Alternative Approaches of Convolution within Network Calculus

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1 Joural of Applied Mahemaics ad Physics Published Olie Ocober 04 i SciRes hp://wwwscirporg/joural/jamp hp://dxdoiorg/0436/jamp04 Aleraive Approaches of Covoluio wihi Nework Calculus Ulrich Klehme Rüdiger Berd Compuer Neworks ad Commuicaio Sysems Friedrich-Alexader-Uiversiä Erlage Germay ulrichklehme@csfaude ruedigerberd@csfaude Received July 04 Absrac Nework Calculus is a powerful mahemaical heory for he performace evaluaio of commuicaio sysems; amog ohers i allows o deermie wors-case performace measures This is why i is ofe used o appoi Qualiy of Service guaraees i packe-swiched sysems like he iere The mai mahemaical operaio wihi his deermiisic queuig heory is he miplus covoluio of wo fucios For example he covoluio of he arrival ad service curve of a sysem which reflecs he daa s deparure Cosiderig Qualiy of Service measures ad performace evaluaio he covoluio operaio plays a cosiderable impora role similar o classical sysem heory Up o he prese day i may cases i is o pracical ad simple o perform his operaio I his aricle we describe approaches o simplify he mi-plus covoluio ad accordigly faciliae he correspodig calculaios Keywords Nework Calculus Mi-Plus Covoluio Algebraic Laws Covex Aalysis Iroducio Timeliess plays a impora role regardig sysems wih real ime requiremes This Qualiy of Service (QoS) requireme ca be foud i may kids of embedded sysems which permaely exchage daa wih heir evirome; like safey-criical auomoive sysems or real ime eworks Sice kowledge of mea values is o sufficie aalyical performace evaluaio of such sysems cao be based o sochasic modelig as applied i radiioal queuig heory For hese sysems wors-case performace parameers like maximum delay of service imes are required I order o specify guaraees of performace figures i erms of boudig values which are valid i ay case a mahemaical ool Nework Calculus (NC) as a ovel sysem heory for deermiisic queuig heory has bee developed [] σ ρ raffic descripio ad his calculus for ework delay Furher seps owards NC were ake by he work of Parekh ad Gallagher [4] o deermie he service curve of Geeralized Processor Sharig (GPS) schedulers The NC framework has bee successfully applied for he aalysis ad dimesioig i various domais icludig idusrial auomaio eworks [5] or auomoive commuicaios bus sysems [6] This aricle is srucured as follows i Secio we describe he basic modelig elemes of NC heory The The developme of his heory has bee sared by Cruz [] [3] o he ( ) How o cie his paper: Klehme U ad Berd R (04) Aleraive Approaches of Covoluio wihi Nework Calculus Joural of Applied Mahemaics ad Physics hp://dxdoiorg/0436/jamp04

2 U Klehme R Berd subseque Secio 3 demosraes he difficulies of covoluio calculaio ad describes possible soluio approaches Fially Secio 4 draws a coclusio ad idicaes fuure seps Basic Modelig Elemes of Nework Calculus The mos impora modelig elemes of NC are he arrival curve (deoed by i he figures) ad he service curve (deoed by β i he figures) ogeher wih he mi-plus covoluio The arrival ad he service curve are he basis for he compuaio of maximum deermiisic boudary values like backlog ad delay bouds [] be a o-egaive o-decreasig fucio Flow F wih ipu Defiiio (Arrival curve) Le ( ) x ( ) a ime is cosraied by or has arrival curve ( ) iff x ( ) xs ( ) ( s) F is also called -smooh Example A commoly used arrival curve is he oke bucke cosrai: rb ( ) b r : > 0 = 0 : 0 for all s 0 Flow Figure shows a arrival curve which represes a upper limi for a give raffic flow x ( ) wih a average rae r ad a isaaeous burs b Therefore: x ( ) xs ( ) ( s) For : = s ad 0 : lim rb { ( ) ( ) lim { x xs r b = b () s 0 Nex he covoluio operaio which plays he mos impora role i Nework Calculus will be defied Defiiio (Mi-plus covoluio) Le f ( ) ad g( ) be o-egaive o-decreasig fucios ha are 0 for 0 A hird fucio called mi-plus covoluio is defied as ( f g)( ) { f ( s) g( s) = if (3) 0 s O his basis we ca characerize he arrival curve ( ) wih respec o a give x ( ) as: x ( ) ( x )( ) I oher words arrival curves describe a upper boud o he ipu sream of a sysem Cosiderig he sysem s oupu we migh be ieresed i service guaraees: like a miimum of oupu y( ) ie he amou of daa ha leaves he sysem The followig defiiio deals wih his problem Defiiio 3 (Service curve) Le a sysem S wih ipu flow x ( ) ad oupu flow y( ) be give The sysem provides a (miimum) service curve β ( ) o he flow iff β ( ) is a o-egaive o-decreasig 0 0 β ie: fucio wih β ( ) = ad if y( ) is lower bouded by he covoluio of x ( ) ad ( ) () Figure Toke bucke arrival curve 988

