Modeling

and

Analysis

of

Channel

Transferability

in

Mobile

Computing

Environments

Ravi

Prakash

and

Mukesh

Singhal

Department

of

Computer

and

Information

Science

The

Ohio

State

University

Columbus

OH

U.S.A.

e-mail

fprakash,singhalg@cis.ohio-state.edu

Abstract

Mobile

computers

use

wireless

channels

to

communicate

with

other

computers

Efficient

channel

allocation

is

at

the

heart

of

an

efficient

mobile

computing

system

The

finite

number

of

channels

should

be

efficiently

allocated

to

maximize

throughput

and

to

avoid

co-channel

interference

Temporal

variations

in

channel

demand

-LRB-

load

-RRB-

require

channel

allocation

to

adapt

dynamically

to

the

changing

demand

The

presence

of

load

imbalance

does

not

automatically

imply

the

usefulness

of

channel

transfer

In

this

paper

we

present

a

probabilistic

analysis

of

the

temporal

imbalance

in

channel

demand

The

effectiveness

of

channel

transfer

depends

on

the

duration

for

which

the

imbalance

persists

This

duration

is

influenced

in

part

by

the

velocity

of

mobile

units

The

wide

variance

in

velocities

of

the

mobile

units

can

be

handled

by

using

a

hierarchical

cellular

layout

Bulk

channel

transfers

can

be

employed

to

reduce

the

overheads

of

channel

transfer

and

to

enable

the

system

to

quickly

adjust

to

spatial

variations

in

channel

demand

The

results

of

our

analysis

provide

guidelines

for

the

design

of

dynamic

distributed

channel

allocation

strategies

Introduction

A

mobile

computing

system

consists

of

a

set

of

mobile

computers

and

fixed

computers

-LSB-

-RSB-

The

fixed

computers

communicate

with

each

other

through

a

fixed

wireline

network

The

mobile

computers

henceforth

referred

to

as

Mobile

Hosts

-LRB-

MH

-RRB-

communicate

with

other

nodes

in

the

network

in

two

ways

By

plugging

into

a

connection

point

of

the

fixed

wireline

network

for

the

duration

of

communication

By

employing

wireless

communication

In

order

to

support

wireless

communication

for

the

mobile

hosts

the

geographical

area

covered

by

the

network

is

divided

into

smaller

regions

called

cells

Each

cell

has

a

fixed

node

referred

to

as

the

Mobile

Service

Station

-LRB-

MSS

-RRB-

The

MSS

is

connected

to

the

wireline

network

and

can

also

communicate

with

the

mobile

hosts

in

its

cell

using

wireless

channels

A

mobile

host

can

have

wireless

communication

with

any

other

node

in

the

network

fixed

or

mobile

only

through

the

MSS

of

the

cell

in

which

it

is

present

-LSB-

-RSB-

If

a

particular

wireless

channel

is

used

concurrently

by

more

than

one

communication

session

in

the

same

cell

or

in

neighboring

cells

the

signals

of

the

communicating

units

will

interfere

with

each

other

Such

an

interference

is

called

co-channel

interference

However

the

same

wireless

channel

can

be

used

to

support

communication

sessions

in

geographically

separated

cells

such

that

their

signals

do

not

interfere

with

each

other

This

is

known

as

frequency

reuse

and

the

corresponding

cells

are

called

co-channel

cells

The

set

of

neighboring

cells

that

are

in

the

co-channel

interference

range

of

each

other

form

a

cluster

At

any

time

a

channel

can

be

used

to

support

at

most

one

communication

session

in

a

cluster

A

cluster

for

frequency

reuse

is

shown

in

Figure

A

cell

and

its

six

neighbors

each

with

a

different

label

belong

to

a

cluster

Figure

A

cluster

of

interfering

cells

with

co-channel

cells

having

identical

labels

With

the

increasing

popularity

of

mobile

computing

the

demand

for

wireless

communication

channels

is

on

the

rise

However

the

communication

bandwidth

is

limited

Therefore

efficient

allocation

of

wireless

channels

to

support

communication

sessions

is

of

vital

importance

An

efficient

channel

allocation

algorithm

should

have

the

following

properties

minimize

connection

set-up

time

maximize

the

number

of

communication

sessions

that

can

be

supported

simultaneously

ability

to

adapt

to

changing

load

distribution

minimize

computation

and

communication

overheads

for

channel

selection

Centralized

channel

allocation

schemes

have

a

single

point

of

failure

and

are

not

robust

-LSB-

-RSB-

Hence

distributed

channel

allocation

schemes

are

preferred

Such

schemes

should

be

able

to

dynamically

adjust

to

changes

in

load

distribution

Load

in

the

context

of

channel

allocation

stands

for

the

demand

for

channels

to

support

communication

sessions

At

the

heart

of

a

dynamic

distributed

channel

allocation

strategy

is

the

channel

transfer

policy

during

periods

of

load

imbalance

if

a

cell

needs

a

channel

to

support

a

communication

session

and

a

neighbor

has

an

idle

channel

then

it

can

be

borrowed

by

the

cell

provided

the

borrowing

does

not

lead

to

co-channel

interference

However

an

important

issue

that

needs

to

be

addressed

is

whether

channel

transfer/borrowing

will

lead

to

an

overall

improvement

in

system

performance

Indiscriminate

channel

transfers

may

lead

to

thrashing

of

channels

between

neighboring

cells

Channel

transfer

may

not

be

possible

in

all

situations

of

load

imbalance

Several

issues

need

to

be

looked

into

If

a

cell

needs

an

extra

channel

what

is

the

probability

that

load

imbalance

exists

and

a

channel

can

be

borrowed

from

a

neighbor

The

demand

for

channels

may

go

down

in

a

heavily

loaded

cell

and

may

go

up

in

the

neighboring

cells

that

were

previously

lightly

loaded

So

given

that

load

imbalance

occurs

how

long

is

it

expected

to

last

Is

it

possible

to

accomplish

channel

transfer

in

this

period

of

time

How

does

the

velocity

of

mobile

units

affect

the

duration

of

load

imbalance

How

can

velocity

be

accounted

for

or

excluded

from

the

analysis

of

load

imbalance

duration

How

can

the

computation

and

communication

costs

of

channel

transfer

be

minimized

or

amortized

without

sacrificing

the

ability

to

dynamically

adjust

the

distribution

of

allocated

channels

to

meet

changes

in

load

distribution

The

answer

to

all

these

questions

should

determine

the

choice

of

channel

allocation

strategies

and

whether

to

pursue

channel

transfer

aggressively

or

not

A

deterministic

analysis

of

the

temporal

issues

of

channel

demand

is

not

possible

Hence

in

this

paper

we

present

a

probabilistic

analysis

of

the

occurrence

of

load

imbalance

and

the

duration

for

which

such

an

imbalance

is

expected

to

persist

We

propose

strategies

to

account

for

the

varying

degrees

of

mobility

of

the

nodes

We

introduce

the

concept

of

bulk

channel

transfer

to

minimize

channel

transfer

overheads

In

Section

we

describe

the

previous

work

done

in

the

area

of

channel

allocation

Section

compares

the

channel

allocation

problem

with

the

problem

of

load

balancing

in

distributed

systems

The

insights

thus

obtained

are

used

in

Section

for

a

probabilistic

analysis

of

the

potential

for

channel

transfer

and

the

duration

of

the

channel

transfer

window

Node

mobility

is

addressed

in

Section

Section

evaluates

the

feasibility

of

bulk

channel

transfer

while

Section

analyzes

the

relative

strengths

of

channel

transfers

and

channel

borrowing

Finally

the

conclusions

are

presented

in

Section

Previous

Work

Channel

allocation

to

support

cellular

communication

has

been

studied

in

-LSB-

-RSB-

These

algorithms

deal

with

the

allocation

of

channels

to

support

analog

signal

communication

like

voice

The

duration

of

a

communication

session

-LRB-

of

the

order

of

minutes

-RRB-

is

significantly

greater

than

the

time

to

allocate

a

channel

-LRB-

fraction

of

a

second

-RRB-

The

system

size

in

terms

of

the

number

of

cells

and

the

mean

number

of

nodes

per

cell

is

assumed

to

show

little

variation

over

time

However

the

distribution

of

the

nodes

across

the

system

may

change

with

time

Channel

allocation

strategies

for

supporting

microcellular

communication

-LSB-

-RSB-

and

picocellular

communication

-LSB-

-RSB-

have

also

been

studied

Several

algorithms

mentioned

above

employ

need

based

channel

borrowing

from

lightly

loaded

to

heavily

loaded

cells

However

these

algorithms

do

not

consider

the

stochastic

nature

of

imbalance

in

channel

demand

There

is

a

likelihood

that

even

though

there

is

a

load

imbalance

the

duration

of

such

imbalance

is

too

small

for

channel

transfers

to

be

effective

in

the

time

that

it

takes

to

transfer

channel

-LRB-

s

-RRB-

between

cells

the

imbalance

triggering

the

transfer

may

have

disappeared

Hence

there

is

a

need

for

a

thorough

analysis

of

the

stochastic

nature

of

load

imbalance

between

cells

in

a

cluster

A

significant

number

of

channel

allocation

algorithms

proposed

in

the

past

are

centralized

in

nature

A

mobile

host

that

needs

a

channel

sends

a

request

to

its

MSS

The

MSS

forwards

the

request

to

a

central

Mobile

Telecommunication

Switching

Office

-LRB-

MTSO

-RRB-

The

MTSO

has

complete

information

about

the

system

state

It

makes

the

decision

about

channel

allocation

and

conveys

it

to

the

MH

through

the

corresponding

MSS

If

channels

are

not

available

to

support

a

communication

session

the

request

is

dropped

The

centralized

algorithms

are

not

robust

The

MTSO

is

a

single

point

of

failure

If

the

MTSO

goes

down

it

can

bring

down

the

entire

network

-LSB-

-RSB-

Also

during

periods

of

high

load

the

MTSO

and

the

communication

subnet

in

its

vicinity

may

become

a

bottleneck

Therefore

centralized

algorithms

are

not

scalable

Hence

a

distributed

channel

allocation

strategy

is

desirable

In

a

distributed

channel

allocation

algorithm

the

MTSO

is

not

needed

The

MSSs

collectively

make

decisions

about

