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Chapter 3. Renewal Processes

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Chapter 3. Renewal Processes

Prof. Ai-Chun Pang

Graduate Institute of Networking and Multimedia,

Department of Computer Science and Information Engineering,

National Taiwan University, Taiwan

(2)

Renewal Processes

} 0 ),

~(

{

= n t t N

} 0 ),

~ (

{

= n t t N

} 0 ),

~(

{

= n t t N

)

~ t( . n

exp

! ) ) (

)

~ (

( k

t k e

t n P

k

t λ

λ

=

=

0

t

? )

~ (

lim =

n t

t

) ?

~ (

lim =

t

t n

t

0 t

1

S~ ~2

S G

0 t

? )

)

~ (

( n t = k =

P

)

~ t( n G

(3)

Outline

• Distribution and Limiting Behavior of ˜n(t) – Pmf of ˜n(t) : P (˜n(t) = k) =?

– Limiting time average : lim

t→∞

n(t)˜

t =? (Law of Large Numbers) – Limiting PDF of ˜n(t) (Central Limit Theorem)

• Renewal Function E[˜n(t)], and its Asymptotic (Limiting) behavior – Renewal Equation

– Wald’s Theorem and Stopping time – Elementary Renewal Theorem

– Blackwell’s Theorem

(4)

Outline

• Key Renewal Theorem and Applications – Definition of Regenerative Process – Renewal Theory

– Key Renewal Theorem

– Application 1: Residual Life, Age, and Total Life – Application 2: Alternating Renewal Process/Theory – Application 3: Mean Residual Life

• Renewal Reward Processes and Applications – Renewal Reward Process/Theory

– Application 1: Alternating Renewal Process/Theory – Application 2: Time Average of Residual Life and Age

• More Notes on Regenerative Processes

(5)

Distribution and Limiting Behavior of ˜n(t)

1

S~ ~2

S ~3

S ~~()1

+ t

Sn t

)

~ t( n

1

~x ~x2 ~x3 ~ tn()

)

~(

~

t

Sn

{˜xn, n = 1, 2, . . .} ∼ F˜x; mean ¯X (0 < ¯X < ∞)

N = {˜n(t), t ≥ 0} is called a renewal (counting) process n(t) = max{n : ˜˜ Sn ≤ t}

(6)

Distribution and Limiting Behavior of ˜n(t)

n(t)˜

1. pmf of ˜n(t) → closed-form

2. Limiting time average [Law of Large Numbers]:

n(t)˜ t

w.p.1

1

X¯ , t → ∞ 3. Limiting time and ensemble average

[Elementary Renewal Theorem]:

E[˜n(t)]

t

w.p.1

1

X¯ , t → ∞

(7)

Distribution and Limiting Behavior of ˜n(t)

4. Limiting ensemble average (focusing on arrivals in the vicinity of t ) [Blackwell’s Theorem]:

E[˜n(t + δ) − ˜n(t)]

δ

w.p.1

1

X¯ , t → ∞ 5. Limiting PDF of ˜n(t) [Central Limit Theorem]:

t→∞lim P

 n(t) − t/ ¯˜ X σ√

t( ¯X)−3/2 < y



=

 y

−∞

1

2πex22 dx ∼ Gaussian( t

X¯ , σ√

t· ¯X32)

(8)

pmf of ˜n(t)

P [˜n(t) = n] = P [˜n(t) ≥ n] − P [˜n(t) ≥ n + 1]

= P [ ˜Sn ≤ t] − P [ ˜Sn+1 ≤ t]

··· x˜i ∼ F,

··

·

x˜i ∼ F (t) ⊗ F (t) . . . ⊗ F (t) ≡ Fn(t)

= Fn(t) − Fn+1(t) n-fold convolution of F (t)

(9)

Limiting Time Average

t→∞lim n(t) =?˜

··· P



t→∞lim n(t) < ∞˜



= P [˜n(∞) < ∞] = P [˜xn = ∞ for some n]

= P





n=1

xn = ∞)



=

 n=1

P [˜xn = ∞] = 0

··

· lim

t→∞ n(t) = ˜˜ n(∞) = ∞ w.p.1

Question: What is the rate at which ˜n(t) goes to ∞ ?

)

~ t( n

?

) ?

