Phase-Type Distributions

The exponential distribution is the natural model for "how long until an event happens", but it forces a constant hazard rate, meaning the process has no memory of how long it's already waited, which is often unrealistic. A phase-type distribution generalizes this by imagining a hidden Markov chain with several transient states plus one absorbing state: the process wanders between the hidden states, spending an exponentially distributed time in each, until it eventually falls into the absorbing state. The total time to absorption is the phase-type random variable.

Because you can choose however many hidden states you like and however they're connected, phase-type distributions can approximate almost any real waiting-time distribution, including ones with the "memory" (rising or falling hazard over time) that a single exponential can't capture, which is why they show up in queueing models, credit-default-time modeling, and risk-of-ruin calculations.

Worked example. Model a loan's time to default as two hidden states, "current" and "early delinquency", each with its own exponential exit rate, before hitting the absorbing "default" state. A loan might spend an average of 24 months in "current" before slipping to "early delinquency," then only 3 more months before default; the resulting total-time-to-default distribution has a rising-then-falling hazard shape that a single exponential distribution could never produce.