Chapter 3. Renewal Processes - 國立中興大學 ch.3 renewal...See next slide. # Laplace...

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Chapter 3. Renewal Processes

Transcript of Chapter 3. Renewal Processes - 國立中興大學 ch.3 renewal...See next slide. # Laplace...

Page 1: Chapter 3. Renewal Processes - 國立中興大學 ch.3 renewal...See next slide. # Laplace Transform of f *g 25 Laplace Transform of f *g (con’t) 26 27 4. Renewal Function

Chapter 3. Renewal Processes

Page 2: Chapter 3. Renewal Processes - 國立中興大學 ch.3 renewal...See next slide. # Laplace Transform of f *g 25 Laplace Transform of f *g (con’t) 26 27 4. Renewal Function

Outline

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Outline

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Page 4: Chapter 3. Renewal Processes - 國立中興大學 ch.3 renewal...See next slide. # Laplace Transform of f *g 25 Laplace Transform of f *g (con’t) 26 27 4. Renewal Function

Distribution and Limiting Behavior of

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… X1

X2

N(t)

S1 S2 SN(t) SN(t)+1 t

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Distribution and Limiting Behavior of

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Distribution and Limiting Behavior of

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1. pmf of N(t)

(fundamental relationship)

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Example

Density of N(t) for Poisson Process Let N be a Poisson process with rate λ. Recall that the sum of i.i.d. Exponential random variables is an Erlang random variable, such that

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…X1

X2

N(t)

S1 S2 SN(t) SN(t)+1t

Exp

What is the pmf of N(t) ?

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2. Limiting Time Average

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Strong Law of Large Numbers

If X1, X2, … is an i.i.d. random sequence with sample mean Mn(X),

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w.p. = with probability

1 2lim 1 n

n

X X X

n

P

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2. Limiting Time Average (con’t)

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Strong Law for Renewal Processes

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Average of the first N(t) inter-arrival times

rate as t -> ∞

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Central Limit Theorem

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3. Central Limit Theorem for

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=

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4. Renewal Function

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4. Renewal Function (con’t)

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4. Renewal Function (con’t)

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4. Renewal Function (con’t)

Proof method in p.100 of the textbook

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4. Renewal Function (con’t)

Big Problem!!!

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4. Renewal Function (con’t)

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4. Renewal Function (con’t)

Why? See next slide.

Page 25: Chapter 3. Renewal Processes - 國立中興大學 ch.3 renewal...See next slide. # Laplace Transform of f *g 25 Laplace Transform of f *g (con’t) 26 27 4. Renewal Function

Laplace Transform of f *g

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Laplace Transform of f *g (con’t)

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4. Renewal Function (con’t)

Elementary Renewal Theorem

Before proving elementary renewal theorem, we first introduce the stopping time (or stopping rule) and Wald’s equation.

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Stopping Time (Rule)

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Yes

Let Xn, n=1, 2,… be independent and let Then, is N is a stopping time? No

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Stopping Time (Rule)

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For a renewal process {N(t), t≥0}, N(t) is not a stopping time for interarrival sequence Xi’s. Why? [Homework]

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Stopping Time (Rule)

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For a renewal process, N(t) + 1 is a stopping time for the interarrival sequence X1, X2, . . . It can be seen that the following events are equivalent: #

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• Examples. To illustrate some examples of random times that are stopping rules and some that are not, consider a gambler playing roulette with a typical house edge, starting with $100: – Playing one, and only one, game corresponds to the stopping time τ = 1, and is

a stopping rule.

– Playing until he either runs out of money or has played 500 games is a stopping rule.

– Playing until he is the maximum amount ahead he will ever be is not a stopping rule and does not provide a stopping time, as it requires information about the future as well as the present and past.

– Playing until he doubles his money (borrowing if necessary if he goes into debt) is not a stopping rule, as there is a positive probability that he will never double his money. (Here it is assumed that there are limits that prevent the employment of a martingale system or variant thereof (such as each bet being triple the size of the last). Such limits could include betting limits but not limits to borrowing.)

– Playing until he either doubles his money or runs out of money is a stopping rule, even though there is potentially no limit to the number of games he plays, since the probability that he stops in a finite time is 1

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Stopping Time (Rule)

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Wald's Equation

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E[N ] is finite

E[X] is finite

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Example (Simple Random Walk)

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Solution

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4. Renewal Function (con’t)

Elementary Renewal Theorem

A simple consequence ?

