Perturbation Methods for Phase-Type Queueing Models

This paper investigates the performance and sensitivity analysis of the finite-capacity \({{M/P}H}/1/N\) queue, where service times follow a phase-type distribution. Using a matrix-analytic formulation of the underlying Markov chain, we derive transition probabilities and stationary distributions for the embedded process. To assess the effect of small perturbations in arrival and service parameters, we develop a Taylor expansion framework, providing both univariate and multivariate approximations of performance measures such as the blocking probability and mean queue length. Theoretical results are complemented by numerical experiments on exponential, Erlang, hyperexponential, and Coxian service-time distributions. In particular, we validate the accuracy of linear and quadratic Taylor approximations through Monte Carlo simulations. The results show that blocking probability exhibits greater sensitivity to parameter variations than mean queue length, offering valuable insights for system design under uncertainty. The proposed methodology provides a tractable and accurate tool for analyzing finite-buffer queueing systems with general service-time variability.