End-to-end window evaluation for packet switched networks based on minimum cut saturation techniques

End-To-End Window Evaluation for Packet Switched Networks Based on Minimum cut Saturation

Techniques

by

Tania Volochine

Center for Communications and Signal Processing Electrical and Computer Engineering

North Carolina State University

CCSP-TR-84/14

ASS7RACT

VOLOCHINE, TANIA. End-to-end window evaluation fo~ packet switched networks based on minimum cut saturation tech-niques. (Under the direction of WUSHOW CHOU.)

lV

TABLE OF CONTENTS

: ~NTRODUCT ION •••••••••••••••••••••••••••••••••••••••••• 1 1.1 Problem Description... 1 l.2 Thesis Organization... 4 2 A SURVEY OF FLOW CONTROL MECHANISMS IN

?AC:{ET SW I TCHED NETWORKS •••••••••••••••••••••••••••••• 6

2.1 Domain of Application 6

2.1.1 Circuit Switched Net~o~ks 6

2.1.2 ?acket Switched Networks. 8

2.1.2.: Datagram Service (JG) 0

2.1.2.2 Virt~al Circuit Se~vice (VC) ...•..•... 1

2.2 ?urpose of Flow Control Mechanisms 3

2.2.1 7hroughput Degradation 3

2.2.2 Deadlock Avoidance 4

2.2.3 :airness

:0

Users 6

2.2.4 Preven~ion of Destination Buffer Overf:ow 6 2.3 ?er:ormance Considerations for Flow Controlled

Networks . . . • . . . :8 2 • 3 . L Throughput .••.•..••••..•.•...•...••..••.••.. :8 2 . 3 . 2 Powe r ••••••••••••••••••••••••••••••••••••••••• 2

a

3 A CLASSIFICATION OF FLOW CONTROL MECHANISMS ...•••.•... 21

3.1 Hop Level Flow Control . . . • . . . 24 3.1.1 Data Link Flow Control Schemes ...•. 0 • • • • • 24

3.1.i.l Sliding-window Flow Control Mechanism ... 25 3.1.2 Network Flow Control . . . • • . . . • . . . 27

3.1.2.1 Channel Queue Limit Scheme 27

3.1.2.2 Struc~ured au:fer Pool ::ow Control 30 3.1.2.3 Virtual C:rc~it Hop-level Flow Control .. 3l 3.2 Entrv-to-Exi~ (ETE) F2.ow Con:rol . . . • . . • . . . 34

3.3 Network Access Flow Control Schemes 36

3.3.1 Isarithmic Flow Control Scheme 36

3.3.2 Input Buffer Limi~ Sc~eme 38

4 WINDOW SIZE EVALUATION 40

4.1 Problem Presentation . . . • . . . 40 4.2 Minimum Cut Maximum Flow Theorem ...•...• ~l ~ WINDOW SIZE EVALUATION FOR FIXED ~OUTING SCHEMES 45

5.1 Location of the Minimum C~t 45

5.2 Wong's Results for Fixed Routing Schemes ...•... 49 5.2.1 Notation •••••..•..•....•••••••..•...•••••.•••. S2

5.2.2 Simplifying Ass~ptions 53

5.2.3 End-to-end delay on a Fixed Path S6

v

5.4.1 Method Description 0 • • • • • • • • • • • • • • • • • • • • • 69

5.4.2 Results of Computations .•..••.... 0 • • • • • • • • • • • • 78

504.201 Conclusions on Results ..• e • • • • • • o o • • • • • • 94 5.403 validation through Simulation . o o o o e o • • • o o • • • o . 99 5.4.3 . 1 Resul ts 0 0 00 • • • • • • Cl 0 • • • • • • • • • e • • • • 0 0•• 0 • 0 132 5.4.3.2 Suffer Space Computations ....•...•...••. 137

S.~.~ The Dual Problem Description .•••• 0•••••••••••• 139 6 WINDOW SIZE EVALUATION FOR ADAP~!VE ROUTING SCn~~ES ••• 141 6.1 Wong's Results for Adaptive ~outing Scheme ....•... 143 6.1.1 End-to-end Delay for Adaptive Routing :44 6.1.2 Number of Class ~ Messages in the ~etNork 0 • • • • 1~6

6.2 Window Size Determination 148

6.2.1 Method Description . . . • . . . . • . . . e . 150 6.2.2 Results of Comoutations ...••.••...••... 153 6.2.3 The Validation·Process . 0 • • • • • • • • • • • • • • • • • • • • • • :63

7 FURTHER D!SCUSSION AND CONCLUSIONS .. 0 0 0 0 0 0 0 0 . 0 0 • • • • • • 0 170

i.l Buffer Overflow Problem ... o O O G . a o o o • • o o • • • o • • • e o o . 170 7.1.1 Minimum Cut 8uffe~ Overflow •...•.•••.•. 0 • • • 0 • • 170

7.1.2 Destination Buffer Overflow ...•...•... 173 7.1.2.1 Destination 3u:fe~ Overflow for

::' i xed Rout ing •.• 0 • • • 0 • • • • • 0 • • • • • • • • • • 0 • • : 73

7.1.202 Destination Euffe~ Overflow for

Adap~ive Routing . o e • • • o • • • • • o • • e • • • • • • • • 175 7.2 A method to Adjust Dynamically the Window Size 0 • • • 177 7.2.1 Presentation of the Concept ..•..•.•.•••••.• 0 • • 177

7.2.2 Discussion on Possible rmclementaticns .•.•••.• 178 7.2.2.1 Statistics Collection on Long-term

window Re-adiustments ...•...•.•.... 183 7.3 Further Extensions and

4 C o n c l u s i o n s

... 0 • • • • • • • • • • • • ~84

- CHAPTER ~

:NTRODUCT:ON

,

.

.

--

.

?~OBLEM ~ESCR:?T!ON.

" C o r n m u nic a t io r i s : " : t h is c o n c e p t i s one o f the mo s t

:mpor~an: and deep :n human history. Since :he Stone Age,

communica~ion has played an impor:ant :ole i~ :he survival

c:

human beings. :rom ,~~man corr~unl~at:on :~e. . .

?i:-s't, =ommunicaticn :nec.ia ~y ~eans of ~ires a~c elec~ra~ag~e::: Naves :~ave:ing :~rough :~e a:~, ~e~e ~sed to allow long distance h~~an co~~unications. No~, ~:~h

the advent of computer technology and techniques t~at need

:ast ~espcnse :imes, the need for ~achine communicat:ons appeared. Due to the increasing amount of informa~:on man generates t.he computer's help, the

machine-informa~ion exchange that excludes the intermediate human interface becomes crucial to :urther work at compute~ speed. ~~is continuous need helped in the generation of computer ~et~orks.

achieved.

2

Due

:0

the high number of ?arame~e~s and ~he constraints to be ~et, network design and its pe~formance

solve anc :~e =apacity-cOSt

. . .

c~:::=ai.

:raceo::s

:e~ ~s ~ecall that because of ~he ~etNo~k:~g cost and t~e

• 41o · •

acc~::o~a~ :~sour:es,

desig~ed :0 meet a~ ex?ec:ed ave~age :~affic ~cac., .

~eac~ congeSt:on.

Such a s:~uation occ~rs Mhenever the amount of ~~affic

exceeds :he availab~e ne~Mork capaci:y. This genera~es

=ottlenecks in a ~a~~ or ~arts ~f :he system ~hicn

.

...

~

~urn, s:o~s dOwn the ~ncongested par~s of :~. As a

~esult, full saturation and deadlock may occur, and the

~esponse time fo~ any que~y is considerably inc~eased. ~ventually, the throughput of the ne~wo~k ~eaches zero. !n sucn a deadlock situation, manual action is :he sole solution :or exiting this state.

3

"'"'he_{... one un er conSlueratlon :or our}d . ., . - _{study is the so-called} "Entry-,:o-E:xit Fl.ow Controi" that ".Me shall also :all

end-an ever :~c~eas:~g amount ~f ~essages from :n:e~:~g t~e

:~us ::~i~:nc ~he ~:sk of ~esour:es

ove~-~ ~ .

a~~cca~lcn. A ~essage ~hose access ~as been ~efused,

~ai~s :~ que~e ~u~s:ce t~e ~e~Nork ~n~il ger~ission :s

issuec ~o :~. ~: :~is end-to-end flow :on~~ol

:5

~nde~-es:::na:ed,

..,~ ~

- .. _5 ::teans _{:or ::1essages :.~side}

:::e ~.e:-,llte:-."< , :,u: a ~_'""'r.:-_- . . . ; -Q_ _r- .::;c-_~_ss- -_ ,_.-~..Q_ _ C""""'se~~~e!1.--...I1&. ~",.,-1 ./_ _ ,

ove:--es::'mated, :he end-~o-end :low control does not :u~fi::

::5

goal a~d =ongestion builds up inside the

~e~wor~ u~der ~eav? :oad.

