Problem related to the compound Poisson process (?)

[gabe_rosser]

Dear all,

I wonder if anyone has come across this problem before and could point me to a relevant ref or tell me what terms I might search for:

I am interested in a continuous time process in which two alternating events (call them A and B) occur. Each event has an exponentially distributed duration, with different rate constants. They occur consecutively and exclusively: A immediately follows B and vice-versa, with no overlapping or gaps.

Our experiment always starts with A. After an exponentially distributed waiting time, B occurs. After a second time wait, exponentially distributed but with different rate constant, A occurs again, etc.

I am seeking the expected number of events (or pairs of events) that occur in a given time interval.

I have attempted this myself, but my approach became complicated quite rapidly so I thought I would check here first to see if anyone had come across this before.

Thanks, and apologies for the lengthy description.

Gabriel

 

[Stephen Tashi]

I think the proper terminology for your model is a "continuous time Markov process". Look it up and see if that fits. If not, let us know.

 

[gabe_rosser]

Thanks for the suggestion. I should have thought to think of it as a Markov process.

I'm still not sure how to use this description to get at my desired result, however. Does anyone know of any results relating to the expected number of transition events for a given Markov chain? I'll look into this as well and post if I find anything useful.