The Virtual Time Function and Rate-based Schedulers for Real-time Communications over Packet Networks

Transcription

1 Westen Austalan Telecommuncatons eseach Insttute (WATI) The Vtual Tme Functon and ate-based Schedules fo eal-tme Communcatons ove Pacet Netwos Tath Navendan Devadason Ths thess s pesented fo the degee of Docto of Phlosophy of The Unvesty of Westen Austala School of Electcal, Electonc and Compute Engneeng July 27

2 To my paents, Muel and Enest And to the late D. Guven Mecanos

3 Acnowledgements I would le to gatefully acnowledge the enomous assstance of my pncpal supevso, D. Guven Mecanos, who conceved ths poject and aded n ts completon. In addton to beng a supevso and teache, he was a mento, colleague and fend. He showed nspatonal couage and stength dung hs fnal llness, and geneously gave of hs tme n hs last months to assst wth pepang ths thess fo examnaton. Hs passng on 4 Januay 27 was a geat loss, and has depved the communty of a thoough eseache and a pncpled man. My sncee thans go to Pof. Antono Canton who, dawng on hs nexhaustble supply of wsdom, povded me wth nsghtful and pactcal advce on nnumeable occasons dung the couse of ths wo. Pof. John Slqun, wth hs gudance and encouagement, tuned completon of ths poject fom an mpossblty to an nevtablty, and has my gattude. He eally s a maste of gettng thngs to happen! Pof. Kevn Fynn, wth almost nfnte patence, has povded oppotuntes and facltes wthout whch the pusut of ths degee would have been dffcult. Hs help s geatly appecated. Those who wthout hestaton mae themselves avalable to help n tmes of need ae tue fends. Bet Wong and Joe Tnle fall nto ths categoy. When tme was unnng ctcally shot, they geneously stepped n to assst wth tme-consumng fomattng tass, and dd so wth Chst-le selflessness. I hope that I shall have the oppotunty soon to epay them fo the effots. It s not possble to oveestmate the contbuton my paents have made to the completon of ths tas. They have povded evey suppot mechansm unstntngly, to gve me tme to spend on ths wo. The wods than you ae nsuffcent. Above all, I than the Lod Jesus Chst. All of Ceaton s sustaned by Hs poweful wod (Hebews 1:3). The completon of ths thess s theefoe also the poduct of Hs gace, extended to me n unbounded measue. Μαραναθα!

4 Abstact The acceleatng pace of convegence of communcatons fom dspaate applcaton types onto common pacet netwos has made qualty of sevce an nceasngly mpotant and poblematc ssue. Applcatons of dffeent classes have dvese sevce equements at dstnct levels of mpotance. Also, these applcatons offe taffc to the netwo wth wdely vaant chaactestcs. Yet a common netwo s expected at all tmes to meet the ndvdual communcaton equements of each flow fom all of these applcaton types. One goup of applcatons that has patculaly ctcal sevce equements s the class of eal-tme applcatons, such as pacet telephony. They eque both the epoducton of a specfed tmng sequence at the destnaton, and nealy nstantaneous nteacton between the uses at the endponts. The assocated delay lmts (n tems of uppe bound and vaaton) must be consstently met; at evey pont whee these ae volated, the netwo tansfe becomes wothless, as the data cannot be used at all. In contast, othe types of applcatons may suffe appecable deteoaton n qualty of sevce as a esult of slowe tansfe, but the goal of the tansfe can stll lagely be met. The goal of ths thess s to evaluate the potental effectveness of a class of pacet schedulng algothms n meetng the specfc sevce equements of eal-tme applcatons n a conveged netwo envonment. Snce the poposal of Weghted Fa Queueng, thee have been seveal schedules suggested to be capable of meetng the dvegent sevce equements of both eal-tme and othe data applcatons. Many of these ae what we call ate-based schedules smla to Weghted Fa Queueng, but wth a dffeent vtual tme functon used n the abtaton algothm. Ths wo nvestgates whethe these schedules, togethe wth some new elated stat-tme schedules that we popose, ae n fact appopate fo eal-tme tansfes, especally when compaed wth what can be acheved wth the smple Fst-Come-Fst-Seved schedule. To acheve these goals, genealzatons ae made fo the class of schedules n tems of the vtual tme functon. Ths wo uses an ognal method, mang use of the

