Optimal

Cell

Loss

Equalization

for

Video

Multiplexers

Ioanis

Nikolaidis

Ian

F.

Akyildiz

GIT?CC

October

Abstract

Video

traffic

multiplexers

in

high

speed

ATM

networks

are

prone

to

fairness

problems

with

respect

to

the

per-connection

cell

loss

ratio

experienced

by

multiplexed

video

sources

The

problem

is

a

result

of

the

random

but

fixed

over

time

relation

of

the

frame

transmission

epochs

that

feed

a

multiplexer

This

paper

presents

a

solution

to

this

fairness

problem

which

is

based

on

the

enforcement

of

controlled

per?connection

delays

The

amount

of

delay

imposed

on

each

source

is

calculated

by

an

optimization

process

at

connection

admission

and

termination

instants

Two

different

optimization

objectives

one

minimax

and

one

minisum

are

considered

Their

performance

and

their

relation

to

buffer

space

constraints

is

examined

The

loss

and

delay

performance

of

the

scheme

is

also

evaluated

through

simulations

In

particular

very

low

per?connection

delay

variance

is

observed

indicating

reduced

jitter

Finally

two

implementation

alternatives

of

the

scheme

on

an

ATM

network

are

presented

-LRB-

a

-RRB-

as

a

protocol

between

multiplexer

and

sources

and

-LRB-

b

-RRB-

as

a

non?work

conserving

service

discipline

for

multiplexers

The

engineering

aspects

and

in

particular

the

buffer

demands

of

the

two

alternative

implementations

are

discussed

in

detail

College

of

Computing

Georgia

Institute

of

Technology

Atlanta

Georgia

Optimal

Cell

Loss

Equalization

for

Video

Multiplexers

Ioanis

Nikolaidis

Ian

F.

Akyildiz

College

of

Computing

School

of

Electrical

and

Georgia

Institute

of

Technology

Computer

Engineering

Atlanta

GA

Georgia

Institute

of

Technology

nikolaid@cc.gatech.edu

Atlanta

GA

ian@armani.mirc.gatech.edu

Abstract

Video

traffic

multiplexers

in

high

speed

ATM

networks

are

prone

to

fairness

problems

with

respect

to

the

per-connection

cell

loss

ratio

experienced

by

multiplexed

video

sources

The

problem

is

a

result

of

the

random

but

fixed

over

time

relation

of

the

frame

transmission

epochs

that

feed

a

multiplexer

This

paper

presents

a

solution

to

this

fairness

problem

which

is

based

on

the

enforcement

of

controlled

per?connection

delays

The

amount

of

delay

imposed

on

each

source

is

calculated

by

an

optimization

process

at

connection

admission

and

termination

instants

Two

different

optimization

objectives

one

minimax

and

one

minisum

are

considered

Their

performance

and

their

relation

to

buffer

space

constraints

is

examined

The

loss

and

delay

performance

of

the

scheme

is

also

evaluated

through

simulations

In

particular

very

low

per?connection

delay

variance

is

observed

indicating

reduced

jitter

Finally

two

implementation

alternatives

of

the

scheme

on

an

ATM

network

are

presented

-LRB-

a

-RRB-

as

a

protocol

between

multiplexer

and

sources

and

-LRB-

b

-RRB-

as

a

non?work

conserving

service

discipline

for

multiplexers

The

engineering

aspects

and

in

particular

the

buffer

demands

of

the

two

alternative

implementations

are

discussed

in

detail

A

significantly

shorter

version

of

this

paper

incorporating

earlier

results

appears

in

the

Proceedings

of

the

ACM

Multimedia

Conference

Techniques

presented

herein

may

be

subject

to

patents

pending

Introduction

Video

sources

are

characterized

by

the

periodic

nature

of

their

frame

generation

epochs

Frame

rates

in

the

range

of

to

frames

per

second

are

typical

for

good

quality

motion

In

packet?based

networks

e.g.

