The non-stationary Erlang-A queue is a fundamental queueing model that is used to describe the dynamic behavior of large scale multi-server service systems that may experience customer abandonments, such as call centers, hospitals, and urban mobility systems. In this paper, we develop novel approximations to all of its transient and steady state moments, the moment generating function, and the cumulant generating function. We also provide precise bounds for the difference of our approximations and the true model. More importantly, we show that our approximations have explicit stochastic representations as shifted Poisson random variables. Moreover, we are also able to show that our approximations and bounds also hold for non-stationary Erlang-B and Erlang-C queueing models under certain stability conditions. Finally, we perform numerous simulations to support the conclusions of our results.