Skorohod–Loynes Characterizations of Queueing, Fluid, and Inventory Processes

Authors:
William L. Cooper;Volker Schmidt;Richard F. Serfozo
Affiliations:
Department of Mechanical Engineering, University of Minnesota, Minneapolis, MN 55455, USA billcoop@me.umn.edu;Department of Stochastics, University of Ulm, D-89069 Ulm, Germany schmidt@mathematik.uni-ulm.de;School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0205, USA rserfozo@isye.gatech.edu
Venue:
Queueing Systems: Theory and Applications
Year:
2001

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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.