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

Hook, James and Broomhead, Dave (2011) Stochastic production trees as products of i.i.d. componentwise exponential max-plus matrices. [MIMS Preprint]

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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
Uncontrolled Keywords: Max-plus algebra, tropical algebra, queues, petri-nets, random matrices, extreme value statistic
Subjects: MSC 2010, the AMS's Mathematics Subject Classification > 05 Combinatorics
MSC 2010, the AMS's Mathematics Subject Classification > 15 Linear and multilinear algebra; matrix theory
MSC 2010, the AMS's Mathematics Subject Classification > 37 Dynamical systems and ergodic theory
MSC 2010, the AMS's Mathematics Subject Classification > 62 Statistics
MSC 2010, the AMS's Mathematics Subject Classification > 92 Biology and other natural sciences
MSC 2010, the AMS's Mathematics Subject Classification > 94 Information and communication, circuits
Depositing User: Mr James Hook
Date Deposited: 18 Aug 2011
Last Modified: 08 Nov 2017 18:18
URI: http://eprints.maths.manchester.ac.uk/id/eprint/1665

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