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The random waypoint model is a commonly used mobility model for simulations of wireless communication networks. In this paper, we present analytical derivations of some fundamental stochastic properties of this model with respect to: (a) the length and duration of a movement epoch, (b) the chosen direction angle at the beginning of a movement epoch, and (c) the cell change rate of the random waypoint mobility model when used within the context of cellular networks. Our results and methods can be used to compare the random waypoint model with other mobility models. The results on the movement epoch duration as well as on the cell change rate enable us to make a statement about the 'degree of mobility' of a certain simulation scenario. The direction distribution explains in an analytical manner the effect that nodes tend to move back to the middle of the system area.