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Poisson Process Review Session 2 EE384X Winter 2004 EE384x 1 Point Processes Supermarket model : customers arrive (randomly), get served, leave the store Arrival Process Server Queue Departure Process Need to model the arrival and departure processes Winter 2004 EE384x 2 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) Winter 2004 EE384x 3 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 Winter 2004 EE384x 4 Notation 0 Winter 2004 EE384x 5 Notation A(t) : Number of points in (0,t] A(s,t) : Number of points in (s,t] Arrival points : Inter-arrival times: Winter 2004 EE384x 6 Poisson Process- Definition 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 t is distributed Poisson(lt) Winter 2004 EE384x 7 Basic Properties Winter 2004 EE384x 8 Stationary Increments The number of arrivals in (t,t+t] does not depend on t Winter 2004 EE384x 9 Orderliness The probability of two or more arrivals in an interval of length t gets small as Arrivals occur “one at a time” Winter 2004 EE384x 10 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 Winter 2004 EE384x 11 Additional Properties Winter 2004 EE384x 12 Inter-arrival times Inter-arrival times are Exponential(l) and independent of each other 0 Winter 2004 EE384x 13 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?) Winter 2004 EE384x 14 Merging two Poisson processes merge Merging two independent Poisson processes with rates l1 and l2 creates a Poisson process with rate l1+l2 A(0)=A1(0)+A2(0)=0 Number of arrivals in disjoint intervals are independent Sum of two independent Poisson rv is Poisson Winter 2004 EE384x 15 Sum of two Poisson rv Characteristic function: So Winter 2004 EE384x 16 Splitting a Poisson process :Poisson process with rate l Split For each point toss a coin (with bias p): With probability p With probability 1-p the point goes to A2(t) A1(t) and the point goes to A1(t) A2(t) are two independent Poisson processes with rates Winter 2004 EE384x 17 proof Define A1(t) and A2(t) such that: A1(0)=0 A2(0)=0 Number of points in disjoint intervals are independent for A1(t) and A2(t) They depend on number of points in disjoint intervals of A(t) Need to show that number of points of A1 and A2 in an interval of size t are independent Poisson(l1t) and Poisson(l2t) Winter 2004 EE384x 18 Dividing a Poisson rv Winter 2004 EE384x 19 Dividing a Poisson rv (cont) So Winter 2004 EE384x 20 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. Unordered variables 0 Ordered variables Winter 2004 EE384x 21 Single arrival case 0 Winter 2004 EE384x 22 General case It is the n order statistics of uniform distribution. Winter 2004 EE384x 23