Stochastic Geometry Model for Multi-Channel Fog Radio Access Networks

The 7 Iteratioal Workshop o Spatial Stochastic Models for Wireless Networks SpaSWiN Stochastic Geometry Model for Multi-Chael Fog Radio Access Networks Mostafa Emara *, Hesham ElSawy, Sameh Sorour, Samir Al-Ghadhba *, Mohamed-Slim Alouii ad Tareq Y. Al-Naffouri * Departmet of Electrical Egieerig, Kig Fahd Uiversity of etroleum ad Mierals KFUM, Saudi Arabia Emails: {mostafaemara, samir} @kfupm.edu.sa CEMSE Divisio, EE program, Kig Abdullah Uiversity of Sciece ad Techology KAUST, Saudi Arabia Emails: {hesham.elsawy, slim.alouii, tareq.alaffouri}@kaust.edu.sa Departmet of Electrical ad Computer Egieerig, Uiversity of Idaho, USA Email: samehsorour@uidaho.edu Abstract Cache-eabled base statio BS desificatio, deoted as a fog radio access etwork F-RAN, is foresee as a key compoet of 5G cellular etworks. F-RAN eables storig popular files at the etwork edge i.e., BS caches, which empowers local commuicatio ad alleviates traffic cogestios at the core/backhaul etwork. The hittig probability, which is the probability of successfully trasmittig popular files request from the etwork edge, is a fudametal key performace idicator KI for F-RAN. This paper develops a schedulig aware mathematical framework, based o stochastic geometry, to characterize the hittig probability of F-RAN i a multi-chael eviromet. To this ed, we assess ad compare the performace of two cachig distributio schemes, amely, uiform cachig ad Zipf cachig. The umerical results show that the commoly used sigle chael eviromet leads to pessimistic assessmet for the hittig probability of F-RAN. Furthermore, the umerical results maifest the superiority of the Zipf cachig scheme ad quatify the hittig probability gais i terms of the umber of chaels ad cache size. Keywords Cachig system, Stochastic geometry, Multichael, F-RAN. I. INTRODUCTION The fifth geeratio 5G of cellular etworks dictates tagible performace leap i terms of etwork capacity fold ad trasmissio rate fold []. Such performace requiremet is expected to be fulfilled by a uprecedeted etwork desificatio phase, i which the small BS SBS desity ca reach hudreds of SBSs per km [], [3]. Deployig more SBSs i the same geographical locatio improves the spatial frequecy reuse ad reduces the umber of users associated to the same BS, thus fosterig light loaded BSs with high data rate liks. However, supplyig this large umber of SBSs with real-time data services imposes a huge burde o the backhaul liks. Cosequetly, backhaulig represets a o-trivial bottleeck to attai the foresee etwork desificatio gais. Exploitig the commo iterests amog users located withi the same geographical locatio, proactive cachig is recetly proposed to solve the backhaulig problem. With the emergece of social etworkig, it is expected that a oegligible percetage of the mobile traffic will be produced from dowloadig/viewig similar cotets [4], deoted as popular files. roactive cachig exploits such pheomea ad brigs popular files to the etwork edge i.e., SBS caches. Cosequetly, popular files requests are directly served from the caches of the SBS ad are ot repeatedly requested from the core etwork via backhaul liks, which reliefs the backhaul ad core etwork cogestio. articularly, a user that requests a popular file is directly served from the SBS that stores the requested file. Such etworkig scheme is curretly kow as fog radio access etworks F-RANs [5] [7]. The performace of F-RAN is maily determied by the hit probability, which is defied as the probability that a popular file request is successfully catered from the etwork edge. Hece, maximizig the hittig probability has bee a focal research poit i the cotext of F-RAN. For istace, the authors i [8] propose a optimal cachig distributio that maximizes the hittig probability i order to alleviate core etwork delay ad achieve miimal file dowloadig time. However, [8] does ot accout for the ucertaities i SBSs ad users locatios. The authors i [9] use stochastic geometry to characterize the hittig probability cosiderig SBSs ad users locatios ucertaities. However, the