Caching in Heterogeneous Networks with Per-File Rate Constraints

Transcription

1 Cachng n Heterogeneous Networks wth Per-Fle Rate Constrants Estefanía Recayte, Student Member, IEEE and Guseppe Cocco, Member, IEEE arxv: v [cs.it] 5 Oct 208 Abstract We study the problem of cachng optmzaton n heterogeneous networks wth mutual nterference and per-fle rate constrants from an energy effcency perspectve. A setup s consdered n whch two cache-enabled transmtter nodes and a coordnator node serve two users. We analyse and compare two approaches: a cooperatve approach where each of the transmtters mght serve ether of the users and a noncooperatve approach n whch each transmtter serves only the respectve user. We formulate the cache allocaton optmzaton problem so that the overall system power consumpton s mnmzed whle the use of the lnk from the master node to the end users s spared whenever possble. We also propose a lowcomplexty optmzaton algorthm and show that t outperforms the consdered benchmark strateges. Our results ndcate that sgnfcant gans both n terms of power savng and sparng of master node s resources can be obtaned when full cooperaton between the transmtters s n place. Interestngly, we show that n some cases storng the most popular fles s not the best soluton from a power effcency perspectve. Index Terms Cachng Networks, Cooperatve Communcatons, Energy Effcency, Interference Channel, Rate Constrants. I. INTRODUCTION Wreless data traffc has grown dramatcally n recent years. It s foreseen that traffc demand n the ffth generaton 5G moble communcaton networks wll ncrease by 000-fold by the year 2020 []. Ths huge demand wll be manly characterzed by hgh-defnton vdeo contents and streamng applcatons whch not only requre a sgnfcant ammount of bandwdth but also have strngent qualty of servce requrements QoS []. In order to offload the network and to reduce the system level energy consumpton, cache-aded networks have been studed extensvely n the recent years [2] [20]. One of the man advantages of cachng s the possblty to spare backhaul resources by storng fles durng perods n whch the network traffc s low and drectly serve the users Estefanía Recayte s wth the Insttute of Communcaton and Navgaton of the Deutsches Zentrum für Luft- und Raumfahrt DLR, Welng, Germany. Emal: estefana.recayte@dlr.de. and wth DEIS, Unversty of Bologna, 4036 Bologna, Italy. Guseppe Cocco s wth the LTS4 Sgnal Processng Laboratory and the Laboratory of Intellgent Systems LIS of the École Polytechnque Fédérale de Lausanne EPFL. Emal: guseppe.cocco@epfl.ch. Guseppe Cocco s partly founded by the European Unon s Horzon 2020 research and nnovaton programme under the Mare Sklodowska-Cure Indvdual Fellowshp grant agreement No Ths work has been accepted for publcaton at IEEE Transactons on Communcatons 208. c 208 IEEE. Personal use of ths materal s permtted. Permsson from IEEE must be obtaned for all other uses, n any current or future meda, ncludng reprntng /republshng ths materal for advertsng or promotonal purposes, creatng new collectve works, for resale or redstrbuton to servers or lsts, or reuse of any copyrghted component of ths work n other works. durng hgh-load perods. Due to the lmted storage capacty of the transmtters, the desgn of effectve cachng polces s requred. Cache-aded networks have been studed n lterature wth the am of mnmzng backhaul congeston [3] and energy consumpton [4] [6] or maxmazng the throughput [7] [8]. The ssue of nterference n cache-aded networks has also been studed n many recent works [7] []. However, most works consder an approach n whch fles are parttoned nto sub-fles and the same transmsson rate over the channel s used for each of them. In ths paper, we focus on a full cooperatve cachng nterference network. By ontly consderng per fle rate requrements and popularty rankng, we determne the content placement and the delvery strateges n order to optmze the energy effcency of the system. A. Related works Cachng optmzaton has been nvestgated n recent works for dfferent knds of heterogeneous networks. In [3] a two layer cachng model that ams at mnmzng the satellte bandwdth consumpton by storng nformaton both at ground statons and at the satellte s proposed. In [4] the authors consder cache-enable unmanned aeral vehcles UAV and study the optmal locaton, cache content and user-uav assocaton wth the am to maxmze users qualty of experence whle mnmzng the transmt power. Most of the proposed solutons explot the statstcs of past users fle requests often referred to as popularty rankng for fllng up local caches wth the most popular contents [3] [5]. Cachng from an energy effcency perspectve has also been extensvely studed. In [6] the authors am at reducng the total energy cost n a heterogeneous cellular network by applyng a cachng optmzaton algorthm for multcast transmsson. Energy mnmzaton n the backhaul lnk s studed n [4], where the authors propose a strategy for ontly optmzng content placement and delvery, whch leads to mnmze the energy consumpton. Energy effcency s also studed n [5], where a proactve downloadng strategy to explot the good channel condtons and mnmze the total energy expendture s proposed. Due to the sgnfcant network densfcaton expected n 5G and beyond, nterference management represents a key challenge. In order to counteract nterference, a combnaton of cachng wth nterference algnment IA and zero-forcng ZF technques has been recently proposed. In ther semnal work [7], Maddah-Al and Nesen leverage on IA and ZF to

2 show that sgnfcant gans n terms of load balancng and degrees of freedom can be obtaned when caches at both the transmtter and the recever sde are used. In [9] and [0] the authors extend the work n [7] consderng a delvery phase n whch dfferent sub-fles are transmtted over dfferent channel blocks. In [] authors propose a fle placement strategy that allows to create cooperaton opportuntes between transmtters through nterference cancellaton and IA. To the best of our knowledge, energy optmzaton n cooperatve nterference networks wth cachng capabltes only at the transmtter assumng a one-shot delvery phase and consderng dfferent per-fle rate requrements and dfferent popularty rankngs has not yet been analysed n lterature. Due to the one-shot delvery phase and to the dfferent per-fle rate requrements, the mnmum sum-power needed to satsfy users requests depends on both the fles rates and the state of the lnks nvolved n the transmsson. Snce the requests are not known durng the placement phase, a statstcal optmzaton algorthm leveragng on the fle popularty rankng has to be used to choose the cache allocaton achevng the mnmum power. Due to the dscrete nature of the optmzaton along the cache allocaton dmenson and the non convexty of the obectve functon n the sum-power dmenson, t s challengng to fnd a closed form soluton or an algorthm wth a complexty that scales well wth the number of fles. B. Contrbutons We optmze the placement and delvery phase for mnmzng the power consumpton of a cache-enabled network under QoS requrements and n the presence of nterference. Our man contrbutons are the followng: Unlke prevous works, fles are delvered over an nterference channel and dfferent PHY rate requrements per fle have to be respected. Whle cachng over nterference channels has been formerly studed [20], n prevous works the rate constrants are relatve to the user rather than to the fle,.e., each fle has the same rate requrement whle the per-user rate over the channel can dffer, n that t accounts for nformaton relatve to possbly more than one fle. Moreover, n [20] only the broadcast BC strategy s consdered and, unlke the present work, recevers are equpped wth a cache, whch allows them to effcently perform nterference cancellaton. Unlke [6], [8], [], we assume a one-shot delvery phase, n whch transmsson takes place n a sngle channel block and all selected fles are transmtted at the same tme. Dependng on cache content and requests, transmtters