The e~d-t~-end :low ~ontrol is applied he~e to a specific =ategory of ~et~orks called packet s~itched t?S)

~e~~orks. More spec:::cally it applies to the subset of those tha~ es:aolish an end-:o-end virtual call. ! t i s

~hus for this restric:ed set that the study is applicable. 7his end-to-end flow control is difficult to consider

analytically due to the complex behavior in a

addressed di~ec~ly. ~h'_{.... lS :'S ~he ~~oblem we consider he~e.} ~he locali=ation of :~e ~otential bOttleneck :n a ?S

~::e s~ar:i:'lg ;Ol~: :or :~e enc-:o-end ~:~dow size

'li~~~al-Circuit (VC-VC) and Jatagram

:=~s:c.e:-ed. ~ean ~a:~e analysis a~d ~he second mcmen:

:~e delay and buf:e~ dis:r:~utions are used .

:::esis

:5

'J~gan:'zed. . - se~len majo r .._{AI . . . . ' - • •}~,

,--~~c:~de ~ 9rese~:ac:on cf backg~ound in:o~m~:ion on ~he :ype of ~e~~ork =~nside~ed, an overview of the ~ajor ::ow con~~ol :ec~niques ex:st:~g :~ =ompu~e~ netMorks, ~he desc=:?~:on of the ~rob:em ~e considered and the ~e~hod ~sed for differen~ =~n~igu~a~ions.

C~ap~er 2 gives a survey of :low =ont~ol illechanisms :or packet switched ne~~orkso :: includes (l) the domain of application of the type

0:

flow control mechanisms ~e investigate, the ?ur?ose of flow control mechanisms :or packet s~itched networks, (3) a brief presentation of the performance criteria ~hich are applied in order

:0

test the effec~iveness of flow cont~ol mechanisms.

5 control ~echanism ~e deal ~ith is presented and its :n~e~action with ~~e other :evels of :low control is ;:ointed ~u:..

~e =ons:dered a~d :~e ~i~:~~'~i""i,.4U __·_'t a~p~oac~t:J ~ ..j, l·S ~es-~i~ec·L... ' - . - . - •

C~ap~e~ 5 presents :~e ~et~cd as :: has been ap?lied :~

:::<ed sc~emes s~itc~ed ~e:wo~ks.

~es~::s

0:

compu~at:o~ for ~~e end-to-end Nindow size

:~ =hap~e~ 7 ~e focus on problems that need to be tak~n

:~:o ac~ount, mainly the problem of buffer over::ow at ~he :~:er~ediate ncdes in :he ~etwork and at ~he desti~ation :"lode. The problem of adjusting the Nindow size

dynamically is also presented and discussed. These points need :0 be addressed Nhen dealing with real computer

networks.

In the last section :urther developments are mentioned and conclusions on the ~esults obtained are given. In

- CHAPTER .,.

3efore .e desc~:be ~he :lo~ =ont~o: nec~anisws a~d

a ,~,..~. ..

.-eel _

(local, feiide ::oea,

-

-,-~;

-

~..--

....

:.

Q . _ • •

-connected, hierarchical, loop 0 • • ) . Such a classif:ca::=n

of differen~ computer ne~Mork :ypes can ~e :ound -

.

.:

..

,

..

ne~works and 9acket s~i~ched (?S) ~etwor~s fo~ ::ow control pu~?ose, due to thei~ diffe~ent behavio~ ~~de~

heavy load. ~he subset of end-to-end virtual ci~~uit (VC}

oriented ?S is presentee. :nain

character:stics of each of ~hese groups are given below.

2.101. Circuit Switched Networks (CS).

~irs~ ~ays of computer ~etwor~i~g ~as :0 ~ake advan~age of

:~ese available ~esour=es. For :hese ~easons, :~e ::rs~

:a~es ~:ace once :~e ~a~~ je:~ee~ :~em has =ee~ se:~p.

:-ecues-:s.

a:-e ·jed.:'=a:e·= :0 _.4~

S~''::::=-__ =5:5. :::e

: :.::".e ::-:e

- ~ ,

_

...,...

'-ot._ . _.,,"'=

.. . .

~es~::-:a::8~

simply :ost and a ~~sy s:g~a: :s ~etur~ed :0 :~e :a __

::1:t.iator. 3ecause of :he :he

addi::ona: ~equests :o~ ~~ansmission are ~ejec:ed. :~e ?roblem of cor~ec: '.:se of ~etNork rescu~:es

:5

a ~aJor =oncern. :: :00 long a:ternate paths are ac=ep~ed, :~e

end-:o-end de2.ay is increased.

:n

add:tion, :t:e

~hroughput drops as fe~e~ calls a~e possible. Alsc a good

evaluation of the expec~ed :raf:ic :s needed in order ~o keep the probability of blocking at a tolerable level. A description of flow control ~echniques for CS networks can

be found in (3]. Although not applicable for shar:

:ransmissions because of the setup time, these ~etworks

are often the only public o~es available some

3

2.:.2. ?acke~ S~i:ched ~et~orks {PS)o

ones ~nde~ cons~de~at~on ~~om ~ow on. ~s ~en:~oned abcve,

sho:-:. t:-ansm:'ss:on

~ransmissions a~~ as:~ct~onous and S~O~~,

. .

_

..

-

..

~.-.. z ~.-..~••.

_-.- ....

~=

...

...

..

~.-..

~he elec~:ical s~itches are ~ep~aced by store-anc-:~~~a~d nodes also called

weE

(Data Communicat~on ~qu:pmen:.). ~hose are computers Mhcse ~ork ~s :0 fo~~a~d a ~essa~e

:=

the next appropriate node~

~ransmission uni~s in such networks are eithe~ ~essages or

packet.s. !f a message is too :ong ~o be tra~smi~~ed

integra:ly, it is sometimes divided into smaller uni:s :alled packets.

Depending on ~he ~outing strategy used (fixed or adaptive), the packets are sent through the same pa~h or may be sent independently of each othe~ acco~ding ~o ~he

9

generally an in:ernal vi~tual c:rcuit service.

routing sc~emes imply an :nternal datagram service. In

packe:s store-and-forNa:d

r:ode

~ases~

(DCS) :0 :~e :1ex~ cne on t.::e

.

....

pat.:1 , l~::t : ..

:.

_::e~' reach

Jue ~o :he ':: ompe ; : : :'.:Jr-. :a~:'::g ~lace :it. ~a·::~ JC=: :J"r

comput :.:--~g ~:ffie

-

..

:;:e node and

c __

access :~ :::e

c .. .u.

:ra~smiss:o~

:ac:::ty,

a~d due ~o ~he stat:s:::a: ~Q'~-~.~_ _ ... 0 " _ - . 1 .r r

:he nessages a~~:v:~g to :~e ~e~~C~K, =~::er s:.~~age

C:mpa~ed :.~ ~:~:~::

net~or~s ~he~~ :je end-:o-end delay on a 9atj:5 :~ns~ant

once ~~e pa~h

:5

established, the delay in packet switched netNorks is 'lariable :or messages of eq~a~ leng:h, even i:

they use :he same route. The Naiting :ime at each DeE can become the ~ajo~ :ac~or in

experiences.

the delay a message

As mentioned previously, the end-:o-end flow =on~rol is applicable to end-to-end vir~ual circuit oriented ?S networks. The difference between virtual circuit (ve) service and datagram (DG) service is discussed next. The

te~minology used in this section as well as in the next

as reference. A decailed :nformation concerning :he functions per:or~ed by each laye~ can be found in [4]0

2.1.2.:. Jatacram Se~~:ce (DG).

~essages recei7ed by :he ne~~ork ~s ~Ot ;ua~an:eed at :~2

~essa~es :~s:de :::e ~e:~ork :a~no~

oe

=c~e. 7~~s, ~=

end-~o-end ~ommunication f~om inside the net~O~K. ~ven if a DG service is inte~~al:y used, ~he T~anspo~~ ~~o~ocol ~ay gi'le a vir~ual =ircuit interface ~o :~e ~~d ~se~. :n such a case, the p~otocol :s ~esponsible fo~ resequenc:ng the messages , for :he des:~~c~:on of cuplicate ~essages

and fo~ the ~ecovery of :he los~ ones. ! f an exte~nal

virtual circuit service is ~rovided, the ne~~o~k looKs

2.2..2.2. Vir~ua: Cir=uit Service \VC).

:n :he :ase an :nter~al:y vc se~vice

:5

provided, :~e

:a~~er'f -_ekes -_ar_~_ o~_- -~e ~es~~ge ~~se_~. .~L::SU .. _ quen'~c: ..g Jr. a

hop-by-hcp basis.

7~e end-:c-end fl~~ =o~t~ol me~hanisrn

both If a DG-JG se~~~:e ~s

:5 :-:0 :1eed i mp l.e:nent ir.g e nc-:'0 - e~..d

~ro:ocol.

::",.e ~ . . . . .

acc :~:cr a , load. ':'his

. ~ _. .

::;.su:::: :'-::"'.:

:0 ~uaran~ee conges~ion avoldance.

Nha~ :he main symptoms of conges~ion are :n a ?S ~etNork

6

5

1

2.2. ?URP0SE OF ::OW C8N~~CL MECHANISMS.

ove rLoad ,

deadloCK a~Qida~ce :~s:de :te ~e:~ork.

2.2.:. :~~ouc~cutd Decraca::cn.

7~roughpu~ degradation occurs ~heneve~ the amoun~ o~ ~eq~es:s ac=e?ted inside :he network is highe~ :han ::5 =apaci:.y. l.ocal resource over-a~~ocat:v:-:

.

