Optmal Cachng Placement for DD Asssted Wreless Cachng Networks Jun Rao, ao Feng, Chenchen Yang, Zhyong Chen, and Bn Xa Department of Electronc Engneerng, Shangha Jao Tong Unversty, Shangha, P. R. Chna Cooperatve Medanet Innovaton Center, Shangha, Chna Emal: {sjtu jun, fenghao, zhanchfexang, zhyongchen, bxa}@sjtu.edu.cn arxv:50.07865v [cs.it] 3 Nov 05 Abstract In ths paper, we devse the optmal cachng placement to maxmze the offloadng probablty for a two-ter wreless cachng system, where the helpers and a part of users have cachng ablty. The offloadng comes from the local cachng, DD sharng and the helper transmsson. In partcular, to maxmze the offloadng probablty we reformulate the cachng placement problem for users and helpers nto a dfference of convex (DC) problem whch can be effectvely solved by DC programmng. Moreover, we analyze the two extreme cases where there s only help-ter cachng network and only user-ter. Specfcally, the placement problem for the helper-ter cachng network s reduced to a convex problem, and can be effectvely solved by the classcal water-fllng method. We notce that users and helpers prefer to cache popular contents under low node densty and prefer to cache dfferent contents evenly under hgh node densty. Smulaton results ndcate a great performance gan of the proposed cachng placement over exstng approaches. I. INTRODUCTION As more and more dfferent types of smart devces are produced and appled n people s daly lfe, wreless traffc demand has experenced an unprecedented growth. Csco s most recent report [] forecasts that the moble multmeda data wll grow at a compound annual growth rate of more than 60%. On the other hand, users demand for multmeda contents s hghly redundant,.e., a few popular contents account for a majorty of all requests []. Therefore, cachng popular contents at varous nodes n the network s a promsng approach to allevate the network bottleneck [3]. For the wreless cachng systems where helpers (WF, femtocells) have hgh storage capacty, the performance depends heavly on the adopted cachng replacement. The cachng placement for helpers s frstly nvestgated n [4] to mnmze the downloadng tme, where both uncoded and coded cases are consdered. It s shown that the optmzaton problem for the uncoded case s NP-hard. In addton, [5] consders the channel fadng factor and develops the cachng placement to mnmze the average bt error rate, where the optmal cachng placement s to balance between the channel dversty gan and the cachng dversty gan. Moreover, the problem of optmal MDS-encoded cachng placement at the wreless edge s nvestgated n [6] to mnmze the backhaul rate n heterogeneous networks. owever, all above analyses [4] [6] are based on the fxed topology between users and helpers. In [7], more realstc network models are adopted to characterze the stochastc natures of geographc locaton and the correspondng optmal cachng placements are derved accordng to the total ht probablty. On the other hand, the potental cache capacty at user sde can also be exploted, e.g., local cache offloadng or DD sharng. Varous works have been done on the cachng placement at user sde. In [8], the DD outage-throughput tradeoff problem s nvestgated and the optmal scalng laws are characterzed. [9] analyzes the scalng behavor of the throughput wth the number of devces per cell under Zpf dstrbuted content request probablty wth exponent γ r, and concludes that the optmal cache dstrbuton s also a Zpf dstrbuton wth a dfferent value γ c. By modelng the moble devces as a homogeneous Posson Pont Processes (PPP), [0] derves the optmal cache dstrbuton resultng n the total probablty of content delvery s maxmzed. owever, the local offloadng probablty s not consdered n ther analyss. In addton, coded cachng s also an effectve approach to explot the content dversty []. By cachng contents partally at user sde accordng to the developed cachng dstrbuton durng the frst phase, a coded multcastng opportunty can be created even for dfferent content requests n the second phase. Moreover, [] further proposes the herarchcal coded cachng to address the jont cachng placement problem at both users and helpers. owever, these analyses [] [3] are based on the fxed topology whch s not sutable for the user moblty scenaro. Despte the aforementoned studes, to the best of our knowledge, the optmal cachng placements for both helpers and users under