The Annals of Probability

Ergodic Theory of Stochastic Petri Networks

Francois Baccelli

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Abstract

Stochastic Petri networks provide a general formalism for describing the dynamics of discrete event systems. The present paper focuses on a subclass of stochastic Petri networks called stochastic event graphs, under the assumption that the variables used for their "timing" form stationary and ergodic sequences of random variables. We show that such stochastic event graphs can be seen as a $(\max, +)$ linear system in a random, stationary and ergodic environment. We then analyze the associated Lyapounov exponents and construct the stationary and ergodic regime of the increments, by proving an Oseledec-type multiplicative ergodic theorem. Finally, we show how to construct the stationary marking process from these results.

Article information

Source
Ann. Probab. Volume 20, Number 1 (1992), 375-396.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176989932

Digital Object Identifier
doi:10.1214/aop/1176989932

Mathematical Reviews number (MathSciNet)
MR1143426

Zentralblatt MATH identifier
0742.60091

JSTOR
links.jstor.org

Subjects
Primary: 05C20: Directed graphs (digraphs), tournaments
Secondary: 60F20: Zero-one laws 60G10: Stationary processes 60G17: Sample path properties 60G55: Point processes 60K25: Queueing theory [See also 68M20, 90B22] 68Q75 68Q90 68R10: Graph theory (including graph drawing) [See also 05Cxx, 90B10, 90B35, 90C35] 93D05: Lyapunov and other classical stabilities (Lagrange, Poisson, $L^p, l^p$, etc.) 93E03: Stochastic systems, general 93E15: Stochastic stability

Keywords
Discrete event systems stochastic Petri networks event graphs queuing networks stationary processes stability stochastic recursive sequences subadditive ergodic theory multiplicative ergodic theory

Citation

Baccelli, Francois. Ergodic Theory of Stochastic Petri Networks. Ann. Probab. 20 (1992), no. 1, 375--396. doi:10.1214/aop/1176989932. https://projecteuclid.org/euclid.aop/1176989932


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