can apply one of the followng strateges: multple nput multple output MIMO broadcast transmsson, the Han- Kobayash rate splttng approach [33], multple nput sngle output MISO transmsson, broadcast and multcast MC transmsson. Up to our knowledge no other paper so far ontly consdered ths set of technques n the context of cachng. Extendng our prelmnary work [8], we establsh a correspondence between the cache allocaton/fle request cache a Tx u a0 a2 master node a20 a2 Tx 2 u 2 cache Fgure : System model of the consdered heterogeneous network wth caches at the transmtters. and the strategy that s mplemented usng the set of cooperatve technques lsted n the prevous pont and we derve the mnmum power cost needed for satsfyng the rate requrements. We propose an teratve algorthm for the Han-Kobayash strategy that asymptotcally converges to the mnmum requred sum-power and s sgnfcantly more effcent than an exhaustve search algorthm usng the same granularty. We propose a low-complexty suboptmal algorthm to solve the cache allocaton problem. The algorthm outperforms benchmark polces based on hghest popularty and hghest rate and has a complexty whch grows lnearly wth the number of fles. Our numercal results show that memorzng the most popular fles s not always the most convenent strategy when transmtters have to respect per-fle rate requrements. Furthermore, thanks to the consdered cooperaton strateges a sgnfcant savng of the coordnator node s resources can be acheved wth respect to a non-cooperatve system. The remander of the paper s organzed as follows. In Secton II we ntroduce the system model. In Secton III the dfferent transmsson strateges are presented and the relatve power optmzaton subproblems are addressed. The cache optmzaton s tackled n Secton IV. V contans the numercal results whle the conclusons are presented n Secton VI. A. Prelmnares II. SYSTEM MODEL We consder a heterogeneous network composed by a transmtter master node whch has access to a collecton of fles eg. vdeo fles, two transmtter nodes wth lmted cachng capabltes and two users u n, n = {, 2}. All nodes n the network are equpped wth a sngle antenna. The analyss presented can be extended to a mult-antenna setup, n whch case a lower power consumpton can be acheved. The study carred out n ths paper can be extended to the case of a generc number of SBSs and users. In dong so, the Han-Kobayash scheme can be replaced wth a transmsson approach smlar to that consdered n [27].

3 a a2 a a2 a22 a a0 b Fgure 2: System model for the non cooperatve approach: a nterference as nose, b broadcast or multcast channel and c orthogonal channel For ease of exposton n the followng we assume a 5G network setup, wth a macro base staton as master node and two small base statons SBS as transmtter nodes. However, the optmzaton problem and the transmsson technques presented n ths work are not lmted to a terrestral archtecture. In fact, n a heterogeneous satellte network a geostatonary GEO satellte connected to a ground staton could act as the master node wth access to all the fles whle low Earth orbt LEO satelltes connected to the respectve cache-enabled ground statons act as transmtter nodes. Another possblty that s ncreasngly recevng attenton both from the research communty and the ndustry s to employ unmanned aeral vehcles UAV as moble base statons. In such context, a hgh alttude UAV, a satellte or an could act as master node whle drones flyng at lower alttude take the role of transmtters. We assume that SBSs and work n two separate frequency bands. The reason for choosng such setup s that t s lkely to be mplemented n the next generaton of moble networks 5G [22]. In 5G, the SBSs wll be assgned frequences n the mmwave that, thanks to the large bandwdth avalable, allow to reach hgh data rates, whle the operate at lower frequency, provdng contnuous coverage to moble users wth lower data rates. An orthogonal bandwdth setup s also foreseen n next generaton heterogeneous satellte networks [26] and could be as well employed n UAVasssted networks. We assume that user u n s assocated to SBS n. Let us denote wth a n0 the channel coeffcent between u n and the whle a nm denotes the channel coeffcent between u n and SBS m, m = {, 2}, as shown n Fg.. We also defne a +0 = max{a0, a 20 } and a 0 = mn{a0, a 20 } whle R + R s the requred rate of the fle requested by the user wth channel coeffcent a +0 a 0. Smlarly P + and P are defned as the power used for fles wth rates R + and R, respectvely. For SBSs we defne a + = max{a, a 2 }, a = mn{a, a 2 }, a +2 = max{a22, a 2 }, a 2 = mn{a22, a 2 } whle R +, R, P + and P are defned for SBS and SBS 2 smlarly as for the. The has access to a lbrary of N fles F={f,..., f N }. We assume that all fles have the same sze. Ths assumpton s ustfable n practce snce fles can be dvded nto blocks of the same sze [9]. Each fle f has a mnmum requred transmsson rate R,.e., the mnmum rate measured n a20 a a20 c bts/s/hz at whch the fle has to be transmtted to the user n order to satsfy the QoS constrants. The probablty that f s requested by a user s called popularty rankng and s ndcated as q. We assume that SBS n s equpped wth a local cache of sze M the content of whch s denoted by M n. The bnary varable x n ndcates whether fle f s present n the memory of SBS n x n = or not x n = 0. We further defne x n x n. We denote wth x and x 2 the allocaton vectors contanng the values for x and x 2, respectvely. Dependng on whether the fles requested by the users are present n the cache or not and on whether a cooperatve scheme s consdered or not, the system mplements a dfferent transmsson approach. The possble approaches are explaned n Secton III. We ndcate wth α the ndcator functon whch takes value f α > 0 and 0 otherwse. We further defne α α. We denote wth P n the power used by the transmtter for sendng f to user u n at rate R. All transmsson rates n the followng are ntended per complex dmenson. We denote wth d = [f f ] the users demand vector. Wthout loss of generalty we assume that f s requested by u and f by u 2. The value c T, ndcates the mnmum power consumed for servng the users when the system mplements the transmsson approach T. Such value s the sum of two transmsson powers c T, = + P 2 f two transmtters are actve whle n case one transmtter takes care of both requests, the power cost s denoted as c T, = P,2,. Fnally, we ndcate wth Q c x, x 2 and Q nc x, x 2 the expected power cost for servng a request n the cooperatve and n the non-cooperatve approach, respectvely. B. System Archtecture We assume that the SBSs are connected to the through a backhaul lnk whle users have a drect rado frequency lnk to both the SBSs and the. In the placement phase the SBSs fll ther own caches wthout determnstc knowledge of the user demands. Our am s to understand whch s the best strategy to choose the fles for each transmtter s cache so that the average power consumpton s mnmzed durng the delvery phase. In the delvery phase each user requests a fle. We assume that the SBSs and the can lsten to the requests of both users and the communcaton s error-free 2. We assume that SBSs and estmate the channel durng the request and the channel does not vary durng the fle transmsson whle changes between two consecutve transmssons. Channel state nformaton at the transmtter CSIT s assumed. The master node has knowledge of all the channel coeffcents, the cache content and the user requests. Accordng to ths knowledge the decdes how the users have to be served and, eventually, coordnates the transmtters. The fle request and the fle transmsson take place n consecutve tme slots and each user Although we consder a constrant n terms of transmsson rate over the channel, ths can be easly translated nto a constrant n terms of delay and qualty, whch s partcularly suted to model vdeo and mage transmsson. 2 In practce ths can be acheved by notng that the sze of the request message s small and assumng a low channel code rate.