realstc network models reman unsolved to date. Thus n ths paper, we consder a two-ter cachng system, where the helpers and users are spatally dstrbuted accordng to two mutually ndependent homogeneous Posson Pont Processes (PPPs) wth dfferent denstes [4]. In order to allevate the traffc load n the cellular network, we am to develop an optmal cachng placement scheme to maxmze the offloadng probablty, where the offloadng ncludes self-offloadng, DD-offloadng and helper-offloadng. More detals along wth the man contrbutons are as follows: We consder a DD asssted two-ter wreless cachng network consstng of users and helpers where the offloadng comes from self-offloadng, DD-offloadng and helperoffloadng. Dfferent from [0], we take self-offloadng events nto consderaton. Moreover, the practcal assumpton that only a part of users has cachng ablty
s consdered. We formulate the total offloadng probablty of cachng placement n the two-ter wreless networks and adopt the DC programmng to solve the non-convex maxmzaton problem. In addton, we notce that users and helpers ought to cache the popular contents whle the densty s low and ought to cache dfferent contents whle the densty s hgh. And our proposed cachng placement can acheve a balance between them. The two extreme cases for one-ter cachng systems are consdered. In absence of user cachng ablty, we formulate the cachng placement for helper-ter as a convex problem, and can be effectvely solved by the classcal waterfllng method; In absence of helper cachng ablty, the cachng placement for users s also formulated nto a convex problem. Furthermore, we combne the solutons of the two cases as a non-jont optmal cachng placement and compare t wth the proposed placement. II. SYSTEM MODEL AND CONTENT ACCESS PROTOCOL In ths secton, we frst ntroduce the two-tered cachng system as llustrated n Fg., where the helpers and users are spatally dstrbuted accordng to two mutually ndependent homogeneous Posson Pont Processes (PPPs) wth densty λ and λ, respectvely. Then the content access protocol s provded. A. System Model ) Content module: The content lbrary conssts of N contents. The popularty dstrbuton vector of the contents s denoted by q = {q,..., q N }, where q s the access probablty for the -th content. In ths paper, we characterze the popularty dstrbuton as a Zpf dstrbuton wth parameter γ []. If we arrange contents n descendng order of popularty, the popularty of the -th ranked content s [5] / γ q = N, () j= /jγ where γ governs the skewness of the popularty. The popularty s unform over contents for γ = 0, and becomes more skewed as γ grows. For smplcty, we assume all the N contents are of equal sze L. ) Network module: In addton to the macro base statons (BSs), the network module also conssts of the helpers wth cachng ablty, where helpers could successfully send the contents n ts local cache to requestng users wthn radus R at relatvely low cost. For smplcty, we assume the cachng capacty for all helpers are the same, denoted by M L, where M < N. Therefore, the helper can cache up to M dfferent contents entrely. Also we assume a content can only be cached entrely rather than partally. Denote the cachng placement at the helpers for each content as P = [p, p,..., p N ], where p s the proporton of helpers cachng the -th content and 0 p for =,,..., N. The cache storage constrant at the helpers can then be wrtten as N M. The helpers p cachng the -th content also follow a PPP wth densty λ p. (a) (c) (d) (d) (b) (a) elper User (c) (b) Cache Fg.. System model of the DD asssted wreless cachng system, where (a), (b), (c) and (d) stand for Self-offload, DD-offload, elper-offload and Celluar-response, respectvely. 3) User module: We assume part of the users havng cachng ablty. Let α denote the proporton of cache-enabled users, where 0 α. The cache-enabled users also follow a thnnng homogeneous PPP wth densty αλ. For smplcty, we assume the cachng capacty for the cacheenabled users are the same, denoted by M L. Therefore, cache-enabled users can cache up to M dfferent contents entrely n ts local cache. Moreover, a devce to devce (DD) communcaton can be establshed f the dstance between the requestng user and the user cachng the desred content s less than R, where R < R due to the transmttng power. Let P = [p, p,..., p N ] denote the cachng placement at the cache-enabled users for each content, where p s the