Device-to-Device Collaboration through Distributed Storage

Transcription

1 Device-to-Device Collaboration through Distributed Storage Negin Golrezaei, Student Meber, IEEE, Alexandros G Diakis, Meber, IEEE, Andreas F Molisch, Fellow, IEEE Dept of Electrical Eng University of Southern California Los Angeles, CA, USA eails: {golrezae,diakis,olisch}@uscedu Abstract Video is the ain driver for the inexorable increase in wireless data traffic In this paper we analyze a new architecture in which device-to-device (DD) counications is used to drastically increase the capacity of cellular networks for video transission Users cache popular video files and - after receiving requests fro other users - serve these requests via DD localized transissions; the short range of the DD transission enables frequency reuse within the cell We analyze the scaling behavior of the throughput with the nuber of devices per cell The user content request statistics, as well as the caching distribution, are odeled by a Zipf distribution with paraeters γ r and γ c, respectively For the practically iportant case γ r < and γ c >, we derive a closed for expression for the scaling behavior of the nuber of DD links that coexist without interference Our analysis relies on a novel Poisson approxiation result for wireless networks obtained through the Chen-Stein Method I INTRODUCTION Wireless data traffic has risen sharply over the past years, and is expected to increase by a further factor of 40 over the next five years [] While the initial increase was ainly driven by web-browsing and associated applications, ephasis is now oving to wireless delivery of video content This threatens to overwhel the already-strained cellular networks Since increasing cellular data capacity by traditional eans (ore spectru, larger nuber of antennas, saller cells) is either reaching its natural liits or becoing very costly, there is a need for fundaentally new cellular architectures that can exploit the special properties of video requests (eg fro YouTube), as well as of odern sartphone applications Our architecture relies on two key observations: (i) Video has a high degree of content reuse, ie, a few popular files are requested by a large nuber of users (ii) Sartphones and tablets have significant storage capacity that is rapidly growing and typically underutilized Our proposed distributed storage schee is based on the fact that if there is enough content reuse, ie, any users are requesting the sae video content, the storage space on cellular devices can be used as a distributed cache [], [3] We allow users to store popular content and use local device-to-device (DD) counication for collaborative disseination when a user in the vicinity requests a popular file Since DD counication occurs over a short range, this enables frequency reuse, ie, the sae tie-frequency resource can be exploited by ultiple DD links within one acrocell This draatically increases the spectral efficiency and is siilar to the constant trend of shrinking cell sizes What is fundaentally different between DD counications and sall cells, however, is that feto and pico-cell base stations require high-bandwidth backhaul links The DD collaborative video content disseination schee we are proposing essentially replaces backhaul with storage, by caching files that have been already requested by users, without relying on additional infrastructure As the nuber of devices in a cell increases, the distance between devices becoes saller, which allows to use less power, and increase the frequency reuse Yet, on the other hand, ore users will request a greater variety of files, which decreases the probability that a device is within counication range of another device that has the desired content Thus, while it sees intuitive that a greater density of devices allows a larger nuber of active and non-interfering links within a cell, the actual functional scaling behavior cannot be obtained siply by inspection In our recent paper [4], we analyzed the scaling behavior of the average nuber of DD links that can coexist without interference when users choose which file to store randoly Throughout this work, the geoetry of our networks follows the standard Rando Geoetric Graph protocol odel [5], [6] We showed that the scaling behavior depends on both the content request statistics and content cache statistics which are odeled by Zipf distributions In particular, our prior work deonstrated that for particular types of content popularity and cache statistics, a linear scaling becoes feasible However, for a different range of paraeters for content statistics, we could only deterine loose upper bounds Our contributions: In this paper we close this gap in our theoretical understanding of the scaling behavior Specifically, we analyze the previously-unknown case where the Zipf distribution for the requests is characterized by a paraeter γ r < and the caching distribution by a paraeter γ c >, and provide a unique scaling law for the nuber of active DD links as a function of the nuber of devices in the cell Our analysis relies on a novel Poisson approxiation result for