proposed paradigm i [9] limits the popular file request to the earest SBS. Hece, cachig diversity amog differet SBSs is ot exploited. The work i [] [] propose optimal file placemet i SBSs ad allow users to dowload popular files from the earest SBS that store them, which is ot limited to the geographically earest SBS to the user. However, oly a sigle chael system is cosidered, which leads to a pessimistic performace assessmet of the hittig probability caused by strog iterferece of the SBSs closer to the user tha its servig SBS. I [3], the authors propose a distributed algorithm to balace the cache-related traffic amog the SBSs i a fair way i order to efficietly utilize the available resources assumig that the user may be covered by multiple SBSs. A joit optimal file placemet ad frequecy reuse is proposed i for multi-cast popular file dowload. However, [4] follows a rigid frequecy reuse scheme, which is well-kow to uderutilize the spectrum resources [5]. I cotrast to prior work, this paper exploits the multichael system which is well-kow i wireless cellular etworks [5] [7]. I the multi-chael system, the total available badwidth is divided ito a set of orthogoal 978-3-988-9-6/7 7 IFI

The 7 Iteratioal Workshop o Spatial Stochastic Models for Wireless Networks SpaSWiN chaels that is shared amog all SBSs. Each SBS assigs a subset of chaels to its associated users accordig to the applied spectrum access policy. I particular, this paper develops a mathematical framework to characterize the hittig probability i multi-chael F-RAN with uiversal frequecy reuse ad dyamic chael assigmet. The proposed framework assumes Zipf distributed file popularity ad allows users to be served by the earest SBS that stores the requested popular file, which is deoted as the file caterig SBS. Furthermore, the caterig SBS exploits opportuistic spectrum access to trasmit the popular file to the user. Specifically, if there are vacat chaels that are ot used by SBSs closer to the user, the caterig SBS radomly chooses oe of these vacat chaels to trasmit the file. Otherwise, the caterig SBS schedules the file o a radomly selected chael from the complete set of chaels. Such trasmissio scheme aims to avoid domiat iterferig SBSs, which sigificatly reliefs the aggregate iterferece ad improves the hittig probability. To this ed, we assess ad compare the performace of two cachig schemes, amely, the uiform ad Zipf cachig. The developed mathematical model is based o stochastic geometry, which is a widely used tool for modelig ad aalyzig cellular etworks [8]. Closed-form expressios are obtaied for the coverage ad hittig probability for a multi-chael cachig sceario. These expressios geeralize the sigle chael sceario that is widely adopted i the literature. All theoretical results i this paper are verified via idepedet Mote Carlo simulatios. The umerical results maifest the pessimistic results of the sigle chael F-RAN ad quatify the hittig probability gais i terms of the umber of chaels, cache size, ad cachig distributio. The model uder study is preseted i Sectio II, where the performace metric is itroduced. I Sectio III, we fid the free chael probability. Service distace distributios are preseted i Sectio IV, which are used to derive the coverage probability expressios i Sectio V. The proposed mathematical paradigm is verified by the umerical aalysis i Sectio VI. Fially, we coclude the paper i Sectio VII. A. Network Model II. SYSTEM MODEL We cosider a F-RAN i which the SBSs are spatially deployed i a give geographic area accordig to a homogeous two-dimesioal D oisso poit process Ψ b = {y i,i =,,3,...