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Polling systems have an important role in performance studies of polling mechanisms in communication and computer networks, where several terminals or users compete for access to a shared resource. In this study we present a polling system with finite queues and investigate its performance. We use the state of the art Markov chain technique to reveal the performance measures of the considered polling system, e.g. mean number of customers in the queue, probabilities that the queue is idle, blocking or loss probabilities for customers in the queue of the polling system.