2 Fig Rando geoetric graph exaple The connectivity radius is r(n) wireless networks obtained through the Chen-Stein Method We note that the case of γ r < is the ost practically relevant one, since prior epirical Youtube studies [7] have estiated this exponent to be in the range of The reainder of this paper is organized as follows: In Section II we setup the proble forulation, defining the popularity distributions and active links Section III contains our ain theore, the scaling behavior of the average nuber of DD interference-free links After soe siulation results, in Section V we discuss future directions, open probles and conclusions II MODEL AND SETUP Consider a cellular network where each cell/base station (BS) serves n users For siplicity we assue that the cells are square, and neglect inter-cell interference, so that we can consider one cell in isolation Users are distributed uniforly in the cell Every user is assued to have a storage capacity called cache, which is filled up with soe video files The base station (BS) ight be aware of the stored files and channel state inforation of the users and control the DD counications We assue that the DD counication does not interfere with counication between the BS and users This assuption is justified if the DD counications occur in a separate frequency band (eg, WiFi) For the deviceto-device throughput, we henceforth do not need to consider explicitly the BS and its associated counications Each user in the cell requests a file randoly and independently fro a library of size The size of the library is a function of the nuber of users n, since a larger nuber of users covers a larger set of files they could be interested in Studies show that there is redundancy in video requests and soe popular files are requested ore [7] We assue that the various file popularities are distributed as per the Zipf distribution, which has been established in nuerous studies as being a good approxiation to the easured popularity of video files [8] In a Zipf distribution, the frequency of the ith popular file, denoted by f i, is inversely proportional to its rank: f i = i γr j=, i, () j γr The exponent γ r characterizes the distribution by controlling the relative popularity of files A large γ r eans a concentrated distribution, ie, the first few popular files account for the ajority of requests Ref [7] confired the Zipf odel for video popularity based on epirical YouTube data and easured the γ r exponent in the range of In this paper we refer to the range γ r > as the high content reuse regie and γ r < as the low content reuse regie Each user (peer) has soe storage capability that we use to cache video files For siplicity in our analysis, we assue that all files have the sae size, and each user can store one file This assuption has the advantage of yielding a clean forulation; however, our work can be easily extended to larger cache size Our counication network is odeled by a rando geoetric graph G(n, r(n)) where two users (assuing DD counication is possible) can counicate if their physical distance is saller than soe collaboration distance r(n) [5], [6] The axiu allowable distance for DD counication r(n) is deterined by the power level for each transission Figure illustrates an exaple of rando geoetric graph (RGG) Our odel works as follows: if a user requests one of the files stored in neighbors caches in the RGG, neighbors will handle the request locally through DD counication; otherwise, the BS should serve the request Thus, to have DD counication it is not sufficient that the distance between two users be less than r(n); users should find their desired files locally in caches of their neighbors Therefore, the probability that DD counications happen depends on what files are stored by the users Caching decisions can be ade either in a distributed or centralized way A central control of the caching by the BS allows very efficient file-assignent to the users [9] However, if such control is not desired, and/or the users are oving around quickly, then the caching has to be done distributedly and randoly In other