Effect of Retransmissions on Optimal Caching in Cache-enabled Small Cell Networks

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1 Effet of Retransmissions on Optimal Cahing in Cahe-enabled Small Cell Networks Shankar Krishnan, Mehrnaz Afshang, and Harpreet S. Dhillon arxiv:66.397v2 s.it] 23 Mar 27 Abstrat Cahing popular ontent in the storage of small ells is being onsidered as an effiient tehnique to omplement limited bakhaul of small elln ultra-dense heterogeneous ellular networks. Limited storage apaity of the small ells rendert important to determine the optimal set of files ahe) to be plaed on eah small ell. In ontrast to prior works on optimal ahing, in this work we study the effet of retransmissions on the optimal ahe plaement poliy for both stati and mobile user senarios. With the popularity of files modeled as a Zipf distribution and a maximum n transmissions, i.e., n retransmissions, allowed to reeive eah file, we determine the optimal ahing probability of the files that maximizes the hit probability. Our losed-form optimal solutions onretely demonstrate that the optimal ahing probabilities are very sensitive to the number of retransmissions. Index Terms Stohasti geometry, small ell ahing, retransmissions, mobility, hit probability, optimal ahe. I. INTRODUCTION Aggressive reuse of spetrum through dense deployment of small ell base stations SCBSs) and ahing popular ontent in their storage an help address growing apaity demands and redue bakhaul loads ]. The basidea behind small ell ahing is to download popular ontent mainly video) automatially in the ahe of SCBSs at off-peak hours, whih an then be delivered to the users during peak hours. However, the key hallenge in designing suh a ahe-enabled network is to determine the ontent that should be ahed at eah SCBS. Among the entire ontent available in the internet, only a small fration of the total ontent, termed popular ontent, is aessed by a large fration of users 2], thus allowing us to fous primarily on the library of popular files to ahe in SCBSs. However due to limited apaities of ahe storage units, one an only ahe a subset of the library on eah SCBS. This neessitates the need to look into optimal ahing of the popular ontent for maximum utilization of these ahes and improve the overall network performane. Prior work: Optimal ahing of popular ontent in the storage of ahe-enabled wireless networks has been studied for quite some time and an be broadly lassified into two ategories. The first line of work onsiders a senario where eah user has aesn overage) to at most one ahe and the optimal ahe plaement resultn storing the most popular filen the storage of all ahes. To give an example, the authorn 3] studied optimal ahe plaement in a Devie-to- Devie D2D) assisted wireless ahing network and showed that the most popular ontents need to be ahed more often in the network to maximize the offloading probabilitien The authors are with Wireless@VT, Department of ECE, Virginia Teh, Blaksburg, VA, USA. {kshank93, mehrnaz, hdhillon}@vt.edu. this two-tier wireless ahing system. The seond line of work onsiders a multi-overage senario user is overed by multiple ahes) and studies optimal ahe plaement while exploiting the multiple overed ahes as a larger distributed ahe. Using tools from stohasti geometry, 4] shows that it is not always optimal to ahe the most popular ontent everywhere in a multi-overage senario and proposes a probabilisti plaement poliy to maximize the user s hit probability. Reently, the authorn 5] also studied optimal ahing in heterogeneous ellular networks with different ahe apabilitien eah tier of the network and showed that an optimal ontent plaement is signifiantly better than storing the most popular ontent everywhere. Despite inreasing interest in the analysis of ahe-enabled networks, all the prior works onsider a single transmission senario, whih is not quite realisti beause of the provision of retransmissionn atual systems. For wireless ad ho networks, it is shown in 6] and 7] that an inrease in the number of allowed retransmissions dereases the link outage probability and thereby improves reliability. In this work, we extend the above argument