Approximation Algorithms for Data Distribution with Load Balancing of Web Servers

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1 Approxmaton Agorthms for Data Dstrbuton wth Load Baancng of Web Servers L-Chuan Chen Networkng and Communcatons Department The MITRE Corporaton McLean, VA Hyeong-Ah Cho Department of Computer Scence The George Washngton Unversty Washngton, DC Abstract Gven the ncreasng traffc on the Word Wde Web (Web), t s dffcut for a snge popuar Web server to hande the demand from ts many cents By custerng a group of Web servers, t s possbe to reduce the orgn Web server s oad sgnfcanty and reduce user s response tme when accessng a Web document A fundamenta queston s how to aocate Web documents among these servers n order to acheve oad baancng? In ths paper, we are gven a coecton of documents to be stored on a custer of Web servers Each of the servers s assocated wth resource mts n ts memory and ts number of HTTP connectons Each document has an assocated sze and access cost The probem s to aocate the documents among the servers so that no server s memory sze s exceeded, and the oad s baanced as equay as possbe In ths paper, we show that most smpe formuatons of ths probem are NP-hard, we estabsh ower bounds on the vaue of the optma oad, and we show that f there are no memory constrants for a the servers, then there s an aocaton agorthm, that s wthn a factor 2 of the optma souton We show that f a servers have the same number of HTTP connectons and the same memory sze, then a feasbe aocaton s acheved wthn a factor 4 of the optma souton usng at most 4 tmes the optma memory sze We aso provde mproved approxmaton resuts for the case where documents are reatvey sma 1 Introducton Internet and Word Wde Web (WWW) traffc has grown exposvey, and ths growth s expected to contnue For a popuar Web ste, network congeston and server overoadng may become serous probems n the future and woud resut n ncreased Web servces deays A number of approaches to sove ths probem has been proposed recenty, ncudng mrrorng, document cachng, and custers of Web servers The frst s to mrror (repcate) popuar Web stes n dfferent ocatons throughout the word The orgna Web ste s homepage woud contan a st of mrror stes Ths aows users to choose a ste based upon ther ocaton One drawback of mrrorng the Web ste s that the user does not typcay have access to nformaton about underyng network and server oad Ths ssue was consdered n a number of papers [16, 11, 1, 14, 9] by takng network atency and server oad nto account, basng decsons on pror performance, or based on erasure codes Web cachng s a mechansm to pace copes of frequenty accessed Web objects coser to the users One dffcuty n Web cachng s the possbty of accessng stae Web objects Cache coherence [10, 4] deas wth probem of keepng Web objects consstent wth the orgna copy Web objects vary n sze, unke tradtona cachng systems, n modern computer memory systems, where a cache-ne s of a fxed sze Repacement agorthms dea wth ths ssue where more than one object mght be removed to repace the current object [13, 6] For the custerng of servers as a snge Web server, Web documents are dstrbuted among servers, and ony one Unversa Resource Locator (URL) s pubshed to the cents Snce many servers are workng together, oad baance s the man ssue Ths has been studed esewhere [2, 12, 15, 5, 8] We w focus on ths approach n ths paper We w present a number of resuts on the compexty of sovng the aocaton optmzaton and decson probems, ether exacty or approxmatey We w consder a number of dfferent formuatons of the probem, snce some seem to be easer to approxmate than others In addton to estabshng that smpe formuaton of the aocaton probem are NP-hard, we present approxmaton agorthms for the cases of no memory constrants, equa memory and HTTP con-