5 concept of a hypothetcal seve opeatng n vtual tme, to extend the undestandng aleady avalable n the lteatue fo ndvdual schedules. Pevous attempts at genealzaton, as well as new mpovements to the schedules, have been focused on faness. Howeve, we move the focus bac to delay n the context of eal-tme applcatons because the undelyng tmng of these flows means that ths s moe petnent, and faness s of lmted value. We use a modfed fom of the pevously poposed latency metc to obtan bounds on delay, whch avods the need fo the leay-bucet chaactezaton of taffc avals used n othe po wo. We then pesent a smulaton analyss of the behavou of both the stat-tme and fnsh-tme vesons of the schedules n tems of the delay behavou they poduce n a test flow. In ths analyss we pesent a novel and apt measue of the solaton povded to eal-tme flows fom the chaactestcs of bacgound taffc. The solaton s detemned by compason to an deal benchma dstbuton. The solaton povded by stat- and fnsh-tme schedules, wth dffeent vtual tme functons, fom bacgound taffc chaactestcs, such as bustness, ate composton and numbe of ndependent flows, s clealy ascetaned. Ths smulaton study also sheds lght on false assumptons that can be made about the solaton poduced by stat-tme and fnsh-tme schedules based on the detemnstc bounds obtaned. The ey contbutons of ths wo ae as follows. We clealy show how the defnton of the vtual tme functon affects both delay bounds and delay dstbutons fo a eal-tme flow n a conveged netwo, and how optmalty s acheved. Despte appaent ndcatons to the contay fom delay bounds, the smulaton analyss demonstates that stat-tme ate-based schedules possess useful chaactestcs fo eal-tme flows that the tadtonal fnsh-tme schedules do not. Fnally, t s shown that all the vtual tme ate-based schedules consdeed can poduce solaton poblems ove multple hops n netwos wth hgh loadng. It becomes appaent that the benchma Fst-Come-Fst-Seved schedule, wth spacng and call admsson contol at the netwo ngesses, s a pefeed aangement fo eal-tme flows (although lowe poty levels would also need to be mplemented fo dealng wth othe data flows). v

6 Table of Contents Chapte 1. Intoducton Bacgound Factos Affectng Qualty of Sevce Types of Applcatons and Qualty of Sevce Metcs Delay Compason fo eal-tme Applcatons Context of eseach Schedules Chaactezaton of ate-based Schedules Impotance of ths Wo Summay of Contbutons Wo fom ths Thess Publshed by Autho Oganzaton of Thess 16 Chapte 2. Confomng Schedules and Confomance Condtons Intoducton Defntons and Notaton Geneal Notaton fo Avng Pacets eal Seve Hypothetcal Seve Sesson Seves Notaton Summay Confomng Schedules Hypothetcal Seve Condton Absolute Gadent Condton Fnsh-Tme Confomng Schedules Compang Fnsh Tmes n the HS and SS Compang Fnsh Tmes n the S and HS Compang Fnsh Tmes n the S and SS Stat-Tme Confomng Schedules elatng Stat Tmes n the HS and SS elatng Stat Tmes n the S and HS elatng Stat Tmes n the S and SS Queueng Latences ove Multple Hops Ealy Pacet Avals Cascadng Two Nodes Cascadng Multple Nodes 58 v

7 2.7 Concluson 59 Chapte 3. The Vtual Tme Functon and ate-based Schedules Intoducton Behavou of ate-based Schedules Queueng Latences fo Confomng Schedules Potected ate of Sevce Confomng Schedule Condtons Elgblty, Steady State and Tanston State Steady State Sevce Non-confomng Schedules Example Schedules VtualCloc Schedules: VS-VCS and VCS GPS Schedules: VS-GPS and PGPS Self-Cloced Fa Queueng (SCFQ) Obsevatons Concluson 79 Chapte 4. Geneal ate-based Schedules Intoducton Outlne of Devaton Notaton Queueng Latency of Stat-Tme Schedules The Poblem No Pacets Seved Out of Ode by S Pacets Seved Out of Ode by S Queueng Latences Confomng Schedules Altenatve Bound Applcaton to Non-Confomng Schedule SFQ Queueng Latency of Fnsh-Tme Schedules The Poblem No Pacets Seved Out of Ode by S Pacets Seved Out of Ode by S Queueng Latency Queueng Latences Confomng Schedules Applcaton to Non-Confomng Schedule SCFQ Obsevatons on Queueng Latences 121 v

