UNIT

E

DE

RECHERCHE

INRIA-SOPHIAANTIPOLIS

Institut

National

de

Recherche

en

Informatique

et

en

Automatique

route

des

Lucioles

B.P.

Sophia-Antipolis

France

Rapports

de

Recherche

Programme

Architectures

parall?eles

Bases

de

donn?ees

R?eseaux

et

Syst?emes

distribu?es

POLLING

ON

A

GRAPH

WITH

GENERAL

ARRIVAL

AND

SERVICE

TIME

DISTRIBUTION

Eitan

ALTMAN

Serguei

FOSS

Avril

Polling

sur

un

graphe

avec

des

temps

d'inter-arriv

ees

et

de

services

avec

distributions

g?en?erales

Eitan

ALTMAN

Serguei

FOSS

INRIA

Centre

Sophia

Antipolis

Route

des

Lucioles

Valbonne

France

Avril

Abstract

Nous

consid?erons

un

syst?eme

de

polling

sur

un

graphe

avec

des

temps

d'interarriv

ees

de

services

et

de

d?eplacements

du

serveur

qui

forment

des

suites

i.i.d.

Nous

pr?esentons

des

conditions

suffisantes

et

n?ecessaires

de

stabilit?e

en

utilisant

une

analyse

de

la

d?erive

de

la

charge

dans

le

syst?eme

et

des

temps

r?esiduels

des

inter-arriv

ees

pendant

les

cycles

This

work

was

supported

by

the

European

Grant

BRA-QMIPS

of

CEC

DG

XIII

Polling

on

a

graph

with

general

arrival

and

service

time

distribution

Eitan

ALTMAN

Serguei

FOSS

INRIA

Centre

Sophia

Antipolis

Route

des

Lucioles

Valbonne

France

April

abstract

We

consider

a

polling

system

on

a

graph

with

general

independent

arrival

service

and

walking

times

We

present

necessary

and

sufficient

conditions

for

stability

based

on

drift

analysis

of

both

the

workload

and

the

residual

interarrival

times

within

some

embedded

times

called

cycles

Introduction

We

consider

a

polling

system

on

a

graph

G

a

single

server

follows

some

fixed

path

L

that

starts

at

some

point

and

returns

to

that

point

after

having

scanned

the

whole

graph

-LRB-

not

necessarily

in

a

cyclic

way

-RRB-

We

consider

some

general

independent

arrival

process

of

customers

The

location

of

the

arrivals

are

i.i.d.

If

the

probability

distribution

of

the

location

of

arrivals

have

some

mass

at

some

points

then

queueing

may

occur

there

The

server

can

serve

one

customer

at

a

time

In

addition

to

the

service

time

the

server

also

requires

some

random

time

-LRB-

that

may

depend

on

its

trajectory

-RRB-

to

move

from

one

customer

to

another

Such

systems

have

been

widely

used

as

models

in

wide

areas

of

applications

in

telecom

munication

systems

in

general

and

for

Local

Area

Networks

in

particular

computer

systems

reader-head

's

movement

in

a

computer

's

hard

disk

flexible

manufacturing

systems

road

traffic

control

repair

and

maintenance

For

overviews

on

polling

systems

see

Takagi

-LSB-

-RSB-

and

-LSB-

-RSB-

Growing

attention

was

given

in

recent

years

to

establish

rigorously

sufficient

and

necessary

stability

conditions

for

such

systems

Several

different

approaches

were

used

to

obtain

ergodicity

as

a

measure

of

stability

see

-LSB-

-RSB-

-LSB-

-RSB-

-LSB-

-RSB-

-LSB-

-RSB-

-LSB-

-RSB-

-LSB-

-RSB-

and

-LSB-

-RSB-

for

the

stability

of

discrete

polling

systems

-LRB-

consisting

of

N

queues

-RRB-

Kroese

and

Schmidt

-LSB-

-RSB-

-LSB-

-RSB-

studied

the

stability

of

polling

on

a

graph

-LRB-

i.e.

continuous

polling

models

-RRB-

and

Altman

and

Levy

-LSB-

-RSB-

studied

the

stability

of

non

cyclic

polling

in

two

and

higher

dimensional

plains

In

-LSB-

-RSB-

and

-LSB-

-RSB-

Altman

et

al.

further

present

sufficient

and

necessary

conditions

for

stronger

notions

of

stability

namely

the

geometric

ergodicity

and

the

geometric

rate

of

convergence

of

the

first

moment

of

the

process

of

queues

length

-LRB-

embedded

at

polling

instants

-RRB-

Sufficient

conditions

for

Central

Limit

Theorems

and

the

Law

of

Iterated

Logarithm

were

given

in

-LSB-

-RSB-

All

the

above

references

assumed

Poisson

arrivals

and

general

independent

walking

times

and

service

times

The

contribution

of

this

paper

is

to

consider

polling

with

general

independent

arrival

process

We

study

several

notions

of

stability

in

this

paper

and

obtain

in

particular

necessary

and

sufficient

conditions

for

the

expected

workload

in

the

system

to

be

uniformly

bounded

in

time

We

show

that

under

some

second

moment

conditions

if

the

rate

ae

at

which

work

arrives

to

the

system

-LRB-

defined

later

-RRB-

is

smaller

than

one

then

the

system

is

stable

An

alternative

sample-path

approach

for

studying

the

stability

of

polling

systems

is

given

by

Altman

et

al.

-LSB-

-RSB-

Different

type

of

conditions

are

imposed

on

a

single

sample

path

which

yields

several

kinds

of

notions

of

stability

under

suitable

assumptions

The

structure

of

the

paper

is

as

follows

In

Section

we

present

the

model

We

define

a

cycle

and

present

a

drift

analysis

of

the

behavior

of

the

workload

-LRB-

Section

-RRB-

and

the

residual

interarrival

times

-LRB-

Section

-RRB-

within

a

single

cycle

The

stability

results

are

finally

presented

in

Section

model

Consider

a

polling

system

on

a

graph

G

We

assume

that

a

single

server

follows

some

fixed

path

L

that

starts

at

some

point

and

returns

to

that

point

after

having

scanned

the

whole

graph

Define

such

a

scan

as

a

cycle

Note

that

in

a

cycle

some

points

-LRB-

including

-RRB-

may

be

scanned

several

times

Moreover

n

consecutive

cycles

can

be

considered

as

one

large

cycle

Customers

arrive

to

the

system

to

arbitrary

locations

on

the

graph

Arriving

customers

are

routed

to

some

location

according

to

a

routing

sequence

assumed

to

be

i.i.d.