4 a a2 a a2 a22 a a0 b a20 a a20 c lterature related to cachng. In [20], for nstance, bounds on the mnmum average requred power n a cachng systems are studed wth no explct lmts n terms of maxmum per-node power. The decdes whch of the strateges descrbed n the followng has to be appled. The chosen strategy depends on whch fle has been requested and on the cache content of the SBSs. From now on, the subscrpts n and P 2 are removed f there s no ambguty. a d a2 Fgure 3: System model for the cooperatve approach: a nterference channel wthout common nformaton or nterference channel wth common nformaton or MIMO broadcast channel, b broadcast or multcast channel from, c orthogonal channel, d multcast or broadcast channel from SBS and e MISO channel. has to wat untl the end of the transmsson slot before placng a new request 3. Two dfferent approaches are consdered: a non-cooperatve approach NCA and a cooperatve approach CA. In the NCA user u n s served by SBS n f the requested fle s present n the cache of SBS n and channel condtons nterference plus nose and fadng levels allow relable communcaton at the requred rate. If ths s not possble, the user s served drectly by the. In the CA user u n can be served by ether the SBSs or by the f none of the SBSs has the requested fle. The CA has two goals. On the one hand t ams at reducng the overall system s energy consumpton and on the other hand t tres to spare the resources as much as possble. As a matter of facts n future heterogeneous terrestral networks a sngle wll serve a large number of SBSs [22]. Thus, n order to avod the beng a bottleneck, prorty s always gven to SBS transmssons whenever possble. a a2 III. CHANNEL MODELS AND POWER MINIMIZATION SUBPROBLEMS In ths secton we ntroduce the dfferent transmsson approaches wth the relatve channel models. For each of them we mnmze the power c T, requred to transmt f to u and f to u 2 at rate R and R, respectvely. In practcal systems, a maxmum per-node power constrant s present due to the physcal lmtatons of the transmtters. Unless otherwse stated, n the followng we assume that such maxmum power s larger than that requred for each consdered setup. Ths allows us to get an nsght on the power effcency and resource sparng, whch s a step towards the full understandng of nterference-lmted cachng systems. Note that usng such assumpton s not new n the 3 Snce fles have the same sze but dfferent rate constrants, the transmsson of the fle wth hghest rate termnates before the other fle s. Ths mples an nterference-free transmsson durng the part of the slot for the latter. In ths paper we look for an upper bound on the mnmum requred power and assume that the nterference s constant through the transmsson a20 e A. Gaussan Interference Channel wth Common Informaton GI c Ths channel s consdered n the CA when the demand vector of the users s d = [f f ] and x =, x 2 =, Fg. 3-a. We consder the nterference channel model n standard form, whch was ntroduced n [23]. Startng from the physcal model, the standard form s obtaned as follows. Let us consder the physcal channel model y = a x + a 2 x 2 + z, y 2 = a 2 x + a 22 x 2 + z 2, where z and z 2 are ndependent Gaussan varables wth zero mean and unt varance. From the pont of vew of the achevable rates such model s equvalent to [23] y = x + c 2 x 2 + z, y 2 = c 2 x + x 2 + z 2, where c 2 = a 2 /a 22, c 2 = a 2 /a. Let us denote wth P 2 the physcal power of x x 2. Gven the power P n n the standard model, the physcal power s: = /a and P 2 = P 2 /a 22. In the GIc the same fle f has to be transmtted to both users at rate R > 0. Usng the results n [24] and notng that n our case only a common message s to be transmtted, t can be easly shown that the power mnmzaton problem can be wrtten as P,P a mnmze subect to + P 2, a 22 2 log log 2 +, P c 2 P 2 R, 2 P 2 + c 2 R, The mnmum power cost c GIc, = + P 2 s derved n Appendx I. B. Gaussan Interference as Nose GIN Ths channel s consdered n the NC approach when x =, x 2 = and α = Fg. 2-a. where α s the ndcator functon and α = a 22 + a2 a2 a 2 2R 2 2R. The channel model s gven by. In ths transmsson scheme, each SBS consders the nterference generated by the other SBS as nose. The mnmzaton problem s presented n

5 [8] and not reported here for a matter of space. The mnmum cost s acheved when P = a 2 2R a 2 P + and P = [ ] 2 2R a2 a 2 2R + a 22 + a2 a2 a 2 2R 2 2R. 3 The value of P n 3 s postve f and only f α > 0. In such case the SBSs are able to successfully delver ther fles to the respectve users. On the contrary, f α 0 no and P 2 exst such that the mnmum requred rate can be acheved. In such case, the SBSs are not able to serve ther users and the wll serve both users ether applyng a broadcast f fles are dfferent or a multcast transmsson f fles are the same. The cost of the GIN channel s [ ] 2 2R a 2 a 2 2R + c GIN, = a 22 + a2a2 a 2 2R 2 2R + [ ] 2 2R a2 2 2R a 2 a 2 2R + a a 22 + a2a2 a 2 2R 2 2R +. C. Multcast Channel Ths channel s consdered when d = [f f ] n the followng cases: n both CA and NCA approaches when requests are satsfed by the x = 0, x 2 = 0, n whch case, snce both users requre the same fle, the sends the fle at the level of power needed to satsfy the u n experencng the worst channel condton, n the CA when both requests are satsfed by one SBS x =, x 2 = 0 or x = 0, x 2 = and n the NCA when α = 0. The CA s represented n Fg. 3-b,c whle the NCA n Fg. 2-a. For the case of multcast or broadcast transmsson the channel model s = max channel s 2 2R a 0 y = a 0 x + z, y 2 = a 20 x 2 + z 2. The mnmum power requred for u s = 22R a 0 whle for u 2 s P 2 = 22R { } a 20. Thus, we have P,2, 22R a 20. The cost of ths = 22R a 0 c MC, = 22R a 0. Smlarly, for a multcast transmsson from SBS n