GTPP: General Truncated Pyramid Peer-to-Peer Architecture. over Structured DHT Networks

Transcription

1 GTPP: Geeral Trucated Pyramid Peer-to-Peer Architecture over Structured DHT Networs Zhoghog Ou, Eri Harjula, Timo Kosela ad Mia Yliattila Abstract Hierarchical Distributed Hash Table (DHT) architectures have bee amog the most iterestig research topics sice the birth of flat DHT architecture. However, most of the previous wor has merely focused o the two-tier hierarchy. I this paper, we study ad aalyze Geeral Trucated Pyramid Peer-to-Peer (GTPP) architecture, the geeralized versio of Partially Vertical Hierarchical Architecture (PV-HA). The idea is to study whether added tiers of hierarchy ca provide added value i performace ad fuctioality. Through mathematical aalysis, we demostrate performace results i compariso to flat architecture, which helps uderstadig the typical characteristics of hierarchical architectures. Firstly, GTPP has slightly higher expected looup hop cout, although it ca be decreased with optimizig the sub-overlay setup. However, GTPP sigificatly decreases the expected looup routig latecy. Secodly, GTPP has clearer ad more reasoable traffic distributio amog all the peers from differet tiers of sub-overlays, ad ca wor with slightly lower maiteace traffic. Thirdly, our studies idicate that -3 tiers are most suitable i most cases for GTPP, cosiderig all the parameters. Keywords geeral trucated pyramid peer-to-peer architecture, hierarchical architecture, partially vertical, fully vertical, horizotal, DHT. I. INTRODUCTION Peer-to-Peer (PP) techologies have attracted extesive attetio of the academia, idustry, ad the media i the past few years due to their decetralized, failure tolerat ad scalable characteristics. A few importat proposals have bee put forward to implemet the distributed looup services utilizig Distributed Hash Table (DHT), icludig Chord [], CAN [], Kademlia [3], Pastry [4], Tapestry [5], just to ame a few. The ey poit of DHT-based PP overlay looup algorithms is that they firstly defie a certai metric for the distace of the peer-peer ad peer-ey, ad the store the ey-value pair i the closest peer accordig to the distace metric defied. Whe looig up the value of the ey, they utilize differet mechaisms to approach the target peer which stores the ey-value pair step by step, ad fially, get the value of the ey i a logarithmic umber of hops. Origially, the aforemetioed DHT looup algorithms are based o flat architecture, which meas the peers of the overlay have the same fuctioalities ad load i routig ad maiteace, etc. O the other had, the hierarchical architecture almost always accompaies the large-scale, complex systems. O the Iteret, routers are grouped ito various autoomous systems (AS). Differet routig protocols are used for the itra-as routig ad iter-as routig; while the former utilizes Routig Iformatio Protocol (RIP) or Ope Shortest Path First (OSPF) protocol, etc., the latter uses Border Gateway Protocol (BGP), etc. Whe describig the desig of a global ame system, Butler Lampso [6] stated the hierarchy as a fudametal method for accommodatig growth ad isolatig faults. I the cotext of DHT desig, hierarchy has This wor was supported i part by Tees, Noia, Ericsso ad Nethaw i the project Decicom. Zhoghog Ou was with Beijig Uiversity of Posts ad Telecommuicatios. He is ow with Departmet of Electrical ad Iformatio Egieerig, Uiversity of Oulu, Filad, ad School of Electroic Egieerig, Beijig Uiversity of Posts ad Telecommuicatios, Beijig, Chia. ( tommy@ee.oulu.fi). Eri Harjula, Timo Kosela ad Mia Yliattila are with Departmet of Electrical ad Iformatio Egieerig, Uiversity of Oulu, FIN-94, Filad. ( firstame.lastame@ee.oulu.fi ).

2 also some otable advatages proved by the previous wor: fault isolatio ad security, effective cachig ad badwidth utilizatio [7], deductio i looup hops ad latecy [8], adaptatio to the uderlyig physical etwor [7] [9], providig admiistratio cotrol ad autoomy [8] [], more adaptable to mobile eviromets []. I our previous studies [] [3], we also foud the same advatage of hierarchical DHT, e.g. decreased looup hop cout. I this paper, we cotiue our exploratio i the hierarchical DHT desig, or more precisely, the multiple-tier hierarchical DHT which is ot limited to two-tier, as was usually doe i the previous wor. We first put forward a geeralized multiple-tier hierarchical DHT, called Geeral Trucated Pyramid Peer-to-Peer (GTPP) architecture. Geeral here meas the architecture is built o the basic fuctioalities of the hierarchical DHT, all of the tiers utilize the same basic DHT looup algorithm (Chord as our istatiatio), o optimizatio is adopted i the architecture, e.g. aycast [], multicast [4], mesh-coectio or sigle-coectio [5] [6] i the lower tier. Thus, we ca treat GTPP as a geeralized versio of multiple-tier hierarchical DHT architecture. With these assumptios, we the aalyze the expected looup hop cout, expected looup routig latecy, traffic distributio of each sigle peer from a differet tier of sub-overlay ad the total traffic of GTPP compared to flat architecture. Fially, we give the experimetal results based o the two-tier hierarchical commuity maagemet system. The mai coclusios i this paper are as follows: GTPP, as the geeralized versio of multiple-tier hierarchical DHT architecture, has slightly higher amout of looup hop cout tha flat architecture. However, the looup latecy of GTPP is sigificatly decreased compared to flat architecture. GTPP has clearer ad more reasoable traffic distributio amog the peers from differet tiers of sub-overlays, i.e. the higher tier a peer locates, the more traffic it shares. The maiteace traffic of GTPP is slightly less tha flat architecture, although the total traffic of GTPP is slightly more tha flat architecture i a highly active overlay etwor where there are a great deal of looup operatios. The total traffic of GTPP is icreased as the umber of tiers icreases. As a whole, GTPP shows its advatages compared to flat architecture, our studies idicate that -3 tiers are most suitable i most cases for GTPP, cosiderig all the parameters. The remaider of this paper is orgaized as follows: Sectio II gives the related wor ad describes the taxoomy of hierarchical architecture. Sectio III itroduces the GTPP architecture. Sectio IV aalyzes the performace of GTPP. Sectio V presets the evaluatio results. I Sectio VI, we elaborate the experimetal results. Sectio VII gives some discussio about the cotet cachig ad iterative routig mechaism. Sectio VIII describes the future wor ad i Sectio IX we coclude the article. II. RELATED WORK AND TAXONOMY Hierarchical PP architecture is resulted from the high popularity of file-sharig applicatios based o ustructured PP etwors, e.g. KaZaA [7], Gutella [8]. Large populatio of KaZaA ad Gutella applicatio results i large scale PP etwors aroud the world, which i tur causes the ever icreasig search traffic across differet admiistrative domais. Meawhile, the search recall ratio, i.e. the ratio of the umber of search results ad the total umber of available copies of the searched object, is