ATM

networks

the

packetization

and

transmission

epochs

of

successive

frames

also

exhibit

the

same

periodicity

That

is

the

cells

related

to

successive

frames

are

available

for

transmission

at

periodic

points

in

time

In

this

paper

the

frame

transmission

period

will

be

denoted

by

T

Consequently

any

periodic

description

will

be

reduced

to

a

time

point

t

in

the

-LSB-

T

-RRB-

interval

with

the

understanding

that

it

occurs

at

all

t

kT

time

points

-LRB-

k

-RRB-

The

focus

of

this

paper

is

the

cell

loss

fairness

problem

which

can

be

stated

as

follows

When

a

number

of

statistically

identical

video

traffic

sources

are

multiplexed

at

a

finite

buffer

multiplexer

they

suffer

different

per?connection

cell

loss

ratios

dependent

on

the

relative

position

of

the

sources

frame

transmission

This

problem

generates

concerns

regarding

the

quality

of

service

-LRB-

QoS

-RRB-

of

separate

channels

utilizing

the

same

network

That

is

certain

channels

may

consistently

receive

better

performance

than

others

potentially

contrary

to

contractual

agreement

A

first

solution

to

the

cell

loss

fairness

problem

is

proposed

in

-LSB-

-RSB-

and

it

is

based

on

a

modification

of

the

multiplexer

's

FCFS

scheduling

discipline

Each

connection

is

associated

with

a

counter

indicating

how

many

cells

of

this

connection

have

been

dropped

so

far

If

upon

its

arrival

a

cell

finds

the

buffer

of

the

multiplexer

full

it

is

not

necessarily

dropped

Instead

an

already

queued

cell

is

dropped

from

the

connection

with

the

lowest

counter

value

of

dropped

cells

The

counter

for

this

connection

is

subsequently

increased

and

the

released

buffer

space

is

used

to

accommodate

a

new

arrival

This

modification

ameliorates

the

fairness

problem

but

it

does

not

solve

it

For

example

even

when

the

modified

scheduling

discipline

is

applied

there

exist

connections

that

receive

an

order

of

magnitude

more

losses

than

others

-LRB-

in

Table

I

of

-LSB-

-RSB-

-RRB-

The

intuitive

explanation

is

that

there

exist

instances

when

the

connection

with

the

lowest

counter

value

has

no

cells

queued

Hence

even

though

its

counter

is

low

we

can

not

victimize

this

connection

Consequently

we

are

forced

to

victimize

a

connection

with

a

higher

counter

to

salvage

the

buffer

space

necessary

to

store

the

incoming

cell

Moreover

when

we

consider

the

call?level

dynamics

we

realize

that

long?lived

connections

gain

a

performance

advantage

over

short?lived

ones

since

short?lived

connections

are

always

associated

with

small

counter

values

Three

alternative

solutions

to

the

problem

of

temporal

placement

of

frame

transmission

epochs

are

presented

in

-LSB-

-RSB-

However

the

objective

in

-LSB-

-RSB-

is

not

cell

loss

fairness

but

the

overall

cell

loss

and

delay

performance

Moreover

control

of

a

source

's

transmission

epoch

is

performed

only

once

at

connection

admission

after

which

it

remains

fixed

In

addition

the

best

of

the

three

alternatives

presented

in

-LSB-

-RSB-

needs

extensive

information

about

the

source

traffic

-LRB-

stationary

distribution

of

each

source

's

traffic

intensity

throughout

the

frame

period

-RRB-

We

take

a

different

approach

by

controlling

the

transmission

epochs

even

after

admission

with

the

intention

to

always

equi?distribute

losses

among

admitted

connections

We

do

not

require

knowledge

about

the

traffic

model

but

we

assume

homogeneous

sources

and

the

same

frame

rate

for

all

sources

The

scheme

we

present

can

be

implemented

either

as

a

protocol

between

the

sources

and

the

multiplexer

or

as

a

non?work

conserving

service

discipline

at

the

multiplexer

In

frame

transmission

epochs

of

a

connection

are

the

periodic

time

points

at

which

the

multiplexer

receives

the

first

cell

of

a

new

frame

Note

also

that

we

use

the

terms

source

and

connection

interchangeably

since

they

are

related

in

a

one?to?one

fashion

the

first

case

space

overhead

is

necessary

at

the

sources

while

in

the

second

the

space

overhead

is

located

at

the

multiplexer

The

objective

is

to

enforce

a

particular

alignment

of

the

frame

transmission

epochs

of

the

multiplexed

sources

An

underlying

optimization

process

minimizes

the

necessary

delays

to

achieve

the

alignment

The

optimization

and

alignment

operations

are

triggered

by

call

connection

and

termination

events

thus

they

can

be

part

of

the

call

admission

control

which

operates

at

the

exact

same

time

points

The

infrequent

nature

of

call

admission

and

termination

events

-LRB-

at

least

in

comparison

to

cell

arrival

events

-RRB-

implies

that

a

very

low

amortized

computational

cost

is

needed

to

support

the

proposed

scheme

Before

we

introduce

the

scheme

in

more

detail

a

few

clarifications

are

necessary

about

the

context

of

the

present

study

First

we

consider

two

example

applications

for

the

subsequent

simulation

and

evaluation

sections

One

is

a

multiplexer

of

video?conferencing

connections

while

the

second

is

a

multiplexer

for

a

video

distribution

service

of

entertainment

video

Notably

the

nature

of

the

first

application

is

real?time

and

delays

are

more

crucial

than

in

the

second

application

Secondly

the

characterization

of

the

number

of

cells

per

frame

K

as

a

random

process

is

a

topic

that

has

attracted

the

attention

of

many

researchers

during

recent

years

-LSB-

-RSB-

Since

there

exists

no

single

conclusive

model

for

all

types

of

video

traffic

our

evaluation

section

is

based

on

simulations

conducted

using

-LRB-

a

-RRB-

a

Gamma

distribution?based

traffic

and

-LRB-

b

-RRB-

an

actual

compressed

video

trace

The

Gamma

distribution

closely

matches

the

traffic

produced

by

videoconference

applications

-LSB-

-RSB-

The

trace

on

the

other

hand

corresponds

to

the

traffic

from

an

actual

entertainment

video

-LSB-

-RSB-

according

to

the

MPEG

compression

standard

-LSB-

-RSB-

Futhermore

we

assume

that

there

exists

a

maximum

Kmax

for

the

number

of

cells

that

can

be

produced

per

frame

This

maximum

typically

corresponds

to

frames

for

which

the

compression

scheme

does

not

result

in

substantial

savings

We

also

assume

that

the

bit

rate

Bin

of

the

link

connecting

the

source

to

the

network

is

sufficient

to

ensure

that

Kmax

cells

can

be

transmitted

in

T

time

units

This

assumption

ensures

-LRB-

a

-RRB-

that

even

in

the

worst

case

the

source

has

no

leftover

cells

from

a

previous

frame

to

transmit

when

the

cells

from

a

new

frame

are

generated

and

-LRB-

b

-RRB-

that

if

the

network

behaves

as

an

ideal

constant

delay

line

all

departing

cells

related

to

a

frame

will

arrive

at

the

destination

within

the

next

T

time

units

to

be

available

for

playout

Finally

once

the

K

cells

representing

a

frame

have

been

generated

there

exists

a

variety

of

ways

to

arrange

their

transmission

in

the

next

T

time

units

One

arrangement

is

to

place

the

transmission

of

all

K

cells

back?to?back

at

the

peak

link

transmission

rate

in

a

continuous

burst

This

arrangement

results

in

more

intense

congestion

at

the

multiplexer

but

it

experiences

better

delay

A

second

arrangement

is

to

smooth

the

variable

length

burst

over

the

next

T

time

units

and

hence

to

avoid

transmission

at

the

peak

rate

Consequently

the

cells

can

be

transmitted

T

K

time

units

apart

This

second

arrangement

results

in

less

intense

congestion

at

the

multiplexer

but

at

the

expense

of

additional

-LRB-

as

much

as

T

-RRB-

delay

For

the

purpose

of

this

paper

we

opt

for

the

back?to?back

peak

rate

transmission

because

it

both

represents

a

pessimistic

scenario

with

respect

to

multiplexer

congestion

and

it

does

not

cause

extra

delays

that

can

harm

real?time

communication

The

remaining

of

the

paper

is

organized

as

follows

in

section

we

present

the

algorithmic

aspects

and

the

operations

necessary

to

implement

the

proposed

scheme

In

section

we

evaluate

the

scheme

in

terms

of

attained

cell

loss

fairness

and

in

terms

of

delay

and

delay

Source

-LRB-

a

-RRB-

-LRB-

b

-RRB-

t

T-t

t

T-t

-LRB-

c

-RRB-

Source

TimeSource

TimeSource

Time

Figure

En

example

of

-LRB-

a

-RRB-

the

worst

-LRB-

b

-RRB-

the

best

and

-LRB-

c

-RRB-

a

random

alignment

of

the

frame

transmissions

epochs

for

the

exact

same

traffic

trace

generated

by

two

video

sources

jitter

while

we

also

provide

the

engineering

guidelines

for

its

implementation

on

an

ATM

network

We

conclude

in

section

with

a

summary

of

the

contributions

of

the

proposed

scheme

and

we

point

out

future

directions

for

related

research

The

Frame

Alignment

Technique

In

the

previous

section

we

pointed

out

that

the

cell

loss

fairness

problem

is

caused

by

the

particular

relative

time

alignment

of

the

frame

transmission

epochs

of

multiplexed

video

sources

In

order

to

solve

the

problem

we

focus

on

ways

to

enforce

a

certain

best

alignment

with

respect

to

perceived

cell

loss

ratio

performance

Without

any

formal

we

appeal

to

intuition

and

distinguish

three

cases

regarding

the

transmission

epoch

alignment

of

multiplexed

sources

ffl

The

Worst

Alignment

The

transmission

epochs

of

all

admitted

sources

are

aligned

at

the

exact

same

time

point

That

is

the

transmission

of

the

frames

from

all

the

sources

starts

at

the

same

point

in

time

This

alignment

is

repeated

for

every

single

frame

It

becomes

clear

why

this

is

the

worst

case

in

terms

of

cell

loss

ratio

by

noting

that

all

the

sources

compete

for

the

multiplexer

buffer

at

exactly

the

same

time

Thus

the

likelihood

to

find

the

buffer

full

is

greater

than

if

they

followed

any

other

time

alignment

ffl

The

Best

-LRB-

Ideal

-RRB-

Alignment

The

transmission

epochs

of

the

N

different

connections

are

spaced

T

N

time

units

apart

Thus

the

transmission

epochs

will

be

at

time

points

kT

N

for

k

N

We

will

call

these

time

points

the

ideal

transmission

epochs

such

proof

would

necessitate

an

analytical

model

for

the

video

traffic

which

is

both

beyond

the

scope

of

this

paper

and

is

still

an

open

research

question

That

is

all

transmission

epochs

are

equi?spaced

within

the

frame

period

Assuming

that

the

sources

are

statistically

identical

the

multiplexer

receives

the

same

-LRB-

in

the

statistical

sense

-RRB-

load

of

arrivals

in

each

T

N

interval

defined

by

any

two

consecutive

ideal

transmission

epochs

Thus

the

load

is

equalized

and

likewise

are

the

cell

losses

There

can

be

no

better

alignment

because

if

the

successive

transmission

epochs

of

a

pair

of

sources

are

moved

closer

than

T

N

time

units

the

contention

between

them

for

the

multiplexer

buffer

increases

The

best

alignment

does

not

discriminate

favorably

for

any

source

ffl

A

Random

Alignment

The

transmission

epochs

are

positioned

randomly

in

the

interval

-LSB-

T

-RRB-

For

the

sake

of

convenience

and

since

no

indication

to

the

opposite

exists

we

assume

that

the

transmission

epoch

si

of

any

source

i

is

generated

from

a

uniform

distribution

in

the

-LSB-

T

-RRB-

interval

The

random

alignment

represents

the

arbitrary

alignment

we

expect

to

find

in

an

actual

system

that

lacks

a

control

mechanism

such

as

the

one

we

propose

Such

alignment

can

suffer

in

terms

of

cell

loss

ratio

fairness

as

well

as

in

terms

of

overall

cell

loss

ratio

compared

to

the

ideal

alignment

Note

also

that

a

random

alignment

can

not

be

precluded

from

being

close

to

the

worst

alignment

To

illustrate

the

effect

of

the

different

alignments

consider

a

scaled

down

example

such

as

that

of

Figure

We

represent

time

in

a

slotted

fashion

and

T

slot

times

Assume

that

the

initial

buffer

occupancy

is

zero

and

that

the

buffer

size

is

while

the

output

link

speed

is

equal

to

one

input

link

's

speed

Thus

whenever

two

arrivals

occur

in

the

same

input

time

slot

the

one

can

be

serviced

while

the

second

can

be

stored

in

the

buffer

provided

there

exists

space

in

the

buffer

otherwise

it

is

lost

Under

these

assumptions

the

worst

alignment

-LRB-

Figure

-LRB-

a

-RRB-

-RRB-

receives

the

worst

loss

performance

a

total

of

losses

in

the

first

cycle

and

in

the

second

The

best

alignment

-LRB-

Figure

-LRB-

b

-RRB-

-RRB-

suffers

no

losses

while

the

random

alignment

-LRB-

Figure

-LRB-

c

-RRB-

-RRB-

with

suffers

losses

in

the

first

cycle

and

in

the

second

That

is

the

exact

same

frame

traffic

of

two

independent

video

connections

will

result

in

markedly

different

cell

loss

performance

under

different

frame

transmission

alignments

This

example

illustrates

the

importance

that

the

best

alignment

has

on

the

overall

cell

loss

statistics

In

addition

the

best

alignment

achieves

cell

loss

fairness

as

we

will

see

in

section

Our

objective

is

to

force

the

random

transmission

epochs

of

the

sources

to

be

aligned

exactly

as

in

the

best

alignment

In

-LSB-

-RSB-

a

similar

observation

was

made

and

it

was

hinted

that

additional

circuitry

is

needed

to

enforce

the

alignment

However

no

details

as

to

how

to

achieve

the

alignment

even

in

circuitry

were

given

In

our

approach

we

observe

that

the

alignment

depends

on

the

number

of

admitted

sources

N

which

is

information

not

necessarily

available

to

the

individual

but

certainly

available

to

the

multiplexer

and

to

the

call

admission

process

Hence

any

such

operation

fits

naturally

as

a

function

of

the

Call

Admission

Control

-LRB-

CAC

-RRB-

Finally

note

that

the

assumption

that

the

transmission

epochs

are

aligned

according

to

the

best

alignment

has

been

used

in

the

past

e.g.

in

the

case

of

a

multiplexer

delay

study

under

different

bandwidth

sharing

schemes

in

-LSB-

-RSB-

That

is

there

seems

to

be

an

agreement

on

the

importance

of

the

ideal

alignment

What

we

provide

in

this

paper

is

a

mechanism

to

achieve

it

least

for

security

concerns

a

connection

should

not

be

aware

of

how

many

other

connections

are

currently

admitted

T

s

s

s

s

next

Ts

s

s

s

.0120.00.00.00.00.00.00.020.020.020.020.00.020.00.0

Random.Aligned

-LRB-

-RRB-

Aligned

-LRB-

-RRB-

-LRB-

a

-RRB-

-LRB-

b

-RRB-

-LRB-

c

-RRB-

s

w

w

Figure

A

random

frame

alignment

for

two

video

sources

-LRB-

a

-RRB-

and

the

two

possible

scenarios

for

the

selection

of

the

reference

connection

-LRB-

b

-RRB-

ref

and

-LRB-

c

-RRB-

ref

The

Optimization

Algorithm

The

solution

we

propose

operates

by

enforcing

a

vector

of

delays

ffi

on

the

admitted

connections

That

is

connection

i

is

delayed

ffi?i

time

units

where

ffi?i

T

The

delays

cause

the

transmission

epochs

of

the

sources

to

coincide

with

the

ideal

transmission

epochs

There

exists

an

infinite

number

of

possible

delay

assignments

that

map

the

N

random

transmission

epochs

to

the

N

ideal

epochs

If

we

assume

that

each

such

delay

assignment

is

represented

by

an

appropriate

ffi

vector

the

issue

becomes

to

find

the

specific

ffi

vector

that

is

optimal

and

which

we

denote

by

ffi

We

consider

two

optimality

criteria

ffl

Minimization

of

the

aggregate

delay

over

all

connections

TMS

-LRB-

ffi

-RRB-

PN

i

ffii

i.e.

a

minisum

-LRB-

MS

-RRB-

formulation

ffl

Minimization

of

the

maximum

delay

over

all

connections

TMM

-LRB-

ffi

-RRB-

maxN

i

fffiig

i.e.