we have that c SBS MC, = 22R a n. D. Broadcast Channel In the broadcast channel, we have one transmtter and two recevers whch request dfferent fles [28]. Ths type of channel s mplemented by the or one of the SBSs when d = [f f ],, n the followng cases: n the CA when fles are not present n SBSs caches x = 0, x 2 = 0, x = 0, x 2 = 0, n ths case the serves both users, n the CA when one SBS has both fles whle the other has none of them, that s when x =, x 2 =, x = 0, 4 x 2 = 0 or x = 0, x 2 = 0, x =, x 2 =, n ths case a SBS serves both users and n NCA when the fle requested by u n s not present n SBS n for n =, 2 thus the serves both users x = 0, x 2 = 0. Fnally, v the broadcast channel s mplemented n the NCA when α = 0. The CA s represented n Fg. 3-b,c whle the NCA n Fg. 2-a. The power mnmzaton problem to be solved, consderng an transmsson, can be found n [8] and thus s not reported here snce t s formally the same as n case of SBS transmsson. Let P, P + be the powers related to the worst and best channel, respectvely, as defned n Secton II-A. The overall cost of ths channel s c BC, = 22R+ a R + a 0 P + a 0. Smlarly, for the broadcast transmsson from SBS n we have: c SBS BC, = 22R+ a +n + 22R + a n P + a n. E. MIMO Broadcast Ths channel s consdered n the CA when d = [f f ], for, and each SBS has both requested fles x =, x 2 =, x =, x 2 =, as depcted n Fg.3-a. In ths case, the coordnates the SBSs transmsson and provdes the nformaton needed e.g. the channel matrces such that each SBS calculates the porton of the MIMO codeword to be transmtted. Ths channel s not consdered n [8] and s analyzed n the followng. We denote wth H = [ a a2 ] T and H 2 = [ a 2 a22 ] T the channel matrces between each SBSs and the two users. In the MIMO BC channel the two sngle-antenna SBSs collaborate for transmttng both fles actng as a sngle multantenna termnal. The drty paper codng DPC approach s used. DPC conssts n a layered precodng of the data so that the effect of the self-nduced nterference can be cancelled out [29]. User messages are sequentally encoded. Let assume that the message for u s encoded frst. The decodng s such that a message experences nterference only from messages encoded after hm. In ths case, the message for u s nterfered by the message for u 2 and u 2 experences no nterference. Let us defne π m wth m = {, 2} as a permutaton descrbng the possble encodng orders such that π = {2, } and π 2 = {, 2} and let π m l ndcate the l-th element of the encodng order m, e.g. π = 2 and π 2 =. The mnmzaton problem to be solved s mnmze S,S 2,π m subect to 2 tr = S 5 2 log 2 I + H T π S H m2 π m2 πm2 = R πm2, 2 log I + n H = πms Hπm 2 I + H T π S H = R πm, m π m2 πm m =, 2

6 where H n s a square channel matrx obtaned by opportunely modfyng H n. Such modfcaton s done as suggested n [30], by ntroducng a vrtual receve antenna at each user wth vrtual channel gan equal to ɛ, wth ɛ much smaller than any of the channel coeffcents. Thus, H = a ɛ a2 ɛ and H 2 = a 2 ɛ a22 ɛ. S and S 2 are the postve sem-defnte nput covarance matrces for u and u 2 respectvely whle I s the nose covarance matrx. To solve 5 we transform the BC optmzaton problem nto a dual MAC optmzaton problem [3]. Then, the equvalent convex optmzaton problem can be wrtten as mnmze tr S M,SM 2,πm subect to 2 = S M 2 log 2 I H S M = H T = R M π m2, 2 log 2 I + H πms M π m H T π m = R M π m, m =, 2, where S M ndcates the dual MAC covarance matrx. 2 The followng holds: tr = S 2 = tr = SM and R M = 2 = R. It s not straghtforward to fnd the soluton to problem 6. A suboptmal soluton S M, S M 2 can be obtaned usng the teratve algorthm suggested n [30]. The mnmum power s readly obtaned as P + P = tr S + S 2 = tr S M + S M 2 and the overall cost of the channel s c MIMO, = P + P. F. Orthogonal Channel Ths channel s consdered when u s u 2 s request s satsfed by the SBS and u 2 s u s request s satsfed by the usng orthogonal channels. In the CA ths occurs when x =, x 2 = 0, x = 0, x 2 = 0, x = 0, x 2 = 0, x = 0, x 2 = or x = 0, x 2 =, x = 0, x 2 = 0, x = 0, x 2 = 0, x =, x 2 = 0. In the NCA such channel s consdered when x =, x 2 = 0 or x = 0, x 2 = Fg. 2-c and Fg. 3-c. Let us assume, wthout loss of generalty, that u s served by SBS and u 2 by the. The channel model s the followng The cost of ths channel s G. MISO Orthogonal y = a x + z, y 2 = a 20 x 2 + z 2. c, = 22R a + 22R a 20. The multple nput sngle output orthogonal channel s consdered n the CA when both SBSs have the fle requested by one user but do not have the fle requested by the other user x =, x 2 =, x = 0, x 2 = 0 and x = 0, x 2 = 0, x =, x 2 = Fg.3-e. Wthout loss of generalty, let us assume that u s served by the two SBSs and u 2 s served by the. We recall that all nodes have a sngle antenna but, thanks to the coordnaton of the, the two transmtters act as a mult-antenna termnal. The channel model s the followng y = a x + a 2 x + z, y 2 = a 20 x 2 + z 2. The problem to be solved s [32] mnmze P,P P + P, subect to 2 log 2 + a + a 2 P R, 2 log 2 + a 20 P R, P, P 0, where the soluton s P = 22R a +a 2, P = 22R a 20 and the cost s c MISO, = 22R + 22R. a + a 2 a 20 H. Gaussan Interference Channel wthout Common Informaton GI wc In ths channel the SBSs mplement the Han-Kobayash rate-splttng strategy [33]. Such cooperatve scheme conssts n splttng each message rate nto two parts. One part of the message s prvate and s decoded only by the ntended user whle the other part of the message s common and s decoded by both recevers. The sum of the rates of such parts s equal to the rate of the orgnal message. Decodng part of the nterferng sgnal allows to expand the capacty regon wth respect to the GIN case. Ths scheme s consdered when d = [f f ], wth, and requests are satsfed by the two SBSs x =, x 2 =, x 2 = 0, x = 0 and x =, x 2 =, x = 0, x 2 = 0. In the latter case, each SBS does not have the fle requested from the correspondng user but has the fle requested by the other user. Thus SBS serves u 2 and SBS 2 serves u. Fg. 3-a. Solvng analytcally the power mnmzaton subproblem for the case of GIwc s a challengng task due to the non convexty of the problem and to the ntrcacy of the condtons see expressons 4-8 and Appendx II. In order to calculate the cost c GIwc = P + P, we developed the algorthm calculate Mn Ptot HK, shown n Algorthm n pseudocode. Let us ndcate wth P tot the granularty at whch the sum power s evaluated, wth P the granularty wth whch the sum power s dvded between the SBSs P and P and wth λ the granularty n whch power s splt between the prvate and the common message. 