Optimizig Codes for Cachig at the Edge Valerio Bioglio, Frédéric Gabry, Igmar Lad Mathematical ad Algorithmic Scieces Lab Frace esearch Ceter, Huawei Techologies Co Ltd Email: {valeriobioglio,fredericgabry,igmarlad}@huaweicom arxiv:50805753v [csit] 8 Sep 05 Abstract I this paper we ivestigate the problem of optimal -ecoded cache placemet at the wireless edge to miimize the backhaul rate i heterogeeous etworks We derive the backhaul rate performace of ay cachig scheme based o file splittig ad ecodig ad we formulate the optimal cachig scheme as a covex optimizatio problem We the thoroughly ivestigate the performace of this optimal scheme for a importat heterogeeous etwork sceario We compare it to several other cachig strategies ad we aalyze the ifluece of the system parameters, such as the popularity ad size of the library files ad the capabilities of the small-cell base statios, o the overall performace of our optimal cachig strategy Our results show that the careful placemet of -ecoded cotet i caches at the wireless edge leads to a sigificat decrease of the load of the etwork backhaul ad hece to a cosiderable performace ehacemet of the etwork I INTODUCTION Cachig cotet at the wireless edge is a promisig techique for future 5G wireless etworks [] Oe of the mai motivatios behid the idea of edge cachig comes from the possibility of sigificatly reducig the backhaul usage ad thus the latecy i cotet retrieval by brigig the cotet closer to the ed users Exploitig the ew capabilities of future multi-tier etworks [], umerous works have recetly ivestigated the potetial beefits of cachig data i desely deployed small-cell base statios ) equipped with storage capabilities [3] I [4] it was show that cachig at the edge ca provide sigificat gais i terms of eergy efficiecy which is cosidered a fudametal metric for future wireless etworks The ivestigatio of cachig from a iformatio-theoretic poit of view is preseted i [5], where cachig metrics are defied ad aalyzed for large etworks I [6] the problem is studied i terms of outage probability Moreover, other embodimets ad advatages of edge cachig have bee discussed i the literature I [7], the idea of usig the mobility of users i the etwork to icrease cachig gais is proposed, while i [8] the authors leverage the possibility of exploitig the storage capabilities of mobile users by cachig cotet directly o the users devices I order to improve the theoretical performace limits of cachig, a ew iterestig perspective stems from usig ideas from codig theory I [9], etwork codig NC) is exploited i a sceario where each user is equipped with a cache For this sceario, the authors show that a etwork codig based cachig scheme decreases the backhaul load i compariso to the usual ucoded schemes However, the extesio of NC to a more complex sceario, where each user ca be served by multiple s, has bee proved to be NP-complete [0] Moreover, the use of NC i a sceario where the iformatio is set via uicast trasmissio does ot give ay advatage Aother promisig related directio is the use of maximumdistace separable ) codes to virtually icrease the storage size i case of overlappig coverage areas I [], cotet is coveyed to distributed helpers usig codes to remedy the limitatios of ucoded delivery i terms of delay Assumig the etwork topology is kow, the cotet placemet strategy is formulated as a covex miimizatio of the overall delay A radom cachig scheme for a cooperative multiple-iput multiple-output MIMO) etwork is combied with ecodig of the cached cotet to icrease the gais resultig from MIMO cooperatio [] Departig from those previous works, i this paper we ivestigate the performace of -ecoded distributed cachig at the s I particular, we cosider a sceario where mobile users ca be served by multiple s I this setup, we derive the oit optimal ecodig ad placemet of the ecoded packets at the s i order to miimize the backhaul rate Our mai cotributios are: We formally defie the problem of cachig -ecoded cotet at the wireless edge for a heterogeeous etwork