words, each user will cache files according to a probability density function (pdf) In order to avoid the difficulties of optiizing a function, we assued that the functional for of the caching pdf is also a Zipf distribution, but with a paraeter γ c that can be different fro the paraeter γ r In our previous work, we showed the surprising fact that the optiu γ c is not equal to γ r [] We assue that all DD links share the sae tie-frequency transission resource within one cell area This is possible since the distance between requesting user and sartphone with the stored file will be sall in ost cases However, there should be no interference of a transission by others on an active DD link We assue that - given that node u wants to transit to node v - any transission within range r(n) fro v (the receiver) can introduce interference for the u v transission Thus, such two links cannot be activated siultaneously This odel is known as protocol odel; while it neglects iportant wireless propagation effects such as fading [0], it can provide fundaental insights and has been widely used in the literature [5]

3 The proble we investigate is: how does the nuber of active DD links scale given that users request files based on a Zipf distribution with exponent γ r and cache files with exponent γ c III ANALYSIS In this section, we investigate the asyptotic behavior of the average nuber of DD links that can exist without introducing interference to other DD links We consider a dense scenario in which the nuber of users in the cell n goes to infinity Recall big O and Ω notations : f(n) = O(g(n)) and f(n) = Ω(g(n)) respectively denote that f(n) cg(n) and f(n) cg(n) where c is a constant Besides, according to Knuth s notations if f(n) = Θ(g(n)), it eans that f(n) = O(g(n)) as well as f(n) = Ω(g(n)) Little-o notation, f(x) ie, f(x) = o(g(x)) is equivalent to li x g(x) = 0 Throughout this paper, c is used to denote deterinistic positive constants not depending on n The scaling law for the high content reuse regie was derived in our recent prior work [4] We showed that if γ r >, the expected nuber of active DD links scales like (a) E[L] = Θ(n) However, for the case low content reuse regie, γ r <, the upper and lower bound scaling laws of [4] have different exponents The ain result of this paper is a precise characterization of E[L] for the low content reuse regie Theore : If γ r <, γ c >, and r(n) = Θ( E[L] = Θ( n ( a)( γr) ), b n ), where a = b γ c and 0 < a < ( γr) γ c+ γ r This is achieved for any value of the storage exponent γ c > We can see that as a user sees on average ore users in its neighborhood (larger a), the nuber of active DD links that can coexist increases with n Proof: First, we establish the lower bound for E[L] We divide the cell into r(n) virtual square clusters Figure (a) shows the virtual clusters in the cell The cell side is noralized to and the side of each cluster is equal to r(n) Thus, all users within clusters can counicate with each other Besides, based on the protocol odel, in each cluster only one link can be activated siultaneously When there is an active link within a cluster, we call the cluster good But, not all good clusters can be activated siultaneously One good cluster can at ost block 6 clusters (see Figure (b)) The axiu interference happens when a user in the corner of a cluster receives a file fro another user Thus, we have E[L] E[G] 7 where E[G] is the expected nuber of good clusters Since we want to find the lower bound for E[L], we can liit users (b) Fig a) Dividing cell into virtual clusters b) In the worst case, a good cluster can block at ost 6 clusters In the dashed circle, receiving is not possible and in the solid circle, transission is not allowed to counicate with users in virtual clusters they belong to Thus, we have: E[G] = n r(n) Pr[good k] Pr[K = k] k=0 r(n) Pr[good k] Pr[K = k], () k A where A = [nr(n) ( δ)/, nr(n) (+δ)/] and 0 < δ < r(n) is the total nuber of virtual clusters K is the nuber of users in the cluster, which is a binoial rando variable with n trials and probability of r(n), ie, K = B(n, r(n) ) Pr[K = k] is the probability that there are k users in the cluster and Pr[good k] is the probability that the cluster is good conditioned on k Define k arg in Pr[good k] Pr[K = k] k A Notice that k is Θ(nr(n) ) Fro (), we have: E[G] r(n) Pr[good k ] Pr[K A] (3) r(n) Pr[good k ]( e nr(n) δ /6 ) (4)