If

the

distribution

of

this

location

has

mass

at

some

points

then

queueing

phenomena

may

occur

there

The

processes

of

inter-arrival

times

and

service

times

of

customers

are

assumed

to

be

independent

and

each

of

them

forms

i.i.d.

sequence

It

takes

some

time

for

the

server

to

move

from

one

point

to

another

on

the

graph

The

total

walking

times

at

different

cycles

are

i.i.d.

and

independent

of

arrival

and

service

processes

The

number

of

customers

that

are

served

at

point

l

L

on

the

graph

at

each

visit

is

determined

by

some

polling

discipline

-LRB-

such

as

the

gated

exhaustive

etc

-RRB-

which

may

be

random

and

may

depend

on

the

history

of

the

processes

Drift

analysis

in

a

single

cycle

We

first

focus

on

drift

analysis

within

a

single

cycle

We

are

interested

to

upper

bound

in

some

sense

the

difference

between

the

work

present

at

the

beginning

of

the

cycle

and

the

end

of

the

cycle

We

introduce

the

following

random

variables

defined

on

some

probability

space

-LRB-

F

P

-RRB-

T

is

a

random

variable

representing

the

time

at

which

the

cycle

begins

V

is

a

random

function

V

G

IR

representing

the

distribution

of

workload

in

the

graph

at

time

T

V

R

V

-LRB-

x

-RRB-

dx

is

the

total

amount

of

work

in

the

system

at

time

T

O

is

the

residual

interarrival

time

-LRB-

i.e.

the

time

between

T

and

the

next

arrival

-RRB-

are

i.i.d.

random

variables

representing

the

interarrival

times

is

the

time

between

the

arrival

that

occurs

at

time

T

O

and

the

following

arrival

We

shall

assume

that

Eoi

are

random

variables

representing

the

service

times

oei

is

the

service

time

required

by

the

ith

arriving

customer

-LRB-

after

T

-RRB-

We

shall

assume

that

Eoei

ae

f

are

random

variables

representing

the

location

of

the

arrivals

D

is

the

total

walking

time

in

the

cycle

We

assume

that

E

-LSB-

D

-RSB-

foei

are

assumed

to

be

i.i.d

although

oei

and

i

may

be

dependent

Define

a

nonnegative

integer-valued

random

variable

representing

the

number

of

arrivals

that

occur

in

the

cycle

Define

F

l

to

be

the

oe-algebra

generated

by

ffoigi

l

foeigi

l

f

igi

lg

For

all

l

f

lg

is

independent

of

F

l

-LRB-

-RRB-

and

f

g

fV

D

Og

We

further

assume

that

-LRB-

T

V

O

D

-RRB-

are

independent

of

-LRB-

-RRB-

G

a

nonnegative

random

variable

-LRB-

-RRB-

and

for

all

sets

A

F

the

event

fG

Ag

f

lg

is

independent

of

F

l.

-LRB-

-RRB-

G

o

a.s.

on

the

event

-LRB-

-RRB-

G

V

D

on

the

event

f

g.

-LRB-

-RRB-

G

is

the

past

interarrival

time

of

the

last

arrival

that

occurs

in

the

cycle

on

the

event

f

g.

Denote

O

-LRB-

so

that

G

o

a.s.

-RRB-

Introduce

T

G+P

i

oi

which

represents

the

random

time

at

which

the

cycle

ends

-LRB-

where

by

convention

oi

-RRB-

o

G

which

represents

the

residual

interarrival

time

at

time

-LRB-

i.e.

the

time

between

and

the

next

arrival

-RRB-

P?j

oej

is

the

total

work

arriving

during

the

cycle

V

out

the

amount

of

work

that

left

the

system

during

the

cycle

It

satisfies

V

out

D

T

the

amount

of

work

at

time

Thus

V

V

out

Lemma

Assume

E

-LSB-

-RSB-

The

following

hold

E

-LRB-

jT

V

O

D

-RRB-

E

-LRB-

jT

V

O

D

-RRB-

a

s

-LRB-

-RRB-

E

-LRB-

X

i

oijT

V

O

D

-RRB-

E

-LRB-

jT

V

O

D

-RRB-

a

s

-LRB-

-RRB-

E

-LRB-

jT

V

O

D

-RRB-

aeE

-LRB-

T

OjT

V

O

D

-RRB-

a

s

-LRB-

-RRB-

Proof

Follows

from

Wald

's

identity

-LRB-

-RRB-

-LRB-

-RRB-

and

the

fact

that

T

O

X

i

oi

Denote

E

-LRB-

jT

V

O

D

-RRB-

aeE

-LRB-

T

jT

V

O

D

-RRB-

Then

it

follows

that

E

-LRB-

OjT

V

O

D

-RRB-

a

s

We

are

now

ready

to

present

the

main

result

of

the

section

Lemma

Assume

E

-LSB-

-RSB-

Then

E

-LRB-

V

-RRB-

V

aeE

-LSB-

D

-RSB-

-LRB-

ae

-RRB-

E

-LRB-

V

outjT

V

-RRB-

aeE

-LRB-

OjT

V

-RRB-

a

s

-LRB-

-RRB-

Proof

Follows

from

-LRB-

-RRB-

Residual

interarrival

times

The

Poisson

case

and

Borovkov

's

conditions

Below

are

some

cases

where

the

term

E

-LRB-

V

-RRB-

can

be

evaluated

A

general

treatment

of

this

issue

is

delayed

to

the

next

section

-LRB-

i

-RRB-

In

the

case

Poisson

arrivals

O

and

are

both

independent

of

T

and

V

and

have

the

same

distribution

with

finite

mean

In

that

case

-LRB-

-RRB-

simplifies

to

E

-LRB-

V

-RRB-

V

aeE

-LSB-

D

-RSB-

-LRB-

ae

-RRB-

E

-LRB-

V

outjT

V

-RRB-

a

s

-LRB-

-RRB-

-LRB-

ii

-RRB-

In

our

case

the

following

holds

The

conditional

distribution

of

given

G

T

V

O

D

depends

on

G

only

i.e.

P

-LRB-

tj

G

T

V

O

D

-RRB-

P

-LRB-

t

x

-RRB-

a

s

on

the

event

f

fG

xg

Assume

now

that

for

some

c

and

for

all

t

x

P

-LRB-

t

x

-RRB-

cP

-LRB-

t

-RRB-

-LRB-

-RRB-

-LRB-

This

condition

is

known

as

Borovkov

's

condition

see

-LSB-

-RSB-

-RRB-

Then

aeE

-LRB-

OjT

V

-RRB-

aeE

-LRB-

V

-RRB-

E

h

E

-LRB-

G

T

V

O

D

-RRB-

T

V

i

cE

-LSB-

-RSB-

a

s

Hence

we

get

E

-LRB-

V

-RRB-

V

aeE

-LSB-

D

-RSB-

-LRB-

ae

-RRB-

E

-LRB-

V

outjT

V

-RRB-

a

s

-LRB-

-RRB-

moment

conditions

Exhaustive

and

Gated

type

policies

In

this

Section

we

present

a

drift

analysis

for

the

residual

interarrival

times

For

simplicity

we

restrict

to

a

cyclic

polling

system

composed

of

N

queues

and

begin

by

analysing

the

case

where

each

of

the

queues

is

surved

by

either

a

gated

or

an

exhaustive

policy

In

order

to

have

a

stable

behavior

of

the

residual

interarrival

times

it

is

not

necessary

to

require

that

ae

However

if

queue

n

is

served

according

to

the

exhaustive

service

discipline

then

we

need

to

assume

that

the

rate

at

which

work

arrives

to

that

queue

is

less

than

or

equivalently

E

-LSB-

-RSB-

E

-LSB-

ng

-RSB-

-LRB-

-RRB-

We

shall

consider

phases

in

the

cycle

at

phase

n

N

queue

n

are

visited

and

served

At

phase

n

N

the

server

moves

-LRB-

walks

-RRB-

from

queue

n

to

the

next

queue

during

a

random

time

Dn

Let

On

be

the

residual

interarrival

time

at

the

beginning

of

phase

n

and

O

-LRB-

t

-RRB-

be

the

the

residual

interarrival

time

at

time

t.