a

minimax

-LRB-

MM

-RRB-

formulation

The

MS

criterion

attempts

to

minimize

the

overall

delay

enforced

on

all

connections

but

does

not

consider

inherent

limitations

on

the

acceptable

maximum

per?connection

delay

In

this

sense

it

may

cause

excessive

delays

on

a

few

connections

thus

jeopardizing

their

real?time

nature

This

case

is

handled

by

the

MM

formulation

which

however

may

inflate

the

delay

values

for

more

connections

than

necessary

compared

to

the

delay

values

produced

by

the

MS

formulation

The

optimal

ffi

for

MS

and

MM

will

be

denoted

by

ffi?MS

and

ffi?MM

The

purpose

of

the

optimization

algorithm

we

present

is

to

calculate

the

optimal

ffi?MS

and

the

optimal

ffi?MM

However

whichever

of

the

two

optimization

criteria

we

pick

the

set

of

candidate

ffi

vectors

can

be

dramatically

reduced

using

the

following

straightforward

lemma

Lemma

Any

delay

assignment

vector

ffi

where

ffii

-LRB-

-RRB-

can

not

be

optimal

with

respect

to

the

minimization

of

either

TMS

-LRB-

ffi

-RRB-

or

TMM

-LRB-

ffi

-RRB-

Proof

The

proof

is

by

contradiction

Suppose

that

we

pick

a

ffi

vector

which

has

all

its

elements

not

equal

to

zero

and

it

is

optimal

under

either

MS

or

MM

optimization

We

the

following

for

the

sake

of

brevity

the

MS

and

MM

subscripts

will

be

omitted

for

statements

that

are

valid

regardless

of

whether

MS

or

MM

is

assumed

can

calculate

ffimin

the

minimum

over

all

ffii

's

Let

us

also

assume

that

ffimin

ffij

for

a

certain

j

We

can

now

construct

a

new

delay

assignment

vector

where

ffii

ffimin

-LRB-

hence

-RRB-

Observe

that

ffi

and

define

the

exact

same

relative

frame

transmission

epochs

and

that

it

is

only

their

absolute

position

in

time

that

we

influenced

by

subtracting

ffimin

Moreover

TMS

-LRB-

-RRB-

TMS

-LRB-

ffi

-RRB-

N

ffimin

-RRB-

TMS

-LRB-

-RRB-

TMS

-LRB-

ffi

-RRB-

and

at

the

same

time

TMM

-LRB-

-RRB-

TMM

-LRB-

ffi

-RRB-

ffimin

-RRB-

TMM

-LRB-

-RRB-

TMM

-LRB-

ffi

-RRB-

Hence

from

a

ffi

with

nonzero

elements

we

can

directly

derive

a

with

at

least

one

zero

element

which

preserves

the

same

relative

position

of

the

ideal

transmission

epochs

but

results

in

less

overall

delay

TMS

-LRB-

-RRB-

and

less

maximum

delay

TMM

-LRB-

-RRB-

Clearly

ffi

can

not

be

the

optimal

choice

Consequently

we

limit

the

search

for

an

optimal

ffi

among

vectors

with

one

or

more

ffii

elements

equal

to

zero

However

the

physical

meaning

of

a

zero

element

e.g.

in

position

j

in

the

ffi

vector

is

the

following

The

frame

transmission

epoch

of

the

j?th

connection

coincides

with

an

ideal

transmission

epoch

Following

this

line

of

thought

we

come

to

the

realization

that

the

optimal

ffi

can

be

derived

by

only

considering

N

possible

configurations

The

i?th

configuration

is

constructed

by

setting

the

i?th

connection

transmission

epoch

to

coincide

with

an

ideal

frame

transmission

epoch

Without

harm

of

generality

we

will

assume

that

this

ideal

transmission

epoch

is

the

first

ideal

transmission

epoch

i.e.

the

one

at

time

of

the

-LSB-

T

-RRB-

interval

For

the

i?th

configuration

all

the

frame

transmission

times

of

the

remaining

N

connections

are

translated

relative

to

the

i?th

frame

transmission

epoch

and

within

the

-LSB-

T

-RRB-

interval

Therefore

the

problem

is

reduced

to

finding

for

each

configuration

the

optimal

assignment

of

the

remaining

N

connections

to

the

remaining

N

ideal

epochs

From

the

N

produced

optimal

assignments

we

select

the

one

that

corresponds

to

the

minimum

TMS

-LRB-

TMM

-RRB-

and

this

will

be

the

optimal

assignment

ffi?MS

-LRB-

ffi?MM

-RRB-

We

will

describe

the

algorithm

in

detail

using

the

notion

of

a

reference

connection

-LRB-

ref

-RRB-

to

identify

the

configuration

being

generated

and

in

a

one?to?one

fashion

the

index

of

the

connection

that

is

aligned

to

the

first

ideal

transmission

epoch

The

concept

of

a

reference

connection

does

not

give

particular

significance

or

power

to

any

connection

It

merely

facilitates

the

description

of

the

control

flow

of

the

optimization

process

For

each

of

the

N

configurations

-LRB-

corresponding

to

assigning

ref

from

up

to

N

-RRB-

the

following

steps

are

performed

The

frame

transmission

epochs

are

normalized

relative

to

the

transmission

epoch

of

connection

ref

in

the

-LSB-

T

-RRB-

interval

The

transformation

preserves

the

relative

position

of

the

frame

transmission

epochs

within

a

cycle

of

period

T

and

at

the

same

time

it

fixes

the

transmission

epoch

of

the

ref?th

connection

to

coincide

with

the

beginning

of

the

frame

transmission

cycle

-LRB-

time

zero

in

the

-LSB-

T

-RRB-

interval

-RRB-

The

result

is

a

vector

which

preserves

the

transformed

position

of

the

random

transmission

within

the

-LSB-

T

-RRB-

cycle

-LRB-

-LSB-

T

-RRB-

-RRB-

The

values

are

calculated

as

follows

si

sref

si

sref

si

sref

T

si

sref

i

ref

ref

N

-LRB-

-RRB-

The

calculation

is

performed

in

the

lines

of

the

algorithm

in

Figure

Figures

-LRB-

b

-RRB-

and

-LRB-

c

-RRB-

show

among

other

information

an

example

of

the

derived

vector

values

from

the

s

vector

of

Figure

-LRB-

a

-RRB-

for

the

two

possible

configurations

The

costs

i.e.

delays

of

aligning

any

of

the

remaining

-LRB-

that

is

except

ref

-RRB-

N

frame

transmission

epoch

to

any

of

the

N

remaining

ideal

transmission

epochs

are

calculated

The

cost

of

aligning

connection

i

to

the

j?th

ideal

epoch

is

the

delay

necessary

to

reach

the

jT

N

time

point

-LRB-

either

in

this

-LSB-

T

-RRB-

cycle

or

in

the

next

-RRB-

starting

from

That

is

the

cost

represented

by

wij

is

the

difference

between

jT

N

and

If

jT

N

and

since

the

delay

has

to

be

positive

the

alignment

has

to

be

performed

with

the

same

ideal

epoch

but

in

the

next

frame

interval

Figure

-LRB-

c

-RRB-

depicts

exactly

this

situation

N

hence

the

delay

must

extend

to

the

next

cycle

Note

that

the

cost

wij

can

not

be

more

than

T

since

from

we

can

reach

any

ideal

epoch

either

in

the

current

frame

interval

or

in

the

next

within

a

delay

of

T

time

units

Summarizing

the

calculation

of

the

costs

is

performed

as

follows

wij

jTN

jTN

T

jTN

jTN

i

ref

ref

N

j

N

-LRB-

-RRB-

The

calculation

of

wij

's

is

performed

in

lines

of

the

algorithm

in

Figure

The

next

step

is

the

calculation

of

the

optimal

assignment

for

the

specific

configuration

i.e.

for

the

specific

reference

connection

We

will

use

bij

as

the

binary

variables

expressing

the

selection

-LRB-

if

set

to

-RRB-

or

not

-LRB-

if

set

to

-RRB-

of

the

respective

wij

Therefore

the

optimization

problem

can

be

stated

as

follows

-LRB-

MS

-RRB-

minimize

TMS

PN

i

ref

PN

j

bijwij

-LRB-

-RRB-

OR

-LRB-

MM

-RRB-

minimize

TMM

max

i

N

ref

j

N

fbijwijg

-LRB-

-RRB-

s.t.

PN

j

bij

PN

i

ref

bij

bij

i

ref

ref

N

j

N

The

MS

optimization

version

is

a

typical

assignment

problem

solvable

by

polynomial

time

algorithms

e.g.

by

the

Hungarian

algorithm

-LSB-

-RSB-

and

similarly

is

the

MM

optimization

version

solvable

by

bottleneck

assignment

algorithms

e.g.

