A Frugal Bidding Procedure for Replicating WWW Content

Transcription

1 A Frugal Bddng Procedure for Replcatng WWW Content Samee Ullah Khan and Cemal Ardl Abstract Fne-graned data replcaton over the Internet allows duplcaton of frequently accessed data objects, as opposed to entre stes, to certan locatons so as to mprove the performance of largescale content dstrbuton systems. In a dstrbuted system, agents representng ther stes try to maxmze ther own beneft snce they are drven by dfferent goals such as to mnmze ther communcaton costs, latency, etc. In ths paper, we wll use game theoretcal technques and n partcular auctons to dentfy a bddng mechansm that encapsulates the selfshness of the agents, whle havng a controllng hand over them. In essence, the proposed game theory based mechansm s the study of what happens when ndependent agents act selfshly and how to control them to maxmze the overall performance. A bddng mechansm asks how one can desgn systems so that agents selfsh behavor results n the desred system-wde goals. Expermental results reveal that ths mechansm provdes excellent soluton qualty, whle mantanng fast executon tme. The comparsons are recorded aganst some well known technques such as greedy, branch and bound, game theoretcal auctons and genetc algorthms. Keywords Internet, data content replcaton, statc allocaton, mechansm desgn, equlbrum. I I. INTRODUCTION N the Internet a magntude of heterogeneous enttes (e.g. provders, servers, commercal servces, etc.) offer, use, and even compete wth each other for resources. The Internet s emergng as a new platform for dstrbuted computng and brngs wth t problems never seen before. New solutons should take nto account the varous new concepts derved from mult-agent systems n whch the agents cannot be assumed to act n accordance to the deployed algorthm. In a heterogeneous system such as the Internet enttles act selfshly. Ths s obvous snce they are drven by dfferent goals such as to mnmze ther communcaton costs, latency, etc. Thus, one cannot assume that agents would follow the protocol or the algorthm; though they respond to ncentves (e.g. payments receved for compensaton). In ths paper, we wll use game theoretcal technques and n partcular auctons to dentfy a bddng mechansm that S. U. Khan s wth the Department of Electrcal and Computer Engneerng, North Dakota State Unversty, Fargo, ND 58102, USA (phone: ; fax: ; e-mal: samee.khan@ ndsu.edu). C. Ardl s wth the Natonal Academy of Avaton, Baku, Azerbajan, (emal: cemalardl@gmal.com). encapsulates the selfshness of the agents, whle havng a controllng hand over them. Ths work s nspred from the work reported n [46] and [54]. In essence, game theory s the study of what happens when ndependent agents act selfshly. A bddng mechansm asks how one can desgn systems so that agents selfsh behavor results n the desred systemwde goals. In ths paper, we wll apply the derved mechansm to the fne graned data replcaton problem (DRP) over the Internet. Ths problem strongly conforms to the selfsh agents noton and has a wde range of applcatons. Replcaton s wdely used to mprove the performance of large-scale content dstrbuton systems such as the CDNs [50]. Replcatng the data over geographcally dspersed locatons reduces access latency, network traffc, and n turn adds relablty, robustness and fault-tolerance to the system. Dscussons n [14], [17], [39], [40], [47], etc. reveal that clent(s) experence reduced access latences provded that data s replcated wthn ther close proxmty. However, ths s applcable n cases when only read accesses are consdered. If updates of the contents are also under focus, then the locatons of the replcas have to be: 1) n close proxmty to the clent(s), and 2) n close proxmty to the prmary (assumng a broadcast update model) copy. For fault-tolerant and hghly dependable systems, replcaton s essental, as demonstrated n a real world example of OceanStore [50]. Therefore, effcent and effectve replcaton schemas strongly depend on how many replcas to be placed n the system, and more mportantly where. Needless to say that our work dffers form the exstng technques n the usage of game theoretcal technques. To the best of the authors knowledge ths s the very frst work that addresses the problem usng such technques. The major results of ths paper are as follows: 1. We derve a specalzed aucton mechansm. Ths mechansm allows selfsh agents to compete n a noncooperatve envronment. 2. We nvestgate ths aucton mechansm, provde some useful propertes and dentfy the necessary condtons of optmalty. 3. As an applcaton we employ the derved mechansm to the DRP. We perform extensve expermental comparsons aganst some well known technques, such as greedy, branch and bound, genetc and game theoretcal auctons. 960

2 Mechansm p 1 x1 b 1 p M x M b M Agent 1... Fg. 1 Frugal bddng mechansm Agent M TABLE I NOTATIONS AND THEIR MEANINGS Symbols M N O k o k S s r k R k w k W k NN k c(,j) P k R k C overall SGRG GT-ITM PR GT-ITM W SGFCGUD SGFCGRD SGRGLND DRP OTC Meanng Total number of stes n the network Total number of objects to be replcated k-th object Sze of object k -th ste Sze of ste Number of reads for object k from ste Aggregate read cost of r k Number of wrtes for object k from ste Aggregate wrte cost of w k Nearest neghbor of ste holdng object k Communcaton cost between stes and j Prmary ste of the k-th object Replcaton schema of object k Total overall data transfer cost Self Generate Random Graphs Georga Tech Internetwork Topology Models Pure Random Georga Tech Internetwork Topology Models Waxman Self Generated Fully Connected Graphs Unform Dstrbuton Self Generated Fully Connected Graphs Random Dstrbuton Self Generated Random Graphs Lognormal Dstrbuton Data replcaton problem Object transfer cost (network communcaton cost) The remander of ths paper s organzed as follows. Secton II descrbes the resource allocaton mechansm. Secton III formulates the DRP. Secton IV concentrates on modelng the resource allocaton mechansm for the DRP, followed by theoretcal proofs n Secton V. The expermental results, related work and concludng remarks are provded n Sectons VI, VII and VIII, respectvely. II.THE FRUGAL