Lev B. Sofman Frontline Communications 2221 Lakeside Blvd Richardson, TX, USA

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1 Lev B. Sfman Frntlne Cmmuncatns 2221 Lakesde Blvd chardsn, TX, USA Cachng s mprtant tl fr reducng Internet access latency, netwrk traffc and cst n cntent delvery netwrks (CD). In case f herarchcal netwrks wth several levels f cachng (e.g., IPTV netwrk), the prblem s where and hw much cache memry shuld be allcated n rder t acheve maxmum cst effectveness. In ths paper, we cntnue study f herarchcal cache ptmzatn and present a mdel f herarchcal cache ptmzatn fr CD wth tw levels f herarchy and unbalanced traffc. Ths mdel depends n several basc parameters: traffc thrughput fr each lw level nde, cache effectveness as a functn f memry sze, and cst parameters. Sme reasnable assumptns abut netwrk cst structure and cache effectveness functn allw us t btan an analytcally ptmal slutn f the prblem. We then analyze the factrs that mpact ths slutn. : cache, traffc, netwrk, cst, ptmzatn. Many CD netwrks have a herarchcal structure. Fr example, n IPTV netwrk n metr area, at the tp f the herarchy s a Vde Hub ffce (VHO) where all vde cntents are stred n vde servers. VHO s cnnected t several Intermedate Offces (IO), every IO s cnnected t several Central Offces (), every s cnnected t several Access des (A), e.g. Dgtal Subscrber Lne Access Multplexers (DSLAMs), and every A/DSLAM s cnnected t a number f subscrbers. In an IPTV netwrk, Vde n Demand and ther vde servces generate large amunt f uncast traffc frm VHO t subscrbers and, therefre, requre addtnal BW/equpment resurce n the netwrk. T reduce ths traffc (and verall netwrk cst), part f vde cntent (mst ppular ttles) may be stred n caches clser t subscrbers, e.g., n DSLAM, n servce swtches at s, r n servce ruters at IO (Fg.1). Fgure 1: Herarchcal cachng n IPTV netwrk. There s a tradeff between cst savngs fr capacty n transmssn lnks and csts fr the caches (Krgfss, Sfman, and Agrawal 28). On the ne hand, cache memry has a cst, but n anther hand, cachng f the cntent allws decreasng amunt f traffc frm upstream levels f ths herarchy and,

2 Sfman therefre decreasng f netwrk cst. Overall, t fnd an ptmal (n terms f netwrk cst) lcatn and amunt f cache memry s a cmplex ptmzatn prblem. In (Krgfss, Sfman, and Agrawal 28; Krgfss, Sfman, and Agrawal 29; Sfman and Krgfss 29) we assumed that netwrk tplgy s a symmetrcal tree and each lw level nde has the same amunt f traffc (balanced traffc case). In ths paper, we cnsder CD netwrk wth unbalanced traffc. We slve the ptmzatn prblem fr ths mdel usng analytcal methds. In the next Sectn, we revew prevus results n cachng fr multmeda servces. In Sectn 3, we ntrduce cncepts f cache effectveness functn and ncremental (L, U)-cache that wll be used n the study. In Sectn 4, we descrbe assumptns abut cst parameters and cache effectveness functn that are used n ur analytcal mdel. tatns and mathematcal frmulatn f the prblem are presented n Sectn 5. We then descrbe methdlgy f slutn and prve sme mprtant relatnshp between ptmal cache parameters n Sectn 6. Senstvty study n Sectn 7 nvestgates an mpact f cache cst and the shape f cache effectveness functn n ptmal cache sze n every level f netwrk. Sectn 8 summarzes results and cntans sme cncludng remarks regardng mprtance f herarchcal cachng fr netwrk cst ptmzatn. ELATED WOK Cntent cachng s ne f the mst effectve slutns t mprve perfrmance and cst effectveness f multmeda servces. evews f cachng technques fr multmeda servces can be fund n a number f publcatns (Ghandeharzadeh and Shayandeh 27; Ghandeharzadeh et al. 26; Chen et al. 23; Cbarzan and Bszrmeny 27; Lee and Park 25). In