8 Chapte 5. Isolaton Chaactestcs of ate-based Schedules Intoducton Study Pocedue Taffc Geneaton Data Collected Scenao Paametes Smulaton Analyss esults and Analyss Benchma Schedule: FCFS Vayng Bustness Vayng Composton Bustness Atefact Stat-tme ate-based Schedules Vtual Space VtualCloc Sevce (VS-VCS) Vtual Space Genealzed Pocesso Shang (VS-GPS) Stat-tme Fa Queueng (SFQ) Fnsh-tme ate-based Schedules VtualCloc Sevce (VCS) Pacet-based Genealzed Pocesso Shang (PGPS) Self-Cloced Fa Queueng (SCFQ) Genealzatons and Conclusons equements of eal-tme Flows Stat-tme vesus Fnsh-tme ate-based Schedules Hgh-ate and Low-ate Flows Isolaton fom Bacgound Bustness Isolaton fom Bacgound Composton Detemnstc and Pobablstc Delay Bounds VCS vesus GPS Vtual Tme Functon Effect of Bacgound Bustness Multple Hops ate-based Schedules vesus FCFS 29 Chapte 6. Concluson and Futue Wo ate-based Schedules fo eal-tme Communcatons An Altenatve Achtectue fo Futue Detaled Study Futue Wo Centalzed ate-based Schedules VS-GPS Futhe Smulatons 219 v

9 6.3.4 Futhe Delay Bound Analyses Faness and Steady State 219 Appendx 22 Glossay 22 efeences 227 v

10 Lst of Fgues Fgue 2.1 epesentaton of the eal seve. 25 Fgue 2.2 epesentaton of the hypothetcal seve n vtual tme. 27 Fgue 2.3 epesentaton of a sesson seve. 28 Fgue 2.4 An llustatve example showng the elatonshp between SS and S quanttes. 3 Fgue 2.5 Compang sesson baclog peods n eal tme n the SS and HS. 38 Fgue 2.6 Example pacet epochs showng devaton between S and HS. 41 Fgue 2.7 Example of data sevce n the HS n an S busy peod. 49 Fgue 4.1a Bound on addtonal delays above those of confomng schedules. 124 Fgue 4.1b Hypothetcal seve sevce ate exceeds ln capacty, whee ρ =. 124 ψ ( v) Fgue 5.1 Bacgound pacet aval fo souce wthn a cycle. 129 Fgue 5.2 FCFS wth Vayng Bustness, One Hop, H = 3%, L= 5%. 142 Fgue 5.3 FCFS wth Vayng Bustness, Ten Hops, H = 3%, L= 5%. 143 Fgue 5.4 FCFS wth Vayng Composton, One Hop, y = Fgue 5.5 FCFS wth Vayng Composton, Ten Hops, y = Fgue 5.6 VS-VCS wth Vayng Bustness, One Hop, H = 3%, L= 5%. 15 Fgue 5.7 VS-VCS wth Vayng Bustness, Ten Hops, H = 3%, L= 5%. 153 Fgue 5.8 VS-VCS-FM wth Vayng Bustness, One Hop, H = 3%, L= 5%. 155 Fgue 5.9 VS-VCS-FM wth Vayng Bustness, Ten Hops, H = 3%, L= 5%. 156 Fgue 5.1 VS-VCS-FM wth Vayng Bustness, Ten Hops, H = 3%, L= 48%, T = 3%. 158 x

