Fighting Fire with Fire: Using Randomized Gossip to Combat Stochastic Scalability Limits

ly, let M be the number of copies of a single multicast that a member receives from the lower dissemination layer. For the Pbcast implementation, since each member gossips a given multicast for O(log(n)) gossip rounds and to randomly chosen targets, we have M = O(log(n)). To maximize the throughput, one would want to minimize the number of copies M of a multicast that a member receives from the multicast dissemination layer, as well as the number of repairs sent by the upper protocol layer, in order to repair ‘holes’ in the message delivery stream from the multicast dissemination layer. The second term is p mln, since p ml is the likelihood of a given member receiving none of the copies of a multicast. The expression to be minimized, is thus: L = p mln + M Setting dL/dM = 0 gives us: M = log(n) + log(1/pml) log(1/pml) (3) or M = O(log(n)), which is exactly what the Pbcast primitive achieves with each member gossiping a given multicast for O(log(n)) gossip rounds. So, in a sense, the Pbcast primitive is a provably optimal multicast dissemination mechanism with respect to the minimum load on participants required to achieve reliable delivery. Not surprisingly, implementations of reliable multicast in the Isis/Horus-style stack, as well as over SRM behavior (Figure 3), can also be characterized by a similar analysis. In Horus, the dissemination layer consists of each member verifying the stability of the message with every other member, with the higher Vsync layer repairing the holes. In SRM, each member receives multiple copies of each multicast (repairs) that, from Figure 3, is linear with the group size. The value of the expression L for this is thus obtained by setting M = O(n). From the above analysis, using any of these two mechanisms at the lower multicast dissemination protocol layer gives a (provably) suboptimal implementation of reliable multicast. We now analyze the achievable throughput of the implementation of virtual synchrony when implemented over Pbcast, as depicted in Figure 5. This analysis adds to the cost of view changes, and its effect on throughput. The optimal throughput T ∗ under this implementation is: