A Theoretical Study of Process Dependence for Standard Two-Process Serial Models and Standard Two-Process Parallel Models

TL;DR

This theoretical study differentiates serial and parallel two-process models by analyzing their dependence structures and distributional behaviors, revealing key distinctions in process dependence and implications for empirical observations.

Contribution

It provides a mathematical characterization of process dependence in standard two-process serial and parallel models without assuming specific distribution forms.

Findings

Serial models do not exhibit positive dependence without specific distributions.

Parallel models have independent total completion times.

Survival functions depend on hazard function properties.

Abstract

In this article we differentiate and characterize the standard two-process serial models and the standard two process parallel models by investigating the behavior of (conditional) distributions of the total completion times and survivals of intercompletion times without assuming any particular forms for the distributions of processing times. We address our argument through mathematical proofs and computational methods. It is found that for the standard two-process serial models, positive dependence between the total completion times does not hold if no specific distributional forms are imposed to the processing times. By contrast, for the standard two-process parallel models the total completion times are independent. According to different nature of process dependence, one can distinguish a standard two process serial model from a standard two-process parallel model. We also find that in standard two-process parallel models the monotonicity of survival function of the intercompletion time of stage 2 conditional on the completion of stage 1 depends on the monotonicity of the hazard function of processing time. We also find that the survival of intercompletion time(s) from stage 1 to stage 2 is increasing when the ratio of hazard function meets certain criterion. Then the empirical finding that the intercompletion time is grown with the growth of the number of recalled words can be accounted by standard parallel models. We also find that if the cumulative hazard function is concave or linear, the survival from stage 1 to stage 2 is increasing.