General Design of Grid-based Data Replication. Schemes Using Graphs and a Few Rules. availability of read and write operations they oer and

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1 General esin of Grid-based ata Replication Schemes Usin Graphs and a Few Rules Oliver Theel University of California epartment of Computer Science Riverside, C , US bstract Grid-based data replication protocols have been proven to be extremely ecient due to the hih availability and the low cost of read and write operations they oer However, althouh each rid protocol uses loical rids, the semantics and interpretin alorithms vary from scheme to scheme This has the disadvantae that switchin from one protocol to another, results in a re-implementation of the new protocol y desinin a eneral framework which separates the policy from the mechanism this drawback can be overcome The purpose of this paper is to present such a eneral framework for rid-based protocols We show how to model various popular rid protocols as instances of the framework, thereby demonstratin that rid-based protocols which look quite dierent at rst lance share a common concept of construction dditionally, we explain how new rid-based protocols with particular properties can be desined by usin the framework in a raphical way Keywords: istributed Systems, ata Replication, Grid Protocols, Graphical esin, Loical Structures 1 Introduction The replication of data within a networkin environment is a commonly accepted technique to improve the data's availability in the presence of failures resultin from node crashes and communication link failures It also increases the eciency of operations by allowin operations to use local replicas instead of remote ones ata replication schemes manae multiple replicas and oer a sinle copy imae of the replicated data object in order not to put this burden on the application prorammers In recent years, several data replication schemes have been developed which use loical structures such as trees [1, 6], nite projective planes [9, 11], acyclic raphs [12], or rids [2, 8, 14] for manain the replicated data Especially rid-based protocols have been proven to be extremely ecient due to the hih This work was partially supported by iital Equipment Corporation's Campus-based Enineerin Center Karlsruhe, Germany Hennin Pania-Koch Technical University of armstadt epartment of Computer Science armstadt, Germany availability of read and write operations they oer and due to the very small number of copies which must be accessed durin an operation execution lthouh all rid-based protocols use loical rids, their semantics and interpretin alorithms vary from scheme to scheme This has the disadvantae that switchin from one kind of rid-based protocol to another one results in the re-implementation of the new protocol The purpose of this work is to present a unifyin framework for rid-based protocols usin a raphical specication in order to achieve a three-folded oal: 1 The framework depends only on a uniform rid This allows the development of a eneral alorithm and therefore a uniform treatment of a wide variety of rid-based data replication schemes Thus, the eneral protocol must only be implemented once and can then be used to model dierent rid-based protocols avoidin time- and money-consumin reimplementations This is of particular interest due to the fact that constantly chanin networkin environments and varyin application requirements make it impossible to favor one particular replication scheme aainst the other 2 eneral protocol which is able to model various rid-based protocols without the need of different alorithms (as a hybrid approach would require) allows the convenient synthesis of new data replication protocols by combinin the construction strateies used for modelin dierent already known rid-based protocols within the framework These new protocols are directly executable once the eneral framework has been implemented y oerin a raph-based method and a small set of easy-to-understand and easy-to-use rules, the development of new and hihly customized replication protocols becomes simple and convenient 3 The uniform rid structure used for modelin replication schemes permits to analyze and compare two or more modeled schemes in a straihtforward way y usin some simple raph classication criteria it is in particular quite easy to decide whether one