-LSB-

-RSB-

We

will

assume

that

the

solution

of

the

MS

formulation

is

provided

by

the

ASSIGNMENT

MS

function

and

the

solution

for

the

MM

formulation

by

the

ASSIGNMENT

MM

function

In

both

cases

the

function

returns

an

array

b

of

the

optimal

b

assignments

as

well

as

the

optimal

value

T

of

the

optimization

pass

ref

to

the

respective

optimization

function

to

indicate

that

the

ref?th

row

and

column

of

w

indexed

by

ref

are

to

be

ignored

begin

T

overall

ffi

for

ref

upto

N

do

for

i

upto

N

do

si

sref

if

-LRB-

-RRB-

then

T

endfor

for

i

upto

N

and

i

ref

do

for

k

upto

N

do

wik

kT

N

if

-LRB-

wik

-RRB-

then

wik

wik

T

endfor

endfor

ASSIGNMENT

MS

-LRB-

N

ref

w

b

T

-RRB-

-LRB-

ASSIGNMENT

MM

-LRB-

N

ref

w

b

T

-RRB-

;-RRB-

if

T

T

overall

then

T

overall

T

for

i

upto

N

and

i

ref

do

ffi?i

PN

k

b?ikwik

endfor

ffi?ref

endif

endfor

return

-LRB-

T

overall

ffi

-RRB-

end

Figure

Overview

of

the

optimization

algorithm

The

algorithm

for

finding

the

optimal

delay

assignment

is

presented

in

Figure

It

initializes

the

optimal

overall

delay

and

delay

assignment

in

lines

Then

it

scans

through

all

N

possible

configurations

constructing

the

and

the

wij

values

for

each

configuration

and

solving

the

resulting

assignment

problem

If

-LRB-

at

line

-RRB-

the

optimal

assignment

for

the

current

configuration

results

in

a

smaller

overall

delay

than

all

the

configurations

until

now

it

is

taken

to

be

the

optimal

and

its

associated

assignment

is

taken

correspond

to

be

the

optimal

delay

assignment

ffi

-LRB-

lines

-RRB-

At

the

end

of

the

run

the

optimal

T

overall

and

ffi

are

returned

Whether

this

is

the

optimal

for

MS

-LRB-

T

MS

-RRB-

or

for

MM

-LRB-

T

MM

-RRB-

is

only

influenced

by

the

selection

of

the

core

optimization

function

at

line

Evidently

that

there

is

no

need

to

perform

any

alignment

when

only

one

connection

is

admitted

Suppose

now

that

two

connections

are

accepted

as

in

Figure

the

resulting

example

is

trivial

but

easy

to

follow

Two

configurations

need

to

be

considered

In

the

first

-LRB-

Figure

-LRB-

b

-RRB-

-RRB-

ref

and

the

only

cost

is

The

resulting

optimal

assignment

Cell

GenerationDelayCell

GenerationDelayCell

GenerationDelaySource

NMUXOptimization

Algorithm

fromCAC

Entity

FTto

the

rest

of

the

Network

SourcesDelay

ControlMultiplexing

Figure

Block

diagram

of

an

ATM

Network

multiplexer

and

the

position

of

the

proposed

protocol

functions

-LRB-

in

dashed

lines

-RRB-

The

circles

around

the

input

links

denote

the

observation

of

the

input

stream

for

SOF

cells

will

be

and

and

thus

T

In

the

second

configuration

the

only

cost

is

The

resulting

optimal

assignment

will

be

and

and

thus

T

Obviously

for

both

MS

and

MM

the

first

configuration

results

in

the

least

cost

and

therefore

ffi?MS

ffi?MM

ffi

ffi?MS

ffi?MM

ffi

and

T

MS

T

MM

T

overall

The

optimization

process

provides

the

optimal

delay

assignment

ffi?i

for

each

source

i

What

remains

is

a

mechanism

that

enforces

these

delays

on

each

source

for

cell

loss

equalization

The

implementation

of

such

a

mechanism

is

addressed

in

the

next

section

The

common

assumption

for

the

proposed

implementations

is

the

existence

of

an

underlying

ATM

network

It

is

therefore

convenient

to

consider

the

delays

in

terms

of

cell

transmission

times

at

the

respective

link

speeds

This

allows

us

to

directly

gain

insight

into

the

space

-LRB-

buffer

-RRB-

overheads

necessary

to

hold

the

cells

in

order

to

enforce

the

delays

For

this

reason

we

will

express

the

delay

values

of

ffi?MS

i

-LRB-

ffi?MM

i

-RRB-

in

terms

of

integer

cell

transmission

times

D?MS

i

-LRB-

D?MM

i

-RRB-

at

peak

input

link

rate

That

is

if

the

input

link

speed

is

Bin

and

D

is

the

size

of

the

cell

-LRB-

octets

for

ATM

networks

-RRB-

then

MS

D?MS

i

d

ffi?MS

i

-LRB-

D

Bin

-RRB-

e

MM

D?MM

i

d

ffi?MM

i

-LRB-

D

Bin

-RRB-

e

-LRB-

-RRB-

Implementation

on

an

ATM

Network

The

block

diagram

of

Figure

presents

the

operations

of

the

proposed

scheme

on

an

ATM

network

The

implementation

depends

on

the

location

of

the

Delay

Control

part

One

approach

is

to

integrate

the

Delay

Control

with

the

sources

because

the

delay

control

for

any

source

can

be

separated

from

all

other

sources

Consequently

a

protocol

must

be

established

between

the

multiplexer

and

the

sources

to

ensure

the

equalized

operation

whereby

the

multiplexer

at

connection

acceptance

and

termination

points

updates

the

delays

enforced

on

each

source

A

second

approach

is

to

integrate

the

Delay

Control

with

the

multiplexing

operation

Therefore

we

can

essentially

modify

the

multiplexer

scheduling

discipline

to

ensure

that

the

per?source

additional

delay

is

enforced

In

this

case

the

sources

are

unaware

of

the

necessary

alignment

operations

and

there

is

no

need

for

the

definition

of

any

protocol

between

sources

and

multiplexer

However

the

multiplexer

is

burdened

with

the

task

of

allocating

additional

buffers

in

order

to

delay

the

incoming

traffic

Protocol?based

Implementation

We

will

describe

the

functions

performed

by

the

protocol

while

distinguishing

them

into

two

groups

-LRB-

a

-RRB-

multiplexer

-LRB-

MUX

-RRB-

functions

i.e.

functions

performed

by

the

multiplexer

and

its

accompanying

call

admission

logic

and

-LRB-

b

-RRB-

source

functions

i.e.

performed

by

the

source

traffic

generator

and

with

the

assistance

of

appropriate

control

information

The

multiplexer

functions

include

the

estimation

of

the

si

time

points

of

new

incoming

connections

the

calculation

of

the

best

delay

vector

ffi

and

the

transmission

of

the

updated

ffi?i

back

to

source

i

for

all

admitted

sources

The

source

functions

include

advertising

the

si

of

the

source

at

connection

setup

time

and

the

reception

of

subsequent

ffi?i

values

that

control

the

delay

imposed

on

the

outgoing

source

traffic

Following

is

the

list

of

the

protocol

functions

and

whether

they

are

part

of

the

source

or

the

multiplexer

functions

ffl

Source

Advertising

-LRB-

Source

-RRB-

Along

with

a

connection

request

sent

to

the

CAC

entity

the

source

starts

sending

to

the

multiplexer

the

regular

traffic

as

it

would

normally

The

first

cell

of

the

cell

burst

related

of

a

frame

transmission

is

marked

by

the

ATM

Adaptation

Layer

-LRB-

AAL

-RRB-

as

a

Start

of

Frame

-LRB-

SOF

-RRB-

cell

The

transmission

of

frames

continues

until

the

connection

terminates

ffl

si

Calculation

-LRB-

MUX

-RRB-

A

continuously

running

Frame

Timer

-LRB-

FT

-RRB-

exists

at

the

multiplexer

Its

purpose

is

to

continuously

count

from

to

T

cycling

back

to

when

it

reaches

T

The

multiplexer

also

keeps

track

of

the

number

of

already

admitted

connections

N

Upon

reception

of

an

SOF

cell

for

a

connection

which

is

not

yet

accepted

it

records

the

FT

value

as

sN

i.e.

