  In this paper, we study a birth and death process $\{N_t\}_{t\ge0}$ on
positive half lattice, which at each discontinuity jumps at most a distance
$R\ge 1$ to the right or exactly a distance $1$ to the left. The transitional
probabilities at each site are nonhomogeneous. Firstly, sufficient conditions
for the recurrence and positive recurrence are presented. Then by the branching
structure within random walk with one-side bounded jumps set up in Hong and
Wang (2013), the explicit form of the stationary distribution of the process
$\{N_t\}_{t\ge0}$ is formulated.