3 ( ) ( β )( ) U Klehme R Berd y x (4) Figure shows ( x β)( ) = if0 s { x( s) β( s) as lower boud of oupu y( ) ad ipu x ( ) Such service curves are fucios of he ime ad describe he service of ework elemes like rouers or schedulers i a absrac maer [7] Example A commoly used service curve of pracical applicaios is he rae-laecy fucio: β ( ) β ( ) R [ T] R { T = = : = max 0 (5) RT This fucio reflecs a service eleme which offers a miimum service of rae R afer a wors-case laecy of T Sice we wa o aalyze he wors-case performace i is possible o absracly model he complex ieral behavior of he ode by jus describig he wors-case service usig his curve This is very impora i pracical uilizaio I Figure 4 he (gree) lie depics he rae-laecy service curve wih rae R ad laecy T Cosider a sysem wih ipu flow x ( ) arrival curve ( ) oupu flow y( ) ad service curve ( ) Accordig o [] he followig hree bouds ca be derrived: Backlog Boud v : Delay Boud d i case of FIFO service: Oupu Boud ( ) : Remark: The complee backlog v ( ) x ( ) y ( ) ( ) The expressio if { τ : ( s) β( s τ) ( ) = ( ) ( ) sup s 0 { ( ) β( ) v x y s s { { τ ( ) β( τ) d sup if : s 0 s s ( ) = : = sup s 0{ ( s) ( s) β β β = a ime wihi a sysem is also called ufiished work or buffer is called virual delay: he miimal ime disace which is ecessary for ipu x for beig served o oupu Thus he backlog ad delay boud are he maximal verical ad horizoal deviaio bewee arrival ad service curve Figure 3 depics d ad v for a give arrival- ad service curve Example 3 Le a sysem wih a oke bucke-smooh ipu ad rae-laecy service be give hus: ad ( ) ( ) ( ) x xs s rb Figure Covoluio as a lower oupu boud 989

4 U Klehme R Berd { RT ( ) ( ) β ( ) y if s x s s b Based o he hree bouds above we ca calculae he delay boud as d T he oupu boud as R = Figure 4 shows hese resuls ( ) = r( T) b ad he backlog boud as v b rt 3 Difficulies of he Covoluio Operaio Wihi he Nework Calculus (NC) heory he covoluio operaio of so-called mi-plus algebra is cosidered he mos esseial operaio This is because based o he arrival- ad service curve i compues he deparure of a ework eleme herefore i is he mai isrume of aalyical performace evaluaio For example i [] his operaio is carried ou by had usig pure aalyical calculaio Take Example 3 for which rb βrt as he lower boud for oupu y: y rb βrt eeds o be compued 3 Aalyical Compuaio of Covoluio Operaor a) 0 T : Figure 3 Backlog ad delay boud Figure 4 Example for he bouds 990

5 U Klehme R Berd b) > T : { ( ) ( )( ) ( ) [ ] { ( ) β = if s R s T = if s 0 = 0 0 = 0 rb RT rb rb rb s s ( rb βrt )( ) = if { rb ( s) Rs [ T] s if { ( ) [ ] if ( ) [ ] if s T T s s= = if { b r( s) 0 if { b r( s) R( s T) { 0 R( T) { { ( ) [ ] rb s Rs T rb s Rs T rb s Rs T = s T T< s< { ( ) { ( ) if {( ) T< s< { b r( T) { b r( T) { R( T) { b r( T) { R( T) { = b r T b r RT R r s R T = = Here A B= mi { AB ad covoluio ( rb βrt ) wih resul (6) is depiced i Figure 5 For he followig covoluio of wo rae-laecy service curves (cocaeaio) a similarly complex calculaio is required β = β β = β (7) R T R T mi ( R R ) T T As we ca see from above his aalyical approach is cumbersome ad error-proe Bu similar o covoluio of classical sysems heory he covoluio is of grea imporace for NC heory This is why we are lookig for oher more elega mehods o compuaio Nex wo aleraive approaches called Use of algebraic laws ad Use of covex aalysis will be iroduced 3 Use of Algebraic Laws I he followig he NC-releva fucios Φ are defied as o-egaive o-decreasig ad passig hrough he origi: { f : f ( ) f ( 0) 0 0 f ( 0) 0 Φ= = (8) Amog ohers he followig algebraic properies for he se of fucios Φ ca be foud i lieraure [] (6) Figure 5 Covoluio of rb βrt 99

