Efficient

Interconnection

Designs

for

Switch-Based

Local

Area

Rose

P.

Tsang

and

David

H.C.

Du

Computer

Science

Department

University

of

Minnesota

Union

Street

SE

Minneapolis

MN

March

Abstract

Today

with

the

proliferation

of

switch-based

local

area

network

standards

such

as

ATM

-LRB-

Asynchronous

Transfer

Mode

-RRB-

HIPPI

-LRB-

HIgh

Performance

Parallel

Interface

-RRB-

and

Fibre

Channel

the

issue

of

the

efficient

interconnection

among

connection-oriented

switches

arises

This

paper

presents

a

study

of

efficient

arrangements

of

switches

intended

to

support

a

local

area

network

environment

This

multi-switch

interconnection

problem

is

described

and

distinguished

from

previously

studied

interconnection

networks

such

as

multistage

interconnection

networks

-LRB-

for

multiprocessors

or

telephone

switching

exchanges

-RRB-

and

wide-area

telephone

networks

Given

a

number

of

station

nodes

attached

to

a

set

of

switches

minimal

hardware

requirements

-LRB-

number

of

dedicated

I/O

ports

for

inter-switch

communications

or

number

of

inter-switch

links

-RRB-

as

well

as

the

interconnection

topology

which

supports

the

maximal

aggregate

network

throughput

are

sought

This

problem

is

shown

to

be

NP-Complete

Network

configurations

based

upon

switch

usage

and

station

node

distribution

are

presented

A

network

configuration

based

upon

switch

usage

and

station

node

distribution

with

superior

performance

to

other

configurations

in

terms

of

maximizing

network

throughput

or

resource

sharing

is

analyzed

Lower

bounds

on

the

necessary

minimal

number

of

inter-switch

links

and

maximal

number

of

nonblocking

station

nodes

are

derived

Finally

in

cases

where

less

than

the

minimal

amount

of

hardware

necessary

to

support

the

maximal

aggregate

network

throughput

is

provided

blocking

probabilities

are

derived

The

results

of

this

paper

are

directly

applicable

to

connection-oriented

switch-based

networks

such

as

ATM

HIPPI

or

Fibre

Channel

networks

supported

in

part

by

the

Army

High

Performance

Computing

Research

Center

-LRB-

AHPCRC

-RRB-

under

DoD

contract

No

Introduction

In

the

past

computer

networks

were

intended

to

support

a

homogenous

traffic

environment

where

the

majority

of

data

packets

were

transmitted

in

infrequent

bursts

Future

high

speed

networks

intended

for

multimedia

traffic

support

as

well

as

traditional

computer

data

are

required

to

support

high

bandwidth

stream

data

as

well

as

bursty

data

This

has

resulted

in

network

architectures

which

support

high

aggregate

bandwidths

-LRB-

switch-based

networks

-RRB-

and

are

in

some

manner

connection-oriented

For

instance

the

ATM

-LRB-

Asynchronous

Transfer

Mode

-RRB-

-LSB-

-RSB-

standard

emulates

a

connection-oriented

environment

in

a

fast

packet-switched

network

and

the

HIPPI

-LRB-

HIgh

Performance

Parallel

Interface

-RRB-

-LSB-

-RSB-

standard

specifies

a

purely

connection-oriented

network

It

is

most

likely

that

future

high-speed

networks

will

consist

of

multiple

connectionoriented

switches

-LSB-

-RSB-

The

purpose

of

this

paper

is

to

examine

ways

of

efficiently

interconnecting

multiple

switches

in

a

local

area

environment

such

that

blocking

is

minimized

-LRB-

i.e.

the

network

can

support

the

maximal

aggregate

network

throughput

-RRB-

Most

LAN

environments

consist

of

a

large

percentage

of

sharable

devices

These

devices

are

typically

servers

-LRB-

file

servers

database

servers

etc.

-RRB-

special

purpose

processors

storage

devices

and

ordinary

workstations

Hereafter

we

will

refer

to

these

devices

as

station

nodes

An

attached

network

device

may

correspond

to

multiple

station

nodes

The

number

of

station

nodes

which

corresponds

to

a

particular

device

is

indicative

of

that

device

's

I/O

requirements

Figure

shows

an

example

of

a

switch-based

network

environment

Given

a

number

of

station

nodes

and

switches

it

is

of

interest

to

know

the

interconnection

topology

which

will

allow

for

the

largest

amount

of

device

-LRB-

resource

-RRB-

sharing

i.e.

the

interconnection

topology

which

will

sustain

the

highest

aggregate

throughput

Arrangements

for

interconnecting

switches

have

been

studied

in

the

following

two

contexts

-LRB-

i

-RRB-

Designing

arrays

of

switches

in

a

multistage

manner

in

order

to

simulate

a

large

crossbar

switch

These

networks

referred

to

as

multistage

interconnection

networks

may

serve

as

interconnection

networks

for

multiprocesors

telephone

switching

exchanges

or

large

general

purpose

switches

-LRB-

e.g.

Fibre

Channel

-RRB-

-LSB-

-RSB-

And

-LRB-

ii

-RRB-

designing

the

placement

and

quantity

of

transmission

lines

-LRB-

or

trunks

-RRB-

between

telephone

switching

exchanges

to

support

telephone

network

connections

-LSB-

-RSB-

In

this

paper

we

define

multiple

switch

interconnections

and

describe

the

characteristics

which

distinguish

them

from

traditional

multi-stage

interconnection

networks

and

widearea

telephone

networks

We

show

that

the

general

problem

of

finding

an

interconnection

topology

consisting

of

a

minimal

number

of

links

and

which

supports

the

maximal

aggregate

network

throughput

is

NP-Complete

We

then

discuss

different

network

configurations

based

upon

the

usage

of

the

switches

and

the

distribution

of

station

nodes

among

the

switches

Optimal

and

sub-optimal

configurations

for

different

network

types

are

derived

and

discussed

From

these

results

we

present

a

configuration

-LRB-

a

Balanced

Direct-Direct

network

-RRB-

with

superior

performance

to

other

networks

Balanced

networks

are

networks

where

station

nodes

are

evenly

divided

among

all

the

switches

and

the

switches

are

interconnected

such

that

the

number

of

I/O

ports/switch

is

approximately

the

same

for

all

switches

Every

switch

contains

direct

connections

to

station

nodes

Bounds

on

the

minimal

number

of

I/O

ports/switch

and

maximal

number

SN

SN

SNSN

SN

station

nodes

switches

SN

inter-switch

links

Figure

A

switch-based

local

area

network

of

nonblocking

station

nodes

are

derived

Finally

in

cases

where

less

than

the

minimal

amount

of

hardware

to

support

nonblocking

is

provided

blocking

probabilities

are

derived

We

also

discuss

the

generality

of

the

balanced

network

topology

with

respect

to

nonbalanced

networks

and

show

its

application

to

a

realistic

network

environment

The

results

of

this

paper

are

applicable

to

networks

consisting

of

connection-oriented

switches

ATM

provides

connection-oriented

service

in

a

packet-switched

environment

via

virtual

circuits

All

cells

using

the

same

virtual

circuit

are

transmitted

in

order

via

the

same

physical

path

An

ATM

switch

consisting

of

M

physical

ports

where

multiple

virtual

circuits

are

multiplexed

on

each

port

can

be

logically

translated

into

a

purely

connectionoriented

switch

consisting

of

N

-LRB-

N

M

-RRB-

logical

ports

where

each

of

the

N

ports

is

used

by

a

single

virtual

circuit

Thus

an

ATM

network

directly

corresponds

to

a

purely

connectionoriented

network

The

results

of

this

paper

can

be

mapped

to

a

virtual

connection-oriented

network

such

as

ATM

or

a

purely

connection-oriented

environment

such

as

HIPPI

This

paper

is

organized

as

follows

Section

presents

an

overview

of

various

properties

of

switch-based

interconnection

networks

and

describes

how

multi-switch

networks

differ

from

multistage

interconnection

networks

and

telephone

networks

In

Section

the

general

problem

is

formulated

and

shown

to

be

NP-Complete

Different

types

of

configurations

based

upon

the

usage

of

the

switches

and

the

distribution

of

station

nodes

among

switches

are

discussed

and

compared

Section

provides

the

construction

of

an

optimal

structure

of

the

type

shown

in

Figure

case

-LRB-

a

-RRB-

and

a

sub-optimal

structure

of

the

type

shown

in

Figure

case

-LRB-

b

-RRB-

In

Section

bounds

for

balanced

Direct-Direct

nonblocking

networks

are

presented

Blocking

probabilities

are

also

derived

Section

provides

an

example

of

how

the

results

of

this

paper

may

be

applied

to

a

realistic

network

Section

provides

the

Conclusion

Multi-Switch

Networks

A

switching

network

consists

of

a

set

of

switches

interconnected

by

links

which

permit

data

communication

between

a

set

of

input

and

output

ports

The

simplest

switching

network

is

a

crossbar

An

N

N

crossbar

is

a

rectangular

array

of

crosspoints

It

allows

for

the

simultaneous

connection

of

all

sets

of

disjoint

pairs

Hereafter

the

term

switch

implies

a

switch

with

the

functionality

of

a

crossbar

switch

Multi-Switch

Networks

versus

other

Switching

Networks

The

interconnection

networks

we

will

be

examining

are

different

from

previously

studied

switching

networks

both

in

structure

and

design

motivations

Hereafter

we

distinguish

them

from

other

switching

networks

by

referring

to

them

as

multi-switch

networks

In

this

sub-section

we

discuss

the

features

which

disinguish

them

from

the

other

previously

studied

switching

networks

Multi-switch

networks

vs.

multistage

interconnection

networks

ffl

Environment

Multi-switch

networks

are

networks

of

switches

interconnecting

station

nodes

in

a

LAN

environment

Multistage

interconnection

networks

are

used

as

interconnection

networks

for

multiprocessors

telephone

switching

system

exchanges

and

general

purpose

large

switches

-LRB-

e.g.

Fibre

Channel

-RRB-

ffl

Switching

elements

Multistage

interconnection

networks

generally

use

a

large

number

of

small

switches

typically

x

switches

If

larger

switches

are

used

-LRB-

e.g.