2 strants, and for aocaton nvovng sma document szes 2 Prevous Work There have been severa studes of oad baancng among a custer of Web servers These are usuay broken nto two broad categores: cent-based oad baancng and serverbased oad baancng Lewontn and Martn [9] mpemented a cent-sde oad baancng agorthm Ther method s based on the past performance of the requests to mnmze response atency performance The performance s measured by the number of bytes transmtted dvded by the tota tme A st of repcated servers performances s mantaned at the cent s proxy server and then uses a drectory servce to map a URL onto one of the servers Many of the server-based oad baancng systems are based on a 2-ter archtecture A front-end server s responsbe for dspatchng an ncomng Web document request to one of the back-end document servers In NCSA servces [7], a round-robn Doman Name Servce (DNS) s used for dstrbutng cent requests to one of the Web servers The drawback of usng DNS s that t does not provde oad baance among the servers, due to the non-unformy document szes and DNS namng cachng DNS does not know the status of Web servers When a server s down or busy (because t has been takng a the arge document sze requests), DNS mght st rotate the request to that server Garand et a [5] overcome NCSA s uneven oad baancng probem by mpementng a mechansm that montors server oad and seects the east oaded server for servng an ncomng request Ther server s oad metrc s determned by the number of Web document requests for the server pus the number of processes currenty actve n the server Narendran et a [12] mpemented a dstrbuted Web servers system based on the combnaton of DNS roundrobn, HTTP redrecton and document s access rate as a mechansm to baance the oad Our modes cosey reated to thers, but ncudes server memory sze mts Exstng research has stressed practca approaches to the probem of achevng oad baance, but there has not been a theoretca anayss of the performance of these agorthms, and very tte work has been done n terms of how Web documents are aocated among servers In ths paper we approach ths oad baance probem from a more theoretca drecton We consder the aocaton of Web documents among a custer of Web servers n order to acheve oad baance Each of the servers s assocated wth resource mts n ther memory and ther number of HTTP connectons Each document has an assocated sze, s j, and access rate, r j Foowng [12], we defne the access rate to be the product of tme needed to access the document and the probabty that the document s requested We prove that even smpe formuatons of ths probem are NP-hard We aso provde smpe approxmaton agorthms for a number of formuatons of ths probem In each case we show that the approxmaton agorthm acheves a fxed performance rato wth respect to the optmum souton Before presentng our resuts, we defne our moden greater deta 3 Probem Formuaton Our modes a generazaton of one proposed by Narendran et a [12] It conssts of M servers and N documents Throughout we use the ndex when referrng to servers and j when referrng to documents Each server s assocated wth a memory sze m and a number of smutaneous HTTP connectons Each document j s assocated wth a document sze s j and an access cost r j as defned earer The tota access cost ˆr s sum of a documents access cost, ˆr = N r j The tota number of HTTP connectons ˆs sum of a servers HTTP connectons, ˆ = M Let r = (r 1, r 2,, r N ), = ( 1, 2,, M ), s = (s 1, s 2,, s N ), m = (m 1, m 2,, m M ) The nput to the aocaton probem s a quadrupe I = r,, s, m The output s an aocaton (or access) matrx, whch s an m n matrx, a j, where 0 a j 1 If a j 0, then document j s aocated to server We permt a document to be aocated to more than one server, and we nterpret a j to be the probabty that a request for document j s to be processed by server Any aocaton must satsfy the foowng aocaton constrant M a j = 1, for 1 j N A speca case, caed a 0-1 aocaton s one n whch a j {0, 1} In such an aocaton each document appears n exacty one server Let D denote the set of documents aocated to server, that s, D = {j a j 0} The sum of document szes n server cannot exceed the memory of ths server From ths we have the foowng memory constrant j D s j m, for 1 j M An aocaton satsfyng these constrants s caed a feasbe aocaton Let R denote the tota access cost for server, that s, R = a j r j

