Timed (or Stochastic) Data Flow Diagrams (TDFD's or SDFD's) introduced in Symanzik and Baker (1996d) are an extension of the Formalized Data Flow Diagrams, defined in Leavens et al. (1996). This extension allows us to assess the quantitative behavior (e. g., performance, throughput, average load of a bubble, etc.) as well as the qualitative behavior (e. g., deadlock, reachability, termination, finiteness, liveness, etc.), eventually depending on different types of transition times, for the system modeled through the TDFD. In this paper, we consider Markovian transition times for the consumption of in--flow items and for the production of items on the out--flow. Moreover, we require the TDFD to be periodic and irreducible and it must have a finite reachability set. For these models, we have been able to apply an aggregation principle of Schassberger (1984), extended for periodic Markov chains by Woo (1993), to efficiently determine stationary probabilities, expected waiting times, and limiting process probabilities.