11 Fgue 5.11 VS-VCS-FM wth Vayng Bustness, Ten Hops, H = 48%, L= 45%, T = 3%. 159 Fgue 5.12 VS-VCS wth Vayng Composton, One Hop, y = Fgue 5.13 VS-VCS wth Vayng Composton, Ten Hops, y = Fgue 5.14 VS-VCS wth Vayng Composton, Ten Hops, y = Fgue 5.15 VS-VCS-Fnte Memoy wth Vayng Composton, Ten Hops, y = Fgue 5.16 VS-GPS wth Vayng Bustness, One Hop, H = 3%, L= 5%. 164 Fgue 5.17 VS-GPS wth Vayng Bustness, Ten Hops, H = 3%, L= 5%. 168 Fgue 5.18 VS-GPS wth Vayng Bustness, Ten Hops, H = 48%, L= 45%, T = 3%. 17 Fgue 5.19 VS-GPS wth Vayng Composton, One Hop, y = Fgue 5.2 VS-GPS wth Vayng Composton, Ten Hops, y = Fgue 5.21 SFQ wth Vayng Bustness, One Hop, H = 3%, L= 5%. 174 Fgue 5.22 SFQ wth Vayng Bustness, Ten Hops, H = 3%, L= 5%. 175 Fgue 5.23 SFQ wth Vayng Bustness, Ten Hops, H = 48%, L= 45%, T = 3%. 176 Fgue 5.24 SFQ wth Vayng Composton, One Hop, y = Fgue 5.25 SFQ wth Vayng Composton, Ten Hops, y = Fgue 5.26 VCS wth Vayng Bustness, One Hop, H = 3%, L= 5%. 181 Fgue 5.27 VCS wth Vayng Bustness, Ten Hops, H = 3%, L= 5%. 187 Fgue 5.28 VCS-FM wth Vayng Bustness, Ten Hops, H = 3%, L= 5%. 188 Fgue 5.29 VCS wth Vayng Composton, One Hop, y = x

12 Fgue 5.3 VCS wth Vayng Composton, Ten Hops, y = Fgue 5.31 VCS-FM wth Vayng Composton, Ten Hops, y = Fgue 5.32 PGPS wth Vayng Bustness, One Hop, H = 3%, L= 5%. 193 Fgue 5.33 PGPS wth Vayng Bustness, Ten Hops, H = 3%, L= 5%. 196 Fgue 5.34 PGPS wth Vayng Composton, One Hop, y = Fgue 5.35 PGPS wth Vayng Composton, Ten Hops, y = Fgue 5.36 SCFQ wth Vayng Bustness, One Hop, H = 3%, L= 5%. 2 Fgue 5.37 SCFQ wth Vayng Bustness, One Hop, H = 3%, L= 5%. 21 Fgue 5.38 SCFQ wth Vayng Composton, One Hop, y = Fgue 5.39 SCFQ wth Vayng Composton, Ten Hops, y = x

13 Lst of Tables Table 2.1 Pacet notaton used n ths chapte. 31 x

14 1.1 Bacgound 1. Intoducton 1.1 Bacgound Snce the ncepton, pacet-swtched netwos have been used to tanspot data fo a vaety of applcatons. One of the majo tends n the feld of communcatons engneeng n ecent tmes has been the polfeaton n the numbe and type of use applcatons that opeate acoss these netwos. Even applcatons, such as telephony, that have tadtonally been the peseve of ccut-swtched netwos le the Publc Swtched Telephone Netwo, ae wdely beng mgated to pacet netwos, an nevtable consequence of the ubquty of IP netwos. Ths convegence of communcatons onto a sngle type of netwo, usng common potocols and nfastuctue, ases an mpotant ssue: how to smultaneously satsfy the dvese equements of vaous applcatons. The tem qualty of sevce (QoS) may be defned as the degee to whch a use of a netwo eceves sevce that satsfes hs, he o ts equements. It follows mmedately that the QoS enjoyed fom a netwo s applcaton-specfc, snce dffeent applcatons can have dffeent equements, and the elatve mpotance of each of these equements can also vay Factos Affectng Qualty of Sevce Thee ae many factos that can affect the QoS expeenced by an applcaton whose pacets ae tanspoted acoss a netwo. These nclude: () () factos that elate to the avalable netwo hadwae; and factos that elate to the way netwo esouces ae allocated among the uses. The chaactestcs and qualty of the netwo hadwae necessaly nfluence the way n whch an applcaton s pacets ae tanspoted to the destnaton. The quantty of esouces avalable n the netwo, fo example ln capacty, buffe capacty, and pocessng speed at ntemedate nodes, affect the numbe of pacets Chapte 1 1

