This paper further develops a stochastic model for rainfall at a single site, in which storms arrive in a Poisson process, each storm generating a cluster of rain cells, with each cell having a random duration and a random intensity. The model is generalized by allowing each generated cell to be of n types. The duration of each cell is an exponential random variable that has parameter dependent on the cell type. The distribution of cell intensity is also dependent on the cell type, so that the generalized model provides a correlation between the intensities and durations of the generated rain cells. The case for two cell types is considered in some detail. The cells are categorized as either `heavy' or `light', where the heavy cells have a shorter expected lifetime than the light cells, which agrees with observational studies on precipitation fields. Properties are derived and used to fit the model to rainfall data at a single site. The adequacy of fit is assessed by considering properties not used in the fitting procedure but which are of interest in applications, e.g. extreme values. Further properties are derived which enable the model to be fitted to multi-site rainfall data.