2 replication scheme is covered by another more eneral scheme or not The remainin parts of the paper are oranized as follows: in the next section we state the basic idea of our approach and relate it to previous work Section 3 describes our underlyin model and in section 4 we present the eneral framework In section 5 we show how rid protocols can be desined by usin some simple transformation rules The specication of hierarchical rid protocols as instances of the framework is iven in section 6 Section 7 contains a comparison of dierent rid protocols in terms of their operation costs and availabilities Finally, section 8 concludes the paper by summarizin the main achievements and describin our onoin and future work 2 asic Idea In the Grid Protocol [2] the replicas of a replicated data item are loically arraned as a rid read operation is performed by readin at least one replica of every column Such a set of replicas is called a c-cover write operation requires to modify at least one c-cover and additionally a complete rid column s a consequence, every two sets of replicas used by concurrent write operations have at least one replica in common The same property holds for concurrent read and write operations This leads to a correct synchronization of the operations and therefore preserves the consistency of the replicated data item accordin to the one-copyserializability correctness criterion Fiure 1 shows the R 5 Fiure 1: 3 3-rid with nine nodes ownin replicas oranization of nine nodes with replicas as a 3 3-rid set of nodes which can be used to consistently perform a read operation is e fr 1 ; R 2 ; For a write operation, e the modication of the replicas located on the nodes R 1 ; R 2 ; ;, and uarantees a correct behavior accordin to the correctness criterion lthouh the raphical oranization of the nodes should support the reader to understand how sets of nodes for readin and writin must be constructed, the rid in Fiure 1 is misleadin in two aspects: 1 The paths used for read and write operations cannot be directly derived from the rid 2 Some paths used for constructin sets of nodes for read operations are not in consistency with the edes of rid The c-cover fr 1 ; R 2 ; for example is derived by startin at node R 1, followin the ede to node R 2 but then, it somehow "jumps" to node There is no ede which suests the correctness of this continuation t this point, it becomes quite obvious that the raphical representation is not in correspondence with the alorithmical procedure of collectin nodes with replicas for read and write operations: the neihborhood relation of the nodes formin a row for example is by no means essential for the alorithm; the horizontal edes are meaninless These lacks cause some confusion in order to understand the underlyin concept of the protocol and to desin new and consistent rid-based replication schemes Even more important, it unnecessarily restricts the applicability of the rid approach In order to overcome these shortcomins, we propose rids with two dierent types of directed edes: horizontal edes and vertical edes Horizontal edes are solely used for ndin a path in a horizontal direction Such a path { a so called horizontal path { "touches" the replicas which must be accessed for a read operation Vertical edes describe the connections amon the nodes which are allowed to be used as part of a vertical path for a write operation In analoy with the oriinal procedure where a c-cover and a complete column of replicas are needed to uarantee the correctness of a write operation, the new approach requires to modify the replicas formin a horizontal path and a vertical path Fiure 2 shows the raphical representation of the re-dened 3 3-rid of Fiure 1 The horizontal edes are shown as solid lines whereas the vertical edes are iven as dotted lines The arrows of the directed edes are not shown in the ure in order to simplify the raph However, the direction of horizontal edes is never from riht to left and vertical edes are never directed from the bottom to the top In the raph, the R 2 R 9 Fiure 2: Graphical representation of the 33-rid usin dierent edes for horizontal and vertical paths nodes havin replicas are shown as boxes Solid lines amon the boxes indicate edes which can be used for

