A Semi-Distributed Axiomatic Game Theoretical Mechanism for Replicating Data Objects in Large Distributed Computing Systems

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1 A Sem-Dstrbuted Axomatc Game Theoretcal Mechansm for Replcatng Data Objects n Large Dstrbuted Computng Systems Samee Ullah Khan and Ishfaq Ahmad Department of Computer Scence and Engneerng Unversty of Texas, Arlngton, TX 76019, U.S.A. {sakhan, ahmad}@cse.uta.edu Abstract Replcatng data objects onto servers across a system can allevate access delays. The selecton of data objects and servers requres solvng a constrant optmzaton problem, whch s NP-complete n general. A majorty of conventonal replca placement technques falter on ssues of scalablty or soluton qualty. To counteract such ssues, we propose a game theoretcal replca placement technque, n whch computatonal agents compete for the allocaton or reallocaton of replcas onto ther servers n order to reduce the user perceved access delays. The technque s based upon sx welldefned axoms, each guaranteeng certan basc game theoretcal propertes. Ths eccentrc method of desgnng game theoretcal technques usng axoms s unque n the lterature and takes away from the desgners the cumbersome mathematcal detals of game theory. The dstnctve feature of these axoms s that when amassed together, ther ndvdual propertes constrct nto one system-wde performance enhancement property, whch n our case s the reducton of access tme. The control of the proposed technque s sem-dstrbuted n nature, wheren all the heavy processng s done on the servers of the dstrbuted system and the central body s only requred to take a bnary decson: (0) not to replcate or (1) to replcate. Ths sem-dstrbuted approach makes the technque scalable and helps solutons to converge n a fast turn-around tme wthout loosng much of the soluton qualty. Expermental comparsons are made aganst: 1) branch and bound, 2) greedy, 3) genetc, 4) Dutch aucton, and 5) Englsh aucton. As attested by the results, the proposed technque mantans superor soluton qualty n terms of lower communcaton cost and reduced executon tme. 1. Introducton Numerous methods have been proposed that address /07/$ IEEE. the data object replcaton problem n large dstrbuted computng systems. (For a recent survey see [20].) However, all of the reported methods operate under the generc assumpton that the servers cooperate wth each other n order to attan system-wde benefts. For nstance, n a content dstrbuton network (CDN) the dstrbuton system moves contents to the replca servers [12], [13], [14]. Ths dstrbuton system acts as a centralzed decson makng body, and makes decsons on where and what to replcate by observng the system s parameters such as server load, storage capacty, communcaton latences, etc. Although the Internet encourages dstrbuted control, t requres that the overall system performance crtera be met. Managng a large dstrbuted system, such as the Internet, through a sngle entty would requre huge amounts of data processng n order to fnd a feasble replca schema and would be vulnerable to system falures [6]. Consequently, we propose a semdstrbuted [1] (a hybrd of both the centralzed and decentralzed approaches) replcaton technque, by whch all the heavy processng s done on the servers of the dstrbuted system and the central body s only requred to take a bnary decson: (0) not to replcate or (1) to replcate. Ths leads to a smple yet an effcent scheme wth enhanced scalablty and low complexty that grows n proporton to the number of servers n the system. For the proposed sem-dstrbuted replca allocaton technque, we abstract the Internet as an agent based model wheren agents contnuously compete for allocaton and reallocaton of data objects. An agent s a computatonal entty that s capable of autonomous behavour n the sense of beng aware of the optons avalable to t when faced wth a decson makng task related to ts doman of nterest [28]. In such a compettve model, there s no a-pror motvaton for cooperaton and the agents due to ther localzed vew of the problem doman are encouraged to take decsons ndvdually that are benefcal only to themselves. To cope wth such ndvdualsm and localzed optmzaton, new mechansms based on novel oracles need to be derved. The objectve of the mechansms should be to allow the agents to take decsons that are based on ther

2 local nformaton, whch essentally translate nto the overall system performance enhancement. Ths paper ams to formally specfy a semdstrbuted game theoretcal replca allocaton mechansm, n whch autonomous agents compete to replcate data objects n a non-cooperatve game. Brefly, the mechansm works as follows. It frst asks all the agents to provde a lst of objects that are benefcal for replcaton onto ther servers. Havng obtaned ths data, the mechansm makes the allocaton and nforms the agents. For every allocaton the mechansm makes a payment to each agent (to compensate for hostng object(s)). Each agent s goal s to provde the mechansm wth a lst of objects that maxmze ts beneft. The mechansm on the other hand, s desgned such that, the agents maxmze ther beneft only f the reported lst contans objects that when replcated, brngs the overall system communcaton cost to the mnmum. Thus, the agents whch are competng aganst each other n a noncooperatve game collaborate unknowngly to optmze the overall system goal. The major contrbutons of ths paper are as follows: 1. We derve a general purpose axomatc game theoretcal mechansm. Ths mechansm ensures that although, the agents use self-benefcal strateges, yet the effect s translated nto a global optmzaton. 