~( lim =

t t n

i.e. t

(10)

Strong Law for Renewal Processes

Theorem. For a renewal process N = {˜n(t), t ≥ 0} with mean inter- renewal interval ¯X, then

t→∞lim

n(t)˜

t = 1

X¯ , w.p.1 Proof.

)

~ t( n

)

~(

~

t

Sn t ~( ) 1

~

+ t

Sn

t

)

~ t( n

(11)

Strong Law for Renewal Processes

··· S˜˜n(t) ≤ t < ˜S˜n(t)+1

S˜˜n(t)

n(t)˜ t

n(t)˜ <

S˜˜n(t)+1 n(t)˜ =

S˜˜n(t)+1

n(t) + 1˜ × n(t) + 1˜ n(t)˜

⇒ lim

t→∞

S˜˜n(t) n(t)˜

= ¯X why?

≤ lim

t→∞

t

n(t)˜ < limt→∞

S˜˜n(t)+1 n(t) + 1˜

= ¯X

× n(t) + 1˜ n(t)˜

=1

··

· lim

t→∞

n(t)˜

t = 1 X¯

(12)

Renewal Function E[˜n(t)]

Let m(t) = E[˜n(t)], which is called “renewal function”.

1. Relationship between m(t) and Fn

m(t) =

 n=1

Fn(t), where Fn is the n-fold convolution of F 2. Relationship between m(t) and F

[Renewal Equation]

m(t) = F (t) +

 t

0 m(t − x)dF (x)

3. Relationship between m(t) and L˜x(r) (Laplace Transform of ˜x) Lm(r) = L˜x(r)

r[1 − L˜x(r)]

(13)

Renewal Function E[˜n(t)]

→ [Wald’s Equation]

4. Asymptotic behavior of m(t) (t → ∞, Limiting)

→ [Elementary Renewal Theorem]

→ [Blackwell’s Theorem]

(14)

Renewal Function E[˜n(t)]

1. m(t) = E[˜n(t)] ←→ F? n (i.e., PDF of ˜Sn)

Let ˜n(t) =

 n=1

In, where In =

1, nth renewal occurs in [0, t];

0, Otherwise;

m(t) = E[˜n(t)] = E





n=1

In



=

 n=1

E[In]

=

 n=1

P [nth renewal occurs in [0, t]]

=

 n=1

P [ ˜Sn ≤ t]

(15)

Renewal Function E[˜n(t)]

··

· m(t) =

 n=1

Fn(t)

or m(t) =

 n=1

P [˜n(t) ≥ n] =

 n=1

P [ ˜Sn ≤ t] = 

n=1

Fn(t)

. . . .

As t → ∞, n → ∞, finding Fn is far too complicated

⇒ find another way of solving m(t) in terms of F˜x(t)

(16)

Renewal Function E[˜n(t)]

2. m(t) ←→ F? ˜x(t) (i.e., PDF of ˜x)

··· S˜n = ˜Sn−1 + ˜xn, for all n ≥ 1, and ˜Sn−1 and ˜xn are independent,

··

· P [ ˜Sn ≤ t] =  t

0 P [ ˜Sn−1 ≤ t − x]dF˜x(x), for n ≥ 2 for n = 1, ˜x1 = ˜S1, P [ ˜S1 ≤ t] = F˜x(x)

··

· m(t) =

 n=1

P [ ˜Sn ≤ t] = F˜x(t) +

 t

0

 n=2

P [ ˜Sn−1 ≤ t − x]dF˜x(x) m(t) = F˜x(t) +

 t

0 m(t − x) · dF˜x(x) ⇒ Renewal Equation

(17)

Renewal Function E[˜n(t)]

3. Lm(r) ←→ L? ˜x(r) (Laplace Transform of ˜x)

(Laplace Transform of m(t) = Lm(r)) Answer:

Lm(r) = L˜x(r) r[1 − L˜x(r)]

<Homework> Prove it.

4. Asymptotic behavior of m(t):

t→∞lim

m(t)

t = lim

t→∞

E[˜n(t)]

t =?

(18)

Stopping Time (Rule)

Definition. ˜N , an integer-valued r.v., is said to be a “stopping time” for a set of independent random variables ˜x1, ˜x2, . . . if event { ˜N = n} is independent of ˜xn+1, ˜xn+2, . . .

Example 1.

• Let ˜x1, ˜x2, . . . be independent random variables,

• P [˜xn = 0] = P [˜xn = 1] = 1/2, n = 1, 2, . . .