3 3 sigificatly decreased as the etwor scale becomes larger [6]. Therefore, KaZaA ad Gutella itroduce the hierarchy i their architectures. KaZaA [7] desigates the more powerful ad available peers as super-odes. The ew odes joiig the KaZaA etwor set up coectios to the closest super-odes though the shortest Roud Trip Time (RTT) metric. The super-odes costruct a top-level overlay etwor by proprietary desig. I the later versio of Gutella [8], the similar mechaism is used which it ames as super-peers. No-super-peer loos up the service provided by the overlay through the gateway super-peers. The top level overlay utilizes the same ustructured algorithm as ormal Gutella. Aother similar architecture is CAP [9], a two-tier ustructured PP etwor focusig o scalability ad stability. Hierarchical PP Systems Structured (DHT-based) Ustructured Vertical Horizotal KaZaA [7] Gutella [8] CAP [9] Partially Fully Cao [7] Cycloe [] mdht [4] Two-tier Brocade [], PPSIP [3], Garces-Erice et al.[8], Mislove et al. [], Zoels et al. [5] [6]. Multipletier Le et al. [], GTPP [] [3] HIERAS [9] Fig.. Taxoomy of hierarchical PP architecture. I the structured PP etwors, or more precisely, DHT-based PP etwors, the same tred ca be foud. Sa chez-artigas et al. [] distiguish horizotal hierarchical ad vertical hierarchical architecture ad summarize the differeces as follows: I a horizotal overlay, all the leaf overlay etwors are coected usig a sigle DHT that cotais the coceptual hierarchy ad optimizes the routig i the whole etwor, while i a vertical hierarchical architecture, every layer or leaf i the hierarchical tree is a self-cotaied DHT overlay etwor. I this paper, we further divide the vertical hierarchical architecture ito fully-vertical hierarchical architecture (FV-HA) ad partially-vertical hierarchical architecture (PV-HA). I FV-HA, all the peers participate i the top-tier sub-overlay as well as its local sub-overlay; therefore, o gateway peers are eeded; while i PV-HA, just partial peers participate i the top-tier overlay, the other peers do ot participate i the top-tier because of firewall security, Networ Address Traslatio (NAT), admiistrative reaso, etc. Therefore, dedicated gateway peers are eeded i PV-HA to route the query o behalf of the peers from lower-tiers of sub-overlays. A illustrated category of hierarchical PP architecture is show i Fig.. I the horizotal hierarchical architecture (H-HA), Cao [7], mdht [4], ad Cycloe [] are typical examples. Cao [7] ad Cycloe [] adopt special Idetifier (ID) desig mechaism to mae the total umber of lis per ode maitaiig remai the same as the flat DHT desig. I mdht [4], the selectio of eighborig lis is more flexible, it treats a group of host computers

4 4 i a subset participatig i the DHT overlay as a sigle ode, ad multiple lis ca be made betwee differet odes. I H-HA, o gateway peers are eeded. The advatages of H-HA iclude: fault isolatio, load balacig, topology awareess, ad hierarchical storage of cotet ad access cotrol [7]. To our best owledge, the most otable advatage of H-HA i compariso to PV-HA ad FV-HA is the optimized umber of coectios each ode maitaiig. The typical disadvatages of H-HA iclude: it supposes all the odes of the overlay etwor ca participate i the top-tier sub-overlay, i cases where may odes caot satisfy this requiremet because of firewall or NAT problem, etc. the H-HA does ot wor; furthermore, H-HA does ot have a clear distictio of the load differet odes are supposed to tae, which maes it o-suitable for highly heterogeeous eviromets. I FV-HA, all the peers participate i all the tiers of sub-overlays. HIERAS [9] is a represetative example. By groupig topologically adjacet peers ito lower level rigs, HIERAS provides the overlay etwor with routig locality. Meawhile, HIERAS has to create ad maitai oe figer table for each tier of the architecture. The advatages ad disadvatages of FV-HA are obvious. The former icludes routig locality, topology awareess, etc. The latter icludes the facts that it is o-suitable for eviromets where lots of peers locate behid firewall or NAT, ad it icreases overlay maiteace cost, etc. I PV-HA, the ey poit is the existece of dedicated gateway peers which participate i more tha oe tiers of the hierarchy ad serve as proxies of lower-tier peers. Besides most of the advatages of H-HA, i.e. fault isolatio, admiistrative autoomy, efficiet cachig ad replica, PV-HA has its ow beefits: it has a clear distictio of the load differet peers are supposed to tae which maes it suitable for highly heterogeeous eviromets, e.g. wireless etwors; it wors i eviromets where a umber of peers locate behid NAT or firewall, etc. A lot of proposals have bee put forward i this category, icludig Brocade [], Le et al. [], PPSIP [3], Garces-Erice et al.[8], Mislove et al. [], Zoels et al. [5] [6]. Our previous studies [] [3] also belog to this category. However, most proposals are focused o two-tier hierarchy, just [] ad our previous studies [] [3] explore the multiple-tier, or more precisely, more tha two-tier hierarchy. Le et al. [] presets a architecture where all the peers at the lower-tier form oly oe loop which maes it lose most of the advatages of PV-HA; therefore, we do ot study the special architecture i this paper. I [], we oly aalyzed the looup hop cout of the GTPP architecture ad did ot cosider the forwardig hop cout (the hop cout betwee differet tiers of sub-overlays); while i [3], we studied a special three-tier hierarchical architecture without ay emphasis o the geeral architecture (GTPP). I this paper, we exted the aalysis from our previous wor by ) cosiderig more performace metrics, icludig expected looup hop cout, expected looup latecy, ad maiteace traffic of multiple-tier PV-HA, ad ) substatiatig the claims regardig the looup hop cout ad maiteace traffic by simulatios. The idea is to study whether more tha two tiers of hierarchy ca provide added value i performace ad fuctioality. To our best owledge, this paper is the first attempt to provide mathematical aalysis to these metrics i a multiple-tier PV-HA. III. GENERAL TRUNCATED PYRAMID PP ARCHITECTURE I this sectio, the GTPP architecture ad the associated service looup procedure are itroduced. We do ot discuss the

