Open Access
2008 Cycle time of stochastic max-plus linear systems.
Glenn Merlet
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Electron. J. Probab. 13: 322-340 (2008). DOI: 10.1214/EJP.v13-488


We analyze the asymptotic behavior of sequences of random variables defined by an initial condition, a stationary and ergodic sequence of random matrices, and an induction formula involving multiplication is the so-called max-plus algebra. This type of recursive sequences are frequently used in applied probability as they model many systems as some queueing networks, train and computer networks, and production systems. We give a necessary condition for the recursive sequences to satisfy a strong law of large numbers, which proves to be sufficient when the matrices are i.i.d. Moreover, we construct a new example, in which the sequence of matrices is strongly mixing, that condition is satisfied, but the recursive sequence do not converges almost surely.


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Glenn Merlet. "Cycle time of stochastic max-plus linear systems.." Electron. J. Probab. 13 322 - 340, 2008.


Accepted: 10 March 2008; Published: 2008
First available in Project Euclid: 1 June 2016

zbMATH: 1190.60021
MathSciNet: MR2386735
Digital Object Identifier: 10.1214/EJP.v13-488

Primary: 60F15 , 93C65
Secondary: 60J10 , 90B15 , 93D209

Keywords: Law of Large Numbers , Markov chains , max-plus , Products of random matrices , stochastic recursive sequences , subadditivity‎

Vol.13 • 2008
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