Markov additive 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.[1]


Finite or countable state space for J(t)[edit]

The process is a Markov additive process with continuous time parameter t if[1]

  1. is a Markov process
  2. the conditional distribution of given depends only on .

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]



A fluid queue is a Markov additive process where J(t) is a continuous-time Markov chain[clarification needed][example needed].


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.


  1. ^ 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.
  2. ^ 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.