2. The essence of ths mechansm s captured n sx well-defned axoms that exhbt propertes of global optmalty, truthfulness, and utltaransm. 3. The dstnctve feature of these axoms s that when amassed together, ther ndvdual propertes constrct nto one system-wde performance enhancement property. 4. We use these axoms to obtan an effcent algorthm for the data replcaton problem. The remander of ths paper s organzed as follows. In Secton 2 we present a formal descrpton of the data replcaton problem. Secton 3 focuses on descrbng the generalzed axomatc game theoretcal mechansm along wth ts propertes. Secton 4 concentrates on modelng the mechansm for the data replcaton problem. The expermental results, related work, and concludng remarks are provded n Sectons 5, 6, and 7, respectvely. 2. The Data Replcaton Problem Consder a dstrbuted system comprsng M servers, wth each server 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 server where 1 M. The M servers of the system are connected by a communcaton network. A lnk between two servers 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 servers S and S j. If the two servers 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 server S to the server S j. Wthout the loss of generalty we assume that c(,j) = c(j,). 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. Our replcaton polcy assumes the exstence of one prmary copy for each object n the network. Let P k, be the server 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 server of the k-th object. Each prmary server P k, contans nformaton about the whole replcaton scheme R k of O k. Ths can be done by mantanng a lst of the servers where the k-th object s replcated at, called from now on the replcators of O k. Moreover, every server S stores a two-feld record for each object. The frst feld s ts prmary server P k and the second the nearest neghborhood server NN k of server S whch holds a replca of object k. In other words, NN k s the server 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 server of O k. Another possblty s that NN k = P k, f the prmary server s the closest one holdng a replca of O k. When a server S reads an object, t does so by addressng the request to the correspondng NN k. For the updates we assume that every server can update every object. Updates of an object O k are performed by sendng the updated verson to ts prmary server P k, whch afterwards broadcasts t to every server n R k. For the data replcaton problem (DRP) under consderaton, we are nterested n mnmzng the total Object Transfer Cost (OTC) due to object movement. 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 server NN k. Ths cost s gven by: Rk = rkokc(, NNk), (1) where NN k = {Server 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 server P k. Ths cost s gven by the followng equaton: k k k k k j Rk, j (, ) (, ) W = w o c P + c P j. (2) 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

3 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: C M N 1 1 ( overall = k R k + = = W k ). (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. Equaton 3 s now refned to: ( 1 X ) k rk ok mn { c(, j) X jk = 1 M N } X = = 1 k= 1 M. (4) x + woc k k ( P, k) + Xk ( w 1 k ) oc k ( P, x= k) Servers 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 server of O k. Servers 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 N capacty constrant: X (1 ) k 1 kok s M, and subject to = the prmary copes polcy: X = 1 (1 k N). Pk k 3. Axomatc Game Theoretcal Mechansm In ths secton we use varous buldng blocks to construct a generalzed axomatc game theoretcal mechansm. We begn by defnng a mechansm [3], whch has two components: a) the algorthmc output, and b) the agents valuaton functons. Defnton 1 ([3]): A mechansm s n whch: 1. The system conssts of M agents. Each agent has some prvate data t R. Ths s termed as the agent s true data or true value. Everythng else n the mechansm s publc knowledge. 2. The algorthmc output maps to each true data vector t = t 1 t M a set of allowed outputs x X, where x = x 1 x M. 3. Each agent s preferences are gven by a real valued functon v (t,x) called valuaton. Remarks: The valuaton of an agent represents the quantfcaton of ts value from the output x, when ts true data s t n terms of some predefned currency. For example, f the output of the mechansm s x and t hands the agent p amount of payment, then ts utlty becomes: u = p + v (t,x). We now focus on dentfyng a mechansm that allows the algorthmc output to optmze a gven objectve functon. Below we gve a refned defnton of an optmzaton (mnmzaton n our case) orented mechansm. Defnton 2 ([3]): An optmzaton orented mechansm s one where: 1. The algorthmc output s gven by a postve real valued objectve functon g(t,x), and 2. a set of feasble outputs X. Thus, we requre an output x X that mnmzes g, such that for any other output x X, g(t,x) g(t,x ). Ths s fne as long as we can fnd a mechansm that can solve a gven problem by assurng that the requred algorthmc output occurs, when all the agents choose strateges that maxmze ther utlty functons (a mn-max procedure). Let a - denote a vector of strateges, not ncludng agent,.e, a - = (a 1,,a -1,a +1,,a M ). We can defne a mechansm that s able to solve a utltaran based mnmzaton problem as: Defnton 3 ([24]): A mechansm (m = (x( ),p( ))) s composed of a) an algorthmc output functon x( ), and b) the payment functon p( ). The mechansm should have the followng propertes: 1. The mechansm allows for each agent a set of strateges A. It s up to the agent what strategy (a A ) to adopt n order to have ts utlty functon optmzed. 