# Making Geo-Replicated Systems Fast as Possible, Consistent when Necessary

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4 Alice in EU S a 1 S 1 b 1 S 2 b 2 S 3 a 2 S 4 a 3 S 5 b 3 S 6 b 4 a 1 a 2 a 3 Alice in EU S 7 (a) RedBlue order O of operations b 1 b 2 b 3 b 4 (b) Causal serializations of O Figure 1: RedBlue order and causal serializations for a system spanning two sites. Operations marked with are red; operations marked with are blue. Dotted arrows in (a) indicate dependencies between operations. 4.1 Defining RedBlue consistency The definition of RedBlue consistency has two components: (1) A RedBlue order, which defines a partial order of operations, and (2) a set of local causal serializations, which define site-specific total orders in which the operations are locally applied. Definition 1 (RedBlue order). Given a set of operations U = B R, where B R = /, a RedBlue order is a partial order O = (U, ) with the restriction that u,v R such that u v, u v or v u (i.e., red operations are totally ordered). Recall that each site is a deterministic state machine that processes operations in a serial order. The serial order executed by site i is a causal serialization if it is compatible with the global RedBlue order and ensures causality for all operations initially executed at site i. A replicated system with k sites is then RedBlue consistent if every site applies a causal serialization of the same global RedBlue order O. Definition 2 (Causal serialization). Given a site i, O i = (U,<) is an i-causal serialization (or short, a causal serialization) of RedBlue order O = (U, ) if (a) O i is a S b 1 S 1 ' a 1 S 2 ' b 2 S 3 ' b 3 S 4 ' a 2 S 5 ' a 3 S 6 ' b 4 S 7 ' 1 float balance, interest =.5; 2 func deposit( float money ): 3 balance = balance + money; 4 func withdraw ( float money ): 5 if ( balance - money >= ) then: 6 balance = balance - money; 7 else print "failure"; 8 func accrueinterest(): 9 float delta = balance interest; 1 balance = balance + delta; Figure 2: Pseudocode for the bank example. linear extension of O (i.e., < is a total order compatible with the partial order ), and (b) for any two operations u,v U, if site(v) = i and u < v in O i, then u v. Definition 3 (RedBlue consistency). A replicated system is O-RedBlue consistent (or short, RedBlue consistent) if each site i applies operations according to an i-causal serialization of RedBlue order O. Figure 1 shows a RedBlue order and a pair of causal serializations of that RedBlue order. In systems where every operation is labeled red, RedBlue consistency is equivalent to serializability [5]; in systems where every operation is labeled blue, RedBlue consistency allows the same set of behaviors as eventual consistency [38, 25, 26]. It is important to note that while RedBlue consistency constrains possible orderings of operations at each site and thus the states the system can reach, it does not ensure a priori that the system achieves all the end-toend properties identified in 2, in particular, state convergence and invariant preservation, as discussed next. 4.2 State convergence and a RedBlue bank In order to understand RedBlue consistency it is instructive to look at a concrete example. For this example, consider a simple bank with two users: Alice in the EU and Bob in the US. Alice and Bob share a single bank account where they can deposit or withdraw funds and where a local bank branch can accrue interest on the account (pseudocode for the operations can be found in Figure 2). Let the deposit and accrueinterest operations be blue. Figure 3 shows a RedBlue order of deposits and interest accruals made by Alice and Bob and causal serializations applied at both branches of the bank. State convergence is important for replicated systems. Intuitively a pair of replicas is state convergent if, after processing the same set of operations, they are in the same state. In the context of RedBlue consistency we formalize state convergence as follows: Definition 4 (State convergence). A RedBlue consistent system is state convergent if all causal serializations of the underlying RedBlue order O reach the same state S. The bank example as described is not state convergent. The root cause is not surprising: RedBlue consistency allows sites to execute blue operations in different orders but two blue operations in the example correspond to non-commutative operations addition (dep- 4

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