• if ˜N = min{n : ˜x1 + . . . + ˜xn = 10}

→ Is ˜N a stopping time for ˜x1, ˜x2, . . .?

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Stopping Time (Rule)

Example 2.

• ˜n(t), X = {˜xn, n = 1, 2, 3, . . .},

• S = { ˜Sn, n = 0, 1, 2, 3, . . .},

• ˜Sn = ˜Sn−1 + ˜xn

1

~x ~x2 ~x3

4

~x ~x5

1

S~ ~2

S 3

S~ ~4

S 5

S~

t

)

~ t( n

→ Is ˜n(t) the stopping time of X = {˜xn, n = 1, 2, . . .}?

(20)

Stopping Time (Rule)

Example 3. Is ˜n(t) + 1 the stopping time for {˜xn}?

Answer:

whether ˜n(t) + 1 = n (→ ˜n(t) = n − 1) depends on ˜Sn−1 ≤ t < ˜Sn

··

· depends on ˜Sn−1 and ˜Sn, i.e., up to ˜xn

··

· n(t) + 1 is the stopping time for {˜˜ xn}, so is ˜n(t) + 2, ˜n(t) + 3, . . .

(21)

Stopping Time - from ˜I

n

Definition. ˜N , an integer-valued r.v. is said to be a stopping time for a set of independent random variables {˜xn, n ≥ 1}, if for each n > 1, ˜In, conditional on ˜x1, ˜x2, . . . , ˜xn−1, is independent of {˜xk, k ≥ n}

Define. ˜In - a decision rule for stopping time ˜N , n ≥ 1 I˜n =

1, if the nth observation is to be made;

0, Otherwise

1. ··· N is the stopping time˜

··

· I˜n depends on ˜x1, . . . , ˜xn−1 but not ˜xn, ˜xn+1, . . . 2. ˜In is also an indicator function of event { ˜N ≥ n},

i.e., ˜In =

1, if ˜N ≥ n;

(22)

Stopping Time - from ˜I

n

Because

• If ˜N ≥ n, then nth observation must be made;

• Since ˜N ≥ n implies ˜N ≥ n − 1 and happily, ˜In = 1 implies I˜n−1 = 1

··

· Stopping time

{ ˜N = n}, is independent of ˜xn+1, ˜xn+2, . . . or

I˜n is independent of ˜xn, ˜xn+1, . . .

(23)

Wald’s Equation

Theorem. If {˜xn, n ≥ 1} are i.i.d. random variables with finite mean E[˜x], and if ˜N is the stopping time for {˜xn, n ≥ 1}, such that

E[ ˜N ] < ∞. Then,

E

N˜

n=1

x˜n

= E[ ˜N ] · E[˜x]

Proof. Let ˜In =

1, if n ≤ ˜N ; 0, otherwise;

E

N˜



n=1

x˜n

= E





n=1

x˜n · ˜In



=



E[˜xn · ˜In] =



E[˜xn] · E[˜In] (why? stopping time)

(24)

Wald’s Equation

= E[˜x]

 n=1

E[ ˜In] = E[˜x]

 n=1

P ( ˜N ≥ n)

= E[˜x]E[ ˜N ]

. . . .

For Wald’s Theorem to be applied, other than {˜xi, i ≥ 1}

1. ˜N must be a stopping time; and 2. E[ ˜N ] < ∞

(25)

Wald’s Equation

Example. (Example 3.2.3 – Simple Random Walk, [Kao])

2 1

{˜xi} i.i.d. with: P (˜x = 1) = p

P (˜x = −1) = 1 − p = q S˜n =

n

x˜k

(26)

Wald’s Equation

• Let ˜N = min{n| ˜Sn = 1}

→ ˜N is the stopping time

E[ ˜SN˜] = E[ ˜N ] · E[˜x] = E[ ˜N ] · (p − q)

··· S˜N˜ = 1 for all ˜N

··

· E[ ˜SN˜] = 1

– if p = q, E[ ˜N ] = ∞ ⇒ Wald’s Theorem not applicable – if p > q, E[ ˜N ] < ∞ ⇒ E[ ˜N ] = p−q1

– if p < q, E[ ˜N ] = ∞ ⇒ Wald’s Theorem not applicable

(27)

Wald’s Equation

• Let ˜M = min{n| ˜Sn = 1} − 1

··

· S˜M˜ = 0 E[ ˜SM˜ ] = 0 assume E[ ˜N ] < ∞, p > q, ··

· E[ ˜M ] < ∞ but E[ ˜SM˜ ]

=0

= E[ ˜ M ] finite

(p − q)

Why?

