Saturday, July 19, 2014
Poisson, Erlang and CLT
Suppose you call some hotline and you are the 56th person in line, excluding the person currently being served. Callers depart according to a Poisson process with a rate of 2 per minute. What is the probability you will have to wait for more than 30 minutes ?
There are at least 3 ways to approach this.
Poisson
P(take > 30 min):
55∑k=0Pλ(k,τ)=55∑k=0(λτ)ke−λτk!=55∑k=0(2⋅30)ke−2⋅30k!≈0.285491 |
Erlang
1−P(take ≤ 30 min):
1−∫300λktk−1e−λt(k−1)!dt=1−∫300256t55e−2t55!dt≈1−0.714509=0.285491 |
CLT
μ=nλ=562=28σ2=nλ2=14P(T>30)=1−P(30−μσ)≈1−ϕ(0.5345)≈0.2981 |
See Example 6.12 of Introduction to Probability, 2nd Edition.