Let

Vn

denote

the

vector

of

workloads

in

the

different

queues

at

the

beginning

of

the

nth

phase

In

particular

we

have

O

and

we

shall

understand

-LRB-

where

O

and

were

defined

in

the

previous

Section

as

the

residual

interarrival

time

in

the

beginning

and

in

the

end

of

the

cycle

respectively

-RRB-

Since

we

consider

gated

and

exhaustive

policies

only

it

is

easily

seen

that

-LRB-

Vn

On

-RRB-

constitute

an

embedded

Markov

chain

We

assume

throughout

the

Section

that

the

interarrival

times

have

finite

second

moments

It

then

follows

from

standard

renewal

arguments

that

there

exists

some

finite

constant

C

such

that

sup

t

E

-LSB-

O

-LRB-

t

-RRB-

-RSB-

C

-LRB-

-RRB-

Lemma

-LRB-

i

-RRB-

For

each

n

there

exist

some

finite

constants

Mn

Cn

and

ffln

such

that

E

-LSB-

On

j

Vn

On

-RSB-

-LRB-

On

ffl

-RRB-

I

fOn

Mng

CnIfOn

Mng

-LRB-

-RRB-

-LRB-

ii

-RRB-

There

exist

some

finite

constants

M

and

C

such

that

E

-LSB-

j

-RSB-

-LRB-

ffl

-RRB-

I

Mg

Mg

where

ffl

n

ffln

Proof

If

phase

n

corresponds

to

a

visit

to

a

station

where

the

service

discipline

is

gated

then

it

follows

from

-LRB-

-RRB-

that

E

-LSB-

On

j

Vn

On

-RSB-

-LRB-

On

Vn

-RRB-

IfOn

Vng

CIfOn

Vng

max

-LRB-

On

C

-RRB-

Hence

-LRB-

-RRB-

is

satisfied

with

ffl

Mn

C

and

Cn

C.

Assume

that

phase

n

corresponds

to

a

walking

time

Di

for

some

i.

Choose

M

such

that

E

-LSB-

-LRB-

Di

C

-RRB-

IfDi

Mg

-RSB-

E

-LSB-

Di

-RSB-

Then

it

follows

from

-LRB-

-RRB-

that

E

-LSB-

On

j

Vn

On

-RSB-

E

-LSB-

-LRB-

On

Di

-RRB-

IfOn

Dig

CIfOn

DigjOn

-RSB-

IfOn

Mg

h

E

-LRB-

max

-LRB-

On

Di

-RRB-

jOn

-RRB-

CP

-LRB-

M

Di

-RRB-

i

IfOn

Mgmax

-LRB-

C

M

-RRB-

IfOn

Mg

On

E

-LSB-

Di

-RSB-

E

-LSB-

-LRB-

Di

C

-RRB-

IfDi

Mg

-RSB-

IfOn

Mgmax

-LRB-

C

M

-RRB-

IfOn

Mg

On

E

-LSB-

Di

-RSB-

IfOn

Mgmax

-LRB-

C

M

-RRB-

Hence

-LRB-

-RRB-

is

satisfied

with

ffln

E

-LSB-

Di

-RSB-

Mn

M

and

Cn

max

-LRB-

C

M

-RRB-

-LRB-

-RRB-

It

remains

to

establish

the

case

that

phase

n

corresponds

to

a

visit

to

a

station

say

k

where

the

service

discipline

is

exhaustive

We

have

If

On

Vn

then

On

On

Vn

-LRB-

-RRB-

Assume

now

that

On

Vn

Denote

tn

the

moment

when

phase

n

begins

-LRB-

and

thus

tn

is

the

moment

when

it

ends

-RRB-

On

n

ol

n

the

-LRB-

l

-RRB-

st

arrival

epoch

after

time

tn

oel

n

the

amount

of

work

brought

by

the

customer

that

is

the

lth

to

arrive

after

tn

n

l

the

queue

to

which

is

routed

the

customer

that

is

the

lth

to

arrive

after

tn

l

on

l

oen

lIf

n

l

kg

Sl

Pli

l.

OE

-LRB-

t

-RRB-

minfl

Sl

tg

for

all

t

Note

that

foi

n

oei

n

i

ng

=d

foi

oei

ig

Then

For

On

Vn

On

SOE

-LRB-

Vn

-RRB-

Vn

-LRB-

-RRB-

It

follows

from

Gut

-LSB-

-RSB-

p.

that

there

exists

some

finite

constant

C

such

that

sup

t

E

-LSB-

SOE

-LRB-

t

-RRB-

-RSB-

t

C

-LRB-

-RRB-

By

combining

-LRB-

-RRB-

and

-LRB-

-RRB-

we

conclude

that

On

C.

Hence

-LRB-

-RRB-

is

satisfied

with

ffl

Mn

C

and

Cn

C

We

establish

-LRB-

ii

-RRB-

for

N

The

proof

then

follows

a

simple

inductive

argument

for

arbitrary

N

Let

M

max

-LRB-

-RRB-

C

max

-LRB-

-RRB-

E

-LSB-

-RSB-

E

-LSB-

-LRB-

-RRB-

I

-RSB-

-LRB-

-RRB-

P

-LRB-

-RRB-

-LRB-

-RRB-

-RRB-

P

-LRB-

-RRB-

-LRB-

-RRB-

Mg

-LRB-

-RRB-

Mgmax

-LRB-

-RRB-

-LRB-

-RRB-

Mg

-LRB-

-RRB-

MgC

The

fact

that

ffl

follows

from

-LRB-

i

-RRB-

and

the

fact

that

E

-LSB-

D

-RSB-

and

from

-LRB-

-RRB-

Remark

The

results

of

the

previous

Lemma

easily

generalize

to

any

mixture

of

gatedtype

and

exhaustive-type

policies

which

are

defined

in

a

way

similar

to

-LSB-

-RSB-

and

-LSB-

-RSB-

A

gated-type

policy

is

characterized

by

a

function

f

IR

IR

such

that

if

upon

arrival

to

a

station

the

amount

of

work

in

that

station

is

V

then

the

amount

of

service

time

given

by

the

server

in

that

station

is

f

-LRB-

V

-RRB-

-LRB-

f

may

even

be

a

random

variable

independent

of

any

other

quantity

-RRB-

A

pure

gated

policy

is

then

characterized

by

f

-LRB-

V

-RRB-

V

An

exhaustivetype

policy

is

characterized

by

a

function

f

IR

IR

such

that

if

upon

arrival

to

a

station

the

amount

of

work

in

that

station

is

V

then

the

amount

of

service

time

given

by

the

server

in

that

station

is

such

that

when

leaving

the

station

the

amount

of

work

remaining

there

is

V

f

-LRB-

V

-RRB-

-LRB-

f

again

may

be

a

random

variable

independent

of

any

other

quantity

-RRB-

A

pure

exhaustive

policy

is

characterized

by

f

-LRB-

V

-RRB-

V

In

the

proof

of

the

previous

Lemma

we

simply

replace

Vn

by

f

-LRB-

Vn

-RRB-

in

-LRB-

-RRB-

and

-LRB-

-RRB-

General

stopping-type

policies

Next

we

generalize

the

last

results

to

a

much

larger

class

of

policies

We

consider

again

a

cyclic

polling

system

composed

of

N

queues

where

in

each

station

the

policy

to

be

defined

as

follows

Assume

that

the

phase

n

corresponds

to

a

visit

to

some

station

say

k

with

an

initial

workload

of

Vn

and

residual

interarrival

time

On

We

shall

use

again

the

notation

introduced

in

the

previous

Subsection

For

each

phase

n

we

define

the

customers

that

arrive

after

the

beginning

of

the

phase

-LRB-

i.e.

after

tn

-RRB-

to

be

new

customers

Consider

phase

n.

