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ME-fication results in much lower state-space dimensionalities for the auxiliary MRMFQ than would be with Erlangization. As an alternative, we propose the so-called ME-fication technique, in which a Concentrated Matrix Exponential (CME) distribution replaces the Erlang distribution for approximating deterministic time horizons. The conventional method to approximately model the deterministic time horizon is Erlangization. This auxiliary MRMFQ is constructed from the original one, using sample path arguments, and has a larger cardinality stemming from the need to keep track of time. The method relies on the observation that these transient measures can be computed via the stationary analysis of an auxiliary MRMFQ. We propose a numerical method to obtain the transient and first passage time distributions of first- and second-order Multi-Regime Markov Fluid Queues (MRMFQ).