3 A server s abty to respond to document requests s affected by two quanttes The tota number of bytes ths server must send s proportona to the server s tota access cost As the number of HTTP connectons ncreases, the server s abty to satsfy mutpe requests ncreases Hence, we defne the oad of server per HTTP connecton to be R Defne objectve functon f(a) to be f(a) = max 1 M ( R ) Our goas to baance the oad by mnmzng the maxmum oad over a servers Aocaton Optmzaton Probem: Gven nput quadrupe I, fnd a feasbe aocaton a that mnmzes f(a) Ca ths optmum aocaton a I, and et f I = f(a I ) be ts optmum vaue When I s cear from context, we w smpy wrte f Aocaton Decson Probem: Gven nput I and vaue f 0, s fi f 0? Our nterest n the decson probem s that, gven an agorthm for the decson probem, we may use t wthn the context of bnary search to search for the optmum vaue for the optmzaton probem 4 Contrbutons Frst we estabsh a ower bound of ˆr/ˆ, on the vaue of the optmum oad f Ths can be acheved when memory s not a constrant, by aocatng every document to every server Ths mmedatey mproves the resuts of Narendran et a [12], snce ther resuts dd not consder bounds on memory Our remanng resuts anvove 0-1 aocatons (n whch each document s assgned to exacty one server) Hardness: We show that even wthout memory constrants and wth a servers havng an equa number of HTTP connectons, the aocaton optmzaton probem s NP-hard We aso show that f memory constrants are present, then even determnng the exstence of a feasbe 0-1 aocaton s NP-hard Ths remans true even f a servers have the same memory sze No memory constrants: We show that f there are no memory constrants for a the servers, then there s a smpe and effcent greedy aocaton agorthm, whch s wthn a factor 2 of the optma souton Equa memory and HTTP constrants: We show that f a servers have the same number of HTTP connectons and the same memory sze, then a feasbe aocaton s acheved wthn a factor 4 of the optma souton usng 4 tmes the optma memory sze Sma document szes: The above hardness resuts rey on the fact that documents can be neary as arge as the memory szes of the servers However, n practce, document szes are typcay much smaer than the servers memory szes We show that f the memory szes for a servers are equa to some vaue m, and the sze of the argest document s at most m/k, then we can compute an aocaton whose oad s at most a factor of 2(1 + 1/k) tmes optma A of our approxmaton agorthms are based on smpe greedy approaches, and are easy to mpement 5 Lower Bounds Consder an nput I = r,, s, m where there are no memory constrants; that s m = Reca that ˆr = N r j and ˆ = M We begn by provdng a ower bound on the optma aocaton cost f Lemma 1 Let r max = max 1 j N r j and max = max 1 M, f { rmax max, max } ˆr ˆ Proof: Assumng memory space s arge enough to aocate a documents for each server Then the memory constrant s trvay satsfed By the pgeon-hoe prncpe, there exsts a HTTP connecton on a server that has to provde servce at east the tota access cost ˆr dvded by the tota number of HTTP connectons ˆ, that s f ˆrˆ The document wth the argest access rate must be assgned to some server, and n the best case t s assgned to the server wth argest number of HTTP connectons, mpyng a cost of at east rmax max We w use the foowng aternatve ower bound n the proof of Theorem 2 beow Lemma 2 Assume r 1 r 2 r N and 1 2 M, then for a j, 1 j mn(m, N), f max 1 j mn(n,m) j r j j Proof: Consder the optma aocaton of the frst j documents to the M servers Let S j denote the servers used n ths aocaton and et fj denote the cost of ths aocaton Ceary S j j Snce we aocate a subset of documents to a the servers we have fj f