} with itesity λ b, where y i is the locatio of the i th SBS. Also, the users are modeled as a idepedet Ψ u with itesity λ u. It is assumed that all SBSs trasmit with the same power ad share a set of chaels S to serve their associated users. It is assumed that the sigal power decays at a rate r with the trasmissio distace r betwee a geeric SBS ad its associated user, where η > is the path loss expoet. The chaels betwee the SBSs ad their tagged users are assumed to be idepedet Rayleigh fadig chaels with uit mea power gais. Without loss of geerality, we will focus our aalysis o a typical user u located at the origi of the geographical area uder study. B. Cachig Model Cosider a fiite set of J popular multimedia files J = {c,c,...,c J }. We assume that all files are of the same legth. However, our aalysis ca still be applied with files of differet legth, by choppig such files ito equal legth packets. The file popularity is assumed to follow the well-kow Zipf s distributio [9], i.e., a j = j γ J,j =,,...,J, i= where a i γ j reflects the probability that a geeric user requests file c j, ad γ is the Zipf parameter that cotrols the skewess of the popularity distributio. Without loss of geerality, it is assumed that the files are idexed accordig to their popularity, i.e., a a a J. Each SBS is equipped with a cache memory of size K < J files. Thus, it stores a combiatio of K differet files out of the total J files. A set X {,,...,X} deotes all possible combiatios with set cardiality X = J K. Cachig copies of a file at the same SBS is avoided because it comes with o beefit but clearly restricts the SBS optios to store ad thus deliver other files. Let p x deotes the probability that a geeric SBS stores a combiatio x X. Thus, the probability that it stores a particular file c j is give by b j = p x x X j where X j is the set of all possible combiatios that have file c j, which is a subset of the set J with cardiality J K. The probability p x is defied accordig to the applied cachig scheme. This paper focuses o the uiform ad Zipf cachig schemes. I the uiform cachig, the K-files combiatios are cached radomly ad uiformly ito the SBSs. Thus, K = K J p x = X ad from, b j = J. O the other had, K J the Zipf cachig follows the popularity of files, i.e., the Zipf distributio. Thus, the probability that a combiatio x X to be stored at a geeric SBS is p x = K j x a j. C. Associatio Model Without loss of geerality, we assume that the idices of SBSs are ordered accordig to their distaces from the typical user. The typical user is assumed to be served by the earest SBS that stores the requested file. We also assume that the other users surroudig the typical user are traditioally associated to their geographically closest SBSs based o the maximum received power. It is assumed that each SBS picks a chael at radom from the pool S to serve each of its associated users. A SBS does ot assig the same chael to two of its associated users to avoid overwhelmig itracell iterferece. Noetheless, differet SBSs ca assig the same chael to oe of their tagged users ad thus itercell iterferece exists betwee BSs with schedulig ties. Accordig to Slivyak-Mecke theorem [], evaluatig the etwork performace for the typical user at the origi is sufficiet ad applicable to ay geeric locatio i the D

The 7 Iteratioal Workshop o Spatial Stochastic Models for Wireless Networks SpaSWiN where κ deotes the probability mass fuctio MF of the umber of chaels used by the SBSs. Similar to [], it ca be expressed by the followig recursive equatio κ = κ v v= κ w=κ v w [N u = w] w κ v w κ v v v κ v, κ S 4 S S where[n u = k] is the MF of the umber of users associated to a geeric SBS. Based o the maximum received power associatio policy, [N u = k] = c c Γc+k λu λ k b Γk+Γcc+ λu, where λ k+c b c = 3.575 is a costat related to the Vorooi cell area distributio. Fig.. A realizatio of the etwork. The red diamod represets the typical user at the origi ad the black dots are the other users. The blue triagles represet the SBSs that store the requested file while the black squares are the others. The red ad black lies show the associatio policies of both the typical user ad the other users i the etwork, respectively. The dashed red circle ecloses the SBSs that the file caterig SBS should avoids their used chaels whe assigig chael to serve the typical user. plae. A realizatio of the cosidered system model is show i Fig.. D. erformace Metric The average hit probability is the mai performace metric of the system. It is defied as the probability that the typical user