BIDDING PROCEDURE We approach the mechansm (Fg. 1) desgn n a stepwse fashon. The Bascs: The mechansm contans M agents. Each agent has some prvate data t R. Ths data s termed as the agent s true data or true type. Only agent has knowledge of t. Everythng else n the mechansm s publc knowledge. Let t denote the vector of all the true types t = (t 1 t M ). Communcatons: Snce the agents are selfsh n nature, they do not communcate to the mechansm the value t. The only nformaton that s relayed to the mechansm s the correspondng bd b. Let b denote the vector of all the bds ((b = (b 1 b M )), and let b - denote the vector of bds, not ncludng agent,.e., b - = (b 1 b -1,b +1, b M ). It s to be understood that we can also wrte b = (b -,b ). Components: The mechansm has two components 1) the algorthmc output x( ), and 2) the payment mappng functon p( ). Algorthmc output: The mechansm allows a set of outputs X, based on the output functon whch takes n as the argument, the bddng vector,.e., x(b) = {x 1 (b),, x M (b)}, where x(b) X. Ths output functon relays a unque output gven a vector b. That s, when x( ) receves b, t generates an output whch s of the form of allocatons x (b). Intutvely t would mean that the algorthm takes n the vector bd b and then relays to each agent ts allocaton. Many sophstcated allocaton outputs can be constructed, but n ths paper we choose to have a smple, ntutve and an optmal allocaton. Monetary cost: Each agent ncurs some monetary cost 961

3 c (t,x),.e., the cost to accommodate the allocaton x (b). Ths cost s dependent upon the output and the agent s prvate data. Payments: To offset c, the mechansm makes a payment p (b) to agent. An agent always attempts to maxmze ts proft (utlty) u (t,b) = p (b) - c (t,x). Each agent cares about the other agents bd only nsofar as they nfluence the outcome and the payment. Whle t s only known to agent, the functon c s publc. Bds: Each agent s nterested n reportng a bd b such that t maxmzes ts proft, regardless of what the other agents bd,.e., u (t,(b -,t )) u (t,(b -,b )) for all b - and b. A closer look at the bds reveal that the agents are postng bds that are domnant n nature. We state the followng lemma from lterature whch says that truth-tellng are domnant strateges,.e., f all the agents report the exact worth of an object, t conforms to domnatng strateges. Lemma 1: Truth-tellng s a domnant strategy. Proof: [46]. The Mechansm: We now put all the peces together. A mechansm m conssts of a par m = (x(b),p(b)), where x( ) s the output functon and p( ) s the payment mappng functon. The objectve of the mechansm s to select an output x, that optmzes a gven objectve functon g(b,x). Below we dentfy the desred characterstcs of the mechansm. Truthfulness: We say that an output functon admts truthful payments f there exsts a payment mappng functon p( ) such that the mechansm m s truthful. Usng Lemma 1, ths would transform to: a mechansm m that s mplemented usng domnant strateges (m = (x(t),p(t))). Voluntary partcpaton: A mechansm s characterzed as a voluntary partcpaton mechansm f for every agent, u (t,(b -,t )) 0,.e., no agent ncurs a net loss. In recent tmes, such mechansms have been appled to the schedulng problems [2], [13], [46], etc. The only practcal work that can be found n the lterature s reported n [13]. However, that work s the exact mplementaton of the work reported n [46]. Moreover, the authors n [13] have faled to reason why the mplementaton works to begn wth. It s to be noted that a truthful mechansm strongly reles on the payment functon,.e., the agents are forced to tell the truth because tellng a le gves them no greater proft. Therefore, they are better off tellng the truth. Our work dffers from others n that we gve a concrete applcaton of the truthful mechansm, and concentrate on the payment functon. We prove that our approach results n payments that are the exact representatons of the monetary costs,.e., the payments are frugal. Objectve: The mechansm defned above leaves us wth the followng two optmzaton problems: 1. Identfy a strategy that s domnant to each agent. 2. Identfy a payment mappng functon that s truthful, and frugal. III. DESCRIPTION OF THE DATA REPLICATION PROBLEM Consder a dstrbuted system comprsng M stes, wth each ste havng ts own processng power, memory (prmary storage) and meda (secondary storage). Let S and s be the name and the total storage capacty (n smple data unts e.g. blocks), respectvely, of ste where 1 M. The M stes of the system are connected by a communcaton network. A lnk between two stes S and S j (f t exsts) has a postve nteger c(,j) assocated wth t, gvng the communcaton cost for transferrng a data unt between stes S and S j. If the two stes are not drectly connected by a communcaton lnk then the above cost s gven by the sum of the costs of all the lnks n a chosen path from ste S to the ste S j. Wthout the loss of generalty we assume that c(,j) = c(j,). Ths s a common assumpton (e.g. see [14], [17], [40], [47], etc.). Let there be N objects, each dentfable by a unque name O k and sze n smple data untes o k where 1 k N. Let r k and w k be the total number of reads and wrtes, respectvely, ntated from S for O k durng a certan tme perod. Our replcaton polcy assumes the exstence of one prmary copy for each object n the network. Let P k, be the ste whch holds the prmary copy of O k,.e., the only copy n the network that cannot be de-allocated, hence referred to as prmary ste of the k-th object. Each prmary ste P k, contans nformaton about the whole replcaton scheme R k of O k. Ths can be done by mantanng a lst of the stes where the k-th object s replcated at, called from now on the replcators of O k. Moreover, every ste S stores a two-feld record for each object. The frst feld s ts prmary ste P k and the second the nearest neghborhood ste NN k of ste S whch holds a replca of object k. In other words, NN k s the ste for whch the reads from S for O k, f served there, would ncur the mnmum possble communcaton cost. It s possble that NN k = S, f S s a replcator or the prmary ste of O k. Another possblty s that NN k = P k, f the prmary ste s the closest one holdng a replca of O k. When a ste S