channel

allocation

Decisions

to

borrow

and

lend

channels

should

be

based

on

the

temporal

fluctuations

in

channel

demand

in

a

cell

and

its

neighbors

Information

about

channel

demand

in

neighboring

cells

can

be

obtained

only

through

messages

and

the

message

propagation

time

between

the

MSSs

over

the

wireline

network

is

finite

but

unpredictable

So

at

any

instant

of

time

an

MSS

does

not

have

the

latest

state

information

about

neighboring

cells

and

has

to

make

channel

allocation/transfer

decisions

based

on

system

information

that

may

be

outdated

The

performance

of

a

distributed

channel

allocation

strategy

will

depend

on

how

outdated

the

state

information

at

the

MSSs

is

A

heavily

loaded

cell

's

decision

to

borrow

a

channel

from

a

neighbor

may

not

always

lead

to

better

utilization

of

resources

if

the

state

of

the

neighbor

has

changed

significantly

since

the

last

time

it

sent

state

information

to

the

cell

Due

to

temporal

changes

in

load

distribution

a

significant

amount

of

time

and

effort

may

be

wasted

transferring

a

channel

from

one

cell

to

another

and

back

resulting

in

channel

thrashing

Dynamic

Channel

Allocation

Lessons

from

Load

Balancing

In

a

mobile

computing

and

communication

system

the

load

distribution

can

exhibit

high

variation

over

time

and

space

For

example

during

the

business

hours

the

downtown

area

may

be

heavily

loaded

During

the

morning

and

evening

commute

periods

the

load

may

be

heavy

along

the

arterial

roads

A

fixed

channel

allocation

strategy

lacks

the

flexibility

to

cope

with

such

wide

fluctuations

in

load

distribution

Dynamic

channel

allocation

strategies

make

decisions

about

channel

distribution

based

on

the

latest

information

available

about

the

existing

load

in

various

cells

An

efficient

channel

allocation

strategy

should

not

only

consider

the

present

load

but

also

the

load

distribution

in

the

recent

past

Based

on

this

information

it

should

be

able

to

project

the

load

distribution

in

the

near

future

A

channel

allocation

strategy

that

takes

into

account

projections

of

future

load

distribution

will

be

able

to

adapt

to

changes

in

load

distribution

quicker

than

the

strategies

that

do

not

This

is

because

the

system

is

prepared

for

the

changes

in

advance

As

long

as

the

predictions

are

within

an

acceptable

accuracy

range

this

strategy

will

work

well

For

example

consider

a

usually

heavily

loaded

cell

Ci

surrounded

by

lightly

loaded

cells

Due

to

channel

transfers

done

in

the

past

Ci

will

have

a

large

number

of

channels

assigned

to

it

The

lightly

loaded

neighbors

will

have

fewer

channels

assigned

to

them

Due

to

statistical

fluctuations

in

the

load

there

may

be

a

short

period

of

time

during

which

Ci

is

lightly

loaded

and

a

neighboring

cell

Cj

is

heavily

loaded

A

naive

dynamic

channel

allocation

strategy

will

try

to

transfer

a

channel

from

Ci

to

Cj

It

is

possible

that

soon

after

the

channel

is

transferred

load

in

Ci

goes

up

and

it

needs

the

channel

that

was

transferred

out

of

it

If

the

channel

is

being

used

in

Cj

the

request

is

denied

in

Ci

Had

the

channel

not

been

transferred

to

Cj

based

on

the

history

and

projections

of

the

future

load

of

Ci

and

Cj

the

channel

request

in

Cj

would

have

been

denied

while

the

request

in

Ci

would

have

been

serviced

Thus

either

way

one

request

would

have

been

denied

However

by

refusing

the

channel

transfer

in

the

first

place

the

channel

transfer

overheads

would

have

been

avoided

For

lightly

loaded

systems

this

saving

may

not

have

much

impact

However

in

highly

loaded

systems

susceptible

to

channel

thrashing

this

overhead

can

be

high

By

making

channel

transfer

decisions

based

on

future

projections

thrashing

can

be

avoided

or

reduced

The

channel

allocation

problem

is

similar

to

the

load

balancing/sharing

problem

in

distributed

systems

-LSB-

-RSB-

The

communication

sessions

between

MHs

and

MSSs

are

like

the

jobs

in

a

distributed

system

The

channels

needed

to

support

the

sessions

are

analogous

to

the

resources

the

jobs

need

to

execute

In

the

load

balancing

problem

typically

if

there

are

more

jobs

than

a

threshold

value

at

a

node

some

of

the

jobs

are

migrated

to

lightly

loaded

nodes

However

in

a

mobile

communication

system

the

problem

is

complicated

due

to

the

following

reasons

The

jobs

-LRB-

communication

sessions

-RRB-

can

not

be

migrated

to

a

cell

with

under-utilized

resources

Instead

resources

-LRB-

channels

-RRB-

have

to

be

migrated

from

lightly

loaded

cells

to

the

cells

where

they

are

needed

A

job

migrates

on

its

own

accord

when

a

communicating

mobile

host

moves

out

of

its

old

cell

into

a

new

cell

Therefore

it

is

possible

that

by

the

time

a

channel

is

transferred

into

a

cell

to

support

a

communication

session

for

a

mobile

host

in

that

cell

the

mobile

host

has

moved

to

another

cell

In

the

load

balancing

problem

the

various

jobs

are

independent

of

each

other

The

only

requirement

for

the

migration

of

jobs

is

the

availability

of

lightly

loaded

nodes

However

in

the

channel

allocation

scenario

co-channel

interference

has

to

be

considered

A

cell

may

not

be

using

all

its

allocated

channels

yet

the

free

channels

of

the

lightly

loaded

cell

can

not

be

transferred

to

a

heavily

loaded

neighboring

cell

because

this

may

lead

to

interference

The

actual

layout

of

the

cells

in

a

region

can

also

affect

the

channel

allocation

decisions

Neighboring

cells

may

overlap

with

each

other

If

a

mobile

node

requesting

a

wireless

channel

lies

in

the

overlapping

region

of

a

heavily

loaded

and

a

lightly

loaded

cell

two

options

are

available

-LRB-

a

-RRB-

the

MSS

of

the

lightly

loaded

cell

may

relinquish

control

over

a

channel

and

the

MSS

of

the

heavily

loaded

cell

may

acquire

control

over

the

channel

and

use

it

to

support

the

session

-LRB-

b

-RRB-

instead

of

channel

transfer

the

mobile

host

connects

to

the

MSS

of

the

lightly

loaded

cell

through

the

available

channel

Even

though

option

-LRB-

a

-RRB-

appears

counter-intuitive

it

may

improve

the

overall

system

performance

for

certain

spatial

distributions

of

the

MHs

the

spatial

distribution

of

the

channel

in

use

in

the

neighboring

cells

and

the

direction

of

movement

of

the

mobile

host

For

example

consider

the

situation

depicted

in

Figure

The

mobile

host

lies

CiCjmh

Figure

Channel

allocation

based

on

direction

of

mobile

host

's

movement

in

the

region

of

overlap

between

cells

Ci

and

Cj

and

it

is

moving

in

the

direction

shown

by

the

arrow

Assuming

Cj

is

highly

loaded

and

Ci

is

lightly

loaded

exercising

option

-LRB-

a

-RRB-

will

be

preferable

This

is

because

of

the

following

reason

Once

a

channel

is

transferred

from

Ci

to

Cj

that

channel

can

be

used

for

supporting

the

communication

of

the

mobile

host

at

least

as

long

as

it

is

in

Cj

If

option

-LRB-

b

-RRB-

is

employed

instead

the

communication

is

controlled

by

the

MSS

of

cell

Ci

In

a

short

while

the

mobile

host

leaves

the

region

of

overlap

and

is

only

in

Cj

As

soon

as

this

happens

a

hand-off

from

Ci

to

Cj

will

be

needed

At

the

same

time

Cj

will

have

to

borrow

a

channel

from

a

lightly

loaded

neighboring

cell

to

support

the

communication

session

Thus

using

option

-LRB-

b

-RRB-

has

the

additional

cost

incurred

in

communication

hand-off

For

load

balancing

in

distributed

systems

the

processing

overhead

and

the

communication

time

needed

for

job

transfer

is

small

as

compared

to

the

job

processing

time

-LSB-

-RSB-

The

same

is

true

for

voice

communication

over

the

cellular

network

where

the

link

set-up

time

is

of

the

order

of

fractions

of

a

second

and

the

subsequent

duration

of

link

usage

is

of

the

order

of

minutes

Hence

in

load

balancing

for

distributed

systems

and

in

telephony

job

migration

and

channel

transfers

respectively

are

carried

out

whenever

an

imbalance

arises

However

in

the

case

of

mobile

computing

there

may

be

a

great

variation

in

the

duration

of

communication

sessions

Typically

the

duration

of

communication

sessions

has

a

bimodal

distribution

The

overheads

involved

in

channel

transfer

may

be

insignificant

compared

to

the

duration

of

the

resultant

communication

session

if

a

large

file

is

to

be

downloaded

or

uploaded

However

sometimes

the

transferred

channel

may

be

used

to

support

a

communication

session

of

a

fraction

of

a

second

during

which

a

short

burst

of

control

signals

is

sent

or

received

Thus

the

overheads

of

channel

transfer

can

significantly

affect

system

performance

Indiscriminate

channel

transfers

without

considering

the

impact

of

transfer

overheads

can

adversely

affect

the

overall

performance

of

the

system

Temporal

Analysis

of

Load

Imbalance

As

described

in

Section

there

are

several

similarities

between

the

problems

of

dynamic

distributed

channel

allocation

for

mobile

computing

and

load

balancing

for

distributed

systems

However

the

differences

mentioned

above

imply

that

load

balancing

strategies

proposed

in

the

past

can

not

be

directly

used

for

channel

allocation

Hence

the

issues

enumerated

in

Section

need

to

be

analyzed

to

obtain

guidelines

for

the

design

of

dynamic

distributed

channel

allocation

strategies

Channel

Allocation

Policy

Let

us

assume

a

C-cell

cluster

system

with

a

total

of

N

communication

channels

Assuming

that

initially

the

channels

are

equally

distributed

among

the

C

cells

the

number

of

channels

available

in

every

cluster

is

equal

to

N

C

n.

