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QUORUMS

QUORUMS. By gil ben-zvi. definition. Assume a universe U of servers, sized n. A quorum system S is a set of subsets of U, every pair of which intersect, each Q belongs to S is called a quorum. EXAMPLES.

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QUORUMS

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  1. QUORUMS By gil ben-zvi

  2. definition • Assume a universe U of servers, sized n. A quorum systemS is a set of subsets of U, every pair of which intersect, each Q belongs to S is called a quorum.

  3. EXAMPLES • Weighted majorities: assume that every server s in the universe U is assigned a number of votes w(s). Then weighted majorities is a quorum set defined by

  4. EXAMPLES • MAJORITIES : a weighted majorities quorum system when all weights are the same. • Singleton: a weighted majorities quorum system when for one server s: w(s)=1, and for each v of the other servers w(v)=0. (only quorum is s)

  5. EXAMPLES • Grid: suppose n is a square of some integer k. arrange the universe in a k x k grid. A quorum is the union of a full row and one element from each row below. • FPP: suppose a projective plane over a field sized q. each point is an element, and each line is a quorum. By projective plane attributes, each quorum intersect.

  6. More definitions • Coterie: a coterie S is a quorum system such that for any Q1,Q2 quorums in S: Q1 isn’t included in Q2 • Domination: coterie S1 dominates coterie S2 if for every quorum Q2 belongs to S2, there exist Q1 in S1, such that S1 is contained in S2. • Strategy: a probability vector representing the probability to access each quorum.

  7. measures • Load: the load L(S) of a quorum system is the minimal access probability minimized over the strategies. • Resilience: resilience is k, if k is the largest number such that for every k server crashes, one quorum remains unhit.

  8. measures • Failure probability: if every server has certain probability to crash (assuming independently here), the probability that each quorum is hit. Usually assuming each server has the same crash probability p.

  9. Measures examples • Singleton: load=1, resilience=0, failure probability=p • Majorities: load is about ½. Resilience about (n-1)/2. failure probability (if p < ½) smaller than exp(e,-n). • Grid: load is O(1/sqrt(n)). Resilience = sqrt(n)-1, failure probability tends to 1 as n grows.

  10. Access protocol • Implements the semantics of a multi-writer multi-reader atomic variable. • Assumes all clients and servers are non byzantine, unique timestamp for a client • Write: a client asks some quorum to obtain a set of value/timestamps pairs, then he writes his value with higher timestamp than each of the timestamps received to each server in the quorum.

  11. Access protocol • Read: a client asks for each server in some quorum to obtain a set of value/timestamp. The client chooses the pair with the highest timestamp. It writes back the pair to each server in some quorum • Server S updates a pair of value/timestamp, only if the timestamp is greater than the timestamp currently in S

  12. Byzantine quorum systems • We will use access protocol to demonstrate the subject • Assuming communication is reliable, clients are correct, servers can be byzantine, assuming that a non-empty set of subsets of U: BAD, is known, some B in BAD contains all the faulty servers.

  13. Masking quorum systems • A quorum system S is a masking quorum system for a fail-prone system BAD if the following properties are satisfied:

  14. Access protocol • write: remains the same • Read: for a client to read the variable x, it queries servers for some quorum Q to obtain a set of value/timestamp pairs

  15. Access protocol • The client chooses the pair with the highest timestamp in C, or null if C is empty.

  16. Access protocol • Claim: a read operation that is concurrent with no write operations return the value written by the last preceding write operation in some serialization of all preceding write operations. • Claim: there exists a masking quorum system for BAD iff is a masking quorum system for BAD

  17. Access protocol • Criterion: there exists a masking quorum system for BAD iff for all

  18. F-masking quorum systems • F-masking quorum system: A masking quorum system where BAD is the set of all groups of servers sized f. • By previous claims: • There exists a masking quorum system for BAD iff n>4f • Each pair of quorums must intersect by at least 2f+1 elements.

  19. examples • For f-masking quorums:

  20. Dissemination quorum systems • Assumes clients can digitally sign the value/timestamp they propagate. • Therefore weaker demands than masking • A quorum system S is a dissemination quorum system for a fail-prone system BAD if the following properties are satisfied:

  21. Dissemination quorum systems • The same way as masking we reach the (different) criterion: • There exists a dissemination quorum system for BAD iff • If no more than f servers can fail, but any set of f servers can fail, then must hold: n>3f

  22. Opaque masking quorum systems • Motivation: We want not to expose the fail-prone system BAD. • done by majority decision. • properties for quorum system to become opaque masking system:

  23. Opaque masking quorum systems • Read: the modification is that the client choose the pair <v,t> that appears most often, if there are multiple such sets, it chooses the newest one. • Claim: Suppose maximum f servers can fail, there exists an opaque quorum system for BAD iff n>=5f, sufficient because quorums sized [(2n+2f)/3] is an opaque quorum system for B.