reads an object, t does so by addressng the request to the correspondng NN k. For the updates we assume that every ste can update every object. Updates of an object O k are performed by sendng the updated verson to ts prmary ste P k, whch afterwards broadcasts t to every ste n ts replcaton scheme R k. For the DRP under consderaton, we are nterested n mnmzng the total network transfer cost due to object movement,.e. the Object Transfer Cost (OTC). The communcaton cost of the control messages has mnor mpact to the overall performance of the system, therefore, we do not consder t n the transfer cost model, but t s to be noted that ncorporaton of such a cost would be a trval exercse. There are two components affectng OTC. The frst component of OTC s due to the read requests. Let R k denote the total OTC, due to S s readng requests for object O k, addressed to the nearest ste NN k. Ths cost s gven by the followng equaton: 962

4 R k = rk okc(, NNk ), where NN k = {Ste j j R k ^ mn c(,j)}. The second component of OTC s the cost arsng due to the wrtes. Let W k be the total OTC, due to S s wrtng requests for object O k, addressed to the prmary ste P k. Ths cost s gven by the followng equaton: W = w o ( c(, P ) + c( NN, j)). (2) k k k k ( j Rk ), j Here, we made the ndrect assumpton that n order to perform a wrte we need to shp the whole updated verson of the object. Ths of course s not always the case, as we can move only the updated parts of t (modelng such polces can also be done usng our framework). The cumulatve OTC, denoted as C overall, due to reads and wrtes s gven by: M N Coverall = = 1 k= 1 ( Rk + Wk ). (3) Let X k =1 f S holds a replca of object O k, and 0 otherwse. X k s defne an M N replcaton matrx, named X, wth boolean elements. Stes whch are not the replcators of object O k create OTC equal to the communcaton cost of ther reads from the nearest replcator, plus that of sendng ther wrtes to the prmary ste of O k. Stes belongng to the replcaton scheme of O k, are assocated wth the cost of sendng/recevng all the updated versons of t. Usng the above formulaton, the DRP can be defned as: Fnd the assgnment of 0, 1 values n the X matrx that mnmzes C overall, subject to the storage capacty N constrant: = X k kok s ( 1 M) 1, and subject to the prmary copes polcy: X P k = 1 (1 k N ) k. The mnmzaton of C overall wll have two mpacts on the dstrbuted system under consderaton: Frst, t ensures that the object replcaton s done n such a way that t mnmzes the maxmum dstance between the replcas and ther respectve prmary objects. Second, t ensures that the maxmum dstance between an object k and the user(s) accessng that object s also mnmzed. Thus, the soluton ams for reducng the overall OTC of the system. In the generalzed case, the DRP has been proven to be NP-complete [40]. IV. MECHANISM APPLIED TO THE DRP We follow the same pattern as dscussed n Secton II. The Bascs: The dstrbuted system descrbed n Secton III s consdered, where each ste s represented by an agent,.e., the mechansm contans M agents. In the context of the DRP, an agent holds two key elements of data a) the avalable ste capacty ac, and b) the cost to replcate (RC k = R k +W k ) an object k to the agent s ste. There are three possble cases: 1. DRP [π]: where each agent holds the cost to replcate RC k = t assocated wth each object k, where as the avalable ste capacty and everythng else s publc knowledge. k (1) 2. DRP [σ]: where each agent holds the avalable ste capacty ac = t, where as the cost to replcate and everythng else s publc knowledge. 3. DRP [π,σ]: where each agent holds both the cost to replcate and the ste capacty {RC k,ac } = t, where as everythng else s publc knowledge. Intutvely, f agents know the avalable ste capactes of other agents, that gves them no advantage whatsoever. However, f they come about to know ther replcaton cost then they can modfy ther valuatons and alter the algorthmc output. It s to be noted that an agent can only calculate the replcaton cost va the frequences of reads and wrtes. Everythng else such as the network topology, latency on communcaton lnes, and even the ste capactes can be publc knowledge. Therefore, DRP[π] s the only natural choce. Communcatons: The agents n the mechansm are assumed to be selfsh and therefore, they project a bd b to the mechansm. In realty the amount of communcatons made are mmense. Ths fact was not realzed n [13], where the authors assume superfluous assumptons on the mplementaton. In the later text we wll reveal how to cope wth ths dlemma. Components: The mechansm has two components 1) the algorthmc output x( ), and 2) the payment mappng functon p( ). Algorthmc output: In the context of the DRP, the algorthm accepts bds from all the agents, and outputs the maxmum benefcal bd,.e., the bd that ncurs the mnmum replcaton cost overall (Equaton 3). We wll gve a detaled descrpton of the algorthm n the later text. Monetary cost: When an object s allocated (for replcaton) to an agent, the agent becomes responsble to entertan (read and wrte) requests to that object. For example, assume object k s replcated to agent. Then the amount of traffc that the agent has to entertan due to the replcaton of object k s exactly equvalent to the replcaton cost,.e., c = RC k. Ths fact s easly deducble from Equaton 4. Payments: To offset c, the mechansm makes a payment p (b) to agent. Ths payment s equvalent to the cost t ncurs to replcate the object,.e., p (b) = c. The readers would mmedately note that n such a payment agent can never get a proft greater than 0. Ths s exactly what we want. In a selfsh envronment, t s possble that the agents bd hgher than the true value, the mechansm creates an lluson to negate that. By compensatng the agents wth the exact amount of what the cost occurs, t leaves no room for the agents to overbd or underbd (n the later text we wll rgorously prove the above argument). Therefore, the voluntary characterstc of the mechansm now becomes a strongly voluntary and we quote from the lterature the followng defnton. Defnton 1: A mechansm s characterzed as a strongly 963

