The effect of energy dissipation on transport and activated rate processes in condensed phase is analyzed in detail within the non-Poissonian collision model (NPCM). The NPCM is a generalized variant of the collision model (CM) describing the instantaneous change of the velocity of probe particles induced by random collisions with particles of a medium. Unlike the conventional CM, the NPCM assumes the non-Poissonian collision statistics. In this work we concentrate on the stationary variant of the NPCM (SNPCM), which differs from the nonstationary NPCM (NNPCM) discussed in previous studies by the proper treatment of the collision statistics ensuring the time homogeneity of the process. The SNPCM is shown to be free of inconsistencies inherent in the NNPCM. In particular, the SNPCM reproduces the physically natural relations between the average parameters (the average displacement and velocity, correlation functions, etc.) well known in the transport theory. The SNPCM describes properly the specific features of the processes under study, for example, the kinetic cage effect predicted earlier. Within the SNPCM the analytical expressions for the rate of passage over a parabolic barrier, valid in the intermediate-to-strong friction limit, are derived for some particular values of the parameters of the model. The expressions obtained are analyzed in detail. (C) 2001 American Institute of Physics.