Denote

by

n

-LRB-

u

-RRB-

the

number

of

the

last

new

customer

being

served

during

the

phase

n

if

policy

u

is

used

Definition

We

say

that

a

policy

u

is

of

the

stopping

type

-LRB-

ST

policy

-RRB-

if

it

satisfies

-LRB-

i

-RRB-

Upon

arriving

to

the

station

some

of

the

initial

work

in

the

system

is

served

The

amount

of

work

served

is

a

function

of

the

amount

of

work

found

at

tn

in

the

station

say

f

-LRB-

Vn

-RRB-

-LRB-

with

f

-LRB-

Vn

-RRB-

Vn

-RRB-

-LRB-

ii

-RRB-

After

that

some

of

the

new

customers

are

served

according

to

the

FCFS

policy

-LRB-

iii

-RRB-

for

each

i

the

event

n

-LRB-

u

-RRB-

i

is

independent

of

the

sequences

-LRB-

fol

ngl

i

foei

ngl

i

f

i

ngl

i

-RRB-

Note

that

the

exhaustive

and

gated

type

policies

are

special

cases

of

ST-policies

Lemma

Assume

that

ST-policies

are

used

in

all

stations

Then

the

statements

of

Lemma

still

hold

Proof

Consider

a

ST-service

u

that

uses

some

function

f

in

step

-LRB-

i

-RRB-

above

Consider

the

exhaustive

type

policy

u

with

the

same

f

-LRB-

i.e.

service

will

be

given

until

the

amount

of

work

in

the

system

decreases

by

f

-LRB-

Vn

-RRB-

-RRB-

Note

that

fOn

Vng

ae

f

n

-LRB-

u

-RRB-

g

and

n

-LRB-

u

-RRB-

n

-LRB-

u

-RRB-

Then

tn

tn

f

-LRB-

Vn

-RRB-

n

-LRB-

u

-RRB-

X

i

si

-LRB-

k

-RRB-

where

si

-LRB-

k

-RRB-

oei

nIf

i

n

kg

and

On

-LRB-

On

f

-LRB-

Vn

-RRB-

-RRB-

I

-LRB-

O

f

-LRB-

Vn

-RRB-

-RRB-

-LRB-

S

n

On

-LRB-

tn

tn

-RRB-

-RRB-

IfOn

f

-LRB-

Vn

-RRB-

g

where

Sl

n

n

ol

n

and

minfl

Sl

n

On

-LRB-

tn

tn

-RRB-

g

The

second

term

in

the

right

hand

side

of

the

last

equality

can

be

rewritten

in

the

form

-LSB-

-LRB-

S

n

On

-LRB-

tn

tn

-RRB-

-RRB-

If

n

-LRB-

u

-RRB-

n

-LRB-

u

-RRB-

g

-LRB-

S

n

On

-LRB-

tn

tn

-RRB-

-RRB-

If

n

-LRB-

u

-RRB-

n

-LRB-

u

-RRB-

g

-RSB-

IfOn

f

-LRB-

Vn

-RRB-

g

But

on

the

event

f

n

-LRB-

u

-RRB-

n

-LRB-

u

-RRB-

g

we

have

n

-LRB-

u

-RRB-

n

-LRB-

u

-RRB-

a

s

and

on

the

event

f

n

-LRB-

u

-RRB-

n

-LRB-

u

-RRB-

g

we

have

n

-LRB-

u

-RRB-

a

s

So

S

n

S

n

-LRB-

u

-RRB-

X

n

-LRB-

u

-RRB-

on

i

Therefore

E

n

-LRB-

S

n

On

-LRB-

tn

tn

-RRB-

-RRB-

If

n

-LRB-

u

-RRB-

n

-LRB-

u

-RRB-

g

On

Vn

o

E

-LRB-

S

n

On

n

-LRB-

u

-RRB-

X

i

si

-LRB-

k

-RRB-

-RRB-

On

Vn

C

where

C

is

given

in

-LRB-

-RRB-

-LRB-

and

the

last

inequality

follows

from

Gut

-LSB-

-RSB-

p.

exactly

as

in

-LRB-

-RRB-

-RRB-

On

the

other

hand

E

n

-LRB-

S

n

On

-LRB-

tn

tn

-RRB-

-RRB-

If

n

-LRB-

u

-RRB-

n

-LRB-

u

-RRB-

g

On

Vn

o

E

n

-LRB-

S

n

On

-LRB-

tn

tn

-RRB-

-RRB-

IfS

n

-LRB-

u

-RRB-

n

On

-LRB-

tn

tn

-RRB-

g

On

Vn

o

E

@E

X

i

n

-LRB-

u

-RRB-

oi

n

F

n

-LRB-

u

-RRB-

On

S

n

-LRB-

u

-RRB-

-LRB-

tn

tn

-RRB-

A

I

n

S

n

-LRB-

u

-RRB-

On

-LRB-

tn

tn

o

On

Vn

-RRB-

CE

n

I

n

S

n

-LRB-

u

-RRB-

On

-LRB-

tn

tn

i

On

Vn

o

C

a

s

Here

FOEn

-LRB-

u

-RRB-

is

the

oe

field

generated

by

Vn

On

foi

n

oei

n

i

n

i

n

-LRB-

u

-RRB-

g

Finally

we

get

E

On

On

Vn

-LRB-

On

f

-LRB-

Vn

-RRB-

-RRB-

IfOm

f

-LRB-

Vn

-RRB-

g

f

-LRB-

Vn

-RRB-

g

We

can

thus

choose

Mn

and

Cn

in

the

same

way

we

did

in

the

proof

of

Lemma

for

the

gated

or

exhaustive

policies

Positive

recurrence

In

this

section

we

consider

quantities

that

correspond

to

the

sequence

of

consecutive

cycles

We

add

the

i

to

the

quantities

defined

in

Section

in

order

to

denote

a

quantity

refering

to

the

ith

cycle

Thus

T

-LRB-

i

-RRB-

is

the

beginning

of

the

ith

cycle

etc.

Let

X

-LRB-

i

-RRB-

-LRB-

T

-LRB-

i

-RRB-

V

-LRB-

i

-RRB-

O

-LRB-

i

-RRB-

-RRB-

Define

U

-LRB-

-RRB-

foj

oej

-LRB-

-RRB-

Let

n

be

the

shift

operator

such

that

-LRB-

on

j

oen

j

n

j

-RRB-

-LRB-

oj

oej

j

-RRB-

ffi

n

-LRB-

-RRB-

a.s.

