  We propose a generalization of the classical M/M/1 queue process. The
resulting model is derived by applying fractional derivative operators to a
system of difference-differential equations. This generalization includes both
non-Markovian and Markovian properties, which naturally provide greater
flexibility in modeling real queue systems than its classical counterpart.
Algorithms to simulate M/M/1 queue process and the related linear birth-death
process are provided. Closed-form expressions of the point and interval
estimators of the parameters of these fractional stochastic models are also
presented. These methods are necessary to make these models usable in practice.
The proposed fractional M/M/1 queue model and the statistical methods are
illustrated using S&P data.
