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Products of Irreducible Random Matrices in the (Max, +) Algebra
- Part of
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- Journal:
- Advances in Applied Probability / Volume 29 / Issue 2 / June 1997
- Published online by Cambridge University Press:
- 01 July 2016, pp. 444-477
- Print publication:
- June 1997
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- Article
- Export citation
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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))= 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
such that 
and the matrices 
have a unique periodic regime.
such that 
and the matrices 
have a unique periodic regime.