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We propose a family of finite approximations for the departure process of a MAP/MAP/1 queue. The departure process approximations are derived via an exact aggregate solution technique (called ETAQA) applied to quasi-birth-death processes (QBDs) and require only the computation of the frequently sparse fundamental-period matrix G. The approximations are indexed by a parameter n, which determines the size of the output model as n-1 QBD levels. The marginal distribution of the true departure process and the lag correlations of the interdeparture times up to lag n+1 are preserved exactly. Via experimentation we show the applicability of the proposed approximation in traffic-based decomposition of queueing networks and investigate how correlation propagates through tandem queues.