Completely unconstrained serial and parallel models can mimic one another in a number of conventional experimental paradigms (e.g., Townsend, 1972). Special theory driven experimental methodologies are required to provide broad and firm tests among serial and parallel models (e.g., see Townsend, Wenger, & Houpt, 2016). However, standard serial models and standard parallel models do not mimic one another and are prototypes of their classes and hence are of especial interest in their own right. In particular, the temporal statistics, intercompletion time and total processing time, tell much about processing of the two opposed systems because intercompletion times are the actual processing times of items (stages, etc.) being processed in a serial system and total completion times are the actual processing times of items (channels, etc.) being processed in a parallel system. Part of the definition of standard parallel models is that their total completion times (i.e., actual processing times) are stochastically independent. Though some special results are available for serial models (Townsend & Evans, 1983), general predictions are lacking. Similarly, part of the definition of serial models is that their intercompletion times (i.e., actual processing time) are stochastically independent. Although parallel models can be designed to have independent intercompletion times, again there are no general theorems about their behavior in the broadest class of standard parallel models. This chapter solves these outstanding puzzles concerning these important characteristics of the canonical serial vs. parallel types of processing.