4 At ntalzaton the algorthm performs the search over a grd of ponts 5 spaced apart by the largest granularty specfed n the nput n order to speed-up convergence. The granularty s progressvely decreased to the mnmum value specfed n the nput to enhance the accuracy 4 The three parameters are bounded as follows: P tot > 0, 0 < P < and 0 < λ <. The algorthm takes as nput the mnmum and maxmum granulartes of P tot, P and λ as well as the requred fle rates R and R and returns the mnmum value of c GIwc that satsfes the rate constrants for the two cached fles. 5 As an example, the frst pont of the grd for a gven P tot s P, P 2, λ, λ 2 = P P tot, P P tot, λ, λ, the second pont s P, P 2, λ, λ 2 = 2 P P tot, 2 P P tot, λ, λ and so on.

7 Algorthm c GIwc = calculate Mn Ptot HK P tot P mn, P max, λ mn, λ max, R, R mn, Pmax, tot : c GIwc = ; P tot = Pmax; tot P tot = Pmax; tot 2: whle P tot > Pmn tot do Go on untl the mnmum granularty for P tot s reached 3: flag = 0 4: P = P max; λ = λ max Reset P and λ 5: whle flag == 0 do 6: create G P, λ Create search grd wth current granularty excludng ponts prevously checked 7: for g G P, λ do Each pont g s defned by coordnates P, P, λ, λ 2 8: f g allows to acheve the par R, R then 9: flag = 0: c GIwc = P tot : P tot = P tot /2 2: P tot = P tot P tot Check f R, R are achevable wth a smaller P tot 3: break for 4: end f 5: end for 6: f flag == 0 then 7: f P > P mn then 8: P = P/2 9: else f λ > λ mn then 20: λ = λ/2 2: else f c GIwc then Check f so far R, R has been acheved 22: P tot = P tot /2 23: P tot = P tot + P tot 24: else 25: P tot = P tot + P tot If R, R s not achevable wth P tot, the same holds for a smaller P tot 26: break nner whle 27: end f 28: end f 29: end whle 30: end whle 3: return c GIwc of the soluton. Note that the mnmum power c GIwc s lower bounded by c and upper bounded by c GIN,. The algorthm s much more effcent than exhaustve search over a grd wth same mnmum granularty. Specfcally, f P tot tot Pmax, the gan n terms of number of operatons s mn on the order of Pmax/ P tot mn tot / λ mn / P mn. The algorthm s based on the fact that, f a gven sum power P tot does not allow to satsfy the rate constrants for any rate- and power-splt, then no P tot < P tot can satsfy such constrants. Ths can be derved from the condtons n Appendx II. The algorthm converges to the absolute mnmum asymptotcally as P tot mn 0, λ mn 0, P mn 0. Note that from condtons n Appendx II follows that a soluton to the mnmzaton problem always exst. Therefore, there s no need to check for the exstence of a soluton as done for the GIN channel n the NCA, where the ndcator functon α > 0 was requred. The proof s not reported here for a matter of space. A. Cooperatve Approach IV. OPTIMIZATION PROBLEM In ths secton we present the cachng optmzaton problem for the CA. In the CA a user can be served by the or any of the SBSs. The expected power cost for servng a request Q c x, x 2 can be wrtten as n 7. In the followng, gven a varable h = {, 2} we ndcate wth h ts complementary value,.e. f h = then h = 2 and vce-versa. Denotng the capacty of a pont to pont Gaussan channel wth sgnalto-nose rato x as Cx = 2 log 2 + x, the optmzaton problem for the CA can be formulated as follows mnmze Q c x, x 2, 8,P 2,x,x 2 subect to: x n M n, n = {, 2}, 9 f F x x 2 x x 2 R Ca 0, P 0 x x 2 x x 2 R + C a +0 P +, x lw x l w x lw x l w R l C a w0 P w, 2 l x lw x l w x lw x l w R l C w = {, 2}, l = {, },, 3 a wk P w l x r x r x r x r R ρ c r r, c rr, x r x r x r x r R ρ 2 c r r, c rr, f l = then k = else k = 2, x r x r x r x r R + R ρ 2 c r r, c rr, x r x r x r x r 2R + R ρ 0 c r r, c rr, x r x r x r x r R + 2R ρ 20 c r r, c rr, 2 P, P, 4 r = {, 2}, 2 P, P 5 P, P P, P 7 2 P, P 8 x x 2 x x 2 R Cπ m, H, H 2, S, S 2, 9 x x 2 x x 2 R Cπ m, H, H 2, S, S 2 20 x z x z x z x z R C a z P / + a z P +, 2 z = {, 2}, x z x z x z x z R + C a +z P +, 22 x x 2 x x 2 R C a + a 2, 23 x x 2 x x 2 R Ca 20 P 2, 24 x x 2 x x 2 R Ca 0, 25 x x 2 x x 2 R C a 22 + a 2 P 2, 26 x p x p R Ca pp P p, p = {, 2}, 27 x p x p R Ca pp P p, 28 x x 2 x x 2 R C a 0, =, 29 x x 2 x x 2 R C a 20 P 2, =, 30 x x 2 R C x x 2 R C + P 2 + c 2 c 2 P 2 P 2, =, 3 2, =, 32 0, P 2 0, 33 P 0, P + 0, 34