Define

recursively

U

-LRB-

i

-RRB-

U

-LRB-

i

-RRB-

ffi

-LRB-

i

-RRB-

foj

-LRB-

i

-RRB-

oej

-LRB-

i

-RRB-

j

-LRB-

i

-RRB-

-LRB-

-RRB-

As

before

we

assume

-LRB-

i

-RRB-

to

be

a

stopping

time

-LRB-

for

each

i

-RRB-

i.e.

we

make

the

following

assumption

-LRB-

A

-RRB-

-LRB-

X

-LRB-

i

-RRB-

foj

-LRB-

i

-RRB-

oej

-LRB-

i

-RRB-

j

-LRB-

i

-RRB-

j

-LRB-

i

-RRB-

g

-LRB-

i

-RRB-

G

-LRB-

i

-RRB-

g

are

independent

of

U

-LRB-

i

-RRB-

for

each

i

As

a

corollary

of

this

assumption

we

have

-LRB-

i

-RRB-

for

each

i

U

-LRB-

i

-RRB-

=d

U

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

ii

-RRB-

each

of

the

sequences

fX

-LRB-

i

-RRB-

g

and

fY

-LRB-

i

-RRB-

-LRB-

V

-LRB-

i

-RRB-

O

-LRB-

i

-RRB-

g

forms

a

time-homogeneous

Markov

chain

Theorem

Assume

that

there

exist

some

finite

constants

M

C

and

ffl

such

that

EfO

-LRB-

i

-RRB-

X

-LRB-

i

-RRB-

g

-LRB-

O

-LRB-

i

-RRB-

ffl

-RRB-

I

-LRB-

O

-LRB-

i

-RRB-

M

-RRB-

C

I

-LRB-

O

-LRB-

i

-RRB-

M

-RRB-

-LRB-

-RRB-

a.s.

for

all

i

Then

-LRB-

i

-RRB-

for

each

initial

condition

X

-LRB-

-RRB-

minfi

O

-LRB-

i

-RRB-

Mg

-LRB-

-RRB-

a.s.

-LRB-

ii

-RRB-

sup

Ef

X

-LRB-

-RRB-

g

where

we

take

a

supremum

on

all

non-random

initial

conditions

X

-LRB-

-RRB-

such

that

O

-LRB-

-RRB-

M

Proof

Follows

directly

from

the

Tweedie

's

criterion

-LRB-

-LSB-

-RSB-

Section

-RRB-

Corollary

Assume

in

addition

to

the

conditions

of

Theorem

that

-LRB-

i

-RRB-

E

-LRB-

i

-RRB-

is

finite

for

all

i

-LRB-

ii

-RRB-

there

exists

a

non-negative

function

h

such

that

h

-LRB-

V

-RRB-

as

V

and

EfV

out

-LRB-

i

-RRB-

X

-LRB-

i

-RRB-

g

h

-LRB-

V

-LRB-

i

-RRB-

-RRB-

-LRB-

-RRB-

a.s.

for

all

i

-LRB-

iii

-RRB-

ae

Then

one

can

choose

constants

L

and

M

such

that

-LRB-

i

-RRB-

for

each

initial

condition

X

-LRB-

-RRB-

minfi

V

-LRB-

i

-RRB-

L

O

-LRB-

i

-RRB-

Mg

-LRB-

-RRB-

a.s.

-LRB-

ii

-RRB-

sup

Ef

X

-LRB-

-RRB-

g

where

we

take

a

supremum

on

all

non-random

initial

conditions

X

-LRB-

-RRB-

such

that

V

-LRB-

-RRB-

L

and

O

-LRB-

-RRB-

M

-LRB-

iii

-RRB-

For

arbitrary

initial

condition

X

-LRB-

-RRB-

introduce

a

sequence

f

ng

n

n

a.s.

where

minfi

V

-LRB-

i

-RRB-

L

O

-LRB-

i

-RRB-

Mg

-LRB-

-RRB-

and

for

n

n

minfi

n

V

-LRB-

i

-RRB-

L

O

-LRB-

i

-RRB-

Mg

-LRB-

-RRB-

Then

there

exists

a

constant

U

such

that

Ef

n

njF

ng

U

-LRB-

-RRB-

a.s.

for

all

n.

Remark

The

statement

of

corollary

means

that

the

set

B

-LSB-

L

-RSB-

-LSB-

M

-RSB-

-LRB-

-RRB-

is

uniformly

positive

recurrent

for

the

Markov

chain

fY

-LRB-

i

-RRB-

g.

Remark

The

condition

-LRB-

-RRB-

is

satisfied

for

ST

policies

with

fn

-LRB-

V

-RRB-

K

V

for

each

phase

n

-LRB-

where

K

is

some

fixed

constant

-RRB-

Proof

of

Corollary

Set

X

-LRB-

T

V

O

-RRB-

-LRB-

-RRB-

Lemma

implies

the

inequality

E

-LRB-

X

-RRB-

V

ae

E

-LRB-

D

-RRB-

-LRB-

ae

-RRB-

KV

ae

C

-LRB-

-RRB-

Therefore

for

each

R

R

Xg

-LRB-

-RRB-

V

R

-LRB-

O

ffl

-RRB-

I

-LRB-

O

M

-RRB-

RC

I

-LRB-

O

M

-RRB-

aeC

aeE

-LRB-

D

-RRB-

-LRB-

ae

-RRB-

h

-LRB-

V

-RRB-

Choose

R

such

that

R

ffl

ae

-LRB-

C

E

-LRB-

D

-RRB-

-RRB-

-LRB-

-RRB-

and

L

such

that

-LRB-

ae

-RRB-

h

-LRB-

V

-RRB-

-LRB-

R

ae

-RRB-

C

ae

E

-LRB-

D

-RRB-

-LRB-

-RRB-

for

all

V

L.

Set

ffi

-LRB-

-RRB-

Therefore

if

either

O

M

or

V

L

then

E

-LRB-

R

X

-RRB-

V

R

O

ffi

-LRB-

-RRB-

a.s.

and

if

O

M

and

V

L

then

E

-LRB-

R

X

-RRB-

L

R

max

-LRB-

C

M

-RRB-

ae

C

ae

E

-LRB-

D

-RRB-

-LRB-

-RRB-

So

we

can

apply

Tweedie

's

criterion

-LSB-

-RSB-

Section

and

this

completes

the

proof

Remark

For

arbitrary

initial

condition

X

-LRB-

-RRB-

introduce

a

sequence

f

ng

n

n

a.s.

where

minfi

V

-LRB-

i

-RRB-

L

O

-LRB-

i

-RRB-

Mg

-LRB-

-RRB-

and

for

n

n

minfi

n

V

-LRB-

i

-RRB-

L

O

-LRB-

i

-RRB-

Mg

-LRB-

-RRB-

It

follows

from

Corollary

-LSB-

-RSB-

that

there

exists

a

constant

U

such

that

Ef

n

njF

ng

U

-LRB-

-RRB-

a.s.

for

all

n.

