Skorohod-Loynes characterizations of queueing, fluid, and inventory processes

We consider queueing, fluid and inventory processes whose dynamics are dete rmined by general point processes or random measures that represent inputs and outputs. The state of such a process (the queue length or inventory lev el) is regulated to stay in a finite or infinite interval-inputs or outputs are disregarded when they would lead to a state outside the interval. The sample paths of the process satisfy an integral equation; the paths have fi nite local variation and may have discontinuities. We establish the existen ce and uniqueness of the process based on a Skorohod equation. This leads t o an explicit expression for the process on the doubly-infinite Lime axis. The expression is especially tractable when the process is stationary with stationary input-output measures. This representation is an extension of th e classical Loynes representation of stationary waiting times in single-ser ver queues with stationary inputs and services. We also describe several pr operties of stationary processes: Palm probabilities of the processes at ju mp times, Little laws for waiting times in the system, finiteness of moment s and extensions to tandem and treelike networks.