Understanding the meaning of "Clock Forward" and "Clock Reset" in Probability?

Question

In Markov Processes, the terms "Clock Forward" and "Clock Reset" are often used to describe the behavior of Markov Chains. In short, "Clock Forward" is used for a Full Markov Process (i.e. Memoryless Property is satisfied) and "Clock Reset" is used for a Semi Markov Process (i.e. Memoryless Property is satisfied).

I am trying to learn more about why these terms (i.e. clock forward, clock reset) are used to describe Markov Processes.

In this research paper (https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0123489), the following note is made:

• With the "clock-forward" approach, the time refers to the time since study start. The clock continues running, independent of the occurrence of intermediate events.

• In contrast, the "clock-reset" approach assumes a reset to zero, every time the subject moves to another state. The current time t then refers to the sojourn time in the present state

• In clock-reset models the time scale itself depends on the time when the present state was reached, therefore the Markov assumption is violated by definition.

While I think I understand this logic, it is a bit confusing to me. For instance, I would have naturally thought the following:

• A Full Markov Process has the memoryless property : Suppose we have currently spent some time $a$ in some state = $i$, then the probability of a transition from this state in time $b$ does not depend on time we have already waited $a$. That is , $P( t > a+b | a) = P(t>b)$. Therefore, the probability has naturally "reset" itself of time having already waited. (Note that the memoryless property only takes places in Exponential Distributions - see the reference below)

• A Semi Markov Process does not have the memoryless property. Therefore, the time keeps ticking forward and does NOT reset itself. As a result, the probability of a transition happening now somewhat depends on the time already waited.

Thus, in my mind : A Semi Markov Process should be called "Clock Forward" whereas a Full Markov Process should be called "Clock Reset".

I am still confused as to how to understand these terms (Clock Forward and Clock Reset) - could someone please help me understand this?

Thanks!

References:

• Proving the Memoryless Property ONLY Applies to Exponential Distributions

Answer 1

Oof, these terms are so stupid and useless, you should try to minimize the amount of time you spend thinking about them.

But, try to think of it this way. A semi-Markov process can have inter-state transition times with arbitrary distribution, so we might have a semi-Markov process whose transition times are uniformly distributed on the interval [3,8]. If we are observing such a process and see that it's in state $10$, and we know it's going to transition next to either state $11$ or state $12$ after its transition timer expires, we might take a look at the wall clock to see how long it's been sitting in state 10. After all, if it's only been sitting in state 10 for 1 time unit, we can probably grab a coffee, since the earliest possible transition time is 2 units from now, but if it's been in state 10 for 7.9 time units, we need to keep our eyes open, because it's guaranteed to transition in the next 0.1 units.

But this only works if we can tell from the wall clock how long the process has been in state 10, like if we wrote down the wall clock time that it first entered state 10 or -- better yet -- if we reset the wall clock to zero when it entered state 10, and keep resetting it every time we change state.

With a clock-reset approach, the two pieces of information we need to completely characterize the future behavior of the semi-Markov process are its current state, and the elapsed time since the last clock reset.

A full Markov process is a little different. Because inter-state transition times are exponentially distributed and so have a memorylessness property, the wall clock isn't important. If we are in state 10 and know we're going to transition after an exponential time with mean 6 time units, it doesn't matter if we've already been in this state for 0.1 units, 100 units, or 5 billion units, the memorylessness of the exponential distribution means that the distribution of the time to the next transition is always exponential with mean 6 as long as we remain in state 10, and we can ignore the wall clock when trying to guess what the process will do next.

We could reset the wall clock, or we could just leave it running, it doesn't matter. So, a "clock-forward" approach is possible with a full Markov process, because the clock does not provide important information about the process. Only its current state is important.

What actually happened here is that the theory and study of Markov processes have used a "clock-forward" approach because that's just the normal, default approach when you're studying stochastic processes, Markov or otherwise. You have a single "time" value that starts at time 0 and keeps running, and you talk about the state of the process at those times. In the study of semi-Markov processes, some idiot came along and decided they could make themselves sound smart by making a big deal out of differentiating between a "clock-forward" approach (i.e., the usual concept of "time" when studying stochastic processes) and a "clock-reset" approach (i.e., a completely useless and overcomplicated way of describing non-exponential inter-transition times), and enough other people decided this sounded smart that the terminology caught on.

Try to ignore it as best you can. If you're using software or reading a paper that's making a big deal out of a "clock-forward" or "clock-reset" approach, all it means is that you need to be careful if the "time" value you're talking about is measured from the very start of the process or measured from the time of the last transition of the process.