3 constructin horizontal paths whereas the dotted lines indicate edes used as parts of vertical paths horizontal path (vertical path) forms a path throuh the rid from its left side (top) to its riht side (bottom) Thus, every c-cover of the oriinal Grid Protocol can be rearded as a horizontal path in this modied approach presented here In particular, the possibility to choose in addition to R 1 and R 2 for a horizontal path, can be derived directly from the raph representin the rid (althouh one can arue that the new raph cannot be called a \rid" any loner) In the presented example, complete columns of nodes which are needed for partially satisfyin the write constraint of the oriinal protocol have their correspondences in vertical paths Whereas the mappin of c-covers to horizontal paths and the mappin of complete columns to vertical paths seems to be somehow irrelevant for the Grid Protocol itself, we will show in section 5 that the use of horizontal and vertical paths dramatically increases the number and kind of replication protocols which can be modeled usin the rid approach In the followin two sections, we describe our model and the new framework for rid-based replication protocols in more detail 3 Model The distributed environment under consideration consists of n independently operatin nodes R 1 ; : : :; R n with independent failure rates Every node either operates accordin to its specication or not at all Let R be the set of all nodes The set G R describes the set of nodes which own a replica of the same replicated data item If v2w holds for two nodes v; w 2 G then they are said to be in a horizontal relation If, on the other hand, v3w is true then both nodes are in a vertical relation v2w (v3w) can also be written as (v; w) 2 2 ((v; w) 2 3) The tuple GS := (G; f2; 3) denes a rid structure Its raph consists of nodes with replicas and two dierent kinds of directed edes: a horizontal relation v2w is represented as a horizontal ede (solid line) from node v to node w vertical relation v3w is shown as a vertical ede (dotted line) from node v to node w The raph of a rid structure is dened to be two-dimensional only, thus edes miht intersect The set S2 G is called horizontal start set and consists of those nodes of the rid structure where horizontal paths throuh the rid are allowed to start naloously, S3 G is called vertical start set and consists of those nodes at which vertical paths start E2 G (E3 G) is called horizontal end set (vertical end set) It is the set of all nodes of the rid structure at which horizontal (vertical) paths terminate s a convention, the left (top) vertices of the raph { which represent the nodes { are those included in the horizontal (vertical) start set The elements of the horizontal (vertical) end set are always the rihtmost (bottom) vertices of the raph Edes endin at a node which is contained in the start set as well as edes beinnin at a node which is contained in the end set are useless and are omitted The rid structure toether with both types of start and end sets are used to evaluate paths throuh the rid horizontal path is a tuple (v 1 ; : : : ; v m ), 1 m n, v i 2 G with v 1 2 S2, v m 2 E2, and v j 2 G n (S2 [ E2); 1 < j < m naloously, a vertical path is a tuple (v 1 ; : : : ; v m ), 1 m n, v i 2 G with v 1 2 S3, v m 2 E3, and v j 2 G n (S3 [ E3); 1 < j < m 4 The Framework fter we have introduced our formal conventions, we demonstrate in this section how access operations on a replicated data item usin rid-based replication protocols are performed The execution of such an operation can be rearded as a service which is oered by certain nodes These nodes are referred to as servers Servers own the rid structure, start sets, end sets etc dened in the previous section as well as the alorithms for dealin with them node which wants to perform an operation submits its request to an arbitrary server who acts on its behalf accordin to the well-known client/serverparadim How the bindin between client and server works and how they exchane information such as operation results is not within the scope of this paper and therefore omitted here We assume that these mechanisms are implemented and ready for use Thus, we focus on how the server performs an operation once it has received a request 41 The Server's ob In the followin, we present an alorithm to be performed at an arbitrary server after receivin a request First, the server uses the rid structure to derive all possible horizontal and/or vertical paths which uarantee a successful access operation The nodes within a horizontal or vertical path dene a read rid set whereas those on a horizontal path in combination with those on a vertical path dene a write rid set Each of these sets contains nodes which are sucient for successfully executin such a cooperation In the next step, the server (also called the coordinator) chooses an arbitrary set and communicates with each of the nodes included in this particular set (the set members are also called participants) More precisely, every participant is asked to lock its replica and to return some status information to the coordinator, e the replica's version number If a participant is not able to do so due to a crash, failures

