Uniqueness of asymptotic solution for general Markov fluid models

We investigate the asymptotic workload distribution of fluid models with input and output rates which are modulated by an irreducible Markov process. An analytical solution is proposed in [H. Nabli, Asymptotic solution of stochastic fluid models, Perform. Eval. 57 (2004) 121–140] for general Markov fluid model. This probability distribution is controlled by a linear differential system with specific boundary conditions. The solution uniqueness is proved under a conjecture in the field of linear algebra. This conjecture is valid for all particular fluid models considered in the literature. In this paper, we will prove this conjecture for general fluid models. The numerical computation of the stationary distribution will be also discussed.