partcular, varus cachng technques fr dfferent ht rat metrcs were dscussed n (Ghandeharzadeh and Shayandeh 27). In (Krgfss, Sfman, and Agrawal 28; Krgfss, Sfman, and Agrawal 29), a herarchcal lgcal tplgy cmpsed f dfferent levels (e.g. A/DSLAM,, IO, and VHO)) was cnsdered, and netwrk cst ptmzatn was perfrmed by chsng the best level where t put each cntent. One apprach t slve ths prblem was presented n mult-level mult-servce cache ptmzatn mdel and mplemented n nternal ptmzatn tl. In ths mdel, netwrk dmensnng was perfrmed n very detal level: fr a gven tplgy (number f As/DSLAMs, s and IOs), traffc thrughput (vde servce characterstcs), cnfguratn and cst parameters, equpment dmensnng n every level f herarchy was dne and ptmal cache archtecture was fund usng sme heurstc algrthm. In anther apprach, a smplfed analytcal mdel fr cache ptmzatn was presented. The ttal netwrk cst n ths mdel was the sum f transprt cst and memry cst, where the transprt cst n every lnk was prprtnal t the rate f traffc that traverses the lnk and the memry cst was a lnear functn f the memry sze. In ths mdel, a mnmum f the ttal netwrk cst assumed t be at a stable pnt (.e., where dervatve f netwrk cst s equal t zer). Ths mdel depends n basc parameters: traffc, tplgy, cache memry lmtatns, and cst parameters, e.g. memry cst per GB and unt netwrk cst per unt f traffc n each level f herarchy. Sme reasnable assumptns allwed us recevng analytcally ptmal slutn f the prblem and analyze the factrs that mpact ths slutn. In (Sfman and Krgfss 29), we refned the mdel (Krgfss, Sfman, and Agrawal 28; Krgfss, Sfman, and Agrawal 29), by cnsderng s-called bundary cases (.e., cases wthut cache r wth maxmum cache sze) at each level f herarchy. etwrk cst mnmum assumed t be at a stable pnt r at a bundary pnt. As we demnstrated n (Sfman and Krgfss 29), bundary cases are mprtant fr understandng why n sme cases ptmal slutn des nt requre cachng at sme levels, r requres usng maxmum allwed cache sze n anther levels. In the same tme, bundary cases ntrduce sgnfcant cmplexty t the prblem because they generate a large number f cases t be cnsdered. In partcular, t was demnstrated that ptmal cache archtecture may use cachng at any cmbnatn f netwrk herarchy DSLAM, and IO dependng n traffc, tplgy, and cst parameters, and therefre herarchcal cachng s an mprtant ptn fr cst savng. In Sfman, Krgfss, and Agrawal (28), the cncept f cntent cacheablty was ntrduced and a fast algrthm that uses cacheablty t parttn ptmally a cache between several vde servces wth dfferent

3 Sfman traffc characterstcs and cntent szes was presented. The gal f the ptmzatn was t serve maxmum (n terms f bandwdth) amunt f subscrbers requests subject t cnstrants n cache memry and thrughput. Cachng algrthm s an mprtant factr defnng ht rat and cache effectveness. T make effectve use f cachng, a decsn has t be made as t whch dcuments shuld be evcted frm the cache n case f cache saturatn. Overvew f varus cachng algrthms was presented n Balamash and Krunz (24). Least ecently Used (LU), Least Frequently Used (LFU), Frst n Frst ut (FIFO) algrthms, varus mdfcatns and cmbnatns f thse algrthms are typcally cnsdered. Cache lgc cnsstng f a mdfed LU cachng algrthm cmbned wth ne f three cnsdered cache cllabratn strateges (herarchcal, brrwng, and federated) were studed n Vleeschauwer and bnsn (211). ASHE EFFECTIVEESS FUCTIO AD ICEMETALLUACHE When we cnsder traffc n CD t s mprtant t dstngush between cacheable and nn-cacheable cntent. Cacheable cntent s statc nfrmatn that des nt change very ften and can be cached. ncacheable cntent s dynamc nfrmatn that changes regularly r fr each user request and serves n purpse f t