> a Poisson process is a memoryless process that assumes the probability of an arrival is entirely independent of the time since the previous arrival. In reality, a well-run bus system will have schedules deliberately structured to avoid this kind of behavior: buses don't begin their routes at random times throughout the day, but rather begin their routes on a schedule chosen to best serve the transit-riding public.
I've never really understood any example involving a poisson process. They always seem to involve bus arrivals or light bulbs burning out, and I can't understand why the memory less property would ever make any sense for these.
Even if the bus system was poorly run, why would it make sense to assume that the expected value of time to arrival doesn't change based on how long you've been waiting?
What is an actual phenomenon that is well modeled by a poisson process?
Phone calls. Say, an office worker gets 8*6 phone calls in an average shift of 8 hours, so one every 10 minutes on average. It doesn't matter how long ago the last call was, since the customers don't coordinate.
This real world example still doesn't perfectly match the theory. For example, if there was no call for a long time, it may indicate that it's some special day or the phone line is malfunctioning or whatever and it could mean that the next call is probably further in the future than the model would say.
> What is an actual phenomenon that is well modeled by a poisson process?
Radioactive decay. Collisions of fluid molecules. Unstimulated (i.e. not in a laser) photon emission due to electron transitions in an atom. Lots of pretty memoryless stuff going on at the microscopic level.
> Even if the bus system was poorly run, why would it make sense to assume that the expected value of time to arrival doesn't change based on how long you've been waiting?
I don't think it's saying anything about how long you've been waiting, and you don't know when was the last arrival.
It's saying that if you pick a random point on the timeline, the expected wait time doesn't change. That's because by taking a random point you have more chances of landing in a larger stretch of wait time than in a smaller one.
> I don't think it's saying anything about how long you've been waiting, and you don't know when was the last arrival.
This is exactly what memorylessness says something about.
Your second paragraph isn't unique to Poisson processes, but the author right at the start says that the expected value of the waiting time is the same as the average interarrival time, which indicates Poisson.
>What is an actual phenomenon that is well modeled by a poisson process?
I used it in a cell tissue simulation where the user could define how frequently the cells divide. If you start with 100 cells and want them each to divide, on average, every N iterations, using a Poisson formula to decide if a cell splits or not based on a random number is ideal, very precise (in the aggregate), and avoids a lot of odd artifacts.
I think your question is deeper than it first appears, at least it was for me cause I forgot to distinguish gaussian from poisson. One phenomenon is retinal response (per card 8 at this nicely designed site https://quizlet.com/124228940/optometry-stevenson-lectures-f...)
I've never really understood any example involving a poisson process. They always seem to involve bus arrivals or light bulbs burning out, and I can't understand why the memory less property would ever make any sense for these.
Even if the bus system was poorly run, why would it make sense to assume that the expected value of time to arrival doesn't change based on how long you've been waiting?
What is an actual phenomenon that is well modeled by a poisson process?