We consider queueing systems with general abandonment. Abandonment times are approximated by a particular Cox distribution with all phase exponential rates being the same. We prove that this distribution arbitrarily closely approximate any non-negative distribution. By explicitly modeling the waiting time of the first customer in line, we obtain a natural bounded jump Markov process allowing for uniformization. This approach is useful to solve, via dynamic programming, various optimization problems where the objectives and/or constraints involve the distributions of the performance measures, not only their expected values. It is also useful for the performance analysis of queueing systems with general abandonment times.