Lawler (1995) Chapter 6 Renewal Processes

AB HELSINKI UNIVERSITY OF TECHNOLOGYNetworking Laboratory

1ch6.ppt S-38.215 – Applied Stochastic Processes – Spring 2004

Lawler (1995)

Chapter 6

Renewal Processes

6.1 Introduction

6.2 Renewal Equation

6.3 Discrete Renewal Processes

6.4 M/G/1 and G/M/1 queues

2

Chapter 6: Renewal Processes

6.1 Introduction

• Definition:

– Let T1, T

2 , … be independent and identically distributed (i.i.d.) nonnegative

random variables with mean µ = E[Ti] > 0. The renewal process

associated with renewal sequence Tiis the process N

twith

– Thus, Ntdenotes the number of events occurred up to time t.

• Definition:

– Let T1, T

2 , … be independent and identically distributed (i.i.d.) nonnegative

random variables with mean µ = E[Ti] > 0. Furthermore, let Y another

nonnegative random variable indepedndent of the sequence Ti. The

renewal process associated with random variable Y and sequence Tiis

the process Ntwith

≥≤++=

<=

11

1

},|,2,1max{

,0

TttTTn

TtN

n

tKK

≥>+++=

<=

YttTTYn

YtN

n

t },|,2,1min{

,0

1 KK (6.1)

3


6.1 Introduction

• Definitions:

– Let Ntbe a renewal process associated with renewal sequence T

i. The age

associated with renewal process Ntis the process A

twith

– The residual life associated with renewal process Ntis the process B

twith

– The total lifetime associated with renewal process Ntis the process C

t

with

• Note:

– Renewal process Ntalone is not a Markov process (unless the interarrival

times Tibe exponential) but the pair (N

t, A

t) constitutes a Markov process

≥++−

<=

11

1

),(

,

TtTTt

Ttt

A

tNt

K

}:inf{1 tsttNt NNsATBt

>=−=++

ttNt BATCt

+==+1

4


6.1 Introduction

• Strong Law of Large Numbers (SLLN) for renewal processes:


i with

mean µ.

– Then, with probability 1,

• Central Limit Theorem (CLT) for renewal processes:


i with

mean µ and variance σ2.

– Then

µ

1lim =

∞→t

N

t

t

)1,0(i.d.

2/12/3

1

N

t

tNt→

−

−

−

σµ

µ

(6.2)

5



• Consider a renewal process Ntassociated with renewal sequence T

i

with mean µ and distribution function F.

• Definition:

– The renewal function U is defined by

• Elementary Renewal Theorem:

• Note:

– It follows that

][1)( tNEtU +=

µ

1)(lim =

∞→t

tU

t

µ

1][lim =

∞→t

NE

t

t

(6.3)

6



• Definition:

– Let X be a nonnegative random variable.

– It has a lattice distribution if there is a > 0 such that

– In this case a is called the period of the distribution.

– Otherwise it has a nonlattice distribution.

• Blackwell’s Theorem:

– If T1, T

2 , … have a nonlattice distribution, then for all r > 0

– If T1, T

2 , … have a lattice distribution with period a, then

1}},2,,0{{ =∈ KaaXP

µ

r

t

tUrtU =−+

∞→

)()(lim

µ

a

n

naUanU =−+

∞→

)())1((lim

(6.4)

7



• Definition:

– Let F and G be distributions of nonnegative random variables.

– The convolution F∗G is another distribution defined by

• Notes:

– Let X and Y be independent nonnegative random variables with

distributions F and G, respectively. Then the convolution F∗G is the

distribution of their sum X+Y,

∫∫ −=−=∗tt

sdFstGsdGstFtGF00

)()()()()]([

)]([}{ tGFtYXP ∗=≤+

8



• Definition:

– Let F be a distribution of a nonnegative random variable.

– The iterated convolution F(n) is defined by

• Proposition:

• Proof:

0 allfor

0 allfor 1)(

)()1(

)0(

≥∗=

≥=

+nFFF

ttF

nn

∑∞

== 0

)(n

nFU

∑∑

∑

∞

=

∞

=

∞

=

=≤+++=

≥+=

+=

0)(

1 1

1

)(}{1

}{1

][1)(

n

n

n n

n t

t

tFtTTP

nNP

NEtU

K

9



• Consider a renewal process Ntassociated with renewal sequence T

i

with distribution function F and renewal function U.