Clos

network

-RRB-

they

must

be

of

a

specified

size

Multi-switch

networks

generally

use

a

smaller

number

of

much

larger

switches

Switches

may

be

of

arbitrary

sizes

e.g.

a

local

area

environment

may

be

interconnected

via

x

and

x

switches

ffl

I/O

terminal

connections

In

a

multi-switch

network

any

switch

may

contain

direct

connections

to

station

nodes

Since

it

is

expected

that

communication

patterns

in

a

LAN

environment

will

exhibit

a

high

degree

of

clustering

the

input

and

output

terminals

of

a

station

node

will

be

assumed

to

be

connected

to

the

same

switch

Nodes

connected

to

the

same

switch

are

expected

to

generate

a

greater

amount

of

traffic

and

hence

should

be

given

a

lower

switching

delay

-LRB-

one

crosspoint

switch

delay

-RRB-

In

multistage

interconnection

networks

-LRB-

MINs

-RRB-

each

station

nodes

-LRB-

network

-RRB-

input

and

output

terminals

are

attached

to

different

switches

-LRB-

i.e.

the

output

terminals

of

the

switches

in

the

last

stage

of

the

MIN

and

the

input

terminals

of

the

switches

in

the

first

stage

of

the

MIN

respectively

-RRB-

ffl

Minimizing

hardware

The

minimization

of

crosspoints

was

an

important

issue

in

designing

multistage

interconnection

networks

In

multi-switch

interconnection

networks

the

minimization

of

the

number

of

inter-switch

links

-LRB-

or

ports

dedicated

to

interswitch

communications

-RRB-

the

maximization

of

the

aggregate

network

throughput

and

the

minimization

of

network

latency

are

the

important

issues

Note

that

a

multi-switch

network

may

consist

of

switches

which

are

internally

multi-stage

interconnection

networks

However

due

to

the

differing

nature

of

the

environments

-LRB-

LAN

vs.

multiprocessor

-RRB-

it

is

not

considered

efficient

for

these

switches

to

be

externally

interconnected

as

a

multi-stage

interconnection

network

-LRB-

see

above

I/O

terminal

connections

-RRB-

Multi-switch

networks

vs.

telephone

networks

ffl

Number

of

connections

-LRB-

users

-RRB-

Multi-switch

networks

are

expected

to

have

to

support

on

the

order

of

several

hundred

to

several

thousand

station

nodes

Telephone

networks

are

expected

to

support

a

far

greater

number

of

users

or

connections

ffl

Environment

Multi-switch

networks

are

local

area

in

range

hence

the

placement

of

inter-switch

trunks

is

not

a

significant

factor

in

designing

the

arrangement

of

interswitch

links

among

the

set

of

switches

Telephone

networks

are

wide

area

networks

The

placement

of

switching

exchanges

and

links

are

determined

by

geographical

and

political

factors

as

well

as

network

design

factors

Hence

the

placement

of

inter-switch

links

is

a

significant

factor

in

the

design

process

Network

Properties

In

this

subsection

we

discuss

the

relationship

between

the

terms

maximum

aggregate

network

throughput

nonblocking

and

blocking

Network

topology

refers

to

the

manner

in

which

network

components

-LRB-

in

this

case

switches

-RRB-

are

interconnected

Switching

networks

are

typically

divided

into

three

categories

blocking

nonblocking

and

rearrangeable

nonblocking

In

a

blocking

network

a

possibility

exists

that

a

connection

between

an

I/O

pair

may

be

unavailable

due

to

previous

existing

connections

In

a

nonblocking

network

a

connection

may

always

be

established

between

any

given

idle

I/O

pair

regardless

of

previous

existing

connections

In

a

rearrangeable

nonblocking

network

a

connection

between

any

idle

I/O

pair

may

be

established

by

rearranging

its

existing

connections

In

this

paper

we

do

not

consider

networks

which

are

rearrangeable

Nonrearrangeable

nonblocking

networks

may

be

divided

into

two

categories

-LSB-

-RSB-

-LRB-

i

-RRB-

nonblocking

in

the

wide

sense

and

-LRB-

ii

-RRB-

nonblocking

in

the

strict

sense

-LRB-

or

strictly

blocking

-RRB-

Strictly

nonblocking

networks

ensure

that

blocking

will

never

occur

regardless

of

the

manner

in

which

connections

are

established

Nonblocking

networks

in

the

wide

sense

ensure

that

blocking

will

never

occur

only

if

a

certain

set

of

connection

establishment

rules

is

obeyed

Nonblocking

networks

in

the

wide

sense

are

of

interest

because

they

typically

employ

much

less

hardware

than

strictly

nonblocking

networks

A

nonblocking

-LRB-

rule

-RRB-

network

is

a

nonblocking

network

where

rule

is

the

algorithm

which

must

be

followed

in

order

to

ensure

nonblocking

In

our

study

we

examine

networks

which

are

nonblocking

in

the

wide

sense

Hereafter

we

use

the

term

nonblocking

to

imply

nonblocking

in

the

wide

sense

By

definition

a

nonblocking

network

supports

the

network

's

maximum

aggregate

throughput

Given

a

blocking

network

to

minimize

the

possibility

of

blocking

the

network

configuration

which

supports

the

largest

possible

network

aggregate

throughput

should

be

used

In

some

cases

it

is

not

feasible

to

find

an

optimal

-LRB-

in

terms

of

topology

consisting

of

a

minimal

number

of

inter-switch

links

-RRB-

nonblocking

network

In

those

cases

we

will

attempt

to

find

the

configuration

which

supports

the

largest

possible

network

aggregate

throughput

and

thus

decrease

the

blocking

probability

Problem

Formulation

In

this

section

we

formulate

the

following

problem

Given

a

set

of

switches

and

a

set

of

station

nodes

distributed

among

the

switches

find

the

overall

network

interconnection

topology

which

supports

the

maximal

aggregate

network

throughput

and

minimizes

the

total

number

of

inter-switch

links

in

the

network

We

use

the

following

notation

to

represent

the

network

G

-LRB-

V

E

-RRB-

represents

a

network

consisting

of

a

set

of

switches

V

vN

and

a

set

of

edges

E

f

-LRB-

eij

cij

-RRB-

j

i

j

N

eij

cij

Z

g.

eij

cij

is

equal

to

the

number

of

links

between

switch

vi

and

switch

vj

where

eij

denotes

whether

an

edge

exists

between

switch

vi

and

switch

vj

and

cij

denotes

the

number

of

links

between

switches

vi

and

vj

The

degree

of

a

switch

node

specifies

the

number

of

I/O

ports

dedicated

to

inter-switch

communications

or

the

number

of

incident

links

of

a

switch

node

which

lead

to

other

switches

Note

that

the

degree

of

a

switch

node

in

G

-LRB-

V

E

-RRB-

may

be

greater

than

N

Each

switch

vi

has

a

set

of

ni

attached

station

nodes

denoted

by

snni

i

g

where

ni

Z

T

denotes

the

set

of

sets

of

station

nodes

i.e.

T

g

g

snnN

N

gg

Tnum

PN

i

ni

A

graph

G

-LRB-

V

E

-RRB-

which

supports

T

will

be

denoted

as

GT

-LRB-

V

E

-RRB-

We

assume

the

following

for

the

remainder

of

the

paper

Each

station

node

may

inject

one

unit

of

data

per

unit

time

through

the

switch

and

into

the

network

Every

switch

port

can

sustain

one

unit

of

data

per

unit

time

Hereafter

we

will

refer

to

one

unit

of

data

per

unit

time

as

unit

transfer

rate

or

unit

capacity

All

inter-switch

links

have

the

same

channel

capacity

unit

capacity

Bidirectional

communication

is

supported

via

a

pair

of

unidirectional

links

each

pair

of

unidirectional

links

will

be

counted

as

one

inter-switch

link

When

the

term

link

is

used

a

bidirectional

link

is

implied

A

network

device

may

correspond

to

a

single

or

multiple

station

nodes

-LRB-

depending

on

the

network

device

's

I/O

requirements

-RRB-

For

example

a

typical

workstation

may

correspond

to

only

one

station

node

while

a

file

server

or

special

purpose

processor

may

correspond

to

multiple

station

nodes

The

I/O

ports

of

all

station

nodes

which

correspond

to

the

same

network

device

are

assumed

to

be

attached

to

the

same

switch

This

assumption

was

discussed

in

Section

under

I/O

terminal

connections

The

degree

of

a

switch

node

is

equal

to

the

number

of

incident

links

which

lead

to

other

switch

nodes

It

does

not

include

incident

links

which

lead

to

station

nodes

Definition

A

connection

set

on

GT

-LRB-

V

E

-RRB-

is

a

pairing

of

station

nodes

-LRB-

set

of

connections

-RRB-

tG

such

that

any

station

node

appears

in

at

most

one

pair

i.e.

tG

f

k

j

k

-LRB-

snri

snsj

-RRB-

g

where

k

i

j

N

r

ni

s

nj

at

time

instance

t.

For

example

Figure

-LRB-

a

-RRB-

depicts

a

graph

consisting

of

switch

nodes

with

station

nodes

attached

to

each

switch

-LRB-

-RRB-

denotes

a

connection

between

station

node

attached

to

switch

a

and

station

node

attached

to

switch

b

An

example

of

a

connection

set

is

-LRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-RRB-

Note

that

the

maximum

number

of

connections

in

a

connection

set

is

bTnum

Definition

GT

is

the

set

of

all

possible

connection

sets

on

GT

-LRB-

V

E

-RRB-

GT

f?tG

j

t

g.

Note

that

for

GT

-LRB-

V

E

-RRB-

to

be

nonblocking

it

must

support

GT

By

definition

a

nonblocking

network

supports

the

maximal

aggregate

throughput

and

a

network

which

always

supports

the

maximal

aggregate

throughput

is

nonblocking

The

following

terminology

holds

for

the

remainder

of

the

paper

The

delay

-LRB-

or

latency

-RRB-

of

a

connection

request

is

the

number

of

crosspoint

switches

the

connection

path

contains

If

a

connection

request

is

between

two

station

nodes

attached

to

the

same

switch

it

is

referred

to

as

a

local

connection

i.e.

a

connection

with

delay

of

If

a

connection

request

is

between

two

station

nodes

attached

to

different

switches

it

is

referred

to

as

a

non-local

connection

i.e.

a

connection

with

delay

Given

a

nonlocal

connection

request

-LRB-

snpi

snqj

-RRB-

if

there

does

not

exist

a

physical

link

between

switch

i

and

switch

j

then

this

connection

request

is

called

an

indirect

connection

Otherwise

it

is

called

a

direct

connection

We

introduce

the

following

variables

necessary

to

formulate

the

problem

csdij

denotes

the

amount

of

traffic

-LRB-

or

number

of

required

links

-RRB-

from

source-destination

switch

node

pair

s

d

V

which

flows

from

node

i

to

node

j

i

j

V

Note

that

P

sd

csdij

cij

esdij

denotes

whether

the

edge

between

nodes

i

and

j

is

used

when

a

connection

between

nodes

s

and

d

is

established

-LRB-

if

so

esdij

if

not

esdij

-RRB-

Note

that

if

P

sd

esdij

then

eij

We

seek

to

minimize

the

following

objective

function

total

-LRB-

c

-RRB-

X

ij

cij

X

ij

X

sd

csdij

-LRB-

-RRB-

subject

to

the

following

constraints

X

d

X

j

cidij

ni

for

i

N

-LRB-

-RRB-

X

sd

X

i

csdij

X

sd

X

i

csdji

P

s

P

i

csdid

if

d

j

P

d

P

i

csdsi

if

s

j

otherwise

-LRB-

-RRB-

X

i

j

esdij

jEj

d

V

-LRB-

-RRB-

esdij

j

V

s

d

V

-LRB-

-RRB-

csdij

j

V

s

d

V

-LRB-

-RRB-

Constraint

forces

the

network

to

support

the

worse

case

-LRB-

maximal

interswitch

traffic

-RRB-

when

all

station

nodes

transmit

nonlocally

Constraint

is

the

flow

conservation

constraint

It

states

that

the

traffic

flow

into

a

node

must

equal

the

traffic

flow

leaving

the

node

-LRB-

except

at

the

source

and

destination

nodes

-RRB-

Constraint

states

that

the

path

length

for

any

source

destination

pair

of

nodes

is

bounded

above

by

the

number

of

edges

in

the

graph

Since

the

variables

in

this

problem

formulation

must

be

integers

the

integer

programming

problem

-LSB-

-RSB-

reduces

to

this

problem

formulation

Thus

this

problem

is

also

NP

Complete

Since

in

most

cases

it

is

not

feasible

to

solve

this

problem

we

seek

to

find

properties

of

possible

network

configurations

which

result

in

better

configurations

i.e.

