A Numerically E cient Method for theMAP / D / 1 / K Queue via Rational

The Markovian Arrival Process (MAP), which contains the Markov Modulated Poisson Process (MMPP) and the Phase-Type (PH) renewal processes as special cases, is a convenient traac model for use in the performance analysis of Asyn-chronous Transfer Mode (ATM) networks. In ATM networks, packets are of xed length and the buuering memory in switching nodes is limited to a nite number K of cells. These motivate us to study the MAP/D/1/K queue. We present an algorithm to compute the stationary virtual waiting time distribution for the MAP/D/1/K queue via rational approximations for the deterministic service time distribution in transform domain. These approximations include the well-known Erlang distributions and the Pad e approximations that we propose. Using these approximations, the solution for the queueing system is shown to reduce to the solution of a linear diierential equation with suitable boundary conditions. The proposed algorithm has a computational complexity independent of the queue storage capacity K. We show through numerical examples that, the idea of using Pad e approximations for the MAP/D/1/K queue can yield very high accuracy with tractable computational load even in the case of large queue capacities.