Difference between sticky channel Poisson-repeat channel
The Poisson-repeat channel is a binary channel where each transmitted bit is repeated at the channel output anywhere from $0$ to $\infty$ times. The repetition distribution is Poisson.

A "sticky channel" is a repeat channel where deletions are forbidden; that is, every transmitted bit is repeated from $1$ to $\infty$ times at the channel output, according to some specified distribution.