2. The mechansm should provde an algorthmc output x derved from the output functon,.e., x = x(a a M ). 3. In order to motvate the agents, the mechansm should provde a payment p = p (a 1 a M ) to each of the M agents. Remarks: Recall that x = x 1 x M. It s then not dffcult to see that agent s algorthmc output can easly be obtaned once the mechansm dentfes the output x. It s possble that agents due to ther selfsh nature may alter the output of the algorthm n order to fervently gather more resources. Hence, we are requred to enforce the mechansm to handle the specal case of selfsh agents by addng the followng property to Defnton 3. Property 4: The mechansm should be a representaton of domnant strateges, and ths s possble f for each agent and each t there exsts a domnant strategy (a A ), such that for all possble strateges of the other agents a -, a maxmzes agent s utlty. Remarks: Lterature survey reveals that the smplest of all the mechansms that exhbts domnant strateges s the one where the agents strateges are to report ther true data. Such types of mechansms are called truthful mechansms, and they are based on the revelaton prncple [11]. Ths prncple reports that for a gven optmzaton problem, f there exsts a truthful mechansm then pareto-optmalty s guaranteed (by default). To algn ourselves wth Property 4 of Defnton 3, we show that truth-tellng s ndeed a domnatng strategy. Lemma 1: For agents to report ther type (truthtellng) s a domnatng strategy. Proof: Let (a,a - ) denote a vector of strateges of all the M agents,.e., (a,a - ) = (a 1 a M ). Truth-tellng would mean that a = t. Then, for every a A, f we defne x =

4 x(a,a - ), x = x(a,a - ), p = p (a,a - ), and p = p (a,a - ), Axomatc Game Theoretcal Mechansm then p +v (t,x) p +v (t,x ),.e., u u. We can use ths result to couple t wth a useful theorem reported n [23]. Ths wll characterze the Axom 1 (Ingredents): A mechansm should have a) an algorthmc output specfcaton, and b) agents utlty functons. mechansm as truthful. Below we state the theorem. Axom 2 (Agent dsposton): Every agent has a prvate value termed Theorem 1 ([23]): A mechansm that mplements a as true data, everythng else s publc knowledge. Ths value along gven problem wth domnant strateges s truthful. wth a valuaton functon should reveal the preferences of the agent. In Defnton 3 we stated (Property 1) that a Axom 3 (Truthful): The mechansm should have agents that project mechansm m = (x( ),p( )) allows the agents to maxmze ther domnant strateges. ther utltes. Snce utltes enable the agents to express ther preferences, t s mportant to dentfy an oracle that does exactly that. Lterature survey revealed the followng Axom 4 (Utltaran): The mechansm s objectve functon should be to sum the agents valuatons. result, whch we append as property 5 to the property lst Axom 5 (Motvaton): The mechansm should reward the agents wth of Defnton 3. a payment. These payments are made n accordance to a specfed Theorem 2 ([24]): A mechansm s called utltaran functon based on the algorthmc output. f ts objectve functon satsfes g(x,t) = v (t,x). Axom 6 (Algorthmc output): The mechansm s algorthmc output Property 5: The mechansm should have an objectve functon that satsfes g(x,t) = v (t should be a functon that ads the agents to execute ther preferences.,x). The above property s so useful that t can help us dentfy the two mportant components of a game Fgure 1: Axomatc Game Theoretcal Mechansm. theoretcal mechansm. We state: Theorem 3 ([10]): A truthful mechansm m = Assume for the tme beng that the feasble output (of the (x(t),p(t)) belongs to the class of mnmzaton utltaran algorthm exsts, and) are all parttons x = x 1 x M of the mechansms f and only f: 1. x(t) argmn x ( v (t objects to the agents, where x s the set of objects,x). 2. p (t) = j v j (t j,x(t)) + h (t - ), where h allocated to agent. Also assume that each agent s utlty ( ) s an arbtrary functon of t - functon u exsts. (In the subsequent text t wll be clear. what exactly x and u are.) Remarks: Note that condtons 1 and 2 of Theorem 3 Agent dsposton (Axom 2): An agent holds two are the exact mathematcal dervatons of Propertes 2 and key elements of data a) the avalable server capacty b, 3 of Defnton 3, respectvely. Moreover, the mechansm and b) the cost of replcaton or valuaton (CoR k ) of stated n Theorem 3 takes n as argument t for both the object O k to the agent s server. There are three possble algorthmc output and the payment functon,.e., m = cases: (x( ),p( )) s now wrtten as m = (x(t),p(t)). Ths s because 1. DRP [π]: Each agent holds the cost to replcate of the mergence of Theorems 1 and 2. For convenence CoR k =t assocated wth each object O k, where as we shall now state the truthful mechansm n the the avalable server capacty and everythng else axomatc form (Fgure 1). s publc knowledge. We am to use the above (dscussed) axomatc game 2. DRP [σ]: Each agent holds the avalable server theoretcal mechansm to fnd solutons for the data capacty b =t, where as CoR k and everythng replcaton problem (Secton 3). The sx axoms defned else s publc knowledge. above wll act as a cast for the data replcaton problem. 3. DRP [π,σ]: Each agent holds both the cost of In essence, we want a replca allocaton mechansm that replcaton and the server capacty {CoR k,b } = solves the data replcaton problem wth the propertes t, where as everythng else s publc knowledge. guaranteed by the sx axoms. Remarks: Intutvely, f agents know the avalable server capactes of other agents, that gves them no advantage whatsoever. However, f they come about to 4. Axomatc Game Theoretcal Replca Allocaton Mechansm (AGT-RAM) The drectons lad down n Secton 3 wll be used to apply the axomatc game theoretcal mechansm to the data replcaton problem. Ingredents (Axom 1): The dstrbuted system descrbe n Secton 2 s consdered and t conssts of M agents. That s, each server s represented by an agent. know ther CoR k then they can modfy ther valuatons and alter the algorthmc output. Note that an agent can only calculate CoR k va the frequency of read and wrtes. Everythng else such as the network topology, latency on communcaton lnes, and even the server capactes can be publc knowledge. Therefore, DRP[π] s the only natural choce. Descrpton of valuaton (CoR k ): We can wrte CoR k as follows:

