Çinlar uses the unique structure of the MAP to prove that, given a gamma process with a shape parameter that is a function of Brownian motion, the resulting lifetime is distributed according to the Weibull distribution.
Kharoufeh presents a compact transform expression for the failure distribution for wear processes of a component degrading according to a Markovian environment inducing state-dependent continuous linear wear by using the properties of a MAP and assuming the wear process to be temporally homogeneous and that the environmental process has a finite state space.
Notes
^ abMagiera, R. (1998). "Optimal Sequential Estimation for Markov-Additive Processes". Advances in Stochastic Models for Reliability, Quality and Safety. pp. 167–181. doi:10.1007/978-1-4612-2234-7_12. ISBN978-1-4612-7466-7.
^Asmussen, S. R. (2003). "Markov Additive Models". Applied Probability and Queues. Stochastic Modelling and Applied Probability. Vol. 51. pp. 302–339. doi:10.1007/0-387-21525-5_11. ISBN978-0-387-00211-8.
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markov, additive, process, this, article, about, bivariate, processes, arrival, processes, queues, markovian, arrival, process, applied, probability, bivariate, markov, process, where, future, states, depends, only, variables, contents, definition, finite, cou. This article is about bivariate processes For arrival processes to queues see Markovian arrival process In applied probability a Markov additive process MAP is a bivariate Markov process where the future states depends only on one of the variables 1 Contents 1 Definition 1 1 Finite or countable state space for J t 1 2 General state space for J t 2 Example 3 Applications 4 NotesDefinition EditFinite or countable state space for J t Edit The process X t J t t 0 displaystyle X t J t t geq 0 is a Markov additive process with continuous time parameter t if 1 X t J t t 0 displaystyle X t J t t geq 0 is a Markov process the conditional distribution of X t s X t J t s displaystyle X t s X t J t s given X t J t displaystyle X t J t depends only on J t displaystyle J t The state space of the process is R S where X t takes real values and J t takes values in some countable set S General state space for J t Edit For the case where J t takes a more general state space the evolution of X t is governed by J t in the sense that for any f and g we require 2 E f X t s X t g J t s F t E J t 0 f X s g J s displaystyle mathbb E f X t s X t g J t s mathcal F t mathbb E J t 0 f X s g J s dd Example EditA fluid queue is a Markov additive process where J t is a continuous time Markov chain clarification needed example needed Applications EditThis section may be confusing or unclear to readers Please help clarify the section There might be a discussion about this on the talk page April 2020 Learn how and when to remove this template message Cinlar uses the unique structure of the MAP to prove that given a gamma process with a shape parameter that is a function of Brownian motion the resulting lifetime is distributed according to the Weibull distribution Kharoufeh presents a compact transform expression for the failure distribution for wear processes of a component degrading according to a Markovian environment inducing state dependent continuous linear wear by using the properties of a MAP and assuming the wear process to be temporally homogeneous and that the environmental process has a finite state space Notes Edit a b Magiera R 1998 Optimal Sequential Estimation for Markov Additive Processes Advances in Stochastic Models for Reliability Quality and Safety pp 167 181 doi 10 1007 978 1 4612 2234 7 12 ISBN 978 1 4612 7466 7 Asmussen S R 2003 Markov Additive Models Applied Probability and Queues Stochastic Modelling and Applied Probability Vol 51 pp 302 339 doi 10 1007 0 387 21525 5 11 ISBN 978 0 387 00211 8 This probability related article is a stub You can help Wikipedia by expanding it vte Retrieved from https en wikipedia org w index php title Markov additive process amp oldid 999294609, wikipedia, wiki, book, books, library,