MAS-I: Poisson arrivals and exponential waiting times share a rate
A homogeneous Poisson process links counts in a fixed time interval with exponentially distributed waiting times. Keep rate and time units consistent. Independence and a constant rate are assumptions, not consequences of the word “arrival.”
Worked example or practice scenario
At rate 3 per hour, the probability of no arrival in 20 minutes is e^(−3/3)=e^−1. The expected first waiting time is 1/3 hour. In two hours the expected count is 6, a different random variable.
Try this next
Calculate the probability of at least one arrival and check it complements the no-arrival probability. Then describe why an arrival rate that varies by time of day requires a different specification.
Reading sources
ActNet editorial guide · October 1, 2026 · Original illustrative examples.
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