Real-Time

Block

Transfer

Under

a

Link

Sharing

Hierarchy

Geoffrey

G.

Xie

and

Simon

S.

Lam

Department

of

Computer

Sciences

The

University

of

Texas

at

Austin

Austin

Texas

fxie,lamg@cs.utexas.edu

June

Abstract

Most

application-level

data

units

are

too

large

to

be

carried

in

a

single

packet

-LRB-

or

cell

-RRB-

and

must

be

fragmented

for

network

delivery

For

many

applications

the

end-to-end

delays

and

loss

rate

of

their

data

units

are

much

more

relevant

measures

of

performance

than

QoS

parameters

specified

for

individual

packets

-LRB-

or

cells

-RRB-

The

concept

of

a

block

-LRB-

or

burst

-RRB-

was

introduced

to

represent

a

sequence

of

cells

which

may

contain

an

entire

application

data

unit

a

portion

of

one

or

multiple

application

data

units

In

this

paper

we

describe

how

a

real-time

VBR

service

with

QoS

parameters

for

block

transfer

delay

block

loss

rate

and

block

delay

variation

can

be

provided

by

integrating

concepts

and

delay

guarantee

results

from

our

previous

work

on

burst

scheduling

together

with

ideas

from

ATM

block

transfer

and

hierarchical

link

sharing

Two

new

contributions

are

presented

First

we

show

how

to

extend

existing

end-to-end

delay

bounds

for

bursts

to

networks

with

hierarchical

link

sharing

Second

we

design

an

admission

control

algorithm

to

provide

the

following

classes

of

service

bounded-delay

block

transfer

with

no

loss

and

bounded-delay

block

transfer

with

a

specified

block

loss

rate

Introduction

For

many

applications

the

end-to-end

delay

of

an

application-level

data

unit

is

a

much

more

relevant

performance

measure

than

the

end-to-end

delays

of

individual

packets

-LRB-

or

cells

-RRB-

within

a

network

For

example

a

video

picture

-LRB-

or

file

-RRB-

being

sent

by

an

application

over

an

IP

network

may

be

segmented

into

a

sequence

of

IP

datagrams

The

delay

incurred

to

deliver

the

entire

video

picture

-LRB-

or

file

-RRB-

is

much

more

important

to

the

application

than

the

delays

of

individual

IP

datagrams

As

another

example

an

IP

datagram

encapsulating

some

email

message

may

be

segmented

into

a

sequence

of

cells

for

delivery

over

an

ATM

network

The

delay

incurred

to

deliver

the

entire

email

message

is

more

important

than

the

delays

of

individual

cells

From

these

observations

we

proposed

the

class

of

burst

scheduling

networks

-LSB-

-RSB-

The

ATM

block

transfer

-LRB-

ABT

-RRB-

capability

being

standardized

by

ITU-T

is

based

upon

a

similar

observation

-LSB-

-RSB-

ABT

has

been

known

for

some

time

as

a

fast

reservation

protocol

The

objective

of

ABT

is

to

allow

a

data

source

to

dynamically

negotiate

its

bandwidth

reservation

on

the

basis

of

a

block

of

cells

Note

that

a

higher-layer

protocol

data

unit

fragmented

into

a

number

of

ATM

cells

is

lost

if

any

one

of

its

cells

is

lost

Therefore

even

a

low

cell

loss

rate

can

cause

a

significant

loss

rate

for

the

higher-layer

protocol

i.e.

the

higher-layer

protocol

's

throughput

may

be

much

less

than

the

protocol

session

's

throughput

measured

in

delivered

cells

The

concept

of

a

block

was

introduced

to

represent

a

sequence

of

cells

which

may

contain

a

portion

of

a

data

unit

a

single

data

unit

or

multiple

data

units

for

the

higher-layer

protocol

A

block

is

bracketed

by

two

RM

cells

A

leading

RM

cell

requests

a

reserved

bandwidth

for

the

block

and

a

trailing

RM

cell

releases

the

reserved

bandwidth

Cells

are

handled

in

blocks

in

particular

a

block

of

cells

is

either

discarded

or

accepted

entirely

by

an

ATM

switch

-LRB-

This

is

similar

to

the

idea

of

early

packet

discard

in

recent

studies

on

IP

over

ATM

-LSB-

-RSB-

-RRB-

For

the

ABT

service

the

concept

of

cell

loss

rate

can

be

generalized

to

block

loss

rate

Such

a

generalization

is

backward-compatible

with

the

existing

ATM

TM

service

architecture

-LSB-

-RSB-

since

a

block

is

a

sequence

of

cells

with

a

single

cell

being

a

special

case

Similarly

the

concept

of

cell

transfer

delay

for

real-time

VBR

services

can

be

generalized

to

block

transfer

delay

which

we

believe

is

a

more

useful

performance

measure

for

some

applications

for

example

if

every

picture

in

a

video

sequence

is

carried

by

a

block

of

cells

then

the

block

transfer

delays

are

the

same

as

picture

delays

In

this

paper

we

describe

how

a

real-time

VBR

service

with

QoS

parameters

for

block

transfer

delay

block

loss

rate

and

block

delay

variation

can

be

provided

by

integrating

concepts

and

delay

guarantee

results

from

our

previous

work

on

burst

scheduling

-LSB-

-RSB-

and

group

priority

-LSB-

-RSB-

together

with

ideas

from

ATM

block

transfer

and

hierarchical

link

sharing

described

below

It

is

envisioned

that

future

integrated

services

networks

will

support

not

only

link

sharing

by

multiple

service

classes

-LRB-

e.g.

real-time

service

best-effort

service

-RRB-

but

also

by

multiple

administrative

classes

-LRB-

e.g.

different

agencies

and

organizations

-RRB-

-LSB-

-RSB-

Specifically

packets

-LRB-

or

cells

-RRB-

from

different

sessions

belonging

to

different

service

classes

and

administrative

classes

interact

with

one

another

when

they

are

statistically

multiplexed

at

an

output

link

of

a

switch

The

switch

's

packet

scheduling

algorithm

plays

an

important

role

in

controlling

such

link

sharing

Hierarchical

link

sharing

has

been

proposed

as

a

solution

-LSB-

-RSB-

There

are

two

new

contributions

in

this

paper

First

we

extend

the

end-to-end

delay

bounds

for

from

networks

in

which

links

are

shared

by

service

classes

only

-LSB-

-RSB-

to

networks

in

which

links

are

hierarchically

shared

-LRB-

e.g.

