I am curious whether there exist continuous time Markov processes for which the times between jumping times (which I call inter arrival times) are not exponentially distributed, but have some other distribution (e.g. uniform distribution).
As people call the version in which inter arrival times occur in an exponential fashion the continuous time Markov chain, it seems me to suggest that continuous time Markov processes can only have exponentially distributed jumping times. Is this true?
Background information for visual thinking mathematicians (reading is optional)
In discrete-time markov chains transitions appear (taking arrows) on integer times. (roughly speaking you could say they happen at t = 1, t = 2, t = 3, t = 4, ...) . The times at which we take arrows in not depicted in the picture. Just think as if you are seeing a dynamical picture in which you are in one of the states of this system and that you change state you are in every second (after this second you are back in the same state, or in an other state, depending on which transition you chose). The time until we transition is a known quantity, and let us say it is 1 for simplicity.
In the picture above you can see such a discrete-time markov chain. Now in the continuous case, transitioning (= taking an arrow), can also appear on non-integer times. E.g. on time 1.22344354, 5.546346, 23.2345234, 28.45 and so on. The time it takes to take the next arrow is not deterministic anymore as in the previous case (it always took us time 1 to jump) but a random time. Thus it can be that we jump after t=1.334, t=3543.1, t=0.345, t=2.3535 etc. The time for the next jump we choose to be exponentially distributed in continuous time-markov chains (see http://en.wikipedia.org/wiki/Continuous-time_Markov_chain). On what ground did we choose for an exponential distribution? Is the exponential distribution the only possible solution?
In my initial question I also spoke about inter arrival times and jumping. With the inter arrival time I mean the time between two transitions. So you take an arrow. Now you start your chronometer, and just before you take again an arrow you stop your chronometer. The inter arrival times tells you after how many seconds you needed to jump to a new state. The jumping times are the times at which the actual transition happened. In the discrete case the jumping times were 1,2,3,4,5, ..., in the text above the jumping times were 1.22344354, 5.546346, 23.2345234, 28.45 and so on.