were cached. Fr example, lnear TV r web pages that return the results f a search are nncacheable, because ther cntents are unque almst all the tme. We can defne a prtn f all traffc that crrespnds t cacheable cntents as P cache. Cache effectveness s defned as the rat f traffc (n terms f bt rate at busy hur) that s served frm the cache t the ttal amunt f requested traffc. Cache effectveness s clsely related t the cncept f ht rat whch s defne as a rat f number f tems served frm the cache t the ttal number f requested tems. Cache effectveness depends n sze and thrughput f the cache, statstcal characterstcs f the traffc (e.g., ppularty dstrbutn f dfferent ttles, ther bt rate, sze n memry) and n cachng algrthm. Under ther equal cndtns (the same statstcal characterstcs f the traffc and the same cachng algrthm) cache effectveness H(m) s a functn f memry sze m. Cache effectveness functn ncreases wth m; H() =, and H(m) tends t P cache when m ncreases (Fg. 2). Cache Effectveness as a Functn f Memry Sze P cache H(m 1 +m 2 ) H(m 1 ) m 1 m 1 + m 2 Memry sze m Fgure 2: Cache Effectveness functn. te that because f effect f dmnshng return, ncrease f memry sze after sme pnt makes nsgnfcant mpact n traffc and equpment cst reductn, and a ttal netwrk cst (cache cst + equpment cst) wll start t grw almst lnearly wth memry sze. Therefre, there shuld be an ptmal memry sze where ttal netwrk cst attans ts mnmum. Subs DSLAM m 1 m 2 T T(1-H(m 1 )) T(1-H(m 1 +m 2 )) Fgure 3: Cumulatve effect f herarchcal cachng. In the herarchcal CD, we can use a cncept f vrtual cache, ntrduced n Vleeschauwer and bnsn (211). The vrtual cache n any nde f the tree s the actual cache at the nde augmented by the cache

4 Sfman szes n a nde deeper (twards the leaves) n the tree, s called cumulatve effect. A cntent resdng n the vrtual cache f the nde (.e., n the nde tself r n a nde further dwn the tree) reduces the traffc n the feeder lnk twards that nde (and n all lnks twards the rt f the tree). Fg. 2 and 3 llustrate a cumulatve effect f herarchcal cachng. If we allcate cache f sze m 1 at DSLAM and cache f sze m 2 at, part f the traffc wll be served frm the caches and amunt f traffc between and DSLAM wll be reduced by T x H(m 1) cmparng wth rgnal traffc demand T frm subscrbers. Vrtual cache at ncludes an actual cache f sze m 2 at augmented by cache f sze m 1 at DSLAM. Upstream frm DSLAM, traffc reductn wll be equal t TH(m 1); upstream frm, traffc reductn wll be equal t TH(m 1 + m 2). Incremental traffc reductn at s equal t T(H(m 1 + m 2H(m 1)) and ncremental cache effectveness at s equal t H(m 1 + m 2H(m 1). In ths example, cache at has tw memry characterstcs: lwer bund L = m 1 and upper bund U = m 1 + m 2, and we wll call t ncremental (L, U)-cache, r smply (L, U)-cache. te that ncremental cache effectveness depends nt nly n memry sze m 2 = U L, but n lw and upper bund characterstcs L and U. Fr the same memry sze = U L, ncremental cache effectveness, H(U) H(L), decreases wth ncrease f L (effect f dmnshng return). In the fllwng sectns, the cncept f (L, U)-cache wll be used n ur mdel f herarchcal cache ptmzatn wth unbalanced traffc t calculate traffc reductn n hgher level f herarchy. T vsualze (L, U)-cache the fllwng smplfed example can be used (Fg. 4). When cache algrthm receves a request fr cntent frm a subscrber t makes a cache mantenance decsn: t add a new cntent t the cache, t evct ld cntent frm the cache (f necessary), etc. Assume, that n ur example the cache algrthm shuld mantan nt ne but tw caches, f szes L and U, L < U. Thse tems that wll be cached n the cache f sze U but nt n cache f sze L shuld belng t ncremental (L, U)-cache. In smple terms, thse tems that are ppular enugh t be cached n cache U f larger sze, but nt ppular enugh t be cached n cache L f smaller sze belng t (L, U)-cache. Fgure 5: Herarchcal CD wth unbalanced traffc. Fgure 4: Explanatn f (L, U)-cache. SSUMPTIOS In ths study, we cnsder a CD wth tw levels f herarchy: several central ffces () n the lw level are dual-hmed t a cuple f ruters n the hgh level (Fg. 5). Each has ts wn amunt f traffc demand (T 1, T 2, ). Every lnk between and the ruter s dmensned fr the whle amunt f traffc

