Amdahl´s law: a generalization under processor failures

Recent advances in VLSI technology make it possible to manufacture computer systems with thousands of processors that can work concurrently on the same problem and improve the running time of programs. Amdahl´s law measures the speedup (the ratio of the running time of a program on a 1-processor system to the running time of the same program on a multi-processor system) under the assumptions that processors are fault free and do not fail. This paper generalizes Amdahl´s law under the assumption that processors are subject to failures. If the failure of processors is modeled as random, the actual number of processors available to a program becomes a random variable as well. This stochastic process is represented by a closed queuing network, and it is completely analyzed under certain assumptions. Numerical results show that value of the degradation factor (the ratio of failure rate of processors to their repair rate) is crucial to the system performance. Amdahl´s law implicitly uses the fact that all of the processors are used by the parallel portion of a program. However, for a real system, processors are subject to failure, and consequently, the number of available processors becomes random. This paper assumes that processors may fail and re-evaluates the expression for the speedup factor in Amdahl´s law. It obtains closed-form expressions for the speedup factor and the PLF (performance loss factor)