Many modern queueing problems involve probability distributions which are heavy-tailed. This means their distribution functions decay more slowly than any exponential function. Analyzing queues with these distributions is difficult since they do not have closed-form, analytic Laplace transforms. This paper investigates a recently proposed method for numerically approximating Laplace transforms, called the Transform Approximation Method (TAM). While TAM can be used to approximate the Laplace transform of a heavy-tailed distribution, one must still invert a Laplace transform to recover the desired probability distribution. This paper investigates using TAM with two numerical methods for inverting Laplace transforms. In particular, we compare the well-known Fourier-series method with a recursion method. We give several benchmark problems and algorithms to compare the methods. In general, the Fourier method is better at finding P (Wq ≤ t) for a single t. For the inverse problem, neither method clearly dominates.