  The Poisson process of order $i$ is a weighted sum of independent Poisson
processes and is used to model the flow of clients in different services. In
the paper below we study some extensions of this process, for different forms
of the weights and also with the time-changed versions, with Bern\v stein
subordinator playing the role of time. We focus on the analysis of hitting
times of these processes obtaining sometimes explicit distributions. Since all
the processes examined display a similar structure with multiple upward jumps
sometimes they can skip all states with positive probability even on infinitely
long time span.
