## Friday, February 4, 2011

### USL Fine Point: Sub-Amdahl Scalability

As discussed in Chapter 4 of my GCaP book, Amdahl's law is defined by a single parameter called the serial fraction, denoted by the symbol α and signifying the proportion of the total workload (W) that is serialized during execution. From the standpoint of parallel processing (where reference to Amdahl's law is most frequent) serialization means that portion of the workload can only execute on a single processor out of N parallel processors. The parallel speedup or relative capacity CA(N) performance metric is given by: $$C_A(N) = \frac{N}{1 + \alpha \, (N-1)}$$ If there is no serialization in the workload, i.e., α = 0, then CA(N) = N, which signifies that the workload scales linearly with the number of physical processors. The important observation made by Gene Amdahl (more than 40 years ago) is that even if α is relatively small, viz., a few percent of the execution time, scalability cannot continue to increase linearly. For example, if α = 5%, then CA(N) will eventually reach a scalability ceiling given by 20 effective processors (1/α), even if there are hundreds of physical processors available in the system.