I have an upper-right triangular transition intensity matrix $Q$ for a 7-state Markov model (with states $X_1,X_2,...,X_7$), from which I numerically calculate the matrix exponential to derive a transition probability matrix $P$.
A number of the transitions into one of the states have equal rates in the intensity matrix, say: $$q_{1,6} = q_{2,6} = q_{3,6}$$ The matrix is otherwise populated such that, after taking the matrix exponential, the derived probabilities of transitioning into $X_6$ from states $X_1,X_2,X_3$ are then different, i.e.: $$p_{1,6} ≠ p_{2,6} ≠ p_{3,6}$$
This is, of course, an expected consequence of taking the matrix exponential; however, for the purposes of a simulation I am trying to run, it is undesirable as it results in different proportions in $X_6$ in two different "arms" of the simulation based on the same Markov structure, but with different initial distributions across $X_1,X_2,X_3$.
Putting concerns around competing risks aside, is there a method by which the intensity matrix can be adjusted (or indeed any other method) to ensure that the derived probabilities of transitioning into a specific state are the same where the underlying rates are the same? (i.e. to ensure that $p_{1,6} = p_{2,6} = p_{3,6}$ when $q_{1,6} = q_{2,6} = q_{3,6}$).