4 of communication links, or because it is overloaded, this is reconized by the coordinator usin a timeout mechanism or by receivin a neative acknowledment If at least one participant disarees to cooperate then the coordinator uses a dierent set of nodes for performin the access operation In this case, the coordinator only communicates with those participants which have not already locked their replicas and delivered the correspondin status information If no complete set has been successfully locked the whole operation cannot be performed and is rejected Thus, the coordinator simply informs the client that the service is temporarily unavailable Otherwise, if a cooperation has been initiated, the coordinator collects the results delivered by the participatin nodes and calculates the result of the entire operation which, in turn, is sent to the client after the cooperation has been terminated The termination of such a cooperation is controlled by the coordinator who informs all the nodes which have been contacted while tryin to execute the access operation In case of a modifyin operation on a replicated data object, the termination procedure is done by usin an atomic commit protocol Otherwise, the coordinator simply sends a termination messae to the participants which then unlock their replica The alorithm as described above has, from dierent perspectives, an advantae and a disadvantae: as already stated, the alorithm calculates in the rst step all possible sets of nodes for a particular access operation before startin to execute the operation usin any of these sets Thus, as a worst-case scenario, the alorithm miht derive a lare number of possible sets of nodes from the rid structure but is successful in communicatin with all the replicas included in the very rst set it has chosen In this case, the calculation of all the remainin sets of nodes was needless On the other hand, rst calculatin all the sets and subsequently choosin a particular set has severe advantaes due to the fact that the choice can now be based on static or dynamic status information For example, a set includin a node which is known to be down a priori is of no use since the access operation will certainly fail If, on the contrary, a set for a read operation usin e the rid of Fiure 2 containin the nodes fr 1 ; R 2 ; proves to be unsuccessful because the coordinator did not receive an answer from then communication costs can be reduced by rst tryin the sets fr 1 ; R 2 ; or fr 1 ; R 2 ; R 9 before usin others which include In this case, only a sinle new node must be contacted since the results delivered by the nodes R 1 and R 2 can be re-used The same arument holds if sets containin nodes which are known to be extremely reliable or inexpensive to reach, are used before tryin sets which do not have these properties In eneral, sortin the possible sets of nodes accordin to some criteria like set cardinality, node reliability or communication costs between nodes oers the chance to specify rid-based protocols ownin the property of raceful deradation [10] In the oriinal Grid Protocol it is not possible to specify some form of raceful deradation In the eneralized approach presented here, this drawback has been eliminated 42 Grid Sets s described above, the server derives sets of nodes from the rid structure We refer to these sets as rid sets The rid sets alorithm shown in Fiure 3 calculates these sets: The followin example models the rid sets (in: 2; 3;S2;E2; S3; E3; mode; top) out: GSETf if (mode = "r") f /* mode = read */ rel := 2 start := S2 end := E2 else f /* mode = write */ rel := 3 start := S3 end := E3 /* end of recursion if start = end */ Q := ffi j i 2 start \ end /* calculate paths without start set elements */ for (a 2 start n end ) do f start 0 := fw j (a; w) 2 rel rel 0 := f(v; w) j (v; w) 2 rel ^ v 6= a if (mode = "r") T := rid sets (rel 0 ; 3; start 0 ;E2; S3; E3; "r",false) else T := rid sets (2; rel 0 ; S2; E2; start 0 ; E3; "w",false) Q 0 := Q Q := Q 0 [ fm [ fa j M 2 T if (top ^ mode = "r") f /* read rid sets: calc horiz and vert paths */ W := rid sets (3;2; S3; E3; S2;E2; "r", false) Q 0 := Q Q := Q 0 [ W else if (top ^ mode = "w") f /* write rid sets: calc horiz [ vert paths */ W := rid sets (2; 3;S2;E2; S3; E3; "w", false) Q 0 := Q Q := f [ j 2 Q 0 ^ 2 W return (Q) Fiure 3: lorithm for derivin rid sets oriinal Grid Protocol [2] for nine replicas of Fiure 1 as an instance of the framework In particular, we ive its formal specication in terms of a rid structure We assume that the network consists of 15 nodes but only nodes R 1 to R 9 own replicas of the same data object ll nine nodes with replicas have identical charac-