the

frame

start

time

for

the

as?yet?not

accepted

connection

All

other

incoming

cells

from

the

new

source

are

simply

ignored

-LRB-

discarded

-RRB-

because

the

connection

is

not

yet

accepted

ffl

Connection

Acceptance

-LRB-

MUX

-RRB-

Upon

acceptance

of

the

connection

by

the

CAC

the

multiplexer

is

informed

that

the

connection

is

accepted

It

increases

the

number

of

admitted

connections

from

N

to

N

and

runs

the

algorithm

that

we

described

in

the

previous

section

for

the

N

frame

start

times

The

ffi

vector

is

determined

and

its

values

are

sent

to

the

corresponding

sources

ffl

Connection

Acceptance

-LRB-

Source

-RRB-

Upon

reception

of

the

first

ffi?i

value

the

source

assumes

that

it

has

been

admitted

It

starts

delaying

the

departing

cells

by

ffi?i

time

units

by

buffering

them

locally

ffl

Connection

Termination

-LRB-

Source

-RRB-

Source

j

terminates

the

connection

by

sending

a

request

to

the

CAC

entity

It

also

ceases

from

transmitting

frames

ffl

Connection

Termination

-LRB-

MUX

-RRB-

The

termination

request

is

processed

by

the

CAC

entity

and

the

indication

to

terminate

source

j

is

received

by

the

multiplexer

The

multiplexer

discards

the

sj

counter

value

and

runs

the

optimization

algorithm

for

the

remaining

sources

It

informs

the

sources

about

the

new

delays

by

sending

the

updated

ffi

vector

ffl

ffi

Updates

-LRB-

Source

-RRB-

If

an

admitted

source

i

receives

a

new

ffi?i

value

it

readjusts

the

delay

imposed

on

each

frame

's

traffic

-LRB-

starting

from

the

next

frame

to

be

generated

-RRB-

to

the

new

value

ffi?i

Refinements

to

the

above

frameworkcan

be

applied

including

for

example

compensation

for

drift

of

the

clocks

between

the

multiplexer

and

the

sources

Note

that

the

destination

must

also

be

informed

whenever

the

ffi

assignment

is

changed

in

order

to

calculate

the

delay

necessary

to

absorb

the

resulting

If

the

network

behaves

as

an

ideal

delay

the

necessary

delay

at

the

destination

is

never

more

than

the

frame

period

interval

Unfortunately

this

is

rarely

the

case

in

actual

networks

where

the

variance

of

delays

inside

the

network

may

require

large

jitter?absorption

delays

at

the

destinations

in

the

order

of

several

frame

intervals

Scheduling?based

Implementation

If

the

Delay

Control

part

of

the

scheme

is

incorporated

into

the

multiplexer

then

the

following

non?work

conserving

scheduling

discipline

is

the

result

whenever

a

cell

from

connection

i

arrives

and

D?i

is

not

zero

it

is

not

directly

forwarded

to

the

shared

multiplexer

buffer

but

instead

it

enters

a

pipeline

of

size

D?i

cells

-LRB-

which

is

functionally

equivalent

to

a

shift?register

where

the

shifted

unit

is

a

cell

-RRB-

The

cell

progresses

through

the

pipeline

clocked

by

the

input

link

cell

rate

When

the

cell

reaches

the

end

of

the

pipeline

it

enters

the

common

multiplexer

buffer

Hence

effectively

a

delay

of

D?i

cell

times

is

enforced

on

each

incoming

cell

The

scheduling

discipline

is

non

work?conserving

because

cells

may

be

in

their

respective

pipelines

although

they

have

arrived

at

the

multiplexer

and

at

the

same

time

the

server

remains

idle

because

no

cell

has

exited

the

pipeline

to

enter

the

common

multiplexer

buffer

The

operation

of

sizing

the

pipeline

is

controlled

by

the

successive

D?i

values

In

every

other

aspect

the

operations

are

as

the

ones

described

in

the

protocol?based

implementation

with

the

exception

that

there

is

no

need

to

communicate

information

to

the

sources

In

the

protocol?based

implementation

we

must

determine

the

size

of

the

necessary

buffer

at

each

source

while

in

the

scheduling?based

implementation

we

must

determine

the

cumulative

size

of

all

the

pipeline

buffer

These

engineering

problems

correspond

respectively

to

the

determination

of

the

maximum

anticipated

per?connection

delay

and

the

maximum

anticipated

aggregate

over

all

connections

delay

Engineering

guidelines

for

dimensioning

these

buffers

are

given

in

the

next

section

along

with

a

delay

and

loss

evaluation

of

the

scheme

Experimental

Evaluation

In

this

section

we

present

the

results

of

simulations

we

conducted

in

order

to

evaluate

four

different

aspects

of

the

proposed

control

mechanism

we

allow

occasionally

the

picture

to

jump

or

freeze

-LRB-

but

never

for

more

than

one

frame

interval

-RRB-

whenever

the

delay

assignments

change

which

is

not

a

frequent

operation

anyway

assume

that

in

the

scheduling?based

implementation

all

connections

utilize

a

common

pool

of

buffer

space

The

per?connection

experienced

cell

loss

ratio

under

the

best

alignment

versus

the

cell

loss

ratio

under

the

worst

and

a

random

alignment

The

random

alignment

is

constructed

by

picking

time

points

according

to

a

uniform

random

distribution

in

the

-LSB-

T

-RRB-

interval

The

random

alignment

is

converted

to

the

best

alignment

through

the

use

of

the

delays

produced

by

the

proposed

optimization

process

Two

sets

of

examples

are

considered

The

first

set

represents

traffic

originating

from

videoconferencing

applications

while

the

second

represents

a

video

distribution

service

based

on

an

actual

video

traffic

trace

The

extent

to

which

the

ffi

and

D

values

are

reasonable

for

a

number

of

configurations

The

optimization

process

guarantees

their

optimality

but

does

not

give

a

direct

idea

about

their

magnitudes

Therefore

we

observe

the

average

and

maximum

delay

assignment

and

the

subsequent

additional

buffering

that

is

necessary

In

addition

we

present

the

relation

of

the

results

produced

by

the

MS

and

the

MM

formulations

of

the

optimization

problem

The

buffer

sizing

demands

for

both

the

protocol?based

and

the

scheduling?based

implementation

The

selection

is

influenced

by

the

maximum

expected

value

of

Di

and

P

i

Di

for

respectively

the

protocol?based

and

the

scheduling?based

implementation

The

extent

to

which

the

queueing

delay

distribution

of

the

connections

is

influenced

by

enforcing

additional

delay

according

to

the

proposed

scheme

In

particular

the

squared

coefficient

of

variation

of

the

queueing

delay

is

used

as

an

indication

of

the

experienced

jitter

Since

the

traffic

is

delayed

for

the

sake

of

alignment

further

delays

at

the

multiplexer

or

more

variable

delays

would

be

an

extra

penalty

that

we

wish

to

avoid

Cell

Loss

Equalization

Examples

In

order

to

illustrate

the

effectiveness

of

the

best

alignment

and

thus

the

effectiveness

of

the

proposed

control

scheme

we

performed

a

number

of

simulations

One

set

of

simulations

closely

followed

the

assumptions

in

the

simulations

presented

in

-LSB-

-RSB-

Our

example

consists

of

sources

feeding

a

multiplexer

Each

source

sends

its

traffic

without

smoothing

over

a

Mbps

link

at

a

frame

rate

of

frames

per

second

The

multiplexer

has

a

finite

buffer

of

cells

and

is

serviced

by

an

output

link

of

Bout

Mbps

equivalent

to

a

link

Thus

the

worst

case

delay

experienced

through

buffering

at

the

multiplexer

is

msec

This

value

is

reasonably

small

for

a

single

node

for

effective

real?time

communication

We

do

not

make

any

particular

assumption

about

the

length

of

the

links

from

the

sources

to

the

multiplexer

which

can

be

potentially

long

without

harming

our

equalization

scheme

The

traffic

of

each

source

is

generated

by

a

Gamma

distribution

with

the

same

parameters

as

in

-LSB-

-RSB-

The

only

difference

with

respect

to

the

Gamma?based

traffic

model

in

-LSB-

-RSB-

is

that

we

consider

a

cells

octets

instead

of

octets

Bursts

of

more

than

cells

per

frame

are

truncated

to

satisfy

the

assumption

that

the

per?frame

burst

of

cells

can

be

transmitted

in

a

single

frame

period

at

maximum

input

link

rate

We

simulate

two

hours

of

source

activity

or

equivalently

frames

per

source

and

we

report

the

results

in

Table

Less

than

frames

per

simulated

MS

MM

CLR

Source

si

D?MS

i

ffi?MS

i

D?MM

i

ffi?MM

i

Best

Random

Worst

-LRB-

ms

-RRB-

-LRB-

cell

times

-RRB-

-LRB-

ms

-RRB-

-LRB-

cell

times

-RRB-

-LRB-

ms

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

Overall

CLR

Table

The

cell

loss

ratio

-LRB-

CLR

-RRB-

experienced

by

each

of

the

sources

under

the

worst

the

best

and

a

random

alignment

for

a

Gamma

distribution

traffic

model

The

random

alignment

corresponds

to

the

random

frame

start

times

under

the

si

column

The

best

alignment

is

achieved

using

the

equalization

protocol

on

the

random

frame

start

times

by

generating

the

respective

delays

under

the

D?MS

i

or

the

D?MM

i

column

depending

on

selected

optimization

objective

source

were

truncated

during

the

simulations

Summarizing

this

set

of

simulations

captures

the

operation

of

a

multiplexer

fed

by

traffic

generated

by

videoconferencing

applications

However

the

diversity

of

traffic

models

for

video

traffic

and

the

need

for

an

experiment

for

video

distribution

services

motivated

a

second

set

of

simulations

where

we

used

actual

traffic

traces

and

in

particular

the

almost

hour

long

Star

Wars

MPEG

sequence

-LSB-

-RSB-

Each

of

the

sources

used

in

this

set

was

sending

out

the

Star

Wars

sequence

starting

at

a

random

frame

of

the

movie

-LRB-

for

the

sake

of

completeness

we

provide

the

starting

frame

indices

for

each

source

on

Table

-RRB-

and

cycling

through

the

entire

trace

of

frames

for

the

duration

of

the

movie

The

input

link

rate

is

set

to

Mbps

and

the

output

link

rate

to

Mbps

The

buffer

size

is

set

to

cells

representing

a

maximum

queueing

delay

of

msec

The

traffic

generated

per

frame

is

fragmented

into

ATM

cells

assuming

a

payload

of

octets

per

cell

The

frame

rate

is

frames

per

second

The

results

of

a

simulation

spanning

the

duration

of

the

trace

are

presented

in

Table

In

both

sets

three

runs

of

the

simulation

were

performed

for

the

best

the

worst

and

a

random

alignment

The

sequence

of

arrivals

for

a

particular

source

was

identical

across

the

three

different

runs

ensuring

the

replication

of

the

exact

same

traffic

demands