5 5 mechaisms to classify N heterogeeous odes ito sub-overlays. There is a great deal of research wor o this already; oe example is [4]. There are also a great umber of DHT algorithms that ca be used to costitute the overlay. I this paper, we utilize Chord [] as oe istatiatio to form the overlay etwor. However, our results ca be easily exteded to other DHT algorithms, e.g. Kademlia [3], Pastry [4]. A. GTPP Architecture GTPP is a multiple-tier PV-HA. To mae it clear, Fig. just illustrates a three-tier hierarchical architecture. I each tier of sub-overlay, all the peers are grouped ito several disjoited sub-sub-overlays (deoted as SSOs, also the otatio of Sx_y i Fig. ). There is oly oe SSO at the highest tier. Because of the autoomy characteristic of the hierarchical architecture, differet SSO ca utilize differet DHT algorithms, or the topologies of differet SSO ca be differet, e.g. fully-mesh, sigle-coectio (each ormal peer just sets up oe coectio to its super-peer), or DHT-based. Without loss of geerality, this paper assumes all the SSOs adopt the same DHT topology ad the same DHT algorithm, i.e. Chord as our istatiatio. Peers from the same SSO ca loo up the eys stored o each other accordig to the associated DHT algorithm. Peers from differet SSO of the same tier caot loo up the eys stored o each other directly; istead, they utilize the odes from the upper tier of sub-overlay as proxies. Each SSO, except the topmost oe, has oe super-peer, which we ame as gateway super-peer to differetiate it from the other ormal peers. Besides participatig i the local SSO, the gateway super-peer also participates i the higher tier of SSO. From the viewpoit of robustess, practical overlay usually has more tha oe gateway super-peers i each SSO; however, this does ot affect our aalysis much. Therefore, for the simplicity of our aalysis, we just assume oe gateway super-peer i each SSO i the rest of this paper. Accordigly, the peers from the first tier of sub-overlay participate i oe SSO, while the peers from the other tiers of sub-overlays participate i two SSOs at the same time. As oe example i Fig., the first tier of sub-overlay cosists of six SSOs, deoted as S_, S_, etc.; the secod tier of sub-overlay cosists of three SSOs, deoted as S_, S_ ad S_3, while the third tier of sub-overlay cosists of oly oe SSO, deoted as S3_. N_5 ad N_4 participate i oly oe SSO, i.e. S_6. N_48 participates i two SSOs at the same time, i.e. S_6 ad S_3. N3_39 participates i two SSOs as well, i.e. S_3 ad S3_. From the top dow, from the vertical perspective, all the peers from differet sub-overlays costitute multiple trees rooted by the peers at the topmost sub-overlay, which maes it loo lie a pyramid beig trucated. Ad this ispires the ame of Trucated Pyramid (TP) i our GTPP architecture.

6 6 3 rd tier of sub-overlay N3_39 N3_9 Leged Request S3_ N3_4 Respose N_48 S_3 N_35 N_8 S_ N_ S_ N_ N_5 d tier of sub-overlay N_36 N_33 S_5 N_3 N_8 S_3 N_3 N_4 N_5 S_6 N_3 N_7 S_4 N_4 N_ S_ N_7 S_ st tier of sub-overlay Fig.. Geeral Trucated Pyramid PP (GTPP) architecture. B. Service looup procedure Whe a peer wats to loo up a service or resource provided withi the GTPP architecture, it follows the followig steps:. The iitiatig peer seds the looup request to its gateway super-peer, ad the the gateway super-peer forwards the request to its ow gateway super-peer at the higher tier, ad so o. This operatio is executed recursively util the request reaches the top-most tier of SSO.. The super-peer locatig at the top-most tier of SSO resolves the request ad loos up the service or resource i its local SSO utilizig the associated DHT algorithm. If o successful respose is got from its local SSO, the request is forwarded to the immediately close lower tier of SSO which is closest to the service or resource ad the looup procedure jumps to Step 3. If oe successful respose is got from the top-most tier of SSO, the looup procedure is halted immediately ad the respose is retured bac to the iitiatig peer. 3. The same procedure is doe recursively as i Step. If o successful respose is received, the looup procedure cotiues util the request reaches the lowest tier of SSO. 4. At the lowest tier of SSO, a respose, either successful or failed, is retured to the iitiatig peer. The respose follows the reverse path of the request (the so-called recursive routig) to the iitiatig peer. I Step, we assume that all the looup requests are resolved at the top-most tier of SSO firstly, which maes sese as the peers at the top-most tier of SSO have the most owledge of the whole overlay etwor. I Sectio VII.A, we will see the advatage of this foutai-lie looup mechaism to the efficiecy of cotet cachig i PV-HA. We also assume recursive routig i the aforemetioed looup procedure. We will aalyze the iterative routig mechaism i Sectio VII.B. I Step 4, if the respose is retured bac to the iitiatig peer directly istead of followig the reverse path of the request, the routig mechaism is so-called semi-recursive routig. The advatages ad disadvatages of semi-recursive routig are still uder debate. The most explicit disadvatages of semi-recursive routig iclude asymmetric routig ad wea capability of