configurations

with

a

minimal

number

of

links

which

support

the

maximal

aggregate

throughput

of

the

network

These

properties

include

the

distribution

pattern

of

station

nodes

among

switches

and

the

usage

of

switches

-LRB-

see

Figure

-RRB-

Note

that

the

least

number

of

links

a

network

must

contain

to

be

non-blocking

is

bTnum

inter-switch

bi-directional

links

This

is

because

the

network

must

be

able

to

support

the

worse

case

i.e.

all

station

nodes

are

communinicating

non-locally

or

Tnum

pairs

Network

topologies

based

upon

switch

usage

Multi-switch

network

configurations

may

consist

of

an

arrangement

where

some

switches

have

direct

connections

to

station

nodes

and

the

remaining

switches

have

connections

only

to

other

switches

-LRB-

Figure

case

-LRB-

a

-RRB-

-RRB-

or

where

every

switch

has

direct

connections

to

station

nodes

-LRB-

Figure

case

-LRB-

b

-RRB-

-RRB-

We

will

case

-LRB-

a

-RRB-

case

-LRB-

b

-RRB-

switches

with

direct

connections

only

to

other

switches

inter-switch

links

only

SN

SN

SN

SN

SN

SN

SN

SN

SN

SN

SN

switchset

ofstation

nodes

Figure

Different

network

configurations

-LRB-

a

-RRB-

Direct-Inter-Direct

-LRB-

b

-RRB-

Direct-Direct

V

V

V

V

Figure

An

optimal

case

refer

to

Figure

case

-LRB-

a

-RRB-

as

the

Direct-Inter-Direct

type

of

configuration

and

Figure

case

-LRB-

b

-RRB-

as

the

Direct-Direct

type

of

configuration

Section

presents

an

analysis

of

the

Direct-Inter-Direct

configuration

An

optimal

network

structure

S

is

derived

-LRB-

Definition

Definition

Figure

and

Figure

-RRB-

and

proved

optimal

-LRB-

in

terms

of

maximizing

the

number

of

nonblocking

station

nodes

for

a

given

set

of

switches

-RRB-

in

Theorem

Section

presents

an

analysis

of

the

Direct-Direct

configuration

A

sub-optimal

network

structure

A

is

presented

-LRB-

Definition

and

Figure

-RRB-

and

shown

to

be

nonblocking

in

Theorem

Theorem

and

Corollary

state

that

an

optimal

configuration

S

in

the

Direct-InterDirect

topology

class

yields

the

minimum

number

of

I/O

ports

per

switch

needed

to

support

Tnum

nonblocking

station

nodes

with

N

switches

i.e.

m

Tnum

N

Tnum

N

-LRB-

-LRB-

Tnum

m

-RRB-

-RRB-

-LRB-

-RRB-

where

m

is

the

number

of

I/O

ports

per

switch

Theorems

and

state

that

a

sub-optimal

configuration

A

in

the

Direct-Direct

topologoy

class

yields

the

minimum

number

of

I/O

ports

per

switch

needed

to

support

Tnum

nonblocking

station

nodes

with

N

switches

i.e.

m

N

-LRB-

-RRB-

Equations

and

imply

that

when

Tnum

m

-LRB-

-RRB-

it

is

more

efficient

to

interconnect

station

nodes

and

switches

in

the

form

of

a

Direct-Direct

topology

In

other

words

a

suboptimal

structure

in

a

Direct-Direct

configuration

is

superior

-LRB-

in

most

cases

Tnum

m

-RRB-

to

an

optimal

structure

in

a

Direct-Inter-Direct

configuration

Note

that

a

multiple

switch

network

consisting

of

several

m

m

switches

almost

always

implies

Tnum

m.

-LRB-

A

single

m?m

switch

supports

Tnum

m

station

nodes

-RRB-

In

terms

of

latency

A

incurs

a

latency

of

for

-LRB-

N

-RRB-

-LRB-

N

-RRB-

of

non-local

requests

and

a

latency

of

for

-LRB-

N

-RRB-

of

non-local

requests

S

-LRB-

m

m

-RRB-

i

incurs

a

latency

of

-LRB-

i

-RRB-

for

every

non-local

request

Thus

in

terms

of

latency

networks

configured

as

Direct-Direct

configurations

are

superior

to

networks

configured

as

Direct-Inter-Direct

configurations

Section

presents

the

derivation

of

the

results

presented

in

the

above

paragraphs

Uniform

vs.

non-uniform

distribution

of

station

nodes

among

switches

Figure

illustrates

an

example

of

an

optimal

configuration

of

four

switches

each

having

and

attached

terminals

respectively

The

resulting

total

number

of

interswitch

links

is

optimal

-LRB-

minimal

-RRB-

This

is

because

the

number

of

station

nodes

is

so

there

must

exist

at

least

enough

links

to

satisfy

the

case

when

all

pairs

of

station

nodes

are

communicating

nonlocally

Thus

there

must

exist

at

least

links

in

the

network

However

switch

must

dedicate

I/O

ports

to

inter-switch

communications

while

the

other

three

switches

need

to

dedicate

significantly

fewer

So

a

non-uniform

distribution

of

station

nodes

among

switches

may

result

in

configurations

where

a

few

switches

need

to

dedicate

a

significantly

greater

number

of

ports

Actually

it

is

intuitively

obvious

that

most

optimal

configurations

based

upon

a

non-uniform

distribution

of

station

nodes

will

result

in

irregular

topologies

Thus

it

is

of

interest

to

study

how

topologies

where

station

nodes

are

distributed

evenly

-LRB-

or

almost

evenly

-RRB-

among

switches

may

be

configured

to

maximize

the

number

of

nonblocking

station

nodes

Analysis

of

Two

Network

Configurations

In

this

section

we

derive

an

optimal

network

in

the

Direct-Inter-Direct

class

of

configurations

-LRB-

Figure

case

-LRB-

a

-RRB-

-RRB-

and

a

suboptimal

network

in

the

Direct-Direct

class

of

configurations

-LRB-

Figure

case

-LRB-

b

-RRB-

-RRB-

Since

these

results

are

only

used

to

show

the

superiority

of

the

Direct-Direct

type

of

configuration

to

the

Direct-Inter-Direct

type

of

configuration

the

reader

may

skip

to

Section

without

a

loss

of

continuity

The

Direct-Inter-Direct

Topology

In

this

sub-section

we

describe

the

Direct-Inter-Direct

interconnection

pattern

-LRB-

shown

in

Figure

Case

-LRB-

a

-RRB-

-RRB-

which

guarantees

nonblocking

while

supporting

a

maximal

number

of

stations

The

network

in

this

section

consists

of

the

following

two

categories

of

switches

-LRB-

i

-RRB-

switches

which

are

used

solely

to

interconnect

other

switches

these

switches

will

be

referred

to

as

intermediate

switches

and

-LRB-

ii

-RRB-

switches

which

interconnect

station

nodes

to

intermediate

switches

Note

that

the

constructed

interconnection

pattern

we

describe

in

this

section

is

only

optimal

in

terms

of

minimal

hardware

and

may

not

be

the

only

optimal

interconnection

pattern

We

construct

it

here

solely

in

order

to

derive

for

a

given

number

of

station

nodes

Tnum

the

minimal

amount

of

hardware

necessary

to

guarantee

nonblocking

Definition

S

-LRB-

m

Tnum

-RRB-

specifies

a

switch

interconnection

pattern

which

interconnects

Tnum

stations

via

a

number

of

m?m

switches

Definition

S

-LRB-

i

-RRB-

S

-LRB-

m

m

-RRB-

i

is

a

network

of

switches

such

that

S

-LRB-

i

-RRB-

consists

of

columns

of

intermediate

switches

Each

column

is

denoted

by

ck

where

k

denotes

the

first

column

of

switches

which

interconnect

the

station

nodes

to

the

first

column

of

intermediate

switches

-LRB-

-RRB-

In

S

-LRB-

i

-RRB-

there

are

switches

in

the

first

column

Each

switch

in

the

first

column

is

directly

connected

to

m

station

nodes

-LRB-

i.e.

m

I/O

ports

dedicated

to

m

station

nodes

-RRB-

Each

switch

in

the

first

column

also

has

two

outgoing

edges

with

each

edge

consisting

of

m

inter-switch

links

All

the

columns

consisting

of

intermediate

switches

consist

of

switches

In

S

-LRB-

i

-RRB-

for

i

the

switch

configuration

S

-LRB-

m

-RRB-

consists

of

two

S

-LRB-

m

-RRB-

sub-networks

of

switches

interconnected

via

two

additional

columns

denoted

by

and

each

of

which

consists

of

switches

The

switches

in

the

last

column

of

each

of

the

sub-networks

have

outgoing

edges

with

each

of

the

edges

connected

to

a

separate

switch

in

Each

edge

consists

of

m

lines

Column

is

similarly

connected

to

column

Column

is

connected

to

the

switches

from

the

first

column

in

the

network

such

that

the

first

half

of

the

switches

in

column

is

connected

to

the

first

half

of

the

switches

from

the

first

column

Each

switch

in

column

has

outgoing

edges

with

each

edge

connected

to

a

different

switch

in

Each

edge

consists

of

m

lines

The

second

half

of

the

last

column

is

similarly

connected

to

the

second

half

of

the

first

column

of

switches

Figures

and

show

the

networks

S

-LRB-

m

-RRB-

and

S

-LRB-

m

-RRB-

respectively

Theorem

Given

Tnum

station

nodes

such

that

Tnum

m

where

i

and

a

number

of

m

m

switches

S

-LRB-

i

-RRB-

S

-LRB-

m

-RRB-

-LRB-

as

given

by

definitions

and

-RRB-

is

the

interconnection

pattern

which

minimizes

the

number

of

switches

necessary

to

ensure

nonblocking

among

Tnum

m

stations

Proof

Proof

by

induction

on

i

for

S

-LRB-

m

-RRB-

Base

case

For

S

-LRB-

m

-RRB-

the

worse

case

occurs

when

all

station

nodes

attempt

to

communicate

nonlocally

An

example

of

a

worse

case

is

when

in

Figure

the

following

occurs

-LRB-

i

-RRB-

m

station

nodes

from

switch

wish

to

establish

m

connections

to

the

station

nodes

attached

to

station

links/edge

C

C

C

BA

Figure

S

-LRB-

m

m

-RRB-

S

-LRB-

m

m

-RRB-

links/edge

station

nodes/switch

C

C

C

C

C

Figure

S

-LRB-

m

m

-RRB-

switch

and

vice

versa

i.e.

