In this paper, we investigate a retrial tandem queueing system with a finite number of queues. Each queue consists of a finite buffer and a single server with phase-type distributed service times. Customers arrive at the system according to a Markovian arrival process (MAP). At any queue, a customer that finds both the server busy and the buffer full enters a common orbit and makes repeated attempts to rejoin the first queue after exponentially distributed time intervals. This system models telecommunication networks with linear topology that implement retransmission protocols for lost data packets. We provide a complete mathematical analysis for the two-queue system with an infinite-capacity orbit, deriving the ergodicity condition, stationary state distribution, and key performance characteristics. For systems with an arbitrary number of queues, we develop a comprehensive solution approach that combines queueing theory methods, discrete-event simulation, and machine learning techniques to predict the mean sojourn time. We implement and compare multiple machine learning methods, evaluating their predictive performance through extensive numerical experiments.