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The Poisson Process

Poisson process, rate parameter e.g. packets/second

• Three equivalent viewpoints of the Poisson process are illustrated in the next 3 slides....

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First Viewpoint

• Behaviour in small time interval– Bernoulli distribution– 1 event with probability t– 0 events with probability 1-t

time t

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Second Viewpoint

• Behaviour over a long time interval– Poisson distribution

time t

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Third Viewpoint

• Behaviour between events– Negative exponential distribution

time t

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Pr{an arrival in time (t, t t)}t O(t)

3 Equivalent Viewpoints1)

and arrivals are memoryless, i.e. independent of what has happened before.

!

} in time arrivals Pr{k

tetk

kt

2) i.e. a Poisson distribution, with parameter t.

3)The probability density function of the times between events (the interarrival times) is negative exponential, with parameter , i.e. t

T etp

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Sums of Poisson Processes

• Consider superposition of several Poisson processes– m independent Poisson Processes, rates

i, i=1,2,...,m

• Sum is also a Poisson process, rate i

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Sums of Poisson Processestime t

Stream 1

Stream 2

Stream 3

Stream m