A Theoretical Study of Process Dependence for Critical Statistics in Standard Serial Models and Standard Parallel Models

TL;DR

This paper investigates the dependence structures of standard serial and parallel models, revealing their tendencies to predict specific dependencies in processing times and demonstrating they cannot perfectly mimic each other.

Contribution

It provides a theoretical analysis of process dependence in serial and parallel models, clarifying their predictive limitations and differences.

Findings

Serial models tend to predict positive dependence for total completion times.

Parallel models often predict increasing intercompletion times, but can be independent under exponential distributions.

Serial and parallel models cannot perfectly replicate each other's behavior.

Abstract

Critical parts of the definitions of standard serial and standard parallel modes refer to stochastic independence. Standard serial models are defined by stochastic independence and identical distributions of their processing times. Processing times in the serial models are identical to the intercompletion time statistics. Similarly, standard parallel models assume stochastically independent and identical processing times. Their processing times are equivalent to the statistic known as total completion times. Little is known about what standard serial models can predict for the total completion time or what standard parallel models can predict for the intercompletion times. Here we demonstrate that standard serial models possess a tendency to predict a positive dependence for the total completion times with that always being true in the case of a single processing order. However, with mixtures of processing orders, standard serial models may predict negative dependence of the total completion times. Comparably, standard parallel models typically predict neither independence of the intercompletion times nor identical distributions. In fact, standard parallel models predict increasing intercompletion times as the individual channels continue to finish. Nevertheless, dramatically increasing hazard functions of the channels can defeat that tendency. And, standard parallel models can predict intercompletion time independence but only when individual channel distributions are exponential. Finally, we use these and ancillary mathematical results to conclude that standard serial and standard parallel models can never perfectly mimic one another. Therefore, our findings set the stage for explicit model testing between these classes.