## 2011.69: Stochastic production trees as products of i.i.d. componentwise exponential max-plus matrices

2011.69:
James Hook and Dave Broomhead
(2011)
*Stochastic production trees as products of i.i.d. componentwise exponential max-plus matrices.*

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## Abstract

We introduce a class of stochastic production tree model, based on Petri nets, which admit a random matrix product description in the Max-plus algebra. With a kind of combinatorial change of variables we are able to simplify the form of the matrices arising from these models. For this class of \emph{Componentwise exponential} matrix we prove a new result relating the (Max-plus) spectrum of the product to the principal (classical) eigenvalue of an associated adjacency matrix by means of a sandwich inequality. This theorem highlights several important theoretical factors in the dynamics of Max-plus linear systems generally and gives us some neat insight into the different production tree models.

Item Type: | MIMS Preprint |
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Uncontrolled Keywords: | Max-plus algebra, tropical algebra, queues, petri-nets, random matrices, extreme value statistic |

Subjects: | MSC 2000 > 05 Combinatorics MSC 2000 > 15 Linear and multilinear algebra; matrix theory MSC 2000 > 37 Dynamical systems and ergodic theory MSC 2000 > 62 Statistics MSC 2000 > 92 Biology and other natural sciences MSC 2000 > 94 Information and communication, circuits |

MIMS number: | 2011.69 |

Deposited By: | Mr James Hook |

Deposited On: | 18 August 2011 |

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