for

each

run

MS

MM

CLR

Source

si

D?MS

i

ffi?MS

i

D?MM

i

ffi?MM

i

Best

Random

Worst

-LRB-

ms

-RRB-

-LRB-

cell

times

-RRB-

-LRB-

ms

-RRB-

-LRB-

cell

times

-RRB-

-LRB-

ms

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

Overall

CLR

Table

The

cell

loss

ratio

-LRB-

CLR

-RRB-

experienced

by

each

of

the

sources

under

the

worst

the

best

and

a

random

alignment

for

the

Star

Wars

trace

traffic

The

random

alignment

corresponds

to

the

random

frame

start

times

under

the

si

column

The

best

alignment

is

achieved

using

the

equalization

protocol

on

the

random

frame

start

times

by

generating

the

respective

delays

under

the

D?MS

i

or

the

D?MM

i

column

depending

on

selected

optimization

objective

Source

Starting

Frame

11 157641

13 125959

14 170511

15 150755

Table

The

starting

frame

index

for

each

of

the

sources

as

used

in

the

trace?driven

simulations

The

random

alignment

was

obtained

by

setting

the

transmission

epochs

to

the

si

's

of

the

second

column

of

Tables

and

The

si

's

were

taken

from

a

uniform

distribution

in

the

respective

-LSB-

T

-RRB-

interval

The

best

alignment

was

achieved

through

the

loss

equalization

scheme

by

delaying

the

traffic

that

is

generated

at

the

si

epochs

The

per?source

delay

necessary

to

achieve

the

best

alignment

is

produced

by

either

the

MS

or

the

MM

optimization

algorithm

The

values

of

D?MS

i

D?MM

i

ffi?MS

i

and

ffi?MM

i

are

presented

under

the

respective

columns

of

Tables

and

Finally

the

worst

alignment

results

are

provided

just

to

illustrate

the

extreme

case

that

a

random

alignment

can

reach

if

no

frame

alignment

control

is

enforced

First

consider

Table

The

reference

connection

according

to

MS

is

number

while

according

to

MM

it

is

connection

Note

how

the

MM

optimization

avoids

the

excessive

delays

that

MS

produced

for

connections

and

while

at

the

same

time

it

inflates

the

delays

of

and

Consequently

the

maximum

delay

under

MM

is

msec

-LRB-

source

-RRB-

compared

to

msec

of

MS

-LRB-

source

-RRB-

The

remarkable

result

is

the

difference

of

the

CLR

values

for

the

random

configurations

It

ranges

from

for

source

to

for

source

a

difference

of

almost

three

orders

of

magnitude

Thus

connection

receives

an

effective

QoS

CLR

of

and

connection

an

effective

QoS

CLR

of

Under

the

best

alignment

that

the

protocol

produces

the

cell

losses

are

equalized

around

the

overall

value

of

Note

also

that

at

the

same

time

the

overall

CLR

of

the

random

alignment

-LRB-

-RRB-

is

worse

than

that

of

the

equalized

frame

start

times

Notably

not

only

is

the

equalized

case

better

for

the

per?connection

performance

but

also

for

the

overall

performance

Finally

consider

the

worst

case

CLR

which

is

of

the

order

of

Moreover

the

losses

are

not

equalized

in

the

worst

case

This

is

an

artifact

of

the

simulation

due

to

the

fixed

way

with

which

ties

for

arrivals

at

the

same

time

point

were

broken

-LRB-

first

was

considered

the

arrival

from

the

source

with

the

smallest

index

-RRB-

A

similar

artifact

can

be

present

in

an

actual

system

when

the

frame

transmission

epochs

of

each

source

differ

only

slightly

-LRB-

less

than

the

time

for

a

cell

transmission

time

on

the

input

link

-RRB-

Without

the

operation

of

the

proposed

scheme

there

is

nothing

to

prevent

the

worst

case

configuration

Thus

a

random

configuration

can

deteriorate

to

a

worst?case

CLR

of

Table

summarizes

the

results

for

the

Star

Wars

sequence

In

the

random

alignment

case

certain

connections

are

fortunate

to

have

no

losses

at

all

This

is

a

consequence

of

the

random

point

at

which

the

respective

connection

started

playing

out

the

video

trace

-LRB-

Table

-RRB-

If

the

frame

transmission

epoch

of

a

connection

is

sufficiently

spaced

apart

from

other

transmission

epochs

and

if

the

source

is

sending

high

activity

frames

when

neighboring

sources

-LRB-

in

the

sense

of

neighboring

transmission

epochs

-RRB-

are

sending

low

activity

frames

then

it

is

possible

for

such

a

connection

to

experience

zero

losses

For

this

reason

under

the

best

alignment

sources

and

suffer

no

losses

at

all

while

source

suffers

the

most

Even

in

this

case

though

the

benefits

of

using

the

equalization

scheme

are

obvious

The

CLR

of

the

random

alignment

ranges

from

to

while

the

overall

average

CLR

is

In

sharp

contrast

the

best

alignment

brings

the

loss

ratio

of

all

sources

closer

together

and

around

the

value

of

Finally

note

that

the

reference

connection

is

number

for

both

the

MS

and

the

MM

configuration

From

our

experimentation

with

the

two

optimization

criteria

we

have

concluded

that

in

the

majority

of

cases

the

optimal

delay

assignment

under

both

MS

and

MM

is

characterized

by

the

same

reference

connection

Moreover

in

both

the

Star

Wars

and

the

videoconference

experiment

the

delay

imposed

on

the

traffic

is

lower

than

the

delay

necessary

for

smoothing

which

is

in

the

range

of

msec

A

detailed

discussion

of

the

dynamics

of

D

follows

in

the

next

subsection

The

Dynamics

of

the

Optimal

Delay

Assignment

Since

D?i

's

are

the

result

of

an

optimization

process

we

can

not

obtain

them

in

a

closed

form

For

this

reason

we

report

on

the

results

generated

by

solving

a

large

number

-LRB-

tens

of

thousands

-RRB-

of

optimization

instances

for

uniformly

generated

si

's

in

the

-LSB-

T

-RRB-

range

where

T

msec

-LRB-

frame

rate

of

frames

per

second

-RRB-

Moreover

we

vary

the

input

link

rate

Bin

from

to

to

Mbps

and

the

number

of

multiplexed

sources

from

to

The

two

different

optimization

formulations

accentuate

two

different

aspects

of

the

alignment

delay

assignment

The

MS

optimization

aims

towards

maintaining

an

overall

low

delay

overhead

with

the

hope

that

no

connection

is

delayed

more

than

absolutely

necessary

The

MM

optimization

tries

to

avoid

victimizing

particular

connections

in

terms

of

delay

more

than

all

other

connections

Intuition

suggests

that

MM

may

tend

to

inflate

the

overall

delay

Surprisingly

this

is

not

the

case

As

Figure

indicates

the

average

value

of

the

overall

delay

-LRB-

E

-LSB-

P

i

D?i

-RSB-

-RRB-

for

both

MS

and

MM

is

almost

identical

for

different

link

rates

and

for

variable

number

of

sources

The

same

holds

true

for

the

average

per?connection

delay

-LRB-

E

-LSB-

D?i

-RSB-

-RRB-

as

Figure

suggests

Increasing

number

of

sources

brings

an

anticipated

increase

in

the

overall

delay

-LRB-

more

sources

more

delays

to

sum

up

-RRB-

But

it

also

brings

the

interesting

observation

that

the

per

source

delay

depicted

in

Figure

is

reduced

dramatically

-LRB-

note

that

in

Figure

the

y?axis

is

logarithmic

-RRB-

An

explanation

is

the

following

urn?like

example

one

can

think

of

the

sources

as

partitions

of

a

box

A

number

of

marbles

equal

to

the

number

of

partitions

is

dropped

60 80 100 120

Cell

Transmission

Times

Multiplexed

Sources

N

Mbps

-LRB-

MS

-RRB-

Mbps

-LRB-

MM

-RRB-

Mbps

-LRB-

MS

-RRB-

Mbps

-LRB-

MM

-RRB-

Mbps

-LRB-

MS

-RRB-

Mbps

-LRB-

MM

-RRB-

Figure

Average

value

of

the

sum

of

alignment

delays

E

-LSB-

P

i

D?i

-RSB-

for

both

MS

and

MM

optimization

at

different

input

link

rates

Bin

The

delay

is

expressed

in

cell

transmission

times

at

the

input

link

rate

Note

that

the

lines

for

MS

and

MM

are

almost

identical

60 80 100 120

Cell

Transmission

Times

Multiplexed

Sources

N

Mbps

-LRB-

MS

-RRB-

Mbps

-LRB-

MM

-RRB-

Mbps

-LRB-

MS

-RRB-

Mbps

-LRB-

MM

-RRB-

Mbps

-LRB-

MS

-RRB-

Mbps

-LRB-

MM

-RRB-

Figure

Average

value

of

the

per?connection

alignment

delay

E

-LSB-

D?i

-RSB-

for

both

MS

and

MM

optimization

at

different

input

link

rates

Bin

The

delay

is

expressed

in

cell

transmission

times

at

the

input

link

rate

Note

that

the

lines

for

MS

and

MM

are

almost

identical

60 80 100 120

Cell

Transmission

Times

Multiplexed

Sources

N

Mbps

-LRB-

MS

-RRB-

Mbps

-LRB-

MM

-RRB-

Mbps

-LRB-

MS

-RRB-

Mbps

-LRB-

MM

-RRB-

Mbps

-LRB-

MS

-RRB-

Mbps

-LRB-

MM

-RRB-

Figure

Average

value

of

the

maximum

per?configuration

delay

E

-LSB-

maxi

D?i

-RSB-

for

both

MS

and

MM

optimization

at

different

input

link

rates

Bin

The

delay

is

expressed

in

cell

transmission

times

at

the

input

link

rate

60 80 100 120

Cell

Transmission

Times

Multiplexed

Sources

N

Maximum

-LRB-

MS

-RRB-

Maximum

-LRB-

MM

-RRB-

Aggregate

-LRB-

MS

-RRB-

Aggregate

-LRB-

MM

-RRB-

Per

Source

-LRB-

MS

-RRB-

Per

Source

-LRB-

MM

-RRB-

Figure

Comparison

of

the

averages

of

-LRB-

a

-RRB-

the

maximum

per?configuration

delay

E

-LSB-

maxi

D?i

-RSB-

-LRB-

b

-RRB-

the

aggregate

per?configuration

delay

E

-LSB-

P

i

D?i

-RSB-

and

-LRB-

c

-RRB-

the

per?source

alignment

delay

E

-LSB-

D?i

-RSB-

for

both

MS

and

MM

optimization

at

an

input

link

rate

of

Mbps

Note

that

the

only

marked

difference

is

in

the

average

value

of

the

maximum

alignment

delay

randomly

into

the

box

The

more

the

partitions

the

more

likely

it

is

that

no

more

than

one

marble

falls

within

each

partition

In

our

example

the

marbles

are

the

si

's

and

the

number

of

partitions

is

equal

to

the

number

of

sources

Hence

the

more

the

sources

the

smaller

the

average

per?source

delay

alignment

time

Another

aspect

of

the

behavior

of

D?i

's

supported

by

the

results

in

Figures

and

is

that

doubling

or

halfing

the

input

link

rate

results

in

a

proportional

increase

or

decrease

of

the

delay

times

Such

behavior

would

seem

to

cause

problems

at

higher

link

speeds

by

requiring

an

ever

increasing

buffer

in

order

to

accommodate

the

delayed

traffic

However

there

always

exists

a

limit

which

is

independent

of

the

input

link

speed

Namely

Kmax

Since

no

frame

start

time

is

delayed

more

than

T

time

units

-LRB-

ffi?i

T

-RRB-

no

more

than

Kmax

cells

-LRB-

a

frame

's

worth

of

cells

at

their

maximum

-RRB-

need

to

be

stored

Unfortunately

the

average

delay

assignment

can

not

be

used

for

engineering

the

buffers

necessary

to

delay

traffic

Instead

we

are

interested

in

the

maximum

i.e.