HetNet) sceario We derive the backhaul rate performace of a cachig scheme based o storig ecoded packets, formulatig the optimal cachig scheme as a covex optimizatio problem We ivestigate the performace of the proposed packet placemet ad delivery scheme for a relevat HetNet sceario Moreover, we compare it to several cachig schemes, aalyzig the beefits of ecodig o the performace of the scheme, as well as the importace of carefully optimizig the placemet of the coded packets Our results show that the use of codes i the placemet of cotet at the wireless edge yields to sigificat offloadig of the etwork backhaul This paper is orgaized as follows I Sectio II, we defie our system model, cachig scheme ad performace measures I Sectio III, we derive the mai theoretical results of the paper I Sectio IV we thoroughly ivestigate the performace of our optimal schemes ad we compare it to other cachig schemes i a heterogeeous etwork sceario Fially, Sectio V cocludes this paper
File Library F F F N Core Network MBS MBS M M 3 M 4 M u u u 3 u 4 u 5 u 6 u 7 u 8 u 9 Fig Heterogeeous etwork Fig Istataeous Heterogeeous etwork topology II SYSTEM MODEL I this sectio we describe our system model ad we formally defie cachig schemes for heterogeeous etworks A Network Model We cosider the etwork illustrated i Fig, where U wireless users sed requests to dowload files from a macrocell base statio The macro-cell base statio MBS) has access to a library of N files F = {F,,F N }, each of size B bits Note that the assumptio of equal size files is ustifiable i practice sice files ca be divided ito blocks of the same size We suppose that the users request etire files stored i the library, ie, if k u is the data request for user u, the k u F Each file has a differet probability to be requested by the users I particular, the probability distributio vector of the files is deoted by p = {p,,p N }, referred to as the file popularity, where file F is requested with probability p ad N = p = Moreover N small-cell base statios are deployed i order to serve user requests withi short distace We assume that each has a cache of size M B bits ie, it ca store up to M < N complete files), ad ca sed data to the users through a error-free lik, ie, we assume that cotet is delivered without errors as log as a user is withi the coverage rage We deote by r the coverage rage of the s Each user requestig for files i F is iitially served by the s it ca cotact If the requested file is ot completely preset i the caches, the MBS has to sed the missig data to the user, usig the backhaul coectio The topology of the overall etwork may vary durig time; i particular at each istat t, the coectio etwork ca be described as illustrated i Fig Each user u {,,U} ca be served by d u s depedig o its locatio i the area see Fig 3) I particular, we call γ i the probability for a user to be served by d u = i s This probability depeds o the s deploymet ad o the desity of the users: i Sectio IV we will show how to evaluate this probability i ay sceario B Cachig Schemes A cachig scheme is costituted of two phases, amely the placemet phase ad the delivery phase ) Placemet phase: I this phase, the caches of the s are filled accordig to the placemet strategy This phase typically occurs at a momet with a low amout of file requests, eg at ight ) Delivery phase: I this phase, the users sed requests to the MBS These requests are iitially served by the s coverig their locatios If a user caot collect eough iformatio to recover the file, the MBS has to deliver the missig iformatio through the backhaul Before the placemet phase, each file i the library is split ito fragmets, ie, F = {f ),,f) } for all N I a coded cachig scheme, these fragmets are used to create E ecoded packets {e ),,e) E } We wat to highlight that a fragmet is a ucoded part of a file, while a packet is a ecoded part of a file I the followig, a part of a file ca deote either a fragmet or a ecoded packet I the placemet phase, each receives m parts of file F, where m = [m m N ] is referred to as the cache placemet The placemet problem cosists of fidig the optimal umber of parts m of each file to be stored i the caches i order to miimize the average backhaul rate, which is defied as the average fractio of files that eeds to be dowloaded from the MBS durig the delivery phase
