Piecewise $α$-stable Ornstein-Uhlenbeck (OU) processes arising in queue networks usually do not have an explicit dissipation, which makes the related numerical methods such as Euler-Maruyama (EM) scheme more difficult to analyze. We develop an EM scheme with decreasing step size $Λ=(η_n)_{n\in \mathbb{N}}$ to approximate their ergodic measures. This approximation does not have a bias and has a rate $η^{1/α}_n$ in Wasserstein-1 distance. We show by the classical OU process that our convergence rate is optimal. We further prove the central limit theorem (CLT) and moderate derivation principle (MDP) for the empirical measure of these piecewise $α$-stable Ornstein-Uhlenbeck processes. In addition, we use the Sinkhorn--Knopp algorithm to compute the Wasserstein-1 distance and conduct simulations for several concrete examples.