by

administrative

classes

first

and

by

then

service

classes

within

each

administrative

class

-RRB-

To

do

so

we

prove

two

theorems

that

are

used

to

derive

packet

delay

guarantees

by

a

fluctuation

constrained

server

from

packet

delay

guarantees

by

a

constant-rate

server

The

theorems

are

general

They

are

proved

for

a

large

class

of

well

known

servers

Second

we

design

an

admission

control

algorithm

for

real-time

block

transfer

which

provides

the

following

service

classes

bounded-delay

block

transfer

with

no

loss

-LRB-

deterministic

delay

guarantee

-RRB-

and

bounded-delay

block

transfer

with

a

specified

block

loss

rate

-LRB-

statistical

delay

guarantee

-RRB-

The

balance

of

this

paper

is

organized

as

follows

In

Section

we

describe

the

basic

ideas

of

ATM

block

transfer

burst

scheduling

and

block-based

admission

control

The

end-to-end

delay

bounds

of

burst

scheduling

are

shown

In

Section

we

introduce

hierarchical

link

sharing

We

model

each

logical

server

in

a

link

sharing

hierarchy

as

a

fluctuation

constrained

-LRB-

FC

-RRB-

server

and

show

how

to

extend

the

endto-end

delay

bounds

shown

to

networks

with

hierarchical

link

sharing

In

Section

we

design

the

admission

control

algorithm

In

Section

we

present

experimental

results

from

a

simulator

driven

by

MPEG

video

traces

Real-time

Block

Transfer

ABT

and

Block-based

Admission

Control

In

ABT

the

dynamic

bandwidth

reservation

and

allocation

for

a

block

of

cells

can

be

carried

out

in

two

ways

-LRB-

i

-RRB-

ABT

with

Delayed

Transmission

-LRB-

ABT/DT

-RRB-

and

-LRB-

ii

-RRB-

ABT

with

Immediate

Transmission

-LRB-

ABT/IT

-RRB-

For

our

discussion

we

focus

on

the

latter

In

ABT/IT

the

block

is

sent

immediately

after

the

preceding

RM

cell

which

contains

a

request

for

a

peak

cell

rate

and

cell

delay

variance

tolerance

-LRB-

This

is

why

ABT

is

also

known

as

the

fast

reservation

protocol

-LSB-

-RSB-

-RRB-

The

block

proceeds

on

a

switch

by

switch

basis

with

each

switch

either

forwarding

the

block

with

guaranteed

QoS

for

every

cell

in

the

block

or

discarding

the

entire

block

if

the

required

bandwidth

is

not

available

In

other

words

the

switches

perform

block-based

admission

control

delay

bounds

for

blocks

are

obtained

by

specifying

a

burst

-LSB-

-RSB-

to

represent

a

block

of

ATM

cells

short

overview

of

ABT

can

be

found

in

-LSB-

-RSB-

With

block-based

admission

control

cell

losses

are

concentrated

over

a

small

number

of

blocks

and

bandwidth

is

not

wasted

on

delivery

of

partial

blocks

Therefore

ABT

is

able

to

avoid

the

situation

in

which

cell

losses

spread

over

a

large

number

of

higher-layer

data

units

causing

throughput

degradation

of

such

data

units

With

block-based

admission

control

ABT

is

also

able

to

offer

QoS

measured

in

terms

of

blocks

In

particular

the

concept

of

cell

loss

rate

can

be

generalized

to

block

loss

rate

Burst

Scheduling

For

Burst

Scheduling

networks

we

have

modeled

a

flow

as

a

sequence

of

bursts

-LSB-

-RSB-

A

burst

is

a

sequence

of

packets

which

corresponds

to

a

block

in

ABT

with

some

small

difference

In

particular

instead

of

two

special

packets

being

used

the

first

packet

of

each

burst

is

marked

and

stored

in

it

the

information

on

the

size

of

the

burst

-LRB-

in

number

of

packets

-RRB-

and

the

average

rate

of

the

burst

Moreover

for

efficient

packet

scheduling

and

delay

jitter

control

packets

of

each

burst

satisfy

a

jitter

timing

constraint

and

a

separation

timing

constraint

-LSB-

-RSB-

The

following

delay

bound

results

are

taken

from

-LSB-

-RSB-

and

-LSB-

-RSB-

End-to-end

delay

bounds

for

Burst

Scheduling

networks

Consider

a

flow

that

traverses

a

sequence

of

nodes

indexed

by

K

where

node

denotes

the

source

node

K

the

destination

and

the

other

nodes

packet

switches

If

the

capacity

of

every

link

on

the

network

path

is

not

exceeded

then

the

end-to-end

delay

of

every

burst

m

of

the

flow

denoted

by

Dm

m

has

the

following

upper

and

lower

bounds

Dm

bm

gm

m

-LRB-

K

-RRB-

max

n

mf

gn

n

g

KX

k

k

-LRB-

-RRB-

Dm

-LRB-

K

-RRB-

gm

m

KX

k

k

-LRB-

-RRB-

where

we

have

bm

size

of

burst

m

-LRB-

packets

-RRB-

m

average

rate

of

burst

m

-LRB-

packets/second

-RRB-

gm

group

size

-LRB-

in

packets

-RRB-

chosen

for

burst

m

by

the

switches

for

efficient

scheduling

gm

bm

k

a

small

constant

associated

with

the

link

from

switch

k

to

k

Our

design

of

Burst

Scheduling

networks

consists

of

specialized

components

such

as

source

regulator

and

the

flow

regulator

But

the

concept

of

burst

scheduling

i.e.

providing

bounded

delays

to

bursts

-LRB-

blocks

-RRB-

when

link

capacity

is

not

exceeded

is

quite

general

It

can

be

realized

by

many

designs

other

than

ours

In

the

balance

of

this

paper

we

refer

to

burst

scheduling

as

a

general

concept

independent

of

the

design

details

Integration

of

ABT

and

Burst

Scheduling

ABT

is

able

to

provide

QoS

measured

by

block

losses

through

the

use

of

block-based

admission

control

Burst

scheduling

networks

provide

bounded

block

transfer

delays

when

link

capacity

is

not

exceeded

Integrating

the

concepts

and

results

from

ABT

and

burst

scheduling

we

define

a

real-time

VBR

service

called

real-time

block

transfer

that

provides

the

following

classes

of

service

-LRB-

i

-RRB-

bounded-delay

block

transfer

with

zero

block

loss

and

-LRB-

ii

-RRB-

bounded-delay

with

a

specified

block

loss

rate

Specifically

real-time

block

transfer

is

based

on

burst

scheduling

For

class

-LRB-

i

-RRB-

peak

rate

reservation

at

the

flow

level

can

be

used

to

ensure

link

capacity

not

exceeded

without

discarding

any

blocks

For

class

-LRB-

ii

-RRB-

overbooking

of

link

capacity

at

the

flow

level

is

allowed

block-based

admission

control

is

used

to

ensure

link

capacity

not

exceeded

by

discarding

blocks

if

necessary

Admission

control

at

the

flow

level

is

also

required

to

control

the

degree

of

link

overbooking

which

is

necessary

for

limiting

the

block

loss

rate

at

a

specified

value

A

flow

level

admission

control

algorithm

for

real-time

block

transfer

is

presented

in

Section

Hierarchical

Link

Sharing

Existing

delay

bounds

for

bursts

-LRB-

blocks

-RRB-

e.g.

the

one

presented

in

Section

were

derived

for

networks

with

a

flat

one-level

link

sharing

structure

Specifically

each

link

in

the

networks

is

modeled

as

a

single

server

In

this

section

we

discuss

how

to

generalize

the

delay

bounds

to

networks

in

which

links

are

hierarchically

shared

Directed

Tree

Model

First

we

describe

a

directed

tree

model

largely

borrowed

from

-LSB-

-RSB-

for

representation

of

a

link

sharing

hierarchy

-LRB-

See

Figure

-RRB-

The

root

node

denoted

by

R

corresponds

to

the

physical

link

each

leaf

node

corresponds

to

a

flow

with

a

queue

of

packets

and

each

non-leaf

node

-LRB-

except

for

the

root

-RRB-

corresponds

to

a

link

sharing

entity

e.g.

an

administrative

agency

a

traffic

type

or

a

service

class

A

non-leaf

node

n

is

called

backlogged

if

at

least

one

flow

in

its

leaf

descendent

node

set

denoted

by

leaf

-LRB-

n

-RRB-

is

backlogged

Conceptually

node

n

is

a

logical

server

for

its

descendents

The

amount

of

work

done

by

n

in

the

time

interval

-LSB-

-RSB-

is

defined

to

be

Wn

-LRB-

-RRB-

-LRB-

n

-RRB-

Wf

-LRB-

-RRB-

where

Wf

-LRB-

-RRB-

is

the

amount

of

flow

f

traffic

served

in

the

interval

-LSB-

-RSB-

To

achieve

link

sharing

each

node

m

is

assigned

a

weight

OEm

fnRnRf

Figure

Link

sharing

tree

model

which

is

a

relative

measure

of

the

link

share

desired

by

entity

m.

For

ease

of

presentation

and

without

loss

of

generality

we

assume

that

OER

-LRB-

-RRB-

X

-LRB-

n

-RRB-

OEm

OEn

non-leaf

node

n

-LRB-

-RRB-

Notation

Let

H

-LRB-

m

-RRB-

be

the

number

of

ancestors

that

node

m

has

Let

ph

-LRB-

m

-RRB-

be

the

ancestor

node

of

m

that

is

h

levels

higher

than

m

in

the

tree

h

H

-LRB-

m

-RRB-

Clearly

-LRB-

m

-RRB-

parent

-LRB-

m

-RRB-

and

pH

-LRB-

m

-RRB-

-LRB-

m

-RRB-

R.

