Observations of meteor showers allow us to constrain several cometary parameter and to retrieve useful parameters on cometary dust grains, for instance the dust size distribution index s. In this first paper, we describe a new model to compute the time and level of a meteor shower whose parent body is a known periodic comet. The aim of our work was to use all the available knowledge on cometary dust to avoid most of the "a priori" hypotheses of previous meteoroid stream models. The ejection velocity is based on a hydrodynamic model. Because of the large amount of particles released by the comet, it is impossible to compute the orbits of all of them. Instead, we link each computed particle with the real number of meteoroids ejected in the same conditions, through a "dirty snowball" cometary model calibrated with the [A f ρ] parameter. We used a massive numerical integration for all the particles without hypotheses about size distribution. The time of maximum is evaluated from the position of the nodes of impacting meteoroids. The model allows us to compute ephemerides of meteors showers and the spatial density of meteors streams, from which a ZHR can be estimated. At the end a fit of our predictions with observations allows us to compute the dust size distribution index. We used 2002 and 2003 leonid meteor showers to illustrate our method. The application of our model to the Leonid meteor shower from 1833 to 2100 is given in Paper II. © ESO 2005.