An alternative to the Markov process diagram is the Gallagher's diagram. The diagram represents the source and channel using state-dependent memoryless channel diagrams.
For our above example, below-left shows the Gallagher's diagram is at time index n−1 the state is S0 while the right hand side shows the diagram if at n−1 the state is S1.
Note that the output y(n) and state S(n) [or S(n) for short] updates simultaneously. That is
From Gallagher's diagram we have
In other words,
Let index i = 0 or 1 but fixed. Then for a fixed S(n−1) we can get the conditional entropy H(Y | S(n−1) = Si). That is
Hence, for fixed state Si = S0
Following determination of the conditional probability H(Y | S(n−1) = Si), the steady-state entropy rate of the output sequence is therefore