5 teristics in terms of reliability or communication costs Thus, they can be arbitrarily arraned as a 3 3-rid The complete scenario can therefore be described as follows: R = fr 1 ; R 2 ; : : : ; R 15 G = fr 1 ; R 2 ; : : : ; R 9 S2= fr 1 ; ; E2= f ; ; R 9 S3= fr 1 ; R 2 ; E3= f ; ; R 9 2 = f (R 1 ; R 2 ); (R 1 ; R 5 ); (R 1 ; ); ( ; R 2 ); ( ; R 5 ); ( ; ); ( ; R 2 ); ( ; R 5 ); ( ; ); (R 2 ; ); (R 2 ; ); (R 2 ; R 9 ); (R 5 ; ); (R 5 ; ); (R 5 ; R 9 ); ( ; ); ( ; ); ( ; R 9 ) 3 = f (R 1 ; ); (R 2 ; R 5 ); ( ; ); ( ; ); (R 5 ; ); ( ; R 9 ) The raph of the rid structure was already presented in Fiure 2 The execution of the procedure rid sets (2; 3; S2; E2; S3; E3; "r"; "true") calculates a set of rid sets for a read operation The set consistin of 30 rid sets with a cardinality of three each, is shown below: GSET r = f fr 1 ; R 2 ; ; fr 1 ; R 2 ; ; fr 1 ; R 2 ; R 9 ; fr 1 ; R 5 ; ; fr 1 ; R 5 ; ; fr 1 ; R 5 ; R 9 ; f ; ; ; f ; ; ; f ; ; R 9 ; fr 1 ; ; ; fr 2 ; R 5 ; ; f ; ; R 9 s expected, every rid set used for a read operation corresponds either to a horizontal path or a vertical path 1 dened by the rid structure For a write operation this is somehow dierent, as we will see next The rid sets for a write operation are derived by combinin the sets of nodes formin a vertical path with those nodes formin a horizontal path: GSET w = ffr 1 ; R 2 ; ; ; ; fr 1 ; R 2 ; ; ; ; fr 1 ; R 2 ; R 9 ; ; ; fr 1 ; R 2 ; ; R 5 ; ; f ; R 5 ; R 9 ; ; ; f ; ; R 9 ; ; 5 esinin Grid Structures The correctness of a rid protocol depends on the choice of the edes If edes are chosen in an inappropriate way, then the resultin protocol does not uarantee an intersection between every read and write rid set which can be derived from the rid structure In order to distinuish between rid structures which lead to a 1 renement of the oriinal Grid Protocol presented in [8] additionally allows whole columns of the rid to be used for read accesses Our approach exploits this renement since it increases the availability of the read operation correct protocol and those which do not, we introduce the term of a correct rid structure enition 1: (Correct Grid Structure) rid structure is correct if every horizontal path and every vertical path which can be derived from the rid structure have at least one node in common 2 Since the paths are used for constructin the rid sets, correct rid structures lead to read and write rid sets which intersect in a proper way The rid structure of the Grid Protocol shown in Fiure 2 is an example of a correct rid structure nother essential attribute of a rid structure is the number of edes Obviously, the more edes a rid structure contains the more paths and therefore the more rid sets can be derived from it ddin an ede to a rid structure leads to a hiher number of read and/or write rid sets and thus to hiher operation availabilities However, addin an ede may result in an incorrect rid structure In order to cope with that, we introduce the followin denition of complete rid structures: enition 2: (Complete Grid Structure) rid structure is complete if no ede can be added to it, such that the resultin rid structure is still correct 2 In the followin, we restrict the discussion to rid structures where horizontal edes do not skip columns and vertical edes do not skip rows in order to keep our model simple enition 3: (Conictin Edes) Two edes are in conict if they belon to different types and cannot be inserted into the raph of the rid structure as straiht lines without crossin each other 2 Note, that two edes of the same type are never in con- ict with one another Since we have forbidden horizontal edes which skip a column and vertical edes which skip a row, every horizontal path includes at least one node of every column and every vertical path contains at least one node of every row Therefore, iven a horizontal path h, every vertical path v either intersects with h in a node or in an ede or both If there is an intersection in a node, the rid structure is correct If a horizontal and a vertical ede intersect, the edes are in conict and the rid structure is incorrect if the two paths do not additionally have some nodes in common This leads to the followin theorem which provides a simple method for checkin whether a rid structure is correct dditionally, it ives us rules for completin a