6 U Klehme R Berd Rule (Commuaiviy of ): f g = g f Rule (Associaiviy of ): ( f g) h = f ( g h) Rule 3 (Disribuiviy of ad ): h ( f g) = ( h f ) ( h g) Rule 4 (Addiio of a cosa): g ( K f ) = K ( g f ) K 0 Rule 5 (Cocave fucios passig hrough he origi): f g = mi { f g Rule 6 (Shifig of f o righ): ( f δt )( ) = f ( T) wih δ T ( ) = 0 for T ad oherwise Now we oly apply hese algebraic rules o compue he covoluio We cosider Example 3 i a eve more geeral form: K β wih ay cosa K ; le λ r = r ( RT) rb ( K βrt ) rb ( Rule) ( Rule6) ( Rule4) ( ) ( ) ( Rule) Rule5 Rule3 (( ) ) = K β = K δ λ = K δ λ RT rb T R rb T R rb ( δ ( )) ( ) ( ( )) ( ) T λr rb δt λr rb ( δt λr) ( δt rb ) = K = K = K ( Rule6) Rule4 Rule6 ( ( )) ( ) ( ) = K b r( T) R( T ) ( ) ( ) ( ( )) = K βrt δt λr b = K βrt δt λr b = K βrt βrt b { { Or ake he cocaeaio example: β = βr T β R T : ( Rule5) Rule6 ( ) ( ) ( ) ( ) { ( ) ( ) ( ) ( ( )) ( Rule ) T T ( ) ( T ( ) T ( )) β = R T R T = R δ R δ = R R δ δ ( ) ( Rule6 ) δ δ { ( ( )) β ( ) = mi R R = mi R R T T = 33 Use of Covex Aalysis T T mi R R T T The ex approach is based o he heory of covex aalysis [8] Similar cosideraios relaed o NC heory ca be foud a [9] However we go aoher way cosiderig icomplee covex fucios I Rockafellar s book [8] all ecessary precodiios for he followig defiiio ad proofs of he ex heorems ca be foud Defiiio 4 (Cojugae fucio) Le f be ay closed covex fucio o R f ( s) : = sup { s f( ) domf is called he cojugae of f or Fechel-rasform wih s R he ier produc i R Theorem (Cojugae f ** ) The cojugae f of f is f : { ( ) ( ) f = sup s f s s domf Theorem (Covoluio) Le f f be proper covex fucios o Remark: I case of he covex NC fucios of Φ : ( ) s = f f f f R R The: R From Theorem ad we ge ( f f) = ( f f ) Example: Agai he aim is he cocaeaio β = βr T β R T wih (9) (0) β Ri Ti ( ) ( ) Ri Ti : Ti i = = 0 : < Ti () 99

7 U Klehme R Berd β RT eeds o be deermied From defii- Firs of all he cojugae of he rae-laecy fucio β RT ie io 4 we kow ha So we ge: Case 0 : s < ( s) ( s) sup{ s ( ) dom sup{ s R( T ) β = β β = () β = Case 0 : s = β ( s) = 0 Ts : 0 s R ad for Case s 0 : β > ( s) = < : s > R Deermiaio of β ( ) as cojugae of ( s) : s < 0 Alogeher: β ( s) = βrt ( s) = Ts : 0 s R : s > R β wih s domβ : : s < 0 β ( ) = sups s β ( s) = sup s s Ts : 0 s R = βrt : s > R { { ( ) This ceraily cofirms Theorem Applyig Theorem : ( β β) β β β β β β For he cocaeaio βr T β R T we obai (le β = β R T ad β = β R T T > T > 0 R > R > 0 ) = ; here ( ) = ad by applyig Theorem : ( ) ( ) ( β β ) ( ) β ( ) β ( ) ( ) β β : s < 0 : s < 0 : s < 0 s = s s = Ts : 0 s R = T s : 0 s R = T T s : 0 s R : s R : s R > > : s > R : s < 0 ( β β ) ( ) = sup s { s ( T T ) s : 0 s R : s > R Bu his is he same srucure as i expressio for β ( ) ad herefore β β = β = β holds Figure 6 depics his operaio ( ) ( ) mi ( ) R T T R R T T (3) (4) (5) (6) Figure 6 Cocaeaio of wo rae-laecy fucios 993