-LRB-

-RRB-

and

-LRB-

ii

-RRB-

m

station

nodes

from

switch

wish

to

establish

m

connections

to

the

station

nodes

attached

to

switch

and

vice

versa

i.e.

-LRB-

-RRB-

Let

-LRB-

x

y

-RRB-

signify

the

edge

-LRB-

all

links

-RRB-

from

switch

x

to

switch

y.

-LRB-

i

-RRB-

implies

the

following

connections

will

have

to

be

used

-LRB-

A

-RRB-

-LRB-

B

-RRB-

-LRB-

A

-RRB-

-LRB-

B

-RRB-

-LRB-

A

-RRB-

-LRB-

B

-RRB-

-LRB-

A

-RRB-

-LRB-

B

-RRB-

-LRB-

ii

-RRB-

implies

the

following

connections

will

have

to

be

used

-LRB-

A

-RRB-

-LRB-

B

-RRB-

-LRB-

A

-RRB-

-LRB-

B

-RRB-

-LRB-

A

-RRB-

-LRB-

B

-RRB-

-LRB-

A

-RRB-

-LRB-

B

-RRB-

Thus

this

worse

case

uses

all

links

in

S

-LRB-

m

-RRB-

From

inspection

of

Figure

is

is

obvious

that

there

is

adequate

amount

of

hardware

-LRB-

inter-switch

lines

-RRB-

to

ensure

blocking

will

never

occur

for

any

permutation

of

requests

-LRB-

this

can

be

seen

by

examining

every

worse

case

connection

pattern

-RRB-

It

is

also

apparent

that

S

-LRB-

m

-RRB-

consists

of

no

more

than

the

necessary

amount

of

hardware

because

in

every

worse

case

of

any

m

pairs

of

nodes

communicating

non-locally

every

inter-switch

line

is

utilized

Induction

hypothesis

Assume

S

-LRB-

m

-RRB-

where

j

i

represents

a

nonblocking

interconnection

network

which

uses

the

minimal

number

of

switches

Induction

step

From

Definition

S

-LRB-

m

-RRB-

consists

of

two

S

-LRB-

m

-RRB-

sub-networks

interconnected

via

two

columns

of

switches

each

A

worse

case

occurs

when

station

nodes

attached

to

sub-network

-LRB-

see

Figure

-RRB-

wish

to

communicate

to

the

station

nodes

attached

to

sub-network

-LRB-

and

vice

versa

-RRB-

At

column

sub-network

can

establish

connections

through

at

most

edges

to

the

bottom

half

of

Each

edge

consists

of

m

lines

Thus

at

this

stage

the

maximum

total

flow

from

sub-network

to

the

bottom

half

of

column

is

Thus

at

least

another

column

of

switches

is

necessary

At

column

the

top

half

of

the

previous

column

can

establish

another

connections

to

the

bottom

half

of

while

the

bottom

half

of

can

continue

to

establish

the

previous

connections

from

sub-nework

-LRB-

described

in

the

above

paragraph

-RRB-

Thus

at

column

all

connections

from

sub-network

can

be

established

to

the

station

nodes

attached

to

sub-network

A

similar

argument

can

be

made

for

any

arbitrary

set

of

m

station

nodes

which

wish

to

establish

connections

to

the

other

set

of

m

station

nodes

Corollary

Given

Tnum

station

nodes

the

minimal

number

of

m?m

switches

necessary

to

ensure

nonblocking

-LRB-

from

S

-LRB-

i

-RRB-

S

-LRB-

m

m

-RRB-

-RRB-

is

-LRB-

m

-RRB-

-LRB-

Tnum

m

-RRB-

-LRB-

-LRB-

Tnum

m

-RRB-

-RRB-

-LRB-

-RRB-

Proof

Theorem

constructed

a

minimal

hardware

nonblocking

configuration

S

-LRB-

m

m

-RRB-

The

total

number

of

switches

in

S

-LRB-

m

Tnum

-RRB-

where

Tnum

i

is

the

number

of

switches

in

the

first

column

plus

the

number

of

switches

in

the

intermediate

stages

Each

intermediate

column

has

Tnum

m

switches

and

there

are

-LRB-

Tnum

m

-RRB-

intermediate

columns

Thus

S

consists

of

a

total

of

m

Tnum

m

-LRB-

-LRB-

Tnum

m

-RRB-

-RRB-

m?m

switches

Interconnection

networks

in

S

are

not

strictly

blocking

That

is

a

certain

sequence

of

S

-LRB-

m

m

-RRB-

S

-LRB-

m

m

-RRB-

C

CC

m

sub-network

sub-network

C

Figure

S

-LRB-

m

-RRB-

Worse

case

scenario

connection

establishments

may

lead

to

a

blocking

state

although

the

hardware

provided

does

provide

for

every

possible

permutation

of

requests

to

be

satisfied

A

blocking

situation

occurs

if

a

connection

call

connx

is

at

switch

vi

and

wishes

to

reach

switch

vk

via

an

intermediate

switch

In

order

to

reach

switch

vk

connx

must

go

through

an

intermediate

switch

If

it

chooses

to

go

through

an

intermediate

switch

which

does

not

have

an

available

open

port

which

leads

to

vk

a

blocking

state

will

occur

When

this

occurs

the

intermediate

switch

must

send

a

signal

back

to

switch

vi

so

that

another

intermediate

switch

can

be

choosen

This

process

continues

until

an

intermediate

switch

which

does

have

an

available

open

port

to

switch

vk

is

found

Theorem

guarantees

that

such

an

intermediate

switch

will

be

found

The

optimally

constructed

interconnection

network

in

this

section

resemble

traditional

multistage

interconnection

networks

However

because

of

the

assumption

that

the

input

and

output

terminals

of

a

station

node

are

attached

to

the

same

switch

the

problem

and

results

differ

from

those

of

the

traditionally

studied

multistage

interconnection

networks

-LSB-

-RSB-

However

this

work

may

be

considered

an

extension

of

previous

work

related

to

MINs

The

assumption

that

station

nodes

input

and

output

terminals

are

attached

to

the

same

switch

is

a

valid

assumption

considering

we

are

studying

local

area

networks

Network

traffic

in

a

local

area

environment

is

expected

to

exhibit

a

high

degree

of

locality

Thus

we

would

want

to

minimize

the

latency

of

the

bulk

of

the

traffic

by

attaching

the

input

and

output

port

-LRB-

s

-RRB-

of

a

device

to

the

same

switch

The

Direct-Direct

Topology

In

this

section

we

derive

a

class

of

nonblocking

graphs

in

the

Direct-Direct

configuration

class

-LRB-

see

Figure

-RRB-

These

nonblocking

graphs

are

sub-optimal

They

are

constructed

here

to

be

used

in

comparison

with

the

Direct-Direct

class

of

networks

In

terms

of

minimal

hardware

-LRB-

required

I/O

ports/switch

-RRB-

and

latency

they

are

superior

to

optimal

networks

in

the

Direct-Inter-Direct

class

Definition

Adeg

T

-LRB-

V

E

-RRB-

denotes

a

network

consisting

of

jV

j

N

vertices

-LRB-

switches

-RRB-

where

every

vertex

has

degree

specified

by

deg

The

following

hold

for

any

Adeg

T

-LRB-

V

E

-RRB-

N

and

N

is

even

deg

-LRB-

N

-RRB-

i

where

i

deg

and

-LRB-

deg

mod

-RRB-

There

exists

exactly

N

pairs

of

vertices

which

are

not

directly

connected

All

other

pairs

of

vertices

are

directly

connected

via

i

links

Figure

shows

Adeg

T

-LRB-

V

E

-RRB-

with

-LRB-

a

-RRB-

deg

and

-LRB-

b

-RRB-

deg

In

both

figures

there

are

exactly

pairs

of

vertices

which

are

not

directly

connected

-LRB-

a

c

-RRB-

-LRB-

b

f

-RRB-

and

-LRB-

d

e

-RRB-

Theorem

Given

Tnum

station

nodes

evenly

divided

among

N

switches

to

ensure

nonblocking

the

least

amount

of

hardware

any

configuration

Adeg

T

-LRB-

V

E

-RRB-

must

use

consists

of

the

following

-LRB-

i

-RRB-

Tnum

inter-switch

links

and

-LRB-

ii

-RRB-

N

I/O

ports/switch

-LRB-

of

which

N

I/O

ports/switch

are

dedicated

to

inter-switch

communications

and

Tnum

N

I/O

ports/switch

are

used

for

direct

connections

to

station

nodes

-RRB-

Proof

We

examine

the

set

of

connections

which

requires

the

greatest

amount

of

channel

capacity

This

occurs

when

N

pairs

of

switches

are

each

injecting

Tnum

N

units

into

the

network

and

all

pairs

must

communicate

indirectly

Assume

the

all

indirect

connections

have

the

same

path

length

or

latency

Let

pathLength

and

numPaths

denote

the

length

of

a

connection

path

and

the

total

number

of

paths

-LRB-

connections

-RRB-

in

Adeg

T

-LRB-

V

E

-RRB-

respectively

Since

all

switches

are

communicating

indirectly

pathLength

Each

path

consumes

degrees

from

the

corresponding

graph

representation

The

total

number

of

degrees

in

a

graph

is

N

deg

so

N

deg

numPaths

pathLength

or

numPaths

N

deg

-LRB-

pathLength

-RRB-

-LRB-

-RRB-

Each

switch

uses

its

outgoing

edges

for

intermediate

traffic

-LRB-

when

the

switch

is

serving

as

an

intermediate

switch

node

-RRB-

and

for

direct

conections

to

station

nodes

-LRB-

i.e.

direct

outgoing

traffic

-RRB-

The

number

of

outgoing

edges

each

switch

must

dedicate

to

direct

outgoing

traffic

is

numPaths

-LRB-

N

-RRB-

deg

pathLength

-LRB-

by

Equation

-RRB-

with

each

edge

having

capacity

Tnum

pathLength

-LRB-

N

deg

-RRB-

Thus

the

total

capacity

-LRB-

number

of

inter-switch

links

-RRB-

the

network

must

support

is

-LRB-

N

deg

-RRB-

-LRB-

Tnum

pathLength

-LRB-

N

deg

-RRB-

or

Tnum

pathLength

units

capacity

a

de

f

b

c

a

b

c

d

e

f

Figure

-LRB-

a

-RRB-

-LRB-

V

E

-RRB-

-LRB-

b

-RRB-

-LRB-

V

E

-RRB-

Each

switch

node

must

use

deg

-LRB-

pathLength

-RRB-

of

its

degrees

when

serving

as

an

intermediate

switch

and

the

other

deg

pathLength

for

direct

outgoing

traffic

So

each

switch

node

must

be

capable

of

injecting

Tnum

pathLength

N

units

into

the

network

Thus

each

switch

node

must

dedicate

Tnum

pathLength

N

input/output

ports

to

interswitch

communications

Given

Tnum

station

nodes

we

choose

pathLength

because

that

requires

the

least

number

of

inter-switch

links

and

I/O

ports

per

switch

As

noted

before

for

any

GT

-LRB-

V

E

-RRB-

to

be

non-blocking

it

must

support

GT

However

j?GT

j

or

even

j?Adeg

T

-LRB-

V

E

-RRB-

j

is

a

very

large

and

unstructured

set

In

order

to

simplify

the

task

of

analyzing

these

non-blocking

networks

we

first

reduce

the

size

of

the

set

of

all

possible

connection

sets

to

include

only

a

special

subset

of

that

set

If

we

can

show

that

this

special

subset

can

be

established

on

Adeg

T

-LRB-

V

E

-RRB-

that

will

imply

that

all

other

connection

sets

may

be

established

on

Adeg

T

-LRB-

V

E

-RRB-

i.e.