6 rid structure as well as for transformin a iven correct rid structure into another correct one Theorem 1: If no conictin horizontal and vertical edes exist in a rid structure then the rid is correct 2 Transformation Rules Rule 1 - correct rid structure can be completed by addin any vertical (horizontal) ede which is not in conict with any existin horizontal (vertical) ede 2 Rule 2 - correct rid structure can be transformed into another correct rid structure by addin a horizontal (vertical) ede and removin all conictin vertical (horizontal) edes 2 y applyin these rules we will show next how the 3 3- Trianular-Lattice Protocol (TLP) [14] can be derived from the 3 3-Grid Protocol The TLP is a replication protocol where the replicas are oranized as a trianular lattice The protocol provides as low access costs as the Grid Protocol but has a hiher write availability Only a sinle type of ede exists for this protocol Read operations must access replicas formin a horizontal or a vertical path usin the dened edes, replicas accessed by a write operation must form both a horizontal and a vertical path The edes are used equally for all operation types: no dierence is made between reads and writes Fiure 4a shows the 3 3-TLP The replicas R 1, R 5, and R 9 are sucient for read as well for write operations since they form both a horizontal and a vertical raph The raph of the rid structure for the 3 3-TLP is shown in Fiure 4b Fiure 5 illustrates the derivation process R 5 (a) R 1 R 2 R 5 R 8 R 9 Fiure 4: The conceptional representation of the 3 3- Trianular-Lattice Protocol (a) and the raph of its rid structure (b) of the TLP startin from the Grid Protocol by applyin the two rules iven above In step 1, we add the vertical edes (,R 5 ) and (R 5, ) In step 2, the con- ictin horizontal edes (R 1, ), (,R 2 ), (R 2,R 9 ), and (b) R 5 Grid Protocol R 5 R 5 fter Step 4 fter Step 2 R 5 fter Step 1 R 5 fter Step 3 R 5 fter Step 5 Fiure 5: erivin the TLP from the Grid Protocol (, ) are removed leadin to a correct rid The rid diaonals (R 1,R 5 ), (R 2, ), (, ), and (R 5,R 9 ) are added as vertical edes to the rid in step 3 In order to reain a correct rid, we remove the conictin horizontal edes (,R 2 ), (R 5, ), (,R 5 ), and (, ) in step 4 fter completin the rid structure in step 5 with all non-conictin edes, we nally obtain the TLP The proposed transformation rules cannot only be used for derivin known protocols like the TLP but also new protocols can be constructed Fiure 6 shows the derivation process for a new rid-based protocol In the rst step we add the two vertical edes (R 2, ) and (, ) The conictin horizontal edes (,R 2 ), (R 5, ), (,R 5 ), and (, ) are removed in the second step In step 3, we add the vertical edes (R 5, ) and (,R 5 ) and remove the conictin horizontal edes (R 1, ), (R 2, ), (R 2,R 9 ), and (, ) in step 4 In a nal step, we complete the rid structure with all nonconictin edes Since the resultin rid consists of alternatin crosses of vertical and horizontal edes similar to a chess-board

7 R 5 Grid Protocol R 5 fter Step 2 R 5 fter Step 4 R 5 fter Step 1 R 5 fter Step 3 R 5 fter Step 5 Fiure 6: erivin a new protocol from the Grid Protocol we call this protocol the Chess-oard Protocol 6 General Hierarchical Grid Protocols The combination of already known replication schemes in order to obtain new schemes with desired properties has already proven to be a powerful desin method [4, 7, 13] In particular, the hierarchical combination of replication schemes is suited for buildin scalable new schemes, e replication schemes which manae a lare number of replicas in an inexpensive manner In [3, 6] this has been shown by the hierarchical layerin of Quorum Consensus Protocols [5] [7] applies this approach to the oriinal Grid Protocol In this section, we show how hierarchical rid-based protocols can be specied within the proposed framework in a homoeneous way First, we model the Hierarchical Grid Protocol of [7] and second, a hierarchical version of the Trianular-Lattice Protocol The basic idea of all hierarchical protocols is to layer several instances of \ordinary" or \at" replication protocols on top of each other and to apply the schemespecic rules for readin and writin of the replicated data item in a recursive manner Thus, the entities on all non-bottom levels of this hierarchy are not physical but loical replicas on which the standard operations like read and write are performed In hierarchical rid-based protocols, a loical replica which must be accessed as part of a horizontal (vertical) path can be accessed by accessin a horizontal (vertical) path formed by its members Physical replicas are accessed directly Thus, in order to perform a read operation on a replicated data item, either a horizontal or a vertical path amon the top-level loical replicas must be accessed For a write operation, a horizontal and additionally a