., .

i~side ~~e ne:Nork ~ay occu~. Jue:o t~e i~abil:~y of a

part of ~he ~e:~ork :0 ~ee~ :he demand,

propaga~es ~o the neighboring nodes, ~aus:ng them

:0

slow down. The ~eso~r=es at ~he ~eighboring nodes are :~en misused. ~ec ~s now consider t~e da~a l:nk leve:, :~e

.. "

-~

:he sending ~ode does not obtain :he expected ACKs. Consequent:y it does not :~ee its ~uffe~ space and is ~sed

as an addi~iona: ~ai::~g ~ocm fc~ :~e c~~ges:~d ~cde.

i:s p~cc~ssur ~escurces a~e ~as~ed .._

...

-

_{·...;se1.:55}

~e~~a~smission and ~~cover¥ sc~emes.

~e:ationship ~et~ee~ :~~ of:e~ed load and :he ef:ec~:ve

"'---give smaller' ::l~o~gnpu~, . a: :'i.gh: ::-le

"'-~~-'""'l~_...'-"... '-'~

:hapter-.

-

~

..

-.._ . \ ,

'-2.2.2. Deadlock Avoidance.

Similar :0 ~he prob~em presen~ in ope~a:in~ sys~e~s, a non controll~c ~esource alloca~ion may ~esul~ in an eventual deadlock situation. Deadlock ~roblems ~ay oc=~~

~wo ~esource users O~ may involve indi~ec~

:..

.~

--'"

e

-=

CONT3.0t::::J

:) E.-\.:JLOCK

offered load Figure 2.2 Tnrougput versus the Offered

2.2.3. Fairness ~ Users.

Any fregly shared resou~:e

:5

subjec::o ~n:a:~~esso

~;n.iess con~r-cl is exer'~:'sed ~n :he

cUtong the use:-s, scme :~em us e :or :::e:.t"'

.

; ~

:Jene_::. Unfai:::ess ~ay a~ise iue ~o ~referen::a: :~eat~ent given :0 ~se~ a~:o~ding :0 geog~aphic :oca::ons

ar,c

. .

:.:np~eme!"le a~:ons •.

. .

:~=~m:ng ~eq~es:s,

... III • _ . 4

available co ~hem and ::1e expe':~ed

of s e r·"ic:e • Thus flo~ con~rol techniques seek a compromise to this si~ua~ion. ~his implies ~hat use~s

having a high t~affic ~ate, Mill get a Norse response t:~e

:~an in an uncon~~olled en7:ro~~e~~.

2.2.4. ?revention of the Destination 3uffe~ Overf:ow. Although the end-to-end flow con~~ol is used to fulfill the previous goals, its primary use 15 the

sr.atus, the receiver ~ill send an ACK ::ght away or delay it for some pe~:od of ~:me i~ order to ~ope Nith a

~ui:di:1g ccnce st ic n •

r ef e r r ed : 0 :.:1 some ::'te~a,:ure as :~e "us e r ::'Ow :cr::r~:"

:on:e:-:1S :~e end-:~-e~d buf:e~s.

:'mp~e~e~ta:ions are ~sed :or s~ch :on~~ol, :te slidi:1g

,.

-

,

~:J j •

2 • 3 • ?ERFCRM~'lC~ c,o~rs:OERATIONS FOR

:.a

F:'OW-CONTRCLLED

~ETWORKSo

:~ the 9~evious sec:io~ ~e ~riefly p~~sen~ed 4hat NOS

, . .

:ne':r:a~:s:n :n

~easures a~e used i~ o~de~ :0 :~s~ ::le ef::c:ie~c:(

2.301. 7hrouahcut.

_ _ _ t

T~e mos~ Nidely used per:ormance meas~re

:s

:~e "th:-oughputry. it is defined as the ~a~e i~ ~ni:s pe~ second at ~hich che ~etwork can deliver 9ackets to :he des~ina~iono The th~oughput versus the offe~ed load for a specific e~ror rate g:ves a good evaluation of the netwo~k capacity used , The critical load for the ~on :low controlled network can be determined.

performance curve is given in figure 202.

=

20

2.3.2. ?o~ero

~he thro~ghput ~er:ormance :~a~ac::~istic 10es ~Ot

give any indicatian abou~ :~e a'le~age de~ay a ~essage

:~roughpu~ as a

2.3 gives such a r~presenta~ion. :: is expec~ed ~o ~ave

:~c~eases. ~his ~easu~emen:

:5

eften ~e:er~ed :0 as

~~c~er". !~ese ~e~:c~~ance issues ha~e =een c~~sice~ed ~y

:~e delay. The pONe~ :5 max:m:zed as sho~n in :igure 203 a: :he point ~he~e a line drawn from ~he origin is a

~angent to the =urv@. Above ~his point, the de~ay

i~creases :as~er :han :~e :nroughput ~hich is an

i~d:cation of increased ~ueuing t:me. 3elow tha~ poin~,

the maximum possi~le through9~t system is under-utilized.

21

CHAPTER III

-A CLASSIFICATION OF F~OW CONTROL MECHANISMS

~he :low control classification presented hereafte~, ~efe~s mai~ly to the work done by L. Kleinrock and M. Gerla on the subject [8]. Thus, it is with respec~ ~o their terminology that ~e give the main techniques available a~ each level of f~ow =cnt~ol. Examples of implementations in eXlst~ng ~etworks are prov:ded Nhe~

possible.

The n etvo rk protocols that are considered for the flow control cl.assification are shown in figure 3 .1. The t.er:ninology used is common to ANS: (American National

S~andards rnstitute) and ISO (International Standards Organization) specifications. The representat.ion of

figure 3.1 shows which layers are presen~ at the DeE level and at the DTE level.

We will use a bottom to top ievel app~oach l~ describing the existing levels

0:

flow control. This approach is chosen since flow control mechanisms ~ere

first implemented at lower levels. Thus, the type of

22

~o matter at which Level flow control applies,

:t

al~ays

seeks to prevent destina~ion buffer overflow. ~hus, at the data link level the data :ink ~uffe~s are considered. At the Transport laye~ t~e ~eassembly ~uf:e~s a~e of

:~portance. Sasically th~ee levels of flo~ centrol ~eed :0 be ~onside~ed.

~he hop-level flow cont~ol, ~hich concer~s both the Data Link Protocol and the Ne~~ork ?rotocolo

2 - 7he E~try-~o-~xit flow cont~ol (E7E), ~hich ~=nce~~s

the sou~ce-destination s~i:c~eso

3 - 7he Net~ork Access :evel, Mhich can ~h~Ottle the exte~nal ::affic.

(

. _

... ~':'J':'~

ZJ

E

j ( :~1-:~3.~r0 D=:)

,< ~,- -

-?ACK:::r

:.:~r~ ac.:ass

<

24

3.1. 30P-LEVEL FLOW CONTROL.

The objective of ~op flow control schemes is ~o prevent congestion :0 build up ac each switching nodeo

7~is =ongestion oc=urs Mhenever ~he s~o~e-and-for~ard

buf:ers start ~o o7e~:lcw.

ove~flows is perfo~med ~y 9ut~:ng t~resho~ds on the ~umber

of outstanding messages :cr t~ansmissicns, and on the number of buffers allocar;ed for s~ore-and-for~ard

pu=poseso ~essages ~:scarc:ng a~c ~equeSt ~o the sou~ce

:0 s~ow down are the ~ain :echniques ~sed.

T~o :eve~s ~f flow cont:ol a~e dis:i~guished:

- :he ~ata :~nk flow con~rol

- the Network flow control.

3.:.10

_{- - - -}

Data Link Flew Cont~ol Schemes.

Above the physical layer :s t~e data link layer ~hich

is ~~ charge of the correct transmission over a :i~k. !t

~rovides at best a smooth flow along the ~ink and an optimum use of ~esources. Let us recall t~a~ the firs~ data link protocols implemented used Stop-and-Wait ARQ scheme. Once a message was sent over the link, the source waited for an acknowledgement to come back befo~e i: could send :he next message. Such a scheme ~as ~o~ ef:icient and the data link access ~as a bottleneck. _{~e'A/ schemes}

25

~essage. One of the schemes :s commonly ~nown as window

flow control scheme.

3.1.1.1. Slidi~c-~i~dow ~:ON Con~r~l Mec~anis~.

In the slidi~g-~indo~mectanism, the source node ₁₅

permitted to send ~9 ~o W messages, ~he MindoN s:ze, :0 the dest:nation node oefore any _{acknowledgement is} received. ~his is :mplemenced in the followi~g way:

- ~~e sou~=e node maintains a cou~~er i~itialized to W

:~e data link ~:ndow.

eac~ :ime a message is sen:, :~e ~ouncer :5 decremented

by one. Once the counter ~eaches zero, the sending node ~aits until new permission for a message is

obtained from t~e receiving station. The permit :s an ACK message on a message previously :ransmitted.

- if errors occur and a message is not received or not acknowledged, the transmission is timed out and the message is resent. The message retransmission is not

limited by the Nindow size scheme.