to ahe-enabled small ell networks by retransmitting the pakets reeived in error to inrease the hanes of a ahe hit. In partiular, we determine the optimal ahing strategies that maximize hit probability. It should be noted that retransmissions may impat other metris, e.g., it may inrease energy onsumption, inrease lateny, and lower spetral effiieny. However, the investigation of the effet of retransmissions on these metris not in the sope of this paper. More details are provided next. Contributions: In this work, we determine the optimal ahing poliies that maximize hit probability for a typial user stati or mobile), whih tries to reeive its file of interest from the ahe of SCBSs within a predefined number of maximum transmissions n. Here hit probability is the probability that the user is able to suessfully reeive the file within n transmissions. As expeted, it is seen that the hit probability inreases with the number of retransmissions. In ontrast to prior works on optimal ahing, we also determine the optimal ahing poliien stati and mobile senarios, and show that the optimal solutionn the two ases are signifiantly different. While the stati ase tends to ahe the most popular files from the library on eah SCBS, mobility de-emphasizes the importane of popularity of files and allows the SCBSs to ahe ontent in a more balaned way. II. SYSTEM MODEL We onsider an ultra-dense ahe-enabled SCBS network as illustrated in Fig. and model the loations of the SCBSs by a

2 2 homogeneous Poisson Point Proesses PPP) Φ with density λ 8]. We assume eah SCBS an ahe a maximum of L files, and the total number of filen the library is denoted by K. We denote by P Ri the probability that the i th file, F i, will be requested. We order the files based on their popularity, i.e., i and i K orrespond to the most popular and least popular files, respetively. The popularity of the filen the library is assumed to follow Zipf s law 2], i.e. P Ri where γ > is the Zipf parameter whih governs the skewness of the popularity distribution. We also assume that eah SCBS ahes file F i with probability b i independently of the other i γ / K j j γ, SCBSs. Therefore, K i b i L. Retransmissions: We onsider a typial user that attempts to download its file of interest from the ahe of SCBSn an ultra dense network as shown in Fig.. It is quite likely that a single attempt to download the file is unsuessful, either beause none of the SCBSn the user s viinity has ahed the file eah SCBS ahes only a fration of the library) or due to poor hannel onditions. In this work, we hene study the effet of retransmissions on the network performane for both stati same user loation during all retransmission attempts) and mobile user senarios. For the analysis of mobile users, we onsider the popular infinite mobility model 9] ] in whih a user is assumed to experiene an independent realization of the point proess for eah transmission. This model is quite relevant for ultra dense networks where even a small displaement of a user may take it to a ompletely new loal neighborhood of SCBSs. For the setup studied in this paper, we assume that a typial user an be mobile or stati) tries to reeive its file of interest F i from the ahe of SCBSs for a maximum of n transmissions or n retranmissions). The signal to interferene ratio SIR) reeived at the typial user in the k th h transmission an be expressed as SIR i,k xk x α h yk y, α where {h xk, h yk } exp) denote the Rayleigh fading hannel gains from the serving devie x Φ and interferer {y} in the k th transmission, and α is standard powerlaw pathloss with exponent α > 2. Here, we assume that the fading gains are independent aross transmission attempts. Cell Seletion Poliy: To hoose a serving SCBS for the typial user, a straightforward hoie would be to onnet to the SCBS that maximizets average reeived power at the reeiver, agnosti of the ahed file in that SCBS. This orresponds to the losest SCBS to the typial user. We term this poliy as poliy ahe-agnosti poliy) and denote it by P. While this poliy is meaningful in general, it suffers from a disadvantage in ahe-enabled networks: the losest SCBS is not guaranteed to have the file requested by the user. To