For

ease

of

presentation

we

also

set

-LRB-

m

-RRB-

m.

Assume

that

the

link

bandwidth

is

C

-LRB-

bits/second

-RRB-

Ideally

the

link

share

of

each

logical

server

-LRB-

non-leaf

node

-RRB-

n

at

any

time

should

be

at

least

rn

OEn

C

that

is

for

any

time

interval

-LSB-

-RSB-

in

which

n

is

continuously

backlogged

the

following

should

hold

Wn

-LRB-

-RRB-

rn

-LRB-

-RRB-

-LRB-

-RRB-

We

refer

to

-LRB-

-RRB-

as

the

ideal

link

sharing

condition

for

logical

server

n.

With

a

link

sharing

hierarchy

packets

of

each

flow

are

scheduled

jointly

by

all

ancestor

nodes

of

the

flow

before

being

served

by

the

link

-LSB-

-RSB-

Therefore

what

schedulers

to

use

at

the

logical

servers

is

key

to

achieve

a

good

link

sharing

performance

We

illustrate

this

point

with

the

following

example

Example

Consider

the

simple

link

sharing

hierarchy

shown

in

Figure

Assume

that

all

logical

server

nodes

are

currently

backlogged

For

the

link

to

serve

one

of

the

ABR

flows

from

agency

next

both

of

the

following

conditions

have

to

be

met

-LRB-

i

-RRB-

flowsflowsflowsflows

flowsflows

Figure

An

example

link

sharing

tree

it

is

determined

by

the

scheduler

of

the

root

logical

server

that

it

is

agency

's

turn

to

receive

service

and

-LRB-

ii

-RRB-

it

is

determined

by

the

scheduler

of

the

logical

serve

that

it

is

traffic

class

ABR

's

turn

to

receive

service

It

is

straightforward

to

show

that

if

every

logical

server

is

a

Fluid

model

Fair

Queueing

-LRB-

FFQ

-RRB-

server

then

ideal

link

sharing

will

be

achieved

for

every

logical

server

Unfortunately

FFQ

is

not

feasible

Therefore

scheduling

algorithms

that

are

good

approximations

of

FFQ

should

be

used

for

the

logical

servers

Prior

Work

Work

by

Bennett

and

Zhang

In

-LSB-

-RSB-

Bennett

and

Zhang

proposed

a

new

approximation

of

FFQ

named

Virtual

Start-time-based

Fair

Queueing

-LRB-

VSFQ

-RRB-

for

providing

tight

delay

bounds

to

flows

under

a

link

sharing

hierarchy

They

defined

a

metric

for

a

logical

server

called

Normalized

Bit

Worst-case

Fair

Index

-LRB-

NB-WFI

-RRB-

Definition

A

logical

server

node

n

is

said

to

guarantee

a

Normalized

Bit

Worstcase

Fair

Index

-LRB-

NB-WFI

-RRB-

of

m

n

for

its

child

m

if

during

any

time

interval

-LSB-

-RSB-

in

which

m

is

continuously

backlogged

the

following

holds

Wm

-LRB-

-RRB-

OEm

OEn

Wn

-LRB-

-RRB-

m

n

-LRB-

-RRB-

referred

as

General

Processor

Sharing

-LRB-

GPS

-RRB-

server

in

the

literature

-LSB-

-RSB-

They

then

showed

that

delay

of

flow

f

packets

are

bounded

by

oef

rf

H

-LRB-

f

-RRB-

X

h

ph

-LRB-

f

-RRB-

rph

-LRB-

f

-RRB-

-LRB-

-RRB-

where

ph

-LRB-

f

-RRB-

is

the

NB-WFI

guaranteed

to

node

ph

-LRB-

f

-RRB-

by

node

ph

-LRB-

f

-RRB-

if

the

flow

is

constrained

by

a

leaky

bucket

-LRB-

aef

oef

-RRB-

with

aef

rf

Based

on

this

result

Bennett

and

Zhang

claimed

that

small

NB-WFI

values

for

the

logical

servers

are

necessary

for

providing

tight

delay

bounds

to

the

flows

They

also

showed

that

among

all

proposed

packet

fair

queueing

servers

a

VSFQ

server

offer

the

smallest

NB-WFI

the

length

of

one

packet

when

a

fixed

packet

length

is

used

by

all

flows

Work

by

Pawan

et

al.

In

-LSB-

-RSB-

Pawan

et

al.

also

studied

the

problem

of

scheduling

packets

for

flows

under

a

link

sharing

hierarchy

They

proposed

a

new

fair

queueing

algorithm

called

Starttime

Fair

Queueing

-LRB-

SFQ

-RRB-

SFQ

is

very

similar

to

the

well

known

Self

Clocked

Fair

Queueing

-LRB-

SCFQ

-RRB-

-LSB-

-RSB-

except

that

in

SFQ

the

virtual

start

tag

instead

of

the

virtual

finish

tag

of

a

packet

is

used

as

the

priority

value

of

the

packet

Pawan

and

et

al.

observed

that

the

intermediate

logical

servers

no

longer

have

constant

service

rates

for

their

children

Therefore

they

analyzed

the

performance

of

the

SFQ

algorithm

in

the

context

of

Fluctuation

Constrained

servers

first

defined

by

Lee

-LSB-

-RSB-

Definition

A

server

is

said

to

be

Fluctuation

Constrained

-LRB-

FC

-RRB-

with

parameters

-LRB-

C

ffi

-RRB-

or

simply

a

FC

-LRB-

C

ffi

-RRB-

server

if

for

all

intervals

-LSB-

-RSB-

in

a

busy

period

of

the

server

the

work

done

by

the

server

denoted

by

W

-LRB-

-RRB-

satisfies

W

-LRB-

-RRB-

C

-LRB-

-RRB-

ffi

-LRB-

-RRB-

In

addition

they

showed

that

the

service

received

by

a

flow

from

a

FC

SFQ

server

is

also

fluctuation

constrained

Thus

if

the

logical

servers

-LRB-

called

virtual

servers

in

-LSB-

-RSB-

-RRB-

are

all

SFQ

servers

the

packet

delay

bound

for

a

flow

under

a

link

sharing

hierarchy

can

be

recursively

computed

The

exact

bound

is

not

given

in

-LSB-

-RSB-

A

General

Approach

While

the

authors

cited

above

have

made

important

contributions

to

the

design

of

VSFQ

and

SFQ

their

approach

is

not

the

best

Specifically

they

put

too

much

emphasis

on

one

good

fair

queueing

algorithm

for

both

link

sharing

and

packet

scheduling

constant

server

is

also

FC

with

ffi

Such

an

approach

does

not

take

into

account

the

high

likelihood

that

heterogeneous

packet

scheduling

algorithms

may

be

required

at

different

parts

of

the

link

sharing

hierarchy

to

achieve

multiple

design

goals

For

example

if

implementation

complexity

is

of

concern

scheduling

algorithms

like

Deficit

Round

Robin

-LSB-

-RSB-

may

be

more

desirable

than

others

Also

there

are

a

large

number

of

performance

results

in

the

literature

for

one-level

servers

It

is

not

obvious

from

-LSB-

-RSB-

how

these

results

e.g.

