We study a double-sided queueing model with batch Markovian arrival processes and abandonment, which arises in various stochastic systems such as perishable inventory systems and financial markets. In the first part of the paper, we propose an approach to obtain the joint stationary distributions of the age processes via the stationary analysis of a multi-layer Markov modulated fluid flow (MMFF) process. In the second part of the paper, using the joint stationary distribution of the age processes, we derive a number of queueing quantities related to matching rates, fill rates, sojourn times and queue length for both sides of the system. Numerical examples are presented to demonstrate the effectiveness of the approach and to show an application in inventory control of vaccine in healthcare.