Let X(t) be a non-homogeneous birth and death process. In this paper w
e develop a general method of estimating bounds for the state probabil
ities for X(t), based on inequalities for the solutions of the forward
Kolmogorov equations. Specific examples covered include simple estima
tes of Pr(X(t) <j\X(0)=k) for the M(t)/M(t)/N/0 and M(t)/M(t)/N queue-
length processes.