Mutually

disjoint

sets

of

channels

are

allocated

to

different

cells

in

the

cluster

The

channels

allocated

to

a

cell

will

henceforth

be

referred

to

as

the

nominal

channels

of

the

cell

A

channel

can

be

used

concurrently

to

support

communication

sessions

in

two

cells

as

long

as

these

cells

are

separated

by

at

least

their

reuse

distance

Figure

shows

a

cluster

system

where

channels

can

be

reused

at

a

distance

of

cells

We

assume

that

all

the

channels

in

the

Spectrum

can

be

completely

ordered

based

on

their

frequency

band

As

described

in

-LSB-

-RSB-

we

assume

that

in

order

to

support

new

communication

sessions

in

a

cell

channel

assignment

is

done

in

ascending

order

from

among

the

set

of

free

nominal

channels

allocated

to

the

cell

and

channel

borrowing

is

done

in

descending

order

of

free

nominal

channels

of

the

neighbors

Also

when

a

communication

session

terminates

channels

are

reallocated

according

to

the

following

rules

When

a

nominal

channel

of

a

cell

becomes

free

and

there

is

a

communication

session

in

progress

in

the

cell

using

a

higher

order

channel

-LRB-

nominal

or

borrowed

-RRB-

then

the

communication

session

is

switched

to

the

newly

freed

lower

order

nominal

channel

When

a

session

using

a

cell

's

nominal

channel

terminates

and

there

is

another

session

in

progress

using

a

borrowed

channel

the

ongoing

session

is

switched

to

the

newly

freed

nominal

channel

of

the

cell

The

borrowed

channel

is

returned

to

its

owner

cell

When

a

session

using

a

borrowed

channel

terminates

and

there

is

another

session

using

a

lower

order

borrowed

channel

then

the

ongoing

session

is

switched

to

the

newly

freed

higher

order

borrowed

channel

The

lower

order

borrowed

channel

is

returned

to

its

owner

cell

As

a

result

of

this

strategy

we

can

assume

the

following

ffl

all

the

free

channels

in

a

cell

are

clustered

towards

the

higher

end

of

the

spectrum

ffl

between

a

pair

of

co-channel

cells

Ci

and

C

i

if

their

sets

of

free

nominal

channels

are

F

reei

and

F

then

F

reei

F

or

F

F

reei

Potential

for

Channel

Transfer

The

need

to

borrow

a

channel

arises

in

cell

Ci

if

Ci

has

all

its

nominal

channels

in

use

and

Channel

allocation

request

-LRB-

s

-RRB-

are

pending

in

Ci

A

channel

can

be

borrowed

by

Ci

from

a

neighboring

cell

if

the

neighbor

has

a

free

nominal

channel

such

that

its

use

in

Ci

will

not

lead

to

co-channel

interference

When

a

channel

is

borrowed

by

Ci

from

Cj

two

co-channel

cells

of

Cj

closest

to

Ci

can

not

use

the

same

channel

concurrently

Also

this

channel

should

be

free

in

those

co-channel

cells

at

the

time

of

borrowing

For

example

in

Figure

suppose

cell

-LRB-

highlighted

in

the

figure

-RRB-

needs

a

channel

and

cell

to

its

left

has

a

free

channel

The

channel

can

not

be

borrowed

by

cell

if

the

same

channel

is

being

used

in

any

of

the

two

co-channel

cells

of

cell

close

to

cell

These

cells

are

also

highlighted

in

the

figure

The

probability

of

occurrence

of

a

situation

where

channel

transfer

is

possible

is

referred

to

as

channel

transfer

potential

Unrestrained

channel

transfers

can

lead

to

channel

starvation

in

some

cells

Hence

a

threshold

is

maintained

by

each

cell

A

cell

will

not

lend

channels

to

its

neighbor

-LRB-

s

-RRB-

if

the

number

of

its

free

nominal

channels

is

below

the

threshold

Before

we

proceed

with

a

probabilistic

analysis

of

the

potential

for

channel

transfer

a

few

important

terms

need

to

be

defined

First

passage

time

has

been

defined

in

-LSB-

-RSB-

as

follows

Definition

-LRB-

First

Passage

Time

-RRB-

Let

fX

-LRB-

t

-RRB-

t

g

be

a

stationary

stochastic

process

with

state

space

consisting

of

the

non-negative

integers

Let

m

and

n

be

two

distinct

nonnegative

integers

The

First

Passage

Time

between

states

m

and

n

denoted

by

T

-LRB-

m

n

-RRB-

is

the

following

T

-LRB-

m

n

-RRB-

infft

X

-LRB-

t

-RRB-

n

X

-LRB-

-RRB-

mg

if

m

n

otherwise

So

T

-LRB-

m

n

-RRB-

is

a

random

variable

which

measures

the

minimum

amount

of

time

to

make

a

transition

from

state

m

to

state

n

As

X

-LRB-

t

-RRB-

is

a

stationary

stochastic

process

translating

the

above

events

by

a

fixed

amount

of

time

has

no

effect

upon

the

probability

distribution

of

T

-LRB-

m

n

-RRB-

Definition

-LRB-

High

Load

-RRB-

A

cell

is

said

to

be

heavily

loaded

if

all

its

nominal

channels

are

in

use

and

there

is

at

least

one

pending

channel

request

Definition

-LRB-

Low

Load

-RRB-

A

cell

is

said

to

be

lightly

loaded

if

at

least

threshold

number

of

its

nominal

channels

are

free

A

cell

that

is

neither

heavily

loaded

nor

lightly

loaded

will

henceforth

be

referred

to

be

in

a

state

of

normality

For

simplicity

we

assume

that

threshold

is

equal

to

Otherwise

the

analysis

that

follows

becomes

intractable

With

threshold

set

to

one

a

cell

is

said

to

be

lightly

loaded

if

it

has

at

least

one

nominal

channel

free

Probability

Analysis

Let

the

arrival

rate

of

channel

requests

for

each

cell

be

We

assume

Poisson

distribution

for

the

arrival

of

channel

allocation

requests

-LRB-

from

the

MHs

-RRB-

at

the

MSS

in

each

cell

We

also

assume

that

once

a

channel

is

allocated

to

a

communication

session

the

channel

usage

time

for

the

session

is

exponentially

distributed

with

mean

In

order

to

satisfy

a

local

channel

allocation

request

in

a

cell

any

of

the

available

nominal

channels

can

be

used

Hence

each

cell

corresponds

to

an

M

M

n

queue

The

n

nominal

channels

of

the

cell

correspond

to

the

n

servers

for

the

queue

At

most

n

communication

sessions

can

be

simultaneously

supported

in

the

cell

if

idle

channels

are

not

borrowed

from

neighboring

cells

As

channels

can

be

transferred

between

cells

the

queues

interact

with

each

other

Hence

in

the

context

of

the

channel

allocation

problem

the

mobile

computing

environment

corresponds

to

a

network

of

M

M

n

queues

The

state

transitions

due

to

the

arrival

of

channel

requests

and

completion

of

communication

sessions

can

be

modeled

as

the

birth-death

system

shown

in

Figure

-LRB-

-RRB-

nnnn

Figure

State

transition

diagram

for

a

cell

However

there

are

some

important

differences

between

an

ideal

network

of

M

M

n

queues

and

the

channel

allocation

scheme

of

the

mobile

computing

environment

Some

of

the

important

differences

are

as

follows

An

M

M

n

queue

has

an

infinite

buffer

space

When

jobs

arrive

in

the

system

and

find

all

the

servers

busy

they

get

queued

up

until

they

are

serviced

However

channel

allocation

decisions

can

not

be

delayed

indefinitely

These

requests

have

to

be

processed

in

real

time

The

maximum

duration

of

time

a

channel

allocation

request

will

be

queued

up

at

an

MSS

is

equal

to

a

prespecified

period

of

time

called

Ttransfer

This

is

a

parameter

of

the

channel

allocation

algorithm

A

channel

allocation

request

will

time-out

if

the

request

is

not

satisfied

within

this

period

of

time

Unlike

a

network

of

M

M

n

queues

borrowing

of

channels

is

restricted

by

the

interference

constraints

of

the

two

co-channel

cells

of

the

lending

cell

that

are

closest

to

the

borrowing

cell

The

MSS

will

start

its

effort

to

borrow

channels

as

soon

as

the

highly

loaded

state

is

reached

Due

to

ordered

allocation

of

channels

-LSB-

-RSB-

the

free

channel

sets

in

the

cell

lending

a

channel

and

in

its

co-channel

cells

closest

to

the

borrower

are

related

by

the

relation

Hence

the

interference

constraints

of

the

co-channel

cells

during

borrowing

-LRB-

item

above

-RRB-

can

be

ignored

The

steady-state

probability

of

state

x

represented

by

P

-LRB-

x

-RRB-

can

be

expressed

as

-LSB-

-RSB-

P

-LRB-

x

-RRB-

P

-LRB-

-RRB-

-LRB-

nae

-RRB-

x

x

if

x

n

P

-LRB-

-RRB-

-LRB-

ae

-RRB-

xnn

n

if

x

n

where

ae

n

As

the

probabilities

must

sum

to

the

probability

of

state

P

-LRB-

-RRB-

is

equal

to

P

-LRB-

-RRB-

-LSB-

n

X

x

-LRB-

nae

-RRB-

x

x

-LRB-

-LRB-

nae

-RRB-

n

n

-RRB-

-LRB-

ae

-RRB-

-RSB-

Hence

channel

transfer

potential

for

a

heavily

loaded

cell

Ci

-LRB-

CTPi

-RRB-

with

respect

to

its

neighbors

in

the

cluster

can

be

formally

defined

as