Next

we

restrict

to

the

setting

and

assumptions

of

Subsection

In

particular

we

restrict

to

a

cyclic

polling

system

composed

of

N

queues

Proposition

Assume

that

ST

policies

are

used

in

all

station

Assume

moreover

that

ae

and

conditions

-LRB-

ii

-RRB-

of

Corollary

holds

Then

-LRB-

-RRB-

is

satisfied

E

-LRB-

i

-RRB-

and

hence

the

statements

of

Theorem

and

Corollary

hold

Moreover

sup

EfT

-LRB-

>=-RRB-

T

-LRB-

-RRB-

X

-LRB-

-RRB-

g

-LRB-

-RRB-

where

we

take

a

supremum

on

all

non-random

initial

conditions

X

-LRB-

-RRB-

such

that

V

-LRB-

-RRB-

L

and

O

-LRB-

-RRB-

M

Proof

The

result

follows

from

the

comparison

with

the

exhaustive

policy

Indeed

for

each

i

the

r.v.

-LRB-

i

-RRB-

can

be

represented

as

a

sum

-LRB-

i

-RRB-

-LRB-

i

-RRB-

-LRB-

i

-RRB-

-LRB-

-RRB-

where

-LRB-

i

-RRB-

is

a

number

of

arrivals

during

the

service

phase

on

the

station

k

k

N

Then

we

can

use

the

induction

arguments

Assume

that

for

fixed

k

a

r.v.

-LRB-

i

-RRB-

has

a

finite

expectation

Consider

an

exhaustive

policy

on

this

phase

-LRB-

with

the

same

initial

conditions

-RRB-

and

denote

for

it

by

-LRB-

i

-RRB-

a

number

of

arrivals

during

the

service

phase

Then

-LRB-

i

-RRB-

-LRB-

i

-RRB-

a.s.

and

-LRB-

i

-RRB-

minfn

-LRB-

i

-RRB-

Xn

j

j

-LRB-

i

-RRB-

Xn

j

jg

minfn

Xn

j

-LRB-

j

j

-RRB-

-LRB-

i

-RRB-

-LRB-

i

-RRB-

g

Since

E

-LRB-

-RRB-

and

-LRB-

i

-RRB-

then

E

-LRB-

i

-RRB-

-LRB-

see

e.g.

-LSB-

-RSB-

-RRB-

and

therefore

E

-LRB-

i

-RRB-

and

-LRB-

i

-RRB-

The

similar

arguments

can

be

used

to

prove

E

and

-LRB-

-RRB-

Remark

Note

that

if

ae

then

V

-LRB-

i

-RRB-

a.s.

as

i

Indeed

for

each

t

denote

-LRB-

t

-RRB-

minfn

O

-LRB-

-RRB-

on

tg

A

-LRB-

t

-RRB-

P

-LRB-

t

-RRB-

n

the

amount

of

work

the

customers

bring

to

the

system

during

time

interval

-LSB-

t

-RRB-

S

-LRB-

t

-RRB-

the

amount

of

work

is

served

during

the

time

interval

-LSB-

t

-RRB-

By

definition

S

-LRB-

t

-RRB-

t

for

all

t

and

renewal

theorem

implies

-LRB-

t

-RRB-

t

-LRB-

-RRB-

-LRB-

-RRB-

a.s.

Since

T

-LRB-

i

-RRB-

T

-LRB-

i

-RRB-

D

-LRB-

i

-RRB-

a.s.

and

ED

-LRB-

-RRB-

then

-LRB-

T

-LRB-

i

-RRB-

-RRB-

T

-LRB-

i

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

and

A

-LRB-

T

-LRB-

i

-RRB-

-RRB-

T

-LRB-

i

-RRB-

fA

-LRB-

T

-LRB-

i

-RRB-

-RRB-

-LRB-

T

-LRB-

i

-RRB-

-RRB-

g

f

-LRB-

T

-LRB-

i

-RRB-

-RRB-

T

-LRB-

i

-RRB-

g

ae

-LRB-

-RRB-

a.s.

But

V

-LRB-

i

-RRB-

V

-LRB-

-RRB-

A

-LRB-

T

-LRB-

i

-RRB-

-RRB-

S

-LRB-

T

-LRB-

i

-RRB-

-RRB-

Therefore

lim

inf

V

-LRB-

i

-RRB-

T

-LRB-

i

-RRB-

lim

A

-LRB-

T

-LRB-

i

-RRB-

-RRB-

T

-LRB-

i

-RRB-

lim

sup

S

-LRB-

T

-LRB-

i

-RRB-

-RRB-

T

-LRB-

i

-RRB-

ae

-LRB-

-RRB-

and

V

-LRB-

i

-RRB-

-LRB-

-RRB-

a.s.

as

i

Remark

If

we

assume

the

Borovkov

's

condition

on

the

inter-arrival

times

then

the

condition

-LRB-

-RRB-

is

always

true

-LRB-

even

for

an

arbitrary

graph

G

-RRB-

Ergodicity

There

is

a

family

of

well-known

additional

sufficient

conditions

when

positive

recurrence

implies

the

ergodicity

-LRB-

and

moreover

coupling

ergodicity

-RRB-

of

the

sequence

fX

-LRB-

i

-RRB-

g

and

of

the

continuous-time

processes

-LRB-

see

e.g.

Asmussen

-LSB-

-RSB-

Lindvall

-LSB-

-RSB-

Meyn

and

Tweedie

-LSB-

-RSB-

and

others

-RRB-

We

present

two

examples

for

sets

of

such

sufficient

conditions

We

restrict

to

the

setting

and

assumptions

of

Subection

Example

Assume

that

ffl

-LRB-

i

-RRB-

A

r.v.

has

a

non-lattice

distribution

ffl

-LRB-

ii

-RRB-

The

ST-policy

on

each

station

is

fixed

-LRB-

i.e.

it

is

the

same

for

all

cycles

-RRB-

and

such

that

f

-LRB-

V

-RRB-

V

for

all

V

ffl

-LRB-

iii

-RRB-

There

exist

some

constants

and

such

that

P

-LRB-

-RRB-

E

-LRB-

D

-LRB-

-RRB-

-RRB-

-LRB-

-RRB-

and

P

-LRB-

L

max

-LRB-

C

-RRB-

-RRB-

-LRB-

-RRB-

ffl

-LRB-

iv

-RRB-

There

exist

some

constants

a

b

and

p

such

that

P

-LRB-

D

-LRB-

-RRB-

G

-RRB-

p

-LRB-

G

-RRB-

-LRB-

-RRB-

for

each

Borel

set

G

-LSB-

a

b

-RSB-

-LRB-

here

is

a

Lebesgue

measure

-RRB-

Proposition

Under

the

above

conditions

one

can

construct

a

stationary

ergodic

sequence

fY

ig

on

the

same

probability

space

with

fY

-LRB-

i

-RRB-

g

such

that

P

-LRB-

Y

-LRB-

j

-RRB-

Y

j

i

-RRB-

-LRB-

-RRB-

as

i

Proof

For

each

integer

m

we

can

represent

a

set

B

-LSB-

L

-RSB-

-LSB-

M

-RSB-

as

a

union

of

the

sets

B

-LSB-

m

l

-LSB-

m

r

-LSB-

L

-LRB-

l

-RRB-

m

L

l

m

-RSB-

-LSB-

M

-LRB-

r

-RRB-

m

M

r

m

-RSB-

-LRB-

-RRB-

Since

the

set

B

is

uniformly

positive

recurrent

for

the

Markov

chain

fY

-LRB-

i

-RRB-

g

and

a

nonlatticity

of

implies

irreducibility

and

aperiodicity

of

fY

-LRB-

i

-RRB-

g

then

for

each

m

one

can

choose

at

least

one

pair

-LRB-

lm

rm

-RRB-

such

that

a

set

B

-LRB-

m

-RRB-

-LSB-

L

-LRB-

lm

-RRB-

m

L

lm

m

-RSB-

-LSB-

M

-LRB-

rm

-RRB-

m

M

rm

m

-RSB-

-LRB-

-RRB-

is

uniformly

positive

recurrent

Choose

m

such

that

m

-LRB-

b

a

-RRB-

Then

P

-LRB-

Y

-LRB-

i

-RRB-

-LRB-

-RRB-

jY

-LRB-

i

-RRB-

-RRB-

p

-LRB-

-RRB-

-LRB-

-RRB-

a.s.