s request of a popular file is successfully fulfilled by the closest SBS that caches that file. Assumig idepedet cachig at the SBSs, the hit probability depeds o both the coverage ad the free chael existece probabilities ad it ca be expressed by H = a j b j b j C free, + C free, j J = where a j is the probability that the typical user requests the file c j ; b j b j idicates that the desired file is available at the th SBS ad is ot stored i the eclosed SBSs; ad C deotes the coverage probability whe a free chael is available to serve the typical user, while C is the coverage probability whe there is o available free chael. free, represets the probability that there is a chael that is ot used by closer SBSs to the user tha the caterig SBS. III. FREE CHANNEL EXISTENCE ROBABILITY The probability of free chael existece at the file caterig th SBS is equivalet to the probability that the umber of occupied chaels by the SBSs that are closer to the typical user tha its file caterig SBS is less tha the total umber of available chaels S. The free chael existece probability free, is give by free, = S κ= κ = S 3 IV. SERVICE DISTANCE DISTRIBUTION The first step to aalyze the coverage probability of the multi-chael F-RANs is to characterize the service distace which highly affects the sigal-to-iterferece-plus-oise-ratio SINR. As we assume i the system model, the file caterig SBS is the th earby SBS to the typical user at the origi with a servig distace r. Exploitig the ull property of the, the probability desity fuctio DF of the servig distace r is give by [, Lemma 3] f r r = πλ br e πλ br, r 5 rγ As aforemetioed, the typical user is served over a radomly chose used chael if there is o available free chael at its file caterig SBS. Thus, it suffers from κ iterferig SBSs that use the same chael where κ, κ. It is worth otig that i the sigle-chael sceario κ =. To quatify this iterferece, the coditioal distributio f ri x r is eeded, where r i is the distace betwee the typical user ad the iterferig SBS. f ri x r is give by [3] f ri x r = x r, x r 6 V. THE COVERAGE ROBABILITY The coverage probability is defied as the probability that the typical user ca successfully achieve a specified SINR threshold β. Uder the adopted system model, it depeds o whether a free chael exists at the caterig file SBS or ot. Firstly, we eed to fid the set of SBSs that use the same chael as the servig chael of the typical user. By exploitig the radom chael selectio policy at the SBSs ad the idepedet thiig property of the, the set of SBSs that iterfere with the typical user forms a thiig Ψ b, with itesity λ b. The thiig parameter is the probability that a geeric SBS radomly selects a particular chael from S, which is give by [4]: S = [N u = k] S k S k= The, the coverage probabilities C ad C are give by 7

The 7 Iteratioal Workshop o Spatial Stochastic Models for Wireless Networks SpaSWiN C = [SINR β] = [ h r σ + y i Ψ b,,r i>r h i r i = [ h r σ β ] +I out C = [ β ] h r σ + y i Ψ b,,r i<r h i r β ] i +I out = [ h r σ β ] +I i +I out where h resp. h i is the chael gai betwee the typical user ad its file caterig SBS resp. the i th iterferig SBS. r resp. r i is the Euclidea distace betwee the typical user ad its file caterig SBS resp. the i th iterferig SBS. σ is the servig chael oise.i i adi out represet the aggregate iterferece of the SBSs closer to ad farther to the typical user from its file caterig SBS, respectively. Coditioig o the distace r betwee the typical user at the origi ad its file caterig SBS, the coditioal coverage probabilities are give by Cr = e βr η σ βr η L Iout C r = e βr η σ βr η η L LIiβr Iout where L I t = E[e ti ] deotes the Laplace trasform LT of I. L Ii ad L Iout are the Laplace trasforms LTs of I i ad I out, respectively. the LT of the iterferece from the outside r SBSs ca be obtaied similar to [3] with cosiderig oly the SBSs that use the same chael of the typical user. Thus, L Iout is give by βr η L Iout = exp πλ b rβ η dz z=β η +z η O the other had, the LT of the iterferece from the SBSs iside r L Ii if there is o available chael is give by the