6.262.Lec11

6.262.Lec11 - DISCRETE STOCHASTIC PROCESSES Lecture 11...

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Lecture 11 - 3/10/2010 Discrete Stochastic Processes 1 DISCRETE STOCHASTIC PROCESSES Lecture 11 Renewal Processes - Chapter 3 Review: Strong Law for Renewal Processes Time Averages vs. Ensemble Averages General Renewal-Reward Processes Time-Averaged Renewal-Reward Theorem Stopping Times Wald's Theorem The Stopping Time (N(t) +1) A Lower Bound on E[N(t)] The Renewal Equation for N(t) An Approach to the Renewal Equation Using Laplace Transforms The Elementary Renewal Theorem Blackwell's Theorem
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Lecture 11 - 3/10/2010 Discrete Stochastic Processes 2 S TRONG L AW FOR R ENEWAL P ROCESSES Theorem 1 : For a renewal process with mean interarrival time X (possibly not finite) , lim t Nt t X →∞ = af 1 w. p. 1. ( This result describes a long-term time average, not an expectation.
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Lecture 11 - 3/10/2010 Discrete Stochastic Processes 3 ( ) ( ) lim / t Nt t →∞ Ensemble Averages vs. Time Average s We now understand (?) the time average , which is 1/ X WP1. What's Next? We will also want to understand the ensemble average, E N t () / t [ ] . Is this 1/ X for all t ? (Answer is no except for Poisson) Is this 1/ X in limit t → ∞ ? (Answer is yes, but it doesn't follow from time average. E Nt t t lim →∞ F H G I K J a f doesn't mean the same as lim t E t F H G I K J a f . ) t
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Lecture 11 - 3/10/2010 Discrete Stochastic Processes 4 ( ) ( ) ( ) ( ) , R tZ t X t =ℜ ( ) Nt General Renewal Reward Processes Given a renewal process, N(t), t 0, let R(t), t 0 be a collection of random variables; R ( t ) is called the reward rate at time t . R ( t ) is restricted to be a function only of the duration of the current renewal interval and the location of t within that interval, i.e., that is . R ( t ) is independent of the arrival epochs before S N(t) and after S N(t) + 1 . If is embedded in some more complex stochastic process, R ( t ) can depend on other random variables, so long as the values of R ( t ) in one inter-renewal interval are independent of the values in all other inter-renewal intervals.
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Lecture 11 - 3/10/2010 Discrete Stochastic Processes 5 () {} 0 1 lim , t n t ER Rd tX ττ →∞ = Time Average Renewal Reward Theorem Let ( ) R t be a renewal reward function for any renewal process with interrenewal intervals i X . Define R n as R n = R τ d = S n 1 S n . Then with probability 1, provided at least one of the numerator and denominator is finite. .
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Lecture 11 - 3/10/2010 Discrete Stochastic Processes 6 Stopping Times Stopping time J is random, and can depend on observations up to and including X N . If experiment has proceeded to X n , the decision to stop immediately following n (i.e., to choose J = n) should be independent of X n + 1 , X n + 2 ,... . Therefore, for any fixed t, J =N(t) is not a valid stopping time, since N(t) = n means that X 1 + X 2 + - - - + X n t and X 1 + X 2 + - - - + X n + X n+1 > t, which depends on X n+1.
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6.262.Lec11 - DISCRETE STOCHASTIC PROCESSES Lecture 11...

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