Asymptotic Behaviors of the Loss Probability for a Finite Buffer Queue with QBD Structure

We are concerned with a finite buffer queue with arrival and service processes that are generated by a continuous time Markov chain with finitely many states. This model includes a many-server queue, and is described by a truncated QBD, where QBD stands for a quasi-birth-and-death process. It is shown that the loss probability of this queue geometrically decays with a constant prefactor as the buffer size goes to infinity if the corresponding queue with an unlimited buffer is stable. For this, we first consider asymptotic behaviors of the truncated QBD. We then specialize this result for a many-server queue with a finite buffer and Markovian arrival and service processes of Neuts. Using the asymptotic results, we also numerically consider an approximation of the loss probability.