on

the

event

fY

-LRB-

i

-RRB-

B

-LRB-

m

-RRB-

g

where

p

const

and

is

a

probability

measure

on

-LSB-

-RRB-

-LSB-

-RRB-

and

p

and

can

be

represented

in

terms

of

p

So

the

Markov

chain

fY

-LRB-

i

-RRB-

g

is

Harris

ergodic

and

this

completes

the

proof

Corollary

Let

Vt

be

a

vector

of

workloads

on

the

stations

at

time

t.

Under

the

above

conditions

there

exists

a

probability

distribution

P

-LRB-

-RRB-

such

that

the

distributions

P

-LRB-

Vt

-LRB-

-RRB-

-RRB-

converge

weakly

to

P

-LRB-

-RRB-

as

t

Example

We

present

below

conditions

that

imply

not

only

ergodicity

but

also

the

finiteness

of

the

first

moment

of

the

workload

in

steady

state

and

the

convergence

of

the

first

moments

of

the

workload

to

the

steady

state

moment

ASSUMPTIONS

ffl

-LRB-

Ai

-RRB-

In

all

stations

a

ST

policy

is

used

ffl

-LRB-

Aii

-RRB-

ae

and

the

following

stronger

version

of

conditions

-LRB-

ii

-RRB-

of

Corollary

holds

There

exists

some

K

such

that

EfV

out

-LRB-

i

-RRB-

Y

-LRB-

i

-RRB-

g

KV

-LRB-

i

-RRB-

-LRB-

-RRB-

a.s.

for

all

i

ffl

-LRB-

Aiii

-RRB-

P

-LRB-

o

D

-LRB-

-RRB-

D

-LRB-

-RRB-

oe

M

and

D

-LRB-

-RRB-

M

-RRB-

where

M

is

given

in

-LRB-

-RRB-

-LRB-

which

is

satisfied

according

to

Proposition

-LRB-

-RRB-

ffl

-LRB-

Aiv

-RRB-

Let

be

a

Borel

set

in

IR

P

-LRB-

f

-LRB-

-RRB-

D

-LRB-

-RRB-

D

-LRB-

-RRB-

D

-LRB-

-RRB-

oe

Lg

fD

-LRB-

-RRB-

Mg

-RRB-

implies

inf

l

M

P

-LRB-

fl

D

-LRB-

-RRB-

D

-LRB-

-RRB-

fo

D

-LRB-

-RRB-

D

-LRB-

-RRB-

oe

fD

-LRB-

-RRB-

Mg

-RRB-

where

L

is

given

in

-LRB-

-RRB-

Conditions

-LRB-

Ai

-RRB-

and

-LRB-

Aii

-RRB-

will

ensure

the

positive

recurrence

Condition

-LRB-

Aii

-RRB-

will

imply

moreover

the

existence

of

first

moments

of

the

workloads

Conditions

-LRB-

Aiii

-RRB-

and

-LRB-

Aiv

-RRB-

are

required

to

construct

a

small

set

as

will

be

required

in

Theorem

below

A

simple

sufficient

condition

for

both

-LRB-

Aiii

-RRB-

and

-LRB-

Aiv

-RRB-

to

hold

is

both

D

-LRB-

-RRB-

and

have

both

positive

probabilities

to

belong

to

Borel

sets

having

finite

measures

Proposition

Under

the

above

assumptions

fY

-LRB-

i

-RRB-

g

is

ergodic

and

V

-LRB-

i

-RRB-

has

finite

expectation

for

all

i.

Let

ss

be

the

steady

state

distribution

of

the

workload

Then

E

-LSB-

V

-RSB-

lim

i

-LSB-

V

-LRB-

i

-RRB-

jX

-LRB-

-RRB-

x

-RSB-

E

-LSB-

V

-RSB-

-LRB-

-RRB-

for

all

x

X.

To

prove

the

proposition

we

shall

use

the

following

Theorem

-LRB-

-LSB-

-RSB-

-RRB-

Theorem

Consider

a

strongly

aperiodic

Markov

chain

Xn

on

a

state

space

X

with

transition

probabilities

P

X

B

-LRB-

X

-RRB-

-LSB-

-RSB-

Assume

that

there

exists

a

set

K

X

and

a

functions

g

f

X

IR

g

-LRB-

-RRB-

f

-LRB-

-RRB-

such

that

-LRB-

i

-RRB-

there

exists

some

ffl

such

that

E

-LSB-

g

-LRB-

Xn

-RRB-

g

-LRB-

Xn

-RRB-

jXn

-RSB-

fflf

-LRB-

Xn

-RRB-

for

Xn

Kc

-LRB-

ii

-RRB-

E

-LSB-

g

-LRB-

Xn

-RRB-

jXn

-RSB-

for

Xn

K

-LRB-

iii

-RRB-

K

is

a

small

set

Then

-LRB-

a

-RRB-

Xn

is

ergodic

-LRB-

b.

-RRB-

The

steady

state

expectation

exists

E

-LSB-

f

-LRB-

X

-RRB-

-RSB-

-LRB-

b.

-RRB-

E

-LSB-

f

-LRB-

Xn

-RRB-

x

-RSB-

E

-LSB-

f

-LRB-

X

-RRB-

-RSB-

for

all

x

X.

