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6. The Bernoulli and Poisson Processes

6. The Bernoulli and Poisson Processes - 6....

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1 Wireless Network Lab, NCTU, Taiwan 1 6.  The Bernoulli and Poisson Processes Prof. Sin-Horng Chen TEL: 03-5712121 ext 31822 Rm: ED211, ED805

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2 Wireless Network Lab, NCTU, Taiwan 2 Outline Introduction The Bernoulli Process The Poisson Process
3 Wireless Network Lab, NCTU, Taiwan 3 Introduction 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.

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4 Wireless Network Lab, NCTU, Taiwan 4 Emphasis Dependencies For example, how do future prices of a stock  depend on past values? Long-term 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 Arrival-type 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.

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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 ).

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8 Wireless Network Lab, NCTU, Taiwan 8 Properties of the Bernoulli Process Fresh-Start 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 Priority task arrives with probability  and require one

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