1. Winter 2004EE384x1 Poisson Process Review Session 2 EE384X

2. 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

3. 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)

4. 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

5. Winter 2004EE384x5 Notation 0

6. 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:

7. 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

8. Winter 2004EE384x8 Basic Properties

9. Winter 2004EE384x9 Stationary Increments  The number of arrivals in (t,t+  ] does not depend on t

10. Winter 2004EE384x10 Orderliness  The probability of two or more arrivals in an interval of length  gets small as  Arrivals occur “one at a time”

11. 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

12. Winter 2004EE384x12 Additional Properties

13. Winter 2004EE384x13 Inter-arrival times  Inter-arrival times are Exponential ( ) and independent of each other 0

14. 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?)

15. 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

16. Winter 2004EE384x16 Sum of two Poisson rv  Characteristic function:  So

17. 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

18. 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  )

19. Winter 2004EE384x19 Dividing a Poisson rv

20. Winter 2004EE384x20 Dividing a Poisson rv (cont)  So

21. 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

22. Winter 2004EE384x22 Single arrival case 0

23. Winter 2004EE384x23 General case  It is the n order statistics of uniform distribution.