4 We cam that f S j then we may assume that 1 S j If not, then move a the documents from server to server 1 Snce 1 s not n S j t contans no documents Snce 1, ths can ony decrease the overa cost Based on ths and the fact that S j j t foows that we may assume that S j {1, 2,, j } For 1 j, et R be the sum of r j for the documents assgned to server By defnton ( ) fj = max R 1 j Thus for 1 j, Therefore, j fj R, fj R = r j j j f fj r j j j Narendran et a [12] present an aocaton agorthm under a smar mode to ours, but wthout memory constrants Now we show that t s trva to acheve an optma aocaton by seectng a j appropratey Theorem 1 If for a, m N s j, then an optma aocaton s acheved by settng a j = ˆ, for a, j Proof: Snce a j > 0, t mpes that each server must have copes of a documents, thus { } R f(a) = max 1 M { N = max ( r } ja j ) 1 M { ( = max ˆ ) N r } j 1 M { } = max 1 M ˆrˆ = ˆrˆ f Therefore, a s an optma souton 6 NP-Competeness We observe that even smpe formuatons of ths optmzaton probem are NP-hard Consder the foowng probems: 0-1 Aocaton: Gven an nput quadrupe I = r,, s, m, does there exst a 0-1 aocaton? Ths probem s NP-compete even f we set a memory szes equa m 1 = m 2 = = m M = m The reason s that satsfyng the memory constrants s equvaent to the bn packng probem where s denotes the szes of the objects and the bns are of sze m If we choose to gnore memory constrants atogether then the probem s st NP-hard for 0-1 aocatons as we show beow 0-1 Aocaton wth No Memory Constrants: Gven an nput quadrupe I = r,, s, m wth m = for 1 M, does there exst an aocaton wth oad vaue f 1? We show that ths probem s NP-compete even f a servers have an equa number of HTTP connectons, 1 = 2 = = M = As before we may reduce the bn packng probem to ths probem by ettng be the bn sze and ettng r denote the szes of the objects to be packed A 0-1 aocaton of vaue at most 1 s equvaent to a bn packng nto M bns, snce for each server 1 M we have R / 1 mpyng that the tota sze of objects R assgned to bn s at most the bn sze of These resuts mpy that the probem s ony nterestng when there are memory constrants or mts on the number of servers to whch a document can be aocated Henceforth we ony consder 0-1 aocatons 7 Approxmaton Agorthms Throughout the remander of the paper we consder ony 0-1 aocatons 71 No Memory Constrant Consder an nstance of the document aocaton probems n whch there are no memory constrants, that s m =, 1 M Consder Agorthm 1 shown n Fg 1 We w show that t produces an aocaton that s wthn a factor 2 of the optma souton Note that for a 0-1 aocaton wth no memory constrants we may assume that N M, snce otherwse the optma assgnment s acheved by pacng one document n each of the N servers wth the argest vaues of Theorem 2 Let f 1 be the objectve functon vaue for the Agorthm 1 for no memory constrant Then f 1 2f Proof: Suppose not Let j be the frst document aocated to some server such that > 2f, where R

