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Consider a Markov modulated fluid queue with multiple layers separated by a finite number of boundaries, where each layer is characterized by its own set of matrices. In the past, matrix analytic methods have been devised to determine the stationary behavior of such a fluid queue for no-resistance, sticky and repellent boundaries. In this paper we extend this approach by allowing general phase transitions at the boundaries. As an application, we analyze the MMAP[K]/PH[K]/1 queue with general, customer type dependent impatience, where customers remain impatient while being served. We show that the steady state distribution of the age process of this queue can be expressed via the steady state distribution of a multi-layered fluid queue with phase transitions at the boundary. Based on the analysis of the age process, expressions for the sojourn time distribution and for the probability of abandonment are presented.