5 k CoRk = w k ( 1) o k c Pk j = + rocnn k ( = 0) + X k (, X k ) j Rk, j k k k k (, ) wocp k (, k), (5) whch mples that f an agent replcates O k (denoted n Equaton 5 as k X k =1), then the cost ncurred due to reads s 0 = r k o k c(,nn k ) snce NN k =. The cost ncurred due to local wrtes (or updates) s equal to zero snce the copy resdes locally, but whenever O k s updated anywhere n the network, agent has to contnuously update O k s contents locally as well. Therefore, the aggregate cost of wrtes s equvalent to w k o k Σ (j Rk), j c(p k,j). On the other hand f an agent does not replcate O k (denoted n Equaton 5 as k X k =0), then the cost ncurred due to reads s equal to r k o k c(,nn k ), and the cost ncurred due to wrtes s equal to w k o k c(,p k ) snce t only has to send the update to the prmary server whch then broadcasts the update based on R k to the agents who have replcated the object. Remarks: Equaton 5 (above) captures the dlemma faced by an agent when consderng replcatng O k. If replcates O k then t brngs down the read cost to zero, but now t has to keep the contents of O k up to date. If does not replcate O k, then t reduces the overhead of keepng the contents up to date, but now t has to redrect the read requests to the nearest neghborhood server whch holds a copy of O k. Truthful (Axom 3): From Lemma 1, we know that truth-tellng s a domnate strategy. From Axom 2 (above) we know that t = CoR k. Utltaran (Axom 4): We proceed n two steps. 1. Let y = {x,o} and y = {x,o }. In the context of the data replcaton problem, the valuaton of an agent s gve as v (x,t ) = k x CoR k. Ths means that when an agent s asked to express ts preference over two objects o and o, t can do so by calculatng the object mpact factor,.e., the agent expresses ts preference as mn { k y CoR k, k y CoR k }. 2. A utltaran mechansm s one that has an objectve functon that s the sum of all agents valuatons,.e., g(x,t) = v (x,t ) = k x CoR k. We can see that Axom 4 (Step 2) represents the data replcaton problem n ts exact form as descrbed n Equaton 4. For a mnute, let us ponder over both the representatons. Equaton 4 expresses the fact that n the data replcaton problem we have to fnd object allocatons such that the object transfer cost (OTC) s mnmzed. Step 2 (n conjuncton wth Step 1) of Axom 4 does exactly the same,.e., fnd the object allocatons x such that the total cost of replcaton CoR k s mnmzed Motvaton (Axom 5): The motvatonal payment for each agent s defned as p (t) = k x(t) mn t = k x(t) mn CoR k,.e., for each object allocated to t, the agent s gven payment equal to the overall second best cost of replcaton of any object to any server (a very strong ncentve). The payment procedure also answers one of the pendng questons of Axom 1 (utlty functon),.e., u = k x(t) mn t +v (t,x). Remarks: Ths motvatonal payment s need by the agents to cover the cost of hostng the object onto ther server. Ths payment also ensures that the agents do ndeed report true data. We justfy ths payment by analyzng the followng cases: 1. Over projecton: Agents n antcpaton of more revenue over project ther true data, but ths does not help, as the agent who s allocated the object gets the second best payment. 2. Under projecton: If every agent under projects ther true data, that does not help ether as the revenue would drop n proporton to the under projecton. 3. Random projecton: In ths case the deservng agent would be at loss. Therefore, t s unlkely that a selfsh agent would agree to project random true data. For more detals on the optmalty of such type of payment procedure see [27]. In that paper, the authors have dentfed many such scenaros, but all fal to explot ths payment opton. Algorthmc output (Axom 6): We now defne an algorthm that actually ads the agents to execute ther preferences. In the context of the data replcaton problem, the mechansm after gatherng the true data from every agent, decdes whch object to be allocated to whch agent. The mechansm s descrbed n Fgure 2. Ths also answers the pendng queston of Axom 1 (algorthmc output functon). Before we descrbe the algorthm, the data replcaton problem n the axomatc game theoretcal mechansm form s gven as follows: Fnd all the feasble outputs of the mechansm whch are all the parttons x={x 1 x M }, where x s the entre replca allocaton of the dstrbuted system, and x s the set of replcas allocated to server. 1. The objectve functon of the mechansm s g(x,t) = v (x,t ). 2. Agent s valuaton s v (x,t ) = k x t. 3. Agent s true data s t = CoR k. 4. Agent s payment s p (t) = k x(t) mn t. 5. Agent s utlty functon s u = p + v (t,x). Remarks: In the above problem formulaton we dd not menton that: 1) the agent s valuaton s actually to obtan the object mpact (mnmum cost of replcaton), 2) the agent s server capacty constrant, and 3) the prmary object constrants (both 2 and 3 are captured by x ), but t s to be understood that they are ndrectly embedded nto the problem formulaton.