5 Agorthm 1 Input: A quadrupe I = r,, s, m, where m = for 1 M Output: A 0-1 aocaton of documents to servers 1 Sort documents by decreasng access cost, r j 2 Sort servers by decreasng port connectons, 3 for 1 M do { 4 set R = 0; } 5 for 1 j N do { 6 Choose that mnmzes R+rj for 1 M 7 Aocate document j to server 8 R += r j } Fgure 1 The 0/1 approxmaton agorthm for no memory constrant R = j a jr j Let S {1, 2,, M} denote the set of servers whch have receved at east one of the frst j documents by the agorthm Note that S j By ne 2 of the agorthm servers are sorted n descendng order by We cam that f S then 1 S To see ths, consder the frst document j aocated to Just pror aocaton of j, we have R + r j > R 1 + r j 1 By ne 8 of the agorthm, f 1 / S then R 1 = 0 and snce ths s the frst document aocated to server, R = 0 Thus r j < r j, 1 whch mpes > 1, a contradcton Thus t foows that we have S {1, 2,, j } Because each of the frst j documents has been aocated to one server n S we have j a jr j = r j Consder the stuaton just after the aocaton of document j By the choce of n ne 6 of the agorthm we have for 1 M, By our hypothess, whch mpes R + r j R + r j R + r j R + r j R R + r j > 2 f Summng over 1 j we have j j R + r j > > 2f, j 2 f j j j R + j r j > 2f j j a j r j + j r j > 2f j j r j + j r j > 2f Snce r j r j for 1 j j, we have j j r j + j r j Ths mpes 2 j j r j > 2f f < j r j j r j + j r j However ths contradcts the ower bound of Lemma 2 It s easy to see that a straghtforward mpementaton of Agorthm 1 runs n O(N og N + NM) tme, where nes 1 and 6 domnate the tota tme If there are L dstnct vaues of t s possbe to acheve a runnng tme of O(N og N + NL), whch s no worse snce L M To do ths we partton the servers nto L groups accordng to the vaue of For each group we mantan a bnary heap [3] whch s sorted by the vaue R For each group we can determne the mnmum R vaue n O(1) tme and hence can determne the server on ne 6 n O(L) tme by nspectng each heap For the seected heap we update the vaue of R n O(og N) tme Thus each teraton of the oop of ne 5 takes O((og N) + L) tme for a tota runnng tme of O(N og N + LN) 72 Equa Memory and Load Constrants In ths secton we w show how to reax the assumpton on memory sze made n the prevous secton Reca that we are gven an nput quadrupe I = r,, s, m, N = r = s, M = = m Assume that we have homogeneous servers wth a servers havng the same number of HTTP connectons and equa memory szes, that s, = and m = m for 1 M Assume there exsts a 0-1 aocaton a wth an objectve functon vaue f such that both memory and oad baancng constrants are satsfed for a, that s, s j a j m, r j a j f

6 For 1 j N, we normaze each document s access cost r j and each document s sze s j as foows: r j = r j f, s j = s j m Ths mpes that N r j a j 1, and N s j a j 1 In genera we do not know f, so our approach w be to conduct a bnary search to fnd the smaest vaue of f such that we can aocate a the documents nto servers wth memory 4m wth tota cost at most 4f Ths w provde us wth the desred approxmaton bounds Agorthm 2 Input: A quadrupe I = r,, s, m,where =, m = m for 1 M and target cost f Output: A 0-1 aocaton of documents to servers and ndcaton of success 1 / Intazaton / L 1, L2, M 1, M 2 = 0 for 1 M, normaze r j, s j, r j = rj f, s j = sj m, for a j 2 Spt the documents nto two sets, D 1, D 2, where D 1 = {j r j s j }, D2 = {j r j < s j } 3 ca Agorthm 3; 4 f a documents have been assgned to some server then output yes, ese output no Agorthm 3 Subroutne used n Agorthm 2 // Phase 1: Assgn documents of D 1 to // servers such that R f and M m 1 j = 1; 2 for ( = 1 to M and j N) do { 3 whe (j D 1 and L 1 < 1) do { 4 Aocate document j to server 5 L 1 += r j ; 6 M 1 += s j ; 7 j ++; } } // Phase 2: Assgn documents of D 2 to // servers such that R f and s m 1 j = 1; 2 for ( = 1 to M and j N) do { 3 whe (j D 2 and M 2 < 1) do { 4 Aocate document j to server 5 L 2 += r j ; 6 M 2 += s j ; 7 j ++; } } Fgure 3 The 0/1 approxmaton agorthm for both memory and oad constrants (cont) Cam 2 At any pont n the agorthm, Fgure 2 The 0/1 approxmaton agorthm for both memory and oad constrants We spt the documents nto two sets, D 1, D 2 D 1 conssts of the documents whose (normazed) access cost s bgger than ts (normaze) document sze and D 2 conssts of documents whose