Multi-Step Birth-Death Processes

The quasi-stationary solution to (4.100) should first be determined corresponding to the solution (4.72) for the single-step case. It is easily verified, however, that the condition of detailed balance (4.40) is not fulfilled in the general multi-step case and so, in fact, the quasi-stationary solution to the present problem cannot be derived in the form (4.39). 

Instead of this a return to the conservation laws is made which remain valid in the multi-step case Starting from the general form of the master equation 

Setting yWi = 0 leads to a stable distribution qs ( ) forn = 1, 2. exactly as for the single-step master equation. 

The form (4.110 b) yields a finite continued fraction representation of the beginning with 

Inserting this condition for n into the continued fraction representation (4.110 b) and assuming slowly varying i] and Aji, i.e. Aa g A and + Vn = U 

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