CTPi

probability

that

cell

Ci

can

borrow

a

channel

P

-LRB-

Ci

has

all

its

n

channels

in

use

and

at

least

one

neighbor

Cj

has

threshold

number

of

free

nominal

channels

and

two

co-channel

cells

of

Cj

closest

to

Ci

have

threshold

free

nominal

channels

-RRB-

As

free

nominal

channels

of

one

cell

are

assumed

to

form

a

subset

of

the

set

of

free

channels

of

their

co-channel

cells

or

vice-versa

and

threshold

is

equal

to

the

channel

transfer

potential

can

be

expressed

as

follows

CTPi

probability

that

cell

Ci

can

borrow

a

channel

P

-LRB-

Ci

has

all

its

n

channels

in

use

and

at

least

one

neighbor

Cj

has

free

nominal

channel

and

two

co-channel

cells

of

Cj

closest

to

Ci

have

free

nominal

channel

-RRB-

The

probability

of

all

the

n

nominal

channels

of

a

cell

being

busy

i.e

the

probability

that

newly

arriving

channel

requests

will

be

queued

up

is

denoted

by

P

nqueue

P

nqueue

can

be

expressed

as

-LSB-

-RSB-

P

nqueue

-LRB-

-LRB-

nae

-RRB-

n

n

-RRB-

-LRB-

ae

-RRB-

Pn

x

-LRB-

nae

-RRB-

x

x

-LRB-

-LRB-

nae

-RRB-

n

n

-RRB-

-LRB-

ae

-RRB-

As

the

arrival

of

channel

allocation

requests

and

their

service

at

Ci

and

its

neighbors

are

independent

of

each

other

CTP

for

Ci

can

be

expressed

as

CTPi

P

nqueue

P

-LRB-

at

least

one

neighbor

and

its

co-channel

cells

closest

to

Ci

have

at

least

one

free

channel

-RRB-

For

convenience

let

the

C

neighbors

of

Ci

in

a

C-cell

cluster

system

be

numbered

from

to

C

and

ni

number

of

nominal

channels

allocated

to

ith

neighbor

number

of

nominal

channels

allocated

to

one

co-channel

cell

of

the

ith

neighbor

closest

to

Ci

number

of

nominal

channels

for

the

other

co-channel

cell

of

the

ith

neighbor

closest

to

Ci

Therefore

the

probability

that

a

channel

is

available

for

borrowing

from

the

ith

neighbor

out

of

the

C

neighbors

in

a

C-cell

cluster

can

be

expressed

as

Pi

-LRB-

Pni

queue

-RRB-

-LRB-

queue

-RRB-

-LRB-

queue

-RRB-

The

probability

that

a

channel

is

available

for

borrowing

from

at

least

one

of

the

C

neighbors

in

the

cluster

is

a

union

of

probabilities

of

a

channel

being

available

for

borrowing

in

each

of

these

C

cells

From

set

theory

we

know

that

if

P

-LRB-

A

-RRB-

and

P

-LRB-

B

-RRB-

are

the

probabilities

of

occurrence

of

events

A

and

B

respectively

then

P

-LRB-

A

-LSB-

B

-RRB-

P

-LRB-

A

-RRB-

P

-LRB-

B

-RRB-

P

-LRB-

A

B

-RRB-

Also

if

events

A

and

B

are

independent

of

each

other

P

-LRB-

A

B

-RRB-

P

-LRB-

A

-RRB-

P

-LRB-

B

-RRB-

Applying

these

results

to

the

case

of

channel

availability

in

the

C

neighbors

of

Ci

channel

transfer

potential

for

Ci

can

be

expressed

as

CTPi

P

nqueue

-LRB-

C

X

X

Y

X

Y

-LRB-

-RRB-

C

PC

-RRB-

where

the

subscripts

etc.

range

from

to

C

The

expression

for

channel

transfer

potential

derived

above

gives

an

upper

bound

on

the

probability

that

a

channel

transfer

can

be

carried

out

to

support

communication

sessions

This

is

because

we

have

made

a

simplifying

assumption

about

the

free

nominal

channels

of

co-channel

cells

being

related

to

each

other

by

the

function

The

actual

transfer

potential

for

channel

transfer

may

be

lower

than

the

derived

value

However

CTP

gives

us

a

good

idea

of

whether

channels

are

available

for

transfer

Channel

Transfer

Window

Even

if

channels

are

available

for

transfer

at

a

given

instant

of

time

as

indicated

by

the

channel

transfer

potential

all

attempts

at

channel

transfer

may

not

be

productive

and

may

lead

to

channel

thrashing

thus

degrading

the

system

performance

This

leads

to

two

interesting

questions

-LRB-

i

-RRB-

Given

that

there

exists

a

potential

for

channel

transfer

how

much

time

is

available

to

carry

out

such

a

transfer

before

the

system

state

changes

rendering

such

a

transfer

unnecessary

or

counter

productive

-LRB-

ii

-RRB-

Is

it

possible

to

transfer

more

than

one

idle

channel

from

lightly

loaded

neighbors

to

a

heavily

loaded

cell

If

so

what

are

the

constraints

and

what

are

the

advantages

Let

there

exist

a

pair

of

neighboring

cells

Ci

and

Cj

such

that

Ci

is

highly

loaded

while

Cj

is

lightly

loaded

Due

to

temporal

fluctuations

in

channel

demand

it

is

possible

that

after

a

period

of

time

Ci

no

longer

needs

to

borrow

the

channel

and/or

Cj

can

no

longer

lend

the

channel

Ci

will

no

longer

need

to

borrow

channel

-LRB-

s

-RRB-

if

it

makes

a

transition

from

an

overloaded

state

with

x

pending

channel

requests

to

a

state

of

normality

with

no

pending

requests

Let

T

-LRB-

x

-RRB-

be

the

corresponding

first

passage

time

from

x

pending

channel

requests

to

no

pending

requests

in

cell

Ci

Let

yj

be

number

of

free

channels

in

Ci

's

neighboring

cell

Cj

and

be

the

number

of

free

channels

in

the

two

co-channel

cells

of

Cj

close

to

Ci

If

a

nominal

channel

of

Cj

is

not

being

used

in

Cj

but

is

being

used

to

support

a

communication

session

in

at

least

one

of

Cj

's

co-channel

cells

close

to

Ci

the

channel

can

not

be

transferred

from

Cj

to

Ci

Such

a

transfer

will

lead

to

co-channel

interference

as

already

described

This

interference

characteristic

along

with

the

fact

that

the

sets

of

free

channels

in

co-channel

cells

are

related

by

the

relation

implies

that

the

number

of

channels

that

Ci

can

borrow

from

Cj

is

upper

bounded

by

min

-LRB-

yj

-RRB-

Let

T

-LRB-

yj

-RRB-

be

the

first

passage

time

from

a

state

of

min

-LRB-

yj

-RRB-

available

channels

in

cell

Cj

-LRB-

for

lending

to

Ci

-RRB-

to

a

state

of

no

available

channels

at

Cj

for

lending

to

Ci

Therefore

the

time

window

during

which

a

channel

can

be

borrowed

by

Ci

from

Cj

depends

on

the

first

passage

time

of

the

borrowing

cell

-LRB-

Ci

-RRB-

from

a

highly

loaded

state

to

a

state

of

normality

and

the

first

passage

time

of

the

lending

cell

-LRB-

Cj

-RRB-

and

its

two

co-channel

cells

nearest

to

Ci

from

a

lightly

loaded

state

to

a

state

of

normality

Wmax

the

upper

bound

on

the

channel

transfer

window

is

a

random

variable

that

can

be

expressed

as

follows

Wmax

-LRB-

x

yc

-RRB-

min

-LRB-

T

-LRB-

x

-RRB-

maxallCj

-LRB-

T

-LRB-

yj

-RRB-

-RRB-

-RRB-

If

min

-LRB-

yj

-RRB-

then

T

-LRB-

yj

-RRB-

Similarly

Wmin

the

lower

bound

on

the

channel

transfer

window

is

a

random

variable

that

can

be

expressed

as

follows

Wmin

-LRB-

x

yc

-RRB-

min

-LRB-

T

-LRB-

x

-RRB-

T

-LRB-

-RRB-

T

-LRB-

yc

-RRB-

-RRB-

The

probability

distribution

function

HWmax

of

the

upper

bound

on

CTW

can

be

expressed

as

HWmax

-LRB-

t

x

yc

-RRB-

P

-LRB-

Wmax

-LRB-

x

yc

-RRB-

t

-RRB-

for

t

HWmin

can

be

expressed

in

a

similar

fashion

Henceforth

we

shall

use

HW

to

refer

to

the

probability

distribution

function

of

the

channel

transfer

window

when

we

do

not

wish

to

distinguish

between

HWmax

and

HWmin

Similarly

W

will

be

used

to

refer

to

the

channel

transfer

window

when

we

do

not

wish

to

distinguish

between

Wmax

and

Wmin

The

exact

expression

for

HW

depends

upon

the

following

arrival

rates

of

channel

allocation

requests

at

the

cells

in

a

cluster

and

their

closest

co-channel

cells

distribution

of

channel

usage

time

-LRB-

the

length

of

a

communication

session

-RRB-

and

the

queuing

model

for

channel

allocation

requests

in

each

cell

However

irrespective

of

the

arrival

rate

and

channel

usage

time

distribution

the

empirical

formula

for

HW

can

be

expressed

as

follows

HW

-LRB-

t

x

yc

-RRB-

P

-LRB-

W

-LRB-

x

yc

-RRB-

t

-RRB-

for

t

Let

us

assume

that

in

cell

Ci

and

all

its

neighbors

Cj

the

transitions

from

high

load

state

to

a

state

of

normality

and

from

low

load

state

to

a

state

of

normality

is

independent

Then

the

probability

distribution

function

for

the

lower

bound

on

CTW

can

be

expressed

as

HWmin

-LRB-

t

x

yc

-RRB-

-LSB-

-LRB-

-LRB-

t

x

-RRB-

-RRB-

-LRB-

-LRB-

t

-RRB-

-RRB-

-LRB-

-LRB-

t

yc

-RRB-

-RRB-

-LRB-

-LRB-

t

-RRB-

-RRB-

-LRB-

-LRB-

t

-RRB-

-RRB-

-LRB-

-LRB-

t

-RRB-

-RRB-

-LRB-

-LRB-

t

-RRB-

-RRB-

-RSB-

where

F

-LRB-

t

x

-RRB-

probability

distribution

function

of

the

first

passage

time

from

x

pending

channel

allocation

requests

to

no

pending

requests

in

cell

Ci

i.e.

