One of the conditions of the simple Poisson process is that "events occur singly", i.e. that the probability of two events occurring in a time interval is disapearingly small as the length of the interval tends to zero.
To introduce a model which gets round this restriction, consider a (simple) Poisson process, where each event consists of a random number of "sub-events", being at least one sub-event to each event. The distribution of the number of sub-events per event would have to be specified. Typically we would be interested in the total number of sub-events in a given time period.