In this paper, we consider a finite-buffer MX/GY/1/K + B queue with setup times, which has a wide range of applications. The primary purpose of this paper is to discuss both exact analytic and computational aspects of this system. For this purpose, we present a modern applied approach to queueing theory. First, we present a set of linear equations for the stationary queue-length distribution at a departure epoch based on the embedded Markov-chain technique. Next, using two simple approaches that are based on the conditioning of the system states and discrete renewal theory, we establish two numerically stable relationships for the stationary queue-length distributions at three different epochs: departure, random, and arrival. Finally, based on these relationships, we present useful performance measures of interest such as the moments of the stationary queue lengths at three different epochs, the blocking probability, the mean delay in queue, and the probability that the server is busy, with computational experience.
