## Abstract

We consider queueing, fluid and inventory processes whose dynamics are determined by general point processes or random measures that represent inputs and outputs. The state of such a process (the queue length or inventory level) 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 finite local variation and may have discontinuities. We establish the existence and uniqueness of the process based on a Skorohod equation. This leads to an explicit expression for the process on the doubly-infinite time axis. The expression is especially tractable when the process is stationary with stationary input-output measures. This representation is an extension of the classical Loynes representation of stationary waiting times in single-server queues with stationary inputs and services. We also describe several properties of stationary processes: Palm probabilities of the processes at jump times, Little laws for waiting times in the system, finiteness of moments and extensions to tandem and treelike networks.

Original language | English (US) |
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Pages (from-to) | 233-257 |

Number of pages | 25 |

Journal | Queueing Systems |

Volume | 37 |

Issue number | 1-3 |

DOIs | |

State | Published - Jan 1 2001 |

## Keywords

- Fluid models
- Inventory
- Little's law
- Loynes' representation
- Palm probability
- Queues
- Skorohod equation