  Let $\omega=(\omega_i)_{i\in\mathbb
Z}=(\mu^{L}_i,...,\mu^{1}_i,\lambda_i)_{i\in \mathbb Z}$, which serves as the
environment, be a sequence of i.i.d. random nonnegative vectors, with $L\ge1$ a
positive integer. We study birth and death process $N_t$ which, given the
environment $\omega,$ waits at a state $n$ an exponentially distributed time
with parameter $\lambda_n+\sum_{l=1}^L\mu^{l}_n$ and then jumps to $n-i$ with
probability ${\mu^i_n}/(\lambda_n+\sum_{l=1}^L\mu^{l}_n),$ $i=1,...,L$ or to
$n+1$ with probability ${\lambda_n}/(\lambda_n+\sum_{l=1}^L\mu^{l}_n).$ A
sufficient condition for the existence, a criterion for recurrence, and a law
of large numbers of the process $N_t$ are presented. We show that the first
passage time $T_1\overset{\mathscr D}{=}\xi_{0,1}+\sum_{i\le
-1}\sum_{k=1}^{U_{i,1}}\xi_{i,k}+\sum_{i\le -1}\sum_{k=
1}^{U_{i,1}+...+U_{i,L}}\tilde{\xi}_{i+1,k},$ where
$(U_{i,1},...,U_{i,L})_{i\le0}$ is an $L$-type branching process in random
environment and, given $\omega,$ $\xi_{i,k},\ \tilde\xi_{i,k},\ i\le 0,\ k\ge
1$ are mutually independent random variables such that
  $P_\omega(\xi_{i,k}\ge t)=e^{-(\lambda_i+\sum_{l=1}^L\mu^{l}_i)t},\ t\ge 0.$
This fact enables us to give an explicit velocity of the law of large numbers.