those

for

WFQ

servers

-LSB-

-RSB-

or

even

FIFO

servers

can

be

extended

to

networks

with

hierarchical

link

sharing

Next

we

describe

an

approach

in

which

link

sharing

and

packet

scheduling

concerns

are

separated

In

particular

a

link

sharing

hierarchy

is

considered

an

extension

of

a

one-level

constant

rate

server

Consider

a

particular

flow

f

under

the

link

sharing

hierarchy

Even

with

hierarchical

link

sharing

the

flow

is

in

essence

scheduled

by

a

one-level

server

its

parent

node

but

with

a

variable

service

rate

The

impact

of

all

non-parent

ancestors

of

f

is

indirectly

accounted

for

by

the

service

rate

fluctuation

of

the

parent

node

Therefore

the

analysis

of

flow

f

performance

can

be

carried

out

in

two

steps

-LRB-

-RRB-

characterization

of

the

service

rate

fluctuation

of

the

parent

server

-LRB-

-RRB-

extension

of

the

performance

results

for

one-level

servers

to

account

for

service

rate

fluctuations

-LRB-

characterized

in

the

previous

step

-RRB-

For

step

we

characterize

the

parent

server

n

parent

-LRB-

f

-RRB-

as

a

FC

server

That

is

there

exists

a

constant

ffin

such

that

for

any

time

interval

-LSB-

-RSB-

in

which

n

is

backlogged

-LRB-

busy

-RRB-

throughout

the

work

-LRB-

service

-RRB-

done

by

n

satisfies

Wn

-LRB-

-RRB-

rn

-LRB-

-RRB-

ffin

-LRB-

-RRB-

Note

that

the

smaller

ffin

is

the

less

service

rate

fluctuation

for

n

If

all

ancestors

of

n

are

FFQ

servers

ffin

Therefore

good

approximations

of

FFQ

should

be

used

for

ancestors

of

n

to

ensure

a

small

value

for

ffin

Let

us

look

at

two

examples

Example

Assume

that

all

ancestors

of

n

are

VSFQ

servers

and

all

packets

have

a

fixed

length

of

L

bits

From

Definition

and

the

fact

that

VSFQ

guarantees

a

NB-WFI

of

L

-LSB-

-RSB-

it

can

be

shown

that

ffin

is

at

most

-LRB-

H

-LRB-

n

-RRB-

X

h

rn

rph

-LRB-

n

-RRB-

-RRB-

L

-LRB-

-RRB-

Let

us

make

the

conservative

assumption

that

rph

-LRB-

n

-RRB-

rph

-LRB-

n

-RRB-

h

H

-LRB-

n

-RRB-

-LRB-

The

ratio

is

usually

much

smaller

in

reality

-RRB-

Then

it

is

easy

to

show

that

ffin

Example

Assume

that

all

ancestors

of

n

are

SFQ

servers

and

all

packets

have

a

fixed

length

of

L

bits

Applying

Theorem

of

-LSB-

-RSB-

it

can

be

shown

that

ffin

is

at

most

-LRB-

-LRB-

H

-LRB-

n

-RRB-

X

h

rn

rph

-LRB-

n

-RRB-

-LRB-

Q

-LRB-

h

-RRB-

-RRB-

-RRB-

L

-LRB-

-RRB-

where

Q

-LRB-

h

-RRB-

is

the

number

of

siblings

that

ph

-LRB-

n

-RRB-

has

plus

one

Let

us

assume

that

rph

-LRB-

n

-RRB-

rph

-LRB-

n

-RRB-

and

Q

-LRB-

h

-RRB-

Q

h

H

-LRB-

n

-RRB-

Then

ffin

could

be

as

large

as

-LRB-

Q

-RRB-

L.

Therefore

in

general

SFQ

causes

larger

service

rate

fluctuations

than

VSFQ

On

the

other

hand

SFQ

appears

to

have

a

smaller

implementation

cost

For

step

we

next

show

how

to

extend

delay

guarantee

results

from

constant

rate

servers

to

FC

servers

Notation

In

the

balance

of

this

paper

for

a

FC

-LRB-

C

ffi

-RRB-

server

we

will

often

refer

to

its

corresponding

constant

rate

server

which

is

defined

to

be

identical

to

the

FC

server

except

that

it

has

a

constant

service

rate

of

C

bits/second

Consider

a

class

of

work

conserving

servers

called

the

priority

class

which

can

be

described

in

general

as

follows

A

priority

value

is

computed

and

assigned

to

every

packet

upon

its

arrival

and

queued

packets

are

scheduled

for

service

in

the

order

of

increasing

priority

values

Ties

between

packets

are

broken

arbitrarily

Also

within

a

particular

flow

the

priority

of

each

packet

is

non-decreasing

in

packet

arrival

time

We

focus

on

Service

rate

Independent

Priority

-LRB-

SIP

-RRB-

servers

defined

below

Definition

A

server

belongs

to

the

Service

rate

Independent

Priority

-LRB-

SIP

-RRB-

class

if

it

belongs

to

the

priority

class

and

in

addition

its

packet

priority

computation

is

independent

of

the

work

done

by

the

server

Virtual

Clock

-LSB-

-RSB-

Delay-EDD

-LSB-

-RSB-

and

Leave-in-Time

-LSB-

-RSB-

servers

belong

to

the

SIP

class

So

do

all

the

packet

fair

queueing

servers

that

approximate

a

hypothetical

constant

rate

Fluid

model

Fair

Queueing

-LRB-

FFQ

-RRB-

server

which

include

WFQ

-LSB-

-RSB-

PGPS

-LSB-

-RSB-

and

VSFQ

-LSB-

-RSB-

FIFO

servers

also

belong

to

the

SIP

class

since

they

in

effect

use

the

arrival

time

of

a

packet

as

the

packet

priority

which

is

independent

of

the

work

done

the

server

Lee

obtained

some

delay

bound

results

for

FIFO

FC

servers

with

leaky

bucket

constrained

sources

-LSB-

-RSB-

In

particular

we

consider

two

types

of

SIP

servers

preemptive-resume

with

no

overhead

and

nonpreemptive

For

a

SIP

server

that

is

preemptive-resume

with

no

overhead

it

will

immediately

stop

the

service

of

a

packet

and

serve

a

newly

arrived

packet

if

the

new

arrival

has

a

lower

priority

value

But

no

work

will

be

lost

because

of

the

preemption

i.e.

when

the

server

resumes

service

for

the

preempted

packet

it

will

start

from

where

it

stopped

For

a

SIP

server

that

is

nonpreemptive

the

service

of

a

packet

can

not

be

preempted

once

it

is

started

Next

we

present

two

theorems

on

extension

of

delay

guarantees

for

SIP

servers

-LRB-

All

proofs

are

deferred

to

the

appendix

-RRB-

They

deal

with

preemptive

and

nonpreemptive

SIP

servers

respectively

We

say

that

a

delay

guarantee

has

the

firewall

property

if

the

guarantee

to

packets

of

a

flow

is

independent

of

the

arrival

sequence

of

other

flows

Theorem

Consider

two

versions

of

a

SIP

server

that

is

preemptive-resume

with

no

overhead

They

are

identical

except

that

one

has

a

constant

service

rate

of

C

bits/second

while

the

other

is

a

FC

-LRB-

C

ffi

-RRB-

server

If

the

constant

rate

server

a

departure

deadline

of

P

-LRB-

p

-RRB-

to

every

packet

p

where

P

-LRB-

p

-RRB-

is

the

priority

value

of

packet

p

and

is

a

constant

and

the

guarantee

has

the

firewall

property

then

the

FC

server

guarantees

to

every

packet

p

a

departure

deadline

of

P

-LRB-

p

-RRB-

ffiC

Corollary

A

FIFO

FC

-LRB-

C

ffi

-RRB-

server

guarantees

a

delay

bound

of

ffiC

to

every

packet

p

if

its

corresponding

constant

rate

server

guarantees

a

delay

bound

of

to

every

packet

p.

For

nonpreemptive

servers

the

service

may

be

out-of-order

sometimes

It

happens

when

some

newly

arrived

packets

even

with

a

smaller

priority

value

can

not

preempt

the

packet

being

served

Theorem

Consider

two

versions

of

a

nonpreemptive

SIP

server

They

are

identical

except

that

one

has

a

constant

service

rate

of

C

bits/second

while

the

other

is

a

FC

-LRB-

C

ffi

-RRB-

server

If

the

constant

rate

server

guarantees

a

departure

deadline

of

P

-LRB-

p

-RRB-

to

every

packet

p

and

the

guarantee

has

the

firewall

property

and

accounts

for

the

occurrence

of

out-of-order

service

then

the

FC

server

guarantees

to

every

packet

p

a

departure

deadline

of

P

-LRB-

p

-RRB-

ffiC

Note

that

both

theorems

and

Corollary

are

quite

general

They

do

not

depend

on

specific

admission

control

conditions

or

source

control

mechanisms

In

contrast

most

of

Lee

's

analysis

on

FIFO

FC

servers

-LSB-

-RSB-

were

for

leaky

bucket

constrained

sources

Example

This

example

illustrates

that

the

extended

deadline

is

tight

if

the

original

deadline

is

Consider

a

Virtual

Clock

server

-LSB-

-RSB-

that

is

FC

-LRB-

-RRB-

It

is

currently

shared

by

two

flows

flow

A

has

a

reserved

rate

of

and

flow

B

Assume

that

both

flows

use

a

fixed

packet

length

of

Assume

that

the

first

packet

of

flow

A

arrived

at

time

and

the

first

packet

from

flow

B

denoted

as

p

arrived

at

time

A

constant

rate

Virtual

Clock

server

with

a

unit

service

rate

guarantees

to

p

a

note

that

delay

guarantee

generalizations

for

other

types

of

servers

-LRB-

e.g.

