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An operational calculus for matrix-exponential disributions, with applicaions to a Brownian (q,Q) inventory model

Asmussen, Sören LU and Perry, David (1998) In Mathematics of Operations Research 23(1). p.166-176
Abstract
distribution G on [math not displayed] is called matrix-exponential if the density has the form αeTz s where α 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, where 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-crosses some level q. Explicit formulas for the stationary density are found under the assumption that the leadtime Z has a matrix-exponential distribution, and involve expressions of the form... (More)
distribution G on [math not displayed] is called matrix-exponential if the density has the form αeTz s where α 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, where 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-crosses some level q. Explicit formulas for the stationary density are found under the assumption that the leadtime Z has a matrix-exponential distribution, and involve expressions of the form f(T) where [math not displayed]. (Less)
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author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Measurement, Performance, Theory, Verification, (s, S) model, Brownian motion, EOQ model, computer algebra, inventory system, matrix-exponential distribution, operator calculus, phase-type distribution, stochastic decomposition, storage model
in
Mathematics of Operations Research
volume
23
issue
1
pages
166 - 176
publisher
Informs
ISSN
0364-765X
language
English
LU publication?
yes
id
f4449990-3da7-4aa9-ae92-b7f830684bfb (old id 1273198)
date added to LUP
2008-12-09 13:59:28
date last changed
2016-04-16 05:49:49
@article{f4449990-3da7-4aa9-ae92-b7f830684bfb,
  abstract     = {distribution G on [math not displayed] is called matrix-exponential if the density has the form αeTz s where α 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, where 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-crosses some level q. Explicit formulas for the stationary density are found under the assumption that the leadtime Z has a matrix-exponential distribution, and involve expressions of the form f(T) where [math not displayed].},
  author       = {Asmussen, Sören and Perry, David},
  issn         = {0364-765X},
  keyword      = {Measurement,Performance,Theory,Verification,(s,S) model,Brownian motion,EOQ model,computer algebra,inventory system,matrix-exponential distribution,operator calculus,phase-type distribution,stochastic decomposition,storage model},
  language     = {eng},
  number       = {1},
  pages        = {166--176},
  publisher    = {Informs},
  series       = {Mathematics of Operations Research},
  title        = {An operational calculus for matrix-exponential disributions, with applicaions to a Brownian (q,Q) inventory model},
  volume       = {23},
  year         = {1998},
}