5 Sfman (prtected mde) r fr the half f the traffc (unprtected mde). As n (Krgfss, Sfman, and Agrawal 28), we assume that equpment cst s prprtnal t amunt f traffc that traverses the equpment. Mre specfcally, we estmate the equpment cst based n amunt f traffc receved by ths equpment frm upstream levels f netwrk herarchy (upstream traffc) and n amunt f traffc send by ths equpment dwnstream (dwnstream traffc). In rder t calculate a ttal equpment cst, we defne unt cst C f netwrk fr the traffc between subscrbers and level, unt cst C - fr the traffc between and ruter levels, and unt cst C fr the traffc upstream frm the ruter level. There s a cache at each as well as n ruter level; we assume that cst f the cache s lnear functn f ts sze (see mre detals n the next sectn). Ttal netwrk cst ncludes equpment and cache cst. Our gal s t fnd ptmal szes f caches at each and at the ruter level that mnmzes the netwrk cst. We assume that all s have the same demgraphcal prfle n terms f traffc dstrbutn and cntent ppularty; n partcular, all s have the same cache effectveness functn H (m). Usually the sze f cache memry ccuped by ne ttle s small cmpared wth the sze f the whle cache, s we can neglect a granularty factr and assume that functn () s smth,.e. has a cntnuus dervatve (m) and ths dervatve decreases (effect f dmnshng return). Ths cndtn mples that the functn H (m) s strctly cncave see Fg. 6. Functn H(m) n sme deal case (when cache algrthm has cmplete nfrmatn abut statstcal characterstcs f the traffc) s equvalent t ppularty curve, whch, ndeed, decreases as a functn f the ttle s rank..7.6 Cache Effectveness H(m) H(m) H'(m). Cache Memry m Fgure 6: Cache effectveness functn and ts dervatve. OTATIO AD MATHEMATICAL FOMULATIO a ttal number f s n the netwrk T traffc demand fr -th (Gbs), = 1,, P prtectn factr: P = 2 n case f prtected mde n - lnk, therwse, P = 1 M maxmum cache sze fr -th (GB), = 1,, ptnal parameter Trp maxmum cache thrughput fr -th (Gbs), = 1,, ptnal parameter M maxmum cache sze fr ruter (GB) ptnal parameter Cst parameters: C netwrk cst per unt f traffc n the lnk dwnstream frm ($/Gbs) C - netwrk cst per unt f traffc n the lnk between and ruter ($/Gbs) C netwrk cst per unt f traffc n the lnk upstream frm ruter ($/Gbs) CS memry cst per unt f strage at level ($/GB) CF fx memry cst at level ($) CS memry cst per unt f strage at the ruter level ($/GB) CF fx memry cst at the ruter level ($) H(m) cache effectveness functn

6 Sfman Decsn varables: m cache memry sze fr -th (GB), = 1,, L and U lw and upper level f ncremental (L, U)-cache at the ruter level (GB) A ttal netwrk cst, twkcst, may be calculated as twkcst( m, m PC where: 1 1 2,..., m, L, U ) C 1 T (1 H ( m )) C([ L, U ]) C T 1 1 C( m ) T (1 H ( m,[ L, U ])) (1) C(m ) s a cst f cache m at -th : C(m ) = CF + CS m,fm >, C(m ) =, f m =, = 1,, C([L,U]) s a cst f cache at the ruter (n the fllwng, we assume that L U): C ([L,U]) = CF + CS (U L), f L < U; C ([L,U]) =, f L = U H(m, [L,U]) s a factr f traffc T reductn upstream frm the ruter level, t can be calculated as H ( m,[ L, U ]) H ( m ) H ( U ) H ( L), f m H ( U ), f H ( m ), L m U f U m L (2) The ratnale fr