5 Frugal Aucton (FA) Mechansm Intalze: 01 LS, L, T k, M, MT 02 WHILE LS NULL DO 03 OMAX = NULL; MT = NULL; P = NULL; 04 PARFOR each S LS DO 05 FOR each O k L DO 06 T k = compute (t ); /*compute the valuaton correspondng to the desred object*/ 07 ENDFOR 08 t = argmax k(t k ); 09 SEND t to M; RECEIVE at Mt n MT; 10 ENDPARFOR 11 OMAX = argmax k(mt); /*Choose the global domnate valuaton*/ 12 P = 1/OMAX; /*Calculate the payment*/ 13 BROADCAST OMAX; 14 SEND P to S ; /*Send payments to the agent who s allocate the object OMAX*/ 15 Replcate O OMAX; 16 ac =ac - o k; /*Update capacty*/ 17 L = L - O k; /*Update the lst*/ 18 IF L = NULL THEN SEND nfo to M to update LS = LS - S ; /*Update mechansm players*/ 19 PARFOR each S LS DO 20 Update NN OMAX /*Update the nearest neghbor lst*/ 21 ENDPARFOR /*Get ready for the next round*/ 22 ENDWHILE Fg. 2 Frugal Aucton (FA) Mechansm voluntary partcpaton mechansm f for every agent, u (t,(b -,t )) = 0 [54]. We want to emphass that each agent s ncentve s to replcate objects so that queres can be answered locally, for the sake of users that access the agent s ste. If the replcas are made avalable elsewhere, the agent may lose the users, as they mght dvert ther accesses to other stes. Bds: Each agent reports a bd that s the drect representaton of the true data that t holds. Therefore, a bd b s equvalent to 1/RC k. That s, the lower the replcaton cost the hgher s the bd and the hgher are the chances for the bd b to wn. In essence, the mechansm m(x(b),p(b)), takes n the vector of bds b from all the agents, and selects the hghest bd. The hghest bdder s allocated the object k whch s added to ts allocaton set x. The mechansm then pays the bdder p. Ths payment s equvalent to the cost ncurred due to entertan requests from object k by users. The mechansm s gven n Fg. 2. Descrpton of Algorthm: We mantan a lst L at each server. Ths lst contans all the objects that can be replcated by agent onto ste S. We can obtan ths lst by examnng the two constrants of the DRP. Lst L would contan all the objects that have ther sze less then the total avalable space ac. Moreover, f ste S s the prmary host of some object k, then k should not be n L. We also mantan a lst LS contanng all stes that can replcate an object,.e., S LS f L NULL. The algorthm works teratvely. In each step the mechansm asks all the agents to send ther preferences (frst PARFOR loop). Each agent recursvely calculates the true data of every object n lst L. Each agent then reports the domnant true data (lne 09) to the mechansm. The mechansm receves all the correspondng entres, and then chooses the globally domnant true data. Ths s broadcasted to all the agents, so that they can update ther nearest neghbor table NN k, whch s shown n Lne 20 (NN OMAX). The object s replcated and the payment s made to the agent. The mechansm progresses forward tll there are no more agents nterested n acqurng any data for replcaton (Lne 18). The above dscusson allows us to deduce the followng two results about the mechansm. Theorem 1: FA requres O(MN 2 ) tme. Proof: The worst case scenaro s when each ste has suffcent capacty to store all objects. In that case, the whle loop (Lne 02) performs MN teratons. The tme complexty for each teraton s governed by the two PARFOR loops (Lnes 04 and 19). The frst loop uses at most N teratons, whle the send loop performs the update n constant tme. Hence, we conclude that the worst case runnng tme of the mechansm s O(MN 2 ). Theorem 2: In the worst case FA uses O(M 3 N) messages. Proof: Frst we calculate the number of messages n a sngle teraton. Frst, each agent sends ts true data to the mechansm, whch consttutes M messages. Second, the mechansm broadcasts nformaton about the object beng allocated, ths consttutes M messages. Thrd, the mechansm sends a sngle message about payment to the agent to whom the replca was assgned, whch we can gnore snce t has lttle mpact on the total number of messages. The total number of messages requred n a sngle teraton as of the order of M 2. From Theorem 1, we can conclude that n the 964

6 worst case the mechansm requres O(M 3 N) messages. there must be an agent - who has a bd that supersedes b. It s evdent from Theorem 2 that the mechansm requres tremendous amounts of communcatons. Ths mght be reduced by usng some sophstcated network protocols. In the But that s not possble as we began wth the assumpton that all other bds are fxed so there can be no other agent -. If = -, then that s also not possble snce we assumed that b > future generaton dstrbuted computng systems, the b. partcpatng agents mght actually be moble,.e., they can travel from node to node n the network. In such a scenaro the agents can converge to a specfc ste and execute the We now extend the result obtaned n Theorem 3 and state: Theorem 4: A decreasng output functon admts a truthful payment scheme satsfyng voluntary partcpaton f and only mechansm. In that case the number of messages wll be f x ( b, u) du < for all, b -. In ths case we can take the 0 reduced astronomcally. In our expermentatons we mmc ths scenaro and use IBM Pthreads to mplement the payments to be: p ( b, b ) = b x ( b, b ) + x ( b, u) du. 0 mechansm. Other possble mprovements are left as future research ssues. Proof: The frst term b x (b -,b ) compensates the cost ncurred by agent to host the allocaton x. The second term x ( b, u) du represents the expected proft of agent. If agent V. SUPPLEMENTARY RESULTS Here, we present some results that strengthen our clam on the optmalty of the derved bddng mechansm. We begn by makng the followng observatons. Assume that the mechansm m = (x(b),p(b)) s truthful and each payment p (b -,b ) and allocaton x (b -,b ) s twce dfferentable wth respect to b, for all the values of b -. We fx some agent and derve a formula for p, allocaton x, and proft to be the functons of just agent s bd b. Snce agent s proft s always maxmzed by bddng truthfully (Lemma 1), the dervatve s zero and the second dervatve s nonpostve at t. Snce ths holds no matter what the value of t s, we can ntegrate to obtan an expresson for p. We state: p (b ) = p (0)+b x (b )- b 0 x (u)du. Ths s now the bass of our extended theoretcal results. Lterature survey revealed the followng two mportant characterstcs