Poisson Arrivals, explained.
A homogeneous Poisson process models event counts with stationary independent increments and a constant arrival rate.
Why it happens
Independent exponential gaps generate arrivals within ten model minutes. Revealing time exposes the realised count beside rate times elapsed duration and reciprocal-rate expected waiting time.
A homogeneous Poisson process models event counts with stationary independent increments and a constant arrival rate.
Read the result
Event rate is an expectation, not a timetable. The count revealed so far and expected count over that duration can differ substantially.
A worked example
Reveal a constant-rate arrival stream
Choose two expected arrivals per minute before revealing the first minute.
Reveal successive minutes and inspect uneven gaps in the same seeded event stream.
Compare the realised ten-minute total with expectation twenty without demanding exact equality.
OPTIONAL DEEPER DETAILGo deeper: inside the model
Inside this model
Independent exponential gaps generate arrivals within ten model minutes. Revealing time exposes the realised count beside rate times elapsed duration and reciprocal-rate expected waiting time.
Where this idea is useful
A practical use
Distinguish a constant average arrival rate from regularly spaced arrivals.
A common misconception
“A constant arrival rate means equal spacing between events.”
The homogeneous Poisson model has random exponential gaps; its rate describes average events per unit time, not periodic scheduling.
What this explanation leaves out
- A constant rate and independent increments are essential. Rush hours, batches, self-excitation and scheduled arrivals are not represented.
Would this model represent a scheduled bus exactly?
No. Schedules, time-varying service and dependence between arrivals can violate the stationary independent-increments assumptions.
Is your arrival assumption about a mean rate or an actual schedule?
Associated thinkers
Associations marked provisional are awaiting source review.
Further reading
Explore the original research or the teaching reference behind this experiment.