such calculatn f the functn H (m, [L, U]) s llustrated n Fg. 7. If, fr example, m < L (Fg.7 (a)), traffc reductn n the ruter level ncludes tw cmpnents: traffc reductn H(m ) caused by cache at, and traffc reductn H(U) H(L) caused by cache at the ruter. In tw ther cases presented n Fg.7 (b) and Fg.7 (c), traffc reductn n the ruter level s defned by vrtual cache f sze U and m, crrespndngly. H(m) H(m) H(m) m L U L m U L U m (a) (b) (c) Fgure 7: Calculatn f traffc reductn upstream frm the ruter level. The gal s t mnmze ttal netwrk cst subject t cnstrants n cache memry sze: twkcst m, m,..., m, L, U ) mn (3) T H ( m ) Trp and L U ( 1 2 such that m M,, 1,2,..., (4)

7 Sfman Cnstrant ptmzatn prblem (3-4) has the fllwng cnstrant factrs: maxmum sze f memry M, maxmum thrughput Trp, parameters L and U. Fr fxed values f L and U, netwrk cst (1) s a functn f varables m m,..., m ) n a plyhedral defned by equatn (4). ( 1 ETHODOLOGY OF SOLUTIO Ttal netwrk cst (1) can be parttned nt several cmpnents: twkcst ( m1, m2,..., m, L, U ) C( m, L, U ) C([ L, U ]) where 1 (5) C( m, L, U ) C T C( m ) PC T (1 H ( m )) C T (1 H ( m,[ L, U ])) (6) s a cst cmpnent crrespndng t -th ( = 1, 2,, ) and C([L,U]) s a cst f cache at the ruter. te that each -th cst cmpnent (6) depends nly n ne memry parameter m, memry sze n the -th. Therefre, n rder t mnmze (5) fr a fxed values f L and U we can mnmze each cmpnent (6) ~ mn{, 1 M H by m. Let ~ m be an ptmal value,.e. C m, L, U ) mn{ C( m, L, U ) : m M }, where ( M ( Trp / T )}. We wll use the fllwng result frm calculus. Let f(x) be a lwer sem-cntnuus functn n an nterval [a, b], and has dervatve f '( x) n (a, b). Then ne f the fllwng shuld take place (Fg. 8): (a) f(x) attans ts mnmum at x = a (b) f(x) attans ts mnmum at x = x mn, a < x mn < b, and f '( xmn) (.e. x mn s a stable pnt) (c) f(x) attans ts mnmum at x = b y=f(x) y=f(x) y=f(x) a b x a x mn b x a b x Fgure 8: Mnmum f functn n segment. Applyng ths therem t functns C (m, L, U) (15) we can deduce that fr every = 1, 2,, the ptmal value f m s ether - equal t ne f the bundary values, r M ~, r st - ne f the stable pnt(s) m whse value can be estmated frm equatn: ( C( m, L, U ) m Equatns (2) and (7) allw us estmatng a dervatve f functn H at a stable pnt: ' H ( m ) CS /( PC T ), f L m U ' H ( m ) CS /(( PC C ) T ), f m L r U m (7) (8) (9)

8 Sfman Frm (8) and (9), we can calculate crrespndng values f 1 st and 2 nd st stable pnts, m 1 and st 2 m. st1 st 2 Dependng n relatnshp between m, m, L and U, t can be ne, tw r n stable pnts at all. As a result, usng analytcal apprach, fr a gven values f L and U, we can fnd ptmal values ( m1, m2,..., m ) that mnmze the netwrk cst (3) subject t cnstrants (4). If ( m, m,..., m, L, ) s an ptmal slutn f the prblem (3-4) that uses ncremental cache,.e. 1 2 U L U, then cncavty f functn H mples the fllwng tw statements regardng parameters U : 1. L shuld be equal t ne f the ptmal values fr memres m, 1,2,...,. 2. U shuld be equal t ne f the fllwng values: - ptmal values fr memres m, 1,2,..,, - stable pnts,.e. such values f U that twkcst( m1, m2,..., m, L, U ) U ~ - bundary values, r U mn( U, L M ), where U ~ s defned frm equatn: max ' ~ H ( U ) CS /( C 1 T ) L and Usng these bservatns, we can defne a fnte set f (L, U) pars (canddate set) amng whch the ptmal ( L, U ) par can be fund. We can calculate the ptmal slutn fr each feasble (L, U) par frm the canddate set and then select the best slutn ver all such pars. ESITIVITY STUDY As a reference case, we cnsder a netwrk wth the fllwng parameters: = 4 ttal number f s n the netwrk wth the fllwng traffc demands: T 1 = 5 Gbs; T 2 = 1 Gbs; T 3 = 2 Gbs; T 4 = 3 Gbs. P = 