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Unformatted text preview: ECE 5510: Random Processes Lecture Notes Fall 2008 Lecture 16 Today: (1) Poisson Processes HW 7 due today at 5pm. OH today 1-2:40. Exam 2 is Tue. Nov. 11. Todays material is the last new material covered in Exam 2. Tue (Nov. 4) is a review class. AA #4 now due 5pm Wed. Nov. 5th. 1 Poisson Process Today, we will cover the right-hand column of this table: Discrete Time Continuous-Time What is this counting process called? Bernoulli Poisson How long until my first ar- rival/success? Geometric p.m.f. Exponential p.d.f. After a set amount of time, how many arrivals/successes have I had? Binomial p.m.f. Poisson p.m.f. 1.1 Last Time This is the marginal pmf of Y n during a Binomial counting process: P Y n ( k n ) = parenleftbigg n k n parenrightbigg p k n (1- p ) n- k n 1.2 Derivation of Poisson pmf Now, time is continuous. I want to know, how many arrivals have happened in time T . Eg, how many packets have been transmitted in my network. Heres how but I could translate it to a Bernoulli process question: Divide time into intervals duration T/n (eg, T/n = 1 minute). Define the arrival of one packet in an interval as a success in that interval. ECE 5510 Fall 2008 2 Assume that success in each interval is i.i.d. Sum the Bernoulli R.P. to get a Binomial process. Problem: Unless the time interval T/n is really small, you might actually have more than one packet arrive in each interval. Since Binomial can only account for 1 or 0, this doesnt represent your total number of packets exactly. 1.2.1 Let time interval go to zero Long Story : Lets define K ( T ) to be the number of packets which have arrived by time T for the real, continuous time process. Lets define Y n as our Binomial approximation, in which T is divided into n identical time bins. Lets show what happens as we divide T into more and more time bins,...
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- Fall '08