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Here is the question:

Assume that we have $N$ Poisson processes, with arrival rates $\lambda_n, n=1...N$. At the start, we randomly choose, e.g. with equal probability, one Poisson process. Then, when there is an (the first) arrival in this process, we switched to another link, also with equal probability. This switching operation is then repeated.

So, what is the resulted process? I'm thinking if it is still a Poisson process? What is the arrival rate then?

Could some one give me the answer? Thanks a lot.

user64946
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  • I assume you mean by switching to another Poisson process that you wait until a new arrival happens there (and not if the n-th arrival has already happened you take 0 for the arrival times in between). This may be a special case of a Cox-Process, but I'm not sure if the $\lambda(t)$ there can depend on the arrival times. In any case it is no longer a Poisson process. By the way this question does not appear to be about the software Mathematica. – Jacob Akkerboom Mar 26 '14 at 16:16
  • More specifically it seems to be a Poisson hidden Markov model. The process is not Markov chain. – Jacob Akkerboom Mar 26 '14 at 16:28
  • When switching, can one stay at the same link or must one choose a different one? – Did Mar 29 '14 at 17:24

1 Answers1

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This is Markov Switched Poisson Process, a special case of Markovian Arrival Process.

Specifically, Markov Switched Poisson Process is a point process consisting of geometric runs of intervals that are exponentially distributed with rates depending on the state of the underlying discrete time Markov chain with m transient states and with transition probability matrix P.

This is a more general set up.

Srinivas Chakravarthy
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