7 7 traversig NAT. We do ot cosider this routig mechaism i this paper. Oe example sceario is illustrated i Fig., where N_5 i S_6 wats to fid the ey (ID=) stored at N_3 i S_. The requests ad resposes are show as the blac dashed lies ad the red dash-dotted lies respectively. The looup procedure is as follows:. N_5 seds the looup request to its gateway super-peer, i.e. N_48;. N_48 forwards the request to its ow gateway super-peer, i.e. N3_39; 3. N3_39 resolves the request i its local SSO, i.e. S3_, by utilizig the Chord algorithm, ad the the request is routed to N3_9 which is the closest peer to the requested ey i S3_; 4. N3_9 loos up the requested ey i its lower tier of SSO, i.e. S_, by utilizig Chord, ad the the request is routed to N_5 which is closest to the target ey i S_; 5. Fially, N_5 resolves the request i S_ ad fids the ey stored at N_3. The successful respose with the ey-value pair is retured bac to N_5 alog the reverse path of the request. I the looup procedure metioed above, it is oteworthy that Step ad Step oly cost oe looup hop, while Step 3-Step5 each costs multiple looup hops accordig to the adopted DHT algorithm. Except the forwardig hops to the gateway super-peers, we ca see that through oe SSO (S3_), the eys stored at the top-most tier of SSO (S3_) ca be foud; through two SSOs, the eys stored at the d tier of sub-overlay ca be foud; while through three SSOs, the eys stored at the st tier of sub-overlay ca be foud. Therefore, i a -tier GTPP architecture, at most SSOs are eeded to cover the whole overlay architecture. IV. PERFORMANCE ANALYSIS Le et al. [] adopted expoetial distributio (ED) to divide all the peers of the whole PP overlay ito multiple tiers of sub-overlays. The basic idea is that, accordig to the maximum iformatio etropy priciple [5], the expoetial distributio with a mea /λis the maximum etropy distributio amog all the cotiuous distributios supported i [, ) that have a mea of / λ. Physically, this also maes sese. As i real world, the majority of peers have relatively wea capability, while the much powerful peers are i the miority. By groupig the relatively wea peers ito lower tiers, ad the powerful peers ito the upper tiers, the limited physical capability of the whole overlay etwor ca be optimized. Therefore, Le et al. [] utilized the ED classifier to group the peers of the overlay etwor ito differet tiers. I our previous wor [], we foud the similarity betwee ED ad Geometric Distributio (GD), i.e. the GD has almost the same curve shape as ED except GD is i discrete domai ad the ED is i cotiuous domai, thus, GD ca roughly be treated as the discrete couterpart of ED. Therefore, we utilized GD to distribute all the peers of the overlay ito differet tiers approximately. I this paper, we cotiue to utilize the GD to group all the peers ito differet tiers. It meas from the bottom up, the umber of peers at each tier of sub-overlay follows GD. We argue here that the adopted distributio is ot so importat, as the mai goal of this paper focuses o the qualitative aalysis istead of quatitative aalysis. Differet distributio may have a ifluece o the quatitative result; however, its ifluece o qualitative aalysis is limited. The otatios used are show i Table.

8 8 TABLE I SYMBOLS AND THEIR MEANING Symbol Descriptio Symbol Descriptio N The total umber of peers i the whole overlay, atural umber. D(,) The average looup latecy for the whole GTPP overlay etwor. p The probability of oe peer beig located at the first tier of sub-overlay, <p<. ( ) Z i The matrix of the looup traffic distributio for a sigle looup which is iitiated by the i th tier of sub-overlay, i=,,,. The j th colum vector of ( ) Z i, j=,,3,,. The looup The total umber of tiers of GTPP, =,,3,, atural umber. ( ) z i j traffic distributio amog each tier of sub-overlay for the looup halted through j SSOs ad iitiated by the i th tier of sub-overlay, i=,,,. N i The umber of peers at the i th tier of sub-overlay, i=,,3,,. F Iterim matrix, with the size of *. ζ i The probability of oe ey beig stored at the i th tier of sub-overlay, i=,,3,,. f i The i th colum vector of F. P i The probability of oe peer beig located at the i th tier of sub-overlay, i=,,,. α i The looup traffic distributio of the peer at the i th tier of sub-overlay. H i, j The hop cout of oe successful looup through j SSOs, j=,,,, for the request iitiated at the i th tier of sub-overlay, i=,,3,,. β i The maiteace traffic distributio of the peer at the i th tier of sub-overlay. L i, j The latecy of oe successful looup through j SSOs, j=,,,, for the request iitiated at the i th tier of sub-overlay, i=,,3,,. γ i The republish traffic distributio of the peer at the i th tier of sub-overlay. ω i The average roud trip time (RTT) of oe looup hop at the i th tier of sub-overlay, i=,,3,,. ϑ The average umber of items shared by each peer. Γ i The expected hop cout of oe successful looup for peers at the i th tier of sub-overlay, i=,,,. m i The total traffic distributio of the peer at the i th tier of sub-overlay. Ω i The expected latecy of oe successful looup for peers at the i th tier of sub-overlay, i=,,,. T The time iterval. E(,) The average looup hop cout for the whole GTPP overlay etwor. To mae the aalysis clearer, we mae the followig assumptios:. The overlay is i stable state, which meas all the N peers are located at the proper sub-overlay tiers accordig to their physical capability. (The effect of chur to the performace of PV-HA is our future wor). The IDs of peers ad eys are geerated ad distributed uiformly, which meas all the peers from differet tiers have the same probability to store a ey. (This assumptio ca be easily satisfied utilizig the existig load balacig mechaisms) 3. Each peer from a differet tier of sub-overlay has equal probability to loo up a radom ey.

9 9 A. Looup hop cout Accordig to the assumptios ad otatios, we ca see that the umber of peers at each tier of sub-overlay has the followig equatio: N = p N N = p ( p) N.... () ( ) N = p ( p) N ( ) = ( ) N p N Here the topmost tier of sub-overlay has the differece betwee the total umber of peers ad the accumulated umber of peers from the other tiers of sub-overlays. That is to say, ( ) = i = ( ) i= N N N p N. Therefore, N p = = N p p. () Eq. () stads for the ratio of the umber of peers at the (-) th tier of sub-overlay ad the umber of peers at the th tier of sub-overlay. From Eq. (), we ca see that as log as p >.5, the N - >N. It meas the (-) th tier of sub-overlay has more peers tha the th tier of sub-overlay. It is ituitively reasoable i the sese that the peers at the lower tiers of sub-overlays have less physical capability ad accout for the majority of the whole overlay etwor. It also coforms to the maximum iformatio etropy priciple aforemetioed. From Eq. (), we ca also see that, if i, the N / N = / ( p ). As <p<, the value of / ( p) is always larger tha. It i i meas that the umber of peers at the ( i ) th tier of sub-overlay is / ( p) times of the umber of peers at the i th tier of sub-overlay. It also meas that the umber of peers i each SSO (except the topmost oe) is / ( p ) (see Fig. ). More precisely, the umber of peers i each SSO (except the topmost oe) is + / ( p ), as the gateway super-peer participates i two SSOs at the same time. To mae it clear, we deote / ( p) as, the Eq. () ca be expressed as follows: N = ( ) N N = ( ) N... N ( ) N = ( ) N ( ) = ( ) ( ) N. (3) We ca see that the SSO from the topmost tier of sub-overlay has N peers, while the other SSOs from the lower tiers of sub-overlays have the same (+) peers. As a example i Fig., the SSO from the topmost tier of sub-overlay, i.e. S3_, has three peers (N3_9, N3_4, ad N3_39); while the other SSOs from the first ad secod tiers of sub-overlays have the same three peers. S_6 icludes peers N_4, N_5, ad N_48. S_3 icludes peers N_35, N_48, ad N3_39. Gateway super-peers, e.g. N_48 ad N3_39, participate i two SSOs

