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Winter 2004EE384x1 Poisson Process Review Session 2 EE384X
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Winter 2004EE384x2 Point Processes Supermarket model : customers arrive (randomly), get served, leave the store Need to model the arrival and departure processes Server Queue Arrival Process Departure Process
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Winter 2004EE384x3 What does Poisson Process model? Start time of Phone calls in Palo Alto Session initiation times (ftp/web servers) Number of radioactive emissions (or photons) Fusing of light bulbs, number of accidents Historically, used to model packets (massages) arriving at a network switch (In Kleinrock’s PhD thesis, MIT 1964)
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Winter 2004EE384x4 Properties Say there has been 100 calls in an hour in Palo Alto We expect that : The start time of each call is independent of the others The start time of each call is uniformly distributed over the one hour The probability of getting two calls at exactly the same time is zero Poisson Process has the above properties
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Winter 2004EE384x5 Notation 0
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Winter 2004EE384x6 Notation A(t) : Number of points in (0,t] A(s,t) : Number of points in (s,t] Arrival points : Inter-arrival times:
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Winter 2004EE384x7 A(0)=0 and each jump is of unit magnitude Number of arrivals in disjoint intervals are independent For any the random variables are independent. Number of arrivals in any interval of length is distributed Poisson ( ) Poisson Process- Definition
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Winter 2004EE384x8 Basic Properties
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Winter 2004EE384x9 Stationary Increments The number of arrivals in (t,t+ ] does not depend on t
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Winter 2004EE384x10 Orderliness The probability of two or more arrivals in an interval of length gets small as Arrivals occur “one at a time”
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Winter 2004EE384x11 Poisson Rate Probability of one arrival in a short interval is (approx) proportional to interval length Poisson process is like a continuous version of Bernoulli IID
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Winter 2004EE384x12 Additional Properties
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Winter 2004EE384x13 Inter-arrival times Inter-arrival times are Exponential ( ) and independent of each other 0
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Winter 2004EE384x14 Points to the left and right is a fixed point closest point to the right (left) of Apparent Paradox: Inter-arrival = sum two exp (why?)
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Winter 2004EE384x15 Merging two Poisson processes Merging two independent Poisson processes with rates 1 and 2 creates a Poisson process with rate 1 + 2 A(0)=A 1 (0)+A 2 (0)=0 Number of arrivals in disjoint intervals are independent Sum of two independent Poisson rv is Poisson merge
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Winter 2004EE384x16 Sum of two Poisson rv Characteristic function: So
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Winter 2004EE384x17 Splitting a Poisson process For each point toss a coin (with bias p ): With probability p the point goes to A 1 (t) With probability 1-p the point goes to A 2 (t) A 1 (t) and A 2 (t) are two independent Poisson processes with rates Split :Poisson process with rate
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Winter 2004EE384x18 proof Define A 1 (t) and A 2 (t) such that: A 1 (0)=0 A 2 (0)=0 Number of points in disjoint intervals are independent for A 1 (t) and A 2 (t) They depend on number of points in disjoint intervals of A(t) Need to show that number of points of A 1 and A 2 in an interval of size are independent Poisson ( 1 ) and Poisson ( 2 )
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Winter 2004EE384x19 Dividing a Poisson rv
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Winter 2004EE384x20 Dividing a Poisson rv (cont) So
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Winter 2004EE384x21 Uniformity of arrival times Given that there are n points in [0,t], the unordered arrival times are uniformly distributed and independent of each other. 0 Ordered variables Unordered variables
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Winter 2004EE384x22 Single arrival case 0
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Winter 2004EE384x23 General case It is the n order statistics of uniform distribution.
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