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Rate Conservation Law for Stationary Semimartingales

Published online by Cambridge University Press:  27 July 2009

Indrajit Bardhan  and
Karl Sigman
Indrajit Bardhan
Affiliation:
Department of Industrial Engineering and Operations ResearchColumbia University New York, New York 10027-6699
Karl Sigman
Affiliation:
Department of Industrial Engineering and Operations ResearchColumbia University New York, New York 10027-6699
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Abstract

The Rate Conservation Law (RCL) of Miyazawa [18] is generalized to what we call a General Rate Conservation Law (GRCL) to cover processes of unbounded variation such as Brownian motion and more general Levy processes. The general setup is that of a time-stationary semimartingale Y = [Yt: t ≥ 0], which is allowed to have jumps. From an elementary application of Ito's formula together with the Palm inversion formula, we obtain a law that includes Miya-zawa's RCL as a special case. A variety of applications and connections with the RCL are given. For example, we show that using the GRCL, one can immediately obtain the noted steady-state decomposition results for vacation queueing models, including those obtained by Kella and Whitt [13] for Jump-Levy processes. Other examples include state-dependent diffusion processes such as the Ornstein-Uhlenbeck process.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 7 , Issue 1 , January 1993 , pp. 1 - 17
Copyright
Copyright © Cambridge University Press 1993

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