10 at the same time. Actually, as we assume oe gateway super-peer i each SSO, each peer from the upper tiers of sub-overlays is a gateway super-peer, therefore, participates i two SSOs at the same time. I this way, by adjustig the value of, we ca get differet overlay topologies with variable peers at each tier of sub-overlay. From Eq. (3), we ca see that the probability of oe peer beig located at the i th tier of sub-overlay, i=,, ca be calculated by the followig equatios: N P = = N N P = = ( ) N... N P = = ( ) ( ) N N ( ) P = = ( ) N ( ). (4) With the Assumptio, the IDs of eys ad peers are distributed uiformly. Therefore, the probability of oe ey beig stored at the i th tier of sub-overlay, i=,, equals to the probability of oe peer beig located at the i th tier of sub-overlay, i.e. ζ = i therefore, the followig results exist: P, i N ζ = = N N ζ = = ( ) N... N ζ = = ( ) ( ) N N ( ) ζ = = ( ) N ( ). (5) Now, let us first cosider the situatio where the peer from the topmost tier of sub-overlay wats to loo up the eys stored by the other peers i GTPP. I this situatio, the followig results exist: H, = log( N ) = log( N) ( ) log( ) H, = log( ) + log( + ) N... H, = log( ) + ( ) log( + ) N H, = log( N ) + ( ) log( + ). (6) I Eq. (6) above, we assume the overlay is fully populated. Accordig to [], o average, log N hops are eeded to resolve the resposible peer i a fully populated DHT overlay with N peers. Tae H as a example, it meas the looup request is iitiated, from the th tier of sub-overlay, ad eeds two SSOs to complete the looup procedure. The first part of log( N ) meas the looup

11 hop cout i the topmost, i.e. th, tier of sub-overlay (the first SSO has N odes), while the secod part of log( + ) meas the looup hop cout i the (-) th tier of sub-overlay (the secod SSO has + odes). Thus, the expected hop cout for oe successful looup iitiated at the topmost tier of sub-overlay has the followig result: Γ = H, j ζ + j j= = log( N) ( ) log( ) + ( ) log( + ) ( ) j j= ( j). (7) From the looup procedure metioed i Sectio III.B, we ow that the looup procedure from the lower tiers of sub-overlays just adds a small umber of additioal forwardig hops to reach the topmost tier of sub-overlay. Thereafter, the looup procedure is the same as the looup iitiated from the topmost tier of sub-overlay. Thus, compared to the looup iitiated from the th tier of sub-overlay, the looup iitiated from the (-) th tier of sub-overlay adds oe more looup hop, while the looup iitiated from the (-) th tier of sub-overlay adds two more looup hops, ad so o. Based o Eq. (7), we ca get the followig equatio: Γ = Γ + Γ = Γ +... Γ = Γ + Γ = log( N) ( ) log( ) + ( ) log( + ) j ( ) j= ( j). (8) After we get the expected hop cout for the looup iitiated from each tier of sub-overlay ad the probability of oe peer located at each tier of sub-overlay, the expected hop cout for oe successful looup i the whole GTPP overlay ca be calculated as follows: i i i= E(, ) = P Γ. (9) B. Looup latecy Followig the similar aalysis procedure, based o Eq. (6), we ca get the values of L, j as follows by multiplyig the expected looup hop cout by the associated looup latecy for each hop: L, log( ) ω log( ) ( ) log( ) ω = N = N L, log( ) ω log( ) ω = N L, log( ) ω log( ) ω = N + + i j= L, = log( N ) ω + log( + ) ω i j=. () The expected latecy of oe successful looup for the peers at the top-most tier of sub-overlay is show as: Ω = L, j ζ + j. j=

12 The other values of Ω i ca be calculated as follows: Ω = Ω + ω j j= Ω = Ω + ω j j=... Ω = Ω + ω Ω = L ζ, j + j j=. () Tae Ω as a example, it meas the looup request is iitiated from the (-) th tier of sub-overlay. Compared to the looup request which is iitiated from the th tier of sub-overlay, oe more forwardig hop taig place at the (-) th tier of sub-overlay is eeded, ad thus, the associated RTT latecy is ω. After we get the expected latecy for the looup iitiated from each tier of sub-overlay, ad the probability of oe peer located at each tier, the expected latecy for oe successful looup i the whole GTPP architecture has the followig equatio: i i i= D(, ) = P Ω. () C. Traffic distributio I this sectio, we aalyze the traffic distributio of each sigle peer from differet tier of sub-overlay. We begi our aalysis with the looup traffic distributio. The, the maiteace traffic is aalyzed. Fially, the traffic geerated by republishig the resources is aalyzed. Looup traffic distributio We first aalyze the looup request iitiated from the topmost ( th ) tier of sub-overlay. From the aalysis procedure i Sectio IV.A, we ow that, if the looup procedure is halted at the topmost tier of sub-overlay, oe SSO is eough, i.e. / log( N ) looup hops; if the looup procedure is halted at the (-) th tier of sub-overlay, two SSOs are eeded, oe from the th tier of sub-overlay, ad the other from the (-) th tier of sub-overlay, / log( N ) ad / log( + ) looup hops, respectively. For the umber of messages set ad received, as oe hop eeds two messages set ad two messages received, the umber of messages set ad received is twice of the looup hop cout, respectively. Meawhile, it is oteworthy that the messages set ad received by the gateway super-peer should be grouped ito the upper tier of sub-overlay. Therefore, the followig equatio exists (just shows the umber of set messages): z ( ) T = [,,,log( N )] ( ) T z = [,,,log( + ),log( N ) + ] ( ) T z = [,log( + ),,log( + ),log( N ) + ] - ( ) T = [log( + ),log( + ),,log( + ),log( ) + ] z N. (3)

