A Competitive Replica Placement Methodology for Ad Hoc Networks

Transcription

1 World Academy of Scence, Engneerng and Technology A Compettve Replca Placement Methodology for Ad Hoc Networks Samee Ullah Khan and Cemal Ardl Abstract In ths paper, a mathematcal model for data object replcaton n ad hoc networks s formulated. The derved model s general, flexble and adaptable to cater for varous applcatons n ad hoc networks. We propose a game theoretcal technque n whch players (moble hosts) contnuously compete n a non-cooperatve envronment to mprove data accessblty by replcatng data objects. The technque ncorporates the access frequency from moble hosts to each data object, the status of the network connectvty, and communcaton costs. The proposed technque s extensvely evaluated aganst four well-known ad hoc network replca allocaton methods. The expermental results reveal that the proposed approach outperforms the four technques n both the executon tme and soluton qualty. Keywords Data replcaton, auctons, statc allocaton. I. INTRODUCTION HERE s contnung nterest to study and mprove T technologes related to ad hoc networks, as they are constructed from only moble hosts and requre no partcular nfrastructure [1]. In such a setup, every moble host acts as a router and communcates wth other moble hosts. As a consequence even f the source and destnaton moble hosts are not n each other s communcaton range, data s stll forwarded to the destnaton moble host by relayng transmsson through other moble hosts whch exst between the two moble hosts. Due to the contnuous moblty of hosts, an ad hoc network suffers from frequent dsconnectons. Ths phenomenon s undesrable when moble hosts are accessng data from each other. Snce one cannot control network dsconnectons, an alternatve soluton to ths problem s to replcate data onto moble hosts so that when dsconnectons occur, moble hosts can stll access data [7]. The decson where to place the replcated data must trade off the cost of accessng the data, whch s reduced by addtonal copes, aganst the cost of storng and updatng the replcas. These costs have severe mplcatons n ad hoc networks snce moble hosts have poor resources (storage and processng power). In general the moble hosts would experence reduced access latences provded that data s S. U. Khan s wth the Department of Electrcal and Computer Engneerng, North Dakota State Unversty, Fargo, ND 5812, USA (phone: ; fax: ; e-mal: samee.khan@ ndsu.edu). C. Ardl s wth the Natonal Academy of Avaton, Baku, Azerbajan, (emal: cemalardl@gmal.com). 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 moble hosts, and 2) n close proxmty to the prmary (assumng a master replcaton envronment [8]) copy. Therefore, effcent and effectve replcaton schemas strongly depend on how many replcas to be placed n the system, and more mportantly where [1]. Ad hoc data replcaton problem (ADRP) was frst ntroduced by Hara [5], whch was further extended ([6], [7]) to ncorporate varous network connectvty related ssues. Although the above mentoned works are plausble n the sense that they advance the study of ADRP, yet none nvolve reasonng va a concrete mathematcal model. Thus, t s mperatve to derve and understand an optmzaton model that s general, flexble and modfable to cater for varous applcatons of ADRP. In ths paper, we wll frst focus on dervng a mathematcal model for ADRP and show that ths problem n general s NP-complete. We then follow t up by proposng Mosac-Net (acronym of game theoretcal method for selecton and allocaton of replcas n ad hoc networks), whch exhbts fast executon tme and guarantees excellent soluton qualty. The remnder of ths paper s organzed as follows. In Secton II, we gve a bref descrpton of the related works and dscuss how our approach s dfferent from others. Secton II focuses on dervng a mathematcal model for ADRP. Secton IV depcts the Mosac-Net technque for ADRP, followed by expermental evaluatons n Secton V. Fnally, n Secton VI, we summarze ths paper. II. RELATED WORK Several strateges for cachng data contents n moble computng envronments have also been proposed [1], [2], [8], [9], [11], [13]. Most of these strateges assume an envronment where moble hosts access contents at stes n a fxed network, and cache data on the moble hosts because wreless communcaton s more costly than wred communcaton. Such strateges address the ssue of keepng consstency between orgnal data and ts replcas or caches wth low communcaton costs. They are consdered to be smlar to our approach, because both approaches replcate data on moble hosts. However, these strateges assume only one-hop wreless communcaton, and thus, they are 128

