Point process models for rainfall are constructed generally based on Poisson cluster processes. Most commonly used point process models in the literature were constructed either based on Bartlett–Lewis or Neyman–Scott cluster processes. In this paper, we utilize a class of Cox process models, termed the Markov modulated Poisson process (MMPP), to model rainfall intensity. We use this class of models to analyse rainfall data observed in the form of tip time series from rain gauge tipping buckets in a network of gauges in Somerset, southwest England, recorded by the Hydrological Radar Experiment (HYREX). Univariate and multivariate models are employed to analyse the data recorded at single and multiple sites in the catchment area. As the structure of this proposed class of MMPP models allows us to construct the likelihood function of the observed tip time series, we utilize the maximum likelihood methods in our analysis to make inferences about the rainfall intensity at sub-hourly time scales. The multivariate models are used to analyse rainfall time series jointly at four stations in the region. Properties of the cumulative rainfall in discrete time intervals are studied, and the results of fitting three-state models are presented.