Deficit

Round

Robin

-LSB-

-RSB-

SCFQ

-LSB-

-RSB-

-RRB-

may

be

carried

out

in

a

similar

form

that

the

server

needs

to

perform

some

form

of

admission

control

to

provide

meaningful

deadline

guarantees

We

consider

such

admission

control

a

component

of

the

server

departure

deadline

of

P

-LRB-

p

-RRB-

-LSB-

-RSB-

p

actually

will

depart

from

the

constant

rate

server

at

From

Theorem

the

deadline

guaranteed

to

p

by

the

FC

server

is

The

FC

server

could

be

idle

from

time

to

and

in

this

case

p

would

not

depart

from

the

FC

server

until

time

The

burst

-LRB-

block

-RRB-

delay

bounds

presented

in

Section

were

derived

from

a

delay

guarantee

of

Virtual

Clock

server

that

has

the

firewall

property

and

accounts

for

out-of-order

service

-LSB-

-RSB-

From

Theorem

it

is

straightforward

to

extend

them

to

networks

with

hierarchical

link

sharing

Corollary

Consider

the

end-to-end

burst

delay

bounds

presented

in

Section

If

at

each

switch

k

the

packets

of

the

flow

are

served

by

a

FC

-LRB-

Ck

ffik

-RRB-

logical

server

instead

of

a

constant

rate

link

with

capacity

Ck

then

the

end-to-end

delay

of

burst

m

of

the

flow

is

bounded

as

follows

Dm

bm

gm

m

-LRB-

K

-RRB-

max

n

mf

gn

n

g

KX

k

-LRB-

k

ffik

Ck

-RRB-

-LRB-

-RRB-

Dm

-LRB-

K

-RRB-

gm

m

KX

k

k

-LRB-

-RRB-

Admission

Control

In

this

section

we

design

a

flow

level

admission

control

algorithm

for

real-time

block

transfer

Consider

a

flow

whose

source

requests

for

real-time

block

transfer

service

We

assume

that

at

the

time

of

connection

setup

the

source

the

network

two

sets

of

parameters

-LRB-

i

-RRB-

QoS

parameters

block

loss

probability

BLP

block

transfer

delay

BTD

and

block

delay

variation

BDV

and

-LRB-

ii

-RRB-

traffic

parameters

sustained

cell

rate

SCR

peak

cell

rate

PCR

and

cell

rate

variation

CRV

Whether

or

not

to

admit

the

flow

is

a

decision

made

collectively

by

all

switches

in

the

path

of

the

flow

Specifically

each

switch

in

the

path

accepts

the

flow

only

if

doing

so

will

not

cause

violation

of

QoS

guarantees

to

other

flows

the

network

admits

the

flow

only

if

all

the

switches

accept

the

flow

System

Model

Our

system

model

for

admission

control

by

a

particular

switch

is

shown

in

Figure

There

are

M

classes

of

real-time

block

transfer

services

They

share

one

logical

present

the

design

in

the

context

of

ATM

networks

because

ATM

QoS

and

traffic

parameters

are

well

defined

Our

design

should

be

applicable

to

other

types

of

networks

assume

that

some

connection

setup

protocol

is

in

place

for

the

communication

between

the

source

and

the

network

flowsflowsflowsClass

Class

MClass

-LRB-

C

-RRB-

Figure

System

model

link

that

is

FC

-LRB-

C

ffi

-RRB-

Each

class

is

associated

with

a

weight

OEs

for

class

s

which

is

a

relative

measure

of

its

share

of

the

bandwidth

of

the

logical

link

For

ease

of

presentation

and

without

loss

of

generality

we

assume

that

PMs

OEs

Therefore

class

s

has

a

share

of

the

logical

link

equal

to

rs

OEs

C

-LRB-

bits/second

-RRB-

Each

class

offers

a

specific

block

loss

rate

ps

for

class

s.

Without

loss

of

generality

we

assume

that

pM

In

other

words

class

provides

deterministic

service

i.e.

the

class

-LRB-

i

-RRB-

service

defined

in

Section

and

the

others

provide

different

levels

of

statistical

service

i.e.

the

class

-LRB-

ii

-RRB-

service

defined

in

Section

Block

transfer

delay

parameters

such

as

BTD

and

BDV

are

not

considered

because

they

are

dependent

on

the

individual

burst

scheduling

algorithm

implemented

by

each

class

We

also

assume

that

an

appropriate

scheduling

algorithm

-LRB-

e.g.

VSFQ

-LSB-

-RSB-

-RRB-

is

used

by

the

logical

link

to

provide

a

firewall

between

the

service

classes

and

minimize

their

service

fluctuations

With

the

firewall

admission

control

for

each

class

can

be

performed

independently

Statistical

Multiplexing

For

class

flows

the

admission

control

should

be

based

on

peak

rate

reservation

This

is

necessary

to

ensure

that

the

link

share

of

the

class

never

be

exceeded

without

discarding

blocks

Consider

a

statistical

class

s.

Flows

of

this

class

can

tolerate

a

certain

level

of

block

losses

Therefore

the

admission

control

for

them

should

be

more

aggressive

to

the

switch

has

a

priori

knowledge

of

flow

behaviors

peak

bandwidth

reservation

may

not

be

necessary

But

it

is

unrealistic

with

today

's

networks

increase

link

utilization

of

the

class

by

taking

advantage

of

statistical

multiplexing

Next

we

derive

a

set

of

conditions

that

are

sufficient

to

limit

the

block

loss

probability

of

the

class

at

ps

Assume

that

the

service

of

the

class

is

currently

shared

by

a

set

of

N

flows

-LRB-

labeled

by

N

-RRB-

Let

BLPi

BTDi

BDVi

SCRi

PCRi

and

CRVi

be

the

QoS

and

traffic

parameters

of

flow

i

that

are

supplied

to

the

switch

-LRB-

Note

that

BLPi

is

the

probability

of

burst

losses

through

one

switch

-LRB-

i.e.

a

single

hop

-RRB-

required

for

flow

i

We

will

discuss

how

to

distribute

an

end-to-end

burst

loss

requirement

to

individual

switches

in

a

future

report

-RRB-

At

any

time

because

of

the

block-based

admission

control

at

most

one

block

from

each

flow

is

active

Denote

i

-LRB-

t

-RRB-

the

reserved

cell

rate

for

the

block

of

flow

i

that

is

active

at

time

t.

-LRB-

i

-LRB-

t

-RRB-

if

there

is

no

active

block

for

flow

i

at

t.

-RRB-

In

our

analysis

below

we

consider

i

-LRB-

t

-RRB-

's

as

independent

random

variables

Define

Ys

-LRB-

t

-RRB-

NX

i

i

-LRB-

t

-RRB-

rs

-LRB-

-RRB-

Consider

a

block

that

becomes

active

at

time

t.