6 Axomatc Game Theoretcal Replca Allocaton Mechansm (AGT-RAM) Intalze: LS, L, T k, Mech, MT 01 WHILE LS NULL DO 02 OMAX = NULL; MT = NULL; P = NULL; 03 PARFOR each S LS DO 04 FOR each O k L DO 05 T k = compute (t k ); /*compute the valuaton correspondng to the desred object*/ 06 ENDFOR 07 t k = argmax k (T k ); 08 SEND t k to Mech; RECEIVE at Mech t k n MT; 09 ENDPARFOR 10 OMAX = argmax k (MT); /*Choose the global domnate valuaton*/ 11 DELETE k from MT; 12 P = argmax k (MT); /*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 b =b - o k ; /*Update capacty*/ 17 L = L - O k ; /*Update the lst*/ 18 IF L = NULL THEN SEND nfo to Mech 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 Fgure 2: Pseudo-Code for Axomatc Game Theoretcal Replca Allocaton Mechansm (AGT-RAM). Descrpton of Algorthm: We mantan a lst L at each server. Ths lst contans all the objects that can be replcated by agent onto server 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 b. Moreover, f server S s the prmary host of some object k, then k should not be n L. We also mantan a lst LS contanng all servers 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 08) to the mechansm. The mechansm receves all the correspondng entres, and then chooses the best 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 payments made to the agent. The mechansm progresses forward tll there are no more agents nterested n acqurng any data for replcaton. The above dscusson allows us to deduce the followng two results about the mechansm. Theorem 4: In the worst case AGT-RAM takes O(MN 2 ) tme. Proof: The worst case scenaro s when each server has suffcent capacty to store all objects. In that case, the whle loop (Lne 01) performs MN teratons. The tme complexty for each teraton s governed by the two PARFOR loops (Lnes 03 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 5: AGT-RAM s a truthful mechansm. Proof: The algorthmc output s an allocaton that mnmzes the utltaran functon v (x,t ). Let h -I be k mn t, then v (x,t )+h - s exactly the mechansm s payment functon. It s also evdent that truth-tellng s the only domnate strategy. For smplcty let us consder the case for only one object. The argument for k>1 s smlar. Let d denote the declaratons and t ther real types. Consder the case where d t (=1,2). If d >t, then for d 3- such that d >d 3- >t, the utlty for agent s t - d <0, whch should have been zero n the case of truthtellng. 5. Experments, Results and Dscussons We performed experments on a 440MHz Ultra 10 machne wth 512MB memory. The expermental evaluatons were targeted to benchmark the placement

7 polces. AGT-RAM was mplemented usng Ada and Ada GNAT s dstrbuted systems annex GLADE [25]. To establsh dversty n our expermental setups, the network connectvely was changed consderably. We used GT-ITM [2] for the network topologes, the procedure for whch s as follows: A random graph G(M,P(edge = p)) wth 0 p 1 contans all graphs wth nodes (servers) M n whch the edges are chosen ndependently and wth a probablty p. The pure random topologes were obtaned wth p = {0.4, 0.5, 0.6, 0.7, 0.8}. In each of these topologes the dstance between two serves was reversed mapped to the communcaton cost of transmttng a 1kB of data and the latency on a lnk was assumed to be m/s (copper wre). To evaluate the replca allocaton methods under realstc traffc patterns, we used the access logs collected at the Soccer World Cup 1998 web server [2]. Each expermental setup was evaluated thrteen tmes,.e., only the Frday (24 hours) logs from May 1, 1998 to July 24, (The Frday logs have the heavest traffc compared to any other day of the week.) To process the logs, we wrote a scrpt that returned: only those objects whch were present n all the logs (25,000 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 server was mmcked by choosng random locatons. The capactes of the servers C% were generated randomly wth range from 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 mscellanes to benchmark object updates. The updates were randomly pushed onto dfferent servers, 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. Snce the access logs are of the year 1998, we frst use Inet [5] topology generator to estmate the number of nodes n the network. Ths number came up to be 3718,.e., there were 3718 AS-level nodes n the Internet at the tme when the Soccer World Cup 1998 was beng played. Therefore, we set the upper bound on the number of servers n the system at M = 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. For fne-graned replcaton, the algorthms proposed n [15], [16], [19], [21], and [26] are the only ones that address the problem doman smlar to