# What probability distribution or method could I use to model this phenomena?

my probabilistic and statistics skills are just average. However I am working on a project and I was wondering if any of you could help me understanding which process would better model the following.

So basically there is a system that transitions between $n$ states, for example say $4$. The are "linear" (I just made this term up, don't know if it is in the jergon) meaning that from state 1 I cannot go to state 3 say, but I have to go to state 2 first and then state 3. Now the point is that I don't know what makes it change the states.

So I was thinking to assume that the time that it takes to go from one state to the next is somewhat random.

Now, I would like to know, considering that the passage between one state to another is random, what is the "distribution" of the total time it takes to get to the last state (say, state $4$).

For example, suppose $t_{ij}$ is the time to go from $i$ to $j$. Then suppose I have $t_{12}=1s$, $t_{23}=20s$, $t_{34}=0.1s$. Then the total time would be $T_{tot}=1+20+0.1=21.1s$

So I guess this is a random variable, so how can I model this distribution?

I was thinking that I studied the Poisson Process/Distribution which was indeed related to time. Although it counts how many events happen in a certain time interval, which is basically the opposite of what I want to do. Also, I recall that the exponential distribution shows the time between events. So is there a way to construct a model of this phenomena?

So far my best ideas are indeed the above, Poisson, Exponential or Randomwalk. However I don't know random walks that well, so it's very hard to follow what I find on the internet.

I know surely people have done this before, cause I guess to model queues that is what you do. So can you give me a start and key points on how to proceed at least? I'm studying maths in my second year, so my overall maths skills are good, although my probabilistic skills aren't that refined.

• one thing to notice: looking on Wikipedia, we have that for the Exponential (and Poisson?) there is a constant average rate between events. Well that is not my case. Like the events occur at random! – Euler_Salter Feb 11 '17 at 18:55