This paper considers nonhomogeneous M(t)/M(t)/1 queues which ran model systems such as communications networks. For such systems, bounds on moment-generating functions and on the tail distribution of the queue process ore obtained, These bounds are useful for characterizing the quality of service a system run provide to its users. An approach utilizing the theory of differential equations is adopted. The bounds given in this paper are tighter than those previously available. In fact, the bounds can he made arbitrarily tight given sufficient computational effort.