A hidden Markov model (HMM) with a special structure that captures the 'semi'-property of hidden semi-Markov models (HSMMs) is considered. The proposed model allows arbitrary dwell-time distributions in the states of the Markov chain. For dwell-time distributions with finite support the HMM formulation is exact while for those that have infinite support, e.g. the Poisson, the distribution can be approximated with arbitrary accuracy. A benefit of using the HMM formulation is that it is easy to incorporate covariates, trend and seasonal variation particularly in the hidden component of the model. In addition, the formulae and methods for forecasting, state prediction, decoding and model checking that exist for ordinary HMMs are applicable to the proposed class of models. An HMM with explicitly modeled dwell-time distributions involving seasonality is used to model daily rainfall occurrence for sites in Bulgaria.