(28)

Corollary

Before proving lim

t→∞

m(t)

t 1

X¯ ,

Corollary. If ¯X < ∞, then

E[ ˜S˜n(t)+1] = ¯X[m(t) + 1]

Proof.

E[ ˜S˜n(t)+1] = E

˜n(t)+1

n=1

x˜n

= ¯X · E[˜n(t) + 1] = ¯X · [m(t) + 1]

Why?

(29)

The Elementary Renewal Theorem

Theorem.

m(t)

t 1

X¯ as t → ∞ Proof.

To prove 1

X¯ ≤ lim

t→∞ inf m(t)

t

1

≤ lim

t→∞sup m(t)

t 1 X¯

2

1. ··· S˜˜n(t)+1 > t ··

· from Cor., ¯X[m(t) + 1] > t m(t)

t 1

X¯ 1

t ·

·· lim

t→∞inf m(t)

t 1 X¯

(30)

The Elementary Renewal Theorem

2. Consider a truncated renewal process

x˜n =

x˜n, If ˜xn ≤ M; n = 1, 2, . . . M, otherwise

Let ˜Sn =  x˜n, and ˜N(t) = sup{n : ˜Sn ≤ t}. We have that S˜ ˜N(t)+1 ≤ t + M

From the corollary,

[m(t) + 1] ¯X ≤ t + M, where ¯X = E[ ˜xn]

··

· lim

t→∞ sup m(t)

t 1

X¯

(31)

The Elementary Renewal Theorem

But since ˜Sn ≤ ˜Sn N˜(t) ≥ ˜N (t), m(t) ≥ m(t)

··

· lim

t→∞ sup m(t)

t 1 X¯

Let M → ∞, ¯X → ¯X

··

· lim

t→∞ sup m(t)

t 1 X¯

(32)

Blackwell’s Theorem

• Ensemble Average.

– to determine the expected renewal rate in the limit of large t, without averaging from 0 → t (time average)

• Question.

– are there some values of t at which renewals are more likely than others for large t ?

+4

t t+8 t +12

t

– An example. If each inter-renewal interval {˜xi, i = 1, 2, . . .} takes on integer number of time units, e.g., 0, 4, 8, 12, . . . , then

expected rate of renewals is zero at other times. Such random variable is said to be “lattice”.

(33)

Blackwell’s Theorem

– Definitions.

∗ A nonnegative random variable ˜x is said to be lattice if there exists d ≥ 0 such that

 n=0

P [˜x = nd] = 1

∗ That is, ˜x is lattice if it only takes on integral multiples of some nonnegative number d. The largest d having this property is said to be the period of ˜x. If ˜x is lattice and F is the distribution

function of ˜x, then we say that F is lattice.

• Answer.

– Inter-renewal interval random variables are not lattice

⇒ uniform expected rate of renewals in the limit of large t.

(34)

Blackwell’s Theorem

Theorem. If, for {˜xi, i ≥ 1}, which are not lattice, then, for any δ > 0,

t→∞lim [m(t + δ) − m(t)] = δ X¯ Proof. (omitted)

For non-lattice inter-renewal process {˜xi, i ≥ 1},

1. ··· x˜i > 0 ⇒ No multiple renewals (single arrival)

2. From Blackwell’s Theorem, the probability of a renewal in a small interval (t, t + δ] tends to δ/ ¯X + o(δ) as t → ∞,

··

· Limiting distribution of renewals in (t, t + δ] satisfies

t→∞lim P [˜n(t + δ) − ˜n(t) = 1] = δ

X¯ + o(δ)

(35)

Blackwell’s Theorem

t→∞lim P [˜n(t + δ) − ˜n(t) = 0] = 1 − δ

X¯ + o(δ)

t→∞lim P [˜n(t + δ) − ˜n(t) ≥ 2] = o(δ)

(36)

Blackwell’s Theorem

single arrival Stationary Independent Increment Increment

Poisson yes yes yes

Renewal yes yes no

Process

(Non-lattice)

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