8 U Klehme R Berd I coclusio he use of covex aalysis seems o be a furher approach o deermie he mi-plus covoluio operaio a leas i he coex of covex fucios like he se of piecewise liear fucios where β RT belogs Quesio: How o deal wih o-covex fucios f Φ Take Example 3 wih ipu x = rb ad service curve β The mi-plus covoluio realizes a guaraeed lower boud for he oupu: y( ) ( β )( ) RT is o covex because he relaio i: Bu fucio rb ( ) (( ) ) ( ) ( ) ( ) r b : > 0 = 0 : = 0 rb RT rb λ 0 λ λ rb 0 λrb for 0 = 0 R ad 0< λ < is o fulfilled Therefore Theorem ad are o readily applicable However i coras o [9] oher ways are possible Here he followig oe is suggesed: makig fucios covex by redefiig heir values a cerai pois eg where uaural discoiuiies appear (eg = 0 ) For example we chage rb ( ) o rb ( ) such ha rb ( ) is covex ad sill reflecs he arrival curve feaure well: ( ) rb = r b for 0 (7) Now rb is covex ad he priciples of covex aalysis are applicable Alhough rb Φ (sice rb ( 0) 0) i represes he impora burs feaure : = ( s) 0 of he arrival curve rb (of formula ): { ( ) ( ) 0{ 0 ( ) 0 ( ) lim x xs lim r b = lim = lim = b s rb rb Sice rb rb he covoluio of rb βrt = { b r( T) differs from β ( ) { b r T The expressio ( ) { { ( ) rb RT = b r T R T is ideical i boh formulas Whe cosiderig rae-laecy fucios (cf Example ) where a icomig ipu is served wih miimal rae R afer a maximal delay T he oupu y { { { is lower bouded o oly by b r( T ) bu by mi b r( T) R( T) for he erm b r( T ) 4 Coclusios is he domia oe as you ca see i Figure 5 Of course i he log ru The mos impora operaio wihi he Nework calculus is he mi-plus covoluio Bu he calculaio of his fudameal operaio sill is complex ad error-proe For his reaso we iroduced oher compuaio approaches o perform he covoluio: here called Use of algebraic laws ad Use of covex aalysis I order o apply he covex aalysis we rasform o-covex io covex fucios (eg keepig he oke-bucke arrival curve propery) which is differe for example from he approach of [9] This aricle reflecs work i progress ad he preseed examples ough o demosrae hese echiques I fuure work we will coiue o ivesigae he procedures of subsecio 3 ad 33 o aswer he followig quesios: Ca we exed he priciples o some classes of o-covex fucios or eve o mixed classes of covex ad o-covex fucios? Which kids of applicaios are suied cosiderig algebraic laws ad covex aalysis? Are here eve cases where a combiaio of boh mehods is beeficial? To all of hese we wa o fid aswers i fuure work Refereces [] Le Boudec J-Y ad Thira P (00) Nework Calculus Spriger Verlag LNCS

9 U Klehme R Berd [] Cruz R (99) A Calculus for Nework Delay Par I: Nework Elemes i Isolaio IEEE Trasacios o Iformaio Theory [3] Cruz R (99) A Calculus for Nework Delay Par Ii: Nework Aalysis IEEE Trasacios o Iformaio Theory [4] Parekh AK ad Gallager RG (993) A Geeralized Processor Sharig Approach o Flow Corol i Iegraed Service Neworks: The Sigle-Node Case IEEE/ACM Trasacios o Neworkig hp://dxdoiorg/009/ [5] Kerschbaum S Hielscher K Klehme U ad Germa R (0) Nework Calculus: Applicaio o a Idusrial Auomaio Nework I: MMB ad DFT 0 Workshop Proceedigs (6h GI/ITG Coferece) Kaiserslauer March 0 [6] Herpel T Hielscher K Klehme U ad Germa R (009) Sochasic ad Deermiisic Performace Evaluaio of Auomoive CAN Commuicaio Compuer Neworks hp://dxdoiorg/006/jcome [7] Fidler M (00) A Survey of Deermiisic ad Sochasic Service Curve Models i he Nework Calculus IEEE Commuicaios Surveys & Tuorials [8] Rockafellar RT (970) Covex Aalysis Priceo Uiversiy Press [9] Padi K Schmi J Kircher C ad Seimez R (004) Opimal Allocaio of Service Curves by Exploiig Properies of he Mi-Plus Covoluio Techical Repor 995