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A local proof of the Swiss Army formula of Palm calculus

Part of: Special processes Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Takis Konstantopoulos
Takis Konstantopoulos*
Affiliation:
University of Texas at Austin
*
∗Postal address: Department of Electrical and Computer Engineering, University of Texas, Austin, TX 78712, USA. email:takis@alea.ece.utexas.edu
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Abstract

The so-called ‘Swiss Army formula', derived by Brémaud, seems to be a general purpose relation which includes all known relations of Palm calculus for stationary stochastic systems driven by point processes. The purpose of this article is to present a short, and rather intuitive, proof of the formula. The proof is based on the Ryll–Nardzewski definition of the Palm probability as a Radon-Nikodym derivative, which, in a stationary context, is equivalent to the Mecke definition.

Keywords

STATIONARY RANDOM MEASURESPOINT PROCESSESPALM CALCULUSQUEUEING THEORY

MSC classification

Primary: 60G10: Stationary processes
Secondary: 60K25: Queueing theory
Type
Short Communications
Information
Journal of Applied Probability , Volume 33 , Issue 3 , September 1996 , pp. 909 - 914
Copyright
Copyright © Applied Probability Trust 1996 

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Footnotes

Research supported in part by NSF grants NCR 9211343, NCR 9502582, and by grant ARP 224 of the Texas Higher Education Coordinating Board.

References

[1] Bacelli, F. and Bremaud, P. (1994) Elements of Queueing Theory. Springer, New York.Google Scholar
[2] Bremaud, P. (1993) A Swiss Army formula of Palm calculus. J. Appl. Prob. 30, 40–51.Google Scholar
[3] Daley, D. J. and Vere-Jones, D. (1988) An Introduction to the Theory of Point Processes. Springer, New York.Google Scholar
[4] Konstantopoulos, T., Zazanis, M. and De Veciana, G. (1995) Conservation laws and reflection mappings with an application to multiclass mean value analysis for stochastic fluid queues. Stoch. Proc. Appl. (submitted).Google Scholar
[5] Papangelou, F. (1974) On the Palm probabilities of processes of points and processes of lines. In Stochastic Geometry , ed. Harding, E. J. and Kendall, D. G. Wiley, New York. pp. 114–147.Google Scholar
[6] Ryll-Nardzewski, C. (1961) Remarks on processes of calls. Proc. 4th Berkeley Symp. Math. Stat. Prob. 2, 455–465.Google Scholar