document sze s bgger than ts access cost Let L 1 denote the cumuatve oad for documents that are n D 1 and are assgned to server Let M 1 denote the cumuatve memory for documents that are n D 1 and are assgned to server Defne L 2 and M 2 smary for D 2 Agorthm 3 shown n Fgures 3 conssts of two phases We try to assgn as many documents whch are n D 1 as possbe n phase 1 and then assgn the remanng documents whch are n D 2 n phase 2 The frst phase guarantees that servers are we utzed wth respect to access cost and the second phase guarantees utzaton wth respect to sze Cam 1 At any tme n the executon of Agorthm 2, M 1 L 1, L2 M 2 Proof: Ths foows from defntons of D 1 and D 2 max(max(l 1, L 2, M 1, M 2 )) 2), where the max s over 1 M Proof: The proof s by nducton on the number of documents Intay ths s ceary satsfed Suppose that the cam hods just pror to nserton of document j 0 Let be the server to whch j 0 s aocated Case 1: j 0 D 1 Pror to nserton, L 1 1 (for otherwse j 0 woud not be paced here) So after nserton ts oad s L 1 + r j r j 0 2 (snce r j 0 1, j) Thus after nserton j 0, we have L 1 2, and by the prevous cam, M 1 2 Case 2: j 0 D 2 Pror to nserton, M 2 1 (for otherwse j 0 woud not be paced here) So after nserton ts oad s M 2 + s j s j 0 2 (snce s j 0 1, j) Thus after nserton j 0, we have M 2 2, and by the prevous cam, L 2 2

7 Cam 3 If there exsts an optma aocaton a wth vaue f satsfyng both the memory constrant N s ja j m, and the oad baance constrant, N r ja j f, then Agorthm 2 succeeds n assgnng a documents Proof: Suppose not Let j 0 be the frst document whch fas to ft Consder L, M just pror to the nserton of j 0 Case 1: j 0 D 1 Just pror to nserton of j 0, we cam that L 1 > 1, If ths were not so for some, L 1, then we woud have assgned j 0 to server From ths, we have M < M L 1 = M r j a j < r j, snce each document s assgned to at most one server Ths mpes that the number of servers M N r j Ths contradcts the exstence of an aocaton of vaue f Case 2: j 0 D 2 Just pror to the nserton of j 0, we cam that M 2 > 1, If ths were not so for some, M 1, then we woud have assgned j 0 to server From ths, we have M < M M 2 = M s ja j < s j, snce each document s assgned to at most one sever Ths mpes that M N s j, contradctng the exstence of a feasbe aocaton Theorem 3 Under the assumptons of Cam 3, the aocaton gven by Agorthm 2 assgns a documents, and ts cost s ess than 4 tmes the optma aocaton n memory and oad constrants, that s r j a j 4f, s j a j 4m, for 1 M Proof: By Cam 2 and Cam 3 we have for 1 M, r j a j = j D 1 r j a j + j D 2 r j a j = L 1 + L = 4, s j a j = s j a j + s j a j j D 1 j D 2 = M 1 + M = 4 By normazaton, r j = rj f, s j the orgna formuaton we have r j a j = s j a j = = sj M, and so returnng to N r j a jf = r j a jf 4f, s ja j m 4m Now we descrbe the compete agorthm ower bound from Lemma 1 f ˆr M Reca the Snce = here, we have f M We can derve an easy upper bound by observng that n the worst case a documents are aocated to a snge server, and hence ˆr ˆr M f ˆr Assumng anput quanttes are ntegers, observe that Mf s an nteger n the nterva [ˆr, ˆrM] By appyng bnary search to ths nterva we can determne a mnmum vaue of Mf, and hence a mnmum vaue for f, such that Agorthm 2 succeeds n aocatng a the documents Ths nvoves O(og(ˆrM)) cas to Agorthm 3 It s straghtforward to show that Agorthm 3 runs n O(N +M) tme (The key observaton s that each teraton of the oop of ne 3 ether fnshes a document or fnshes a server) Thus the tota runnng tme s O((N + M) og(ˆrm)) Note that snce the nput sze (n bts) s at east Ω(N +M +og ˆr) the agorthm runs n O(n og n) tme, where n s the nput sze Prevous sectons consdered the document to be as arge as server memores In practce, document szes are typcay much smaer than the server s memory The foowng emma shows that f each server can hod at east k documents, then we can acheve a better resut That s 2(1 + 1/k) tmes optma souton Theorem 4 If there exsts an optma aocaton a of vaue f satsfyng both the memory constrant, N s ja j m, and the oad baancng constrant, N r ja j f, then the aocaton gven by Agorthm 2 s at most 2(1 + 1 k ) tme the optma souton, where k s number of documents that a server can hod For exampe, f r j 1/4, we have 2(1 + 1/4) = 5/2 tmes optma