Either

the

block

has

just

arrived

and

no

other

block

from

the

same

flow

is

active

or

another

block

from

the

same

flow

and

preceding

it

has

just

left

the

switch

In

either

case

the

switch

immediately

performs

admission

control

on

the

block

and

will

discard

it

if

-LRB-

Ys

-LRB-

t

-RRB-

-RRB-

Therefore

Pr

-LRB-

Ys

-LRB-

t

-RRB-

-RRB-

is

a

good

approximation

of

the

block

loss

probability

of

the

class

The

goal

of

our

analysis

is

to

find

conditions

sufficient

for

Pr

-LRB-

Ys

-LRB-

t

-RRB-

-RRB-

ps

-LRB-

-RRB-

From

a

generalized

version

of

the

central

limit

theorem

-LSB-

-RSB-

-LRB-

included

in

the

appendix

as

Theorem

-RRB-

we

have

Ys

-LRB-

t

-RRB-

E

-LSB-

Ys

-LRB-

t

-RRB-

-RSB-

q

V

ar

-LSB-

Ys

-LRB-

t

-RRB-

-RSB-

N

-LRB-

-RRB-

-LRB-

-RRB-

where

N

-LRB-

-RRB-

is

the

standard

normal

distribution

Therefore

we

can

approximate

Pr

-LRB-

Ys

-LRB-

t

-RRB-

-RRB-

with

Pr

-LRB-

X

E

-LSB-

Ys

-LRB-

t

-RRB-

-RSB-

q

V

ar

-LSB-

Ys

-LRB-

t

-RRB-

-RSB-

-RRB-

-LRB-

-RRB-

where

X

N

-LRB-

-RRB-

The

following

condition

is

sufficient

for

-LRB-

-RRB-

E

-LSB-

Ys

-LRB-

t

-RRB-

-RSB-

q

V

ar

-LSB-

Ys

-LRB-

t

-RRB-

-RSB-

Zs

-LRB-

-RRB-

where

Zs

is

the

constant

that

satisfies

Pr

-LRB-

X

Zs

-RRB-

ps

We

have

E

-LSB-

Ys

-LRB-

t

-RRB-

-RSB-

NX

i

E

-LSB-

i

-LRB-

t

-RRB-

-RSB-

C

-LRB-

-RRB-

V

ar

-LSB-

Ys

-LRB-

t

-RRB-

-RSB-

NX

i

V

ar

-LSB-

i

-LRB-

t

-RRB-

-RSB-

-LRB-

-RRB-

Define

Z

rs

PNi

E

-LSB-

i

-LRB-

t

-RRB-

-RSB-

qPNi

V

ar

-LSB-

i

-LRB-

t

-RRB-

-RSB-

-LRB-

-RRB-

Combining

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

and

-LRB-

-RRB-

we

have

the

following

approximate

sufficient

condition

for

-LRB-

-RRB-

Zs

Z

-LRB-

-RRB-

We

refer

to

the

value

of

-LRB-

Zs

Z

-RRB-

as

the

current

Statistical

Multiplexing

Intensity

-LRB-

SMI

-RRB-

of

class

s

It

should

never

exceed

the

threshold

of

In

practice

it

is

difficult

to

obtain

the

exact

value

of

Z.

However

Z

can

be

estimated

as

follows

Z

rs

PNi

SCRi

qPNi

CRVi

-LRB-

-RRB-

Note

that

the

source

of

flow

i

may

not

have

a

good

estimate

of

CRVi

at

the

time

of

connection

setup

In

such

a

case

an

upper

bound

on

CRVi

SCRi

-LRB-

PCRi

SCRi

-RRB-

can

be

used

as

a

pessimistic

estimate

-LRB-

See

Theorem

in

the

appendix

-RRB-

Admission

Control

Algorithm

The

following

admission

control

algorithm

results

from

the

analysis

in

the

previous

section

Algorithm

Specification

Admission

Control

-LRB-

BLP

PCR

SCR

CRV

-RRB-

s

find

class

-LRB-

BLP

-RRB-

if

-LRB-

s

-RRB-

then

if

-LRB-

PCR

-RRB-

then

accept

the

flow

PCR

else

reject

the

flow

else

if

-LRB-

CRV

-RRB-

then

CRV

SCR

-LRB-

PCR

SCR

-RRB-

SMI

Zs

-LRB-

sqrt

-LRB-

Vs

CRV

-RRB-

-LRB-

Bs

SCR

-RRB-

-RRB-

if

-LRB-

SMI

-RRB-

then

accept

the

flow

Bs

Bs

SCR

Vs

Vs

CRV

else

reject

the

flow

The

variables

Bs

and

Vs

are

used

to

store

respectively

the

available

bandwidth

and

the

total

cell

rate

variance

of

class

s.

Initially

Bs

rs

and

Vs

We

assume

that

if

a

source

does

not

have

a

good

estimate

of

CRV

at

the

time

of

connection

setup

it

will

let

the

network

know

by

setting

CRV

The

algorithm

is

simple

and

straightforward

First

BLP

is

used

to

find

the

service

class

desired

by

the

source

Assume

the

class

found

is

s

If

s

is

i.e.

the

source

requests

for

deterministic

service

Therefore

the

admission

decision

is

based

upon

PCR

of

the

flow

and

the

bandwidth

currently

available

for

class

Otherwise

the

source

requests

for

statistical

service

The

admission

decision

is

then

based

on

the

projected

SMI

for

class

s.

Experimental

Results

We

have

performed

a

set

of

experiments

to

evaluate

the

performance

of

the

admission

control

algorithm

We

present

the

results

in

this

section

The

simulated

network

is

shown

in

Figure

The

nodes

labeled

by

VS

are

video

sources

and

VD

video

destinations

Each

video

source

generates

packets

from

a

trace

file

obtained

from

a

MPEG

video

sequence

The

profile

of

the

MPEG

sequences

used

in

our

experiments

are

given

in

Table

Figure

Simulated

network

MPEG

encoding

pattern

picture

bit

rate

-LRB-

Mbps

-RRB-

sequence

minimum

maximum

average

Airwolf

-LRB-

-RRB-

Canyon

-LRB-

-RRB-

Energizer

-LRB-

-RRB-

Hardboiled

-LRB-

-RRB-

Jamie

-LRB-

-RRB-

RedsNightmare

-LRB-

-RRB-

-LRB-

-RRB-

Terminator

-LRB-

-RRB-

UnderSiege

-LRB-

-RRB-

Table

Profile

of

MPEG

sequences

Each

video

source

makes

reservation

with

the

network

before

sending

out

data

packets

-LRB-

The

admission

control

algorithm

is

implemented

for

channel

and

it

is

assumed

that

all

video

sources

requests

for

the

same

class

of

service

-RRB-

To

evaluate

the

admission

control

algorithm

we

varied

the

link

bandwidth

of

as

well

as

the

targeted

loss

rate

for

the

service

class

We

ran

each

experiment

for

seconds

of

simulated

time

Channel

utilization

-LRB-

-RRB-

Targeted

loss

rate

capacity

Mbps

capacity

Mbps

capacity

Mbps

Figure

Channel

utilization

improvement

Channel

Utilization

In

Figure

we

show

the

channel

utilization

as

a

function

of

the

targeted

loss

rate

Compared

with

deterministic

guarantees

the

network

channel

is

used

more

efficiently

for

statistical

guarantees

the

price

to

pay

is

a

small

non-zero

burst

loss

probability

The

utilization

increase

is

more

significant

with

higher

channel

capacity

from

below

to

above

in

the

case

where

the

capacity

of

is

Mbps

This

is

because

the

improvement

is

due

to

statistical

multiplexing

gain

which

is

larger

with

higher

channel

capacity

Actual

loss

rate

Targeted

loss

rate

Mbps

Mbps

Mbps

Figure

Actual

vs.

targeted

loss

rate

Mbps

Mbps

Mbps

Targeted

Actual

Targeted

Actual

Targeted

Actual

Table

Actual

loss

rate

Actual

Loss

Rate

In

Table

we

present

the

actual

burst

-LRB-

picture

-RRB-

loss

rate

which

is

the

average

over

five

simulation

runs

using

different

random

seeds

From

the

data

we

conclude

that

our

admission

control

algorithm

predicts

the

actual

loss

rate

well

when

the

channel

capacity

is

high

compared

to

the

average

flow

rate

-LRB-

In

our

experiments

the

ratio

is

when

the

channel

capacity

of

is

Mbps

-RRB-

This

agrees

with

our

analysis

the

central

limit

theorem

converges

given

a

large

number

of

flows

sharing

the

channel

The

comparison

of

the

actual

and

targeted

loss

rates

is

also

illustrated

in

Figure

Note

that

the

solid

degree

line

represents

a

perfect

prediction

by

the

admission

control

algorithm

Conclusions

To

many

applications

the

performance

of

higher

layer

data

units

which

are

usually

fragmented

into

blocks

of

packets

-LRB-

or

cells

-RRB-

are

more

relevant

than

the

performance

of

individual

packets

-LRB-

or

cells

-RRB-

We

demonstrated

that

a

real-time

VBR

service

called

real-time

block

transfer

with

QoS

parameters

for

block

transfer

delay

block

loss

rate

and

block

delay

variation

can

be

provided

by

integrating

existing

designs

In

particular

we

designed

an

admission

control

algorithm

to

provide

the

following

classes

of

service

bounded-delay

block

transfer

with

no

loss

and

bounded-delay

block

transfer

with

a

specified

block

loss

rate

We

also

showed

how

to

extend

existing

end-to-end

delay

bounds

for

blocks

to

networks

with

hierarchical

link

sharing

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C.R.