Adeg

T

-LRB-

V

E

-RRB-

is

nonblocking

We

state

the

following

definitions

in

general

terms

-LRB-

with

respect

to

GT

-LRB-

V

E

-RRB-

-RRB-

They

still

apply

directly

to

Adeg

T

-LRB-

V

E

-RRB-

Definition

A

full

connection

set

on

GT

-LRB-

V

E

-RRB-

is

the

largest

connection

set

-LRB-

which

consists

of

bTnum

connections

-RRB-

on

GT

-LRB-

V

E

-RRB-

Definition

GT

is

a

mixed

connection

set

if

there

exists

a

-LRB-

snpi

snqj

-RRB-

GT

and

-LRB-

sn?pi

snrk

-RRB-

G

where

j

k

p

p

and

p

p

ni

q

nj

r

nk

Definition

GT

is

a

non-mixed

connection

set

if

for

all

switches

vi

all

station

nodes

-LRB-

which

request

a

connection

-RRB-

attached

to

vi

request

connections

to

-LRB-

i

-RRB-

a

set

of

nonlocal

station

nodes

which

are

all

connected

to

the

same

switch

or

-LRB-

ii

-RRB-

only

local

station

nodes

-LRB-

i.e.

station

nodes

attached

to

vi

-RRB-

Definition

GT

is

a

non-mixed

indirect

full

connection

set

if

it

is

a

non-mixed

full

connection

set

which

consists

of

only

indirect

connections

e.g.

for

any

-LRB-

snpi

snqj

-RRB-

GT

there

is

no

direct

connection

between

switch

i

and

switch

j.

Example

We

examine

connection

sets

for

the

network

shown

in

Figure

-LRB-

a

-RRB-

The

number

in

the

circle

indicates

the

number

of

attached

station

nodes

An

example

of

a

non-mixed

full

connection

set

is

f

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

g

An

example

of

a

mixed

connection

set

is

f

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

g

An

example

of

a

non-mixed

indirect

full

connection

set

is

f

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

g

With

respect

to

the

overall

inter-switch

traffic

patterns

Adeg

T

-LRB-

V

E

-RRB-

has

only

one

nonmixed

indirect

full

connection

set

-LRB-

see

Example

-RRB-

We

refer

to

this

set

as

Adeg

T

-LRB-

V

E

-RRB-

or

if

the

context

is

clear

as

the

non-mixed

indirect

full

connection

set

The

set

Adeg

T

-LRB-

V

E

-RRB-

is

special

because

it

has

the

property

that

if

a

network

Adeg

T

-LRB-

V

E

-RRB-

supports

it

then

the

network

will

be

able

to

support

all

possible

sets

of

connections

Adeg

T

-LRB-

V

E

-RRB-

-LRB-

We

prove

this

in

Lemma

below

-RRB-

Note

that

there

does

not

exist

a

mixed

indirect

full

connection

set

for

any

member

of

Adeg

T

Lemma

If

a

network

Adeg

T

-LRB-

V

E

-RRB-

supports

the

non-mixed

indirect

full

connection

set

-LRB-

Adeg

T

-LRB-

V

E

-RRB-

-RRB-

then

it

can

support

any

connection

set

on

Adeg

T

-LRB-

V

E

-RRB-

Proof

If

we

show

that

all

full

sets

consisting

solely

of

non-local

connection

requests

can

be

supported

by

Adeg

T

-LRB-

V

E

-RRB-

it

follows

that

all

full

sets

consisting

of

non-local

connections

and

local

connection

requests

can

also

be

supported

by

Adeg

T

-LRB-

V

E

-RRB-

Hence

we

assume

all

connection

requests

are

non-local

full

sets

Recall

deg

-LRB-

N

-RRB-

i.

Also

i

is

the

number

of

links

between

any

two

vertices

which

are

directly

connected

Any

Adeg

T

-LRB-

V

E

-RRB-

has

Tnum

N

full

connection

sets

We

divide

the

Tnum

N

full

sets

into

Rounds

At

each

Round

we

want

to

show

that

if

any

indirect

full

connection

set

can

be

established

then

all

other

types

of

connection

sets

may

also

be

established

-LRB-

at

that

Round

-RRB-

Let

ZA

-LRB-

i

-RRB-

be

the

graph

with

the

links

used

in

Round

to

Round

i

removed

We

want

to

show

that

any

full

connection

set

denoted

by

A

may

be

established

on

ZA

-LRB-

i

-RRB-

at

Round

i

for

all

i

Tnum

N

A

consists

of

either

only

direct

connections

or

direct

and

indirect

connections

Note

that

A

consists

of

only

un-mixed

connections

pairs

since

at

each

Round

each

switch

only

has

at

most

one

station

node

requesting

a

connection

To

show

that

A

may

be

established

on

ZA

-LRB-

i

-RRB-

first

establish

all

the

direct

connections

In

order

to

show

that

the

remaining

indirect

connections

may

be

established

it

is

enough

to

show

that

no

link

used

in

a

direct

connection

needs

to

be

used

to

establish

any

of

the

remaining

indirect

connections

This

is

easily

shown

by

the

following

argument

Say

link

-LRB-

x

y

-RRB-

between

switch

x

and

switch

y

was

used

by

a

direct

connection

and

it

blocks

an

indirect

connection

request

between

switch

m

and

switch

n

In

Theorem

we

show

that

A

attains

the

minimum

hardware

bounds

given

in

Theorem

and

is

nonblocking

Thus

we

can

assume

any

indirect

connection

path

is

of

length

Thus

either

switch

m

is

the

same

switch

as

switch

x

or

switch

n

is

the

same

switch

as

switch

y.

Neither

case

can

be

true

because

each

station

node

may

only

occur

in

only

one

connection

pair

and

at

each

Round

there

is

only

one

requesting

station

node

per

switch

For

example

in

Figure

say

link

-LRB-

b

d

-RRB-

is

used

for

a

direct

connection

between

a

station

node

attached

to

switch

b

and

a

station

node

attached

to

switch

d.

Link

-LRB-

b

d

-RRB-

is

not

necessary

for

the

establishment

of

any

remaining

indirect

connections

in

the

same

connection

set

because

neither

switch

b

nor

switch

d

may

request

another

connection

e.g.

a

station

node

attached

to

switch

e

can

not

request

a

connection

to

the

station

node

attached

to

switch

d

-LRB-

via

link

-LRB-

b

d

-RRB-

because

that

station

node

attached

to

switch

d

was

used

in

a

previous

connection

pair

in

the

same

connection

set

or

Round

-RRB-

We

state

a

connection

establishment

algorithm

which

must

be

followed

in

order

to

ensure

nonblocking

The

following

apply

to

the

connection

establishment

algorithm

d

-LRB-

vi

-RRB-

denotes

the

number

of

links

incident

to

vi

-LRB-

and

which

lead

to

other

switch

nodes

-RRB-

which

are

taken

Initially

d

-LRB-

vi

-RRB-

for

all

i.

Note

that

d

-LRB-

vi

-RRB-

N

deg

V

complete

denotes

the

set

of

vertices

which

have

established

connections

for

all

their

attached

station

nodes

If

vi

V

complete

then

d

-LRB-

vi

-RRB-

deg

Since

we

will

be

examining

only

non-mixed

connection

sets

we

simplify

the

notation

used

to

specify

a

connection

pair

to

-LRB-

sni

snj

-RRB-

which

denotes

that

all

station

nodes

attached

to

switch

i

and

switch

j

request

-LRB-

or

have

established

-RRB-

a

connection

-LRB-

s

-RRB-

The

input

to

RuleA

is

any

connection

set

in

Adeg

T

-LRB-

V

E

-RRB-

Basically

the

rule

states

which

set

of

switch

nodes

to

choose

as

the

intermediate

nodes

-LRB-

when

no

direct

connection

is

available

-RRB-

RuleA

Each

indirect

connection

may

include

only

one

intermediate

switch

node

Direct

connections

must

use

the

direct

path

If

there

is

no

direct

connection

between

sni

and

snj

choose

deg

intermediate

switch

nodes

such

that

-LRB-

a

-RRB-

If

vi

is

choosen

then

vj

must

be

choosen

such

that

-LRB-

sni

snj

-RRB-

is

in

the

connection

request

set

For

example

in

Figure

-LRB-

a

-RRB-

given

the

following

set

of

connection

pair

requests

f

-LRB-

sna

snc

-RRB-

-LRB-

snb

snf

-RRB-

-LRB-

sne

snd

-RRB-

g.

-LRB-

sne

snd

-RRB-

must

establish

two

A

B

C

D

E

-LRB-

b

-RRB-

-LRB-

a

-RRB-

CBA

D

E

pairnode

X

X

X

X

X

X

X

X

X

X

Figure

A

switch

network

connection

paths

They

must

use

either

switch

a

and

switch

c

or

switch

b

and

switch

f

as

intermediate

nodes

-LRB-

b

-RRB-

The

intermediate

nodes

must

be

picked

according

to

the

following

priorities

i.

Vertex

vi

V

complete

such

that

d

-LRB-

vi

-RRB-

deg

ii

Vertex

vi

V

complete

such

that

min

-LRB-

V

-RRB-

vi

where

V

vN

and

the

function

min

-LRB-

V

-RRB-

returns

the

vertex

with

the

least

d

-LRB-

vj

-RRB-

for

all

j.

