Fast solvers for queueing systems with negative customers

Abstract

In this paper, we are interested in solving Markovian queueing systems having Poisson batch arrivals, exponential servers and negative customers. Preconditioned conjugate gradient method is applied to solving the steady-state probability distribution of queueing system. Preconditioner were constructed by exploiting near-Toeplitz structure of the generator matrix and the Gohberg-Semumcul formula. We proved the preconditioned system has its spectrum clustered around one. Numerical results are given to demonstrate the efficient of our method.