This is a recent article that describes a mathematical model closely related to something I'm trying to work out for Battletech. The probability distribution that describes how long a Battlemech will survive during play is very complex, which is no surprise, but then I discovered I was still underestimating the problem. The article above describes a type of situation similar to what occurs in Battletech, and demonstrates a nice matrix based approach to formulating the problem.
The "phase" in Phase-Type distribution refers to the particular state of the system. In Battletech terms, the starting phase would be an undamaged mech, the end phase (absorbing state) would be the head, center torso, or engine destruction, and the phases in between would represent various states of destruction in between. The diagram below (borrowed from the PhD Thesis: Aggregate Matrix-analytic Techniques and their Applications of Alma Riska, PhD), shows an example. Here the undamaged starting "phase 0" would be state "0,0" on the left, phase 1 would be any single (non-fatal) section destroyed) such as either arm destroyed (not both), which might correspond to the "0,1" and "0,2" states. Phase 2 would be any legal combination of two destroyed locations, 3 for phase 3, and so on.
The arrows connecting the states represent the probability of moving from one state to another The whole thing can be written as a matrix giving the probability of moving from one state to another at a given time. Battletech has at least 150 states, and a complex interconnections. It's complicated, but now I know it can be done, and I have a new line of study to help me figure out how to do it.
That problem is on the back burner for a while though. I am presently working on something simpler that might have more immediate and practical application. Work and home have been very busy, so progress is slow. Soon, I hope.