address this, we also study poliy 2 ahe-aware poliy), denoted by, where the user onnets to the losest SCBS that has the file it needs. This requires the knowledge of the ahe of nearby SCBSs, whih an be obtained by a entralized mehanism where maroells an assist the users to onnet to the SCBS whih ontaints file of interest. However sine this SCBS may not always be in the viinity of the user, the file transfer may not neessarily sueed due to poor hannel onditions. Therefore, there are tradeoffs P SCBS ahes file of interest P 3 P 4 2 P P 5 Fig.. User tries to obtain the file of interest from the network using Poliies and 2 P and ) while moving from loation through 5. Under P, user onnets to the SCBS providing highest average reeived power losest). Under, user onnets to the losest SCBS that hats file of interest. involved in hoosing the ell seletion poliies. For this system model, we introdue the optimal ahing problem that maximizes the hit probability under the two ell seletion poliies P and in the next setion. III. HIT PROBABILITY In a given transmission, a file is suessfully reeived only when the user in overage of a SCBS whih has the file of interest in its ahe. For a given modulation and oding threshold T, the overage probability of file F i in the k th transmission is given by PSIR i,k > T ). Let S i be the event that file F i is suessfully reeived within n transmissions with the suess probability denoted by p n PS i ). We assume i.i.d. fading over the n transmissions and hene the suess probability of file F i in eah transmission is the same and denoted by p si. If all the n transmissionn reeiving the file are unsuessful, the user is said to be in outage from file F i. Let the outage probability of file F i in n transmissions be given by p n o i p n. The performane metri of interest is the hit probability HP), whih we define mathematially as the sum of the probabilities of suessfully reeiving eah file in the library within n transmissions, weighted by their orresponding request probabilities and is expressed below. P hit i P Ri p n i P Ri p n o i ). ) Our goal is to maximize the hit probability whih will be defined for two senarios - i) mobile user and ii) stati user in the next two subsetions. The optimal ahing probabilities {b i } that maximize the HP for the two ell seletion poliies P and will also be disussed for both senarios. A. Mobile user Under infinite mobility assumption, the probability of file suess or outage) in eah transmission independent of the previous transmissions. Hene, the outage or suess) probability of file F i in n transmissions simply given as the produt of outage or suess) probabilitien eah transmission i.e. p n o i p si ) n. The optimization problem to maximize the hit probability an be hene formulated as: P Ri p si ) n ), 2) max i

3 3 s.t. b i L and b i, i,... K. i ) Poliy : As desribed in setion II, under poliy user onnets to the losest SCBS as that maximizets average reeived signal strength. A suessful reeption of the file hene depends on the probability that it is available in the ahe of the losest SCBS and that the user in overage of the losest SCBS. Mathematially, the probability of suessfully reeiving file F i in a given transmission is thus the produt of its ahing probability b i and overage probability under poliy denoted by p ) i.e. p si b i p ). It is worth noting that this overage probability p ) when user onnets to the losest SCBS) independent of the density of SCBSs under interferene limited regime and has been derived in 2] for a similar ellular downlink problem. The probability of suessfully reeiving file F i in a given transmission under poliy is therefore, p si b i p ) b i + ρ T, α), 3) where ρ T, α) T 2/α T 2/α du. 4) + uα/2 It an be shown that the optimization problem 2) is onave and hene the Karush-Kuhn-Tuker KKT) onditions provide neessary and suffiient onditions for optimality. The Lagrangian funtion orresponding to Problem 2) is Lb, ν, µ, w) i i P Ri b i p ) ) n ) + ν µ i b i + w i b i ), i where µ, w R K + and ν R. b i L) i Let b, ν, µ and w be primal and dual optimal. The KKT onditions for Problem 2) state that b i L, 5) i b i, µ i, w i, µ i b i, i,... K 6) w i