Fig 3 Small cells topology 4 3 III PEFOMANCE ANALYSIS OF CACHING SCHEMES I this paper, we propose to store coded packets at the s istead of fragmets or etire files We iitially preset i detail our scheme, ad calculate the average backhaul rate of a coded cachig scheme Afterwards, we prove that, give a placemet scheme m = [m m N ], storig ecoded packets always provides a advatage over storig fragmets i terms of miimizatio of the backhaul rate Fially, we formulate the coded cachig as a optimizatio problem, ad show that fidig a optimal solutio for the problem is tractable A Coded Cachig Scheme Maximum-distace separable codes are codes matchig the sigleto boud I practice, usig a a, b) code it is possible to create a ecoded packets from b iput fragmets, such that ay subset of b ecoded packets is sufficiet to recover the data The most kow example of codes are the eed-solomo S) codes [3] Lately, sub-optimal codes were proposed, like foutai codes [4], where b+ǫ) ecoded packets are eeded for the recovery I the proposed scheme, each receives m differet coded packets for each file F to be stored i its cache, with m = [m m N ] Moreover, the MBS stores m ecoded packets for each file F i order to serve the requests Formally, we propose the followig: ) Placemet phase: The MBS creates E = +N )m ecoded packets usig a E,) code The MBS keeps m ecoded packets, ad seds the other oes so that each stores m uique packets ) Delivery phase: A user requestig file F cotacts d u s, receivig m d u differet ecoded packets If m d u, the user ca recover the file due to the -property of the code Otherwise the MBS seds the remaiig m d u m ecoded packets Sice the MBS kept m ecoded packets, the user does ot receive replicated packets, ad ca decode the file I the followig, we study the performace of the proposed coded scheme i terms of average use of the backhaul lik, ad we compare the results with the ucoded scheme B Performace Aalysis To begi with, we calculate the average backhaul rate of the proposed coded cachig scheme Propositio The average backhaul rate for a ecoded cachig placemet scheme Cm defied by the placemet m = [m m N ] ca be calculated as C m ) = γ i p mi, dm )), ) d= = where S N is the maximum umber of s servig a user Proof: Let u be a user served by d u s If u requests file F, exactly packets have to be collected i order to retrieve F Each stores m packets of the file F, hece the user ca collect d u m differet ecoded packets If d u m the MBS does ot sed ay packet, otherwise d u m packets are set through the backhaul The rate for user u requestig for file F ca cosequetly be calculated as F ) = mi d=, d um ) ) A user has the probability γ d to be served by d s, hece the expected rate for the file F is give by F ) = γ d mi, dm )) Fially, each file has a differet probability to be requested by a user Averagig over the request probability distributio p = [p p N ], The average backhaul rate for the ecoded cachig placemet scheme is give by C m ) = d= = γ d p mi, dm )) I order to evaluate the gai give by the use of codes, we calculate the backhaul rate of a radom cachig scheme, where oly file fragmetatio is exploited I practice, give a placemet m = [m m N ], the MBS seds to each m differet fragmets radomly draw amog the fragmets of file F Propositio The average backhaul rate for a radom cachig placemet scheme C m defied by the placemet m = [m m N ] ca be calculated as Cm) = d= = γ d p m ) d 3) Proof: The proof is similar to the oe proposed for Propositio The mai differece is the calculatio of the
