Stochastic Process → Poisson Process → Definition → Examples → Arrival and Inter-Arrival Times

Probability of the first Arrival
Let N(t) be a Poisson process with rate . Let X₁ be the time of the first arrival. Then probability of the first arrival is given by

Example An office receive telephone calls according to a Poisson process at mean rate of five per hour. The office timing is from 9:00 to 5:00.
(a) What is the probability that the first call receive after 9:30?
(b) The lunch break is from 12:00 to 12:30. What is the probability that after the lunch break, the first call receive after 1:30?
(c) What is the probability that after 12:30, the first message arrive by 2:00?

Solution
(a) Here λ = 5 and t = 0.5 hour. So, using the above formula

(b) Here λ = 5 and t = 1 hour (starting from 12:30). Hence,

(c) Here λ = 5 and t = 1.5 hour (after 12:30). Hence,

Remember: “Arrive by” means less than or equal to (≤).

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