On the dynamics and performance of stochastic fluid systems

Abstract

A (generalized) stochastic fluid system Q is defined as the one-dimensional Skorokhod reflection of a finite variation process X (with possibly discontinuous paths). We write X as the (not necessarily minimal) difference of two positive measures, A, B, and prove an alternative ‘integral representation’ for Q. This representation forms the basis for deriving a ‘Little's law’ for an appropriately constructed stationary version of Q. For the special case where B is the Lebesgue measure, a distributional version of Little's law is derived. This is done both at the arrival and departure points of the system. The latter result necessitates the consideration of a ‘dual process’ to Q. Examples of models for X, including finite variation Lévy processes with countably many jumps on finite intervals, are given in order to illustrate the ideas and point out potential applications in performance evaluation.