Bennett

and

Hui

Zhang

Hierarchical

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communication

networks

with

variable-rate

links

In

Proceedings

of

ACM

SIGCOMM

pages

-LCB-

August

-LSB-

-RSB-

Michel

Lo?eve

Probability

Theory

I.

New

York

ed

-LSB-

-RSB-

Abhay

K.

Parekh

and

Robert

G.

Gallager

A

generialized

processor

sharing

approach

to

flow

control

in

integrated

services

networks

The

single

node

case

IEEE/ACM

Trans

on

Networking

-LRB-

-RRB-

-LCB-

June

-LSB-

-RSB-

A.

Romanow

and

S.

Floyd

The

dynamics

of

TCP

traffic

over

ATM

networks

In

Proceedings

of

ACM

SIGCOMM

London

England

August

-LSB-

-RSB-

M.

Shreedhar

and

George

Varghese

Efficient

fair

queueing

using

deficit

round

robin

In

Proceedings

of

ACM

SIGCOMM

Cambridge

MA

August

-LSB-

-RSB-

J.S.

Turner

Maintaining

high

throughput

during

overload

in

ATM

networks

pages

-LCB-

Proceedings

of

IEEE

INFOCOM

-LSB-

-RSB-

Geoffrey

G.

Xie

and

Simon

S.

Lam

Delay

guarantee

of

Virtual

Clock

server

IEEE/ACM

Trans

on

Networking

-LRB-

-RRB-

-LCB-

December

Presented

at

IEEE

Workshop

on

Computer

Communications

October

-LSB-

-RSB-

Lixia

Zhang

VirtualClock

A

new

traffic

control

algorithm

for

packet

switching

networks

In

Proceedings

of

ACM

SIGCOMM

pages

-LCB-

August

Appendix

Proof

of

Theorem

Consider

an

arbitrary

sequence

of

packet

arrivals

to

the

FC

server

Denote

A

-LRB-

p

-RRB-

the

arrival

time

and

l

-LRB-

p

-RRB-

the

length

-LRB-

in

bits

-RRB-

of

an

arbitrary

packet

p

We

feed

the

same

sequence

to

the

constant

rate

server

It

is

given

that

for

every

packet

p

that

departs

the

constant

rate

server

its

departure

time

denoted

by

L

-LRB-

p

-RRB-

satisfies

L

-LRB-

p

-RRB-

P

-LRB-

p

-RRB-

-LRB-

-RRB-

Denote

S

-LRB-

p

-RRB-

the

set

of

packets

in

the

arrival

sequence

with

priority

values

less

than

or

equal

to

that

of

p.

Denote

Wp

-LRB-

-RRB-

and

W

p

-LRB-

-RRB-

respectively

the

work

done

by

the

constant

rate

server

and

the

FC

server

in

the

time

interval

-LSB-

-RSB-

for

packets

in

S

-LRB-

p

-RRB-

For

the

FC

server

if

p

departs

before

P

-LRB-

p

-RRB-

ffiC

we

are

done

Otherwise

we

first

show

there

exists

a

packet

q

S

-LRB-

p

-RRB-

such

that

W

p

-LRB-

A

-LRB-

q

-RRB-

-RRB-

Wp

-LRB-

A

-LRB-

q

-RRB-

-RRB-

-LRB-

-RRB-

W

p

-LRB-

A

-LRB-

q

-RRB-

P

-LRB-

p

-RRB-

ffi

C

-RRB-

W

-LRB-

A

-LRB-

q

-RRB-

P

-LRB-

p

-RRB-

-RRB-

-LRB-

-RRB-

where

W

-LRB-

A

-LRB-

q

-RRB-

P

-LRB-

p

-RRB-

-RRB-

is

the

work

done

by

the

constant

server

in

the

time

interval

-LSB-

A

-LRB-

q

-RRB-

P

-LRB-

p

-RRB-

-RSB-

Note

that

if

the

FC

server

is

busy

with

packets

from

S

-LRB-

p

-RRB-

throughout

the

interval

-LSB-

A

-LRB-

q

-RRB-

P

-LRB-

p

-RRB-

ffiC

-RSB-

then

-LRB-

-RRB-

holds

as

follows

W

p

-LRB-

A

-LRB-

q

-RRB-

P

-LRB-

p

-RRB-

ffi

C

-RRB-

C

-LRB-

P

-LRB-

p

-RRB-

ffi

C

A

-LRB-

q

-RRB-

-RRB-

ffi

-LRB-

-RRB-

C

-LRB-

P

-LRB-

p

-RRB-

A

-LRB-

q

-RRB-

-RRB-

-LRB-

-RRB-

W

-LRB-

A

-LRB-

q

-RRB-

P

-LRB-

p

-RRB-

-RRB-

-LRB-

-RRB-

We

show

that

q

exists

by

searching

for

it

First

if

there

is

no

other

S

-LRB-

p

-RRB-

packet

in

the

FC

server

when

p

arrives

we

set

q

p.

Otherwise

we

check

the

S

-LRB-

p

-RRB-

packet

that

arrived

just

ahead

of

p

If

there

was

no

other

S

-LRB-

p

-RRB-

packet

in

the

FC

server

when

it

arrived

we

set

it

to

be

q.