• Definition:

– Let h be a nonnegative function defined on [0,∞). Function φ defined on

[0,∞) satisfies the renewal equation if

– Equivalently, for all t,

• Proposition:

– There is a unique solution to the renewal equation,

• Note:

– If F is nonlattice, h is bounded and ∫0∞ |h(t)|dt < ∞, then

Fh ∗+= φφ

∫ −+=t

sdFsttht0

)()()()( φφ (6.6)

Uh*=φ

∫∫∞

∞→∞→

=−==∞0

1

0)()()(lim)(lim)( dsshsdUstht

t

ttµ

φφ (6.8)

10



• Consider then the age process Atof the renewal process N

t.

• Denote its steady state distribution by

• Proposition:

– Probabilities P{At≤ x} satisfy the following renewal equation:

• Proposition:

– Steady state distribution (t → ∞) is as follows:

– It is a continuous distribution with density

}{lim)( xAPx tt

A ≤=Ψ

∞→

∫ ≤+−=≤−

t

stxt sdFxAPtFtxAP0],0[ )(}{))(1)((1}{ (6.5)

∫ −=Ψx

A dyyFx0

1 ))(1()(µ

))(1()( 1xFxA −=

µψ

11



• Consider then the residual life process Btof the renewal process N

t.


• Proposition:

– Probabilities P{Bt≤ x} satisfy the following renewal equation:

• Proposition:

– Steady state distribution (t → ∞) is the same as for the age process:

– It is a continuous distribution with density

• Note:

– Starting with Y that has distribution ΨBresults in stationary increments

}{lim)( xBPx tt

B ≤=Ψ

∞→

∫ ≤+−+=≤−

t

stt sdFxBPtFxtFxBP0

)(}{))()((}{

∫ −=Ψx

B dyyFx0

1 ))(1()(µ

))(1()( 1xFxB −=

µψ

12



• Consider finally the total lifetime process Ctof the renewal process N

t.


• Proposition:

– Probabilities P{Ct≤ x} satisfy the following renewal equation:

• Proposition:

– Steady state distribution (t → ∞) is as follows:

– If F is a continuous distribution with density f, then the steady state distribution of the total lifetime is a continuous distribution with density

}{lim)( xCPx tt

C ≤=Ψ

∞→

∫ ≤+−=≤−

t

stxt sdFxCPtFxFtxCP0],0[ )(}{))()()((1}{

∫ −=Ψx

C dyyFxFx0

1 ))()(()(µ

)()( 1 xxfxC µψ =

13



• Consider a discrete-time renewal process Nj associated with renewal

sequence Ti with mean µ and a lattice distribution function F with

period a = 1 and F(0) = 0.

• Thus, we have

• Denote the point probabilities by

• Proposition:

– Age process Aj is a discrete-time Markov chain with transition probabilities

where

)1()(}{ −−=== nFnFnTPp in

≥≤++=

<=

11

1

},|,2,1max{

,0

TjjTTn

TjN

nj

KK

11 1)1,( ,)0,(++

−=+=nn

nnpnp λλ

)1(1

)(1

)1(11 ,

−−

−

−−

=−=

nF

nFnnF

pn

nλλ

14



• Proposition:

– Age process Aj is irreducible, aperiodic, and positive recurrent with steady

state probabilities

• Blackwell’s Theorem:

• Proposition:

– Residual life process Bj has steady state distribution

• Proposition:

– Total life time process Cj has steady state distribution

K,2,1 )),(1()( ,)0( 11=−== nnFnAA µµ

ψψ

µψ

11 )0(}0{lim)()1(lim ====−++

∞→∞→

Ajjj

APjUjU

K,2,1 )),1(1()1()( 1=−−=−= nnFnn AB µ

ψψ

K,2,1 ,)( 1== nnpn nC µ

ψ

15


6.4 M/G/1 and G/M/1 queues

• M/G/1 queue

– Renewal every time the queue becomes nonempty

• Cycle = busy time + idle time

• Busy time and idle time independent

• Idle time exponentially distributed

– Renewal every time the queue becomes empty

• Cycle = idle time + busy time

• Busy time and idle time independent

• Idle time exponentially distributed

• G/M/1 queue

– Renewal every time the queue becomes nonempty

• Cycle = busy time + idle time

• Busy time and idle time dependent

– No renewal when the queue becomes empty

16


The End of Chapter 6

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Lawler (1995) Chapter 6 Renewal Processes