2 World Academy of Scence, Engneerng and Technology completely dfferent from our approach that assumes multhop ad hoc communcaton. Another closely related research topc s of push-based nformaton systems n whch a server repeatedly broadcasts data to clents usng a broadband channel. Several cachng strateges have been proposed to mprove user access [4], [12]. In these strateges, clents are typcally moble hosts, and the cache replacement s determned based on several parameters such as the access frequency from each moble host to each data tem, the broadcast frequency of each data tem, and the tme remanng untl each tem s broadcast next. They are also consdered to be smlar to our approach, because both approaches replcate data on moble hosts. However, comparng the strateges for cachng or replcatng, both approaches are completely dfferent because the strateges n push-based nformaton systems do not assume that the clents cooperatvely share cached data n ad hoc networks. Hara s work ([5], [6], [7]) s the closest among all the related works on ADRP compared to ths paper. However, our work dffers from Hara s n: 1) dervng a mathematcal problem formulaton for ADRP, 2) provng that the generalzed form of ADRP s NP-complete, 3) proposng an optmzaton technque that allocates replcas so as to mnmze the network traffc under storage constrants wth read from the nearest and push based update through the prmary moble host polces, 4) extensvely evaluatng the technques under varyng system parameters, 5) usng game theoretcal technques. III. PROBLEM FORMULATION Consder an ad hoc network comprsng m moble hosts, wth each moble host havng ts own processng power and storage. Let M and s be the name and the total storage capacty (n smple data unts e.g. blocks), respectvely, of moble host where 1 m. The m moble hosts can communcate wth each other usng a wreless communcaton network. A wreless channel between two moble hosts M and M j (f t exsts) has a postve nteger c(,j) assocated wth t, gvng the communcaton cost for transferrng a data unt between moble hosts M and M j. If the two moble hosts are not one-hop connected by a wreless channel then the above cost s gven as the sum of the costs of all the wreless channels (mult-hop) n a chosen path from ste M to the ste M j. Wthout the loss of generalty we assume that c(,j) = c(j,). (Such an assumpton can be relaxed when the upstream and downstream bandwdths vary for a wreless channel.) Let there be n data 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 u k be the total number of reads and updates, respectvely, ntated from M for O k durng a certan tme perod. In other words, r k and u k are the read and update frequences, respectvely. Our replcaton polcy assumes the exstence of one prmary copy for each object n the system. Let P k, be the moble host 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 moble host of the k-th object. Each prmary moble host P k, contans nformaton about the whole replcaton scheme R k of O k. Ths can be done by mantanng a lst of the moble hosts where the k-th object s replcated at, called from now on the replcators of O k. When a moble host M ntates a read request r k for a data object O k, the request s redrected to the nearest neghbor NN k moble host that holds ether the orgnal or the copy of the data object O k. In other words, NN k s the moble host for whch the reads from M for O k, f served there, would ncur the mnmum possble communcaton cost. It s possble that NN k = M, f M s a replcator or the prmary moble host of O k. Another possblty s that NN k = P k, f the prmary moble host s the closest one holdng a replca of O k. For the updates we assume that only the moble host P k (that owns the data object(s)) can perform such operatons. Such an assumpton s justfable when consderng sensor networks, whch s a specal class of ad hoc networks. P k updates a data object O k by sendng broadcasts to the