13 3 The probability of oe successful looup through i SSOs, i=,, 3, is show i Eq. (5). Thus, we ca get the value of f as follows: f ( ) ( ) log( ) + ( ) ( ) ( ) log( ) log( ) + + = i. (4) log( + ) log( + ) log( + ) ( ) log( ) log( ) + log( ) + log( ) + N N N N If the looup request is iitiated from the (-) th tier of sub-overlay, most of the looup procedure is the same as the request iitiated from the th tier, just oe more looup hop (forwardig the request to the top-most tier of sub-overlay) is eeded at the begiig of the looup procedure. Therefore, the value of f is as follows: f ( ) ( ) log( ) + ( ) ( ) ( ) log( ) log( ) = i. (5) log( + ) log( + ) + log( + ) + ( ) log( ) + log( ) + log( ) + log( ) + N N N N With the same processes, we ca get the other colum vectors off. The probability of oe peer beig located at the i th tier of sub-overlay is show i Eq. (4). Thus, we ca get the looup traffic distributio of oe sigle peer from differet tier of sub-overlay as follows: [ f f f ][ ] [ α, α,, α ] =,,, P, P,, P, (6) T T α = αi i=, (7) wherei α is the expected umber of messages set (or received) for oe looup of the whole GTPP architecture. As oe looup hop eeds two messages set, we ca also get the relatioship betwee α ad E(, ) as: Maiteace traffic distributio α = E(, ). (8) I additio to the traffic geerated by the service looup, each peer at each SSO also has to eep some state iformatio to mae the overlay etwor fuctio correctly. I Chord [], each peer has to ru the STABILIZE ad FIXFINGER protocols periodically to detect the failed peers ad esure the cosistecy of the figer tables. Each STABILIZE procedure icludes three messages: the iitiator seds the REQUESTPREDECESSOR message to its successor, the successor respods with RESPONSEPREDECESSOR message which icludes its ew (probably) predecessor, ad fially, the iitiator seds the NOTIFY message to this ew (probably) predecessor of the

14 4 successor. As the gateway super-peers participate i two SSOs at the same time, so they have to ru the STABILIZE algorithm i each of the two participatig SSOs periodically. Thus, the umber of messages geerated by the STABILIZE procedure for each peer from the i th tier of sub-overlay is as follows: β β β β, STAB, STAB, STAB, STAB = 3 = = 6 = = 6 = = 6. (9) Each peer from the i th tier of sub-overlay also has to ru the FIXFINGER algorithm periodically to update its figer table. Origially, each FIXFINGER operatio is a looup operatio for the ID of the required figer []. However, we do ot cosider the chur effect i this paper ad assume all the peers are i steady state; therefore, each peer just has to sed KEEPALIVE message to its figers periodically. Similar to STABILIZE procedure, the gateway super-peers also have to ru the FIXFINGER algorithm i each of the two participatig SSOs periodically. I a SSO with peers, the size of the figer table is log(), ad each FIXFINGER operatio eeds two messages, oe from the iitiator, the other from the respoder. Therefore, we ca get the followig results: β β β β, FIX, FIX, FIX, FIX = log( + ) = log( + ) + log( + ) = log( + ) + log( + ) = log( + ) + log( N ). () Republish traffic distributio I order to mae the stored service or ey-value pair up-to-date, peers also have to operate the republish procedure periodically for each of their shared items. The republish procedure is almost the same as the looup operatio. The oly differece is the republish time iterval. Therefore, the relatioship betwee γ i ad α i is: γ = ϑ α i i. Summary I recursive routig, as the umber of messages set ad received are the same, therefore, the total traffic distributio of oe sigle peer from each tier of sub-overlay has the followig result: α β β γ mi = T T T T i i, STAB i, FIX i ( ) LKP STAB FIX REP. () The total umber of messages set ad received by the whole overlay etwor is made up of two parts, the first part is the looup traffic ad republish traffic, the secod part is the maiteace traffic, i.e. the STABILIZE traffic ad the FIXFINGER traffic. Therefore, we ca get the followig results for the total traffic of the whole overlay etwor of GTPP ad Chord: α ϑ α β, β, (, ) = ( + ) + i STAB i FIX M N ( + ) Pi TLKP TREP i= TSTAB TFIX M Chord log( N) ϑ log( N) 3 log( N) = N ( + ) + ( + ) TLKP TREP TSTAB TFIX. () The maiteace traffic for GTPP ad Chord per time uit is as follows:

15 5 βi, STAB βi, FIX Mait eace(, ) = N ( + ) Pi i= TSTAB TFIX. (3) 3 log( N) Mait eace( Chord ) = N ( + ) T T STAB FIX V. RESULTS ANALYSIS I this sectio, we aalyze the expected looup hop cout, expected looup routig latecy, traffic distributio of oe sigle peer from a differet tier of sub-overlay ad the total traffic of GTPP architecture i compariso to flat architecture, based o the equatios i Sectio IV. I the aalysis procedure, we use three differet etwor scales, 4, 6 ad 8. However, we observe the similar curves with differet etwor scales. Therefore, the subsequet sectios just show the results with etwor scale 6. I the followig figures, we loo through all the itegers betwee the lower ad upper limit of. The lower limit of is two, which meas there are at least two peers i each SSO from the lower tiers of sub-overlays to mae the SSO fuctio as a distributed ( ) system. The upper limit of ca be roughly determied by the followig formula: N = ( ) N. Tae = as a example, the upper limit of is roughly 5,, actually 499,999 is the exact umber. I this case, the upper tier of SSO has peers, while each SSO from the lower tier of sub-overlay has equally 499,999 peers. Therefore, oce the total umber of the whole overlay N is fixed, a differet depth of GTPP () will result i a diverse upper limit of. The larger the value of is, the smaller the upper limit of is. Thus, the spa of x-axis is differet i the followig figures. A. Expected looup hop cout I this part, the effect of differet value of ad (the depth of GTPP) to the expected looup hop cout is studied. We aalyze four differet overlay depths ( equals to, 3, 4, ad 5, respectively), ad the results are show i the Fig. 3(a) - 3(d). The figures are based o Eq. (9) i Sectio IV. From the Fig. 3(a) 3(d), o the oe had, we ca see that the value of does ot affect the expected looup hop cout much. O the other had, as the value of becomes larger, the expected looup hop cout become slightly higher as well. The variatio is proportioal to the depth of GTPP, i.e. two tiers add aroud oe more looup hop compared to flat architecture (Chord as oe istatiatio), three tiers add two more hops, ad so o. The primary reaso for this is the additioal forwardig hops set to the top-most gateway super-peer by the peers locatig at the lowest tier of sub-overlay. As i the example i Fig., two more hops are eeded for the request from N_5 to reach the topmost (3 rd ) tier of sub-overlay. Because the peers from the lowest tier of sub-overlay accout for the majority of GTPP, they have the primary impact o the expected looup hop cout of the whole overlay etwor. The expected looup hop cout ca be further decreased by cachig the top-most gateway super-peer iformatio (ode ID, IP address, port) i each peer at the lower tiers of sub-overlays. I that case, the looup request is set to the topmost gateway super-peer directly from the lower tiers. Oly oe more forwardig hop is eeded for all the peers from the lower tiers of sub-overlays. The resultig figures are always similar to Fig. 3(a) ad are idepedet of the overlay depth (). Therefore, from the figures ad aalysis, we ca coclude that GTPP has a slightly higher amout of looup hop cout compared