ours. We select from [26] the greedy approach (Greedy) for comparson because t s shown to be the best compared wth 4 other approaches (ncludng the proposed technque n [19]); thus, we ndrectly compare wth 4 addtonal approaches as well. Algorthms reported n [16] (the effcent branch and bound based technque Aε-Star), [15] (Dutch (DA) and Englsh auctons (EA)) and [21] (Genetc based algorthm (GRA)) are also among the chosen technques for comparsons. Unfortunately, space lmtatons do no permt us to provde the detaled workngs of these algorthms; however, we encourage the readers to obtan an nsght on the comparatve technques from the referenced papers. Performance metrc: The soluton qualty was measured n terms of network communcaton cost (OTC percentage) that was saved under the replca scheme found by the replca allocaton methods, compared to the ntal one,.e., when only prmary copes exsts. Comparatve analyss: We observe the effects of ncrease n storage capacty. 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 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 Fgure 3, whch shows the performance of the algorthms. GRA once agan performed the worst. The gap between all other approaches was reduced to wthn 15% of each other. AGT-RAM and Greedy 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 (49%) followed by Greedy wth 44%. Further experments wth varous update ratos (5%, 10%, and 20%) showed smlar plot trends. It s also noteworthy (plots not shown n ths paper due to space restrctons) that the ncrease n capacty from 10% to 18%, resulted n 4 tmes (on average) more replcas for all the algorthms. Next, we observe the effects of ncrease n the read and wrte frequences. Snce these two parameters are complementary to each other, we descrbe them together. To observe the system utlzaton wth varyng read/wrte frequences, we kept the number of servers and objects 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

8 OTC Savngs (%) 90% 85% 80% 75% 70% 65% 60% 55% 50% 45% 40% 35% 30% 25% 20% 15% 10% 5% M=3718; N=25,000; R/W=0.95 Legend GRA Aε-Star Greedy AGT-RAM DA EA 0 10% 15% 20% 25% 30% 35% 40% Increase n Server Capacty Fgure 3: OTC savngs versus server capacty. OTC Savngs (%) 95% 90% 85% 80% 75% 70% 65% 60% 55% 50% 45% 40% 35% 30% 25% 20% 15% 10% 5% M=3718; N=25,000; C=45% Legend GRA Aε-Star Greedy AGT-RAM DA EA R/W (Rato) Fgure 4: OTC savngs versus read/wrte rato. Table 1: Runnng tme of the replca placement methods n seconds [C=45%, R/W=0.85]. Problem Sze Greedy GRA Aε-Star AGT-RAM DA EA Improvement brought by AGT-RAM (%) M=2500, N=15, [=(( )/310.14) 100] M=2500, N=20, [=(( )/330.75) 100] M=2500, N=25, [=(( )/357.74) 100] M=3000, N=15, [=(( )/452.22) 100] M=3000, N=20, [=(( )/467.65) 100] M=3000, N=25, [=(( )/469.86) 100] M=3718, N=15, [=(( )/613.27) 100] M=3718, N=20, [=(( )/630.39) 100] M=3718, N=25, [=(( )/646.98) 100] Table 2: Average OTC (%) savngs under some randomly chosen problem nstances. Problem Sze Greedy GRA Aε-Star AGT-RAM DA EA Improvement brought by AGT-RAM (%) M=100, N=1000 [C=20%,R/W=0.75] [=(( )/88.12) 100] M=200, N=2000 [C=20%, R/W=0.80] [=(( )/84.95) 100] M=500, N=3000 [C=25%, R/W=0.95] [=(( )/72.15) 100] M=1000, N=5000 [C=35%, R/W=0.95] [=(( )/88.21) 100] M=1500, N=10,000 [C=25%, R/W=0.75] [=(( )/90.25) 100] M=2000, N=15,000 [C=30%, R/W=0.65] [=(( )/73.25) 100] M=2500, N=15,000 [C=25%, R/W=0.85] [=(( )/83.21) 100] M=3000, N=20,000 [C=25%, R/W=0.65] [=(( )/83.01) 100] M=3500, N=25,000 [C=35%, R/W=0.50] [=(( )/72.15) 100] M=3718, N=25,000 [C=10%, R/W=0.40] [=(( )/77.12) 100] the replcas be placed as close as to the prmary server 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 plot n Fgure 4 shows the results of read/wrte rato aganst the OTC savngs. A clear classfcaton can be made between the algorthms. AGT-RAM and Greedy ncorporate the ncrease n the number of reads by replcatng more objects and thus savngs ncreased up to 88%, whle GRA ganed the least of the OTC savngs of up to 42%. To understand why there s such a gap n the performance between the algorthms, we should recall that GRA specfcally depends on the ntal selecton of gene populaton (for detals see [21]). 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, AGT-RAM, DA, EA, Aε-Star and Greedy never tend to devate from

