A walk with a floor

Topic: Markov chains · more questions on this topic
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A walk on \{0,1,2,\ldots\} steps up with probability p=0.3 and down with probability q=0.7, except that from 0 it stays at 0 instead of stepping down. Find the limiting probability that the walk is at 0, and its long-run mean position.

Worked answer (click to show)

With r=p/q=3/7<1, the balance equations give \pi_j=(1-r)r^j, a geometric distribution on \{0,1,2,\ldots\}. So

\pi_0=1-\tfrac37=\tfrac47=0.571,

and the long-run mean position is \sum_jj\pi_j=r/(1-r)=(3/7)/(4/7)=0.75.