16 6 to flat architecture. The expected looup hop cout ca be further decreased by cachig the topmost gateway super-peer iformatio at the lower tiers of sub-overlays E 8 6 E=E(,=) Chord E 8 6 E3=E(,=3) Chord x 5 (a) (b) E 8 6 E4=E(,=4) Chord E 8 6 E5=E(,=5) Chord (c) (d) Fig. 3 (a). Expected looup hop cout (=, N= 6 ); (b). Expected looup hop cout (=3, N= 6 ); (c). Expected looup hop cout (=4, N= 6 ); (d). Expected looup hop cout (=5, N= 6 ). B. Expected looup routig latecy This sectio aalyzes the impacts of differet value of ad to the expected looup routig latecy. We first fix the values of the RTT latecy of oe sigle hop at differet tiers of sub-overlays. I HIERAS [9], Xu et al. foud that i a two-tier hierarchical overlay architecture, with simple proximity approach, the average li delay i the lower rig was oly 35.3% of the li delay of the higher layer. I real life, however, the li latecy differece betwee local area etwor (LAN) ad wide area etwor (WAN) ca be eve larger. Without loss of geerality, we assume that the average RTT latecy for oe looup hop i flat Chord architecture is ms. I GTPP, we assume the same ms RTT latecy for oe hop at the top-most ( th ) tier of sub-overlay, while the RTT latecy for oe hop at the (-) th tier of SSO is half of that, i.e. 5 ms, ad so o. Therefore, the average RTT latecy for oe hop at the lowest ( st ) tier of sub-overlay is ( / ) ms. There is also some proximity approach that ca be used i flat architecture to decease the looup routig latecy. As we focus o the geeralized versio of PV-HA without optimizatios i this paper, we just compare GTPP with geeralized flat architecture without optimizatio either. The expected looup latecy of GTPP is show i Fig. 4(a) - 4(d).We ca see that both the values of

17 7 ad have effects o the expected looup routig latecy. Expected looup routig latecy (ms) D=D(,=) Chord x 5 (a) Expected looup routig latecy (ms) D3=D(,=3) Chord (b) Expected looup routig latecy (ms) D4=D(,=4) Chord Expected looup routig latecy (ms) D5=E(,=5) Chord (c) (d) Fig. 4(a). Expected looup routig latecy (=, N= 6, time uit=ms). (b). Expected looup routig latecy (=3, N= 6, time uit=ms). (c). Expected looup latecy (=4, N= 6, time uit=ms). (d). Expected looup latecy (=5, N= 6, time uit=ms). From the perspective of, the treds i each of Fig. 4(a)-4(d) are the same. As log as the value of becomes larger, the expected looup routig latecy of GTPP becomes shorter. From the aalysis procedure i Sectio IV.A, we ow that (or more precisely +) stads for the umber of peers i each SSO at the lower tiers of sub-overlays (except the topmost SSO). As the value of icreases, more peers are located i the lowest tier of sub-overlay, which has the shortest average RTT latecy i GTPP, therefore, the expected looup routig latecy of the GTPP architecture decreases as the value of icreases. From the perspective of, the treds are also clear. As the value of icreases, the optimal expected looup routig latecy decreases. It meas the more tiers GTPP has, the shorter expected looup routig latecy it ca achieve. The results are reasoable as the more tiers the GTPP architecture has, the shorter RTT routig latecy it has at the lowest tier of sub-overlay. From Fig. 4(a) - 4(d), we ca also see that, compared to flat architecture, two-tier GTPP architecture ca achieve aroud 3% shorter routig latecy compared to flat architecture for most of the values of (see Fig. 4(a), from the poit where the value of y-axis is 7 ms to the upper limit of ). Whe the value of varies from to 3, 4, the optimal value of routig latecy varies from aroud 6ms to 5ms, ad 4 ms, respectively. However, whe we cotiue to icrease the value of from 4 to 5, the optimal value of routig latecy varies slightly.

18 8 Therefore, from the viewpoit of routig latecy, we ca coclude that GTPP ca achieve better performace compared to flat architecture. Whe the depth of GTPP icreases from to 4, the expected looup routig latecy becomes eve shorter. However, after that, we caot achieve eve better performace just by icreasig the overlay depth. Thus, -4 tiers are optimal for GTPP. The traffic distributio of oe sigle peer The traffic distributio of oe sigle peer m(,=) m(,=) x (a) m(,=4) m(,=4) m3(,=4) m4(,=4) The traffic distributio of oe sigle peer The traffic distributio of oe sigle peer m(,=3) m(,=3) m3(,=3) (b) m(,=5) m(,=5) m3(,=5) m4(,=5) m5(,=5) (c) (d) Fig. 5(a). Traffic distributio of a sigle peer (umber of messages set ad received by each peer per secod) from differet tier of sub-overlay (=, N= 6 ). (b). Traffic distributio of a sigle peer (umber of messages set ad received by each peer per secod) from differet tier of sub-overlay (=3, N= 6 ). (c). Traffic distributio of a sigle peer (umber of messages set ad received by each peer per secod) from differet tier of sub-overlay (=4, N= 6 ). (d). Traffic distributio of a sigle peer (umber of messages set ad received by each peer per secod) from differet tier of sub-overlay (=5, N= 6 ). C. Traffic distributio of each sigle peer from differet tier of sub-overlay I this sectio, we aalyze the traffic distributio of each sigle peer from differet tier of sub-overlay. We use the followig experimetal parameter values which are the same as i [6], uless otherwise metioed. T = 6 s, T = 5 s, T = 3 s, T = 3 s, ϑ = 5.The results are based o the Eq. () i Sectio IV.C. I Fig. 5(a)-5(d), the y-axis, i.e. m i (, ), stads for the REP average umber of messages set ad received by each sigle peer per secod from the i th tier of sub-overlay. From Fig. 5(a)-5(d), we ca see that both the values of ad have iflueces o the results. From the perspective of, the treds of all the curves are the same, i.e. as log as the value of icreases, the traffic of each sigle peer from the lower tiers of sub-overlays icreases, while the traffic of oe sigle peer from the topmost tier of sub-overlay decreases. LKP STAB FIX