set of moble hosts that hold the replcas of O k,.e., M R k. For the ADRP under consderaton, we are nterested n mnmzng the total network transfer cost (NTC) due to data object movement,.e. the data object movement due to the read and update accesses. It s to be noted that the mnmzaton of NTC n turn leads to ncreased data accessblty. There are two components affectng NTC. The frst component of NTC s due to the read requests. Let R k denote the total NTC, due to M s readng requests for object O k, addressed to the nearest ste NN k. Ths cost s gven by the followng equaton: R = rocnn (, ), k k k k (1) where NN k = {Moble host j j R k and mn c(,j)}. The second component of NTC s the cost arsng due to the updates. Let U k be the total NTC, due to P k s updates requests for object O k. Ths cost s gven by the followng equaton: U = ukok (, ) k cp. k (2) R k The cumulatve NTC, denoted as C overall, due to reads and wrtes s gven by: C m n = 1 1 ( overall = k R + = k U k ). (3) Let X k =1 f M holds a replca of object O k, and otherwse. X k s defne an m n replcaton matrx, named X, wth boolean elements. Moble hosts whch are not the replcators of data object O k create NTC equal to the communcaton cost of ther reads from the nearest replcator. Stes belongng to the replcaton scheme of O k, are assocated wth the cost of recevng all the updated versons of t. Usng the above formulaton, the ADRP can be defned as: Fnd the assgnment of, 1 values n the X matrx that mnmzes C overall, subject to the storage capacty constrant: n k k, X o s (1 m) k= 1 129

3 World Academy of Scence, Engneerng and Technology and subject to the prmary copes polcy: X = 1 (1 k n). Pk k IV. PROPOSED TECHNIQUE In ths secton we wll frst descrbe the bascs of our proposed game theoretcal model followed by extendng the model to ADRP. The Setup: The ad hoc network descrbed n Secton 3 s consdered, where each moble host s represented by a player,.e., the mechansm contans m players. In the context of the ADRP, a player holds two key elements of nformaton: a) the avalable moble host storage, as, and b) the read access frequency, r k, for a data object k Intutvely, f players know the avalable storage of each other, that gves them no advantage whatsoever. However, f they come about to know the read frequency, then they can easly modfy ther bds and alter the algorthmc output, as the read frequency drectly represent the popularty of data objects [1]. It s to be noted that a player can only calculate a data object k s beneft (that t brngs by reducng the cost of read accesses due to replcaton) by makng use of the access frequency, and thus, everythng else such as the network topology, latency on communcaton lnes, and even the moble host capactes can be publc knowledge. Therefore, DRP[π] s the only natural choce. Communcatons: The players n the mechansm are assumed to be greedy (or selfsh) and therefore, they project a bd b (as apposed to t ) to the mechansm. At ths moment we want to clarfy that the mechansm has no way of knowng that b s t or some other value. However, n the subsequent text we wll rgorously prove that f the mechansm provdes proper ncentve (payments to compensate for holdng replcas of data objects) to the players, then they are n fact forced to bd b = t. 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 ADRP, the algorthm accepts bds from all the players, and outputs the maxmum benefcal bd,.e., the bd that ncurs the mnmum NTC due to object movement (Equaton 3). (We wll gve a detaled descrpton n the subsequent text.) Monetary cost: When a data object s allocated (for replcaton) to a player, the player becomes responsble to entertan (read and wrte) requests to that object. For example, assume object k s replcated to player, then the amount of traffc that the player has to entertan due to the replcaton of object k s exactly equvalent to the NTC cost,.e., c = R k + U k. Ths fact s easly deducble from Equaton 3. Payments: To offset c, the mechansm makes a payment p (b) to player. Ths payment s equvalent to the second hghest bd that s submtted to the mechansm for a data object k. The readers would mmedately note that n such a payment functon, a player can never obtan a net proft greater than. Ths s exactly what we want. In a selfsh (or greedy) envronment, t s