Another approach to modeling parallel system performance is characterized by the use of statistical modeling tools as variable distributions and Markov models. It differs from the scalar parameter approach in the sense that the ``parameters'' are statistical tools.
A measurement-based model of the execution of computationally-bound parallel applications is presented in [Dimpsey and Iyer1995, Dimpsey and Iyer1991]. The constructed model is a statically-clustered Markov model with rewards and costs associated with each state. The rewards in each state quantify the actual system resource available to an application. Monte Carlo simulation is used to solve the model and predict the completion time distribution of the given application with the measured workload. The constructed model is used to evaluate scheduling policies, performance effects of multiprogramming, and scalability of real workloads (extracted from PERFECT benchmarks). The idea here is similar to the approximation of functions without descriptive value (Brewer in 2.2.1), but they use Markov models to express the effects of execution parameters.
We should note that these statistical parameters try to represent the asymptotic behavior of the modeled systems. They are usually used to analyze parallel systems whose workload characteristics are well known, being particularly useful for evaluating architectures across a wide range of applications. In capacity planning, for example, we are generally interested in being able to compare and evaluate different scenarios quickly [Menascé et al. 1994]. Accuracies at the 10 to 30% level are acceptable for this purpose, so statistical tools are preferable because of their efficiency and flexibility. However, depending on the tool used, the user is required to have statistical expertise that is not common to parallel programmers. Markov models and Petri nets, for example, are not as simple to use as mathematical functions or variable distributions. As with other analytical techniques, statistical models do not give any explanation of system behavior, and also require the user to express his understanding about the parallel system behavior as parameters to the statistical tool.