19 9 At some poits, they itersect. We ca also see that the curves of the itermediate tiers of sub-overlays itersect with the curve of the topmost tier before the curve of the lowest tier does. It meas the traffic of the itermediate tiers of sub-overlays icreases promptly with the SSO size of the lower tiers icreasig. Geerally, i hierarchical architectures, the peers from upper tiers of sub-overlays are assumed to have more physical capabilities. Therefore, the crossover poits i the figures above ca be used to decide the overlay size of the lower tiers of sub-overlays to avoid overloadig the peers from itermediate tiers of sub-overlays. Furthermore, i order to avoid overloadig the peers from the top-most tier of sub-overlay, some other parameters should also be defied to get certai optimal operatig poit for the whole overlay etwor. The settig of these parameters will be icluded i our future wor. From the perspective of, hierarchical architectures also show their beefits compared to flat architecture or H-HA. I reasoable fuctioal domais (the overlay size less tha the value of the crossover poits), the higher tier oe peer locates, the more traffic it shares. This maes PV-HA distribute traffic amog the peers from differet tiers of sub-overlays i a clearer ad more reasoable way. From Fig. 5(a) ad 5(b), it is clear that, whe the depth of GTPP icreases from to 3, the traffic shared by the lowest tier of sub-overlay decreases sigificatly. Whe we cotiue icreasig the depth of GTPP from 3 to 4, ad 5, the alleviatio of the traffic load from the lowest tier of sub-overlay is ot so distict. Although i Fig. 5(c) ad 5(d), the peers from the middle tiers of sub-overlays almost share the same traffic; this is maily because we assume each SSO from the lower tiers of sub-overlays have the same umber of peers for the simplicity of aalysis. This effect also proves, from aother perspective, that hierarchical architectures should ot have too may tiers to mae a clear distictio of traffic distributio. From the aalysis above, we ca coclude that, from the perspective of traffic distributio amog differet tiers of sub-overlays, GTPP has a clearer ad more reasoable way tha flat architecture ad -3 tiers are optimal to eep this beefit. D. Total traffic of the whole overlay etwor I the sectio, we aalyze the total traffic ad maiteace traffic of GTPP i compariso to Chord. The results are based o Eq. () ad Eq. (3) i Sectio IV.C. We use two differet looup rates T = 6s ad T = s for the total traffic aalysis. The other parameters are the same as i Sectio V.C., i.e. T = 5 s, T = 3 s, T = 3 s, ϑ = 5. We firstly show the results of the STAB FIX maiteace traffic per secod (umber of messages set ad received per secod for maiteace) i Fig. 6(a)-6(d). From Fig. 6(a)-6(d), it is clear that the maiteace traffic of GTPP is smaller tha Chord. As log as the overlay depth of GTPP icreases, the overall maiteace traffic decreases as well. This tred is evidet whe we vary the value of from to 3, ad 4. The primary reaso for this is that, whe we icrease the overlay depth of GTPP, the peers are partitioed ito smaller sizes of SSOs, which have fewer eighbors to maitai compared to larger SSOs. Therefore, the maiteace decreases remarably. Now, let us cosider the total traffic (umber of messages set ad received per secod for all the peers) of GTPP compared to Chord. The results with T = 6s are show i Fig. 7(a)-7(d), while the results with T = s are show i Fig. 8(a)-8(d). LKP LKP REP LKP LKP

20 4 x 6 4 x 6 Maiteace traffic Maiteace(,=) Maiteace(Chord) Maiteace traffic Maiteace(,=3) Maiteace(Chord) x 5 (a) (b) 4 x 6 4 x Maiteace(,=4) Maiteace(Chord) Maiteace(,=5) Maiteace(Chord) Maiteace traffic.5.5 Maiteace traffic (c) (d) Fig.6 (a).maiteace traffic GTPP vs. Chord (=, N= 6 ). (b). Maiteace traffic GTPP vs. Chord (=3, N= 6 ). (c). Maiteace traffic GTPP vs. Chord (=4, N= 6 ). (d). Maiteace traffic GTPP vs. Chord (=5, N= 6 )..5 x 7.5 x 7. M(,=) M Chord. M(,=3) M Chord Total traffic.5. Total traffic x (a) (b)

21 .5 x 7 M(,=4).5 x 7. M Chord. M(,=5) Total traffic.5. Total traffic.5. M Chord (c) (d) Fig. 7(a). Total traffic per secod GTPP vs. Chord (=, N= 6, T LKP=6s). (b). Total traffic per secod GTPP vs. Chord (=3, N= 6, T LKP=6s). (c).total traffic per secod GTPP vs. Chord (=4, N= 6, T LKP=6s). (d).total traffic per secod GTPP vs. Chord (=5, N= 6, T LKP=6s). 7 x 7 7 x Total traffic 4 3 M(,=) M Chord Total traffic 4 3 M(,=3) M Chord x 5 (a) (b) 7 x x Total traffic 4 3 M(,=4) M Chord Total traffic 4 3 M(,=5) M Chord (c) (d) Fig.8(a). Total traffic per secod GTPP vs. Chord (=, N= 6, T LKP=s. (b). Total traffic per secod GTPP vs. Chord (=3, N= 6, T LKP=s). (c). Total traffic per secod GTPP vs. Chord (=4, N= 6, T LKP=s). (d). Total traffic per secod GTPP vs. Chord (=5, N= 6, T LKP=s). From Fig. 7(a)-7(d), we ca see that the total traffic of GTPP slightly decreases whe the depth icreases from to 3, ad the sigificatly icreases whe the depth varies from 3 to 4, 5. Whe the depth is 3, the total traffic of GTPP is optimal ad is slightly smaller tha Chord. From Fig. 8(a)-8(d), it is clear that the total traffic of GTPP is greater tha Chord. However, the depth of or 3 still show its beefits as the differece of total traffic betwee GTPP ad Chord is ot so big. The primary reaso behid these pheomea is that the total traffic is made up of two parts, the looup ad republish traffic, ad the maiteace traffic. The looup