b i ), i,... K, 7) P Ri n b i p ) ) n p ) +ν µ i + w i, i,... K. 8) The optimal ahe plaement under poliy is given next. Theorem. The optimal ahing probability of file F i denoted by b i, that maximizes the HP with a maximum n transmissions) for a mobile user under poliy, is given by,, ν < P Ri np ) b i, ν > P Ri np ) p ) ) n ] 9) v n p ) ) P Ri np ) where ν P Ri n b i p) ) n p ) an be obtained as the solution of the onstraint K i b i L. Proof. From 6) and 8), we have w i b i P Ri n b i p ) ) n p ) ν ], ) whih when inserted into 7) gives b i b i ) P Ri n b i p ) ) n p ) ν ]. ) From ), we see that < b i Sine we know that b i < only if, ν P Ri n b i p ) ) n p ). 2), thimplies that ν P Ri np ), P Ri np ) p ) ) n ]. 3) For the above interval, solving for ν using the onstraint L, we get, K i b i i p ) P Ri np ) v v ) np ) n ] L ) n a) K Lp ) K ) n j P Rj 4) where a) results by using p ) p ), i,... K and rearranging few terms. Also, it an be seen that for ν < P Ri np ), b i and if v > P Ri np ) p ) ) n, b i. In order to provide intuition, we speialize the above result to the simple ase of 2 filen the library K 2) and unitary storage spae L ) in the SCBS. Corollary. The optimal value b, b 2) obtained by solving the optimization problem 2) for K 2 is, n < + γ ) b log 2 p ), 5) a +p ) a+)p ) where a 2 γ n, γ is the Zipf parameter and b 2 b. Proof. See Appendix A. From Corollary, we an see that it is optimal to ahe only the most popular file F till a ertain number of transmissions, however an optimal ahe ahe both F and F 2 ) exists as the number of transmissionnrease. Also, for n single transmission), it is always optimal to ahe the most popular file in a 2-file library senario. For the simpliity of exposition, we will heneforth onsider L while deriving the optimal solutions. A similar approah an be undertaken to derive the optimal solutions for the generi L storage system. 2) Poliy 2: In this poliy, the user onnets to the losest SCBS whih has the file of interest in its ahe. The key differene in the mathematial formulation under this poliy ompared to P is that the suess probability is not weighted by ahing probability of the file as the user is always onneted to the SCBS that has the file of interest in its ahe. Hene, the suess probability of obtaining file F i under poliy

4 4 2 is the same ats overage probability denoted by p 2), whih has been derived for a similar senario in 3, Theorem ] as: p si p 2) b i b i + ρ T, α) + b i )ρ 2 T, α), 6) T 2/α where ρ 2 T, α) T 2/α du, 7) + uα/2 and ρ T, α) is defined in 4). The solution of the optimization problem 2) under poliy 2 an be determined on the same lines as Theorem poliy ) by using the suess probability p si defined above. The optimal solution is stated below in Theorem 2. Theorem 2. The optimal ahing probability of file F i denoted by b i, that maximizes the hit probability with a maximum n transmissions) for a mobile user under poliy 2, is given by,, ν < PR i n b C i, ν > PR i ncb+c )n, 8) B+C) n+ φν ) where φv ) is the solution over b i of P Ri ncb )b i + C) n Bb i + C) n+ + ν, 9) B ρ 2 T, α), C ρ T, α) + ρ 2 T, α) and ν an be obtained as the solution of the onstraint K i b i. The suess probability p si for poliy 2 given by 6)), being more ompliated than poliy given by 3)) makes the optimal solution in this ase harder to obtain. This requires solving the polynomial equalities of the form 9), whih may not give simple losed form solutions. Therefore, we limit our further disussion in providing the optimal solution only for the extreme ases n and n ) for a user following. Corollary 2. Single transmission - Poliy 2) The optimal ahing probability of file F i denoted by b i, that maximizes the HP for a mobile user under with a maximum of one attempt n ), is given by, b i P Ri ɛ ρ T, α) + ρ 2 T, α)) ] +, i,... K, 2) ρ 2 T, α) K i PRi K )ρ T,α)+K ρ 2T,α)+ where x] + max, x), ɛ and K, K K, satisfies the onstraint that b i. Here ρ T, α) and ρ 2 T, α) are defined in 4) and 7) respetively. Proof. The result follows by substituting n single attempt) in Theorem 2, solving for v using the onstraint K i b i and simple mathematial manipulation. The detailed proof is skipped due to spae