8 Q c x, x 2 = q q f F f F f f [ c, t=, n=,2 { [ c GIc, t=, n=,2 x tn x tn x t n x t n ]} c MIMO, [x x 2 x x 2 x tn x tnx t n x t n n=,2 ] +c SBS [ BC, n=,2 ] ] x n x n x n x n +c BC, [ x x 2 x x 2 + ] [ ] [ ] +c GIwc, x n x n x n x n +c MISO, x t x t2 x t x t2 + + q f F f =f t=, { 2 c MC, [ x x 2 ]+c SBS MC [x x 2 +x 2 x ]+c GIc, [x x 2 ] }. 7 x n {0, }, x n {0, }, letr 35 f, f F. 36 Inequalty 9 ensures that the total amount of data stored n a cache does not exceed ts sze, 0- denote the capacty regon of the Gaussan broadcast channel of the, whle 2-3 represent the condtons for the case of orthogonal channels, 4-8 are the Han-Kobayash condtons and defne the best known achevable rate of the GI wc the explct expressons for such condtons are gven n the Appendx II, 9-20 defne the capacty regon for the MIMO broadcast channel, 2-22 denote the capacty regon for the broadcast channel of SBS n, and defne the capacty regon of the orthogonal channel when the SBSs mplement a MISO channel to serve u u 2 whle u 2 u s served by the, denote the capacty regon of the multcast channel of SBS n, 3-32 defne an achevable rate regon of the GIc when both SBSs have to transmt the same fle, defne the capacty regon of the multcast channel of the, 33 s mposed to guarantee the non negatvty of the transmt power, whle 35 represents the cachng choce and accounts for the dscrete nature of the optmzaton varable. Note that the power varables n constrants 4-8 and refer to the normalzed channel model P, P. Snce we are nterested n evaluatng the physcal power consumpton, the power values n the calculaton of the cost have to be scaled as = P /a and P 2 2 = P /a 22. As a sde remark, note that there s a sgnfcant dfference between the MISO and the GI c channel, despte the fact that n both cases the same fle s transmtted by both SBSs. In fact, n the GI c channel SBSs transmt to both users and powers are adusted such that both users acheve the QoS requred whle n the MISO channel the SBSs transmt only to one user. B. Non-Cooperatve Approach In the NCA each user s served ether by the SBS to whch t s assocated or by the. Ths approach consders for all transmssons the nterference of the other SBS as nose. The transmts to the user when the fle requested by u n s not present n the cache of SBS n or when the channel condtons do not allow to have a relable transmsson. The expected power cost n the NCA Q nc x, x 2 can be wrtten as n 37. The optmzaton problem to be solved s the followng mnmze Q nc x, x 2, 38,P 2,x,x 2 subect to: 9-, x kg x kg x kḡ x kḡ R + C a +0 P +, 39 g = {, 2} f g = then k = else k = x kg x kg x kḡ x kḡ R C a 0 P / + a 0 P +, 40 x 2 x x x 2 R C a 0 P / + a 0 P +, 4 x 2 x x x 2 R + C a +0 P +, 42 x lw x l w x lw x l w R l C a w0 P w, 43 l w = {, 2} f w = then l = else l = x lw x l w x lw x l w R l C a ww P w l, 44 x tm x tm x t m x t m R t Ca tt P m t, 45 t = {, }, m = {, 2}, x tm x tm x t m x t m R t Ca t0 P m t, 46 x tm x t m x tm x t m R t Ca mm P m t, 47 x tm x t m x tm x t m R t Ca m0 P m t, 48 α x x 2 x 2 x R C a / + a 2 P 2, 49 α x x 2 x 2 x R C a 22 P 2 / + a 2, 50 α x x 2 x x 2 R C a / + a 2 P 2, 5 α x x 2 x x 2 R C a 22 P 2 / + a 2, 52 α x x 2 x x 2 R C a 0 P 2, =, 53 α x x 2 x x 2 R C a 20 P 2, =, 54 α x x 2 x 2 x R C a 0, =, 55 α x x 2 x 2 x R C a 20 P 2, =, 56 α x x 2 x x 2 R C a 0 P / + a 0 P + 57 α x x 2 x x 2 R + C a +0 P + 58 α x x 2 x 2 x R C a 0 P / + a 0 P +, 59 α x x 2 x 2 x R + C a +0 P Condtons 9- and are n common wth the CA. Condtons denote the capacty regon of

9 Q nc x, x 2 = { q q c GIN, [ ] α x x 2 +c, [ ] x x 2 + x x 2 +c BC, [ α x x 2 f f { 2 c GIN, [ ] ] α x x 2 +c MC, [ x x 2 + α x x 2 +c, [ x ] } 2 x + x x f F f F ] } + x x 2 + q f F f =f Algorthm 2 [x x 2] = low complexty allocatonr, q : x = 0 and x 2 = 0 0 s an N-dmensonal all-zero vector 2: W = {w,..., w k,..., w N } w k = 2 2Rk q k s the k th element of the set 3: for l =,..., M do 4: w m = arg max{w l } 5: x m = 6: W = W \ {w m} 7: w m = arg max{w l } 8: x m2 = 9: W = W \ {w m} 0: end for : return [x x 2] l l the Gaussan broadcast channel, defne the capacty regon of the orthogonal channels, whle condtons represent the constrants of the GIN channel. Constrants and denote the capacty regon of the Gaussan multcast channel consdered when the SBSs are not capable to delver the fles at the requested rates. C. Implementaton The optmal cache allocaton can be derved by mnmzng the cost functon n 7 ndependently along the power dmenson and the cache allocaton dmenson. Frst, the mnmum average power for a gven cache allocaton s calculated. Let us recall that N s the number of fles. For each of the N 2 possble requests the power has to be mnmzed. Each request, accordng to whether the fle s present or not n a cache, requres to solve one of the power mnmzaton subproblems presented n Secton III. The optmal cache allocaton s obtaned by exhaustve search wthn the set of possble allocatons. The complexty for mnmzng the average power for a gven cache allocaton s ON 2 whle the cost for fndng the optmal cache allocaton s ON!. Although such algorthm s useful to evaluate the performance lmt of the consdered setup, ts complexty makes t unsuted for a practcal applcaton when the number of fles s large. Therefore, we propose n the followng a suboptmal algorthm wth reduced complexty. Let us consder equaton 7. The cost of each channel depends n a non-lnear way on the channel coeffcents a nm and on the rate of both requested fles R, R. It also depends lnearly on the probabltes of the fles requests q, q. Let us now consder gene-aded recevers havng access to the message destned to the other user. In ths case, the nterference can be entrely removed and each user sees an nterferencefree channel. In such channel, the cost of transmttng f s equal to ts popularty rankng q tmes a term proportonal to 2 R. Snce we consder a setup n whch the SBS-user channel s on average better than that between -user one, havng fles wth hgher cost transmtted by SBSs rather than the mnmzes the overall power expendture. Let us defne the utlty functon w k R k, q = 2 2Rk q k assocated to f k. The proposed algorthm, descrbed n Algorthm 2, caches the fles wth hghest wr, q. The complexty of the algorthm grows lnearly wth N. In secton V, we compare ths algorthm wth the optmal one and wth two benchmarks. V. NUMERICAL RESULTS In ths secton we compare CA and NCA n two dfferent scenaros, namely: statc scenaro and moble transmtters