4 We apply a Chernoff bound to derive equation (4) [] Fro the range of r(n), the second ter in (4), ie, ( e nr(n) δ /6 ) is O() Thus, to prove the theore, we should show that Pr[good k nr(n) ] c( ) ( a)( γr ) Let us define an indicator function u for each user within a cluster u is if user u can find its requested file within the cluster; otherwise it is 0 Since users cache files randoly, the set of available files through DD links in the cluster is also rando Thus, u and u for u u are dependent rando variables In that case, given that user u requests according to the soe popularity distribution and it cannot find it locally, it is ore likely that files currently within the cluster cache are not popular files So, if user u requests independently fro user u but according to the sae popularity distribution, it is ore probable that user u also cannot find its request file locally The cluster is good if at least one of users within the cluster finds its desired file via DD counication Thus, Let us define Pr[good k ] = Pr[ u = 0] k u= W u (5) k u= By using the Chen Stein ethod in lea 3, we show that W can be approxiated by a Poisson rando variable with average of λ E[ k u= u] We also show that the probability that W is zero is asyptotically equal to e λ where λ = k E[ u ] = k i= f i ( ( p i ) k ) (6) p i is the probability that a user caches file i based on Zipf distribution with exponent γ c In the above equation we consider the possibility of self-requests, ie, a user ight find the file it requests in its own cache; in this case clearly no DD counication will be activated by this user Then, we have: Pr[good k ] = e λ (7) nr(n) In lea, we show that λ = Θ( ) = o() Thus, ( a)( γr ) Pr[good k nr(n) ] = Θ(λ) = Θ( ( a)( γr) ), fro which the lower bound follows Now, we prove the upper bound The probability that a user cannot find its desired file locally is the probability that either he has the file itself or all his neighbours do not have the desired files Thus we have: E[L] n Pr[no DD] n = n f j p j + n Pr[K = k] f j ( ( p j ) k ) j= j= k= k= j= n n f j p j + n Pr[K = k] [ a f j + k n j= j= j= f j p j + n [ a f j + πnr(n) j= a + j= a + f j p j ], f j p j ] where K is the nuber of a users that counicate K is a binoial rando variable, ie, K = B(n, πr(n) ) In the second equation above, the first suation is the probability that a user requests a file that he has already stored The second suation is the probability that the user requests a file and all users in its neighborhood in RGG have not stored the desired n file Using lea, E[L] = Ω( ) ( a)( γr ) Lea : i) If γ > and a = o(b), H(γ, a, b) = Θ( a ii) If γ <, a = o(b), and a = Θ(), iii) If γ r <, γ c >, γ ) H(γ, a, b) = Θ(b γ ) f j p j = Θ( j= γr ), where the haronic function H(γ, a, b) = x γ b j=a i γ Proof: We first prove the parts i and ii of the lea is onotonically decreasing Thus, H(γ, a, b) b x=a We also have the following inequality: H(γ, a, b) a γ = Thus, H(γ, a, b) satisfies: b ( γ+) a γ+ γ + x γ = b( γ+) a γ+ (8) γ + b j=a+ b x=a j γ x γ = b( γ+) a ( γ+) (9) γ + H(γ, a, b) b( γ+) a γ+ γ + + a γ (0) Therefore, if γ >, H(γ, a, b) = Θ( a γ ) Besides, if γ < and a = Θ(), then H(γ, a, b) = Θ(b γ )

5 For part iii, using (), we have: jγr +γc f j p j = j= j γr j= γ c = H(γ c + γ r,, ) H(γ c,, )H(γ r,, ), () j= j= When γ c > and γ r <, H(γ c + γ r,, ) and H(γ c,, ) are both Θ() and H(γ r,, ) = Θ( γr ) which follows the result Lea : If γ c >, γ r <, k = Θ( b ), p u E[ u ] = Θ( ) () ( a)( γr) p uu E[ u u ] = O( ), (3) ( a)( γr) where a = b γ c Proof: We first prove the first part of the lea E[ u ] is given in (6) E[ u ] = f j ( ( p j ) k ) + a j= j= a + f j + (k ) a j= f j ( ( p j ) k ) j= a + f j p j H(γ r,, a ) H(γ r,, ) + (k ) H(γ c + γ r, a +, ) H(γ r,, )H(γ c,, ), (4) where the haronic function H is defined in lea If we apply the results of lea, we can show that both ters in (4) are Θ( ) ( a)( γr ) For the lower-bound, we have: E[ u ] f j ( ( p j ) k ) a j= f j ( e (k )pj ) a j= Using lea, we know p j = Θ( j γ ) Then, considering that j = O( a ), the exponent (k )p j is Ω() Therefore, a E[ u ] c j= f j = H(γ r,, a ) H(γ r,, ) = Θ( ) (5) ( a)( γr) Now, we show the second part of the lea p uu = Pr[ u = 0 and u = 0] = f i f j ( ( p i p j ) k ) + i= j=,i j f i ( ( p i ) k ) i= i= j= f i f j ( ( p i p j ) k ) f i f j + + a a i= j= i= a + j= a + ( a a ) f i + i= Then, we have: a i= j= a + f i f j k(p i + p j ) i= j= a + f i f j f i f j + k i= a + f i p i p uu H(γ r,, a ) H(γ r,, ) + H(γ r,, a )H(γ r, a +, ) H(γ r,, ) (6) + k H(γ c + γ r, a +, ) H(γ r,, )H(γ c,, ) (7) According to the scaling results of the lea, the first, the second and the third ters are respectively Θ( ), Θ( ) and Θ( ) Thus, ( a)( γr ) ( a)( γr ) ( a)( γr ) p u,u = O( ) ( a)( γr ) Lea 3: W defined in (5) is asyptotically a Poisson rando variable with ean λ = E[W ] and Pr[W = 0] asyptotically equals e λ if k = Θ( aγc ) where 0 < a < ( γ r) γ c+ γ r Proof: To prove the lea, we use the Chen Stein ethod (the following theore) Theore : [] Let W be the nuber of occurrence of dependent events, and Z be a Poisson rando variable with E[Z] = E[W ] = λ Then and D(W ) D(Z) (b + b ), (8) Pr(W = 0) e λ < ( λ )(b + b ), (9) where D(W ) D(Z) is the total variation distance between the distribution of W and Z, λ is the iniu of and λ, and k k b = p u p u (0) b = u= u = k u= u = k p uu ()