onstraints. Corollary 3. Large number of transmissions) For the mobile user senario poliy and 2) with the number of transmissions approahing infinity asymptotially), it is optimal to ahe the filen the library evenly i.e. lim b i n K, where K is the total number of popular filen the library. Proof. As evident from 3) and 6), p si is a monotonially inreasing funtion of b i for both ahing poliies, say p si fb i ). The optimization funtion 2) an be hene generalized as: P Ri fb i )) n ), s.t. b i L. 2) max i i Taking the derivative of 2) w.r.t. {b i } i...k, we obtain P Ri n fb i )) n ) i,...k, i.e., P Ri n fb i )) n ) P Rj n fb j )) n ), i j fb i ) fb j ) f b j )P ) n R j f b i )P, i j R i For n, we hene obtain fb i ) fb K ), or equivalently b i b j. With K i b i L, it is therefore optimal to ahe evenly i.e. b i L/K for a mobile user with a large number of retransmissions. B. Stati user In this senario, the user is stati at a given loation in the network and tries to reeive the file of interest within n transmissions. As the user is stati, it remainn the same loal neighbourhood of transmitters aross the n transmissions resulting in temporal orrelation of the interferers. As a result, the probability of suessfully reeiving the file in a given transmission depends on the suess probability in the previous transmissions. Let S i,k be an event that denotes that file F i in overage during the k th transmission attempt. The probability that file F i in overage atleast) one in n transmissions, denoted by p n i is given as p n P n ks i,k ) P n ksir i,k > T ) a) n ) n ) k+ P i,k, 22) k k where a) follows from the inlusion-exlusion priniple and P i,k P k j SIR i,j > T ) is defined as the joint overage probability of file F i in k transmissions. As was the ase in the mobile user senario, we define an optimization problem for maximizing the hit probability for the two ahing poliies P and for a stati user. ) Poliy : Taking into aount that the file of interest F i is ahed only with probability b i in the losest SCBS, we multiply overage probability p n by b i to obtain the suess probability in n transmission.e. p n b i p n. Using 22) in the above result and substituting in ), we obtain the following optimization problem. n n max P Ri k i k ) ) k+ b i P ) i,k, s.t K i b i, 23) where P ) i,k, the joint overage probability of file F i in k transmissions under P is derived in Appendix B and given below, i.e. P ) exp i,k 2πλ r ) ) u α ) k udu f T r α + u α R r )dr, 24)

5 5 where R denotes the distane of the losest SCBS from the typial user or distane of the losest point of the PPP Φ of intensity λ). The distribution of R is hene given from the null probability of a PPP as f R r ) 2λπr e λπr2 4]. 2) Poliy 2: In poliy 2, as the user is always onneted to the SCBS that has the file of interest in its ahe, the suess probability of obtaining F i is the same ats overage probability i.e p n p n. Thereby, simply using 22) in ), we obtain the following optimization problem for poliy 2. max i P Ri n k ) n ) k+ P 2) i,k k, s.t K i b i, 25) where P 2) i,k, the joint overage probability of file F i in k transmissions under poliy 2 is derived by proeeding similar to poliy and is stated below. P 2) i,k exp 2π b i)λ ) ) u α ) k udu T r 2α + u α exp 2πb iλ ) ) u α ) k udu f T r 2α + u α R2 r 2)dr 2, r 2 where R 2 denotes the distane of the losest SCBS that has the file of interest F i in its ahe. As file F i is ahed with probability b i in the network, the distribution of R 2 is thus given by the losest point of the PPP of intensity b i λ and its distribution is given by f R2 r 2 ) 2b i λπr 2 e biλπr2 2. The key differene in the analysis of poliy 2 is that the interferene field is now divided into two regions: i) Interferene from those SCBSs with the file of interest F i in their ahe, onstituting a PPP of intensity b i λ outside a radius r 2 losest distane of F i ) and ii) Interferene from all other SCBSs not having F i ) onstituting a PPP of intensity b i )λ in R 2. The optimal ahing probabilities of a stati user under poliy and 2 i.e. the solutions of the optimization problem 23) and 25) an be obtained by proeeding similar to the mobile user senario Theorem and 