rate for a user requestig for a file For the give scheme, if a user u requestig file F is served by d u s, the average backhaul rate is m ) du 4) This is due the fact that i this case the fragmets are ot uique sice they ca be replicated i differet caches Therefore the probability that a sigle fragmet is ot preset i ay of the caches is m /) du The rest of the proof is similar to the oe of Propositio Now we ca prove that, give a cache placemet m = [m m N ], it is preferable i terms of backhaul usage to store coded packets rather tha fragmets Propositio 3 For ay placemet m = [m m N ], it holds that C m ) Cm) 5) Proof: As stated i the proof of Propositio, the differece betwee Equatios ) ad 3) is give by ) ad 4) respectively I order to prove the propositio, we have to prove that 4) is larger tha or equal to ) Sice d ad 0 m /, from Beroulli s iequality ad obviously m m ) d d m, ) d 0 We coclude the proof by oticig that mi, dm ) = mi 0, dm ) m ) d C Optimal Coded Cachig I the followig, we study the placemet problem of ecoded packets as a optimizatio problem, ad show its tractability Based o the average backhaul rate of Propositio, we have the followig result Propositio 4 Fidig the optimal coded placemet scheme Copt) defied by m opt) = [m m N ], which miimizes the average backhaul rate C m ), is a covex optimizatio problem Proof: The placemet problem ca be recast as the followig optimizatio problem: mi m,,m N st d= = m = M = γ d p mi, dm )) 0 m =,,N 6) We recall that the umber of fragmets ca be chose, hece we ca reformulate the optimizatio problem as mi q,,q N st d= = q = M = γ d p mi,dq )) 0 q [,N] where q = m / It ca be easily show that 7) is a covex optimizatio problem, which cocludes the proof IV NUMEICAL ILLUSTATIONS I this sectio we ivestigate the performace of the optimal coded cachig scheme i terms of backhaul rate i a HetNet topology of particular relevace We emphasize that our umerical results ca be further geeralized to ay etwork topology I particular the challegig problem of the optimizatio of the locatios of the small-cell base statios is of cosiderable practical ad theoretical iterest, but outside the scope of this paper A Network Topology I the followig we cosider the etwork depicted i Fig, where the s are deployed accordig to a regular grid with a distace d = 60 meters betwee each The macro-cell base statio coverage area has a radius of D = 500 meters I order to reach ay user i, each has a coverage area of radius r such that d/ r d, which meas that the coverage areas are overlappig as show i the highlighted square i Fig 3 If we deote by A d ad ρ d the total area of where a user ca be served by d s ad its average desity, respectively, the probability γ d that a user is served by d s ca be formally calculated as γ d = ρ da d S i= ρ ia i We ote that for this particular deploymet, the coefficietsa d ca be theoretically approximated usig simple geometrical calculatios I the followig, we cosider the users to be uiformly distributed i, with desity ρ d = ρ = 005 users/m These umbers correspod to 36 small-cell base statios beig deployed, coverig U = 3, 45 mobile users The request probability of the files p = [p p N ] is distributed accordig to a Zipf law of parameter α, ie, p = /α /α, where α represets the skewess of the distributio ad ca takes values i [05;5], see eg [5] 7)
Backhaul ate 09 08 07 uif) 06 prop) 05 pop) opt) 04 0 5 0 5 0 Cache Size M Fig 4 Backhaul rate as a fuctio of the cache size M, with N= 00 files, α = 07 ad r = 60 meters The lies without markers depict the use of codes ecoded packets) i the placemet, while the lie with markers depict ucoded fragmets) placemet B Cachig Strategies I order to evaluate the performace of the optimal coded placemet scheme, we compare the optimal achievable backhaul rate opt) to three other practical placemet schemes, with ad without ecodig of the files: ) most popular placemet C pop) ad Cpop) : each stores the M most popular files Sice etire files are stored for this scheme, we ote that C pop) ad Cpop) are equivalet Hece we call pop) the rate achieved usig this scheme ) uiform placemet C uif) ad Cuif) : the file is divided ito fragmets, which are placed uiformly at radom at each, ie, each stores M/N fragmets of