proporton of users cachng the -th content, and 0 p for =,,..., N. The cache storage constrant at the cache-enabled users can then be wrtten as N p M. Therefore, the users cachng the -th content also follows a PPP wth densty αλ p. B. Content Access Protocol As ndcated n Fg., the content access protocol s as follows: (a) Self-offloadng: When a content request occurs, the user frst checks ts local storage whether the desred content has been stored n t. The request wll be satsfed and offloaded mmedately f the user has cached the desred content n ts local storage space. We term t as Selfoffloadng. (b) DD-offloadng: If the exact content has not been cached n the local storage or the requestng user does not have cache ablty, the user wll turn to search near devces for the desred content. If there s at least one users have stored the desred content wthn the radus R. The request would be met and offloaded by establshng a DD communcaton, termed as DD-offloadng. (c) elper-offloadng: In addton to DD-offloadng, f there s at least one helper have stored the desred content wthn R, the request would be satsfed and offloaded
by the helper transmsson, termed as elper-offloadng. (d) Cellular-response: If the request could not be offloaded va local cache, DD or the helpers then t need to be forwarded to the cellular base staton and the cellular network transmts the requested content n response. III. OFFLOADING PROBABILITY AND PROBLEM FORMULATION In ths paper, n order to allevate the traffc load from the cellular network, our goal s to fnd the optmal cachng placement to maxmze the offloadng probablty. Therefore, we frst analyze the offloadng probablty for the DD asssted wreless cachng network. Then, the optmal cachng placement problem s formulated. A. Offloadng probablty analyss For a PPP dstrbuton wth densty λ, the probablty that there are n devces n the area wthn a radus r s: F(n, r, λ) = (πr λ) n n! e πr λ Therefore, for a reference user located at the orgn, the probablty of at least another user cachng the -th content wthn the transmsson range s P DD,off = F(0, R, αλ p ) = e παλp R. (3) Smlarly, the probablty of at least one helper cachng the -th content wthn the radus R s () P,off = F(0, R, λ p ) = e πλp R. (4) The offloadng probablty for the th content of cacheunabled users,.e the probablty at least one helper or one user cachng the -th content s: P,NC = ( P DD,off )( P,off) = e (παλp R +πλ p R). The correspondng offloadng probablty of the cache-enabled users for the -th content s P,C = p (5) + ( p )P,NC. (6) Therefore, the offloadng probablty for the -th content becomes P,off =αp,c + ( α)p,nc = ( αp )e (παλp R +πλ p R). The total offloadng probablty for the DD asssted wreless cachng system becomes P off = (7) q P,off, (8) whle more data offloaded by the wreless cachng network, less data needs to be sent va the cellular network, allevatng the traffc load for the cellular network. B. Problem Formulaton Let P = [P P ] denotes the cachng placement at helper and user sdes. The optmal cachng placement that maxmzes the offloadng probablty for the wreless cachng network can be formulated as max q P,off (9) P p M s.t. M. (0) p 0 p, {,..., N} 0 p, {,..., N} IV. DC PROGRAMMING FOR CACING PLACEMENT OPTIMIZATION In ths secton, we adopt the dfference of convex (DC) program to solve the above problem. The maxmzaton problem s equvalent to the followng mnmzaton problem: mn P s.t. q P,off () p p M M 0 p, {,..., N} 0 p, {,..., N}. () Let F (P) = N q P,off denote the objectve functon n problem () and t can be easly verfed that the hessan matrx of F (P) s not postve defnte and hence F (P) s non-convex. Let (P) = n q h, where h = απλ R (p + p ). Denote G(P) = F (P) + (P), we then have the followng proposton. Proposton : (P) and G(P) are both convex of P. Proof: Let A denote the hessan matrx of h A = h h (p ) p p h h p p (p ) [ απλ R = 0 0 απλ R ence the matrx A s postve defnte and h s convex. Snce the lnear combnaton of convex functons s also convex, (P) s convex. Smlarly, we have the hessan matrx of G(P) s defnte and G(P) convex of P. ence, F (P) can be wrtten as a dfference of the followng two convex functons: ]. F (P) = G(P) (P). (3) Therefore, we adopt the DC programmng to solve ths problem. DC programmng s a quck convergence programmng whch can obtan a partal optmal soluton and sometmes the global optmal soluton of a non-convex functon [6]. Snce (P) P s contnuous and the constrant of problem () s a