1,.e. unprtected - lnk. C - = $6.5/Mbs unt netwrk cst n - lnk. C = $4/Mbs unt netwrk cst n the lnk upstream frm ruter. CS = CS = $22/GB cst f memry per unt f strage at and at the ruter levels. CF = CF = $5, fx memry cst at and at the ruter levels. M = 5, GB maxmum memry sze per and per ruter. Cache effectveness functn H (m) was mdeled n a such a way that ts dervatve, H (m), represents a cntnuus versn f Zpf-Mandelbrt dstrbutn K H ( m), ( m q) truncated n nterval [, M ZM ], where < M ZM -Mandelbrt pwer parameter > characterzes cache effectveness steepness at, shft parameter q >, and K s nrmalzatn ceffcent: M ZM H ( m) dm Fr the reference case, we chse q = 1, M ZM = 1,,, P cache = 1 and varated ZM pwer parameter. Larger crrespnd t mre steep cache effectveness curve and hgher cntent reuse. P cache In the fllwng, we cnsder tw cases: (a) Uncnstraned case, when there s n lmt n cache thrughput and

9 Sfman (b) Cnstraned case, when cache thrughput shuld nt exceed sme threshld. Frst, we nvestgate a senstvty f cache parameters,.e. ptmal cache szes as well as ptmal parameters (L, U) f ncremental cache at the ruter, t a ZM pwer parameter (Fg. 9). We can make the fllwng bservatns: 1. cache sze became small when pwer parameter s very small r very large. Indeed, f pwer parameter s very small, cache effectveness functn becmes very flat, amunt f traffc fflad becmes smaller fr the same memry sze,.e. cachng becmes less effectve because t cannt justfy the cst f the memry. If, n the ther hand, pwer parameter s very large, cache effectveness functn becmes very steep, amunt f traffc fflad becmes larger fr the same memry sze,.e. cachng becmes very effectve and ptmal pnt wll be btaned at a smaller memry sze. 2. Fr uncnstraned case, the mre traffc per the mre ptmal sze f cache at. E.g., n ur reference scenar, traffct1 T2 T3 T4, and crrespndng ptmal cache szes m 1 at -1, m 2 at -2, m 3 at -3 and m 4 at -4 fllw the same pattern ( m1 m2 m3 m4 ) fr each pwer parameter. Indeed, n uncnstraned case, the ptmal value f cache sze s defned by stable pnts f (8) and (9), and the values f these stable pnts ncrease wth ncrease f traffc. Fr cnstraned case, hwever, ths rule des nt hld Cache sze parameters as a functn f ZM pwer (uncnstrant thrughput f cache at ) Cache sze parameters as a functn f ZM pwer 12, 12, 1, 1, Cache sze parameters (GB) 8, 6, 4, L (GB) U (GB) pt. cache at _1 pt. cache at _2 pt. cache at _3 pt. cache at _4 Cache sze parameters (GB) 8, 6, 4, L (GB) U (GB) pt. cache at _1 pt. cache at _2 pt. cache at _3 pt. cache at _4 2, 2, ZM pwer ZM pwer Fgure 9: Senstvty f cache parameters t ZM pwer. any mre as sn as traffc thrughput became a lmtng factr fr a partcular ( when > 1.1). m4 m1 3. There s a range f values f a pwer parameter when cache at the ruter s utlzed fr the ptmal slutn; utsde f ths range we d nt use cache at the ruter,.e. U = L =. The sze f ths range depends n whether we have uncnstraned r cnstraned case. ext, we nvestgate a senstvty f cache parameters t the cst f memry at (Fg. 1). In all fllwng scenars we assume that pwer parameter =.9. The fllwng bservatn can be made: Cache sze parameters as a functn f memry cst (uncnstrant thrughput f cache at ) Cache sze parameters as a functn f memry cst 12, 7, 1, 6, Cache sze parameters (GB) 8, 6, 4, L (GB) U (GB) pt. cache at _1 pt. cache at _2 pt. cache at _3 pt. cache at _4 Cache sze parameters (GB) 5, 4, 3, 2, L (GB) U (GB) pt. cache at _1 pt. cache at _2 pt. cache at _3 pt. cache at _4 2, 1, Cst per unt f strage at ($/GB) Cst per unt f strage at ($/GB) Fgure 1: Senstvty f cache parameters t memry cst at. 1. Wth ncrease f memry cst, ptmal memry sze at s cnstant r decreases.