8 Proof: The proof s based on the proof of Cam 2 and Theorem 3 From Cam 2, max(max(l 1, L2, M 1, M 2 )) 1 + r j 0, where the max s over 1 M If r j 0 1/k, then max(max(l 1, L2, M 1, M 2 )) k From Theorem 3, Agorthm 2 n Fg 2 s ess than L 1 + L2 and M 1 + M 2 tmes optma aocaton Therefore, aocaton gven by 0/1 Approxmaton Agorthm 2 s 2(1 + 1 k ) tme the optma souton 8 Concusons We have consdered the probem of baancng the oad among a group of Web servers We showed that even wthout memory constrants and wth a servers havng an equa number of HTTP connectons, the aocaton optmzaton probem s NP-hard We aso showed that f memory constrants are present, then even determnng the exstence of a feasbe 0-1 aocaton s NP-hard We have presented a number of approxmaton agorthms, ncudng the cases where there are no memory constrants for a the servers and where servers have equa memory and HTTP constrants A of our approxmaton agorthms are based on smpe greedy approaches, and are easy to mpement References [1] J Byers, M Luby, and M Mtzenmacher Accessng mutpe mrror stes n parae: Usng tornado codes to speed up downoads In INFOCOM 99, voume 1, pages , March 1999 [2] V Carden, M Coajann, and P S Yu Dns dspatchng agorthms wth state estmators for scaabe Web-server custers Word Wde Web Journa, pages , 1999 [3] TH Cormen, CE Leserson, and RL Rvest Introducton to Agorthms McGraw-H Book Company, New York, 1990 [4] A Dnge and T Part Web cache coherence Computer Networks and ISDN Systems, 28(7): , 1996 [5] Mchae Garand, Sebastan Grassa, Robert Monroe, and Sddhartha Pur Impementng dstrbuted server groups for the Word Wde Web Technca Report MU-CS , Schoo of Computer Scence, Carnege Meon Unversty, Pttsburgh, Pennsyvana, 1995 [6] Sandy Iran Page repacement wth mut-sze pages and appcatons to Web cachng In Proceedngs of the Twenty-Nnth Annua ACM Symposum on Theory of Computng, pages , May 1997 [7] Erc Dean Katz, Mchee Buter, and Robert Mc- Grath A scaabe HTTP server: The NCSA prototype In Computer Networks and ISDN Systems, voume 27, pages , 1994 [8] Das Ksh, Mukherjee, and Renu Tewar A scaabe and hghy avaabe Web server In COMPCON 96, pages 85 92, 1996 [9] Steve Lewontn and Ezabeth Martn Cent sde oad baancng for the Web In 6th nternatona Word Wde Web Conference Conference, 1997 [10] Chengje Lu and Pe Cao Mantanng strong cache consstency n the Word-Wde Web IEEE Transactons on Computer, 47(4): , Apr 1998 [11] A Myers, P Dnda, and H Zhang Performance characterstcs of mrror servers on the Internet In INFO- COM 99, pages , March 1999 [12] B Narendran, Sampath Rangarajan, and Shan Yajnk Data dstrbuton agorthms for oad baanced faut-toerant Web access In Proc 16th IEEE Sympos Reabe Dstrbuted Systems, pages , 1997 [13] L Rzzo and L Vcsano Repacement poces for a proxy cache IEEE/ACM Transactons on Networkng, 8(2): , Apr 2000 [14] M Saya, Y Bretbart, P Scheuermann, and R Vngraek Seecton agorthms for repcated Web servers ACM Performance Evauaton Revew, 26(3):44 50, December 1998 [15] Rahu Smha, B Narahar, H-A Cho, and L-Chuan Chen Fe aocaton for a parae Webserver In IEEE Int Conf Hgh Performance Computng, pages 16 21, December 1996 [16] E W Zegura, H Ammar, Z Fe, and S Bhattacharjee Appcaton-ayer anycastng: A server seecton archtecure and use n a repcated Web servce IEEE/ACM Transactons on Networkng, 8(4): , 2000

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