1 ,j (cid:54) = k (1 − ( P hj ) B )( P hk ) B . As aresult, the probability of n ik = m ( ≤ m ≤ K ) when a hitfile group only contains G k is C mK ( P k ) m (1 − P k ) ( K − m ) /P hk .Then, the average number of Coop users in the cases where only one hit file group exists can be derived as ¯ N c = K (cid:88) k =1 K (cid:89) j =1 ,j (cid:54) = k (1 − ( P hj ) B )( P hk ) B · B K (cid:88) m =1 C mK ( P k ) m (1 − P k ) ( K − m ) P hk m = K (cid:88) k =1 K (cid:89) j =1 ,j (cid:54) = k (1 − ( P hj ) B )( P hk ) B − BP k K. (C.4)The probability that there only exist two hit file groups , G k and G k , can be obtained from (3) as (cid:81) Kj =1 ,j (cid:54) = k ,k (1 − ( P hj ) B )( P hk ) B ( P hk ) B . Consequently, the probability of n ik = m and n ik = m (where ≤ m ≤ K , ≤ m ≤ K and ≤ m + m = m ≤ K ) when the hit file groups onlycontain G k and G k is C m m C m m − m ( p k ) m ( p k ) m P hk P hk . Then, theaverage number of Coop users in the case where only two hitfile groups exist can be derived as ¯ N c = (cid:88) Φ k k K (cid:89) j =1 ,j (cid:54) = k ,k (cid:16) − ( P hj ) B (cid:17) ( P hk ) B ( P hk ) B · B K (cid:88) m =2 (cid:88) Φ m m C m m C m m − m ( p k ) m ( p k ) m P hk P hk m = (cid:88) Φ k k K (cid:89) k =1 ,k (cid:54) = i,j (cid:16) − ( P hk ) B (cid:17) ( P hi P hj ) B − · B K (cid:88) m =2 (cid:88) Φ m m mm ! m ! m ! ( p k ) m ( p k ) m , (C.5)where Φ k k = { k , k | ≤ k , k ≤ K, i (cid:54) = j } , Φ m m = { m , m | m + m = m } , whose cardinality are K and m ( m ≤ K ), respectively.By combining (C.4) and (C.5), the average number of Coopusers is approximated as ¯ N c ≈ ¯ N c + ¯ N c , (C.6)which can be obtained much easier than (C.3).This proves Proposition 3.A PPENDIX DP ROOF OF P ROPOSITION ¯ R ni from ¯ R ci and considering ¯ I i (cid:29) σ , we can obtain that ¯ R ci − ¯ R ni = E { log (1 + P (cid:80) Bj =1 r − αij Bσ ) }− E { log (1 + P r − αii ¯ I i ) } = E { log ( ¯ I i Bσ ) } . When log ( ¯ I i Bσ ) ≥ (i.e., ¯ I i ≥ Bσ ), ¯ R ci − ¯ R ni ≥ .This proves Proposition 4. R EFERENCES[1] K. Doppler, M. Rinne, C. Wijting, C. B. Ribeiro, and K. Hugl, “Device-to-device communication as an underlay to LTE-advanced networks,”
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