The s~apping-window flow control scheme does ~ot apply ~o

the data link level as it requires more information on the buffer status. Thus, more processing time is necessa~y

26

~evel. Additional safeg~ards are often ~rovided at the data link level. RNR (Receive ~lot ~eady) is used both in SOLe (Synchronous ~ata ~inK Cont~ol) and HDLe H:gh Level

~ata Link Cant~ol) ?rococols. XCN/XOF: is an option available :or async~~onous :e~~ina:s. ~hese ·:ommands have an ef:ec: of stopping the sende~ until a ge~~it :~ se~d

:s

ini~iated.

~he inc~ease obtained :~ ~~e data ~ink th~oughpu~

:~creases ~he ~isk

0:

=u::~~ overflo~ a~ each DC~. ~xperie~ce s~owed :ta~ deaclock si~uations easily occur :: an ~ncon:~olled shari~g of buffers

:5

allo~ed. A~ ou:pu~

Link ~ay use all the buffers, thus sta~ving :he o:her communication links and creating a deadlock situation. A cont:ol on the :uffer allocation is necessary and is

27 3.1.2. Net~ork Flow Cont~ol.

Flow control :e~hniques applied to the buffer management put a l:~i:acion on the number of resour=es

per~i:~ed. 7h~esholds are se~ :n order to con~rol the

~uf:er occupa~cy. ~ a =lass

0:

users ~eached i~s quota,

~o ~ore ~esour:es are 9rovided to and the :ncoming

~eques~s are discarded. ~hree schemes are ~riefly

presen~ed as t~e most

respec~:vely:

:m91emen~ed. 7hese

- a s~atic ~axim~~ buffer a:locacion for eac~ c~annel, :e~e~~ed to as "C~anne~ Queue ~~~i: Sche~e~.

- a ~~f:e~ a:location scheme according

:0

the class of :he incomi~g message, called "3uffer Class Scheme".

- a vi~~ua: =:r~~i: orien:ed ai:ocation scheme, ~hich is refer~ed to as "Vi~tual C:~=uit Hop Level Schemen •

3.1.2.l. Chan~el Queue ~imit Scheme (CQL).

Ei~her a static or a dynamic adjustment on ~he upper

28

allocates the a buf:ers equally to che N OU~?U~ links. Thus, a ~aximum queue length of ~ is cermitted .oe~

N ·

output l ·_{.lne. Such a scheme is very as the} demand :or a speci::c output queue is ~outing dependent.

:~ addition :~e links a~e ~eve~ used equally and o~rsts of

:~affic ~equ:re more buff2~ space. Simula~ion results cf s~c~ buffer ~anagement sc~eme can be found in [31.

~ue ~o the s~a~:stical behavior of ar~ivals to a node and ~hei~ r~q~est far a specific ~:nk. a bu:fe~ ove~-se:lin~ scheme ~as considered and is ~efe~~ed ~o as

"Sha~ing wi~h ~axim~~ Queues" (SMXQ).

rn

such a scheme, an upper bound is set to each link such tha~ the upper

:i~:~

£5 greater than

~

mentioned above, provided that the

maxi~um numbe~ of messages allowed at ~he s~itch is nc~ sreate~ than the real buffe~ space 30

A :nird version of such type of scheme ~as conside~ec

in o~der to imp~ove the buffer allocationo This scheme is

refe~red to as nSharing with Minimum Alloca~ion and Maximum Queue~ and guarantees a ~inimum buffer space to each ou~going link. The ~emaining buffer space is shared in the s~e ~ay as in the SMXQ schemeo _Such

configurations have been studied by !rland [9]. ~

heuristic upper bound was fixed to 3 ~ith n. being the

n. l

l

29

As an example the ARPANET netNo~k ~~ovides both minimum and maximum buffer limits *ith over-selling. A minimum of ten buffers and a maximum of :~enty are reserved ~o t~e

message r e as s emb1y for :he Session pro~ocol. The

i~ternode i~put queue, :.e :he one receiving ~~af:ic :~~~

the ~eighboring DCEs, has a minim~~ and a ~aximum l:mit of

tNO. The in~ernode output queue, one per outgoing link,

7a~ies f~orn a minimum of one buffer to a maximum of eight.

O~e of :te ~ajor :once~ns in dropping ~ackets al~eady acce?~ed in~o :~e ~etwork is the ~esource Nas~e l~

:-epresen:s. ~~e ~ore a packet ~~aveled :~side the networK, t.he more resour~es would have been NaStea 1" F

.

;

.-

~ happens to be discarded. Such considerations have been

·Jith t he "Structured 3uffer Poo11l

:10"'1

30

301.2.2. Structured Suffer Pool Flow Control.

This flow cont:-ol scl1eme has ::een tJ~oposed by

Raubold, et ale A. mOt""e de~ailed descr'ip~ion of

.

.~ may :,e

"-found i:1 [ ;.0 ] • This f ~O'J ~ont:-ol sc::erne all.ocates a class

:0 each message according :0 ::5 hop coun~. Such a scheme =an ~e ~:ther used in a VC at"" in a DG envi~onmen~.

;acket ~i~h a low hop counter- is ~refe~entially disca~dec :~ one ~aving a higher hop councero The algorithm ~O~KS

as f~llows:

A ~essage tha~ jus~ enter~d ~he ne~~ork ==om the hos~ :s al:oca:ed the class O. 7~e highest class Hmax :s depencen: on the net~ork :opolcgy and is defined according to

.

-

-

'-

.

C:ass

a

~essages are allowed to access class

u

buffe~s

only. Once those are filled, new class 0 messages a~e

discarded. Class ~ ~essages use class 0 and 1 buffe~s,

:he highest class may use all the buffe~s. !~ orce~ :0

discard class

a

messages prefe~entially to any othe~s ~hen

congestion builds up, all the messages use :he class C buffers as long as ~here :s some available. Thus, class

a

~essages will be starved f:om buffer space before h:ghe~

classes that can use additional buffers at higher class level. This scheme guarantees that the fi~st messages ~o

be discarded are the junior messages. When ith class buffers at a DeE are full, the a4riving messages :0 ~his

32.

This :ype of buffer allocation has been developed and implemented on the experimental PS network GMDNET.

The schemes presen:ed ea~~ie~ applied to datagram service as ~ell as to vir:ual c~rcuit service.

C:~=~:t env:ronment, addi:ional information on :~e path a

~essage Nil: travel on is available and can be ef:icient:y

~sed ~y ::ow ~on~~al schemes. Such schemes implemented at :he ho? :evel cont~ol are presented next.

3.:.2.3. ~:~:~al C:rcu~~ Reo-level F~o~ Control.

-...

-~ten a v:rtual ::rcui~ service :5 ?rovided in a ?acke~ s~i~ched ~etNork, a virtua~ pa:~ is set up before :ransmission takes place. Once :he pa~h is de:ermined, there is no re-routing of the messages to another pa:h even if congestion builds up. An attempt is made ~o provide such ~ecovery schemes but

:5

noe yet a cammon ?rocedu~e. A f:xed ~outing scheme is then used as soon as a session is established. Due to this incapacity :0

32

:0

the ~ongestion. The basic principle for such a flow control is to set a l:mit ~ on the number of messages of a

specific virtual circuit that are allowed a~ each

in~er~ediate node. Such a lirni: is ei:her fixed a~ se:~?

~ime adjusted ac=o~ding

:0

:he ~oad at each

inte~~edia:e node. If more messages t~an ~ermit~ed arrive

~o a ~C~, :~ey are discarded. rn orde~ to limit :he

cisca~ding, informacion on the buffer status is sent to

:~e ~~ighbor:ng nodes. 7his back?ressu~e scheme ge~~its

:he i~forma~:on :0 ;rcpagace ~o ~he neig~bor:ng noces. 7te

~u~ber ~~ ~:~~di~s~ :ef: ~O~ eac~ session ~ay :hus be

~nown.

Such a scheme is implemented in TYMNET (11] by means of a =ackpressure vecto~. This vector acts as a ON/OFF scheme

:o~ each virtual channel. :f the message quota :s reached at a DCE, the sending neighboring node ~eceives a backpressure vec~or in which the en~ry for this vir~ual circui~ is set to o. Upon recep~ion of a a-bit for a specific transaction, the neighboring node dec~eases its

"c~edit" by one. This credit is the maximum number of

33

a :air service to cornmodi~ies that are not responsible for

l~. The problem of buffer overflow is :hus preven~ed

which in turn limits the ~umber of discarded ~essages at each DeE.

