Theoretical Nabladot Analysis of Amdahl’s Law for Agent-Based Simulations
Amdahl’s Law states that execution speedup S is nonlinearly proportional to the percentage of parallelizable code P and the number N of processors. Additional terms must be added to Amdahl’s Law when applied to agent-based simulations, depending on how synchronization is implemented. Since P is continuous but N is discrete, traditional multivariate operators based on nabla or del ∇ are applicable only for P, not for N, regardless of synchronization architecture (linear, logarithmic, constant, among other). Moreover, relatively low values of N (bound by Miller’s number 7 ±2) are common in some cases. Here I apply a novel and exact operator, called “nabladot” and denoted by the symbol “nabladot”, that is defined for hybrid function such as Amdahl’s Law. The main results show how exact solutions using nabladot differ from traditional approximations, particularly in the logarithmic case that is characteristic of hierarchical synchronization. Improvements in precision are inversely proportional to P and N, converging to 0.8 as N → 2.
KeywordsAmdahl’s Law nabladot multivariate vector analysis distributed systems concrete mathematics hybrid functions
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