On the exponentiality of stochastic linear systems under the max-plus algebra
Cheng‐Shang Chang · IEEE Transactions on Automatic Control · 1996
Considers stochastic linear systems under the max-plus algebra. For such a system, the states are governed by the recursive equation X/sub n/=A/sub n//spl otimes/X/sub n-1//spl oplus/U/sub n/ with the initial condition condition X/sub 0/=x/sub 0/. 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/sub n/ under the independently identically distributed (i.i.d.) assumption on {(A/sub n/,U/sub n/)/sub 1/n/spl ges/1} and a couple of regularity conditions on (A/sub 1/,U/sub 1/) and the initial condition x/sub 0/. 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 A/sub 1/. 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 A/sub 1/ is lower triangular. These bounds are applied to some commonly used systems to derive new results or strengthen known results.