~~is scheme opera~es at a hop level, i.e be~~een

ne:ghbor:~g ~odes. Higher levels of flow cont~ol that

ado?~ the same type of scheme are described in :he next

34

3.2. ENTRY-TO-~ (ETE) FLOW CONTROL.

rn the ~~ev:ous section some flow control schemes between the neighbo~ing nodes ~ere conside~edo ~n this section :he entry-eo-exit :l~w ~~n~~ol level:s examined. As for :o~e~ levels, the ~ain ?urpose of ~~is ::a~ =ontrol is ~o 9~event the receivina buffer congestion, :hus

~revent:ng a loss of messages and therefo~e ~~eventing a waste of resourceso Such a scheme concerns the en:ry DC~

and :~e destination DeE, ~hich are ~he en~~y and ~~e exi~

;oints ~: ~he network. 3y :~o~ control~i~g ~he en~~ance

to :he ~e~Nork, :~e numbe~ of messages accepted to the

net~o~k is :hus limi~ed. ~ue to this limitation of

~essages inside t~e net~ork, ~he ent:y-to-exi~ :low

~ont~ol has also been used to control che net-ork :n:erna~

=ongestion. ~hese schemes a~e implemented ~it~ ~i~ccw :low

cont~ol mec~anisms. Such mechanisms are either

siiding-~indow protocols O~ s~a9ping-~indow p~o~ocols. The

sliding-~indow flow con~rol mechanism ~as described for the hop-level f~ow control. The s~apping-~indcw :low cont~ol nechanism is similar to the sliding-window mechanism. Up to W messages are allowed to enter the network ~efore an acknowledgement is received. The main difference ~ith a sliding-~indow mechanism is that such a Nindow can be dynamically adjusted. ~ credit of

w

35 receiving buffer status, the destination DeE sends an additional credit of W to the source DeE. As a diffe~ence with the sliding-window mechanism, this window size W is adjustable and =an be either increased O~ decreased upon

~he destina~ion buffer status. When the source node used all the credits it had available, it is prevented from send:ng new messages unti: a new c~edit is issued to

:t.

Such a scheme is imolemented in DECnet at the

_-

~ranscort

-level according to the ISO terminology (Network or End Control Level in DECnet terminology). As an additional possibility provided in DECnet, the destination node can take back c~edits it already gave by sending a ~egative

cred:t to the source. DeE. This possibility is mainly used under exceptional conditions when a source needs to be s~opped from transmitting faster than what would a normal scheme provide. However such a scheme creates other problems. The message reassembly ti~e may be prohibitive which implies a ~isk of timeout at the Session layer for the transaction and a waste of bandwidth due to retransmission. The second advantage of swapping-window mechanisms is the considerable decrease in acknowledgement traffic. These acknowledgements are the one generated at the Transport protocol level. In a swapping-window mechanism, a block of messages is acknowledged instead of

36 Such schemes :: designed p4ope41y, guarantee that no ~ore ~essages than the system can handle ~ill be allowed to enter the net~oLk. ~owever the main 9urpose of ~~ese sc~emes is ~o avoid the cestina~ion buffe~ ~ve~:low.

:t

is :his level of f:ow control :~at ~e conside~ ~i~h as a goal, to ~ont~ol the internal net~oLk ccngestion.

Additional f:ow ccnt~ol schemes ~ere :nvest:ga~ec ~hac tend to give p4eferen~ial ~reat~en: :0 ~essages

already accepted inside the network. The i~~e~~edia:;

=uf:=~ ~~nges~ion problem *as considered Ni~h ~~ose sc~emes ~~at are refer~ed :0 as ~etwork access flow :ont~ol s~hemes and are briefly described in the next

sec~ion.

3.3. ~E7~ORK ACCESS :LOW CONTROL SCH~~ES.

The objective of the network access :lo~ cont~ol

schemes is ~o prevent the external t~affic from en~ering

the net~ork. As opposed to othe~ schemes desc~ibed

earlier, the network access flow cont~ol scheme has an overall effect on the network. The most common example of such flow control is the so-called isa~ithmic schemeo

3.3.1. rsarithmic Flow Control Scheme.

37 messages are permit:ed :0 ~e5ide at the same time :n ~he

network. This is done by having permits circulating around. Whe~ a permit 15 seized for :he input a

message, the :otal number of credi~s for the net~ork :s

~ec~eased by one. ~ten :he message leaves ~he system, t~e

~~eci: :s given back and can be used for another message.

7~:s :s a similar approach to :he one ~sed on ring

~e:works ~:~h the buffer insertion scheme where ~he buffer

Nas se:zed by ~he node which had a messa~e to ~~a~smit. ~pon ~ece?~:on of the ~essage by ~he ces~:~ation ncde, :~e =uf:e~ Nas :~eed again. As a diffe~ence, a new ?~ob:em a~ises ~hen ~si~g such a type of sc~eme in topologies Jther :han ri~gs: it is difficult to insure :hat a =~edi: wi:l be equally distributed to each node in the system.

Suc~ equal distribution :s not ?ossible, therefore

~~fairness a~:ses. ~epending on its location, a user may ~c~ain a bet~er service than another user. ~he credits can be prevented from accumulating at :he same location by Quttina limi:s on :he maximum number of such credits

-

~

38 3.302. rnout Buffer Limi~ Scheme.

In this scheme, ~~eferential ~reatment is given to the transic t~affic comparing to the i~pu: ~~affi~. ~his

is similar to the Struc~~red Buffer Pea: scheme =esc~ibed aarlie~. As a diffe~ence, this scheme :akes care of :he input traffic, :hro~tling it if necessary. 7~e s:atus of the ~uffers for the in-transit traffic is no~ :aken :n~o ac:cun~. This scheme preven~s new t~affic f~om en:e~:ng :~e nec~ork ~hen a ~onges:ion ~uilds upo :~ :his sc~eme :~e ~umbe~ of bU::2~S ~ade available to :he i~pu~ :~a:::c

:5

li~ited :0 a maximum size. S~C~ a sc~eme was =evelo?ed ~y the GMO group in West-Germany [10].

All these schemes seek ~he des~:'nation buffe:-overflow prevention. All of these levels are ve~y related :0 each o~her and mus~ be carefully analyzed. ~he~e:5 ~o

point to consider :hem separately ~hen design:~g a

~et~ork, ~hich makes ~he problem mo~e diffic~lt. or :_{•. a}

bottom to top approach is considered, the flow cont~ol

39

40

- CHAPTER IV ~

WINDOW SIZE EVALUATION

4.1. PRCBL~~ PRESENTATION.

A classification of flow control mechanisms a~

various protocol levels was given in the previous chapte~e It was pointed out that these levels are ve~y ~elated ~o

each other and that several of them need to be used. A

ca~eful choice is thus necessary ~hen flow cont~ol schemes are implemented. ~ 90o~ buffer allocation at a node ~i:l seve~ely limit the system throughput and ~he end-to-e~d ~indow size will be inefficient. Vice-versa, a too sma:l end-to-end ~indow size ~ill limit the system possibili:: and the buffers at each node ~ill be unde~-utilizedo

As ~entioned previously, the need for flow con~rol

mechanisms is directed by the limited ~esources in terms of buffers at each DeE. ! f too many users are accepted, buffers start to fill up and overflow, causing indesi~able

41 evaluated for each source-destination pair.

The buffer space available at each DeE has a direc~ impact on the amount of messages that can be allowed in the network. Consequently, the buffer sizes provided at each DeE must be ~aken into account when we determine :he end-to-end window sizes.

It was poin~ed out that the end-to-end Nindow flow cont~ol

:5

originally implemented in order to take care of the destination buffer overflow. Thus, ~he windows that give an optimum use of ~esour=es at the DCEs along the ~inimum

cut, must also avoid :he destination buffe~ overf:ow. Inversely, if the end-to-end ~indows have been des:gned

:0

take care of ~he receiving buffer at the destination node, information on the number of buffers that are necessary at each DCE should be deduced. Thus we need to answer the following basic questions: Starting ~i~h a given buffer size at each node, what should the end-to-end window size be ? Vice-versa, given an end-to-end window size, ~hat

should the buffer sizes be ?

4.2. MINIMUM-CUT-MAX:ML~-FLOWTHEOREM.

MINIMUM-CUT-MAXIMUM-FLOW theorem defined in graph theory, gives a Nay to locate such bot:lenecKs in a PS network. A computer ne~work is defined in graph ~heory as a set of nodes and a set of arcs connecting ~he nodes. rn such an environment,

a "CUT" is ~y definition a set of ar=s such ~ha~ when removed, the ne~work is disconnec~ed in tWO sets.

Depending on the ~e~work topology, a large number of such cuts may be determinedG Given ~hat each arc has a fini~e

capacity, the :ocal capacity of each of the cuts is

~omputable. This capacity represents the maximum amoun~

of :nfo~ma~ion in units per second ~ha: can go through it.

~ong those cu~s, the one ~ith :he mi~im~~ capacity is refer~ed to as the MINIMUM CUT. When those links reach saturation, no more traffic increase is possible on the minimum cut. Thus the source-destination pairs conce~ned ~y this one have their t~affic limited by this MAXIMUM FLOW the minimum cu~ can handle. Since we are in a Packet Switched envi~onment, the maxim~~ flow allowed is defined

~3

also on the traffic mat~ix Ne consider. I t may a~so happen that more tha~ one mini~um cut is located. ~n such a situation, the network can be decomposed into separate sets of sour=e-desti~at:on pairs and :he ~rob~em serialized.

T~e m:nimum C~~ :5 ~~early ~he bottle~eck for ou~ ?S network and

:5

:~e depar:ure point for our end-:o-end Nindow evalua~:o~. ~he set of source-destination pairs involvec :~ ~he :ut can be obtained f~om the rou~:ng mat~ix. T~o cases ~ill be considered, the fixed rou~ing

sc~eme and ~he adap~ive rou~ing scheme.

Since the end-to-end delay is one of the major ~aramete~s

to be considered for a PS ~etwork, the end-to-end window size is de~erm:ned w:t~ this constrain: :n ~ind. :or each of :he rou:ing s~racegies considered:

1) The minimum cut is determined.