P

-LRB-

T

-LRB-

x

-RRB-

t

for

t

G

-LRB-

t

yj

-RRB-

probability

distribution

function

of

the

first

passage

time

from

yj

free

channels

in

Cj

to

no

free

channel

in

Cj

G

-LRB-

t

-RRB-

and

G

-LRB-

t

-RRB-

are

the

corresponding

probability

distribution

functions

for

the

two

co-channel

cells

of

Cj

closest

to

Ci

It

is

to

be

noted

that

if

x

then

F

-LRB-

t

x

-RRB-

and

if

yj

then

G

-LRB-

t

yj

-RRB-

for

t

If

x

and

yj

then

F

-LRB-

t

x

-RRB-

and

G

-LRB-

t

yj

-RRB-

are

undefined

respectively

The

probability

distribution

function

for

the

upper

bound

on

CTW

can

be

expressed

as

HWmax

-LRB-

t

x

yc

-RRB-

-LSB-

-LRB-

-LRB-

t

x

-RRB-

-RRB-

-LRB-

QC

j

-LRB-

G

-LRB-

t

yj

-RRB-

G

-LRB-

t

-RRB-

G

-LRB-

t

-RRB-

-RRB-

-RRB-

-RSB-

The

probability

distribution

function

of

the

Channel

Transfer

Window

makes

no

assumptions

about

the

arrival

rate

and

service

rate

and

distributions

of

channel

requests

at

the

cells

Ci

and

Cj

So

two

neighboring

cells

can

have

different

number

of

channels

allocated

to

them

and

different

arrival

rates

Still

the

expression

derived

above

will

continue

to

hold

Solving

for

probability

distribution

of

first

passage

times

As

already

mentioned

F

-LRB-

t

x

-RRB-

is

the

probability

distribution

of

the

first

passage

time

from

the

highly

loaded

state

of

x

pending

channel

allocation

requests

to

a

state

where

no

requests

are

pending

Let

F

-LRB-

t

p

q

-RRB-

P

-LRB-

T

-LRB-

p

q

-RRB-

t

-RRB-

Then

from

-LSB-

-RSB-

the

following

expression

is

obtained

F

-LRB-

t

p

q

-RRB-

>=p

Z

t

F

-LRB-

t

x

p

q

-RRB-

e

-LRB-

>=p

p

-RRB-

xdx

p

Z

t

F

-LRB-

t

x

p

q

-RRB-

e

-LRB-

>=p

p

-RRB-

xdx

where

>=p

is

the

arrival

rate

of

channel

allocation

requests

and

p

is

the

rate

at

which

communication

sessions

are

completed

in

a

cell

in

state

p

During

the

downward

transitions

from

the

heavily

loaded

state

p

towards

the

state

of

normality

-LRB-

no

pending

requests

-RRB-

all

the

n

nominal

channels

allocated

to

a

cell

are

in

use

Hence

p

n

for

p

x

where

is

equal

to

the

reciprocal

of

the

mean

duration

of

communication

sessions

-LRB-

request

service

rate

for

a

one

channel

system

-RRB-

During

the

time

that

a

heavily

loaded

cell

is

making

downward

jumps

new

channel

allocation

requests

continue

to

arrive

at

the

MSS

from

the

MHs

in

the

cell

at

a

rate

Hence

>=p

for

p

x.