followig Lemma Lemma : The LT of the iterferece from the ier SBSs is give by L Ii βr η = r r 8 9 r= +βr η r rdr+ roof: See Appedix A By averagig over the DF of the servig distace r betwee the typical user at the origi ad its file caterig SBS, f r r, the ucoditioal coverage probabilities are give by. I the special case of iterferece-limited etwork i.e., σ = ad path loss expoet η = 4 which is commo for wireless etworks, ad with itegral maipulatios, the ucoditioal coverage probabilities i tur to the simple closed-form expressios explaied i the followig corollary Corollary : The closed-form for the geeral expressios of the coverage probabilities are give by C = + β arcta β [ C = β arcta ] β + β arcta β 3 Fially, by substitutig from ad 3 ito, we ed up with the geeral ad closed-form expressios of the hit probability for the proposed multi-chael system. It is worth otig that the hit probability of the siglechael F-RAN, which is widely used i the literature ca be derived from the multi-chael system. I the sigle-chael sceario, all the SBSs use the same chael to serve their associated users. Therefore, the typical user suffers from iterferece from all SBSs except its file caterig SBS. Followig the previous aalysis, is omitted ad k is determiistic with value. Thus, the sigle-chael system hit probability H sigle is give by the followig Lemma. Lemma : The hit probability of the sigle-chael system is give by H sigle = a j b j b j C sigle j J = C sigle = πλ b v v Γ v w= +βv η w wdw exp πλ b v β η dz e βv η σ z=β η +z η e πλ bv dv 4 I the special case of iterferece-limited etwork ad η = 4, the coverage probability i 4 turs to a closed-form expressio that is give by C sigle = + β arcta β β arcta VI. NUMERICAL RESULTS β 5 I this sectio, the proposed multi-chael system model is compared with the sigle-chael system. The parameters λ b = 4 SBSs/Km ad λ u = 4 users/km were chose to coduct the aalysis. All the SBSs have the same trasmitted power = watt. The path loss expoet η = 4 is cosidered. Both the uiform cachig ad the Zipf cachig are cosidered. Fig. illustrates the hit probability versus the SINR threshold for both the multi-chael ad the sigle-chael systems usig uiform ad Zipf cachig schemes cosiderig the requested file is c j =. A clear match ca be oticed betwee our derived expressios ad the simulatio results, for both sigle ad multi-chael systems. Furthermore, Fig. quatifies the gai of the multi-chael system over the sigle-chael oe ad reveals the sigificat improvemet of system performace usig the Zipf cachig compared with the uiform cachig scheme. However, focusig o the first popular file of the highest popularity a results i a ufair study.

The 7 Iteratioal Workshop o Spatial Stochastic Models for Wireless Networks SpaSWiN C = πλ b Γ C = πλ b Γ v e βv η σ v e βv η σ { + v v w= exp πλ b v β η exp πλ b v β η +βv η w wdw z=β η z=β η +z η +z η } dz e πλ bv dv dz e πλ bv dv Therefore, Fig. 3 shows the hit probability versus the requested file idex. It ca be observed that the hittig probability is idepedet from the file idex for uiform cachig scheme. This is because the uiform cachig scheme distributes the files ito SBSs radomly ad uiformly irrespective to their popularity. O the other had, the hit probability of the Zipf cachig scheme highly depeds o the requested file popularity. The hittig probability dramatically degrades whe the typical user requests ay other files rather tha the first oe i the case of high Zipf parameter. This figure also idicates that the multi-chael system always outperforms the sigle-chael sceario. I additio, the uiform cachig slightly outperforms the Zipf cachig whe the files with low popularity are requested. The hit probability.5.45.4.35 The hit probability vs. c β = db & J=5 & K=5 & λ / λ = j u b = & γ=.5 = & γ=.5 Uiform: N ch = Uiform: N ch = = & γ=.9 = & γ=.9 = & γ=.8 = & γ=.8.3.5. 