The Poisson process
Random arrivals at a constant rate: number of arrivals over a period, time between two arrivals, memorylessness, merging and splitting streams, and the cases where the model fails.
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Poisson process: what is it?
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A model of random arrivals at constant rate λ, where arrivals are independent across disjoint intervals and never simultaneous.
Three assumptions: constant rate, independence across intervals, no batch arrivals. If one fails (peak hour, group arrivals), the model no longer holds.
A customer arrives every 2 minutes on average in a Poisson process. What is the arrival rate per minute?
- 0.5
- 2
- 0.2
- 1.5
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0.5
The average time between arrivals is 1/λ = 2 minutes, so λ = 0.5 per minute. Taking 2 confuses the mean time with the rate.
If two independent Poisson streams have rates λ₁ and λ₂, their … is a Poisson stream with rate ….
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If two independent Poisson streams have rates λ₁ and λ₂, their merger is a Poisson stream with rate λ₁ + λ₂.
The arrivals of both streams add up, and the independence properties are preserved.
Arrival rate: what is it?
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The average number of arrivals per unit of time, written λ.
It only gives the average: 2 requests per second does not mean exactly 2 each second, but 2 on average, with some seconds at 0 and others at 5.