Products of irreducible random matrices in the (max, +) algebra

Products of irreducible random matrices in the (max, +) algebra

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Article ID: iaor1998964
Country: United Kingdom
Volume: 29
Issue: 2
Start Page Number: 444
End Page Number: 477
Publication Date: Jun 1997
Journal: Advances in Applied Probability
Authors:
Abstract:

We consider the recursive equation x(n + 1) = A(n) ⊗ x(n), where x(n + 1) and x(n) are ℝk-valued vectors and A(n) is an irreducible random matrix of size k × k. The matrix-vector multiplication in the (max, +) algebra is defined by (A(n) ⊗ x(n))i = maxj (Aij(n) + xj(n)). This type of equation can be used to represent the evolution of stochastic event graphs which include cyclic Jackson networks, some manufacturing models and models with general blocking (such as Kanban). Let us assume that the sequence {A(n), n ∈ ℕ} is i.i.d. or, more generally, stationary and ergodic. The main result of the paper states that the system couples in finite time with a unique stationary regime if and only if there exists a set of matrices 𝒞 such that P{A(0) ∈ 𝒞} > 0 and the matrices C ∈ 𝒞 have a unique periodic regime.

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