C.L. Liu and J.W. Layland (1973) provided a utilization bound test which is applicable to rate-monotonic priority assignment, when all tasks deadlines are at the end of their periods. Subsequently, J. Lehoczky et al. (1989) provided an exact schedulability criterion which is applicable to any arbitrary priority assignment scheme, with no restriction on task deadlines, which can be used when all task computation times are known exactly. In this work, we fill an important gap between these two, by presenting a technique for deriving utilization bounds, based on linear programming, that is applicable to similarly generalized situations, but does not require knowledge of exact task computation times.