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Campbell's Formula and Applications to Queueing

(Advances in Queueing, ed. J. H. Dshalalow, 225-242, CRC Press, 1995)

Volker Schmidt
Department of Stochastics
University of Ulm
D-89069 Ulm, Germany

and

Richard F. Serfozo
School of Industrial and Systems Engineering
Georgia Institute of Technology
Atlanta, Georgia 30332-0205, U.S.A



Campbell's formula for Palm probabilities is a basic tool for deriving properties of stationary queueing systems and stationary processes in general. This study reviews Campbell's formula and gives new insights into its versatility by establishing several equivalent versions of it. A few of these are known (e.g. the exchange formula and H=(lambda)G). Also included are applications involving integrals with respect to random product-measures, waiting times in systems, rate conservation laws, sojourn times of processes, travel times in networks, ladder heights in risk processes and virtual delays in queueing systems.

MSC 1991 Subject Classification: Primary 60K25, secondary 60G57, 60G55, 90A46

Key Words & Phrases: Stationary processes, point processes, random measures, Campbell's formula, queueing, Little's laws, sojourns, travel time, ladder height distributions, virtual waiting time, attained waiting time, rate conservation laws, exchange formulas.

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