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We introduce a new approximation procedure to improve the accuracy of matrix analytic methods when using truncated queueing models to analyse infinite buffer systems. This is accomplished through emulating the presence of unobserved waiting customers beyond the finite buffer that are able to immediately enter the system following an observed customer's departure. We show that this procedure results in exact steady-state probabilities at queue lengths below the buffer for truncated versions of the classic M/M/1, M / M / 1 + M , M / M / ∞ , and M/PH/1 queues. We also present two variants of the basic procedure for use within a M / P H / 1 + M queue and a N-queue polling system with exhaustive service, phase-type service and switch-in times, and exponential impatience timers. The accuracy of these two variants in the context of the polling model are compared through several numerical examples. [ABSTRACT FROM AUTHOR]