2) From this minimum cut, and from t~e routing matrix :~ use, the amount of traffic flowing through it is

determined for each source-destination pair.

3) The first and 2nd moment for the delay distribution

1 ~ . . . [,~,

can be evaluated using the resu ts aeve~opea l~ -~J

44

4) The two moments for the number of messages that belong to a specific source-destination pair in the system can be de~er~ined (mean and variance).

5) Depending on ~he real buffer occupancy along the minimum cut, the :raffic given by :he t~aff:c

matrix may O~ may ~o~ ~e increased. This allows us to obtain an upper bound for the window size. This value can be tuned depending on the acceptable buffer overflow a~ong

:ne

minimum c~to

6) the Nindow s:zes obtained for each sour=e-dest:nation pair are such t~a: :he average end-to-end de~ay

constrain~ is s~:ll met.

~hese are the ~oints we consider in the next sec~ions.

The organization is as :ollows:

First the window size determination for fixed ~ou~:ng st~ategies is ~resented. Results obtained a~e given and validation through simulation is provided. Then in ~he

CHAPTER V

-~!~DOW S:ZE SVA~UATION FOR F!X~D

~OUTrNG SCHEMES

The me~hod Ne ~se in order to deter~ine :he ~i~dow sizes is essent:a~ly ~ased on tie ~ini~um cut location in a ?S ne~work. Thus :~e organization of ~his chapter is as follows: (1) the ~ethod used :n order

:0

find a minimum sut is ~riefly desc~ibed. (2) ~he analytica: deve:opments

~e cons:der :or :he determination of :he probabili:y

dist~ibu~ion functions necessary to our evaluat~cn are presentee. (3) the approximation method applied in orde~

to evaluate ~he ?robability distribution :~nctions is given. (4) the determination of the end-to-end windows based on the min:mum cut saturation tec~n:que is ?resented and the results obtained for such configuraticns are provided. (5) validation through simulation of the technique used is given.

-

,

~

.

-

.

LOCATION OF ~HE MINIMUM CUT.

~sed in o~der :0 obtain :hese utilization fac~ors. The input parameters Mere ~espec~ively:

- ~he network :=pology.

- ~~e ~~affic ~atri~.

- ~he pe~c=~tage of overhead in :e~~s of ~eade~s and

acknowledgemen~s specified by the ~se~.

7he compu~ation of ~~:liza:ion facto~s

- :~e maximum end-:o-end delay specified by the use~.

The minimum cut could then be deduced by lock:ng at ~he

set of utilization fac~ors obtained.

The minimum cu~ saturation technique is oased on adap~ive ~ou~ing schemes0 The traffic ~atrix is :nc~eased O~

dec~eased unifo~mly by ~ultiplying it ~y a factor k. :irst ~he shortest paths are used to carry the t~affico Once no more t~affic can be sent through :he shor~est

throughput c~arac~er:=ed ~y the coefficien~ ~_~. and the .. laX

traffic ma:rix, such :ha~ ~o more increase is poss:bie :cr some ccrnrnodi::es.

:~~ ~ay be :ound.

H~e~ s~c~ an event cc:~rs, :~e ~:~:~um

Jbta:~ing :~e ~inim~~ :~~ ~as ;e~:or~ed

._....~

ar: _{: irne}

at a ~ode :an s:~~~a:e :~e !:~::e juf:~~ space case. ~it~e~ ~:~e, ~~::e~ space C~ ~he utiliza~ion :ac:crs ~cst

be used as a c r i t e ricn ..d order to def ine 'N'hethe:- or ~o~ a

pa~j

:5

sa~u~ated.

:5

cbr.ained ::-orn

:::2 s~ls:e!!1 : : x ec

-

. . =-~'':t ::-:g. ~ ~ew se~~: ~~:::za~:c~ :ac~ors is attained using :he ~~evious ~rog~am :or f:xed ~outing schemes. ~he

routing ma~~:x is =ased on :~e sho~~est ?ath i~ ~~~be~

0:

hops. ~hese ~t:::za~ion :ac~ors are :~e da:a ~e need :0 consider for :~e Nindo~ determina:ions.

:na+:r:x, :he se~ of source-destina~:on ~airs conce~ned by

the ~ini~um cu~ is known. ~hese sou~ce-destinat:on =ai~s are ~~e one that ~ay sat~rate the cut. However, :he ot~e~ sou~ce-des:ination ?a:rs ~eed ~o be considered

computing t~e ~indows. If no~ done this Hay, a major par~

of resou~ces ~i:l be al:ocated to :he minimum :u~

commodities, :~us s~arving ~he remaining ones.

system per source-desc:na~:on ~air. On ~he o~her hand, the end-to-end delay cons~raint mus~ be ~e:. 7~is implies

tha~ ~he probabi:ity density f~nc~ic~ =tarac:eri:ation :o~

these t~o va~ia~les must be ~eter~ined. :'h:"s can be obtained

Wang.

~9

5.2. WONG'~ RESULTS :OR ::XED ~OUT:~G SCHEMES.

A fixed routing scheme can be :epresented basically as a set of servers conne~:ed

:0

each o~her in a ~andem s~ruc:ure, :his :s ~epresen:ed as shown in :igure

Suc~ a sys~ern can be ~ode:ed as an open queueing ~e~NorK. ~he ~epresenta~:on of sucj modeling is given

:n

figure ,.. ....

:> •~ • The source ~~af:ic is the average :raf:i~ sent jy so~rce S to des~i~ac:on Q. S:nce the resources a~e shared Ni:h ac~er =~mmodi~ies or source-desti~atio~pairs, an addi~iona~ ~~af:i: en~ers each node.

pic~~~e __ as a ~n:que exter~al traffi~ flow.

i : is a sum of indepe~dent external craffic flows, each of :hem entering and leaving the path as a function of the

~ol;t.ing used. ~he :otal ~tilization :actor :or link ,

.

p:-50

~!~ (Source)

D!!

(desci:tacion)

Dr!

ot::

OTt

=

=

=

0

-a

= ....

~

13

~

=

0

::: ~

-'

:n

=

51

-:J

52

5.2.1. Notation.

:o~ t:xed ~ouci~g sc~eme, the following notation ~i~l

be ~sed:

~ ~he c~ass of a ~essage. One class is allocaced :~

each source-des::na~ion pai~.

at~): the ~oute ~sec ~y class : ~essages and is

desc~ibed as an o~de~ed se~

0:

c~annels.

\~_ mean ar~:7al :a~e ~f c:ass r messages :~ chan~el i.

~;~ :~e ~~:liza::on fac~o~ ~n =~annel _ ~ue :0 class ~

messages.

~i :he to~al u~iliza~ion factor on channel ~.

I •

5.2.2. Simclifving ASSUffiDtions.

:n order to be able :0 model such a configuration,

t~e following siillplif:·~~g assumptions are necessary.

~~e ~rccessl~g ::~e and che ~ropagation a: a ~ode ar~

~ac~ of ~he nodes in the ~pen q~eueing ~ode~ is

rep~esen~ed as a si~gle ser7er ~~eue ~i:~ a

3 - ~he b~f:e~ space a: eac~ ~ode is assumed i~:i~i:e and

:~e :~ansmiss:ons e:~o~ :~ee.

4 - ~he arrival process :or messages of a (s,d) pair is

~ode:ed as a ?oisson process Ni:h a mean !(r).

:he ~essages inside :he ne~~ork are assumed ~o ~ave ~heir length drawn from the same exponential

1 dist:ibution ~i~~ ~ean ;.

6 - Kleinrock's indepe~dency assumption is considered.

:t

states that each time a message ente~s a DeE, a

The assump~ions ~e :onsldered above are discussed and

:~ei~ meaning 90in:ed out.

As sump eicns 1 and :2 ~:-c'."ide a simplis~:'c ~lie'.v of -Jnat a

:eal system is. S~ch assumptions are va:ic ~hen ~ea~l~; ~i:~ a gene~al net~cr~ design. !f ccm9ute~ ;e~:o~=ance ~as ~nder ~onside~ation, such approximations Nou:d not ~e

valid. Cur approach is clearly a ~acroscopic one.

~he ~ean ser~ice :ime a~ each ~cde can be adjus~ed :n

~~de~ ~o :aKe i~t~ ac=oun~ t~ese ave~age ~ai~ing times.

~nde~ s~eacy s~a:e :=ndi::~ns. ~o =acklog ef:ec: due :0

:0 ~ne ne~vobk is :he one specified :n i~pu~. ~he error f~ee environment also guarantees ~hat :~e a~~ival ~a~= a~

the ent~y points is t~e one specif:ed a~ :~e so~:~e ~oces. 7~ere is ~o addi~ional t~affic bet~een ~.O ~odes due :0

~e~:-ansmiss:'on sc:temes. This also f:-ees us from

conside~ations on :he re:~ansrnission scheme and the ~y?e

of additional distr:bu:ion ~hey have compa~ing:o :he

ncrma; traf:ic.