2). Further insights on the optimal ahing probabilities for a stati user under both ahing poliies are provided in setion IV. IV. RESULTS AND DISCUSSION For the purpose of numerial results, we onsider the Zipf parameter γ.2 and an SIR threshold β db.. It is to be noted that in Fig. 2, the asterisk denotes the optimal ahing probability. ) Comparison of poliies P and : Fig. 2 left.) ompares the two poliien terms of HP for different ahing probabilities of file, F and varying number of transmissions n. It an be seen that the optimal ahing probabilities are shifted slightly towards the left in ase of poliy and has a slightly higher HP ompared to poliy P. This behaviour an be attributed to the poliy mehanism itself, with exhibiting a higher HP due to the available information of the ahe loations. Also, the shift in optimal ahing probabilities assert that it is not neessary to ahe the most popular file file F ) predominantly in the network and allows the SCBSs to ahe the lesser requested files to a omparatively larger extent. Finally, as expeted, the HP for both poliies P and inrease with the inreasing number of transmissions n. Hit probability Hit probability : Mobile user P: Mobile user Inreasing n n to Cahing Probability for file : Mobile user : Stati user Inreasing n n to Cahing Probability for file Fig. 2. Effet of ahe gathering poliy poliy and 2) and mobility stati and mobile user) on the hit probability for varying number of transmissions. K 2, L ) 2) Comparison of the mobile and stati ases as a funtion of n for poliy : Fig. 2 right.) ompares the HP for the mobile and stati ases when a user following poliy has a maximum n transmissions to obtain the file of interest in a 2-file library ase K 2) with unitary storage L ). While in the mobile ase the retransmissions are made at different loations, in the stati ase the loation of the user is not hanged and hene all the retransmissions are made to the same user loation. However, we assume that the fading gains are independent aross time slotn whih these retransmissions are made. The temporal diversity due to independent fading gainnreases the HP with n, even for the stati ase. This trend is evident in Fig. 2 right.), where the stati ase is shown by the dotted lines. As expeted, the HP for the mobile user is signifiantly higher ompared to the stati user and the gap widens further with the inreasing n. Similar observations an be for poliy P as well. 3) Effet of retransmissions on optimal ahe plaement strategy: Considering a library of K 3 files, Fig. 3 depits the optimal ahing probabilities for a SCBS network with ahe size L. For smaller number of retransmissions small n), Cahe the most popular ontent strategy seems to be the optimal ahe plaement strategy. In other words, it is optimal to ahe the L most popular files out of the K filen the library for smaller number of retransmissions. In ase of large retransmissions, the optimal ahing probabilities asymptotially approahes L/K, as we disussed in Corollary 3.

6 6 Optimal ahing probability of file Fi F F 2 F 3 /K Optimal Hit Probability, P hit K 4 K 3 K Number of transmissions, n Number of transmissions, n Optimal ahing probability of file Fi F F2 F3 L/K Number of transmissions, n Fig. 3. Effet of retransmissions on the optimal ahe plaement strategy for a library of K 3 files with ahe storage L left.) and L 2 right). 4) Effet of library size K) on the hit probability: Fig. 4 depits the optimal hit probability in a network for a mobile user following poliy ) with K filen the library and ahe size L. As evident from the figure, the hit probability dereases with inrease in number of filen the library. Larger the number of filen the library, fewer the hanes of a file hit from the SCBS ahe with a ertain ahe size. Also, one an observe that the optimal hit probability inreases with the number of retransmissions as disussed before. V. CONCLUSIONS In this paper, we have studied the effet of retransmissions on the the network performane in terms of hit probability) for both stati and mobile user senarion a ahe-enabled SCBS network. Intuitively, if the user is allowed to aess the ahes multiple times due to retransmissions), it beomes more likely