Cycle time of stochastic max-plus linear systems

We analyze the asymptotic behavior of the sequence of random variables (x(n, x 0 )) n∈N defined by:

x(0, x 0 ) = x 0 x i (n + 1, x 0 ) = max j (A ij (n) + x j (n, x 0 )) ,

where (A(n)) n∈N is a stationary and ergodic sequence of random matrices with entries in R ∪ {-∞}. Moreover, we assume that A(n) has at least one finite entry on each row, which is a necessary and sufficient condition for x(n, x 0 ) to be finite. (Otherwise, some coefficients can be -∞.)

Such sequences are best understood by introducing the so-called max-plus algebra, which is actually a semiring.

Definition 1.1. The max-plus semiring R max is the set R ∪ {-∞}, with the max as a sum (i.e. a ⊕ b = max(a, b)) and the usual sum as a product (i.e. a ⊗ b = a + b). In this semiring, the identity elements are -∞ and 0.

We also use the matrix and vector operations induced by the semiring structure. For matrices A, B with appropriate sizes, (

, and for a scalar a ∈ R max , (a ⊗ A) ij = a ⊗ A ij = a + A ij . Now, Equation (1) x(n + 1, x 0 ) ⊗ A(n)x(n, x 0 ). In the sequel, all products of matrices by vectors or other matrices are to be understood in this structure.

For any integer k ≥ n, we define the product of matrices A(k, n) := A(k) • • • A(n) with entries in this semiring. Therefore, we have x(n, x 0 ) = A(n -1, 0)x 0 and if the sequence has indices in Z, which is possible up to a change of probability space, we define a new random vector y(n, x 0 ) := A(-1, -n)x 0 , which has the same distribution as x(n, x 0 ). Sequences defined by Equation 1 model a large class of discrete event dynamical systems. This class includes some models of operations research like timed event graphs (F. Baccelli [1]), 1-bounded Petri nets (S. Gaubert and J. Mairesse [10]) and some queuing networks (J. Mairesse [15], B. Heidergott [12]) as well as many concrete applications. Let us cite job-shops models (G. Cohen et al. [7]), train networks (H. Braker [6], A. de Kort and B. Heidergott [9]), computer networks (F. Baccelli and D. Hong [3]) or a statistical mechanics model (R. Griffiths [11]). For more details about modelling, see the books by F. Baccelli and al. [2] and by B. Heidergott and al. [13].

The sequences satisfying Equation (1) have been studied in many papers. If a matrix A has at least one finite entry on each row, then x → Ax is non-expanding for the L ∞ norm. Therefore, we can assume that x 0 is the 0-vector, also denoted by 0, and we do it from now on.

We say that (x(n, 0)) n∈N defined in (1) satisfies the strong law of large numbers if 1 n x(n, 0) n∈N converges almost surely. When it exists, the limit in the law of large numbers is called the cycle time of (A(n)) n∈N or (x(n, 0)) n∈N , and may in principle be a random variable. Therefore, we say that (A(n)) n∈N has a cycle time rather than (x(n, 0)) n∈N satisfies the strong law of large numbers. Some sufficient conditions for the existence of this cycle time were given by J.E. Cohen [8], F. Baccelli and Liu [4,1], Hong [14] and more recently by Bousch and Mairesse [5], the author [16] or Heidergott et al. [13].

Bousch and Mairesse proved (Cf. [5]) that, if A(0)0 is integrable, then the sequence 1 n y(n, 0) n∈N converges almost-surely and in mean and that, under stronger integrability conditions, 1 n x(n, 0) n∈N converges almost-surely if and only if the limit of 1 n y(n, 0) n∈N is deterministic. The previous results can be seen as providing sufficient conditions for this to happen. Some results only assumed ergodicity of (A(n)) n∈N , some others independence. But, even in the i.i.d. case, it was still unknown, which sequences had a cycle time and which had none.

In this paper, we solve this long standing problem. The main result (Theorem 2.4) establishes a necessary and sufficient condition for the existence of the cycle time of (A(n)) n∈N . Moreover, we show that this condition is necessary (Theorem 2.3) but not sufficient (Example 1) when (A(n)) n∈N is only ergodic or mixing. Theorem 2.3 also states that the cycle time is always given by a formula (Formula (3)), which was proved in Baccelli [1] under several additional conditions.

To state the necessary and sufficient condition, we extend the notion of graph of a random matrix from the fixed support case, that is when the entries are either almostsurely finite or almost-surely equal to -∞, to the general case. The analysis of its decomposition into strongly connected components allows us to define new submatrices, which must have almost-surely at least one finite entry on each row, for the cycle time to exist.

To prove the necessity of the condition, we use the convergence results of Bousch and Mairesse [5] and a result of Baccelli [1]. To prove the converse part of Theorem 2.4, we perform an induction on the number of strongly connected components of the graph. The first step of the induction (Theorem 3.11) is an extension of a result of D. Hong [14].

The paper is organized as follows. In Section 2, we state our res

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