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]

PDF Products_of_iid_componentwise_exponential_maxplus_matricies[1].pdf Download (1MB)

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:	https://eprints.maths.manchester.ac.uk/id/eprint/1665

Actions (login required)

View Item