of a frugal payment mechansm. We state them below. Defnton 2: Wth the other agents bd b - fxed, consder x (b -,b ) as a sngle varable functon of b. We call ths the allocaton curve or the allocaton profle of agent. We say the output functon x s decreasng f each of the assocated allocaton curves s decreasng,.e., x (b -,b ) s a decreasng functon of b, for all and b -. Based on the above defnton, we can state the followng theorem. Theorem 3: A mechansm s truthful f ts output functon x(b) s decreasng. Proof: We prove ths for the DRP mechansm. For smplcty we fx all bds b -, and focus on x (b -,b ) as a sngle varable functon of b,.e., the allocaton x would only change f b s altered. We now consder two bds b and b such that b > b. In terms of the true data t, ths conforms to RC k > RC k. Let x and x be the allocatons made to the agent when t bds b and b, respectvely. For a gven allocaton, the total replcaton cost assocated can be represented as C = k x RC k. The proof of the theorem reduces to provng that x < x,.e., the allocaton computed by the algorthmc output s decreasng n b. The proof s smple by contradcton. Assume that that x x. Ths mples that 1/(C -RC k ) < 1/(C -RC k ) 1/(C -RC k ). Ths means that 0 bds ts true value t, then ts proft s = u (t,(b -,t )) = t x ( b, t ) + x ( b, x) dx t x ( b, t ) = x ( b, x) dx t t If agent bds ts true value, then the expected proft s greater than n the case t bds other values. We explan ths as follows: If agent bds hgher (b >t ), then the expected proft s = u (t,(b -,b )) = b ' x ( b, b ') + ' x ( b, x) dx t x ( b, b ') b = ( b ' t ) x ( b, b ') + x ( b, x) dx. Because b ' b ' x ( b, x) dx < and b > t, we can express the proft when agent bds the true value as b ' follows: x ( b, x) dx + ' x ( b, x) dx. Ths s because x s t b decreasng n b and b > t, we have the followng b ' t equaton: ( b ' t ) x ( b, b ') < x ( b, x) dx. From ths relaton, t can be seen that the proft wth overbddng s lower then the proft wth bddng the true data. Smlar arguments can be used for underbddng. VI. EXPERIMENTAL SETUP AND DISCUSSION OF RESULTS A.Setup We performed experments on a 440MHz Ultra 10 machne wth 512MB memory. The expermental evaluatons were targeted to benchmark the placement polces. The resource allocaton mechansm was mplemented usng IBM Pthreads. To establsh dversty n our expermental setups, the network connectvely was changed consderably. In ths paper, we only present the results that were obtaned usng a maxmum of 500 stes (nodes). We used exstng topology generator toolkts and also self generated networks. In all the 965

7 TABLE II PARAMETER INTERVAL VARIANCE CHARACTERIZATION FOR TOPOLOGIES WITH 100 NODES Topology Mathematcal Representaton Parameter Interval Varance SGRG Randomzed layout wth node degree (d*) and Eucldan dstance (d) d={5,10,15,20}, (12 topologes) between nodes as parameters. d*={10,15,20}. GT-ITM PR [4] (5 topologes) Randomzed layout wth edges added between the randomly located p={0.4,0.5,0.6,0.7,0.8}. vertces wth a probablty (p). GT-ITM W [4] P(u,v)=αe -d/(βl) α={0.1,0.15,0.2,0.25}, β={0.2,0.3,0.4}. (9 topologes) SGFCGUD Fully connected graph wth unform lnk dstances (d). d 1=[1,10],d 2=[1,20],d 3=[1,50], d 4=[10,20], d 5=[20,50]. (5 topologes) SGFCGRD Fully connected graph wth random lnk dstances (d). d 1=[1,10],d 2=[1,20],d 3=[1,50], d 4=[10,20], d 5=[20,50]. (5 topologes) SGRGLND (9 topologes) Random layout wth lnk dstance havng a lognormal dstrbuton [9]. μ={8.455,9.345,9.564}, σ={1.278,1.305,1.378}. 10 km Step 1: Network s Physcally layerd usng GT-ITU or other methods km km 1000 km 100 km km 10 km 4 10 km 1000 km km 100 Mbps Step 2: Comuncaton lnk costs such as bandwdth and latency measures are added. Latency s assumed to be 2x10 8 m/s km 10 Mbps 1 10 km 1Mbps km 10 Mbps 10 km 1 Mbps 1000 km 1 Mbps 8 10 km 100 Mbps 4 (1.16) 100 km 100 Mbps 1000 km 10 Mbps 6 7 Step 3: Process access log fles to get users' acess patterns, object sze, object sze varance, tme of access, etc. (0.12) Step 4: Use nformaton from step 3 to determne: a. System capacty. b. Locatons of P k. c. Mappng randomly 3 users onto each server (8.04) (11.57) (0.44) (0.12) 4 (4.38) (4.38) (8.04) (The package denotes P k ) (1.16) (8.04) (11.57) (0.12) (0.44) Step 5: Employ agents for the (0.12) 4 system. B (4.38) 3 (4.38) (8.04) 6 7 Note that not all agents are shown due to space restrctons. 5 A Fg. 3 A walk through the necessary steps nvolved n an expermental setup C Step 6: Extract from step 3: a. Read frequences from the access logs. b. Wrte frequences requests n the range of the read requests. topologes, the dstance of the lnk between nodes was equvalent to the communcaton cost. Table II summarzes the varous technques used to gather forty-fve varous topologes for networks wth 100 nodes. It s to be noted that the parameters vary for networks wth lesser/larger number of nodes. To evaluate the chosen replcaton placement technques on realstc traffc patterns, we used the access logs collected at the Soccer World Cup 1998 webste [3]. Each expermental setup was evaluated thrteen tmes,.e., only the Frday (24 hours) logs from May 1, 1998 to July 24, Thus, each expermental setup n fact represents an average of the 585 (13 45) data set ponts. To process the logs, we wrote a scrpt that returned: only those objects whch were present n all the logs (2000 n our case), the total number of requests from a partcular clent for an object, the average and the varance of the object sze. From ths log we chose the top fve hundred clents (maxmum expermental setup). A random mappng was then performed of the clents to the nodes of the topologes. Note that ths mappng s not 1-1, rather 1-M. Ths gave us enough skewed workload to mmc real world scenaros. It s also worthwhle to menton that the total amount of requests entertaned for each problem nstance was n the range of 1-2 mllon. The prmary replcas orgnal ste was mmcked by choosng random locatons. The capactes of the stes C% were generated randomly wth range from 966