that it will obtain the requested file, either due to better hannel onditions or a new neighbourhood of ahed SCBSs due to user mobility). The analytial expressions developed also agree with the intuition and show an inrease in hit probability with retransmissions. For a maximum n transmission attempts, we also determine the optimal ahe probabilities for both stati and mobile user senarios. The optimal solutions demonstrate that while it is optimal to ahe the most popular ontent for a stati user, SCBSs an ahe ontent in a more balaned way for mobile users. Fig. 4. Effet of library size K on the hit probability for a mobile user following poliy L ). A. Proof of Corollary APPENDIX Using Equation 4) for the ase of K 2 and L, we obtain, ) v n 2 p ) np ) ) n P R + ) 26) n P R2 Rearranging a few termn the intervals of Theorem, we obtain, ν ) np ) n > P n R b, ν ) np ) n < P n R p ) ) p ) P R ) n v ) ] n np ), p ) > PR P R2 ) n a), < PR p ) P R2 ) n 2 p ) ] b) p ) + P R P R2 ) n, n < +, n < + a +p ) a+)p ) γ log 2 p ) ) γ log 2 p ) ) where a) follows by using Eq. A) and rearranging a few terms. Step b) is obtained by using the Zipf s law P Ri i γ / K j j γ, where γ > is the Zipf parameter and using a 2 γ n. The final result follows by ignoring the interval orresponding to b at happens only when the number of transmissions n <, whih is not possible. B. Proof of Equation 24) For a stati user senario under poliy, the typial user onnets to the losest SCBS loated at distane R during all transmissions. From definition, the joint overage probability of file F i in k transmissions given as P ) i,k E R P j {...k} h xj r α h yj y α > T r )]

7 7 a) k E R b) E R j exp T r α + T r α y α h yj y α )] ) k ] where a) results from the fat that h xj exp) and the i.i.d. fading assumption aross the k attempts resulting in simply the produt of eah term, b) follows from h yj exp). The final result follows from the PGFL of PPP Φ, onverting from Cartesian to polar oordinates and deonditioning w.r.t. R. REFERENCES ] M. K. E. Bastug, M. Bennis and M. Debbah, Cahe-enabled small ell networks: modeling and tradeoffs, EURASIP Journal on Wireless Communiations and Networking, vol. 25, no., pp.. 2] M. Cha, H. Kwak, P. Rodriguez, Y.-Y. Ahn, and S. Moon, I Tube, You Tube, Everybody Tubes: Analyzing the world s largest user generated ontent video system, in Pro. of the 7th ACM SIGCOMM onf. on Internet measurement, 27, pp. 4. 3] J. Rao, H. Feng, C. Yang, Z. Chen, and B. Xia, Optimal ahing plaement for D2D assisted wireless ahing networks, submitted to ICC 26, available online: 4] B. Blaszzyszyn and A. Giovanidis, Optimal geographi ahing in ellular networks, in Pro. IEEE ICC, June 25. 5] B. Serbeti and J. Goseling, On optimal geographial ahing in heterogeneous ellular networks, submitted to ISIT 26, available online: 6] P. H. Nardelli, M. Kaynia, P. Cardieri, and M. Latva-aho, Optimal transmission apaity of ad ho networks with paket retransmissions, IEEE Trans. on Wireless Commun., vol., no. 8, pp , 22. 7] M. Kaynia, N. Jindal, and G. Øien, Performane analysis and improvement of ma protooln wireless ad ho networks, IEEE Trans. on Wireless Commun., vol., no., pp , 2. 8] H. S. Dhillon, R. K. Ganti, F. Baelli, and J. G. Andrews, Modeling and analysis of k-tier downlink heterogeneous ellular networks, IEEE Journal on Sel. Arean Commun., vol. 3, no. 3, pp , 22. 9] M. Haenggi, The loal delay in Poisson networks, IEEE Trans. on Info. Theory, vol. 59, no. 3, pp , 23. ] D. C. Chen, T. Q. S. Quek, and M. Kountouris, Bakhauling in heterogeneous ellular networks: Modeling and tradeoffs, IEEE Trans. on Wireless Commun., vol. 4, no. 6, pp , June 25. ] C. Jarray and A. Giovanidis, The effets of mobility on the hit performane of ahed D2D networks, in Pro. WIOPT-SPASWIN, Arizona, USA, 26. 2] J. G. Andrews, F. Baelli, and R. K. Ganti, A tratable approah to overage and rate in ellular networks, IEEE Trans. on Commun., vol. 59, no., pp , November 2. 3] S. Krishnan and H. S. Dhillon, Distributed ahing in devie-to-devie networks: A stohasti geometry perspetive, in Pro., IEEE Asilomar, 25. 4] M. Haenggi, Stohasti Geometry for Wireless Networks. New York: Cambridge University Press, 23.