Remark

In

the

above

Lemma

a

set

K

is

said

to

be

small

if

there

exists

some

positive

measure

OE

on

X

such

that

for

any

B

ae

X

with

OE

-LRB-

B

-RRB-

there

exists

j

such

that

inf

jX

n

Pn

-LRB-

x

B

-RRB-

where

P

-LRB-

x

B

-RRB-

is

the

transition

probabilities

of

the

Markov

chain

Proof

of

Proposition

Let

Y

-LRB-

V

O

-RRB-

We

apply

Theorem

by

identifying

g

-LRB-

Y

-RRB-

with

g

-LRB-

Y

-RRB-

V

RO

where

R

is

given

in

-LRB-

-RRB-

and

Y

-LRB-

V

O

-RRB-

f

-LRB-

Y

-RRB-

-LRB-

ae

-RRB-

KV

K

f

-LRB-

v

O

-RRB-

v

O

Mg

The

measure

OE

related

to

the

small

set

K

is

defined

as

follows

Let

fOj

-LRB-

O

-RRB-

Bg

Then

OE

-LRB-

B

-RRB-

P

-LRB-

f

-LRB-

-RRB-

D

-LRB-

-RRB-

D

-LRB-

-RRB-

D

-LRB-

-RRB-

Lg

fD

-LRB-

-RRB-

Mg

-RRB-

With

the

above

choice

of

g

and

f

-LRB-

-RRB-

implies

conditions

-LRB-

Ai

-RRB-

and

-LRB-

Aii

-RRB-

of

Theorem

Next

we

show

that

K

is

small

Assume

that

we

are

at

the

beginning

of

a

cycle

say

cycle

in

some

state

-LRB-

V

-LRB-

-RRB-

O

-LRB-

-RRB-

-RRB-

K

Then

ffl

-LRB-

-RRB-

D

-LRB-

-RRB-

M

implies

that

an

arrival

will

occur

during

cycle

since

O

-LRB-

-RRB-

K

and

hence

smaller

or

equal

to

M

ffl

-LRB-

-RRB-

D

-LRB-

-RRB-

D

-LRB-

-RRB-

implies

that

the

next

arrival

will

not

occur

in

the

current

cycle

nor

in

the

next

one

Indeed

in

that

case

the

total

length

of

cycles

and

is

V

-LRB-

-RRB-

D

-LRB-

-RRB-

D

-LRB-

-RRB-

D

-LRB-

-RRB-

D

-LRB-

-RRB-

since

V

-LRB-

-RRB-

K.

Hence

-LRB-

-RRB-

and

-LRB-

-RRB-

imply

that

V

-LRB-

-RRB-

O

-LRB-

-RRB-

is

then

given

by

O

-LRB-

-RRB-

O

-LRB-

-RRB-

-LSB-

V

-LRB-

-RRB-

D

-LRB-

-RRB-

D

-LRB-

-RRB-

-RSB-

Assume

now

that

B

is

such

that

OE

-LRB-

B

-RRB-

i.e.

P

-LRB-

f

D

-LRB-

-RRB-

D

-LRB-

-RRB-

D

-LRB-

-RRB-

D

-LRB-

-RRB-

fD

-LRB-

-RRB-

Mg

-RRB-

Then

for

any

Y

-LRB-

-RRB-

K

P

-LRB-

Y

-LRB-

-RRB-

BjY

-LRB-

-RRB-

-RRB-

P

-LRB-

Y

-LRB-

-RRB-

B

D

-LRB-

-RRB-

D

-LRB-

-RRB-

fD

-LRB-

-RRB-

MgjY

-LRB-

-RRB-

-RRB-

P

-LRB-

O

-LRB-

-RRB-

D

-LRB-

-RRB-

D

-LRB-

-RRB-

fD

-LRB-

-RRB-

MgjY

-LRB-

-RRB-

-RRB-

P

-LRB-

O

-LRB-

-RRB-

-LSB-

V

-LRB-

-RRB-

D

-LRB-

-RRB-

D

-LRB-

-RRB-

-RSB-

D

-LRB-

-RRB-

D

-LRB-

-RRB-

fD

-LRB-

-RRB-

MgjY

-LRB-

-RRB-

-RRB-

inf

l

M

P

-LRB-

l

-LSB-

D

-LRB-

-RRB-

D

-LRB-

-RRB-

-RSB-

D

-LRB-

-RRB-

D

-LRB-

-RRB-

fD

-LRB-

-RRB-

MgjY

-LRB-

-RRB-

-RRB-

uniformly

for

all

Y

-LRB-

-RRB-

K

This

follows

from

the

definition

of

the

measure

OE

and

from

condition

-LRB-

Aiv

-RRB-

References

-LSB-

-RSB-

E.

Altman

S.

Foss

E.

R.

Riehl

and

S.

Stidham

Pathwise

Stability

and

Performance

Bounds

for

Generalized

Vacation

and

Polling

Systems

Manuscript

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-RSB-

E.

Altman

and

D.

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Path

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Measures

of

Token

Rings

submitted

to

Globcom

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E.

Altman

P.

Konstantopoulos

Z.

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Stability

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and

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General

Polling

Systems

Queuing

Systems

special

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Models

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H.

Takagi

et

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Boxma

pp

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-RSB-

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Altman

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Queueing

in

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Submitted

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Journal

of

Applied

Probability

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-RSB-

E.

Altman

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stability

characteristics

of

asynchronous

traffic

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FDDI

token

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the

international

conference

on

High

Speed

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Chapel

Hill

USA

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-RSB-

E.

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F.

Spieksma

Geometric

Ergodicity

and

Moment

Stability

of

Station

Times

in

Polling

Systems

Technical

Report

No

Leiden

University

Holland

-LSB-

-RSB-

S.

Asmussen

Applied

Probability

and

Queues

Wiley

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-RSB-

T.

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Lecture

on

the

Coupling

Method

Wiley

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A.

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Limit

theorems

for

queueing

networks

Theory

Prob

Appl.

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-RSB-

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and

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Ergodicity

of

a

polling

system

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to

Stochastic

Processes

and

Applications

-LSB-

-RSB-

A.

Gut

Stopped

Random

Walks

Springer-Verlag

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-RSB-

L.

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W.

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of

Token

Passing

Rings

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Systems

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on

Polling

Models

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H.

Takagi

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-LSB-

-RSB-

C.

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and

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Department

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University

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-RSB-

D.

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and

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A

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Polling

System

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-RSB-

D.

P.

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and

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Queues

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Spatially

Distributed

Arrivals

Submitted

-LSB-

-RSB-

H.

Levy

M.

Sidi

O.

J.

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Dominance

Relations

in

Polling

Systems

Queueing

Systems

pp.

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-RSB-

S.

P.

Meyn

and

R.

L.

Tweedie

Stability

of

Markovian

processes

I

criteria

for

discrete

time

chains

Advances

in

Applied

Probability

No.

pp.

-LSB-

-RSB-

E.Nummelin

General

Irreducible

Markov

Chains

and

Non-Negative

Operators

Cambridge

Univ.

Press

-LSB-

-RSB-

J.

A.

C.

Resing

Polling

systems

and

multi-type

branching

processes

Report

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C.W.I.

Amsterdam

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-RSB-

H.

Takagi

Analysis

of

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Systems

The

MIT

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H.

Takagi

Queuing

Analysis

of

Polling

Models

an

Update

in

Stochastic

Analysis

of

Computer

and

Communications

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-LRB-

H.

Takagi

Ed

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Elsevier

Science

Pub.

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Tweedie

Criteria

for

classifying

general

Markov

Chains

Adv.

Appl

Prob

pp.

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R.

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Criteria

for

rates

of

convergence

of

Markov

chains

with

application

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queuing

and

storage

theory

Probability

Statistics

and

Analysis

ed

J.F.C.

Kingman

and

G.E.H.

Renter

London

Math

Soc

Lecture

Notes

Series

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pp.

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-RSB-

R.

L.

Tweedie

The

existence

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Markov

chains

J.

Applied

Probability

pp.

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-RSB-

V.

S.

Zhdanov

E.

A.

Saksonov

Conditions

of

Existence

of

Steady-State

Modes

in

Cyclic

Queuing

Systems

Automation

and

Remote

Control

No.

pp.

general arrival
polling systems
service and walking times