10 Sfman 2. Fr uncnstraned case, the mre traffc per the mre ptmal sze f cache at, untl cache sze became equal t a maxmum allwed memry amunt per (5, GB n ur case). 3. When memry cst s small (belw sme threshld), ptmal memry sze s equal t a maxmum allwed memry amunt per (5, GB n ur case). Wth ncrease f memry cst, ptmal memry per begn t decrease. The threshld depends n and ncreases wth traffc. 4. Optmal parameters ( L, U ) f the cache at the ruter fllw the bservatns made n Sectn 5: 4.1. L s equal t ne f the ptmal values fr memres, m, 1,2,3, 4 as cst f memry ncreases, L jumps frm m 4 t m3 t m2. ; n ths partcular scenar, 4.2. When memry cst s belw sme threshld (n ur case, belw ~$32/GB), U Umax. When memry cst s abve the threshld, U start decreasng, but a ttal memry at the ruter, U L, cntnue t be equal t a maxmum allwed memry at the ruter (5, GB). 5. Fr a cnstrant case, traffc thrughput became a lmtng factr fr -4, that s why ptmal cache sze fr -4, m 4, s fxed and fr sme values f memry cst s less than ptmal cache sze fr -1, -2 and Parameter L f the cache at the ruter fllws the bservatns made n Sectn 5: L s equal t ne f the ptmal values fr memres m, 1,2,3, 4 ; n ths partcular scenar, as cst f memry ncreases, L jumps frm m 4 t m A ttal memry at the ruter, U L, s equal t a maxmum allwed memry at the ruter (5, GB). ext, we nvestgate a senstvty f cache parameters t the cst f memry at the ruter (Fg. 11). The fllwng bservatn can be made: Cache sze parameters as a functn f memry cst (uncnstrant thrughput f cache at ) Cache sze parameters as a functn f memry cst 12, 7, 1, 6, Cache sze parameters (GB) 8, 6, 4, L (GB) U (GB) pt. cache at _1 pt. cache at _2 pt. cache at _3 pt. cache at _4 Cache sze parameters (GB) 5, 4, 3, 2, L (GB) U (GB) pt. cache at _1 pt. cache at _2 pt. cache at _3 pt. cache at _4 2, 1, Cst per unt f strage at ruter ($/GB) Cst per unt f strage at ruter ($/GB) Fgure 11: Senstvty f cache parameters t memry cst at the ruter. 1. Wth ncrease f ruter memry cst, ptmal memry sze at the ruter, U L, s cnstant r decreases. 2. Fr uncnstraned case, cache at the ruter s utlzed fr ptmal slutn f memry cst at the ruter s belw sme threshld (~$32/GB). Fr ths range f memry cst, parameter L f the cache at the ruter

11 Sfman fllws the bservatns made n Sectn 5: L s equal t the ptmal values f memry at -3 and -4: L m3 m4 3. If memry cst at the ruter s belw sme threshld (~$23/GB), an ptmal memry at the ruter, U L, s equal t a maxmum allwed memry (5, GB). Wth further ncrease f memry cst at the ruter, U (and sze f the memry at the ruter) decreases t zer. 4. Optmal cache sze per n ths scenar des nt depend n cst f memry at the ruter. Optmal sze f cache at -3 and -4 are lmted by maxmum allwed cache sze at : m 3 m 4 5, GB. As fr ptmal cache at -1 and -2, the rule abut drect dependency f cache sze and traffc hlds: the mre traffc per the mre ptmal sze f cache at,.e. m3 m4 m2 m1, where m2 and m1 crrespnds t stable pnts (8) r (9). 5. Fr cnstraned case, traffc thrughput became a lmtng factr fr -4, that s why ptmal cache sze fr -4, m 4, s fxed and s less than ptmal cache sze fr -1, -2 and (fr sme values f cst f memry at the ruter) fr Parameter L f the cache at the ruter fllws the bservatns made n Sectn 5: f memry cst at the ruter s belw sme threshld (~$6/GB), L s equal t the ptmal values fr -4 memry: L m4. Wth further ncrease f memry cst at the ruter, L decreases t zer. 7. If memry cst at the ruter s belw sme threshld (~$34/GB), an ptmal memry at the ruter, U L, s equal t a maxmum allwed memry (5, GB). Wth further ncrease f memry cst at the ruter, U (and sze f the memry at the ruter) decreases t zer. 8. As memry cst at the ruter ncreases, at sme mment each f ptmal cache at -1, -2 and -3 make a jump t the hgher level, frm the 1 st stable pnt (8) t the 2 nd stable pnt (9); ths jump s crrelated wth sharp decrease f U. Fr example, ptmal cache at -3, m 3, s equal t ~4,67 GB fr the memry cst belw ~$34/GB and m3 s equal t 5, GB fr memry cst abve $34/GB. At the same tme, U decreases frm ~5,991 GB (at $34/GB) t ~3,948 GB (at $35/GB). OCLUSIO In ths paper we presented an analytcal mdel fr herarchcal cache ptmzatn n CD wth tw levels f herarchy, s n lw level and ruters n hgh level. We assume that all s have dfferent traffc demands (unbalanced traffc). The mdel uses several key parameters number f s, traffc per each, cache effectveness functn, unt memry and unt netwrk cst t calculate an ptmal cache sze n each and at the ruter. A cncepts f cache effectveness functn and ncremental (L, U)-cache at the ruter have been ntrduced. We presented analytcal slutn fr ptmal cache parameters and nvestgated relatnshp between these parameters. In partcular, we demnstrated that fr ptmal netwrk cnfguratn f caches at and ruter levels: 1. Optmal sze f cache at each may be calculated analytcally fr a gven (L, U) par. 2. A lwer bund ncremental cache characterstc L shuld be equal t ne f the ptmal cache szes at. 3. There s a fnte number f (L, U) pars that are canddates fr ptmal slutn f the prblem. We can calculate ptmal slutn (as descrbed n 1) fr each (L, U) par frm canddate set and then select the best slutn ver all such pars.

