One consequence of a quantum theory in which resonances are mathematically unified with decaying states is an asymmetry in time evolution, even for closed quantum mechanical systems. This time asymmetry is different from the environmentally induced, dynamical irreversibility that is experienced only by open quantum systems and that is quantified by an increase in entropy. To investigate the connection between time asymmetry and dynamical irreversibility, we study open systems within a time asymmetric theory. We find that, when using a time asymmetric theory for open systems, one must relax the assumption that measurements are perfectly repeatable. To treat this problem, we develop a framework in which one can incorporate the interference from multiple environmental systems affecting a single experiment. We also study a kinematic effect of indistinguishability that affects only open systems, and we show how it leads to a monotonic increase in entropy without requiring an active measurement. Finally, within our framework we develop two models that reproduce for open systems the expected and observed phenomena. One is a model of photons scattering inefficiently from a beam splitter. The other is a model of systems undergoing Rabi oscillations and suffering environmental interference. We find that the kinematic effect of indistinguishability can explain for such systems the generally measured Excitation Induced Dephasing, which has previously been treated dynamically. / text