scenaro. In the statc scenaro no fadng s present,.e., all channel coeffcents are fxed. In the moble transmtters scenaro the transmtter-user lnks experence fadng whle the master node-user lnks are AWGN channels. Ths models a heterogeneous satellte network n whch two LEO satelltes act as transmtters whle a GEO satellte acts as master node. Due to the LEOs orbtal moton and the presence of obstacles such as tall buldngs, fadng s present n the LEO-user channels 6 whle the lnk to the GEO, assumed to be lne of sght, does not change sgnfcantly over tme, whch s the case n clear-sky condtons. Ths setup also models a hybrd network wth a GEO satellte actng as master node whle hoverng UAVs play the role of transmtters. Also n ths case, the UAV-user lnks are affected by fadng due to UAVs moton [34]. Table I: Rate and popularty rankngs for each fle n the drect probablty and nverse probablty setups. F R [bts/sec/hz] q drect q nverse f f f f f We model the channel coeffcents n the lnks affected by fadng as Lognormal random varables,.e., gven a channel coeffcent a, we have loga N µ, σ 2, log. beng the Naperan logarthm. The parameters µ and σ have been chosen so that n the three scenaros we have E{a } = E{a 22 } = and E{a 0 } = E{a 20 } = 0.0, E{.} beng the expectaton operator. We plot our results aganst the mean value of the 6 In practce the equvalent of an SBS would be a LEO GEO satellte together wth the respectve ground staton, where the cache would be physcally located. Ths s because on-board storage n satelltes s not common practce.

10 Qx, x2 n db 20 5 M = {f 4, f 5}M 2 = {f 4, f 5} M = {f, f 2}M 2 = {f, f 2} M = {f, f 4}M 2 = {f 2, f 3} M = {f, f 2}M 2 = {f 3, f 4} M = {f 4, f 5}M 2 = {f 4, f 5} M = {f, f 2}M 2 = {f, f 2} Qx, x2 n db M = {f, f 4}M 2 = {f 2, f 3} M = {f, f 2}M 2 = {f 4, f 5} M = {f 2, f 5}M 2 = {f 3, f 4} M = {f 4, f 5}M 2 = {f 4, f 5} M = {f, f 2}M 2 = {f, f 2} M = {f 4, f 5}M 2 = {f 4, f 5} c 2 = c 2 Fgure 4: Mnmum average power cost n db n the statc case plotted aganst the nterference coeffcent wth drect fle probabltes. The result for the CA s represented wth sold curves whle dashed curves are used for the the NCA. The optmzaton s done for each average nterference coeffcent. In the smulatons we set a 0 = a 20 = 0.0 and a = a 22 = c 2 = c 2 Fgure 5: Mnmum average power cost n db n the statc scenaro plotted aganst the nterference coeffcent wth nverse probabltes. The optmzaton s done for each average nterference coeffcent. In the smulatons we set a 0 = a 20 = 0.0 and a = a 22 =. nterferng lnk coeffcents as presented n the GIc equvalent model c 2, c 2, where drect channel coeffcents have unt mean. The parameters for such coeffcents are chosen so that E{c 2 } = E{c 2 } take values n [0, ]. Ths allows to compare dfferent scenaros aganst a sngle parameter. In order to do a far comparson, ths s done for both the non-cooperatve as well as the cooperatve scheme. We consder the system depcted n Fg. 2 for the NCA and Fg. 3 for the CA where each SBS has a memory of sze M = 2 and the lbrary has cardnalty N = 5. Two relevant setups are consdered. In the frst one, fles wth hgher rate have a hgher popularty rankng whle n the second one fles wth hgher rate have a lower popularty rankng. We refer to these setups as drect probablty and nverse probablty, respectvely. The correspondng values for R and q are shown n Table I. We plot for each scenaro the mnmum power cost Qx, x 2 of the most sgnfcant cache allocatons n the CA sold curves and n the NCA dashed curves versus the mean value of the nterference coeffcents, assumed to be the same for both users 7. Note that, by the symmetry of the problem, the results do not change f the content of the cache of SBS and that of SBS 2 are swtched,.e., the performance of the soluton M = {f, f }, M 2 = {f k, f l } and M = {f k, f l }, M 2 = {f, f } concde. In Fg. 4 the most sgnfcant cache allocatons for the statc scenaro wth drect probabltes are plotted. When no cooperaton s n place the optmal cachng soluton conssts n each SBS storng the fles wth hghest probabltes/transmsson rate M = M 2 = {f, f 2 } whch s ndcated wth the orange dashed curve. In ths case, when mutual nterference s low the only transmts fles wth low probablty and low transmsson rate. On the contrary, when the level of nterference ncreases the SBSs are not able to serve ther own users even f they have the requested fles and transmssons are delegated to the. Thus, the usage of the becomes more frequent and more power s consumed due to the hgh 7 Note that, whle n the statc case ths s equvalent to havng a symmetrc channel matrx, ths s not necessarly the case n the moble transmtter case due to the ndependence of fadng. rate of the fles and lower channel gan. Ths explans the step of roughly 4 db from c 2 = c 2 = 0.2 to c 2 = c 2 = 0.4. For c 2 = c 2 larger than or equal to 0.4 the curve flattens out because, from that pont on, the mutual nterference s so strong that the requred rates cannot be acheved by the SBSs and the takes over all transmssons. In the CA the best performance s obtaned when one of the SBS stores the fles wth hghest probablty/transmsson rate and the two caches store dfferent fles. The result s shown n the lght blue curve wth damonds. A slght ncrease n power from 0.5 db up to db s observed when the caches store dfferent fles but the two most probable fles are n the same cache M = {f, f 2 }, M 2 = {f 3, f 4 }. In both CA and NCA approaches, the most power consumng soluton concdes and corresponds to each SBS storng the two fles wth lower probabltes/rates M = M 2 = {f 4, f 5 }. The flatness of the curves n both the CA and the NCA are due to the fact that the power consumed by the has a strong weght n the overall mean, snce