Let $N(t) :N \in \mathbb{N}, t \in \mathbb{R}$ a stochastic jumping process over the time. $N(t)$ is characterized by a unknown pmf (probability mass function) and represents the number of jumps happened in the interval $[0, t]$. On other hand, the time $t$ between a jump $N_i$ and the next one $N_{i+1}$ is random an characterized by unknown pdf (probability density function).
I wanted to find out such pmf for jumps and pdf for time but I came across the next problem. The time is recorded in minute-by-minute way, so I don't know how it affects to my model. Should I assume that time ($t$) belongs to real $\mathbb{R}$ or natural $\mathbb{N}$ numbers? How should I face this issue? Should I change my model?
Just for adding information, 80% of data time is between 0 and 50 minutes, the other 19% between 50 and 150, and the other 1% are outliers and rare times.