8 Total Prmary Object Szes/2 to 1.5 Total Prmary Object Szes. The varance n the object sze collected from the access logs helped to nstll enough dversty to benchmark object updates. The updates were randomly pushed onto dfferent stes, and the total system update load was measured n terms of the percentage update requests U% compared that to the ntal network wth no updates. A bref overvew on how the expermental setups are obtaned s depcted n Fg. 3. B.Comparatve Algorthms For comparson, we selected fve varous types of replca placement technques. To provde a far comparson, the assumptons and system parameters were kept the same n all the approaches. The technques studed nclude effcent branch-and-bound based technque (Aε-Star [17]). For fnegraned replcaton, the algorthms proposed n [39], [40], and [47] are the only ones that address the problem doman smlar to ours. We select from [47] the greedy approach (Greedy) for comparson because t s shown to be the best compared wth 4 other approaches (ncludng the proposed technque n [39]); thus, we ndrectly compare wth 4 addtonal approaches as well. Algorthms reported n [21] (Dutch (DA) and Englsh auctons (EA)) and [40] (Genetc based algorthm (GRA)) are also among the chosen technques for comparsons. Due to space lmtatons we wll only gve a bref overvew of the comparatve technques. Detals for a specfc technque can be obtaned from the referenced papers. Performance metrc: The soluton qualty s measured n terms of network communcaton cost (OTC percentage) that s saved under the replcaton scheme found by the algorthms, compared to the ntal one,.e., when only prmary copes exsts. 1. Aε-Star: In [17] the authors proposed a 1+ε admssble A-Star based technque called Aε-Star. Ths technque uses two lsts: OPEN and FOCAL. The FOCAL lst s the sub-lst of OPEN, and only contans those nodes that do not devate from the lowest f node by a factor greater than 1+ε. The technque works smlar to A-Star, wth the excepton that the node selecton (lowest h) s done not from the OPEN but from the FOCAL lst. It s easy to see that ths approach wll never run nto the problem of memory overflow, moreover, the FOCAL lst always ensures that only the canddate solutons wthn a bound of 1+ε of the A-Star are expanded. 2. Greedy based technque: We modfy the greedy approach reported n [47], to ft our problem formulaton. The greedy algorthm works n an teratve fashon. In the frst teraton, all the M stes are nvestgated to fnd the replca locaton(s) of the frst among a total of N objects. Consder that we choose an object for replcaton. The algorthm recursvely makes calculatons based on the assumpton that all the users n the system request for object. Thus, we have to pck a ste that yelds the lowest cost of replcaton for the object. In the second teraton, the locaton for the second ste s consdered. Based on the choce of object, the algorthm now would dentfy the second ste for replcaton, whch, n conjuncton wth the ste already pcked, yelds the lowest replcaton cost. Observe here that ths assgnment may or may not be for the same object. The algorthm progresses forward tll ether one of the DRP constrants are volated. The readers wll mmedately realze that the bddng mechansm reported n ths paper works smlar to the Greedy algorthm. Ths s true; however, the Greedy approach does not guarantee optmalty even f the algorthm s run on the very same problem nstance. Recall that Greedy reles on makng combnatons of object assgnments and therefore, suffers from the ntal choce of object selecton (whch s done randomly). Ths s never the case n the derved bddng mechansm, whch dentfes optmal allocatons n every case. 3. Dutch aucton: The auctoneer begns wth a hgh askng prce whch s lowered untl some agent s wllng to accept the auctoneer's prce. That agent pays the last announced prce. Ths type of aucton s convenent when t s mportant to aucton objects quckly, snce a sale never requres more than one bd. In no case does the auctoneer reveal any of the bds submtted to hm, and no nformaton s shared between the agents. It s shown that for an agent to have a probablstcally superor bd than n-1 other bds; an agent should have the valuaton dvded by n. 4. Englsh aucton: In ths type of aucton, the agents bd openly aganst one another, wth each bd beng hgher than the prevous bd. The aucton ends when no agent s wllng to bd further. Durng the aucton when an auctoneer receves a bd hgher than the currently submtted bds, he announces the bd value so that other agents (f needed) can revse ther currently submtted bds. In [44] the dscusson on EA reveals that the optmal strategy for a bdder s to bd a value whch s drectly derved from hs valuaton. 5. GRA: In [40], the authors proposed a genetc algorthm based heurstc called GRA. GRA provdes good soluton qualty, but suffers from slow termnaton tme. Ths algorthm was selected snce t realstcally addressed the fne-graned data replcaton usng the same problem formulaton as undertaken n ths artcle. C.Comparatve Analyss We study the behavor of the placement technques when the number of stes ncreases (Fg. 3), by settng the number of objects to 2000, whle n Fg. 4, we study the behavor when the number of objects ncrease, by settng the number of stes to 500. We should note here that the space lmtatons restrcted us to nclude varous other scenaros wth varyng capacty and update rato. The plot trends were smlar to the ones reported n ths artcle. For the frst experment we fxed C=20% and U=75%. We ntentonally chose a hgh workload so as to see f the technques studed successfully handled the 967

