A family of Markov Chains

Hello,

I have run into a certain family of Markov chains, and I would like to
know if these have been studied before (that is, if there is some
literature dealing with them).

The state space is the non-negative integers, and 0 is an absorbing
state. For all other states i>0, the only transitions possible are the
transitions to 0,1,2,...,i-1, and i+1. My hope is that it is possible
to infer about the probability of being absorbed.

An example of such a Markov chain could be as follows. Let 0<p<1 and
let i>0 be a state. Define the transition probabilities by:

i --> i+1 : i * p / ( 1 - p^i + ip )
i --> j (j<i) : b(i,j) / ( 1 - p^i + ip )

where b(i,j) = binomial(i,j) * p^j * (1-p)^(i-j).

--
Michael Knudsen

.