For

upward

transitions

from

a

lightly

loaded

state

p

-LRB-

the

integer

denotes

the

number

of

free

nominal

channels

in

the

cell

-RRB-

towards

the

state

of

normality

>=p

and

p

-LRB-

n

p

-RRB-

for

p

y

Once

again

n

is

the

number

of

nominal

channels

assigned

to

the

cell

We

assume

that

the

arrival

of

channel

allocation

requests

and

completion

of

communication

sessions

in

a

cell

constitute

a

birth-death

process

Thus

the

upward

and

downward

transitions

occur

one

step

at

a

time

For

such

a

system

the

first

passage

time

from

state

p

to

state

q

can

be

expressed

as

T

-LRB-

p

q

-RRB-

Pq

k

=p

T

-LRB-

k

k

-RRB-

if

p

q

Pp

k

q

T

-LRB-

k

k

-RRB-

if

p

q

Let

Fp

q

-LRB-

s

-RRB-

be

the

Laplace-Stieltjes

Transform

-LRB-

LST

-RRB-

of

F

-LRB-

t

p

q

-RRB-

Then

taking

the

LST

on

both

sides

of

the

equation

for

F

-LRB-

t

p

q

-RRB-

using

the

convolution

property

-LSB-

-RSB-

Fp

q

-LRB-

s

-RRB-

can

be

written

as

Fp

q

-LRB-

s

-RRB-

>=p

Fp

q

-LRB-

s

-RRB-

p

Fp

q

-LRB-

s

-RRB-

s

>=p

p

Fp

q

-LRB-

s

-RRB-

n

Fp

q

-LRB-

s

-RRB-

s

n

Similarly

the

LST

Gp

q

-LRB-

s

-RRB-

of

G

-LRB-

t

p

q

-RRB-

-LRB-

the

probability

distribution

of

the

first

passage

time

of

upward

transition

from

a

state

with

p

free

channels

to

a

state

with

q

free

channels

p

q

-RRB-

can

be

represented

as

Gp

q

-LRB-

s

-RRB-

>=p

Gp

q

-LRB-

s

-RRB-

p

Gp

q

-LRB-

s

-RRB-

s

>=p

p

Gp

q

-LRB-

s

-RRB-

-LRB-

n

p

-RRB-

Gp

q

-LRB-

s

-RRB-

s

-LRB-

n

p

-RRB-

It

is

to

be

noted

that

the

terms

in

the

numerator

have

been

switched

as

compared

to

Fp

q

-LRB-

s

-RRB-

This

is

because

for

the

upward

transitions

p

and

q

in

Gp

q

-LRB-

s

-RRB-

denote

the

number

of

free

channels

An

increase

in

the

values

of

p

and

q

corresponds

to

a

decrease

in

the

number

of

communication

sessions

in

progress

and

vice-versa

This

is

opposite

from

the

meaning

of

p

and

q

for

downward

transitions

As

the

individual

T

-LRB-

k

k

-RRB-

and

T

-LRB-

k

k

-RRB-

values

exhibit

the

Markovian

property

and

are

independent

of

each

other

we

have

Fp

q

-LRB-

s

-RRB-

Qp

k

q

Fk

k

-LRB-

s

-RRB-

where

p

q

Gp

q

-LRB-

s

-RRB-

Qp

k

q

Gk

k

-LRB-

s

-RRB-

where

p

q

As

>=p

and

p

remain

unchanged

during

downward

transitions

from

a

highly

loaded

state

towards

the

state

of

normality

Fk

k

-LRB-

s

-RRB-

-LRB-

s

-RRB-

Hence

Fx

-LRB-

s

-RRB-

-LRB-

-LRB-

s

-RRB-

-RRB-

x

As

the

birth

rates

and

death

rates

remain

constant

during

the

downward

transitions

from

the

heavily

loaded

state

towards

the

state

of

normality

from

-LSB-

-RSB-

the

following

expression

of

-LRB-

s

-RRB-

is

obtained

-LRB-

s

-RRB-

s

n

q

-LRB-

s

n

-RRB-

Using

this

expression

Fx

can

be

computed

On

the

other

hand

Gp

q

-LRB-

s

-RRB-

can

be

recursively

expressed

in

terms

of

Gp

q

-LRB-

s

-RRB-

and

Gp

q

-LRB-

s

-RRB-

Hence

to

solve

the

recursive

expression

we

need

values

of

the

two

boundary

terms

of

the

recursion

namely

-LRB-

s

-RRB-

and

-LRB-

s

-RRB-

As

described

earlier

G

-LRB-

t

-RRB-

for

t

Hence

it

is

a

unit

step

function

for

t

The

corresponding

LST

-LRB-

s

-RRB-

is

equal

to

R

e?stdG

-LRB-

t

-RRB-

Hence

-LRB-

s

-RRB-

For

a

birth-death

system

the

time

until

the

next

transition

does

not

stochastically

depend

on

the

transition

type

When

a

queue

is

in

state

p

the

probability

that

time

until

the

next

transition

is

greater

than

t

is

equal

to

e

-LRB-

>=p

p

-RRB-

t

where

>=p

is

the

birth

rate

and

p

is

the

death

rate

in

state

p

-LSB-

-RSB-

When

one

nominal

channel

out

of

n

is

free

p

-LRB-

n

-RRB-

and

>=p

Hence

G

-LRB-

t

-RRB-

P

-LRB-

time

until

next

transition

is

greater

than

t

-RRB-

which

is

equal

to

e

-LRB-

>=p

p

-RRB-

t

e

-LRB-

-LRB-

n

-RRB-

-RRB-

t.