Otherwise

we

check

the

S

-LRB-

p

-RRB-

packet

just

ahead

of

it

We

continue

this

process

until

we

find

q

The

process

will

terminate

with

a

q

found

because

-LRB-

i

-RRB-

for

the

S

-LRB-

p

-RRB-

packet

that

arrived

first

in

the

FC

server

busy

period

involving

packet

p

denoted

as

e

we

have

W

p

-LRB-

A

-LRB-

e

-RRB-

-RRB-

Wp

-LRB-

A

-LRB-

e

-RRB-

-RRB-

and

-LRB-

ii

-RRB-

the

preemptive

server

is

always

busy

with

S

-LRB-

p

-RRB-

packets

as

long

as

they

are

backlogged

Combining

-LRB-

-RRB-

and

-LRB-

-RRB-

we

have

W

p

-LRB-

P

-LRB-

p

-RRB-

ffi

C

-RRB-

Wp

-LRB-

A

-LRB-

q

-RRB-

-RRB-

W

-LRB-

A

-LRB-

q

-RRB-

P

-LRB-

p

-RRB-

-RRB-

-LRB-

-RRB-

Consider

the

constant

rate

server

and

a

modification

to

the

arrival

sequence

as

follows

In

particular

those

packets

in

S

-LRB-

p

-RRB-

that

arrive

at

or

after

A

-LRB-

p

-RRB-

l

-LRB-

p

-RRB-

C

in

the

original

sequence

will

have

their

arrival

times

moved

forward

in

the

modified

sequence

such

that

they

all

arrive

in

the

time

interval

-LRB-

A

-LRB-

p

-RRB-

A

-LRB-

p

-RRB-

l

-LRB-

p

-RRB-

C

-RRB-

and

the

original

sequence

of

each

flow

is

preserved

Note

that

the

priority

value

as

well

as

the

guaranteed

deadline

of

p

is

unaffected

by

the

modification

Moreover

since

the

priority

value

of

each

packet

within

a

flow

is

non-decreasing

in

packet

arrival

time

-LRB-

by

definition

of

a

SIP

server

-RRB-

the

set

of

packets

in

the

modified

sequence

with

priority

values

less

than

or

equal

to

that

of

p

is

identical

to

S

-LRB-

p

-RRB-

Since

the

delay

guarantee

in

-LRB-

-RRB-

holds

for

the

modified

sequence

because

of

the

firewall

property

and

the

guarantee

is

independent

of

the

tie

breaking

method

we

have

Wp

-LRB-

A

-LRB-

q

-RRB-

-RRB-

W

-LRB-

A

-LRB-

q

-RRB-

P

-LRB-

p

-RRB-

-RRB-

X

-LRB-

p

-RRB-

l

-LRB-

e

-RRB-

-LRB-

-RRB-

Combining

with

-LRB-

-RRB-

we

have

W

p

-LRB-

P

-LRB-

p

-RRB-

ffi

C

-RRB-

X

-LRB-

p

-RRB-

l

-LRB-

e

-RRB-

-LRB-

-RRB-

Therefore

the

departure

time

of

p

in

the

FC

server

is

bounded

by

P

-LRB-

p

-RRB-

ffi

C

-LRB-

-RRB-

Proof

of

Corollary

For

the

FIFO

servers

P

-LRB-

p

-RRB-

A

-LRB-

p

-RRB-

where

A

-LRB-

p

-RRB-

is

the

arrival

time

of

packet

p.

Therefore

a

delay

bound

guarantee

of

is

equivalent

to

a

deadline

guarantee

of

P

-LRB-

p

-RRB-

The

proof

for

the

deadline

guarantee

generalization

is

identical

to

the

one

for

Theorem

except

for

a

simpler

reasoning

for

-LRB-

-RRB-

While

a

deadline

guarantee

by

a

FIFO

server

does

not

have

the

firewall

property

there

is

no

need

for

the

condition

All

packets

in

S

-LRB-

p

-RRB-

arrive

no

later

than

A

-LRB-

p

-RRB-

Therefore

we

can

reason

for

-LRB-

-RRB-

with

the

original

arrival

sequence

Proof

of

Theorem

Consider

an

additional

SIP

server

It

is

identical

to

the

constant

rate

server

except

that

it

is

preemptive-resume

with

no

overhead

Consider

an

arbitrary

sequence

of

packet

arrivals

to

the

FC

server

We

feed

the

same

sequence

to

the

other

two

servers

For

an

arbitrary

packet

p

in

the

sequence

denote

Lpre

-LRB-

p

-RRB-

the

departure

time

of

p

from

the

preemptive

constant

rate

server

and

W

pre

-LRB-

-RRB-

the

work

done

by

the

preemptive

constant

rate

server

in

the

time

interval

-LSB-

-RSB-

Since

the

deadline

guarantee

by

the

nonpreemptive

constant

rate

server

accounts

for

the

occurrence

of

out-of-order

service

we

have

for

every

packet

p

Lpre

-LRB-

p

-RRB-

P

-LRB-

p

-RRB-

lmax

C

-LRB-

-RRB-

Note

that

the

above

guarantee

also

has

the

firewall

property

If

-LRB-

p

-RRB-

P

-LRB-

p

-RRB-

ffiC

we

are

done

Otherwise

similar

to

the

proof

for

Theorem

we

can

show

that

there

exists

q

S

-LRB-

p

-RRB-

such

that

W

p

-LRB-

A

-LRB-

q

-RRB-

-RRB-

W

pre

p

-LRB-

A

-LRB-

q

-RRB-

-RRB-

-LRB-

-RRB-

W

p

-LRB-

A

-LRB-

q

-RRB-

P

-LRB-

p

-RRB-

ffi

C

-RRB-

C

-LRB-

P

-LRB-

p

-RRB-

A

-LRB-

q

-RRB-

-RRB-

lmax

-LRB-

-RRB-

The

extra

term

lmax

in

-LRB-

-RRB-

is

due

to

the

fact

that

part

of

the

work

done

by

the

FC

server

in

-LSB-

A

-LRB-

q

-RRB-

P

-LRB-

p

-RRB-

ffiC

-RSB-

may

be

for

out-of-order

service

of

a

packet

not

in

S

-LRB-

p

-RRB-

Similar

to

the

proof

for

Theorem

we

have

W

pre

p

-LRB-

A

-LRB-

q

-RRB-

-RRB-

W

pre

-LRB-

A

-LRB-

q

-RRB-

P

-LRB-

p

-RRB-

lmax

C

-RRB-

X

-LRB-

p

-RRB-

l

-LRB-

e

-RRB-

-LRB-

-RRB-

Since

W

pre

-LRB-

A

-LRB-

q

-RRB-

P

-LRB-

p

-RRB-

lmax

C

-RRB-

C

-LRB-

P

-LRB-

p

-RRB-

lmax

C

A

-LRB-

q

-RRB-

-RRB-

-LRB-

-RRB-

C

-LRB-

P

-LRB-

p

-RRB-

A

-LRB-

q

-RRB-

-RRB-

lmax

-LRB-

-RRB-

we

have

W

pre

p

-LRB-

A

-LRB-

q

-RRB-

-RRB-

C

-LRB-

P

-LRB-

p

-RRB-

A

-LRB-

q

-RRB-

-RRB-

lmax

X

-LRB-

p

-RRB-

l

-LRB-

e

-RRB-

-LRB-

-RRB-

Combining

-LRB-

-RRB-

-LRB-

-RRB-

and

-LRB-

-RRB-

we

have

W

p

-LRB-

P

-LRB-

p

-RRB-

ffi

C

-RRB-

W

p

-LRB-

A

-LRB-

q

-RRB-

-RRB-

W

p

-LRB-

A

-LRB-

q

-RRB-

P

-LRB-

p

-RRB-

ffi

C

-RRB-

-LRB-

-RRB-

W

pre

p

-LRB-

A

-LRB-

q

-RRB-

-RRB-

C

-LRB-

P

-LRB-

p

-RRB-

A

-LRB-

q

-RRB-

-RRB-

lmax

-LRB-

-RRB-

X

-LRB-

p

-RRB-

l

-LRB-

e

-RRB-

-LRB-

-RRB-

Therefore

the

departure

time

of

p

in

the

FC

server

is

bounded

by

P

-LRB-

p

-RRB-

ffi

C

-LRB-

-RRB-

Theorem

-LRB-

-LSB-

-RSB-

-RRB-

Assume

that

are

independent

random

variables

centered

at

expectations

with

d.f.

's

and

variance

Let

Sn

nX

k

Xk

be

their

consecutive

sums

with

variance

nX

k

n

Then

Sn

sn

N

-LRB-

-RRB-

-LRB-

-RRB-

if

and

only

if

for

all

n

-LRB-

-RRB-

nX

k

Z

jxj

sn

-LRB-

-RRB-

as

n

Theorem

For

any

random

variable

X

that

can

take

value

between

and

Xmax

the

following

holds

V

ar

-LSB-

X

-RSB-

-LRB-

h

-RRB-

E

-LSB-

X

-RSB-

-LRB-

-RRB-

where

h

is

the

peak-to-average

ratio

Xmax

E

-LSB-

X

-RSB-

Proof

By

definition

the

variance

of

X

denoted

by

V

ar

-LSB-

X

-RSB-

satisfy

V

ar

-LSB-

X

-RSB-

Z

-LRB-

x

-RRB-

E

-LSB-

X

-RSB-

-LRB-

-RRB-

Since

Xmax

x

we

have

V

ar

-LSB-

X

-RSB-

Z

-LRB-

Xmax

x

-RRB-

dF

-LRB-

x

-RRB-

E

-LSB-

X

-RSB-

-LRB-

-RRB-

Xmax

Z

xdF

-LRB-

x

-RRB-

E

-LSB-

X

-RSB-

-LRB-

-RRB-

XmaxE

-LSB-

X

-RSB-

E

-LSB-

X

-RSB-

-LRB-

-RRB-

-LRB-

h

-RRB-

E

-LSB-

X

-RSB-

-LRB-

-RRB-

admission control
atm block transfer
burst scheduling
delay guarantee
hierarchical link sharing