The usual constraint on transition probabilities in Hidden Markov Models is

\sum_{j=1}^N a_{ij} =1

for all states 1<=i<=N.

However, when used as generators, this makes impossible to exit the model naturally, and some arbitrary length has to be externally imposed to the generated sequences.

I feel that a more appropriate constraint would be

\sum_j=1^N a_{ij} < 1

for those states i that can be the last state visited. This matches the interpretation of a_{ij} as the probability of visiting state j from state i. In finite sequences, no other state in the model is reached when we are done, so accordingly the sum does not have to reach 1. The model can be exited naturally from states with some probability left for leaving, thus effectively generating variable length finite sequences.

Has this approach already been developed?