6 p u and p uu are respectively defined in () and (3) Using the results of lea b = k Θ( ) () ( a)( γr) b = k O( ) (3) ( a)( γr) Then, fro (8), W is asyptotically a Poisson rando variable if b + b = o() which iplies that k should be o( ( a)( γr)/ ) Considering that k = Θ( aγc ), we can ( γ conclude that a in the lea should be less than r) γ c+ γ r Beside, fro (9), we can see when b + b = o(), the probability that W is zero asyptotically approaches e λ IV SIMULATION RESULTS In this section, we provide soe nuerical results to verify our theoretical analyses We consider there are n active users which are randoly distributed across the entire cell There is a library of size = 0 4 log(n) For the siulation, we use Monte Carlo ethod In every iteration, we place n users randoly in the cell Each user stores/requests one file randoly by sapling fro the Zipf distribution with exponents γ c /γ r Based on that, we deterine all good DD links and axiu nuber of DD links that can be activated siultaneously without introducing interference for each other Figure 3 illustrates the average nuber of active DD links versus the nuber users in the cell with γ c = and for different value of γ r The paraeter a in the theore is 0, ie, r(n) = 0 n We can see that the theoretical results atch the siulations As expected, for larger γ r (ore content reuse) there exist ore non interfering DD links Siulation results deonstrate that the scaling law, though defined for large node density, is experientally verified for 00 obiles per cell, which is a realistic scale for today s systes We furtherore find that even though the scaling behavior of the nuber of non-interfering collaborative links grows sub-linearly with the nuber of users, the deviation fro linearity is sall for a wide range of paraeters γ r and γ c This theoretically justifies what we would intuitively expect: device-to-device video sharing, enabled by local storage can bring treendous benefits in future wireless systes VI ACKNOWLEDGEMENTS This work was financially supported by the Intel/Cisco VAWN project Helpful discussions with Chris Raing and Jeff Foerster (Intel), as well as Giuseppe Caire, Michael Neely, Antonio Ortega, and Jay Kuo (USC) are gratefully acknowledged REFERENCES [] /ns705/ns87/white paper c-5086htl [] N Golrezaei, A F Molisch, and A G Diakis, Base station assisted device-to-device counications for high-throughput wireless video networks, subitted for publication [3] N Golrezaei, K Shanuga, A G Diakis, A F Molisch, and G Caire, Fetocaching: Wireless video content delivery through distributed caching helpers, in INFOCOM IEEE, 0 [4] N Golrezaei, A G Diakis, and A F Molisch, Wireless device-todevice counications with distributed caching, subitted for publication [5] P Gupta and P Kuar, The capacity of wireless networks, Inforation Theory, IEEE Transactions on, vol 46, no, pp , 000 [6] M Penrose and O U Press, Rando geoetric graphs Oxford University Press Oxford, 003, vol 5 [7] [8] M Cha, H Kwak, P Rodriguez, Y Ahn, and S Moon, I tube, you tube, everybody tubes: analyzing the world s largest user generated content video syste, in Proceedings of the 7th ACM SIGCOMM conference on Internet easureent ACM, 007, pp 4 [9] N Golrezaei, A G Diakis, and A F Molisch, Asyptotic throughput of base station assisted device-to-device counications, pp , to be subitted for publication [0] A F Molisch, Wireless counications IEEE Wiley, 0 [] M Mitzenacher and E Upfal, Probability and coputing: Randoized algoriths and probabilistic analysis Cabridge Univ Pr, 005 [] R Arratia, L Goldstein, and L Gordon, Two oents suffice for poisson approxiations: the chen stein ethod, The Annals of Probability, vol 7, no, pp 9 5, 989 Fig 3 The average nuber of active DD links versus the nuber users in the cell, γ c = and a = 0 V CONCLUSIONS We discussed a novel schee for increasing the efficiency of video content delivery in cellular counications systes We found the scaling behavior of nuber of interferencefree DD links as a function of the statistics of requests, the statistics of the caching, and the collaboration distance