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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 message task 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...
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 Spring '11
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