On the solution of algebraic Riccati equations arising in fluid queues

New algorithms for solving algebraic Riccati equations (ARE) which arise in fluid queues models are introduced. They are based on reducing the ARE to a unilateral quadratic matrix equation of the kind AX2 + BX + C = 0 and on applying the Cayley transform in order to arrive at a suitable spectral splitting of the associated matrix polynomial. A shifting technique for removing unwanted eigenvalues of modulus 1 is complemented with a suitable parametrization of the matrix equation in order to arrive at fast and numerically reliable solvers based on quadratically convergent iterations like logarithmic reduction and cyclic reduction. Numerical experiments confirm the very good performance of these algorithms.