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Understanding Poisson Processes: Model Arrivals and Departures in Random Events

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jaynesh
22 diapositives · 718 Views · Jan 04, 2020

This session explores the characteristics and applications of Poisson processes, highlighting their role in modeling random events such as customer arrivals at supermarkets, phone calls, and radioactive emissions. We will cover the properties of Poisson processes, including independence of arrival times, uniform distribution, and exponential inter-arrival times. We'll demonstrate how to merge and split Poisson processes and understand their statistical foundations using real-world examples, ensuring a comprehensive understanding of how to analyze and predict such events systematically.

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  1. Poisson Process Review Session 2 EE384X EE384x

  2. Point Processes • Supermarket model : customers arrive (randomly), get served, leave the store • Need to model the arrival and departure processes Server Queue Departure Process Arrival Process EE384x

  3. 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) EE384x

  4. 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 EE384x

  5. Notation 0 EE384x

  6. Notation • A(t) : Number of points in (0,t] • A(s,t) : Number of points in (s,t] • Arrival points : • Inter-arrival times: EE384x

  7. 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 tis distributed Poisson(lt) EE384x

  8. Basic Properties EE384x

  9. Stationary Increments • The number of arrivals in (t,t+t] does not depend on t EE384x

  10. Orderliness • The probability of two or more arrivals in an interval of lengtht gets small as • Arrivals occur “one at a time” EE384x

  11. 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 EE384x

  12. Additional Properties EE384x

  13. Inter-arrival times • Inter-arrival times are Exponential(l) and independent of each other 0 EE384x

  14. merge Merging two Poisson processes • 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 EE384x

  15. Sum of two Poisson rv • Characteristic function: • So EE384x

  16. Split Splitting a Poisson process :Poisson process with rate l • For each point toss a coin (with bias p): • With probability p the point goes to A1(t) • With probability 1-p the point goes to A2(t) • A1(t) and A2(t) are two independent Poisson processes with rates 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 tare independent Poisson(l1t) and Poisson(l2t) EE384x

  18. Dividing a Poisson rv EE384x

  19. Dividing a Poisson rv (cont) • So 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 EE384x

  21. Single arrival case 0 EE384x

  22. General case • It is the n order statistics of uniform distribution. EE384x