Taking

the

LST

we

get

-LRB-

s

-RRB-

-LRB-

n

-RRB-

-LRB-

n

-RRB-

s

G

-LRB-

t

-RRB-

been

equal

to

for

t

only

-LRB-

excluding

the

time

instant

t

-LRB-

s

-RRB-

would

have

been

equal

to

Look-up

tables

can

be

used

to

obtain

the

inverse

transforms

of

LST

to

compute

the

values

of

F

-LRB-

t

x

-RRB-

and

G

-LRB-

t

y

-RRB-

for

the

appropriate

values

of

x

and

y

The

values

so

obtained

can

then

be

plugged

in

the

expressions

for

HWmax

and

HWmin

the

upper

and

lower

bounds

on

the

probability

distribution

of

the

Channel

Transfer

Window

Alternatively

LSTs

of

both

sides

of

the

equation

for

HWmax

and

HWmin

can

taken

This

will

give

us

expressions

of

HWmax

and

HWmin

in

terms

of

Fx

-LRB-

s

-RRB-

and

Gy

-LRB-

s

-RRB-

Then

look-up

tables

can

be

used

to

obtain

the

inverse

transforms

of

HWmax

and

HWmin

giving

HWmax

and

HWmin

respectively

Feasibility

of

Channel

Transfers

Having

determined

the

potential

for

channel

transfer

and

the

probability

distribution

of

the

Channel

Transfer

Window

we

need

to

determine

if

channel

borrowing

is

feasible

under

situations

of

load

imbalance

The

ability

to

transfer

a

channel

from

one

cell

to

its

neighbor

depends

on

the

relation

between

the

time

needed

to

transfer

a

channel

and

the

expected

duration

of

the

Channel

Transfer

Window

The

time

needed

to

transfer

channel

-LRB-

s

-RRB-

is

dependent

on

the

inter-MSS

network

The

neighboring

MSSs

are

connected

to

each

other

by

a

wired

network

When

a

cell

Ci

needs

to

borrow

a

channel

from

a

neighbor

Ci

has

to

broadcast

a

probe

to

all

the

neighboring

MSSs

Only

when

a

neighboring

MSS

responds

to

the

probe

indicating

that

it

has

a

channel

to

lend

can

the

transfer

take

place

Therefore

the

time

needed

for

the

transfer

has

three

components

time

taken

for

probes

to

return

from

neighboring

cells

indicating

their

channel

load

at

the

time

of

responding

to

the

probe

time

for

Ci

to

determine

a

neighbor

Cj

from

which

it

can

borrow

a

channel

and

send

a

request

time

for

Cj

to

lend

a

channel

to

Ci

in

response

to

the

request

Let

us

denote

the

sum

of

the

three

components

as

If

Ttransfer

channel

transfer

is

not

possible

because

all

channel

allocation

decisions

have

to

be

made

within

Ttransfer

amount

of

time

If

is

less

than

or

equal

to

Ttransfer

and

greater

than

the

Channel

Transfer

Window

channel

transfer

is

either

not

needed

or

not

possible

or

both

In

such

situations

attempts

to

transfer

channel

-LRB-

s

-RRB-

are

likely

to

degrade

the

overall

performance

of

the

system

Hence

channel

transfers

should

be

attempted

only

if

is

less

than

both

the

Channel

Transfer

Window

and

Ttransfer

However

as

the

Channel

Transfer

Window

is

a

random

variable

the

condition

can

be

expressed

as

follows

Given

that

Ttransfer

transfer

a

channel

from

a

lightly

loaded

cell

Cj

to

a

highly

loaded

neighbor

Ci

only

if

the

probability

of

the

Channel

Transfer

Window

between

Ci

and

Cj

W

being

greater

than

is

above

a

threshold

value

where

The

threshold

can

be

selected

as

a

system

tuning

parameter

For

aggressive

channel

transfers

between

neighbors

the

condition

can

be

further

refined

to

be

transfer

a

channel

if

P

-LRB-

Wmin

-RRB-

A

more

conservative

policy

would

attempt

transfers

if

P

-LRB-

Wmax

-RRB-

Node

Mobility

and

Channel

Transfer

Window

As

described

in

Section

the

duration

of

the

Channel

Transfer

Window

depends

on

the

first

passage

times

of

the

highly

loaded

cell

and

its

lightly

loaded

neighbors

to

their

respective

states

of

normality

Besides

the

arrival

rate

of

channel

requests

and

expected

duration

of

a

communication

session

the

channel

demand

-LRB-

load

-RRB-

in

a

cell

depends

on

the

rate

of

migration

of

partially

completed

communication

sessions

into/out

of

cells

due

to

the

mobility

of

the

communicating

mobile

nodes

Hence

the

intrinsic

mobility

of

the

communication

sessions

adds

a

new

dimension

to

the

problem

of

determining

the

probability

distribution

of

the

Channel

Transfer

Window

The

situation

is

complicated

by

the

fact

that

different

mobile

units

move

at

different

speeds

there

is

an

order

of

magnitude

difference

between

the

velocity

of

personal

data

assistants

-LRB-

PDAs

-RRB-

carried

by

pedestrians

and

their

counterparts

inside

cars

Moreover

in

heavily

builtup

urban

areas

the

Channel

Transfer

Window

and

handoff

time

for

communication

sessions

can

be

extremely

small

when

a

mobile

unit

turns

around

a

building

block

from

one

street

to

another

The

shadowing

effects

of

buildings

can

sometimes

require

rapid

migration

of

a

communication

session

from

one

cell

to

another

even

for

a

low

speed

mobile

unit

The

problem

can

be

handled

in

two

possible

ways

-LRB-

a

-RRB-

analyzing

the

velocity

of

the

mobile

units

and

its

impact

on

the

first

passage

times

between

high

and

low

load

conditions

of

the

cells

-LRB-

b

-RRB-

minimizing

the

impact

of

node

mobility

on

the

Channel

Transfer

Window

by

appropriate

cell

planning

Constantly

monitoring

the

velocity

of

all

the

mobile

units

in

a

mobile

computing

system

and

predicting

their

future

locations

is

difficult

It

will

get

more

difficult

in

the

future

because

the

number

of

mobile

units

is

increasing

rapidly

Hence

the

second

option

of

cell

planning

looks

more

promising

Umbrella

cells

overlaying

smaller

microcells

in

urban

areas

-LSB-

-RSB-

are

one

solution

Faster

moving

units

can

be

connected

to

the

umbrella

cells

and

slower

moving

units

can

be

connected

to

the

smaller

microcells

Sung

and

Wong

-LSB-

-RSB-

extend

the

solution

by

proposing

a

hierarchical

organization

of

cells

and

MSSs

Equal

sized

cells

are

grouped

into

layers

with

cell

size

increasing

from

lower

layers

to

higher

layers

The

channel

demands

are

partitioned

into

categories

depending

on

the

velocity

of

the

mobile

unit

from

which

they

emanate

ffl

Channel

requests

from

slow

moving

mobile

units

are

handled

by

smaller

cells

and

those

from

faster

mobile

units

are

handled

by

larger

cells

ffl

As

long

as

the

ratio

between

the

velocity

of

the

mobile

unit

and

the

diameter

of

the

cell

handling

the

mobile

unit

's

communication

is

within

a

prespecified

range

the

impact

of

mobility

on

the

Channel

Transfer

Window

is

small

and

can

be

ignored

from

the

calculations

If

the

ratio

goes

above

or

below

the

range

the

communication

session

can

be

handed

over

to

a

cell

at

a

higher

or

a

lower

layer

respectively

ffl

As

it

is

difficult

to

accurately

measure

the

velocity

of

a

mobile

unit

the

velocity

is

indirectly

determined

by

measuring

the