Theorem

Adeg

T

-LRB-

V

E

-RRB-

is

a

nonblocking

network

which

uses

the

minimal

number

of

interswitch

links

-LRB-

as

stated

in

Theorem

-RRB-

Proof

We

examine

the

cases

deg

N

and

deg

-LRB-

N

-RRB-

i

for

i

separately

N

deg

By

Lemma

in

order

to

show

that

Adeg

T

-LRB-

V

E

-RRB-

is

nonblocking

we

only

need

to

show

that

Adeg

T

-LRB-

V

E

-RRB-

can

support

its

non-mixed

indirect

connection

set

Since

its

non-mixed

indirect

connection

set

consists

of

connection

requests

such

that

all

the

station

nodes

attached

to

a

switch

comminicate

only

with

station

nodes

from

another

single

switch

we

specify

the

connection

requests

as

-LRB-

vi

vj

-RRB-

which

denotes

that

all

station

nodes

attached

to

vi

communicate

only

with

station

nodes

attached

to

vj

And

vi

and

vj

are

not

directly

connected

Thus

there

are

N

connection

request

sets

of

this

type

We

will

show

that

these

sets

can

be

established

on

the

corresponding

network

by

dividing

them

into

Rounds

In

Round

k

-LRB-

k

N

-RRB-

-LRB-

vi

vj

-RRB-

establishes

Tnum

N

-LRB-

or

deg

-RRB-

connections

through

Tnum

N

intermediate

nodes

Note

that

the

term

Round

in

this

proof

has

a

different

meaning

than

the

same

term

used

in

the

proof

of

Lemma

Figure

shows

a

switch

network

where

Tnum

On

the

first

Round

node

pair

A

establishes

two

connections

with

each

node

pair

B

and

C

The

darkened

lines

on

Figure

-LRB-

b

-RRB-

show

the

connections

after

Round

The

X

in

the

box

signifies

that

the

node

pair

has

completed

establishing

all

their

connections

The

number

in

the

box

indicates

the

number

of

available

pairs

that

the

node

pair

may

use

as

intermediate

nodes

-LRB-

at

that

Round

-RRB-

Recall

that

RuleA

says

either

both

or

no

vertices

in

a

node

pair

must

be

used

as

intermediate

vertices

The

number

under

the

boxes

indicate

the

number

of

incident

edges

each

pair

-LRB-

per

node

-RRB-

has

available

-LRB-

at

that

round

-RRB-

The

darkened

boxes

indicate

which

node

pairs

were

choosen

as

intermediate

nodes

in

the

prior

round

Now

using

RuleA

we

can

enumerate

the

least

number

of

available

node

pairs

-LRB-

intermediate

vertices

-RRB-

at

each

round

for

the

switch

pair

whose

turn

it

is

to

establish

connections

-LRB-

for

N

-RRB-

Round

N

Round

N

Round

N

Round

N

Round

dN

N

bN

Round

dN

N

bN

Round

N

-LRB-

-RRB-

Round

N

-LRB-

-RRB-

For

the

example

in

Figure

Round

Round

Round

Round

and

Round

Thus

we

can

see

that

at

the

last

round

the

last

node

pair

has

-LRB-

N

-RRB-

pairs

of

nodes

to

select

which

is

enough

to

establish

the

required

Tnum

N

deg

-LRB-

N

-RRB-

connections

through

Note

that

at

each

round

N

edges

are

used

There

a

total

of

N

rounds

Thus

there

are

no

unused

edges

deg

-LRB-

N

-RRB-

i

for

i

Divide

the

set

of

connection

requests

into

i

subsets

Each

of

the

i

subsets

can

be

established

-LRB-

according

to

the

deg

N

case

-RRB-

An

an

example

Figure

-LRB-

a

-RRB-

depicts

a

nonblocking

graph

consisting

of

station

nodes

Suppose

we

need

a

switch

network

to

support

twice

as

many

station

nodes

and

still

remain

nonblocking

Doubling

each

of

the

existing

inter-switch

links

will

result

in

a

nonblocking

network

because

any

set

of

connection

requests

can

be

divided

into

nonblocking

subsets

Corollary

The

latency

of

any

indirect

connection

in

Adeg

T

-LRB-

V

E

-RRB-

is

-LRB-

crosspoint

switches

-RRB-

i.e.

any

indirect

connection

path

in

AT

deg

-LRB-

V

E

-RRB-

consists

of

two

links

Proof

This

follows

from

the

proof

of

Theorem

and

Theorem

Balanced

Direct-Direct

Networks

In

Section

it

was

shown

that

it

is

impractical

-LRB-

in

most

cases

-RRB-

to

find

an

optimal

solution

to

the

minimal

link

nonblocking

problem

We

observed

that

a

non-uniform

distribution

of

station

nodes

may

result

in

irregular

-LRB-

unbalanced

-RRB-

configurations

-LRB-

albeit

optimal

in

terms

of

number

of

links

-RRB-

We

also

observed

that

in

most

cases

it

is

more

efficient

to

interconnect

switches

such

that

all

switches

are

directly

connected

to

station

nodes

-LRB-

i.e.

the

Direct-Direct

topology

Figure

-LRB-

b

-RRB-

-RRB-

Thus

it

is

of

interest

to

study

how

uniform

-LRB-

or

balanced

-RRB-

regular

topologies

may

be

configured

to

maximize

the

number

of

nonblocking

station

nodes

In

balanced

regular

interconnections

the

total

number

of

I/O

ports

needed

for

inter-switch

communications

will

be

evenly

divided

-LRB-

or

nearly

evenly

divided

-RRB-

among

the

switches

Hereafter

we

assume

a

Balanced

Direct-Direct

network

implies

the

following

Station

nodes

are

uniformly

distributed

among

the

N

switches

The

number

of

switches

N

is

even

This

is

to

simplify

the

derivations

of

the

bounds

in

Section

Once

Theorem

is

derived

the

generalization

to

the

case

where

N

is

odd

is

straightforward

All

switches

-LRB-

vertices

-RRB-

have

the

same

degree

deg

We

denote

a

Balanced

Direct-Direct

network

by

Gdeg

T

-LRB-

V

E

-RRB-

When

deg

-LRB-

i.e.

the

number

of

incident

links/switch

-RRB-

is

greater

than

N

then

multiple

links

will

be

distributed

evenly

among

the

switch

nodes

Each

edge

consists

of

either

bdeg

-LRB-

N

-RRB-

c

links

or

bdeg

-LRB-

N

-RRB-

c

links

Figure

-LRB-

a

-RRB-

shows

the

edge

distribution

for

node

in

-LRB-

V

E

-RRB-

Bounds

for

balanced

nonblocking

networks

In

order

to

determine

the

property

that

a

network

topology

must

have

to

be

nonblocking

we

examine

the

case

when

the

maximum

amount

of

inter-switch

communications

occurs

This

case

occurs

when

all

station

nodes

attempt

to

communicate

nonlocally

i.e.

