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What is the difference between a Poisson process and a Markov process, in terms of applications?

Ferdi
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M.T
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1 Answers1

8

Poisson process is a counting process -- main use is in queuing theory where you are modeling arrivals and departures. The distribution of the time to next arrival is independent of the time of the previous arrival (or on how long you've waited since the last arrival).

Markov processes depend only on the current value only $P(X_{t+1}|X_{t},X_{t-1}...) = P(X_{t+1}|X_{t})$

You can model a Poisson Process as a Markov Process: its just a pure-birth chain.

So, Poisson process is a type of Markov process. However, there are some Markov Processes that are bounded/finite state space. For example, you want to model the weather with choices $\{rainy, sunny, cloudy\}$.

Basically -- if you're modelling discrete arrivals/events then go with Poisson, if you're going with transitions amongst a finite or countably infinite number of states, then Markov, if the states are continuous then you're talking about stochastic processes like Brownian Motion.