each file I the case that codes are employed, the s store M/N ecoded packets for each file istead of fragmets We call uif) ad uif) the rates achieved usig this scheme with ad without ecodig, respectively 3) proportioal placemet C prop) ad Cprop) : for each file F, fragmets are created, alog with a cogruous umber of ecoded packets The amout of parts m of filef stored at the s is proportioal to its popularity p while satisfyig the size costraits I particular, the umber of parts cached at the is iteratively calculated as i = m i = mi,p i m ) M We call prop) ad prop) the rates achieved usig this scheme with ad without ecodig, respectively C Numerical esults I Fig 4 we depict the backhaul rates as a fuctio of the cache size M To begi with, it ca be oticed that the use of codes gives a advatage i terms of reductio of the backhaul usage, sice placemet always outperforms Backhaul ate 085 08 075 07 065 06 055 05 045 uif) prop) pop) opt) 04 44 46 48 50 5 54 56 58 60 Coverage age r Fig 5 Backhaul rate as a fuctio of the coverage radius r, with N= 00 files, α = 07, ad M = 0 files the ucoded placemet for each of the schemes This behavior cofirms the result preseted i Propositio 3 Cosequetly, i the remaider of our aalysis we oly show the rates of cachig schemes Secodly, we observe that, as it ca be expected, the backhaul rates are decreasig whe the storage capacity of the s icreases ad that the optimal cachig scheme outperforms the three other schemes Furthermore, we observe that the differece betwee opt) ad the rate of the other schemes icreases as the cache size icreases Fially, we ca otice that the pop) scheme performs the closest to the optimal scheme for small values of M, gettig worse as M icreases We guess that this scheme is ot able to really take advatage of the ature of future HetNets, ie, of the overlappig betwee coverage areas of the s I Fig 5 we cofirm the previous coecture by represetig the backhaul rates as a fuctio of the coverage radius r of the s We idetify i the figure a strikig behavior for the placemet scheme, as its rate does ot deped o the coverage radius This behavior is due to the fact that the most popular etire files are stored at the s for this scheme: beig served by more s does ot icrease the probability of havig access to ew files sice the same files are stored i Cpop) every cache I cotrast to the Cpop) scheme, the performace of other schemes icreases as the coverage area icreases, ad oticeably the optimal scheme sigificatly outperforms the other schemes as r grows As a umerical illustratio of this fact, we ca imagie the sceario where the N = 00 files are videos of size 00Mbits Whe r = 60, the rate differece betwee opt) This represets a differece of backhaul load of 7Mbits/s if there is demad per secod i the macro cell, which is a cosiderable gai for curret maximal backhaul capacities Fially i Fig 6 we illustrate the backhaul rates as a fuctio of the library size N while the size of the edge caches ad the secod best scheme prop) is 007 remais costat For small library sizes, the Cpop) scheme is outperformed by the other schemes as a spreadig of the library files over the caches is more efficiet tha oly storig a few umber of etire files O the other had, as the library size icreases, the performace of the C pop) scheme becomes
Backhaul ate 08 07 06 05 04 03 0 0 uif) prop) pop) opt) 0 0 40 60 80 00 0 40 60 80 00 Library Size N Fig 6 Backhaul rate as a fuctio of the library size N, with α = 07, M= 0 files ad r = 60 meters better i compariso to the other schemes The Cpop) scheme, however, is still outperformed by the Cprop) scheme, sice we cosider the overlappig sceario with coverage radius r = 60 meters, which is ufavorable to the Cpop) scheme Moreover, we ca otice that iitially the Cuif) scheme is really close to the optimal oe This is due to the fact that if the library size N is small compared to the cache size M, the optimal scheme is well approximated by the scheme Cuif) that stores the packets uiformly across the files As the library size