  Delayed processes are ubiquitous in biological systems and are often
characterized by delay differential equations (DDEs) and their extension to
include stochastic effects. DDEs do not explicitly incorporate intermediate
states associated with a delayed process but instead use an estimated average
delay time. In an effort to examine the validity of this approach, we study
systems with significant delays by explicitly incorporating intermediate steps.
We show by that such explicit models often yield significantly different
equilibrium distributions and transition times as compared to DDEs with
deterministic delay values. Additionally, different explicit models with
qualitatively different dynamics can give rise to the same DDEs revealing
important ambiguities. We also show that DDE-based predictions of oscillatory
behavior may fail for the corresponding explicit model.