Assumption ~ scates t~a~ :he arr:val process ~o a DC~ is a ?oisson process. ~his ~eans that :he arrivals are independent from each othe~. Such a conside~ation is valid for ~he :ncomi~g ~raffic. Howeve~ is an approximation for :he in-t~ansit traffic. =>ue to ~he

~hese arrivals be:ng inde?endent :rom :ac~ othe~, :he Poisson plocess is a qood appr~ximation to our model.

Assump~:on 5 ~ay or ~ay ~c: =e valid depending on :~e type serv:ce prov:ded jy :~e system.

~easonab~e :c assume ~~a: :~e sources gene~~te a :~a!::~

~~at has a s:~i:ar ~ean ~eng~~ fer t~e messages. I:1 crder :0 be ao':'e - ,"""I

-...,) :node:' a :out.e a :andem

:~e assumpt:on 6 is necessa~y. -: :he

~s no~ ass~mec, :::e

:.~.3. ~nd-To-End Delav on

_--

a :ixed

_-

?a:~.

56

As mentioned before. ~ach JC! :s mode~ed as a MIMll

se:-ve~. The serve~s a~e ass~~ed i~dependen~ from each

ot ae r , :_t.:()

:u~c~:on) ~: ~~e end-co-end-delay of ::ass ~ ~essages.

~~e :aplace :~ansfo~m ~ep~esentation

:5

given t~~ough the

us~al exp~ess~cn:

j s :

:~:- ::c:: ~: :~e :-1/~1/: :!1ode2.s, :he mean del-ay is :he sum

c:

:oliowi~g equa:ion:

~_jp I ~

=

~C. J,&C~

l

-~here ~r is t~e ave~age ~~ber of ~essages i~ :~e syste~. Af:e~ s:mplifica~ion ~e ge~ :he well known ~esu:t:

;;; =

1

~ci (1 - Pi)

Let us r-ecall that :or an MIMI!. model, ~he servi ce time

distribu~ion is of the exponential form and given by

b(t)

=

jJ.e-JAt wit.h _J.I.1 being the mean service timeo !ts representation in Laplace transfo~m domain is given by:

..

-~

: I

:t

Nas proved :n [lSj :hat t~e end-to-end delay ~e have :or a single se~ver is of :he :orrn presented above, ~hic~ means that t ae e!"'~c.-':.o--e~~d :ie:ay is in fact. mode2.ed as

expone~tial distr:=u::o~. Conseq~ently in our case, :~e

Laplace :~ans:orm :e~resen:ation :or ~he delay encoun~e~ed

uCi (1 - ;Ji )

=

5 .,.. ~c ~ t 1 p . )

1

( 5 • 5 )

~:~h i~depe~dency ~e:~ee~ ~~e servers, ::'aplace

:~ans:or~ ~e?r:se~~a::on ~f :~e pdf of ~ie ~hole path is

=

n

a(r}

j.LCi (l - ;; ~ )

s .. jJ.C. ( 1 p ~ )

1 •

( 5 .6)

7hus, the mean and :~e 7ariance can ~e easily der:ved using ~~e expression above.

:;:; ₌ ~ 1

~

r _{ieatr )} )J.Ci (1

-

p :. )

2 1 _] 2

"r =

,

.

JJ,C. ( 1 P ; )

itoa ( r) 1 4.

( 5 • -; )

( 5 .8)

~he end-:o-end delay 0btained is obvious:y the same :or all the commodities since :he lengths have ~he same

1

average value of -. JJ.

58 502.4 ~umber of Class

r

Messaaes in the Net~ork.

The number of class ~ messages in the network can be derived in the following way: The states of each channel

a~e considered and denoted by 51' 52' 0 • • , SM :or a pat~

n ) ..,;-~ n ..o

J. the .~T~m·c·e_~ iR ... ';'., 04._

of class j ~essages a~ channel i (queue ~ transmission) ~epresen~s the state of channel i. Saskett et al ~17J developed analytical results for open and closed que~eing

models. These ~esults =nay be applied :0 this

configura~ion. The equilibrium state probabilities are:

This shows that the state at channel i is independent of the states at the other channels. At steady state and

~it~ the independency assumption ~entioned previously, this is exact. The probability of the sta~e in channel: :'s expressed by:

R R

P(s.)

=

(1 - P.)( s n. )! n

1 l ~~l l r r=l

1

- n ·,

· 1r·

po

lr n.

1:- _{( 5 .9)} ~hich can be viewed as the probability of having ~

messages at channel i with N

=

nil ~ ni2 •

k=O ~ P;(S~)zk5i

59

This exo.ression can be sl'mp_'lof,.'ed ana·- °Je.. ge~ ~~e fo··o'·~·~~'- I..H .L~ ...11.~

equation:

~. ,

-...

(::)

=

, - ;J,

1

1 - p._{_} T _{P i r \ l}• - ZI ( 5 . II )

7he proof of :h:s ~esu:~

:5

given in appendix.

Jue :0 the independency assumption, the z-transform f~r

~he whole pa~h is :he produc: af :he :ndivid~al z-:rans:orrns :or each c~annel.

~ (z)

r =

.,

.l.. - P;

fI ., 1

iea trJ> - P; - Pir\ - z )

(5.12) This expression is derived for the fixed ~outing scheme. However, it can also be used for adaptive rou~ing scheme as it ~ill be shown in the next chapter.

From the z-transform ~epresentation, the mean value and the variance can be computed. After simpiificat:on :rom dN (z)

r with z -> 1, the mean is obtained as: dz

Pir

- p.

1

00

7he variance is direc:ly ob~ained by deri~ing the usual expression from :he z-transform ~ith z->l:

'rhus vi th z = ::

eN

(z)

r

dz dz

2

2

=

[ 1 -Pirp.

1

2

(

-

_:.~~~

-

)

Now, ~ith these analytical ~esul:s in hand, the delay

dis~~ibu~ion and the message dis~ribu~ion can ~e computed

:o~ each source-destina~ionpair. !n o~de~ :0 :i~d a~

end-:o-end ~indow size f~om these ~esul~s, :he ?DF

(p~obability distribution funccion) is :lecessa:-y.

exac~ ?OF can be found bu: is computat:ona::y ~oo com;lex.

~easonable apprcxima:ions ~sing the :irs~ and 2nd ~oment5

of the pdf are Su_tlClent.;:,...

.

The ?~F app~ox:mation

5.3 PDF APPROXIMATION US:NG T~E MET~OD OF ST~GES.

~he ~eed in many a?pli:ations of ~he ?DF given t~e

charac~eristics of t~e.• pd:_,.:._:'·ac· 0 a 'n i nc r e a s ed. .i nce r e st

to approxima~ive methods on :hat s~bjec-:. First developments of such ap?rox:~at:ons a~e due to ~~lang

(19:7). The ~ype of approx:rna::on ~e ceve:cped is based

on what is now ~alled :he ~?HASE CONCEP~rt. Jue ~o t~e

memoryless property of ~he expcnen~ial dis!.ribution,

E~lang approximated general sys~ems by a ~ombina~ion of sums of exponen~ially dist~:bu~ed ~andom var:ab:es. 7~e pa~ame:ers :or such combinations Ne~e computed such :~a~

:~e equiva:ent pdf of :he fictive sys~em did closely approximate ~he "~eal" pdf. Once the combination of stages known, the PDF =ould then be de~ived f~om the pdf or :rom itS Saplace :ransforrn.

generalized the :esul~s obtained by ~rla~g any

distribu~:on ~hose Laplace Stieltjes :rans:orrn is of a

rational form. More recently such phase concep~ ~e~hods have been studied by Neuts (1973). Approximate numerical methods based on :hese formulations have ~een developed ~y

w.

Bux in [19]. Approximate methods are also conside~ed

by T. Altiok in [20] ~ho used the first three moments. In our case the first tWO moments are sufficient in order to

dist~ibution is hypoexponential.

62

The time distribution was found to be always hypoexponential. The message distribution is eithe~ hype~exponential

hypoexponential. However ~e ~ill limit ourself ~o the use of the first and 2nd moment since the t~i~d momen~ ~as a =omputational complexity ~hich would be time c~nsum:ng for a la~ge n~ber of nodes in the network. rn our :ase -e apply this method mainly to app~oximate the wessage

dist~ibu~ion. ay doing so, we model a discre~e ~:~e

function for the buffers by a con~inuous time function.

~.lo~ Aco~oximation of :he PDF, Saue~ and Chandv.

The method developed by che authors is based on :he approximation of the probability distribution function :rom the fi~st and 2nd momen~s of the probability densi:y function. 7he mean and the coefficient of ~a~ia~ion a~~

used for the app~oxima~icn. !n ~hat follows t~e

coefficient of varia~ion ~ill be denoted by _{kc and is the}

ratio of a the standard deviation and ~(xl the mean of

the distribution. The coefficient of variation dete~mines

the type of dis~ribution the "realft

system will be modeled with. For k_c less than unity, a generalized E~lang

distribution is used in the approximation process.

greater than unity, a two-stage hype~ex?onen~ial

63

~·l·l.l. ~odel for k < l.

-For such type of distribution, generalized Erlang

dist~ibutions ~ith ~rede~er~ined paramete~s are used for

the ?DF approximation. The general ~rlang dis:ribution is shown :n figure 5.3. _{The mean service :imes at each} fictive station are equal. The equivalent pdf in Laplace transform domain is given by:

*

3 (s ) = _c JL .. (1 - ?) nI l J.L

" ' 5 " J.L ;=,5 ~ J.L (5.16)

~~us ~he pdf is expressed by

n - 1

b(x)

=

~pe - ~x + (1 - p)~n x e - ~x (5.:7) ( n - 1 )!