9 82% Performance N=2000, C=20%, U=75% Performance M=500, C=20%, U=25% 78% 76% 74% 72% 70% 68% Legend Greedy GRA Aε-Star DA EA FA No. of Objects 72% 64% 56% 48% 40% 32% 24% 16% Legend Greedy GRA Aε-Star DA EA FA 66% No. of Stes Fg. 4 OTC savngs versus number of stes 8% Fg. 5 OTC savngs versus number of objects 90% Performance N=2000, M=500, U=5% 90% Performance N=2000, M=500, C=45% 85% 84% 78% 75% 72% 70% 65% 60% 55% 50% Legend Greedy GRA Aε-Star DA EA FA 66% 60% 54% 48% 42% Legend Greedy GRA Aε-Star DA EA FA 45% 9% 12% 15% 18% 21% 24% 27% 30% 33% 36% 39% 42% Capacty of Stes Fg. 6 OTC savngs versus capacty 36% 20% 22% 24% 26% 28% 30% 32% 34% 36% 38% 40% Re ads Fg. 7 OTC savngs versus reads extreme cases. The frst observaton s that FA and EA outperformed other technques by consderable amounts. Second, DA converged to a better soluton qualty under certan problem nstances. Some nterestng observatons were also recorded, such as, all but GRA showed ntal loss n OTC savngs wth the ntal number of ste ncrease n the system, as much as 7% loss was recorded n case of Greedy wth only a 40 ste ncrease. GRA showed an ntal gan snce wth the ncrease n the number of stes, the populaton permutatons ncrease exponentally, but wth the further ncrease n the number of stes ths phenomenon s not so observable as all the essental objects are already replcated. The top performng technques (DA, EA, Aε-Star and FA) showed an almost constant performance ncrease (after the ntal loss n OTC savngs). Ths s because by addng a ste (server) n the network, we ntroduce addtonal traffc (local requests), together wth more storage capacty avalable for replcaton. All four equally cater for the two dverse effects. GRA also showed a smlar trend but mantaned lower OTC savngs. Ths was n lne wth the clams presented n [17] and [40]. To observe the effect of ncrease n the number of objects n the system, we chose a softer workload wth C=20% and U=25%. The ntenton was to observe the trends for all the algorthms under varous workloads. The ncrease n the number of objects has dverse effects on the system as new read/wrte patterns (users are offered more choces) emerge, and also the ncrease n the stran on the overall capacty of the system (ncrease n the number of replcas). An effectve algorthm should ncorporate both the opposng trends. From the plot, the most surprsng result came from GRA. It dropped ts savngs from 58% to 13%. Ths was contradctory to what was reported n [40]. But there the authors had used a unformly dstrbuted lnk cost topology, and ther traffc was based on the Zpf dstrbuton [55]. Whle the traffc access logs of the World Cup 1998 are more or less double-pareto n nature. In ether case the explots and lmtatons of the technque under dscusson are obvous. The plot also shows a near dentcal performance by Aε-Star, DA and Greedy. The relatve dfference among the three technques s less than 2%. However, Aε-Star dd mantan ts domnaton. From the plots the supremacy of EA and FA s observable. Both the technques showed hgh performance, wth a slght edge n favor of FA. Next, we observe the effects of system capacty ncrease. An ncrease n the storage capacty means that a large number of objects can be replcated. Replcatng an object that s already extensvely replcated, s unlkely to result n sgnfcant traffc savngs as only a small porton of the servers wll be affected overall. Moreover, snce objects are 968

10 72% 64% 56% 48% 40% 32% 24% Performance N=2000, M=500, C=60% Legend Greedy GRA Aε-Star DA EA FA 3% Mscellaneous Executon Tme Analyss 21% Replca Placement 7% Shortest Paths Data Gatherng 16% 40% 42% 44% 46% 48% 50% 52% 54% 56% 58% 60% Update s Fg. 8 OTC savngs versus updates Fg. 9 Executon tme components 68% TABLE III RUNNING TIME IN SECONDS [C=20%, U=45%] (SMALL PROBLEM INSTANCES) Problem Sze Greedy GRA Aε-Star DA EA FA M=20, N= M=20, N= M=20, N= M=30, N= M=30, N= M=30, N= M=40, N= M=40, N= M=40, N= TABLE IV RUNNING TIME IN SECONDS [C=45%, U=15%] (LARGE PROBLEM INSTANCES) Problem Sze Greedy GRA Aε-Star DA EA FA M=300, N= M=300, N= M=300, N= M=300, N= M=300, N= M=300, N= M=400, N= M=400, N= M=400, N= M=400, N= M=400, N= M=400, N= M=500, N= M=500, N= M=500, N= M=500, N= M=500, N= M=500, N= not equally read ntensve, ncrease n the storage capacty would have a great mpact at the begnnng (ntal ncrease n capacty), but has lttle effect after a certan pont, where the most benefcal ones are already replcated. Ths s observable n Fg. 5, whch shows the performance of the algorthms. GRA once agan performed the worst. The gap between all other approaches was reduced to wthn 7% of each other. DA and FA showed an mmedate ntal ncrease (the pont after whch further replcatng objects s neffcent) n ts OTC savngs, but afterward showed a near constant performance. GRA although performed the worst, but observably ganed the most OTC savngs (35%) followed by Greedy wth 29%. Further experments wth varous update ratos (5%, 10%, and 20%) showed smlar plot trends. It s 969

11 90% Load Varance (Medan) N=2000, M=500, C=25% 90% Load Varance (Mean) N=2000, M=500, C=25% 85% 85% 75% 75% 70% 65% 60% 55% 50% 45% 40% Maxmum Mnmum 75% 25% Medan Outlers Extremes Grand medan (0.849) 70% 65% 60% 55% 50% 45% 40% Mean+Std Dev Mean-Std Dev Mean+0.5*Std Dev Mean-0.5*Std Dev Mean Outlers Extremes Grand mean (0.8088) Algorthms (L. to R.) [Greedy, GRA, Aε-Star, DA, EA, FA] Fg. 10 Comparatve load varance (Medan) Algorthms (L. to R.) [Greedy, GRA, Aε-Star, DA, EA, FA] Fg.11 Comparatve load varance (Mean) 96% Capacty Varance (Medan) N=2000, M=500, U=25% 104% Capacty Varance (Mean) N=2000, M=500, U=25% 88% 96% 88% 72% 64% 56% 48% 40% 32% 24% 16% Maxmum Mnmum 75% 25% Medan Outlers Extremes Grand medan (0.8847) 72% 64% 56% 48% 40% 32% 24% 16% Mean+Std Dev Mean-Std Dev Mean+0.5*Std Dev Mean-0.5*Std Dev Mean Outlers Extremes Grand mean (0.8549) Algorthms (L. to R.) [Greedy, GRA, Aε-Star, DA, EA, FA] Fg. 12 Comparatve capacty varance (Medan) Algorthms (L. to R.) [Greedy, GRA, Aε-Star, DA, EA, FA] Fg. 13 Comparatve capacty varance (Mean) TABLE V AVERAGE OTC (%) SAVINGS UNDER SOME PROBLEM INSTANCES Problem Sze Greedy GRA Aε-Star DA EA FA N=150, M=20 [C=20%,U=25%] N=200, M=50 [C=20%,U=20%] N=300, M=50 [C=25%,U=5%] N=300, M=60 [C=35%,U=5%] N=400, M=100 [C=25%,U=25%] N=500, M=100 [C=30%,U=35%] N=800, M=200 [C=25%,U=15%] N=1000, M=300 [C=25%,U=35%] N=1500, M=400 [C=35%,U=50%] N=2000, M=500 [C=10%,U=60%] also noteworthy (plots not shown n ths paper due to space restrctons) that the ncrease n capacty from 10% to 17%, resulted n 4 tmes (on average) more replcas for all the algorthms. Next, we observe the effects of ncrease n the read and update (wrte) frequences. Snce these two parameters are complementary to each other, we descrbe them together. In both the setups the number of stes and objects were kept constant. Increase n the number of reads n the system would mean that there s a need to replcate as many object as possble (closer to the users). However, the ncrease n the number of updates n the system requres the replcas be placed as close as to the prmary ste as possble (to reduce the update broadcast). Ths phenomenon s also nterrelated wth the system capacty, as the update rato sets an upper bound on the possble traffc reducton through replcaton. Thus, f we consder a system wth unlmted capacty, the replcate everywhere anythng polcy s strctly nadequate. The read and update parameters ndeed help n drawng a lne between good and margnal algorthms. The plots n Fgs. 6 and 7 show the results of read and update frequences, respectvely. A clear classfcaton can be made between the algorthms. 970