9 ther global (or socal) vew of the problem. Lastly, we compare the termnaton tme of the algorthms. Varous problem nstances were recorded wth C=45% and R/W=0.85. The entres n Table 1 made bold represent the fastest tme recorded over the problem nstance. It s observable that AGT-RAM termnated faster than all the other technques, followed by Greedy, DA, EA, Aε-Star, and GRA. Table 2 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 server and objects, wth varyng storage capacty and update rato. For each row, the best result s ndcated n bold. The proposed AGT-RAM steals the show n the context of soluton qualty, but Greedy and Aε-Star do ndeed gve a good competton. In summary, based on the soluton qualty alone, the replca allocaton methods can be classfed nto four categores: 1) Hgh performance: AGT-RAM; 2) Medum-Hgh performance: Greedy; 3) Medum performance: Aε-Star and DA; 5) Low performance: EA and GRA. Consderng the executon tme, AGT-RAM and Greedy dd extremely well, followed by DA, EA, Aε- Star and GRA. 6. Related Work The data replcaton problem s an extenson of the classcal fle allocaton problem (FAP). Chu [6] studed the fle allocaton problem wth respect to multple fles n a multprocessor system. Casey [3] extended ths work by dstngushng between updates and read fle requests. Eswaran [8] proved that Casey s formulaton was NPcomplete. In [21] Mahmoud et al. provde an teratve approach that acheves good soluton qualty when solvng the FAP for nfnte server capactes. Recently, game theory has emerged as a popular tool to tackle optmzaton problems especally n the feld of dstrbuted computng. However, n the context of data replcaton t has not receved much attenton. We are aware of only three major works whch drectly or ndrectly deal wth the data replcaton problem usng game theoretcal technques. The frst work [8] s manly on cachng and uses an emprcal model to derve Nash equlbrum. The second work [17] focuses on mechansm desgn ssues and derves an ncentve compatble aucton for replcatng data on the Web. The thrd work [18] deals wth dentfyng Nash strateges derved from synthetc utlty functons. Our work dffers from all the game theoretcal technques n: 1) dentfyng a non-cooperatve non-prced based replca allocaton method to tackle the data replcaton problem, 2) usng game theoretcal technques to study an envronment where the agents behave n a self-nterested manner, and 3) dervng pure Nash equlbrum and pure strateges for the agents. Readers are encouraged to see [20] for a comprehensve survey on the conventonal replca placement technques. 7. Conclusons Ths paper proposed a sem-dstrbuted axomatc game theoretcal replca allocaton mechansm (AGT- RAM) for object based data replcaton n large dstrbuted computng systems such as the Internet. AGT- RAM s a protocol for automatc replcaton and mgraton of objects n response to demand changes. It ams to place objects n the proxmty of a majorty of requests whle ensurng that no hosts become overloaded. The nfrastructure of AGT-RAM was desgned such that, each server was requred to present a lst of data objects that f replcated onto that server would brng the communcaton cost to ts mnmum. These lsts were revewed at the central decson body whch gave the fnal decson as to what object are to be replcated onto what servers. Ths sem-dstrbuted nfrastructure takes away all the heavy processng from the central decson makng body and gves t to the ndvdual servers. For each object, the central body s only requred to make a bnary decson: (0) not to replcate or (1) to replcate. To complment our theoretcal results, we compared AGT-RAM wth fve conventonal replca allocaton methods namely: (1) branch and bound, (2) greedy, (3) genetc, (4) Englsh, and (5) Dutch auctons. The expermental setups were desgned n such a fashon that they resembled real world scenaros. We employed GT- ITM and Inet to gather varous Internet topologes and used the traffc logs collected at the Soccer World Cup 1998 webste for mmckng user access requests. The expermental study revealed that the proposed AGT-RAM technque mproved the performance relatve to other conventonal methods n four ways. Frst, the number of replcas n a system was controlled to reflect the rato of read versus wrte access. To mantan concurrency control, when an object s updated, all of ts replcas need to be updated smultaneously. If the wrte access rate s hgh, there should be few replcas to reduce the update overhead. If the read access rate s overwhelmng, there should be a hgh number of replcas to satsfy local accesses. Second, performance was mproved by replcatng objects to the servers based on localty of reference. Ths ncreases the probablty that requests can be satsfed ether locally or wthn a desrable amount of tme from a neghborng server. Thrd, replca allocatons were made n a fast algorthmc turn-around tme. Fourth, the complexty of the data replcaton problem was decreased by multfold. AGT-RAM lmts the complexty by parttonng the complex global problem of replca allocaton, nto a set of smple ndependent sub problems. Ths approach s well suted to the large dstrbuted

