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## High Performance Computing

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**High Performance Computing**Lecture 5 Scalability of Parallel Systems 9/21/2014**Scalability**• Scalability is widely used in parallel computing. However, it is complicate to give scalability a precise mathematical definition. • Performance depends upon both communication patterns of the algorithm and the infrastructure provided by the machine so a good measure of scalability should adequately reflect the interaction between these two aspects. • Intuitively, scalability is the ability of a parallel system to effectively utilize an increasing number of processors.**Parallel Systems**• Whether a parallel system is or not scalable depends on the definition of scalability and the metric used. • Need of a realistic metric to measure scalability • Impact on design of architectures and applications. • Multiple parameters involved • Metrics for grid computing? PARALLEL SYSTEM Parallel Algorithm Parallel Machine**Scalability Models**• Fixed-Problem Size Model • Speedup • Amdhal’s Law • Overhead & Degree of concurrence • Memory-Constrained Model • Scaled speedup (Gustafson) • May lead to unacceptable execution times • Scaled speedup is less than lineal (Flatt & Kennedy) • Isoefficiency (kumar & Gupta) • Fixed-Time Scaling Model • Isospeed (Sun & Rover) • Seeks to determine how much the problem size should be scaled with the system under the constrain that the problem must be solved at the same absolute time Memory-contrained – Storage complexity Fixed-time - computational complexity**Scalability Metrics**• Speedup Ideal Actual # of Processors**Scalability Metrics**• Isoefficiency • Generalization of Flatt & Kennedy’s results • Relation pf problem size and the maximum number of processors which can be used in a cost-optimal fashion • A parallel system is cost optimal iff pTp =O(W). • A parallel system is scalable iff its isoefficiency function exists. If W needs to grow exponentially with respect to p, the parallel system is poorly scalable. On the contrary, if W grows nearly linear with p, the parallel system is highly scalable**Scalability Metrics**• Isoefficiency (Gupta & Kumar) Convergence rate degradation Memory Limit Problem Size Communication overhead speedup Number of Processors**Scalability Metrics**• Effectiveness (MSU-EIRS-ERC-97-6)**Scalability ….**• A scalability metric for a parallel system should reflect the interaction between communication patterns of the application and architecture of the parallel machine. • Scaling based only on problem size represents just a slice of the scalability surface so a scalability metric should take into account other parameters involving architecture, algorithms and the applications. • A Scalability metric should give us information not only about whether a parallel system is scalable or not, but also information regarding the specific values and conditions for scalability**References**• J. Gustafson, “Reevaluating Amdahl’s law.” Communications of the ACM, 31(5):532-543, 1988. • H. Flatt & K. Kennedy, “Performance of parallel processors.” Parallel Computing, 12:1-20, 1989. • A. Gupta & V. Kumar, “Performance properties of large scale parallel systems. ” Journal of Parallel and Distributed Computing, 19:234-244, 1993. • X. Sun & D. Rover, “Scalability of parallel algorithm-machine combinations.” IEEE Transactions on Parallel and Distributed Systems, 5(6):599-613, 1994. • E.A. Luke, I. Banicescu & J. Li, “The optimal effectiveness metric for parallel application analysis.” Technical Report MSSU-EIRS-ERC-97-6.