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Example

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4. Renewal Function (con’t)

Elementary Renewal Theorem

A simple consequence? Not always!!!

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Corollary [Ross Corollary 3.3.3 ]

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Page 40: Chapter 3. Renewal Processes - 國立中興大學 ch.3 renewal...See next slide. # Laplace Transform of f *g 25 Laplace Transform of f *g (con’t) 26 27 4. Renewal Function

Elementary Renewal Theorem

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inf a: greatest lower bound of a sup a: least upper bound of a

(1)

(Corollary 3.3.3)

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(2)

Corollary

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Elementary Renewal Theorem

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=

(Corollary 3.3.3)

Page 43: Chapter 3. Renewal Processes - 國立中興大學 ch.3 renewal...See next slide. # Laplace Transform of f *g 25 Laplace Transform of f *g (con’t) 26 27 4. Renewal Function

Example

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Page 44: Chapter 3. Renewal Processes - 國立中興大學 ch.3 renewal...See next slide. # Laplace Transform of f *g 25 Laplace Transform of f *g (con’t) 26 27 4. Renewal Function

Example (M/G/1 Loss)

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Page 45: Chapter 3. Renewal Processes - 國立中興大學 ch.3 renewal...See next slide. # Laplace Transform of f *g 25 Laplace Transform of f *g (con’t) 26 27 4. Renewal Function

Solution

A renewal occurs every time that a customer actually enters the booth.

If N(t) denotes the number of customers who enter the booth by t, then {N(t) , t≥0} will be a renewal process having mean interarrival time of

Since the time to the next customer follows exponential with rate λ from the memoryless property. So,

(1) Since the rate at which customers enter the booth is just the long-run renewal rate, it is given by

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Page 46: Chapter 3. Renewal Processes - 國立中興大學 ch.3 renewal...See next slide. # Laplace Transform of f *g 25 Laplace Transform of f *g (con’t) 26 27 4. Renewal Function

Solution (con’t)

(2) The loss rate of customers is obtained by

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(3) The utilization of the booth is

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Lattice

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More examples.

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True!!

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Background of Blackwell's Theorem

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Page 50: Chapter 3. Renewal Processes - 國立中興大學 ch.3 renewal...See next slide. # Laplace Transform of f *g 25 Laplace Transform of f *g (con’t) 26 27 4. Renewal Function

Blackwell's Theorem

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(Proof in Feller)

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Example

Solution

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Page 52: Chapter 3. Renewal Processes - 國立中興大學 ch.3 renewal...See next slide. # Laplace Transform of f *g 25 Laplace Transform of f *g (con’t) 26 27 4. Renewal Function

Directly Riemann Integrable

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Page 53: Chapter 3. Renewal Processes - 國立中興大學 ch.3 renewal...See next slide. # Laplace Transform of f *g 25 Laplace Transform of f *g (con’t) 26 27 4. Renewal Function

Directly Riemann Integrable (con’t)

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Key Renewal Theorem

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Blackwell theorem

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Regenerative Processes

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0

0

x1 x2 x3 x4

t

t

i.i.d.

S1 = X1

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Renewal Theory

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i.i.d. cycles

total law of probability

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Renewal Theory (con’t)

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Page 58: Chapter 3. Renewal Processes - 國立中興大學 ch.3 renewal...See next slide. # Laplace Transform of f *g 25 Laplace Transform of f *g (con’t) 26 27 4. Renewal Function

Renewal Theory (con’t)

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Laplace Transform on both sides

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Inverse Laplace transform on both sides

Renewal Theory (con’t)

solution

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Elementary Renewal Theorem

Page 61: Chapter 3. Renewal Processes - 國立中興大學 ch.3 renewal...See next slide. # Laplace Transform of f *g 25 Laplace Transform of f *g (con’t) 26 27 4. Renewal Function

Example

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Page 62: Chapter 3. Renewal Processes - 國立中興大學 ch.3 renewal...See next slide. # Laplace Transform of f *g 25 Laplace Transform of f *g (con’t) 26 27 4. Renewal Function

Proof

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Page 64: Chapter 3. Renewal Processes - 國立中興大學 ch.3 renewal...See next slide. # Laplace Transform of f *g 25 Laplace Transform of f *g (con’t) 26 27 4. Renewal Function

Corollary

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Regenerative Processes

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