3 4 5 6 7 8 9 3 4 5 The requested file c j.9 The hit probability for Zipf cachig γ =.9 vs. uiform cachig N ch =3 Fig. 3. The hit probability vs. the requested file c j with J = 5 & K = 5 & λu λ b = at differet values of Zipf parameter. Hit robability.8.7.6.5.4.3.. Uiform cachig Zipf cachig N ch = Sigle chael 5 5 5 SINR threshold db Fig.. The hit probability vs. the SINR with J = 5 & K = 5 & λu λ b = ad c j = Fig. 4 illustrates the hit probability versus the cache size. It ca be oticed that the performace of the system improves as the cache size icreases. This highlights the tradeoff betwee the F-RAN performace ad the etwork storage resources. Based o the SINR threshold, the hit probability shows differet treds with the cache size. Such SINR threshold depedet behavior ca be explaied by the fact that at most m SBSs ca have sigal-to-iterferece ratio SIR greater tha m for ay positive iteger m [5]. Thus, there are at most SBSs have SIR db ad oly oe SBS with SIR db. This fact explais the o-liearity of the hit probability at db sceario for all the displayed umber of chaels. O the other had, almost all the treds are liear at the db sceario. The depedecy o the umber of chaels comes from the fact that icreasig the umber of chaels ehaces the chace that typical user is served by multiple SBSs ad this combiatio eds up with the o-liear tred. The weighted hit probability.8.7.6.5.4.3.. The weighted hit probability vs. cache size J=5 & γ =.9 & λ u / λ b = = = Uiform: N ch = Uiform: N ch = = Uiform: N ch = β = db β = db 3 4 5 6 7 8 9 Cache size K Fig. 4. The hit probability vs. the cache size K with β = db & J = 5 & γ =.9 & λu λ b = at differet umber of chaels. VII. CONCLUSION This paper cosiders a opportuistic spectrum access approach for F-RANs. We adopt multi-chael sceario with opportuistic spectrum access to improve the F-RANs s overall

The 7 Iteratioal Workshop o Spatial Stochastic Models for Wireless Networks SpaSWiN performace. We develop a tractable mathematical model for the hit probability, which is reduced to closed form expressio i special cases. Our umerical results both verify the derived aalytical paradigms ad quatify the gais of the cosidered multi-chael model. We ed up by the achieved improvemet of the joit multi-chael cached-eabled F-RAN over the sigle-chael system. AENDIX A: ROOF OF LEMMA I the sceario of o available free chael, the typical user is served over a radomly chose chael by the file caterig SBS. Thus, the umber of iside r iterferig SBSs κ is a radom variable with a maximum value. The MF of κ is give by [κ = a] = a a, a 6 a Give that the umber of the iside r iterferig SBSs κ = a, the LT I i,κ is give by L Ii,κ t = E Ii,κ [e ] ti i κ = a [ [ ] ] = E κ E yi Ψ b,κ,r i <r e t i h ir κ i = a [ ] κ a = E κ E r [E h [e thr r = r ]] κ = a [ [ r b = E κ r= ] ] κ +tr fr r rdr κ = a 7 The equality a is by the fact that the cosists of uiformly distributed odes, so ituitively the coditioal DF of the κ odes o the th ode is equivalet to the κ times the coditioal DF of two cosecutive odes. Equality b from the chael gai expoetial distributio with averagig over the coditioal DF f r r r that is give i 6. Replacig t i the above equatio with βrη ad averagig over the radom variable κ, the ucoditioal LT L Ii ca be obtaied. Fially, by applyig the biomial theorem, Lemma ca easily verified. ACKNOWLEDGMENT This research was fuded by a grat from the office of competitive research fudig OCRF at the Kig Abdullah Uiversity of Sciece ad Techology KAUST. The work was also supported by the Deaship of Scietific Research DSR at Kig Fahd Uiversity of etroleum ad Mierals KFUM, Dhahra, Saudi Arabia, through project umber KAUST-. REFERENCES [] J. G. Adrews, S. Buzzi, W. Choi, S. V. Haly, A. Lozao, A. C. Soog, ad J. C. Zhag, What will 5g be? IEEE Joural o Selected Areas i Commuicatios, vol. 3, o. 6, pp. 65 8, 4. [] J. G. Adrews, T. Bai, M. Kulkari, A. Alkhateeb, A. Gupta, ad R. W. Heath Jr, Modelig ad aalyzig millimeter wave cellular systems, arxiv preprit arxiv:65.483, 6. [3] S. G. Larew, T. A. Thomas, M. Cudak, ad A. Ghosh, Air iterface desig ad ray tracig study for 5g millimeter wave commuicatios, i 3 IEEE Globecom Workshops GC Wkshps. IEEE, 3, pp. 7. [4] 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, 4. [5] R. Tado ad O. 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