convex set, the DC programmng can be smply descrbed n Algorthm. The result wll be llustrated n secton VI. 6 5 lnq πλ R p Algorthm DC programmng for cachng placement : ntal value: P 0 = M N, P 0 = M N ; : solve the convex optmzaton problem: mn G(P) (P k ) (P P k ) (P k) P s.t.(); 3: the soluton of step s P k+ ; 4: f F (P k ) F (P k+ ) ε or P k P k+ ε,p k s the optmal soluton of F (P); otherwse,return to step ; 5: RETURN:the result s:f (P k ),the soluton s: P k ; p lnq πλr β4 3 Water Level V. EXTREME CASE ANALYSES In ths secton, we consder the cachng problem under extreme cases where only one ter of the cachng system s consdered and the optmal cachng placement can be calculated. We analyze the cachng placement of the two extreme cases and combne the solutons as a baselne. A. α = 0: helper-ter cachng network ) problem formulaton: In ths case, all users have no cachng ablty and we only need to optmze the cachng placement P at helper sde. The offloadng probablty for the -th content s reduced to P,off = P,off = e πλp R (4) Problem () can be wrtten as mn q ( e πλp R ) (5) P p s.t. M, (6) 0 p, {,..., N} Lemma : (Water-fllng method) The optmal cachng placement of the helpers s ( p = mn (β + ln q ) πλ R )+, (7) for =,,..., N, where x + = max (x, 0) and β s effectvely solved by the bsecton search wth N p = M. Proof: The second dervatve of P,off s P,off = π λ Re 4 πλp R p < 0, (8) thus P,off s convex n p and the objectve functon N q P,off s also convex. Therefore, the cachng placement optmzaton problem s a convex problem. Consder the followng Lagrangan L = q ( e πλp R ) + µ( p M ) (9) 0 0 4 6 8 0 4 6 8 0 Content No. Fg.. Optmal cachng placement at helper sde under the settngs N = 0, γ =, M = 4, λ = 0 π500. where µ s the Lagrange multpler. The KKT condton for the optmalty of a cachng placement s L = 0 f 0 < p < p = πλ R q e πλp R +µ 0 f p = 0. 0 f p = (0) Let β = ln (πλr ) ln µ πλ R and x + = max (x, 0), we then have p ( = mn (β + ln q ) πλ R )+,, () where β can be solved by the bsecton search method under the cache storage constrant. As llustrated n Fg., the water-fllng method allocate more cache probablty to contents wth larger popularty. For nstance, the contents wth larger populartes under the water level,.e., the frst content and the second contents have been cached n all helpers. Whle the contents wth smaller popularty above the water level,.e., the 7-th content to the last content, have not been cached n any helper. Remark : Accordng to Lemma, t s straghtforward that the most popular contents are cached n helper storage under relatvely low helper densty.e., p =... = p M = and p M =... = + p N = 0. Whle under relatvely hgh densty, contents are evenly cached at helper sde,.e, p =... = p N = M N. B. λ M = 0: user-ter cachng network In ths case, no helpers wth cachng ablty partcpate n offloadng the user requests, the optmzaton problem s reduced to the optmal cachng placement of p. We thus rewrtten the functon of offloadng probablty as: P,off = ( αp )e παλp R. ()
TABLE I DEFAULT PARAMETER SETTING Parameters values DD communcaton range:r 5(m) helper transmsson range:r 00 (m) the proporton of cache-enabled users:α the densty of users:λ 5000/π500 the densty of helpers:λ 50/π500 the cache capacty of users and helpers M = ; M = 8 the sze of content lbrary:n 30 the skewness of the popularty:γ TABLE II DIFFERENT CACING SCEMES Schemes cachng schemes of users cachng schemes of helpers popular cache p =, [, M ] p =, [, M ] p j = 0, j [M +, N] p j = 0, j [M +, N] even cache P = M /N P = M /N Non-jont the soluton of Problem (3) the soluton of problem (5) Offloadng Probablty Non jont 0 3 4 elper Densty λ x 0 4 Fg. 3. The mpact of λ on the offloadng probablty Then problem () becomes mn P s.t. p q ( ( αp )e παλp R ) (3) M 0 p, {,..., N}, (4) Proposton : The above problem s also a convex problem. Proof: The second dervatve of the objectve functon becomes P,off = [αb + b ( αp p )]e bp < 0, (5) where b = παλ R. Therefore, P,off s convex n and the objectve functon N q P,off s also convex. p Therefore, the cachng placement optmzaton problem s convex. As a result, we can adopt a nter-pont method to acheve the optmal soluton [7]. VI. SIMULATIONS In ths secton, we provde some numercal results to verfy our analyss and compare the performance of the proposed cachng placement wth other