  24. Opaque masking quorum system • Claim: The load of any opaque system is at least ½. • Proof: if we sum up the load of a certain quorum, we’ll get it’s bigger than it’s size/2. the claim follows. • Example: hadamard matrix, world of size exp(2,l)

  25. Faulty clients • Solves the problem that a client will try to fail the protocol. • The treatment here provides a single-writer multi-reader semantics. • The write operation starts when the 1st server receives update request, and ends when the last server sent acknowledgment.

  26. Faulty clients • Write: for a client c to write the value v, it chooses legal timestamp, larger than any timestamp it has chosen before, chooses a quorum Q, And then it sends <update,Q,v,t> to each server in Q, if after some timeout period it has not received acknowledgment, than it chooses another quorum.

  27. Faulty clients-servers protocol • The servers protocol is as follows: • if a server receives <update,Q,v,t> from a client c, with legal timestamp, then it sends <echo,Q,v,t> to each member of Q. • If a server receives identical echo messages <echo,Q,v,t> from every server in Q, then it sends <ready,Q,v,t> to each member of Q.

  28. Faulty clients-servers protocol 3. If a server receives identical ready messages <ready,Q,v,t> from a set of servers that certainly doesn’t contain faulty server, it sends <ready,Q,v,t> to Q. 4. If a server receives identical ready messages <ready,Q,v,t> from a set Q1 of servers, such that Q1=Q\B for some B in BAD, it sends acknowledgment for c, and update the pair if t is greater than the timestamp it currently has.

  29. Faulty servers-properties • Agreement: if a correct server delivers <v,t> and a correct server delivers <r,t> then r=v • Proof: if a correct server delivers <v,t>, then echo must have been send by all correct servers in Q1. same about Q2, they intersect in a correct server, which doesn’t send different value with the same timestamp

  30. Faulty servers-properties • Claim: Read received last written value if it’s not concurrent with write operations. • Proof: same as masking quorum system. • Propagation: similar ideas to r.b, and byzantine agreement, if server decides to deliver, it is promised that all other decides that too. • Validity: at the end a correct quorum will be accessed, so the write can end.

  31. Load, capacity, availability • Load: we will mark L(S), definition as before • availability: failure probability with the same “p” for all the servers, we will mark it as Fp(S) • Capacity: we’ll define a(S,k) as the maximum number of quorum accesses that S can handle during a period of k time units. Capp(S) is the limit of a(S,k)/k as k tends to infinity.

  32. Load, capacity, availability • Example: majorities • The claim is that cap(S)=1/L(S), and there is a trade off between good availability and good load.

  33. definitions • The cardinality of the smallest quorum is denoted by c(S) • The degree of an element i in a quorum system S is the number of quorums that contain i • Let S be a quorum system. S is a s-uniform if |Q| = s for each Q in S • S is (s,d) fair if it is s-uniform and deg(i)=d foreach i, it is called s-fair if it is (s,d) fair for some d.

  34. LP • We can use a linear programming to calculate the load and the strategy achieving the load.

  35. DUAL LP • Some time we want to use the dual linear program, in which we give probabilities over the elements of the world. It is a known fact that DLP<=LP

  36. The load with failures • A configuration is a vector in which it holds 1 in places representing the failing elements in the world • Dead(x) is the group of elements failed, live(x) is the non failed ones • S(x) is the sub collection of functioning quorums

  37. Load with failiures • The load of quorum system S over a configuration x, if S(x) is empty then L(S(x)) = 1, if there are functioning quorums we define it in similar way as before by linear programming problem. • Let the elements fail with probabilities P=(p1,………,pn). Then the load is a random variable Lp(S) defined by:

  38. Load with fails • Claim: E(Lp(S))>=Fp(S) • Claim: If (configurations) x>=z than L(S(x))>=L(S(z)) • Proof: S(z) contains S(x), strategy for S(x) is for S(z) too. • Claim: E(Lp(S)) is a non decreasing function.

  39. Properties of the load • Claim: L(S)>=c(S)/n • Claim: L(S)>=1/c(S) • Proof: if we choose probability 1/c(S) for every element in c(S) and 0 in the rest, we achieve possible solution for the DLP problem. • Conclusion: L(S)>1/sqrt(n) (achieved when c(S) is close to sqrt(n)

  40. Load/fail probability trade off • Claim: Fp(S)>=exp(p,n*L(S)) • Proof: the probability that all the elements in the smallest quorum will fail, (and therefore the quorum system fails) = exp(p,c(S)). Since c(S)<=nL(S) the claim follows.

  41. examples • Optimal load, optimal load/ failure tradeoff, good failure load – paths system • B-grid system • SC-grid system • AndOr system

  42. Load analyses • Claim: Non dominated coteries have lower bounds. • The claim follows if you choose strategy for the dominator by giving the probability only in quorums which contained by a quorum in the dominated quorum system • Claim: voting systems have high load (more than ½)

  43. Last slide!!!! • Proof: if we define V=the sum of all votes (Vi), then the vector Yi=Vi/V is a solution for DLP larger than ½.

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