Fig. 9.

The X sub-state structure for Example 8 where X dwell time distribution follows the maximum of two Erlang random variables with rates $r_1$ and $r_2$, respectively, and shape parameters $k_i=2$. As in Sect. 3.5.4, where the minimum is assumed instead of the maximum, X can be partitioned using indices based on organizing particles by which events they are awaiting under each Poisson process associated with each Erlang distribution. Upon reaching the target event (here, the 2nd event) under any given Poisson process, particles transition to sub-states with an asterisk in the corresponding index position (e.g., see figure). In general, these sub-states all have a dwell times given by the 1st event time under a Poisson process, but with differing rates (see Example 8): here they follow exponential distributions with either rate $r=r_1+r_2$ (white backgrounds), rate $r_2$ (gray backgrounds), or rate $r_1$ (lined backgrounds)