10 computng systems that are composed of autonomous agents whch do not necessarly cooperate to mprove the system wde goals. All the above mprovements were acheved by a smple, sem-dstrbuted, and autonomous AGT-RAM. As future work, we would extend the semdstrbuted model to regonal autonomous, self-governed and self-reparng mechansms. That s, the current system model would be broadened to ncorporate regonal or herarchcal mechansms. Ths would enable the system to be less vulnerable to the falures of a sngle mechansm, and n turn would open the realms of devsng herarchcal games, where n each level ether a cooperatve or non-cooperatve game could be played to replcate data objects. References [1] I. Ahmad and A. Ghafoor, Sem-Dstrbuted Load Balancng for Massvely Parallel Multcomputer Systems, IEEE Trans. Software Engneerng, 17(10), , [2] M. Arltt and T. Jn, Workload Characterzaton of the 1998 World Cup Web Ste, Tech. report, Hewlett Packard Lab, Palo Alto, HPL (R.1), [3] D. Campbell, Resource Allocaton Mechansms, Cambrdge Unversty Press, [4] R. Casey, Allocaton of Copes of a Fle n an Informaton Network, n Proc. Sprng Jont Computer Conf., IFIPS, 1972, pp [5] H. Chang, R. Govndan, S. Jamn and S. Shenker, "Towards Capturng Representatve AS-Level Internet Topologes," Computer Networks Journal, 44(6), pp , [6] K. Chandy and J. Hewes, Fle Allocaton n Dstrbuted Systems, n Proc. of the Internatonal Symposum on Computer Performance Modelng, Measurement and Evaluaton, 1976, pp [7] W. Chu, Optmal Fle Allocaton n a Multple Computer System, n IEEE Trans. on Computers, C-18(10), pp , [8] B.-G. Chun, K. Chaudhur, H. Wee, M. Barreno, C. Papadmtrou and J. Kubatowcz, Selfsh Cachng n Dstrbuted Systems: A Game-Theoretc Analyss, n Proc. of 23rd ACM Symposum on Prncples of Dstrbuted Computng, 2004, pp [9] K. Eswaran, Placement of Records n a Fle and Fle Allocaton n a Computer Network, Informaton Processng Letters, pp , [10] J. Green and J. Laffont, Characterzaton of Satsfactory Mechansms for the Revelaton of Preferences for Publc Goods, Econometrca, 45(2), pp , [11] T. Groves, Incentves n Teams, Econometrca, vol. 41, pp , [12] S. Hakm, Optmum Locaton of Swtchng Centers and the Absolute Centers and Medans of a Graph, Operatons Research, vol. 12, pp , [13] S. Jamn, C. Jn, Y. Jn, D. Raz, Y. Shavtt and L. Zhang, On the Placement of Internet Instrumentaton, n Proc. of the IEEE INFOCOM, 2000, pp [14] M. Karlsson and M. Mahalngam, Do We Need Replca Placement Algorthms n Content Delvery Networks? n Proc. of Web Cachng and Content Dstrbuton Workshop, 2002, pp [15] S. Khan and I. Ahmad, Internet Content Replcaton: A Soluton from Game Theory, Department of Computer Scence and Engneerng, Unversty of Texas at Arlngton, Techncal Report, CSE , [16] S. Khan and I. Ahmad, Heurstc-based Replcaton Schemas for Fast Informaton Retreval over the Internet, n Proc. of 17th Internatonal Conference on Parallel and Dstrbuted Computng Systems, 2004, pp [17] S. Khan and I. Ahmad, A Powerful Drect Mechansm for Optmal WWW Content Replcaton, n Proc. of 19th IEEE Internatonal Parallel and Dstrbuted Processng Symposum, [18] N. Laoutars, O. Telels, V. Zssmopoulos and I. Stavrakaks, Local Utlty Aware Content Replcaton, to appear n IFIP Networkng Conference, [19] B. L, M. Goln, G. Italano and X. Deng, On the Optmal Placement of Web Proxes n the Internet, n Proc. of the IEEE INFOCOM, 2000, pp [20] T. Loukopoulos, D. Papadas and I. Ahmad, "An Overvew of Data Replcaton on the Internet," n Proc. of Internatonal Symposum on Parallel Archtectures, Algorthms and Networks, 2002, pp [21] T. Loukopoulos, and I. Ahmad, Statc and Adaptve Dstrbuted Data Replcaton usng Genetc Algorthms, Journal of Parallel and Dstrbuted Computng, 64(11), pp , [22] S. Mahmoud and J. Rordon, Optmal Allocaton of Resources n Dstrbuted Informaton Networks, ACM Trans. on Database Systems, 1(1), pp , [23] A. Mas-Collel, W. Whnston and J. Green, Mcroeconomc Theory, Oxford Unversty Press, [24] N. Nsan and A. Ronen, "Algorthmc Mechansm Desgn," n Proc. of 31st ACM Symposum on Theory of Computaton, 1999, pp [25] L. Pautet and S. Tardeu, GLADE: A Framework for Buldng Large Object-Orented Real-Tme Dstrbuted Systems, n 3rd Internatonal Symposum on Object-Orented Real-Tme Dstrbuted Systems, 2000, pp [26] L. Qu, V. Padmanabhan and G. Voelker, On the Placement of Web Server Replcas, n Proc. of the IEEE INFOCOM, 2001, pp [27] S. Saurabh and D. Parkes, Hard-to-Manuplate VCG- Based Auctons, Avalable at: economcs/pubs/hard_ to_manpulate.pdf [28] L. Tesfatson, Agent-based Computatonal Economcs: Growng Economes from the Bottom Up, Artfcal Lfe, vol. 8, no. 1, pp , 2002.