The PDF can be obtained by in~egrating :he 9df and is l~

our case given by:

.ut

k! n - 1 :<

=.1 - pe-~t - (1 - p) E ---e-~t (5.18) k=O

The number of stages for the equivalent distribution 15

de~ermined by the value of the coefficient of variation. In order to have a small number of stages, n is chosen as a function of k with 1 / n1/ 2 < k < 1 / (n - 1 ) 1/2

c c

Once the minimum number of stages known and k given, p

c

the probability of branching can be obtained by solving the eo.uation k = n ~ith a and E[x] computed :or our

64

gene~alized Erlang distribution. Afte~ manipulation, this :eads to the following expression:

2nk 2

~

P

=

(5.19)

~he mean of eac~ stage is ob~ained ~y ~omputing di~ectly ~[x] and equating i~ with ~he "~eal" mean d. ~he

:!lean of each stage is by _{n - ?\!1}d _{- :)0} -:'hus, ~nc~ing the ~ean val~e :or cu~ dis~~i~ution and provided

:~at ~_ :s less chan one, He are able ~o c~mpute the

'-~quivalent ?OF using :hese results.

=

65

~

I

66

5.3.1.2. Model for k~

'-:n :he case K_C :s g~eater than one, t~e system is approximated by a :No-stage hyperex?onential dis~ribu~ion. 7his dist~ibu~io~ is ?ic~ured in figure 5.~. 7he Laplace

c~ansfor~ representation of t~e ~df

:5

g:'le~ ~y:

*

9 Cs )

=

0

-· s of- _J..'l - (1

-JJ. .. :J) .t. • S • JJ.2

~hic~ once inverted gives fo~ ~~e pd::

-J,J. x

- (1 - 9),u

Ze 2

(S.21.)

~e ~hus can obtain ~he ?OF by integrating the

-~lC -~2t

= 1 - ?e - (1 - ~)e

.~

pa .. , which

(5.22)

The relationship bet~een the p~obability of bra~ching a~d

the coefficient of 'lariation of the W~eal" dist~:bution is such :nat:

..

.

k 4 _

2"

+ 1 - ( c 1 )

?

=

Again this ~esult is obtained by solving the equa~ion

a

67

- d/2p for a ~ranching probability of p.

- c/2(1-p) for a ~ranching probability of (l-p).

d ~epresents ~~e ~ean of t~e "~eal" dist~ibu~ion. 7~us,

as :he p~ev~o~s case ~e can compute the ?DF and ~~us

Figure 504 Two-Stage Hyperexponential Dis-trl.bution.

69

5.~ WINDOW SIZE DETE~~INATrON.

5.4.1 Method Desc~~~t:on.

~rom ~he analy~:~a: ~esults presented in the previous sections, the end-:o-end delay and :~S standard deviation can be computed :or each source-destination pair. Simllarly ~he average ~umbe~ cf ~essages in the network and :~S standard deviation are compu~able for each

source-des~ination pair. I~ order ~o perform such

~=~p~~a~l~ns, ~he t~a:fic matrix, :he ~ou~ing matrix and

:~e ne~worK topology are necessary. ~he ~in:murn cut is ~e=essary :~ our ~ethod. As it was pointed out ea~lier,

:ie ~inimlliu cut defines the bottleneck in the network. ~e

~ill use :his information to come with the maximum end-~o-end window size ~hat can be allocated :0 each source-des~ination ~air.

7he ~ethod concerns the commodities sending :raff:c

70

It may happen that :he ena-to-end time constrain~ is ~o~

~et Eor some commodi~ies. :: ~his is the case,

t~affic for :tese commodities :s dec~eased ~ntil they mee~ the end-to-end :ime =onstrain~.

set is obtained :~r ~uf:ers avai lable at each node. Given a =~ffe~ space 3 at each DC~, :he t=affic

~e s~arted with may be :00 high. we conside~ che ~inimum =~~ in orde~ to dete~mine ho~ much bu::er space is

~ecessarv at each DeE for ~he :~affic mat~lx Me have.

~:om :he ~/M/l model ~e can deter~ine ~ow n~=~ ~uffe~

3pace :s necessary on ave~age and ~hi=h bu::e~

~e allocated such tha~ the probability of ove~::ow

:5

e~.Jal:o x , Knowing the utilization :ac~o~s ~n :he minimum cut links and x being chosen, 3 the buffe~ size :s determined.

In the following discussion, ~e assume ~hat :te probability of ove~flow of the ~uffers along :he minimum :ut is small. Probabilities of % and

-=

'Je:-e

=:)nsidered. Due to the fact that the minimum cu~ buffers

a~e :he one ~hat a~e the most highly u~ilized, a

p~obability of overflow of x% of those buffe~s gives in the Morst case a user's probabili~y of retransmission cue :0 buf:er overflow equal to x\. ~his is a first orde~ app~oximation to :he ~eal ?rocability of ~e~~ansmission.

will be mentioned in =hap~er _{bu~ require a}

71

separate study in order ~o obtai~ meaningful results. 7his s~udy

was ~o~ per:or~ed and should be cons:dered as a ?ossible

extension.

In our ter~inology, He shall call:

- "st a rt i nq set", t he set c har act eri z

ec

by the t r aff ic

~a:~:x specified by the use~.

"s:ar::~g :~as'l_·~le sef_n

, ~_~•.e st art i nc se~ -l..,-~ -nects

- ~ - • I". _ _A.L_ _ _ . l Q I " . HL _

:~e e~d-~o-end cons~rai~t specified. ~~:s set may

. . . h

;:,e e:~ner :.e starting set itself or the se~

ob~ained from the starting set and such :~at a~: the commodit:es meet :he time constraint.

"optimized set", the set obtained once :he buffer

space at each DeE was specified. In ou~ st~dy, ~~e

buffer space ~as assumed b~ing the same a~ each DC~.

- "optimized feasible set", the set obtained from :he

.

optimized set such that all the commodi~ies meet t~e

time constraint.

From our method, the window sizes ~ill be obtained froom ~he "optimized :easible setn

72

~indow size we allocate to each commodity is such that:

pC

x > k*mean J -~ 0

4~lcn allows to dete~mi~e the value k. The windo~ ~hosen

~s simply:

W

=

k*mean

Since the traffic ~atrix is such that ~he ne~~ork

~~sources are ~o: over-u~:l:zed and since Ne deter~ined

~~e ~op~imized feasible se~n ~hich uses :~e ~esources at

~:s ~ax:m~, ~he~e is no need:o t~uncate t~e message

d:s:~ibu~iono However, in the results p~esen~ed :n the sections, ~e also considered ?robabili~ies of

:~~ncating the message distribution of and

=

~espectively. ~his ~as di~ected by ~he ~eed

:0

va::cate

~~ose ~esults by simula~ion. Such end-to-end ~~uncations

73 A program was developed in order to compute the first and 2nd moments :or :he delay distribution and for the

~essage distrib~t:on.

~~e :npu~ 9a~amete~s are:

:he ~e:work :cpology, given ~hro~gh [C~ ~

J,

:~e :apaci:y mat r ix •

t he netvo r k traffic given through It r .. ], the

:'j

::-a:::c mat:-ix.

- ~~e ~in:mum cut links defined by ~heir ~~o extremities.

:te u~ilization factor matrix ~hich can be either s;eci!ied in the input or computed from previous

?arame~ers. This allows us to ~ake into ac~oun~ the :raffic overhead.

- the set of source-destination pairs to be considered is specified in order to limit the computation to

a given set of commodities.

From these parameters:

- the time distribution

t

r ..

J

74

- the distribution for the number of messages per source-destination ( [N .. ] , [,J J} is obtained.

l J Ni j

7~e algori~hm is as :ollows:

Sten i:

----

-Compu~e [ CcmpUT:e [

,.,.,.. ')

• i j s ,

N.. },

lJ

(1".. } ] •

Jo i j

,

a .l •

N~ ;

e ~

"s:art.:ng set"o

St.eo 2:

. - - - -

-_

~ _F ".. •. < T

· l J corist r

7hen: - the set is feasible fo~ the time cons~raint.

"Starting feasible set~.

- Check the actual buffe~ oc=upancy at ~he

minimum cut.

* average buffer occupancy 3 * from p[ overflow ]

=

p

deduce 9 for the given po

M/M/l approximation used.

- Finite buffer size chosen is given.

- Increase (decrease) t:affic to the maximum

=::'se:

75 New set [ p . . ] obtained.

: J "optimized set".

Recompu~e :~e ne~ time and message dist:-lbul:ions.

Check t~e :easijili:y of t~e new set obtained 7

If feasible set ~hen ~XIT. "optimized feasible se~"

do s:ep 3.

:'.. > ,"'"

:J constr

Set is not feasible.

Decrease ~he :raffic selectively for :he commodities that do not meet the constraint.

If the starting set was noe feasible, per:orm step 2 once to reach an optimized set.

Else set feasible = optimized se~

EXIT