12 Aε-Star, DA, EA, Greedy and FA ncorporate the ncrease n the number of reads by replcatng more objects and thus savngs ncrease up to 88%. GRA ganed the least of the OTC savngs of up to 67%. To understand why there s such a gap n the performance between the algorthms, we should recall that GRA specfcally depend on the ntal populaton (for detals see [40]). Moreover, GRA mantans a localzed network percepton. Increase n updates result n objects havng decreased local sgnfcance (unless the vcnty s n close proxmty to the prmary locaton). On the other hand, Aε-Star, DA, EA, Greedy and FA never tend to devate from ther global vew of the problem search space. Lastly, we compare the termnaton tme of the algorthms. Before we proceed, we would lke to clarfy our measurement of algorthm termnaton tmngs. The approach we took was to see f these algorthms can be used n dynamc scenaros. Thus, we gather and process data as f t was a dynamc system. The average breakdown of the executon tme of all the algorthms combned s depcted n Fg. 8. There 68% of all the algorthm termnaton tme was taken by the repeated calculatons of the shortest paths. Data gatherng and dsperson, such as readng the access frequences from the processed log, etc. took 7% of the total tme. Other mscellaneous operatons ncludng I/O were recorded to carry 3% of the total executon tme. From the plot t s clear that a totally statc setup would take no less that 21% of the tme depcted n Tables III and IV. Varous problem nstances were recorded wth C=20%, 45% and U=15%, 45%. Each problem nstance represents the average recorded tme over all the 45 topologes and 13 varous access logs. The entres n bold represent the fastest tme recorded over the problem nstance. It s observable that FA and DA termnated faster than all the other technques, followed by EA, Greedy, Aε-Star and GRA. If a statc envronment was consdered, FA wth the maxmum problem nstance would have termnated approxmately n seconds (21% of the algorthm termnaton tme). In summary, based on the soluton qualty alone, the algorthms can be classfed nto four categores: 1) The very hgh performance algorthms that nclude EA and FA, 2) the hgh performance algorthms of Greedy and DA, 3) the medum-hgh performance Aε-Star, and fnally 4) the medocre performance algorthm of GRA. Whle consderng the termnaton tmngs, FA and DA dd extremely well, followed by EA, Greedy, Aε-Star, and GRA. D.Supplementary Analyss Here, we present some supplementary results that strengthen our comparatve analyss provded n Secton VI.C. We show the relatve performance of the technques wth load and storage capacty varance. The plots n Fgs. 10, 11, 12 and 13 show the recorded performances. All the plots summarze the measured performance wth varyng parameters (most of whch could not be ncluded n ths paper due to space lmtatons). We are mostly nterested n measurng the medan and mean performances of the algorthms. Wth load varance FA edges over Aε-Star wth a savngs of 87%. The plot also shows that nearly every algorthm performed well wth grand medan on 84.9%. The graphs are self explanatory and also capture the outlners and extreme ponts. The basc exercse n plottng these results s to see whch algorthms perform consstently. GRA for example, records the lowest extremes, and hardly any outlners. On the other hand the proposed FA s performance s captured n a small nterval, wth hgh medan and mean OTC savngs. Table V shows the qualty of the soluton n terms of OTC percentage for 10 problem nstances (randomly chosen), each beng a combnaton of varous numbers of stes and objects, wth varyng storage capacty and update rato. For each row, the best result s ndcated n bold. The proposed FA algorthm steals the show n the context of soluton qualty, but Aε-Star, EA and DA do ndeed gve a good competton, wth a savngs wthn a range of 5%-10% of FA. VII. RELATED WORK Myrad theoretcal approaches are proposed that we classfy nto the followng sx categores: 1. Faclty Locaton: In [15], the authors employed several technques to address the Internet data replcaton problem smlar to that of the classcal faclty locaton problem. The technques reported are very tedous and have superfluous assumptons. Thus, the problem defnton n [15] does not fully capture the concept of replcatng a sngle object/ste over a fxed number of hosts [41]. 2. Fle Allocaton: Fle allocaton has been a popular lne of research n: dstrbuted computng, dstrbuted databases, multmeda databases, pagng algorthms, and vdeo server systems [1], [8], [37], [42]. All the above referenced artcles ncorporate data replcaton onto a set of dstrbuted locatons (dstrbuted system), whch can easly be modfed to ts equvalent problem n the context of Internet. Under the assumpton of unlmted server memory the authors n [37], provded a guaranteed optmal result for Internet data replcaton, but has lttle practcal use [41], snce the replca placements are based on the belef that the access patterns reman unchanged. 3. Mnmum k-medan: The celebrated NP-complete mnmum k-medan problem captures the coarse-graned replcaton well, as t can tackle wth the problem of dstrbutng a sngle replca over a fxed number of hosts. In [39] the authors studed the problem of placng M proxes at N nodes when the topology of the network s a tree and proposed an O(N 3 M 2 ) algorthm. A more generalzed soluton was presented n [47]. There the authors proposed a greedy algorthm that outperformed other technques ncludng the work reported n [39]. 4. Capacty-constraned Optmzaton: In [16], the 971