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Markov additive process

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This article is about bivariate processes. For arrival processes to queues, seeMarkovian arrival process.

Inapplied probability, aMarkov additive process (MAP) is a bivariateMarkov process where the future states depends only on one of the variables.^[1]

Definition

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Finite or countable state space forJ(t)

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The process $\{(X(t),J(t)):t\geq 0\}$ is a Markovadditive process with continuous time parametert if^[1]

$\{(X(t),J(t));t\geq 0\}$ is aMarkov process
the conditional distribution of $(X(t+s)-X(t),J(t+s))$ given $(X(t),J(t))$ depends only on $J(t)$ .

The state space of the process isR × S whereX(t) takes real values andJ(t) takes values in some countable setS.

General state space forJ(t)

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For the case whereJ(t) takes a more general state space the evolution ofX(t) is governed byJ(t) in the sense that for anyf andg we require^[2]

\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})]

Example

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Afluid queue is a Markov additive process whereJ(t) is acontinuous-time Markov chain^{[clarification needed]}^{[example needed]}.

Applications

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Çinlar uses the unique structure of the MAP to prove that, given agamma process with a shape parameter that is a function ofBrownian motion, the resulting lifetime is distributed according to theWeibull 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 finitestate space.

Notes

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^^a ^bMagiera, 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.

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Discrete time	Bernoulli process Branching process Chinese restaurant process Galton–Watson process Independent and identically distributed random variables Markov chain Moran process Random walk Loop-erased Self-avoiding Biased Maximal entropy
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