This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: 1 Wireless Network Lab, NCTU, Taiwan 1 6. The Bernoulli and Poisson Processes Prof. SinHorng Chen TEL: 035712121 ext 31822 Rm: ED211, ED805 2 Wireless Network Lab, NCTU, Taiwan 2 Outline Introduction The Bernoulli Process The Poisson Process 3 Wireless Network Lab, NCTU, Taiwan 3 Introduction A stochastic process is simply a (finite or infinite) sequence of random variables. We are still dealing with a single basic experiment that involves outcomes governed by a probability law. Example: The sequence of Daily prices of a stock; Scores in a football game; Failure times of a machine; Hourly traffic loads at a node of a communication network. 4 Wireless Network Lab, NCTU, Taiwan 4 Emphasis Dependencies For example, how do future prices of a stock depend on past values? Longterm averages For example, what is the fraction of time that a machine is idle? Boundary events For example, what is the frequency with which some buffer in a computer network overflows with data? 5 Wireless Network Lab, NCTU, Taiwan 5 Two Major Categories of Stochastic Processes Arrivaltype processes Arrival 指指指指指指指指指指指指指指指指指指指 message 指指指指指 task 指指指指指指指 Focus on models in which interarrival times are i.i.d. Bernoulli process (Sec. 6.1) ( discrete time ) Poisson process (Sec. 6.2) ( continuous time ) Markov processes (Ch. 7) The former is memoryless, while the latter has memory. 6 Wireless Network Lab, NCTU, Taiwan 6 6.1 Bernoulli Process The Bernoulli process is a sequence X 1 , X 2 , … of independent Bernoulli random variables X i with P ( X i =1) = P (success at the i th trial) = p , P ( X i =0) = P (failure at the i th trial) = 1 p , for each i . 7 Wireless Network Lab, NCTU, Taiwan 7 Random Variables Associated with the Bernoulli Process S : number of successes in n independent trials ~ binomial( p , n ). T : number of trials up to (and including) the first success ~ geometric( p ). 8 Wireless Network Lab, NCTU, Taiwan 8 Properties of the Bernoulli Process FreshStart For any given time n , the sequence of RV X n +1 , X n +2 , … ( the future of the process ) is also a Bernoulli process, and is independent from X 1 , …, X n ( the past of the process ). Memorylessness n : given time : the time of the first success after time n ~ geometric( p ), and is independent of the RV X 1 , …, X n , i.e., 9 Wireless Network Lab, NCTU, Taiwan 9 Example 6.2 Computer operation: Computer execute two types of tasks: priority and nonpriority in discrete time slots...
View
Full
Document
This note was uploaded on 12/08/2011 for the course ECON 12323 taught by Professor Pubo during the Spring '11 term at Gods Bible.
 Spring '11
 pubo

Click to edit the document details