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In this paper, we consider a finite capacity single server queueing model with two buffers, A and B, of sizes K and N respectively. Messages arrive one at a time according to a Markovian arrival process. Messages that arrive at buffer A are of a different type from the messages that arrive at buffer B. Messages are processed according to the following rules: 1. When buffer A(B) has a message and buffer B(A) is empty, then one message from A(B) is processed by the server. 2. When both buffers, A and B, have messages, then two messages, one from A and one from B, are processed simultaneously by the server. The service times are assumed to be exponentially distributed with parameters that may depend on the type of service. This queueing model is studied as a Markov process with a large state space and efficient algorithmic procedures for computing various system performance measures are given. Some numerical examples are discussed

Two parallel finite queues with simultaneous services and Markovian arrivals