possble that the players bd hgher than the true value. The mechansm creates an lluson to negate that, by compensatng the players wth the payment that s lower than ther ncurred cost. Ths leaves no room for the players to overbd or underbd. For example, 1) If a player overbds for an object k, t gves no advantage snce t receves payment equvalent to the second hghest bd; 2) If a player underbds for an object k, t lessens ts chances to (wn and) replcate k. In lterature, such a payment functon s termed as Vckrey payment [14]. For more detals on the optmalty of such type of a payment functons see [12]. In that paper, the authors have dentfed many such scenaros, but all fal to explot ths (Vckrey) payment opton. We want to emphass that each player s ncentve s to replcate data objects so that queres can be answered locally. If the replcas are made avalable elsewhere, the cost to access data would be much hgher. Bds: Each player reports a bd that s the drect representaton of the true data that t holds,.e., b = r k o k c(,nn k ) = R k. Readers wll mmedately notce that b only represents one-half of the NTC. Ths s because U k s the prvate nformaton for P k. In essence, the mechansm ς(x(b),p(b)), takes n the vector of bds b from all the players, adjusts the bds by njectng the cost of updates and then selects the hghest bd. The hghest bdder s allocated the data object k whch s added to ts allocaton set x. The mechansm then pays the bdder p. Ths payment s equvalent to the second hghest bd. At a relocaton perod, a moble host mght not connect to another moble host whch has an orgnal or a vald replca of a data tem that the host should allocate. In ths case, the memory space for the replca s temporarly flled wth one of the replcas that have been allocated snce the prevous relocaton perod but are not currently selected for allocaton. Ths temporarly allocated replca s chosen from among the possble replcas accordng to Mosac-Net. If there s no replca that can be temporarly allocated, the memory space remans free. When data access to the data tem whose replca should be allocated succeeds, the memory space s flled wth the vald replca as depcted n Fg. 1. Descrpton of Algorthm: At a relocaton perod, each moble host broadcasts ts host dentfer. After all moble hosts complete ther broadcasts, every host knows ts connected moble hosts. In every connected ad hoc network, the Mosac-Net s nvoked. We mantan a lst L at each 13

4 World Academy of Scence, Engneerng and Technology A Compettve Technque for Replca Placement n Ad hoc Networks (Mosac-Net) Intalze: 1 LM, L, T k, ς, MT 2 WHILE LS NULL DO 3 OMAX = NULL; MT = NULL; P = NULL; 4 PARFOR each M LS DO 5 FOR each O k L DO 6 B k = compute (t ); /*compute the valuaton correspondng to the desred object*/ 7 ENDFOR 8 b = argmax k (B k ); 9 SEND t to ς; RECEIVE at ς b and mmedately adjust b to v and save n MT; 1 ENDPARFOR 11 OMAX = argmax k (MT); /*Choose the global domnate valuaton*/ 12 p = argmax k (delete(argmax k (MT))) ; /*Calculate the payment*/ 13 BROADCAST OMAX; 14 SEND P to M ; /*Send payments to the agent who s allocate the object OMAX*/ 15 Replcate O OMAX ; 16 as =as - o k ; /*Update capacty*/ 17 L = L - O k ; /*Update the lst*/ 18 IF L = NULL THEN SEND nfo to ς to update LM = LM - M ; /*Update mechansm players*/ 19 PARFOR each M LM DO 2 Update NN OMAX /*Update the nearest neghbor lst*/ 21 ENDPARFOR /*Get ready for the next round*/ 22 ENDWHILE Fg. 1 Pseudo-code for Mosac-Net moble host. Ths lst contans all the data objects that can be replcated by player onto the moble host M. We can obtan ths lst by examnng the two constrants of the ADRP. Lst L would contan all the data objects that have ther sze less then the total avalable storage as. Moreover, f the moble host M s the prmary host of some object k, then k should not be n L. We also mantan a lst LM contanng all moble hosts that can replcate an object,.e., M LM f L NULL. The algorthm works teratvely. In each step the mechansm (ς) asks all the players to send ther preferences (frst PARFOR loop). Each player recursvely calculates the true data of every data object n lst L. Each player then reports the domnant true data (lne 9) to the mechansm. The mechansm receves all the correspondng entres, njects t wth the update cost and then chooses