BOUNDS ON THE SPEEDUP AND EFFICIENCY OF PARTIAL SYNCHRONIZATION IN PARALLEL-PROCESSING SYSTEMS

In this paper, we derive bounds on the speedup and efficiency of applications that schedule tasks on a set of parallel processors. We assume that the application runs an algorithm that consists of N iterations and before starting its i + 1st iteration, a processor must wait for data (i.e., synchronize) calculated in the ith iteration by a subset of the other processors of the system. Processing times and interconnections between iterations are modeled by random variables with possibly deterministic distributions. Scientific applications consisting of iterations of recursive equations are examples of applications that can be modeled within this formulation. We consider the efficiency of such applications and show that, although efficiency decreases with an increase in the number of processors, it has a nonzero limit when the number of processors increases to infinity. We obtain a lower bound for the efficiency by solving an equation that depends on the distribution of task service times and the expected number of tasks needed to be synchronized. We also show that the lower bound is approached if the topology of the processor graph is ''spread-out,'' a notion we define in the paper.