vertical path of top-level loical replicas are required Note, that the specication of horizontal and vertical paths miht dier from level to level dependin on the particular rid structure used Fiure 7a shows a two-level Grid Protocol consistin of 16 physical replicas These replicas are hierarchically rouped to four loical replicas L 1 ; : : : ; L 4 consistin of four physical replicas each On the top level of this hierarchy, the four loical replicas are manaed by a 2 2-Grid Protocol Within each of these loical replicas, the four physical ones are manaed accordin to exactly the same policy Thus, in order to access a c- cover (complete column) on such a replicated data item, a c-cover (complete column) consistin of top-level loical replicas must be accessed, e fl 1 ; L 4 (fl 1 ; L 3 ) For successfully accessin a c-cover on L 1, for instance, the physical replicas R 1 and are required set of physical replicas to be accessed within a read operation is e fr 1 ; ; R 13 ; R 14 For a write operation, the physical replicas fr 1 ; ; ; R 10 ; R 12 ; R 13 ; R 14 form a possible set of replicas to be accessed Next, we show how to transform the conceptual representation of this hierarchical protocol into a rid structure It is done in two steps The additional index applied to the start and end sets in the followin alorithm indicate the loical replica where this set belons to, e S2L i denotes the horizontal start set of a loical replica L i Transformation lorithm Step - The Grid Protocols on every level are substituted by the correspondin rid structures Step - Every vertical ede (L i ; L j ) between the loical replicas L i and L j is substituted by m n vertical edes (v; w), v 2 E3L i, w 2 S3L j and m = j E3L i j, n = j S3L j j Correspondinly, every horizontal ede (L i ; L j ) between the loical replicas L i and L j is substituted by m n horizontal edes (v; w), v 2 E2L i, w 2 S2L j and m = j E2L i j, n = j S2L j j Fiure 7b shows the resultin \two-level rid structure"

8 ' & % ' & ' ' ' % & % ' & & % % ' & % R 1 R 2 R 5 ' & % & % L 1 L 2 R 9 R 1 R 2 R 5 L 3 L 4 L 3 R 10 R 13 R 14 R 11 R 12 R 15 R 16 L 1 L 2 R 9 R 10 R 13 R 11 L 4 R 12 R 15 R 14 R 16 R 1 R 2 R L LL@@?? L LL L LLL R 9 R 10 R 13 R 11 R 12 R 15 a) b) c) Fiure 7: erivin the rid structure of the Hierarchical Grid Protocol R 14 R 16 derived from the conceptional representation by applyin step of the transformation procedure The resultin rid structure after executin step is iven in Fiure 7c This rid structure can directly be interpreted by the rid sets alorithm iven in Fiure 3 This shows that the proposed framework is capable of modelin arbitrary hierarchical rid protocols, since the transformation procedure can straihtforwardly be eneralized to more than two hierarchical levels fter modelin a known hierarchical rid protocol, we will show next that the framework is also suitable for modelin new hierarchical ones in a simple and convenient way s in the previous example, we arrane 16 physical nodes with replicas as four loical nodes L 1 ; : : : ; L 4 consistin of four physical nodes each The four members of each loical replica as well as the four loical replicas are manaed by a 2 2-Trianular- Lattice Protocol The conceptional representation is iven by Fiure 8a In order to obtain a rid structure which can be interpreted by the rid sets-alorithm, we apply the two steps of the transformation alorithm Fiure 8b shows the intermediate \two-level rid structure" obtained after step Fiure 8c shows the resultin rid structure after applyin step 7 Comparison of the Modeled Protocols In this section we briey analyze the previously modeled rid protocols in terms of operation cost and availability For the analytical investiation we make the followin assumptions: nodes may fail links do not fail node failures occur independently the node reliability of all nodes is identical Note, that these assumptions are not necessary to ensure the correctness of the protocols but simplify the analysis, especially for the operation availability The investiated protocols are the 4 4-Grid Protocol, the 4 4-Hierarchical Grid Protocol, the 4 4- TLP, the 4 4-Hierarchical TLP, and the 4 4-Chess- oard Protocol In the Grid, the H-Grid, and the Chess-oard Protocol at least four nodes must be accessed by a read operation and seven nodes must be accessed by a write operation assumin that all 16 nodes are up In the TLP and the H-TLP at least four nodes are necessary for performin both, a read and a write operation Fiure 9 shows the raphs of write availability of the ve protocols The Grid and the H-Grid Protocol provide vailability Write Operation Chess-oard TLP H-TLP Grid H-Grid Node Reliability Fiure 9: Write availability of dierent 4 4 rid protocols for node reliabilities of hiher than 08 a lower write availability than the other protocols The write availability of the TLP is the hihest of all protocols The Chess-oard Protocol has a slihtly hiher write availability than the H-TLP for node reliabilities of hiher than 08 Surprisinly, the H-TLP { the hierarchical version of the TLP { has no advantaes neither in terms of cost nor in terms of availability compared with the TLP This shows that applyin replication protocols in a hierarchy does not always lead to superior protocols