time

that

the

unit

stays

in

a

cell

The

velocity

is

inversely

proportional

to

the

duration

of

stay

in

the

cell

Bulk

Channel

Transfer

In

existing

channel

allocation

strategies

one

channel

is

borrowed

at

a

time

from

the

neighbors

to

support

a

communication

session

in

a

heavily

loaded

cell

These

strategies

can

respond

more

quickly

to

changes

in

load

distribution

by

employing

bulk

channel

transfer

If

cell

Ci

needs

x

number

of

free

channels

to

support

pending

channel

requests

in

its

region

and

its

neighbor

Cj

can

lend

several

channels

to

Ci

then

Ci

tries

to

acquire

all

these

channels

in

a

single

attempt

rather

than

one

by

one

However

the

time

required

for

bulk

channel

transfer

is

likely

to

be

greater

than

the

time

for

a

single

channel

transfer

The

time

for

a

heavily

loaded

node

to

broadcast

a

probe

and

for

the

probes

to

return

is

the

same

as

in

the

case

of

single

channel

transfer

But

the

cell

that

wishes

to

borrow

multiple

channels

has

to

carry

out

some

additional

computation

to

determine

the

set

of

channels

as

opposed

to

just

one

channel

that

can

be

borrowed

Finally

the

transfer

is

complete

when

the

last

borrowed

channel

is

received

from

a

lending

neighbor

The

additional

computation

for

bulk

channel

transfer

can

be

minimized

by

storing

the

availability

status

of

channels

as

boolean

vectors

Selection

of

appropriate

channels

will

then

require

a

small

number

of

boolean

operations

on

these

vectors

The

cost

of

these

operations

in

terms

of

time

is

small

Hence

x

x

where

x

is

the

time

to

borrow

x

channels

-LRB-

in

bulk

-RRB-

from

the

neighbor

-LRB-

s

-RRB-

while

is

the

time

to

borrow

a

single

channel

Hence

the

condition

for

bulk

channel

transfer

is

P

-LRB-

W

x

-RRB-

Channel

Transfer

vs.

Channel

Borrowing

Until

now

we

have

used

the

terms

channel

borrowing

and

channel

transfer

interchangeably

In

the

existing

dynamic

channel

allocation

algorithms

the

borrowed

channel

is

relinquished

to

its

previous

owner

at

the

end

of

the

communication

session

The

borrowing

is

implemented

by

simulating

dummy

calls

in

the

cells

from

which

the

channels

are

borrowed

and

by

employing

directional

locking

or

cell

sectoring

-LSB-

-RSB-

The

impact

of

channel

transfers

can

be

increased

if

a

channel

acquired

through

transfer

is

not

relinquished

at

the

end

of

the

communication

session

it

was

supporting

in

the

new

cell

Thus

a

channel

is

not

borrowed

from

a

neighbor

just

for

a

single

communication

session

Instead

it

is

transferred

for

future

use

as

well

Over

a

period

of

time

the

number

of

channels

allocated

to

different

cells

in

a

cluster

will

be

proportional

to

the

channel

demand

in

the

cells

-LCB-

highly

loaded

cell

will

acquire

a

large

number

of

channels

and

cells

with

low

load

will

have

fewer

channels

allotted

to

them

This

will

reduce

the

number

of

transfer

attempts

and

the

resultant

network

traffic

as

it

exploits

the

temporal

and

spatial

locality

of

channel

demand

However

it

will

require

a

departure

from

the

concept

of

fixed

sets

of

nominal

channels

assigned

to

the

cell

A

channel

transfer

algorithm

that

exploits

the

locality

of

demand

has

been

presented

in

-LSB-

-RSB-

Conclusions

Distributed

dynamic

channel

allocation

is

suitable

for

mobile

computing

due

to

its

flexibility

robustness

and

scalability

Dynamically

changing

the

channels

allocated

to

various

cells

enables

the

system

to

adapt

to

changes

in

the

temporal

and

spatial

distribution

of

communication

channel

demand

The

channel

allocation

problem

has

some

interesting

similarities

with

the

load

balancing

problem

studied

in

the

context

of

distributed

computing

However

the

two

problems

have

some

fundamental

differences

In

the

classic

load

balancing

problem

the

resources

-LRB-

process

ing

nodes

-RRB-

are

fixed

while

the

jobs

-LRB-

processes

-RRB-

can

be

migrated

between

nodes

In

the

case

of

channel

allocation

both

the

resources

-LRB-

channels

-RRB-

and

the

jobs

-LRB-

nodes

involved

in

mobile

computing

-RRB-

exhibit

mobility

The

mobility

of

the

resources

is

constrained

by

co-channel

interference

while

the

mobility

of

the

jobs

is

dependent

on

the

geometric

layout

of

the

cells

and

the

velocity

of

the

mobile

nodes

Dynamic

channel

allocation

algorithms

improve

the

overall

utilization

of

the

channels

increase

the

number

of

communication

sessions

that

can

be

supported

simultaneously

and

decrease

the

number

of

blocked

communication

sessions

However

channel

transfers

have

computational

and

communication

overheads

due

to

which

in

certain

situations

of

load

imbalance

channel

transfer

may

be

counter-productive

We

derived

an

expression

for

the

potential

for

channel

transfer

in

a

mobile

computing

environment

We

also

presented

a

probabilistic

analysis

for

the

feasibility

of

channel

transfer

given

the

overheads

for

the

transfer

and

the

temporal

fluctuations

in

channel

demand

distribution

The

queuing

model

developed

in

the

paper

can

be

used

to

generate

a

table

of

the

probability

distribution

of

the

Channel

Transfer

Window

A

heavily

loaded

cell

can

do

a

table

look-up

to

decide

if

a

channel

transfer

is

will

be

productive

We

proposed

bulk

channel

transfer

as

a

strategy

to

minimize

the

channel

transfer

overheads

and

to

amortize

the

overheads

over

several

network

communication

sessions

Bulk

channel

transfers

enable

the

system

to

quickly

adjust

to

sudden

changes

in

the

channel

demand

distribution

Channel

demand

exhibits

spatial

and

temporal

locality

Therefore

the

impact

of

channel

transfers

can

be

made

long

lasting

if

a

borrowed

channel

is

not

returned

by

the

borrower

at

the

end

of

the

communication

session

it

was

supporting

By

keeping

the

borrowed

channel

until

it

is

requested

by

a

neighbor

the

number

of

channel

transfer

attempts

can

be

reduced

Also

the

heavily

loaded

cells

will

acquire

a

large

number

of

channels

while

the

lightly

loaded

cells

will

have

fewer

channels

The

variation

in

the

relative

speeds

of

the

different

mobile

units

and

the

frequency

with

which

they

switch

from

one

MSS

to

another

can

be

handled

by

using

a

hierarchy

of

cells

with

varying

size

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-RSB-

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Akaiwa

and

H.

Andoh

Channel

Segregation

-LCB-

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Self-Organized

Dynamic

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Method

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bulk channel transfer
channel allocation
load sharing
mobile computing
transfer potential
transfer window