baselnes. Parameter settng and the three baselnes are descrbed n Table I and Table II. In partcular, we combne the optmal solutons of the two oneter cachng cases as a baselne and named t non-jont optmal cachng placement. Fg. 3 shows that the offloadng probablty ncreases wth helper densty λ. The performance of the proposed cachng placement s better than other three baselnes no matter how λ changes. When λ = 0, the performance of the proposed cachng placement s equal to the non-jont one, because n ths stuaton there are no helpers jonng to offload traffc data. Offloadng Probablty Non jont 0 0.0 0.0 0.03 0.04 0.05 User Densty λ 0.06 0.07 Fg. 4. The mpact of λ on the offloadng probablty Wth the ncreasng of λ, the performance of the proposed cachng placement becomes better than the non-jont one. Whle λ s consderable large, non-jont cachng placement approaches to the proposed scheme agan. That s because the cachng placement of non-jont schemes s also a optmal soluton when there s only helper-ter, and the offloadng s mostly conssted of helper-ter n ths stuaton. From Fg. 4 we can draw a smlar concluson about λ. Furthermore, from Fg. 3 and Fg. 4, we can see that when both of λ and λ are small, the performance of even cache scheme s the worst one. As λ or λ ncreases, the performance of even cache scheme becomes better. When λ = 0 4 and λ =. 0, t exceeds over the popular cache scheme. That s because whle there are few devces partcpatng n the cachng network, users and helpers need to cache popular contents to cope wth the correspondng hgh request probablty, thus the popular cache scheme performs well; When the resource of the cachng network s rch.e the node densty s relatvely hgh, the offloadng probabltes for the most popular contents are easly satsfed, and the surplus
Cachng Probablty p p p 3 p 4 p 5 p p 0.3 p 3 0. p 4 0. p 5 0 0.0 0.0 0.03 0.04 0.05 0.06 0.07 0.08 User Densty λ Offloadng Probablty Non jont 0...4.6.8 Zpf Parameter γ Fg. 5. the cachng placement of the proposed placement Fg. 7. The mpact of γ on the offloadng probablty Offloadng Probablty 5 5 5 5 Non jont Offloadng Probablty Non jont 0 0. Proporton of Cache enable Users α Fg. 6. The mpact of α on the offloadng probablty storage can be used to cache other relatvely less popular contents. So the even cache scheme performs better and the offloadng probablty of popular cache scheme no longer ncreases. When λ s consderable large, the performance of even cache scheme approaches to the optmal cachng placement. In Fg. 5, we demonstrate the proposed cachng placement whch s calculated by DC programmng where N = 5, M =, M = 3. As λ ncreases, we can see that the optmal cachng placement changes from a popular cache scheme to a even cache scheme whch s consstent wth our analyss. Fg. 6 llustrates the mpact of α on offloadng probablty, where α stands for the proporton of cache-enabled users. When α = 0, the system reduces to a helper-ter cachng system hence, the offloadng probablty of the proposed cachng placement s equal to the non-jont one. Whle α s larger, there are more cache-enabled users jonng n the cachng system and therefore the offloadng probablty wll ncrease. And we can see that whle α 0, the performance of the proposed placement s clearly better than the non-jont one. 0 0 30 40 50 60 Content Lbrary Sze N Fg. 8. The mpact of N on the offloadng probablty In ths paper, γ s denoted as the skewness of content popularty. Whle γ s large, the user requests focus on the popular contents and the cachng system may have large probabltes to cache the rght contents. Therefore the offloadng probablty usually ncreases wth γ and we show t n Fg. 7. The performance of popular cache scheme grows rapdly wth ncreasng γ whle the performance of even cache scheme s not affected by γ, because t caches every content wth a same probablty. Fg. 8 llustrates that the offloadng probablty decreases wth N. To expand the sze of content lbrary N, n a sense, s smlar to reduce the cache capacty M, thus the offloadng probablty wll experence a declne accordngly. owever we can notce that the performance of our proposed cachng placement s stll well. It demonstrates that when the system s appled nto a mult-contents stuaton, the proposed cachng placement can fnely adjust the cachng proporton of each content by a jont optmzaton and keep a good performance.
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