S. Asmussen et D. Perry, AN OPERATIONAL CALCULUS FOR MATRIX-EXPONENTIAL DISTRIBUTIONS, WITH APPLICATIONS TO A BROWNIAN (Q, Q) INVENTORY MODEL, Mathematics of operations research, 23(1), 1998, pp. 166-176
Citations number
30
Categorie Soggetti
Operatione Research & Management Science",Mathematics,"Operatione Research & Management Science",Mathematics
A distribution G on (0, infinity) is called matrix-exponential if the
density has the form alpha e(Tz)s where tu is a row vector, T a square
matrix and s a column vector. Equivalently, the Laplace transform is
rational. For such distributions, we develop an operator calculus, whe
re the key step is manipulation of analytic functions f(z) extended to
matrix arguments. The technique is illustrated via an inventory model
moving according to a reflected Brownian motion with negative drift,
such that an order of size Q is placed when the stock process down-cro
sses some level q. Explicit formulas for the stationary density are fo
und under the assumption that the leadtime Z has a matrix-exponential
distribution, and involve expressions of the form f(T) where f(z) = ro
ot 1-2z.