Unreliable Failure Detectors for Reliable Distributed Systems

Unreliable Failure Detectors for Re[iabIe Distributed Systems 2353’ emulatedFIG,2. Transforming 9 into 9‘,T%,+i, need not emulate all the failure detector histories of q‘; what we dorequire is that all the failure detector histories it emulates be histories of 23’.Given a reduction algorithm Tg~9,, any problem that can be solved usingfailure detector Q‘, can be solved using !JI instead. To see this, suppose a givenalgorithm A requires failure detector Q‘, but only 9 is available. We can stillexecute A as follows. Concurrently with A, processes run Tg, +a, to transform ~into ~’. We modify algorithm A at process p as follows: whenever A requiresthat p queries its failure detector module, p reads the current value of outputP(which is concurrently maintained by Ta+q, ) instead. This is illustrated inFigure 2.Intuitively, since TQ +9 is able to use !23to emulate !3’, 9 must provide at leastas much information about process failures as ‘3’ does. Thus, if there is analgorithm Ty, +q, that transforms $3 into 9‘, we write $3 Z 93’ and say that Q‘ isreducible to ‘3; we also say that Qb’is weaker than 9. Clearly, 5 is a transitiverelation. If $2 Z !3’ and 9’ 5 $3, we write !’3 = 9‘ and say that $2 and $3’ areequivalent.Similarly, given two classes of failure detectors % and %‘, if for each failuredetector ’53G % there is a failure detector ‘3’ E %’ such that Q 553’, we write %> ‘%’and say that %’ is weaker than % (note that if Yi 5 %’, then if a problem issolvable using %‘, it is also solvable using %). From this definition, 2 is clearlytransitive. If % > %’ and %‘ > %, we write % - %‘ and say that % and %’ areequivalent.Consider the trivial reduction algorithm in which each process p periodicallywrites the current value output by its local failure detector module into outputP.From this trivial reduction the following relations between classes of failuredetectors are immediate:3. From Weak Completeness to Strong CompletenessIn Figure 3, we give a reduction algorithm T9 +9, that transforms any givenfailure detector $7 that satisfies weak completeness, into a failure detector !21’that satisfies strong completeness. Furthermore, if !2 satisfies one of the four