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| {{About|bivariate processes|arrival processes to queues|Markovian arrival process}}
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| In [[applied probability]], a '''Markov additive process''' ('''MAP''') is a bivariate [[Markov process]] where the future states depends only on one of the variables.<ref name="magiera" />
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| ==Definition==
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| ===Finite or countable state space for ''J''(''t'')===
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| The process {(''X''(''t''),''J''(''t'')) : ''t'' ≥ 0} is a Markov additive process with continuous time parameter ''t'' if<ref name="magiera">{{cite doi|10.1007/978-1-4612-2234-7_12}}</ref>
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| # {(''X''(''t''),''J''(''t'')) : ''t'' ≥ 0} is a [[Markov process]]
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| # the conditional distribution of (''X''(''t'' + ''s'') − ''X''(''t''),''J''(''s'' + ''t'')) given (''X''(''s''),''J''(''s'')) depends only on ''J''(''s'').
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| 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''.
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| ===General state space for ''J''(''t'')===
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| 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<ref>{{cite doi|10.1007/0-387-21525-5_11}}</ref>
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| ::<math>\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)]</math>.
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| ==Example==
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| A [[fluid queue]] is a Markov additive process where ''J''(''t'') is a [[continuous-time Markov chain]].
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| ==Applications==
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| 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]].
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| 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]].
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| ==Notes==
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| {{Reflist}}
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| {{Stochastic processes}}
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| {{probability-stub}}
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| [[Category:Stochastic processes]]
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