transmssons at hgh data rate from are the most probable. Let us now consder the low nterference regme. In the CA, all the cachng schemes at the lowest nterference level consume at least 5 db more wth respect to the other levels. Ths s because, when a sngle SBS serves both users, t apples ether the broadcast or the multcast channel and such channels are penalzed when one of the two channel coeffcents are relatvely small. Later on n ths secton we show that, despte the hgher power consumed n some cases, cooperaton has an advantage n terms of resources sparng. In Fg. 5 the power cost versus nterference level n the statc scenaro and for nverse probabltes.e. fles wth hgher R have lower q s plotted. The plot shows that for both approaches the most power-effcent cache allocaton sold and dashed green curves s not the one ncludng the fle wth the hghest popularty but rather the one wth the hghest rate. The reason for ths s that the weght of transmssons on the overall power budget s hgher wth respect to the SBSs due to the lower channel gans. For the consdered rates and popularty rankng t s better to have the transmttng

11 E{Qx, x2} n db M = {f 4, f 5}M 2 = {f 4, f 5} M = {f, f 2}M 2 = {f, f 2} M = {f, f 4}M 2 = {f 2, f 3} M = {f, f 2}M 2 = {f, f 3} M = {f, f 2}M 2 = {f, f 2} M = {f 4, f 5}M 2 = {f 4, f 5} E{Qx, x2} n db M = {f, f 2}M 2 = {f, f 2} M = {f, f 3}M 2 = {f 2, f 4} M = {f, f 2}M 2 = {f 4, f 5} M = {f 4, f 5}M 2 = {f 4, f 5} M = {f, f 2}M 2 = {f, f 2} M = {f 3, f 5}M 2 = {f 4, f 5} M = {f 4, f 5}M 2 = {f 4, f 5} E{c 2} = E{c 2} Fgure 6: Average power cost n db versus nterference lnk coeffcents for dfferent memory allocatons n the moble transmtters setup n a fadng envronment wth drect probabltes. In the smulatons we set a 0 = a 20 = 0.0 and E{a } = E{a 22} = E{c 2} = E{c 2} Fgure 7: Avarege power cost n db versus nterference lnk coeffcents for dfferent memory allocatons for the moble transmtters setup n a fadng envronment wth nverse probabltes. In the smulatons we set a 0 = a 20 = 0.0 and E{a } = E{a 22} =. low rate messages more often rather than transmttng hgher rate messages wth lower popularty. In the cooperatve case, f the fle wth hghest rate s not cached there s a loss larger than or equal to 2.5 db wth respect to the best curve. Ths holds also for the allocaton ncludng the most popular fle yellow curve, whch confrms that t s better to store the fle wth hghest rate rather than the one wth hghest popularty. As a matter of facts, the worst performance n the both the CA and the NCA case s obtaned when each SBS stores the two fles wth largest probabltes/lowest data rate. We examne now the moble transmtters scenaro fadng on the the SBS-user lnk, no fadng on the the -user lnk. Note that, due to the dfference between two Lognormal random varables at the denomnator of equaton 3, the average transmsson power n the GIN channel s not fnte [35]. Comparng the CA and the NCA n such condtons always leads to an nfnte power gan of the former wth respect to the latter as long as the probablty that the two SBSs transmt together s non-zero. In order to get some nterestng nsght n the presence of fadng, n the smulatons of the moble transmtters scenaro we ntroduced a constrant on the maxmum transmt power. Such constrant s chosen so that the probablty that the SBSs cannot transmt due to t s around 0 5 n the worst case scenaro.e., GIN wth maxmum rates and E{c 2 } = E{c 2 } = In Fg. 6 the mnmum cost E{Qx, x 2 } for dfferent average nterference levels n the moble transmtters case for drect probabltes s plotted. Let us frst consder the CA. Smlarly to the statc scenaro, when nterference s not close to zero the best performance s obtaned when each SBS stores dfferent fles and the most probable ones are n M M 2. The power consumpton s constant 8 db for all levels of nterference when the caches of the SBSs are equal and each SBS stores the fles wth hghest rate/popularty. Note that such cachng soluton s the best for very low levels of nterference. 8 Note that E{c 2 } = E{c 2 } = would lead to a hgher outage probablty for the SBSs n the NCA, but the outage n such case s due almost exclusvely to the lack of a feasble soluton to equaton 3 rather than to the constrant on the maxmum power. Qcx, x Hghest rate Hghest popularty Low complexty algorthm Optmal strategy c 2 = c 2 Fgure 8: Mnmum average power cost n db n the statc scenaro plotted aganst the nterference coeffcent. For N = 0 and M = 2. R k and q k for each fle are descrbed on Table II. The optmzaton s done for each average nterference coeffcent. In the smulatons we set a 0 = a 20 = 0.0 and a = a 22 =. In fact, when both SBSs cache the same fles, broadcast and multcast transmssons, whch are the most expensve ones, are not mplemented. In the NCA the cachng scheme whch consumes less power s the one n whch each SBS stores the most probable fles. Movng from the statc to the moble transmtters scenaro the mnmum requred power ncreases. Ths s not surprsng snce the number of lnks affected by fadng ncreases. However, we note that n the drect fle probablty case, the optmal cache allocaton n both the scenaros remans the same. A smlar observaton s also true n the nverse probablty case. We performed other smulatons for a thrd setup n whch all lnks are affected by fadng, whch models a stuaton where the users are moble, although we do not report here the plots for lack of space. Interestngly, also n ths thrd scenaro the optmal cache allocaton s the same as n the other two. Ths may suggest that mperfect knowledge of channel statstcs may be tolerated at least up to a certan extent. Ths aspect can have mportant practcal mplcatons but due to the lack of space we do not delve further nto ths topc and leave t for future work.