9 ' ' ' ' & & '& % ' ' % & & %& % ' & % % %& % L 1 L 2 R 1 R 2 R 5 R 9 R 10 R 13 R 14 R 11 R 12 R 15 R 16 R 1 R 2 R 5 L 3 L 4 L 3 L 1 L 2 R 9 R 10 R 13 R 11 L 4 R 12 R 15 R 14 R 16 L L LL L LLL R 12? R 15 R 1 R 2 R 5 R 9 R 10 R 13 a) b) c) Fiure 8: erivin the rid structure of the Hierarchical Trianular-Lattice Protocol R 11 R 14 R 16 8 Conclusion and Future Work The purpose of this paper was to present a unifyin framework for rid-based replication protocols which exploits an easy-to-use and easy-to-understand raphical specication method s demonstrated, such a framework oers several advantaes: rst, it allows to conveniently model several instances of rid-based protocols, amon them the oriinal Grid Protocol or the Trianular-Lattice Protocol in a homoeneous manner Second, it oers a raphical way to specify new rid protocols which are uaranteed to be in correspondence with the underlyin correctness criterion, namely the one-copy-serializability criterion Third, the framework is capable of also modelin hierarchical replication strateies like, for instance, the Hierarchical Grid Protocol or the hierarchical version of the Trianular-Lattice Protocol In particular, we presented a eneral transformation alorithm for convertin the conceptional representation of hierarchical rid protocols into so-called rid structures which can directly be interpreted by the framework fter settin up the framework, our current and future work focuses on the identication and application of further transformation rules and meta rules In particular, we are interested in how addin or removin edes impacts the costs and availabilities of rid protocols These eorts will enable us to desin replication protocols with optimal characteristics for a iven application scenario References [1] rawal and El bbadi The Tree Quorum Protocol: n Ecient pproach for Manain Replicated ata In Proc of the 16th VL Conf, paes 243{254, 1990 [2] S Y Cheun, M hamad, and M H mmar The Grid Protocol: Hih Performance Scheme for Maintainin Replicated ata In Proc of the 6th Intern Conf on ata Enineerin, paes 438{445, Feb 1990 [3] Freisleben, H-H Koch, and O Theel The Electoral istrict Stratey for Replicated ata in istributed Systems In Proc of the 5th Intern Conf of Fault- Tolerant Computin Systems, paes 100{111, 1991 [4] Freisleben, H-H Koch, and O Theel esinin Multi-Level Quorum Schemes for Hihly Replicated ata (Extended version) IEICE Trans on Information and Systems, E75-(6):763{770, 1992 [5] K Giord Weihted Votin for Replicated ata In Proc of the 7th CM Symp on Operatin Systems Principles, paes 150{162, 1979 [6] Kumar Hierarchical Quorum Consensus: New lorithm for Manain Replicated ata IEEE Trans on Computers, 40(9):996{1004, 1991 [7] Kumar and S Y Cheun Hih vailable p N Hierarchical Grid lorithm for Replicated ata Inform Proc Letters, 40(0):311{316, 1991 [8] Kumar, M Rabinovich, and R K Sinha Performance Study of General Grid Structures for Replicated ata In Proc of the 13th Intern Conf on istr Comp Sys, Pittsburh, P, paes 178{185, May 1993 [9] M Maekawa p N lorithm for Mutual Exclusion in ecentralized Systems CM Trans on Computer Systems, 3(2):145{159, 1985 [10] S R Mahaney and F Schneider Inexact areement: ccuracy, Precision, and Graceful deradation In Proc of the 4th CM SIGCT-SIGOPS Symp on Principals of istr Comp, paes 237{249 CM, u 1985 [11] S Mullender and P M Vitanyi istributed Match{Makin lorithmica, 3:367{391, 1988 [12] O Theel General Structured Votin: Flexible Framework for Modellin Cooperations In Proc of the 13th Intern Conf on istributed Computin Systems, Pittsburh, P, paes 227{236 IEEE, May 1993 [13] O Theel Meetin the pplication's Needs: esin Study of a Hihly Customized Replication Scheme and its Implementation Computer Systems Science and Enineerin, 9(2):89{97, pril 1994 [14] C Wu and C G elford The Trianular Lattice Protocol: Hihly Fault Tolerant and Hihly Ecient Protocol for Replicated ata In Proc of the 11th Symp on Reliability in istributed Software and atabase Systems IEEE, 1992