the globally domnant true data. Ths s broadcasted to all the players, so that they can update ther nearest neghbor table NN k, whch s shown n Lne 2 (NN OMAX). The object s replcated and the payment s made to the player. The mechansm progresses forward tll there are no more players nterested n acqurng any data for replcaton (Lne 18). V. DISCUSSION OF RESULTS Moble hosts exst n a sze 1 1 flatland. Each host randomly moves n all drectons, and the movement speed s randomly determned from to d. The rado communcaton range of each moble host has a radus of R. The number of moble hosts was set to 2, and the number of data objects was set to 2. The prmary data object s orgnal moble host was mmcked by choosng random locatons. The sze of data objects was obtaned usng the hghly cted [3] hybrd lognormal-pareto dstrbuton, where the dstrbuton s body follows lognormal and tal follows pareto. In all the experments, the basc storage capacty of moble hosts (C%) was proportonal to the total sze of data objects. In order to ensure that the system had moble servers wth dverse enough storng capabltes, the actual storage capacty, s, of a moble host was a random value between (C/2)% and (3C/2)%. For smulaton purposes, the replcas were Moble hosts exst n a sze 1 1 flatland. Each host randomly moves n all drectons, and the movement speed s randomly determned from to d. The rado communcaton range of each moble host has a radus of R. The number of moble hosts was set to 2, and the number of data objects was set to 2. The prmary data object s orgnal moble host was mmcked by choosng random locatons. The sze of data objects was obtaned usng the hghly cted [3] hybrd lognormal-pareto dstrbuton, where the dstrbuton s body follows lognormal and tal follows pareto. In all the experments, the basc storage capacty of moble hosts (C%) 131

5 World Academy of Scence, Engneerng and Technology % 1% 2E+7 Data Accessblty (%) 9% 8% 7% 5% 4% 3% 2% 1% Mosac-Net NTC Savngs (%) 9% 8% 7% 5% 4% 3% 2% 1% Mosac-Net Traffc 1.75E+7 1.5E E+7 1E Mosac-Net Standard Devaton (σ) Fg. 2 (a) Scatter of access and data accessblty Standard Devaton (σ) Fg. 2(c) Scatter of access and NTC Standard Devaton (σ) Fg. 2(b) Scatter of access and traffc Data Accessblty (%) 72% 68% 64% 56% 52% Mosac-Net 48% 44% 4% 36% 32% 28% 24% 2% 16% 12% 8% Zpf (θ) Traffc 7E+7 6.5E+7 6E+7 5.5E+7 5E+7 4.5E+7 Mosac-Net 4E+7 3.5E+7 3E+7 2.5E+7 2E+7 1.5E+7 1E Zpf (θ) NTC Savngs (%) 9% 87% 84% 81% 78% 75% 72% 69% 66% 63% 57% 54% 51% 48% 45% 42% 39% Mosac-Net 36% 33% 3% Zpf (θ) Fg. 3 (a) Updates and data accessblty Fg. 3 (b) Updates and traffc Fg. 3 (c) Update and NTC TABLE I RUNNING TIME IN SECONDS Problem Sze Mosac-Net m=2, n=2 [d=1, R=7, T=256, C=3%, σ=.1, θ=.45] m=2, n=2 [d=1, R=7, T=256, C=3%, σ=.1, θ=.65] m=2, n=2 [d=1, R=7, T=256, C=3%, σ=.1, θ=.75] m=2, n=2 [d=1, R=7, T=256, C=3%, σ=.1, θ=.85] m=2, n=2 [d=1, R=7, T=256, C=3%, σ=.1, θ=.95] was proportonal to the total sze of data objects. In order to ensure that the system had moble servers wth dverse enough storng capabltes, the actual storage capacty, s, of a moble host was a random value between (C/2)% and (3C/2)%. For smulaton purposes, the replcas were the assumptons and system parameters were kept the same n all the approaches. The technques studed nclude: 1) randomzed, 2) extended statc access frequency () [6], 3) extended dynamc access frequency and neghborhood () [6], and 4) extended dynamc connectvty groupng () [6]. Note that the proposed game theoretcal technque has an acronym Mosac-Net. Due to space restrctons, we could not provde the detaled workngs for each of the comparatve technques. Readers are encouraged to fnd more about the technques from the referenced papers. We use three performance metrcs defned as follows: 1. Data accessblty: Percentage number of successful hts/access to data objects. 2. Traffc: The total hop count of data transmsson for allocatng/relocatng replcas. 