worst?case

buffer

size

necessary

to

accommodate

the

assigned

delays

Let

us

first

focus

on

the

expected

value

of

the

per?configuration

maximum

delay

over

all

simulated

configurations

E

-LSB-

maxi

D?i

-RSB-

Figure

shows

that

indeed

the

MM

algorithm

plays

a

crucial

role

in

restricting

the

maximum

delay

and

hence

the

maximum

necessary

buffer

size

That

is

as

Figure

indicates

the

expected

maximum

for

the

MS

optimization

remains

almost

constant

for

increasing

number

of

sources

while

for

MM

optimization

it

is

reduced

drastically

for

increasing

number

of

sources

One

would

therefore

anticipate

that

the

MM

optimization

is

the

obvious

choice

This

is

not

necessarily

true

as

we

will

see

in

the

next

subsection

Finally

Figure

summarizes

the

dynamics

of

the

delay

assignments

for

both

MS

and

MM

for

variable

number

of

sources

at

a

fixed

input

link

rate

of

Bin

Mbps

Engineering

Guidelines

The

average

value

of

the

maximum

anticipated

delay

is

not

sufficient

for

engineering

the

buffer

sizes

for

either

the

protocol?based

or

the

scheduling?based

implementations

Instead

we

would

like

to

know

whether

for

a

certain

fixed

buffer

size

Q

we

are

able

to

support

the

delays

produced

by

the

optimization

process

Alternatively

we

would

like

to

select

the

minimum

buffer

size

Qmin

that

ensures

that

the

necessary

delays

produced

by

the

scheme

can

be

satisfied

of

the

cases

In

the

protocol?based

implementation

the

fixed

size

buffer

is

assigned

to

each

individual

source

In

the

scheduling?based

implementation

it

is

assigned

for

all

sources

and

it

is

located

at

the

multiplexer

For

this

study

we

introduce

a

probability

quantity

that

we

will

call

probability

of

coverage

In

the

protocol?based

implementation

it

is

interpreted

as

Pr

-LSB-

maxi

D?i

Q

-RSB-

while

in

the

scheduling?based

implementation

it

is

interpreted

as

Pr

-LSB-

P

i

D?i

Q

-RSB-

To

this

extent

we

present

the

results

in

Figures

and

for

MS

and

MM

optimization

respectively

for

the

protocol?based

implementation

assuming

an

input

link

rate

of

Mbps

Note

that

the

results

are

influenced

only

by

the

input

link

rate

Bin

and

the

assumption

that

the

si

's

are

uniformly

distributed

in

the

-LSB-

T

-RRB-

interval

The

surface

depicting

the

MM

optimization

flattens

at

faster

than

the

surface

for

the

MS

curve

In

fact

most

of

the

flat

area

of

the

MS

surface

is

not

exactly

at

but

at

lower

values

A

better

look

at

the

same

set

of

experiments

is

presented

in

Figures

-LRB-

sources

but

variable

input

link

rate

-RRB-

and

-LRB-

Mbps

input

link

rate

but

variable

number

of

sources

-RRB-

comparing

the

MS

and

the

MM

results

As

we

can

see

choosing

MM

makes

sense

for

moderate

to

Multiplexed

Sources

N

Buffer

Size

Q

Prob

Coverage

-LRB-

-RRB-

Figure

Probability

of

coverage

for

the

protocol?based

implementation

Pr

-LSB-

maxi

D?i

Q

-RSB-

-LRB-

displayed

as

a

percentage

-RRB-

under

the

MS

optimization

for

variable

number

of

multiplexed

sources

N

and

variable

buffer

size

Q

The

input

link

rate

is

Mbps

large

buffer

sizes

-LRB-

for

larger

buffer

values

than

the

ones

of

the

crossover

points

at

and

-RRB-

while

for

smaller

buffer

sizes

it

is

preferable

to

use

the

MS

optimization

Hence

if

because

of

engineering

constraints

we

have

to

minimize

the

buffer

sizes

allocated

to

delaying

traffic

a

good

choice

for

assigning

delays

is

the

MS

optimization

If

we

are

relatively

flexible

in

selecting

buffer

sizes

then

the

MM

optimization

is

The

results

in

Figures

imply

that

for

decreasing

number

of

admitted

sources

the

necessary

buffer

size

increases

For

example

if

we

maintain

Permanent

Virtual

Circuits

-LRB-

PVC

-RRB-

conveying

video

traffic

and

we

use

MM

optimization

then

any

other

non?permanent

video

connection

established

through

the

same

multiplexer

does

not

need

more

than

cells

allocated

for

the

equalization

scheme

However

if

there

are

no

PVCs

and

the

number

of

admitted

sources

varies

through

time

then

the

necessary

per?source

buffer

can

reach

its

maximum

value

when

the

number

of

admitted

is

The

calculation

of

the

per?source

buffer

necessary

for

the

coverage

assuming

only

two

sources

is

trivial

because

only

one

of

the

two

sources

is

delayed

for

a

maximum

of

T

time

units

Hence

the

minimum

per?source

buffer

that

can

successfully

support

the

scheme

is

Qmin

d

-LRB-

T

-RRB-

-LRB-

D

Bin

-RRB-

e

-LRB-

-RRB-

For

Bin

Mbps

Qmin

is

a

very

modest

cells

Different

steps

are

necessary

for

dimensioning

the

pipeline

buffer

in

the

scheduling?based

implementation

As

Figure

suggests

the

number

of

multiplexed

sources

plays

a

crucial

role

However

this

time

the

transition

to

the

coverage

is

very

steep

and

it

does

not

differ

significantly

for

the

MS

and

the

MM

formulation

-LRB-

therefore

we

only

present

the

graph

for

the

MS

optimization

in

Figure

-RRB-

The

main

feature

is

that

increasing

the

sources

implies

a

source

receives

a

request

to

delay

its

traffic

more

than

what

is

possible

with

its

available

buffer

size

we

assume

that

it

delays

it

for

the

maximum

it

can

that

for

only

one

source

there

is

no

need

for

alignment

delays

Multiplexed

Sources

N

Buffer

Size

Q

Prob

Coverage

-LRB-

-RRB-

Figure

Probability

of

coverage

for

the

protocol?based

implementation

Pr

-LSB-

maxi

D?i

Q

-RSB-

-LRB-

displayed

as

a

percentage

-RRB-

under

the

MM

optimization

for

variable

number

of

multiplexed

sources

N

and

variable

buffer

size

Q

The

input

link

rate

is

Mbps

50 100 150 200

Prob

Coverage

-LRB-

-RRB-

Buffer

Size

Q

Mbps

-LRB-

MS

-RRB-

Mbps

-LRB-

MM

-RRB-

Mbps

-LRB-

MS

-RRB-

Mbps

-LRB-

MM

-RRB-

Mbps

-LRB-

MS

-RRB-

Mbps

-LRB-

MM

-RRB-

Figure

Probability

of

coverage

for

the

protocol?based

implementation

Pr

-LSB-

maxi

D?i

Q

-RSB-

-LRB-

displayed

as

a

percentage

-RRB-

under

both

MS

and

MM

optimization

for

different

input

link

rates

Bin

sources

and

variable

buffer

size

Q.

50 100 150 200

Prob

Coverage

-LRB-

-RRB-

Buffer

Size

Q

N

-LRB-

MS

-RRB-

N

-LRB-

MM

-RRB-

N

-LRB-

MS

-RRB-

N

-LRB-

MM

-RRB-

N

-LRB-

MS

-RRB-

N

-LRB-

MM

-RRB-

Figure

Probability

of

coverage

for

the

protocol?based

implementation

Pr

-LSB-

maxi

D?i

Q

-RSB-

-LRB-

displayed

as

a

percentage

-RRB-

under

both

MS

and

MM

optimization

for

different

number

of

sources

input

link

rate

of

Mbps

and

variable

buffer

size

Q.

increase

of

the

necessary

maximum

pipeline

buffer

Consequently

we

must

dimension

the

pipeline

buffer

based

on

the

maximum

number

of

anticipated

connections

Fortunately

other

concerns

e.g.

cell

loss

ratio

already

place

a

cap

on

the

number

of

connections

that

can

be

supported

For

example

by

inspecting

Figure

we

can

conclude

that

if

a

maximum

of

connections

can

be

admitted

a

pipeline

buffer

of

around

Qmin

cells

is

sufficient

to

support

the

scheduling?based

implementation

of

the

scheme

Finally

for

completeness

consider

Figure

which

displays

the

dependency

of

probability

of

coverage

-LRB-

and

hence

that

of

Qmin

-RRB-

on

the

input

link

rate

Bin

Multiplexer

Queueing

Delay

Our

results

suggest

that

a

small

additional

delay

can

severely

improve

the

cell

loss

fairness

However

the

coupling

that

is

put

between

the

delay

the

cell

losses

and

the

cell

loss

fairness

becomes

an

issue

in

the

context

of

the

overall

Quality

of

Service

-LRB-

QoS

-RRB-

In

particular

our

equalization

scheme

delays

the

traffic

for

the

sake

of

an

ideal

alignment

We

would

anticipate

that

the

traffic

is

not

penalized

any

further

in

terms

of

delay

at

the

multiplexer

The

intuitive

argument

is

that

the

best

alignment

equi?distributes

the

load

of

arrivals

over

the

frame

interval

Thus

the

delay

should

be

lower

than

in

the

random

case

To

illustrate

this

we

return

to

the

simulation

experiments

using

the

Gamma

and

the

Star

Wars

simulations

and

we

evaluate

the

delay

performance

of

the

aggregate

traffic

as

well

as

that

of

an

individual

connection

-LRB-

We

pick

source

to

be

the

individually

observed

source

in

all

our

simulation

experiments

-RRB-

Figures

-LRB-

for

the

aggregate

traffic

-RRB-

and

-LRB-

for

the

traffic

of

connection

-RRB-

present

the

sample

distribution

of

the

delay

at

the

multiplexer

i.e.

not

including

alignment

delays

for

the

worst

a

random

and

the

best

alignment

for

the

Gamma?based

simulation

Evidently

the

Multiplexed

Sources

N

Buffer

Size

Q

Prob

Coverage

-LRB-

-RRB-

Figure

Probability

of

coverage

for

the

scheduling?based

implementation

Pr

-LSB-

P

i

D?i

Q

-RSB-

-LRB-

displayed

as

a

percentage

-RRB-

under

the

MS

optimization

for

variable

number

of

multiplexed

sources

N

and

variable

buffer

size

Q

The

input

link

rate

is

Mbps

Note

that

the

corresponding

graph

for

the

MM

optimzation

is

very

similar

to

this

one

Prob

Coverage

-LRB-

-RRB-

Buffer

Size

Q

Mbps

-LRB-

MS

-RRB-

Mbps

-LRB-

MM

-RRB-

Mbps

-LRB-

MS

-RRB-

Mbps

-LRB-

MM

-RRB-

Mbps

-LRB-

MS

-RRB-

Mbps

-LRB-

MM

-RRB-

Figure

Probability

of

coverage

for

the

scheduling?based

implementation

Pr

-LSB-

P

i

D?i

Q

-RSB-

-LRB-

displayed

as

a

percentage

-RRB-

under

both

MS

and

MM

optimization

for

different

input

link

rates

Bin

sources

and

variable

buffer

size

Q.

Prob

Coverage

-LRB-

-RRB-

Buffer

Size

Q

N

-LRB-

MS

-RRB-

N

-LRB-

MM

-RRB-

N

-LRB-

MS

-RRB-

N

-LRB-

MM

-RRB-

N

-LRB-

MS

-RRB-

N

-LRB-

MM

-RRB-

Figure

Probability

of

coverage

for

the

scheduling?based

implementation

Pr

-LSB-

P

i

D?i

Q

-RSB-

-LRB-

displayed

as

a

percentage

-RRB-

under

both

MS

and

MM

optimization

for

different

number

of

sources

input

link

rate

of

Mbps

and

variable

buffer

size

Q.

