On the exponentiality of stochastic linear systems under the max-plus algebra

Considers stochastic linear systems under the max-plus algebra. For such a system, the states are governed by the recursive equation X _n=A_n⊗X_n-1⊕U_n with the initial condition condition X₀=x₀. By transforming the linear system under the max-plus algebra into a sublinear system under the usual algebra, we establish various exponential upper bounds for the tail distributions of the states X_n under the independently identically distributed (i.i.d.) assumption on {(A_n,U_n)₁n⩾1} and a couple of regularity conditions on (A₁,U₁) and the initial condition x₀. These upper bounds are related to the spectral radius (or the Perron-Frobenius eigenvalue) of the nonnegative matrix in which each element is the moment generating function of the corresponding element in the state-feedback matrix A₁. In particular, we have Kingman´s upper bound for GI/GI/1 queue when the system is one-dimensional. We also show that some of these upper bounds can be achieved if A₁ is lower triangular. These bounds are applied to some commonly used systems to derive new results or strengthen known results