3. NTC savngs: Network transfer cost (n percentage) that s saved under the replcaton scheme found by the algorthm, compared to when only prmary copes exsts. We examne the characterzaton of scatterng the read access frequency of moble hosts. To observe ths the read access frequences of moble hosts are determned based on case 3, and the standard devaton, σ, s changed. When σ =, access frequences to data objects s equvalent to case 1. As σ ncreases, the scatter f read access frequences ncreases. As a consequence the dfference n read access characterstcs among the moble hosts also ncreases. We fx m = 2, n = 2, T = 256, C = 3%, d = 1, R = 7 and θ =.9. Fgs. 2(a)- 2(c) show the smulaton results. Fg. 2(a) shows that as the dfference of read access characterstcs gets larger, the relatve dfference n data accessblty ncreases. Ths s because when the scatter of access characterstcs s larger, each moble host allocates more replcas, and thus, the moble hosts share a wde range of data objects. Ths sharng of data objects n turn generates traffc (for accessng data objects) as depcted n Fg. 2(b). However, notce that wth the ncrease n read access 132

6 World Academy of Scence, Engneerng and Technology characterstc, the traffc generated by Mosac-Net method reduces. Ths phenomenon s attrbuted to the fact that when the access characterstc s larger, the degree of duplcaton s small n Mosac-Net compared to that of. Wth the ncrease n read access characterstcs, all methods showed ncrease n NTC savngs. Mosac-Net showed (Fg. 2(c)) the most savngs followed by. Observe that nearly every method showed almost 9% of ts total mprovement n NTC savngs wth an ntal ncrease n σ (as lttle as.1). Ths s due to the fact that σ njects much more dversty than the number of objects avalable n the system. Thus, wth the later ncrease n σ not much of NTC savngs s observed. We examne the effects of change n updates. We fx m = 2, n = 2, T = 256, C =3%, d = 1, R = 7 and σ =.1. The ncrease n the frequency 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 storage capacty, as the update rato sets an upper bound on the possble traffc reducton through replcaton. Thus, f we consder a system wth unlmted storage capacty, the replcate everywhere anythng polcy s strctly nadequate [1]. Fg. 3(a)-3(c) show the results. Fg. 3(a) shows that as the Zpf parameter (θ) ncreases, the accessblty of every method decreases. The reason behnd ths s that when the update frequency ncreases, the replca contents change wth respect to ther orgnals frequently. Ths n turn shows that when the update frequency s hgher, all the comparatve methods (,, and ) dscussed n ths paper do not perform well. Ths s attrbuted to the fact that the replcas quckly become nvald, and thus, relocaton consderng the tme remanng untl each data object s updated next has rather adverse effects. On the other hand f the update frequency s lower, relocaton consderng the tme remanng untl each data object s updated next s meanngless and t mght also result n adverse effects. Fg. 3(b) shows the relatonshp between traffc generated and the Zpf parameter (θ). method produces the hghest amount of traffc and Mosac-Net method produces the next hghest. Notce that as the update frequency gets hgher the traffc soars. Ths s because each moble host must frequently refresh the replcas that they hold after the orgnals have been updated. Ths fact s also realzed n Fg. 3(c) where all the methods gradually loose NTC savngs as the Zpf parameter ncreases. From the Fgs. 3(a)-3(c) we can observe that the dea value for the Zpf parameter s.9. Fnally, we compare the termnaton tme of the algorthms. We randomly choose the largest sze problem nstances wth varyng parameters. The entres (Table I) n bold represent the fastest tme recorded over the problem nstance. It s observable that and Mosac-Net termnated faster than all the other technques. VI. CONCLUSION The proposed game theoretcal replca allocaton mechansm n ad hoc networks (Mosac-Net) s a protocol for automatc replcaton and mgraton of objects n response to demand changes. Mosac-Net ams to place objects n the proxmty of a majorty of requests whle ensurng that no moble hosts become overloaded. We compared Mosac-Net wth some well-known technques, as: randomzed, extended statc access frequency, extended dynamc access frequency and neghborhood and extended dynamc connectvty group. Mosac-Net compared to other technques exhbted up to 2% better soluton qualty and consderable savngs n termnaton tmngs. As future work, we would lke to extend Mosac-Net to ncorporate the phenomenon of unstable rado lnks [7]. 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