Delay

Probability

msec

Worst

-LRB-

m

-RRB-

Random

-LRB-

m

-RRB-

Best

-LRB-

m

-RRB-

Figure

Sample

distribution

of

the

multiplexer

delay

under

the

best

the

worst

and

the

random

alignment

of

Table

for

Gamma

distribution

traffic

Delay

Probability

msec

Worst

-LRB-

m

-RRB-

Random

-LRB-

m

-RRB-

Best

-LRB-

m

-RRB-

Figure

Sample

distribution

of

the

multiplexer

delay

under

the

best

the

worst

and

the

random

alignment

of

Table

for

the

Star

Wars

trace

traffic

Delay

Probability

msec

Worst

-LRB-

m

-RRB-

Random

-LRB-

m

-RRB-

Best

-LRB-

m

-RRB-

Figure

Sample

distribution

of

the

multiplexer

delay

under

the

best

the

worst

and

the

random

alignment

of

Table

for

Gamma

distribution

traffic

as

experienced

by

connection

number

Delay

Probability

msec

Worst

-LRB-

m

-RRB-

Random

-LRB-

m

-RRB-

Best

-LRB-

m

-RRB-

Figure

Sample

distribution

of

the

multiplexer

delay

under

the

best

the

worst

and

the

random

alignment

of

Table

for

the

Star

Wars

trace

traffic

as

experienced

by

connection

number

Delay

Probability

msec

Random

-LRB-

m

c

-RRB-

Best

-LRB-

MS

-RRB-

-LRB-

m

c

-RRB-

Best

-LRB-

MM

-RRB-

-LRB-

m

c

-RRB-

Figure

Sample

distribution

of

the

total

delay

-LRB-

inclusive

of

alignment

delays

-RRB-

under

the

best

the

worst

and

the

random

alignment

of

Table

for

Gamma

distribution

traffic

Delay

Probability

msec

Random

-LRB-

m

c

-RRB-

Best

-LRB-

MS

-RRB-

-LRB-

m

c

-RRB-

Best

-LRB-

MM

-RRB-

-LRB-

m

c

-RRB-

Figure

Sample

distribution

of

the

total

delay

-LRB-

inclusive

of

alignment

delays

-RRB-

under

the

best

the

worst

and

the

random

alignment

of

Table

for

the

Star

Wars

trace

traffic

average

delay

-LRB-

denoted

with

m

in

the

plots

-RRB-

of

the

aggregate

traffic

greatly

improves

from

the

worst

-LRB-

msec

-RRB-

to

the

random

-LRB-

msec

-RRB-

to

the

best

alignment

-LRB-

msec

-RRB-

and

so

does

the

average

delay

of

the

isolated

connections

-LRB-

from

to

to

msec

-RRB-

Similar

observations

hold

for

the

Star

Wars

simulation

presented

in

Figures

-LRB-

aggregate

-RRB-

and

-LRB-

individual

-RRB-

The

worst

alignment

results

in

a

very

heavy

tail

as

we

would

expect

from

the

heavy

losses

In

addition

it

demonstrates

a

regular

shape

in

the

sample

distribution

plots

The

almost

sawtooth

behavior

is

justified

by

the

following

observation

the

arrivals

at

the

beginning

of

the

frame

interval

can

be

considered

to

occur

in

batches

of

a

fixed

size

equal

to

the

number

of

sources

Therefore

for

our

examples

the

batch

size

is

Until

the

next

batch

arrival

only

Bout

Bin

cells

can

be

serviced

For

Bin

Mbps

and

Bout

Mbps

only

Bout

Bin

cells

can

be

serviced

on

average

For

an

initially

empty

queue

after

cells

arrive

get

serviced

and

the

next

batch

finds

the

queue

with

an

average

of

cells

representing

an

average

delay

of

around

msec

for

the

next

incoming

batch

The

msec

is

the

point

where

the

first

sawtooth

rises

in

Figure

The

next

batch

will

very

likely

find

the

queue

with

an

average

of

cells

or

almost

msec

of

delay

time

-LRB-

where

the

second

sawtooth

rises

-RRB-

and

so

on

for

each

sawtooth

starting

at

k

msec

for

k

etc.

Of

course

the

number

of

batches

of

size

and

the

subsequent

batches

of

lesser

size

during

the

frame

interval

are

not

deterministic

and

this

results

in

the

overall

stochastic

behavior

depicted

by

the

plots

However

the

important

conclusion

from

these

figures

is

that

the

additional

alignment

delay

results

in

an

overall

reduced

delay

at

the

multiplexer

Thus

the

sources

do

not

get

penalized

twice

in

terms

of

delay

i.e.

both

for

the

alignment

and

at

the

multiplexer

The

investigation

of

the

delay

performance

is

not

complete

without

a

look

at

the

jitter

performance

of

the

connections

If

the

alignment

inflates

the

delays

for

the

sake

of

cell

losses

it

should

not

do

so

at

the

cost

of

the

jitter

performance

For

the

study

of

jitter

we

consider

Delay

Probability

msec

Random

-LRB-

m

c

-RRB-

Best

-LRB-

MS

-RRB-

-LRB-

m

c

-RRB-

Best

-LRB-

MM

-RRB-

-LRB-

m

c

-RRB-

Figure

Sample

distribution

of

the

total

delay

-LRB-

inclusive

of

alignment

delays

-RRB-

under

the

best

the

worst

and

the

random

alignment

of

Table

for

Gamma

distribution

traffic

as

experienced

by

connection

number

Delay

Probability

msec

Random

-LRB-

m

c

-RRB-

Best

-LRB-

MS

-RRB-

-LRB-

m

c

-RRB-

Best

-LRB-

MM

-RRB-

-LRB-

m

c

-RRB-

Figure

Sample

distribution

of

the

total

delay

-LRB-

inclusive

of

alignment

delays

-RRB-

under

the

best

the

worst

and

the

random

alignment

of

Table

for

the

Star

Wars

trace

traffic

as

experienced

by

connection

number

the

delay

distributions

that

include

the

alignment

delays

in

addition

to

the

multiplexer

delays

This

setting

is

more

representative

of

the

delay

experienced

by

connections

from

the

traffic

generation

point

and

up

to

the

departure

from

the

multiplexer

In

particular

we

will

consider

the

squared

coefficient

of

variation

-LRB-

denoted

in

the

plots

by

c

-RRB-

as

a

convenient

way

to

quantify

jitter

Removing

from

consideration

the

worst

case

we

focus

on

a

comparison

between

the

results

for

the

best

alignment

under

MS

and

MM

against

the

random

alignment

Figures

-LRB-

aggregate

traffic

-RRB-

and

-LRB-

individual

traffic

from

connection

-RRB-

present

the

results

for

the

Gamma?based

simulation

while

Figures

and

present

the

corresponding

results

for

the

Star

Wars

trace

traffic

simulation

Note

that

the

peaks

of

the

aggregate

distribution

for

the

best

alignment

under

MS

and

MM

correspond

to

the

alignment

delays

under

the

respective

optimization

For

example

the

peak

to

the

right

of

in

Figure

corresponds

to

the

delay

-LRB-

msec

-RRB-

imposed

on

source

under

MS

as

indicated

on

Table

Similarly

the

peak

to

the

left

of

in

Figure

corresponds

to

the

delay

-LRB-

msec

-RRB-

imposed

on

source

under

MM

of

the

same

table

Similarly

the

two

peaks

on

the

extreme

right

side

of

Figure

correspond

to

the

alignment

delays

for

sources

-LRB-

msec

-RRB-

and

-LRB-

msec

-RRB-

The

important

result

is

that

the

s.c.v.

of

delay

is

reduced

for

the

best

alignment

-LRB-

for

either

MS

or

MM

-RRB-

compared

to

the

random

case

In

fact

if

we

focus

on

the

delay

distribution

of

one

particular

connection

-LRB-

as

we

do

in

Figures

and

-RRB-

the

s.c.v.

is

essentially

zeroed

Since

the

jitter

absorption

is

likely

performed

on

a

per?connection

basis

the

s.c.v.

values

for

the

individual

connection

are

more

representative

of

the

actual

jitter

than

the

values

for

the

aggregate

traffic

Hence

the

presented

scheme

achieves

not

only

fair

losses

but

low

jitter

as

well

Conclusions

In

this

paper

we

have

proposed

a

scheme

that

rectifies

the

cell

loss

ratio

fairness

problem

across

multiple

video

connections

that

feed

the

same

finite

buffer

multiplexer

The

scheme

can

be

implemented

either

as

a

protocol

between

sources

and

multiplexer

or

as

a

particular

scheduling

discipline

Through

simulation

experiments

using

both

traffic

models

and

actual

traffic

traces

we

have

verified

the

effectiveness

of

the

scheme

in

terms

of

cell

loss

ratio

fairness

We

also

have

demonstrated

that

the

scheme

results

in

reduced

delay

variability

that

directly

benefits

real?time

video

connections

Moreover

we

have

investigated

the

anticipated

buffer

overheads

necessary

to

support

the

scheme

Finally

we

have

presented

two

underlying

optimization

choices

that

capture

the

sensitivity

to

increased

delays

and

we

have

illustrated

through

examples

the

tradeoff

of

selecting

one

over

the

other

as

it

relates

to

buffer

overheads

Open

problems

in

the

same

context

that

are

worth

particular

attention

include

the

generalization

to

heterogeneous

sources

and

different

frame

rates

and

the

generalization

to

a

cluster

of

networks

elements

instead

of

an

individual

multiplexer

One

can

also

view

the

optimization

of

frame

transmission

epochs

as

a

burst?level

scheduling

scheme

that

strives

to

minimize

delays

inside

the

network

by

enforcing

controlled

delays

at

the

network

access

points

As

such

it

can

have

ramifications

through

suitable

extensions

to

the

scheduling

of

other

types

of

bursty

traffic

apart

from

video

Acknowledgements

We

thank

Professors

James

F.

Kurose

and

Don

Towsley

of

the

Univ.

of

Massachusetts

at

Amherst

for

their

constructive

comments

on

earlier

versions

of

this

paper

We

are

also

grateful

to

Dr.

Mark

W.

Garrett

of

Bellcore

and

Professor

Martin

Vetterli

of

Univ.

California

at

Berkeley

for

making

available

the

Star

Wars

video

trace

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Bandwidth

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