the

network

must

support

a

minimal

capacity

of

Tnum

-LRB-

bidirectional

units

of

data

traffic

-RRB-

If

any

Tnum

station

nodes

were

arbitrarily

choosen

the

number

of

inter-switch

lines

connecting

that

half

of

the

network

to

the

other

half

should

be

at

least

Tnum

bi-directional

lines

Definition

The

cross-sectional

bandwidth

of

a

network

is

the

network

capacity

available

between

any

set

of

one

half

of

the

vertices

to

the

other

half

of

the

vertices

Definition

The

cross-section

consisting

of

the

least

number

of

links

is

the

minimal

cross-sectional

bandwidth

V

V

V

V

V

V

-LRB-

a

-RRB-

-LRB-

b

-RRB-

link

Figure

-LRB-

a

-RRB-

-LRB-

V

E

-RRB-

-LRB-

b

-RRB-

minimal

cross-sectional

bandwidth

of

-LRB-

V

E

-RRB-

Given

Gdeg

T

-LRB-

V

E

-RRB-

we

seek

to

determine

its

minimal

cross-sectional

bandwidth

In

other

words

if

the

set

of

vertices

is

divided

into

two

equal-sized

subsets

we

want

to

find

the

layout

that

will

result

in

the

fewest

number

of

links

interconnecting

one

subset

to

the

other

subset

Figure

-LRB-

b

-RRB-

shows

the

minimal

cross-sectional

bandwidth

of

-LRB-

V

E

-RRB-

-LRB-

jV

j

-RRB-

to

be

links

Lemma

The

minimal

cross-sectional

bandwidth

of

Gdeg

T

-LRB-

V

E

-RRB-

denoted

by

CSB

-LRB-

deg

N

-RRB-

can

be

expressed

by

CSB

-LRB-

deg

N

-RRB-

bdeg

-LRB-

N

-RRB-

c

if

conditionA

is

true

-LRB-

-RRB-

N

b

-LRB-

deg

-LRB-

N

-RRB-

-RRB-

-LRB-

N

-RRB-

c

-LRB-

-LRB-

deg

-LRB-

N

-RRB-

-RRB-

-LRB-

N

-RRB-

-RRB-

bdeg

-LRB-

N

-RRB-

c

otherwise

-LRB-

-RRB-

where

conditionA

deg

-LRB-

N

-RRB-

-LRB-

N

-RRB-

-LRB-

N

-RRB-

is

odd

and

deg

-LRB-

N

-RRB-

is

odd

Proof

Let

degexcess

deg

-LRB-

N

-RRB-

and

degeven

bdeg

-LRB-

N

-RRB-

c

-LRB-

N

-RRB-

Let

Eexcess

-LRB-

Eeven

-RRB-

correspond

to

the

edges

-LRB-

or

links

since

an

edge

may

consist

of

multiple

links

-RRB-

which

interconnect

degexcess

-LRB-

degeven

-RRB-

degrees

from

each

vertex

Thus

deg

degeven

degexcess

-LRB-

-RRB-

E

Eeven

-LSB-

Eexcess

-LRB-

-RRB-

Edges

in

Eeven

contribute

to

the

equal

number

of

links

-LRB-

bdeg

-LRB-

N

-RRB-

c

-RRB-

per

edge

In

Figure

-LRB-

a

-RRB-

the

thick

line

-LRB-

links/edge

-RRB-

corresponds

to

edges

in

Eeven

Edges

in

Eexcess

contribute

to

certain

edges

having

capacity

of

bdeg

-LRB-

N

-RRB-

c

In

Figure

-LRB-

b

-RRB-

the

thin

line

edges

-LRB-

line/edge

-RRB-

correspond

to

edges

in

Eexcess

Figure

-LRB-

a

-RRB-

only

shows

those

links

in

Eeven

and

Eexcess

which

are

directly

incident

upon

If

it

were

to

show

all

the

links

in

Eexcess

and

Eeven

there

would

be

a

total

of

edges

-LRB-

each

consisting

of

links

-RRB-

in

Eeven

and

single

link

edges

in

Eexcess

The

minimal

cross-sectional

bandwidth

is

the

sum

of

the

minimal

cross-sectional

bandwidth

of

links

in

Eeven

and

links

in

Eexcess

or

CSB

-LRB-

deg

N

-RRB-

CSB

-LRB-

degexcess

N

-RRB-

CSB

-LRB-

degeven

N

-RRB-

-LRB-

-RRB-

Contribution

to

the

minimal

cross-sectional

bandwidth

by

edges

in

Eeven

We

observe

that

each

vertex

in

an

equal-sized

subset

has

a

direct

connection

to

each

of

the

N

vertices

in

the

other

subset

and

that

each

connection

consists

of

bdeg

-LRB-

N

-RRB-

c

links

Thus

the

total

contribution

-LRB-

of

links

from

Eeven

-RRB-

from

N

vertices

from

one

subset

to

N

vertices

in

the

other

subset

is

-LRB-

N

-RRB-

-LRB-

N

-RRB-

bdeg

-LRB-

N

-RRB-

c

or

CSB

-LRB-

degeven

N

-RRB-

-LRB-

-RRB-

bdeg

-LRB-

N

-RRB-

c

-LRB-

-RRB-

Contribution

to

the

minimal

cross-sectional

bandwdith

by

edges

in

Eexcess

If

degexcess

N

then

CSB

-LRB-

degexcess

N

-RRB-

This

is

because

if

G

is

divided

into

equal-sized

subsets

each

vertex

in

each

subset

may

be

interconnected

such

that

the

subsets

are

not

connected

When

degexcess

N

to

determine

the

minimal

cross-sectional

bandwidth

divide

G

into

two

subsets

each

consisting

of

N

vertices

Each

vertex

in

each

subset

will

have

degexcess

of

at

least

N

In

other

words

we

attempt

to

maximize

the

connectivity

in

each

subset

in

order

to

minimize

the

links

which

will

contribute

to

the

cross-sectional

bandwidth

Since

edges

must

be

uniformly

distributed

among

the

vertices

each

vertex

from

one

of

the

subsets

will

contribute

-LRB-

deg

-LRB-

N

-RRB-

-RRB-

-LRB-

-LRB-

N

-RRB-

-RRB-

links

to

the

minimal

crosssectional

bandwidth

For

example

in

Figure

-LRB-

b

-RRB-

each

vertex

must

contribute

at

least

link

to

the

minimal

cross-sectional

bandwidth

Each

subset

consists

of

N

vertices

So

the

total

contribution

by

edges

in

Eexcess

is

CSB

-LRB-

degexcess

N

-RRB-

N

-LRB-

-LRB-

deg

-LRB-

N

-RRB-

-RRB-

-LRB-

-LRB-

N

-RRB-

-RRB-

-RRB-

-LRB-

if

degexcess

N

-RRB-

Because

there

is

no

contribution

to

the

minimal

cross-sectional

bandwidth

when

degexcess

N

in

order

to

generalize

the

last

equation

we

add

the

term

b

-LRB-

deg

-LRB-

N

-RRB-

-RRB-

-LRB-

N

-RRB-

c.

Thus

CSB

-LRB-

degexcess

N

-RRB-

N

b

-LRB-

deg

-LRB-

N

-RRB-

-RRB-

-LRB-

N

-RRB-

c

-LRB-

-LRB-

deg

-LRB-

N

-RRB-

-RRB-

-LRB-

-LRB-

N

-RRB-

-RRB-

-RRB-

-LRB-

-RRB-

Equation

follows

Note

the

special

case

when

deg

-LRB-

N

-RRB-

-LRB-

N

-RRB-

N

is

odd

and

deg

-LRB-

N

-RRB-

is

odd

An

example

of

this

case

occurs

in

-LRB-

V

E

-RRB-

and

-LRB-

V

E

-RRB-

when

jV

j

In

this

special

case

only

one

link

in

Eexcess

contributes

to

the

minimal

cross-sectional

bandwidth

Equation

follows

In

the

following

theorem

we

use

the

minimal

cross-sectional

bandwidth

of

a

network

as

the

necessary

condition

in

deciding

whether

a

network

is

nonblocking

Theorem

Given

Gdeg

T

-LRB-

V

E

-RRB-

the

following

equations

hold

Tnum

deg

N

if

deg

N

-LRB-

-RRB-

bdeg

-LRB-

N

-RRB-

c

if

conditionA

is

true

-LRB-

-RRB-

N

b

-LRB-

deg

-LRB-

N

-RRB-

-RRB-

-LRB-

N

-RRB-

c

-LRB-

-LRB-

deg

-LRB-

N

-RRB-

-RRB-

-LRB-

N

-RRB-

-RRB-

bdeg

-LRB-

N

-RRB-

c

otherwise

-LRB-

-RRB-

where

conditionA

deg

-LRB-

N

-RRB-

-LRB-

N

-RRB-

deg

-LRB-

N

-RRB-

is

odd

and

N

is

odd

deg

Tnum

N

if

N

-LRB-

N

-RRB-

Tnum

-LRB-

-RRB-

-LRB-

-LRB-

Tnum

-RRB-

-RRB-

-LRB-

N

-RRB-

if

Tnum

-LRB-

-RRB-

N

if

N

-LRB-

N

-RRB-

Tnum

-LRB-

-RRB-

Proof

Equation

-LRB-

deg

N

-RRB-

See

proof

of

Theorem

Equation

-LRB-

Equation

-RRB-

As

previously

observed

Tnum

is

the

maximal

traffic

amount

which

may

occur

If

the

minimal

cross-sectional

bandwidth

is

less

than

Tnum

blocking

may

occur

Thus

Tnum

CSB

-LRB-

deg

N

-RRB-

-LRB-

-RRB-

Replacing

Equation

-LRB-

Equation

in

the

above

equation

results

in

Equation

-LRB-

Equation

-RRB-

Equation

-LRB-

Tnum

N

-LRB-

N

-RRB-

-RRB-

From

Equation

if

deg

N

then

Tnum

N

deg

N

This

implies

N

-LRB-

N

-RRB-

Tnum

and

deg

Tnum

N

This

makes

intuitive

sense

when

the

following

is

considered

When

the

maximum

network

capacity

is

at

the

minimal

hardware

bound

-LRB-

see

Theorem

-RRB-

Tnum

units

capacity

or

Tnum

inter-switch

links

each

switch

node

must

use

half

of

its

degrees

when

serving

as

an

intermediate

switch

and

the

other

half

for

direct

outgoing

traffic

So

each

switch

node

must

be

capable

of

injecting

N

units

into

the

network

Thus

each

switch

node

must

dedicate

N

input/output

ports

to

inter-switch

communications

Equation

-LRB-

Tnum

-RRB-

We

begin

with

Equation

-LRB-

if

deg

N

N

N

N

:-RRB-

in

order

to

derive

a

bound

for

deg

Setting

b

-LRB-

deg

-LRB-

N

-RRB-

-RRB-

-LRB-

N

-RRB-

c

since

its

maximum

value

is

and

using

the

fact

that

x

y

x

ybx

yc

we

obtain

N

-LRB-

-LRB-

deg

-LRB-

N

-RRB-

bdeg

-LRB-

N

-RRB-

c

-RRB-

-LRB-

N

-RRB-

-RRB-

-LRB-

N

-RRB-

c

Tnum

-LRB-

-RRB-

Using

bdeg

-LRB-

N

-RRB-

c

deg

-LRB-

N

-RRB-

and

bdeg

-LRB-

N

-RRB-

c

deg

-LRB-

N

-RRB-

we

obtain

N

-LRB-

-LRB-

deg

-LRB-

N

-RRB-

-LRB-

deg

-LRB-

N

-RRB-

-RRB-

-RRB-

-LRB-

N

-RRB-

-RRB-

deg

-LRB-

N

-RRB-

Tnum

-LRB-

-RRB-

Solving

for

deg

results

in

deg

-LRB-

-LRB-

Tnum

-RRB-

-RRB-

-LRB-

N

-RRB-

-LRB-

-RRB-

Recall

that

if

N

-LRB-

N

-RRB-

Tnum

then

N

deg

N

This

implies

that

if

Tnum

N

-LRB-

N

-RRB-

then

deg

N

Examining

Equation

we

observe

that

to

find

the

greatest

bound

the

term

must

be

greater

than

implies

Tnum

Equation

-LRB-

N

-LRB-

N

-RRB-

Tnum

-RRB-

This

follows

from

the

above

argument

for

Equation

In

the

above

Theorem

we

derived

bounds

for

the

maximal

number

of

nonblocking

station

nodes

a

particular

network

may

be

able

to

support

and

the

minimal

number

of

I/O

ports/switch

a

network

must

have

in

order

to

be

nonblocking

Example

Say

we

are

given

-LRB-

V

E

-RRB-

where

jV

j

We

wish

to

know

the

upper

bound

on

the

number

of

stations

G

may

be

able

to

support

in

a

nonblocking

manner

Using

Theorem

Equation

results

in

an

upper

bound

of

station

nodes

Thus

-LRB-

V

E

-RRB-

where

jV

j

can

support

at

most

nonblocking

stations

Suppose

we

want

our

network

to

support

station

nodes

in

a

nonblocking

manner

Using

Theorem

Equation

-LRB-

Tnum

-RRB-

results

in

a

lower

bound

of

-LRB-

-RRB-

Thus

for

a

network

consisting

of

switches

in

order

to

support

nonblocking

stations

each

switch

must

devote

at

least

I/O

ports

to

inter-switch

communications

Even

though

Theorem

provides

only

necessary

conditions

it

does

provide

a

very

close

approximate

to

the

nonblocking

property

The

reason

for

this

is

that

the

minimal

crosssectional

bandwidth

measures

the

least

number

of

lines

across

the

cross-section

of

a

network

To

derive

Equations

to

we

force

the

maximal

amount

of

traffic

to

be

less

than

or

equal

to

this

least

network

cross-section

Thus

intuitively

we

can

see

that

if

Equations

to

are

followed

the

overwhelming

majority

of

possible

connection

sets

will

be

able

to

be

established

Thus

Equations

to

are

sufficient

in

most

cases

Section

provides

an

example

of

a

network

which

uses

the

minimal

number

of

I/O

ports/switch

-LRB-

derived

from

Equation

-RRB-

and

exhibits

a

high

percentage

-LRB-

-RRB-

of

non-local

requests

The

resulting

blocking

probability

is

Blocking

Probabilities

In

this

section

we

derive

blocking

probabilities

for

Balanced

Direct-Direct

switch

configurations

which

have

fewer

than

the

necessary

number

of

inter-switch

trunks

to

support

the

maximum

aggregate

network

traffic

Vk

intermediate

direct

direct

direct

intermediate

intermediate

Figure

CM

-LRB-

-RRB-

Each

pair

of

I/O

ports

-LRB-

used

for

inter-switch

communications

-RRB-

of

any

switch

may

be

used

in

the

following

two

ways

-LRB-

a

-RRB-

to

connect

other

switches

-LRB-

i.e.

as

an

intermediate

switch

-RRB-

or

-LRB-

b

-RRB-

for

the

local

stations

-LRB-

directly

attached

to

that

switch

-RRB-

to

communicate

non-locally

Let

CM

-LRB-

m

n

-RRB-

denote

the

usage

of

the

outgoing

links

of

a

particular

switch

node

vk

where

m

is

the

number

of

links

used

in

manner

-LRB-

a

-RRB-

and

n

is

the

number

of

edges

used

in

manner

-LRB-

b

-RRB-

Then

the

blocking

probability

of

vk

is

equal

to

the

summation

of

the

probabilities

of

all

combinations

of

-LRB-

a

-RRB-

and

-LRB-

b

-RRB-

occuring

-LRB-

e.g.