icreases, this aïve scheme gets worse, sice it does ot take ito accout the popularity of the files O the cotrary, the performace of the Cprop) gets better, eve if the gap with the optimal scheme becomes costat This is due to the fact that this scheme does ot truly exploit the possibility a user has to be served by more tha oe I geeral, the presece of a gap betwee the optimal scheme ad sub-optimal oes highlights the importace of implemetig the optimal cachig placemet scheme to exploit the overlappig coverage regios of the HetNet ad thus to effectively decrease the backhaul load of the overall etwork V CONCLUSIONS We cosidered the problem of optimal -ecoded cotet placemet at the cache-equipped small-cell base statios at the wireless edge i order to miimize the overall backhaul load of the etwork We derived the optimal cachig placemet strategy based o file splittig combied with ecodig, which we formulated as a covex optimizatio problem We the deeply ivestigated the performace of this optimal placemet for a relevat HetNet sceario by comparig it to existig cachig strategies ad by measurig the ifluece of the key parameters, such as the capabilities of the small-cell base statios ad the library statistics Our umerical observatios, which ca be easily geeralized for ay geometric topology, showed that optimizig the placemet of ecoded packets at the wireless edge is of crucial importace, sice it yields a sigificat decrease of the load of the etwork backhaul, ad thus achieves a cosiderable improvemet i terms of delay for cotet delivery to the ed users EFEENCES [] X Wag, M Che, T Taleb, A Ksetii, ad V C M Leug, Cache i the air: Exploitig cotet cachig ad delivery techiques for 5G systems, IEEE Commuicatios Magazie, vol 5, o, pp 3 39, February 04 [] J G Adrews, Seve ways that HetNets are a cellular paradigm shift, IEEE Commuicatios Magazie, vol 5, o 3, pp 36 44, March 03 [3] E Bastug, M Beis, ad M Debbah, Livig o the edge: The role of proactive cachig i 5G wireless etworks, IEEE Commuicatios Magazie, vol 5, o 8, pp 8 89, August 04 [4] B Perabathii, E Bastug, M Koutouris, M Debbah, ad A Cotey, Cachig at the edge: a gree perspective for 5G etworks, i IEEE Iteratioal Coferece o Commuicatios ICC), Lodo, Uited Kigdom, Jue 05 [5] U Niese, D Shah, ad G W Worell, Cachig i wireless etworks, IEEE Trasactios o Iformatio Theory, vol 58, o 0, pp 654 6540, October 0 [6] E Bastug, M Beis, ad M Debbah, Cache-eabled small cell etworks: Modelig ad tradeoffs, i IEEE Iteratioal Symposium o Wireless Commuicatios Systems ISWCS), Barceloa, Spai, August 04 [7] K Poularakis ad L Tassiulas, Exploitig user mobility for wireless cotet delivery, i IEEE Iteratioal Symposium o Iformatio Theory ISIT), Istabul, Turkey, July 03 [8] N Golrezaei, AG Dimakis, ad A F Molisch, Wireless device-todevice commuicatios with distributed cachig, i IEEE Iteratioal Symposium o Iformatio Theory ISIT), Cambridge, USA, July 0 [9] M A Maddah-Ali ad U Niese, Fudametal limits of cachig, IEEE Trasactios o Iformatio Theory, vol 60, o 5, pp 856 867, May 04 [0] K Poularakis, V Sourlas, P Flegkas, ad L Tassiulas, O exploitig etwork codig i cache-capable small-cell etworks, i IEEE Symposium o Computers ad Commuicatios ISCC), Fuchal, Portugal, Jue 04 [] K Shamugam, N Golrezaei, AG Dimakis, A F Molisch, ad G Caire, Femtocachig: Wireless cotet delivery through distributed cachig helpers, IEEE Trasactios o Iformatio Theory, vol 59, o, pp 840 843, December 03 [] A Liu ad V Lau, Cache-iduced opportuistic MIMO cooperatio: A ew paradigm for future wireless cotet access etworks, i IEEE Iteratioal Symposium o Iformatio Theory ISIT), Hawaii, USA, July 04 [3] S B Wicker ad V K Bhargava, eed-solomo codes ad their applicatios, Joh Wiley & Sos, 999 [4] D J MacKay, Foutai codes, i IEE Proceedigs-Commuicatios, 005, vol 5, pp 06 068 [5] L Breslau, P Cao, L Fa, G Phillips, ad S Sheker, Web cachig ad Zipf-like distributios: Evidece ad implicatios, i IEEE Joit Coferece of the Computer ad Commuicatios Societies INFOCOM99), New York, USA, March 999