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Renewal_Theory_I_beamer

# Renewal_Theory_I_beamer - Introductory Engineering...

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Introductory Engineering Stochastic Processes, ORIE 361 Instructor: Mark E. Lewis, Associate Professor School of Operations Research and Information Engineering Cornell University Renewal Theory 1/ 30

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Renewal Theory x x x x I T 1 T 2 T 3 T 4 A renewal process can be obtained from a sequence of i.i.d random variables...say { T i , i 1 } . Set S 0 = 0 and S 1 = T 1 . Continuing in this fashion S n = n i =1 T i If N ( t ) represents the number of renewals in (0 , t ] S N ( t ) t < S N ( t )+1 2/ 30
Renewal Theory x x x x I T 1 T 2 T 3 T 4 S 1 A renewal process can be obtained from a sequence of i.i.d random variables...say { T i , i 1 } . Set S 0 = 0 and S 1 = T 1 . Continuing in this fashion S n = n i =1 T i If N ( t ) represents the number of renewals in (0 , t ] S N ( t ) t < S N ( t )+1 2/ 30

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Renewal Theory x x x x I T 1 T 2 T 3 T 4 S 2 A renewal process can be obtained from a sequence of i.i.d random variables...say { T i , i 1 } . Set S 0 = 0 and S 1 = T 1 . Continuing in this fashion S n = n i =1 T i If N ( t ) represents the number of renewals in (0 , t ] S N ( t ) t < S N ( t )+1 2/ 30
Renewal Theory x x x x I T 1 T 2 T 3 T 4 S 3 A renewal process can be obtained from a sequence of i.i.d random variables...say { T i , i 1 } . Set S 0 = 0 and S 1 = T 1 . Continuing in this fashion S n = n i =1 T i If N ( t ) represents the number of renewals in (0 , t ] S N ( t ) t < S N ( t )+1 2/ 30

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Renewal Theory x x x x I T 1 T 2 T 3 T 4 S 4 A renewal process can be obtained from a sequence of i.i.d random variables...say { T i , i 1 } . Set S 0 = 0 and S 1 = T 1 . Continuing in this fashion S n = n i =1 T i If N ( t ) represents the number of renewals in (0 , t ] S N ( t ) t < S N ( t )+1 2/ 30
Renewal Theory x x x x I T 1 T 2 T 3 T 4 t N(t) = 3 S N(t) S N(t)+1 A renewal process can be obtained from a sequence of i.i.d random variables...say { T i , i 1 } . Set S 0 = 0 and S 1 = T 1 . Continuing in this fashion S n = n i =1 T i If N ( t ) represents the number of renewals in (0 , t ] S N ( t ) t < S N ( t )+1 2/ 30

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Renewal Processes Definition Let { T i , i = 1 , 2 , . . . } be a sequence of non-negative i.i.d. r.v.s with cdf F and F (0) < 1 . Let S n = n i =1 T i and S 0 = 0 . Let N ( t ) = max { n : S n t } , then { N ( t ) , t 0 } is a renewal process . { N ( t ) , t 0 } is a counting process 3/ 30
Renewal Processes Definition Let { T i , i = 1 , 2 , . . . } be a sequence of non-negative i.i.d. r.v.s with cdf F and F (0) < 1 . Let S n = n i =1 T i and S 0 = 0 . Let N ( t ) = max { n : S n t } , then { N ( t ) , t 0 } is a renewal process . { N ( t ) , t 0 } is a counting process Some people call { S n , n 0 } the renewal process and { N ( t ) , t 0 } the renewal counting process (it is not standard) 3/ 30

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A Few Facts Proposition Let { N ( t ) , t 0 } be a renewal process with inter-renewal distribution G. Let G n * be the n-fold convolution of G with itself. Then P ( N ( t ) n ) = P ( S n t ) = G ( n ) * ( t ) P ( N ( t ) n ) = P ( S n +1 > t ) = 1 - G ( n +1) * ( t ) This adds to the observation that specifying the renewal process is done equally well by specifying { N ( t ) , t 0 } or { S n , n 0 } 4/ 30
The Renewal Function Define m ( t ) = E N ( t ). Then m ( t ) is called the renewal function .

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