Products of Irreducible Random Matrices in the ( max , + ) Algebra 1

نویسنده

  • Jean Mairesse
چکیده

We consider the recursive equation \x(n + 1) = A(n) x(n)" where x(n + 1) and x(n) are R k-valued vectors and A(n) is an irreducible random matrix of size k k. The matrix-vector multiplication in the (max,+) algebra is deened by (A(n) x(n)) i = max j (A ij (n) + x j (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 fA(n); n 2 Ng is i.i.d or more generally stationary and ergodic. The main result of the paper states that the system couples in nite time with a unique stationary regime if and only if there exists a set of matrices C such that PfA(0) 2 Cg > 0 and the matrices C 2 C have a unique periodic regime.

برای دانلود رایگان متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

Products of irreducible random matrices in the (Max,+) Algebra

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,...

متن کامل

Products of Irreducible Random Matrices in the (Max,+) Algebra - Part I

The study of networks with synchronization, and more particularly of Stochastic Event Graphs has raised an interest for products of random matrices in the (Max; +) algebra. We consider a general model of type \x(n + 1) = A(n)x(n)" where x(n + 1) and x(n) are IR J-valued vectors and A(n) is an irreducible random matrix of size J J. The exogeneous sequence fA(n); n 2 INg is i.i.d or more generall...

متن کامل

On Commuting Matrices in Max Algebra and in Classical Nonnegative Algebra

This paper studies commuting matrices in max algebra and nonnegative linear algebra. Our starting point is the existence of a common eigenvector, which directly leads to max analogues of some classical results for complex matrices. We also investigate Frobenius normal forms of commuting matrices, particularly when the Perron roots of the components are distinct. For the case of max algebra, we ...

متن کامل

Max-Plus algebra on tensors and its properties

In this paper we generalize the max plus algebra system of real matrices to the class of real tensors and derive its fundamental properties. Also we give some basic properties for the left (right) inverse, under the new system. The existence of order 2 left (right) inverses of tensors is characterized.

متن کامل

Memory Loss Property for Products of Random Matrices in the Max-Plus Algebra

Products of random matrices in the max-plus algebra are used as a model for a class of discrete event dynamical systems. This can model a wide range of systems including train or queuing networks, job-shop, timed digital circuits or parallel processing systems. Some stability results have been proved under the so-called memory loss property. When the random matrices are i.i.d, we prove that the...

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

عنوان ژورنال:

دوره   شماره 

صفحات  -

تاریخ انتشار 1995