CM

-LRB-

deg

-RRB-

CM

-LRB-

deg

-RRB-

CM

-LRB-

deg

-RRB-

CM

-LRB-

deg

-RRB-

-RRB-

Let

al

denote

the

average

number

of

links

between

any

two

switch

nodes

Then

al

-LRB-

deg

N

-RRB-

-LRB-

N

-LRB-

N

-RRB-

-RRB-

deg

-LRB-

N

-RRB-

-LRB-

-RRB-

Assumming

all

station

nodes

have

equal

probability

of

making

non-local

connection

requests

among

each

other

let

psn

denote

the

probability

a

station

node

will

attempt

to

establish

a

non-local

connection

Let

psw

denote

the

probabilty

a

switch

node

will

have

an

attached

station

node

which

requests

a

non-local

connection

Thus

psw

-LRB-

Tnum

N

-RRB-

psn

Let

BP

-LRB-

Gdeg

T

-RRB-

denote

the

probabilty

blocking

will

occur

in

Gdeg

T

-LRB-

V

E

-RRB-

then

BP

-LRB-

Gdeg

T

-RRB-

C

-LRB-

N

-RRB-

deg

X

i

CM

-LRB-

deg

i

i

-RRB-

C

-LRB-

N

-RRB-

-LRB-

deg

X

i

CM

-LRB-

deg

i

i

-RRB-

-RRB-

C

-LRB-

N

N

-RRB-

-LRB-

deg

X

i

CM

-LRB-

deg

i

i

-RRB-

-RRB-

N

-LRB-

-RRB-

where

C

-LRB-

N

k

-RRB-

N

-LRB-

N

k

-RRB-

k

Say

vsource

wishes

to

establish

a

connection

to

ndestination

but

there

are

no

direct

links

available

The

probability

that

switch

vk

will

be

used

as

an

intermediate

node

is

-LRB-

N

-RRB-

psw

-LRB-

N

-RRB-

-LRB-

psw

-LRB-

N

-RRB-

-RRB-

al

-LRB-

-LRB-

psw

-LRB-

N

-RRB-

-RRB-

al

-RRB-

pal

sw

=-LRB-

-LRB-

N

-RRB-

-LRB-

N

-RRB-

al

-RRB-

-LRB-

-LRB-

psw

-LRB-

N

-RRB-

-RRB-

al

-RRB-

The

first

term

corresponds

to

the

probability

a

request

will

come

from

one

of

N

switches

stations

to

establish

a

request

from

vk

The

second

term

corresponds

to

the

probability

that

all

direct

links

to

vdestination

have

already

been

used

The

third

term

corresponds

to

the

probability

that

there

are

open

links

to

vk

psw

-LRB-

N

-RRB-

corresponds

to

the

probability

that

an

outgoing

link

of

Vk

will

have

been

used

by

a

station

node

from

Vk

initiating

a

connection

request

-LRB-

N

-RRB-

psw

-LRB-

N

-RRB-

-LRB-

-LRB-

psw

-LRB-

N

-RRB-

-RRB-

al

-RRB-

psw

-LRB-

-LRB-

psw

-LRB-

N

-RRB-

-RRB-

al

-RRB-

corresponds

to

the

probability

that

a

switch

besides

Vk

will

initiate

a

direct

connection

to

Vk

The

first

term

corresponds

to

the

probability

a

request

will

come

from

one

of

N

switches

stations

to

establish

a

request

to

Vk

The

second

term

corresponds

to

the

probability

that

all

direct

links

to

Vk

have

not

been

already

used

Thus

the

blocking

probability

is

BP

-LRB-

GT

-RRB-

NX

k

C

-LRB-

N

k

-RRB-

-LRB-

deg

X

i

-LRB-

-LRB-

pal

sw

=-LRB-

-LRB-

N

-RRB-

-LRB-

N

-RRB-

al

-RRB-

-LRB-

-LRB-

psw

-LRB-

N

-RRB-

-RRB-

al

-RRB-

-RRB-

deg?i

-LRB-

psw

-LRB-

N

-RRB-

psw

-LRB-

-LRB-

psw

-LRB-

N

-RRB-

-RRB-

al

-RRB-

-RRB-

i

-RRB-

k

-LRB-

-RRB-

A

practical

network

application

This

study

was

intended

to

be

of

use

for

the

efficient

interconnection

of

connection-oriented

switches

-LRB-

e.g.

ATM

HIPPI

-RRB-

in

a

local

area

environment

In

this

section

we

describe

a

realistic

high

performance

LAN

environment

We

then

describe

how

the

results

of

our

study

may

be

applied

towards

the

efficient

physical

dimensioning

of

such

a

network

A

typical

LAN

environment

consists

of

a

number

of

sharable

devices

-LRB-

e.g.

workstations

servers

special

pupose

procesors

and

storage

devices

-RRB-

and

a

number

of

non-sharable

devices

Since

we

are

only

interested

in

configuring

our

network

in

such

a

way

as

to

maximize

the

possible

sharing

of

devices

we

can

disregard

the

non-sharable

devices

The

environment

we

consider

consists

of

switches

There

are

station

nodes

attached

to

each

of

switches

station

nodes

attached

to

each

of

switches

and

station

nodes

attached

to

the

remaining

switch

The

duo-fold

goal

is

to

interconnect

these

elements

to

switches

and

interconnect

the

switches

such

that

-LRB-

i

-RRB-

the

number

of

ports

each

switch

must

dedicate

to

inter-switch

communications

is

minimal

and

-LRB-

ii

-RRB-

the

number

of

simultaneous

connections

through

the

interconnection

network

is

maximized

-LRB-

i.e.

the

blocking

probability

is

minimized

-RRB-

Our

results

revealed

that

the

preferable

usage

of

switches

is

of

the

Direct-Direct

configuration

We

wish

to

apply

our

results

in

Section

on

Balanced

Direct-Direct

networks

to

determine

an

efficient

configuration

Since

the

greatest

number

of

station

nodes

attached

per

switch

is

we

can

use

that

when

using

the

results

from

Section

Note

that

by

picking

the

switch

with

the

greatest

number

of

attached

station

nodes

we

can

avoid

redistributing

Blocking

Probability

number

of

I/O

ports/switch

Switches

Station

nodes

p

p

Figure

Blocking

probability

vs.

number

of

I/O

ports/switch

the

station

nodes

among

the

switches

However

we

can

only

do

this

if

the

station

nodes

are

not

severely

unbalanced

According

to

Theorem

in

order

to

preserve

the

nonblocking

property

the

network

must

be

configured

in

the

following

manner

-LRB-

i

-RRB-

each

switch

must

dedicate

at

least

I/O

ports

to

inter-switch

communications

and

-LRB-

ii

-RRB-

the

interconnection

topology

must

be

a

completely

connected

graph

Any

vertex

is

directly

connected

to

all

other

switches

via

either

or

links/edge

of

the

edges

must

consist

of

links

and

edge

must

consist

of

links

If

less

than

the

necessary

number

of

I/O

ports/switch

are

used

it

is

also

of

interest

to

know

the

network

's

blocking

probabilities

The

blocking

probabilities

are

a

function

of

the

probability

a

station

node

will

have

to

establish

a

non-local

connection

Suppose

since

due

to

locality

the

probabilty

a

station

node

will

attempt

a

non-local

connection

is

relatively

low

say

compared

to

the

probibilty

of

a

local

connection

attempt

Thus

each

cluster

-LRB-

i.e.

switch

plus

its

attached

local

nodes

-RRB-

has

an

-LRB-

-RRB-

chance

of

attempting

to

establish

a

connection

to

another

cluster

Using

Equation

we

can

calculate

the

relevant

blocking

probabilities

Figure

shows

that

if

only

I/O

ports/switch

-LRB-

of

the

required

number

-RRB-

are

used

then

the

blocking

probability

is

slightly

less

than

If

the

required

number

-LRB-

I/O

ports/switch

-RRB-

are

used

the

blocking

probability

is

close

to

If

the

probability

is

set

to

then

if

only

ports/switch

are

used

a

blocking

probability

will

occur

if

only

-LRB-

I/O

ports/switch

-RRB-

the

required

number

of

I/O

ports/switch

are

used

a

blocking

probability

of

will

occur

Note

that

if

we

had

choosen

to

redistribute

the

station

nodes

in

a

more

balanced

manner

we

could

obtain

the

same

blocking

probabilities

with

fewer

required

I/O

ports/switch

These

results

correlate

to

the

observation

in

Section

that

Equations

to

are

only

necessary

conditions

However

they

are

very

close

to

sufficient

conditions

in

that

the

majority

of

connection

sets

will

be

able

to

be

established

on

a

network

which

meets

the

bounds

in

Theorem

Conclusion

The

shift

towards

switch-based

local

area

network

architectures

has

caused

new

issues

involved

in

interconnecting

multiple

switches

to

arise

These

switch

networks

aimed

at

LAN

environments

differ

in

form

from

traditional

multistage

networks

aimed

at

multiprocessors

and

telephone

switching

exchanges

as

well

as

from

traditional

wide-area

telephone

networks

However

general

issues

from

previous

studies

of

these

switching

networks

are

also

relevant

to

multi-switch

networks

These

issues

include

minimizing

hardware

requirements

-LRB-

number

of

links

I/O

ports

-RRB-

maximizing

the

aggregate

network

throughput

nonblocking

vs.

blocking

topologies

minimizing

latency

etc.

This

paper

examined

switch

interconnections

in

a

local

area

environment

which

support

the

maximum

network

throughput

while

using

a

minimal

number

of

links

The

problem

of

finding

the

minimal

hardware

necessary

-LRB-

number

of

I/O

switch

links

-RRB-

and

the

interconnection

topology

which

supports

the

maximum

throughput

was

shown

to

be

NP-Complete

We

divided

network

topology

types

into

categories

based

upon

switch

usage

and

the

distribution

of

station

nodes

among

the

switches

We

discussed

the

advantages/drawbacks

of

each

An

analysis

on

balanced

networks

was

presented

Lower

bounds

on

the

maximal

number

of

nonblocking

station

nodes

and

the

minimal

number

of

links

were

derived

For

cases

where

less

than

the

minimal

amount

of

hardware

to

support

the

maximal

aggregate

network

throughput

was

available

blocking

probabilities

were

derived

In

the

last

section

we

examined

an

example

of

a

typical

local

area

environment

We

deduced

the

necessary

amount

of

hardware

to

support

the

maximal

throughput

from

our

previous

results

and

also

the

blocking

probabilities

resulting

from

lesser

numbers

of

dedicated

I/O

ports/switch

Our

example

showed

that

given

a

network

with

a

high

degree

of

device

sharing

if

less

than

the

necessary

number

of

links

were

used

for

a

certain

topology

the

blocking

probability

may

still

be

relatively

low

Specifically

we

found

if

the

probablity

a

switch

will

request

a

non-local

connection

is

-LRB-

-RRB-

a

-LRB-

-RRB-

reduction

in

the

number

of

links

necessary

to

completely

avoid

nonblocking

results

in

a

network

with

a

-LRB-

-RRB-

blocking

probability

Note

that

these

blocking

probabilities

are

highly

dependent

on

the

choosen

interconnection

scheme

These

low

blocking

probabilities

reflect

the

robustness

of

our

proposed

network

design

A

randomly

choosen

interconnection

scheme

with

an

equal

amount

of

dedicated

hardware

would

most

likely

result

in

a

significantly

higher

network

blocking

rate

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asynchronous transfer mode
atm
high performance parallel interface
hippi
local area network
nonblocking networks

blocking probabilties
connection-oriented switches
maxmimal aggregate network throughput