12 Sfman Senstvty studes allwed estmatng an mpact f cache effectveness functn and unt cst f cache n ptmal cache cnfguratn and demnstrated that fr many typcal scenars ptmal cache cnfguratn ncludes cache frm ne () r bth ( and ruter) levels f netwrk herarchy. The authr wuld lke t thank Bll Krgfss f Bell Labs fr valuable techncal dscussns and useful suggestns. Balamash, A., and M. Krunz. 24. An vervew f web cachng replacement algrthms. In Cmmuncatns Surveys & Tutrals, IEEE vl. 6, Issue: 2, pp Chen, H., H. Jn, J. Sun, X. La, and D. Deng. 23. A new prxy cachng scheme fr parallel vde servers. In Prceedngs f the Internatnal Cnference n Cmputer etwrks and Mble Cmputng, pp Cbarzan, C., and L. Bszrmeny. 27. Further Develpments f a Dynamc Dstrbuted Vde Prxy- Cache System. In Prceedngs f the 15th EUOMICO Internatnal Cnference n Parallel, Dstrbuted and etwrk-based, pp Ghandeharzadeh, S., T. Helm, T. Jung, S. Kapada, and S. Shayandeh. 26. An Evaluatn f Tw Plces fr Placement f Cntnuus Meda n Mult-hp Wreless etwrks. In Prceedngs f the Twelfth Internatnal Cnference n Dstrbuted Multmeda Systems, pp Ghandeharzadeh, S., and S. Shayandeh. 27. Greedy Cache Management Technque fr mble Devces. In Prceedngs f the IEEE 23rd Internatnal Cnference n Data Engneerng, pp Krgfss, B., L. Sfman, and A. Agrawal. 28. Cachng archtecture and ptmzatn strateges fr IPTV netwrks. Bell Lab Techncal Jurnal vl. 13, 3, pp Krgfss, B., L. Sfman, and A. Agrawal. 29. Herarchcal cache ptmzatn n IPTV netwrks. In Prceedngs f the IEEE Internatnal Sympsum n Bradband Multmeda Systems and Bradcastng, pp.1-1. Lee, J.P., and S.H. Park. 25. A cache management plcy n prxy server fr an effcent multmeda streamng servce. In Prceedngs f the nth Internatnal Sympsum n Cnsumer Electrncs, pp Sem-cntnuty [Onlne]. Avalable: Sfman, L., B. Krgfss, and A. Agrawal. 28. Optmal Cache Parttnng n IPTV etwrk. In Prceedngs f the 11th Cmmuncatns and etwrkng Smulatn Sympsum, pp Sfman, L., and B. Krgfss. 29. Analytcal Mdel fr Herarchcal Cache Optmzatn n IPTV etwrk. In Prceedngs f the IEEE Transactns n Bradcastng, Vl. 55, 1, pp Vleeschauwer, D., and D. bnsn Optmum Cachng Strateges fr a Telc CD. Bell Labs Techncal Jurnal vl. 16(2), pp Zpf-Mandelbrt Law [Onlne]. Avalable: s a Senr etwrk Archtect n Frnter Cmmuncatns. Hs areas f nterest nclude netwrk/traffc mdelng and cst/perfrmance analyss and ptmzatn. Befre jnng Frnter Cmmuncatns n Octber 215, he wrked wth Alcetel-Lucent/Bell Labs frm 21 t 215. Lev Sfman receved hs Ph.D. degrees n Mathematcs frm Mscw Unversty, ussa, and frm the Insttute f Infrmatn Transmssn Prblems, ussan Academy f Scences. Hs e-mal s lev.sfman@ftr.cm.