Markov fluid queue models have served as one of the main tools for the performance analysis of computer and communication systems and networks. These models have also been used in other disciplines such as insurance risk, finance, inventory control, etc. This thesis focuses on the time dependent (transient) analysis of Markov uid queue models. In particular, a numerical method is proposed to obtain both the transient and first passage time distributions of a Multi-Regime Markov Fluid Queue (MRMFQ). The proposed method is based on obtaining the steady-state solution of an auxiliary MRMFQ that is to be constructed from the original MRMFQ which then leads to the related transient measures of interest. First, in order to model the deterministic time horizon, the Erlangization method is used. Then, as an alternative to Erlangization, ME-fication technique which efficiently replaces the Erlang distribution with a Concentrated Matrix Exponential (CME), is used. As an application of the proposed method, an M/M/S+G queue with generally distributed impatience times is modeled by using MRMFQs and our transient analysis method is subsequently applied to obtain the time dependent distributions. Numerical examples are given to show the effectiveness of the proposed transient analysis method while employing ME-fication.