15 1.1 Bacgound that can be tanspoted by the netwo wthn a gven peod of tme. If the netwo s povsoned moe geneously, each applcaton can expect bette sevce fom t when equed. Smlaly, the chaactestcs of the tanspot medum employed by the netwo can nfluence netwo pefomance. Loss chaactestcs and popagaton delays vay fom medum to medum. Fo a gven netwo, these hadwae factos eman fxed. The othe goup of factos that affect QoS ase because the netwo must be shaed by a numbe of uses, each of whch may smultaneously be executng one o moe applcatons. The quantty of esouces allocated to each applcaton depends on the numbe of uses and applcatons employng the netwo and each patcula ln, and the amount of taffc geneated by them, at the elevant tme. Moeove, the way n whch the avalable esouces ae allocated among these uses and applcatons depends on the chaactestcs of the offeed taffc (fo example, ts egulaty o bustness), and the elatve demand fo esouces by ndvdual applcatons wth espect to each othe (fo example, the ates at whch applcatons ae sendng taffc to the netwo, o whethe any souces ae demandng an excessve amount of sevce). Futhemoe, QoS s mpacted by the topology of the netwo (the numbe of hops equed fo an applcaton to tavese the netwo, and the dstbuton of taffc on lns n the netwo). Ths thess s elevant to the ssue of managng the mpact of ths second goup of factos on QoS Types of Applcatons and Qualty of Sevce Metcs Havng establshed that QoS depends on the type of applcaton beng consdeed, we now boadly classfy types of applcaton taffc and dentfy elevant netwo metcs fo quantfcaton of QoS. The bass of classfcaton that s elevant to ths wo s senstvty to delay. All applcaton taffc tavesng a netwo s senstve to delay to some extent: n most cases ealy delvey of data to the destnaton endpont s at least as good as, f not bette than, late delvey, but n some cases excessvely ealy delvey can also be a poblem (when avalable buffe esouces mpose estctons). Fo ou puposes, senstvty to delay can be consdeed n two contexts. Fst, some applcatons eque tmng specfed at the souce to be epoduced at the Chapte 1 2

16 1.1 Bacgound destnaton. Delays ntoduced n the netwo can ntefee wth achevng ths. Secondly, dffeent applcatons have dffeent equements fo nteacton between the uses at the endponts. Some applcatons eque no nteacton at all, othe than pehaps to nque of the souce f the data s not eceved n a easonable amount of tme. Fo othe applcatons, tanspotaton though the netwo needs to facltate nteacton, but ove longe peods of tme and possbly afte ecevng lage quanttes of data. Fnally, stll othe applcatons eque mmedate nteacton between uses afte ecevng small quanttes of data (possbly as small as a pacet). Data applcatons can have vayng equements acoss both of these classfcatons. Some applcatons, such as FTP fle tansfes, eque no nteacton between end uses; the only equement s fo bul data to be tansfeed n as shot a tme as possble. If delays n complete tansfe ncease above use expectaton, thee s a declne n QoS expeenced, but the applcaton can stll functon and meet ts most mpotant goal: the eventual elable tansfe of the fle. Othe applcatons, such as emal, web bowsng, nstant messagng and emote temnal access, eque data tansfe to facltate use nteacton ove vaous tme scales, wth emal geneally allowng fo the slowest esponse tmes (pehaps hous o even days), and web bowsng, nstant messagng o emote temnal access cayng wth t the expectaton of moe mmedate esponse tmes (seconds to pehaps mnutes). Hee nceased delays have a moe dect detment on QoS, but once agan the man goal of the applcaton can stll be acheved: nteactve communcaton that can be modfed based on the QoS expeenced. epoducng the tmng specfed by the souce, but wthout use nteacton, s necessay fo steamng applcatons such vdeo-on-demand. Snce thee s no use nteacton, thee s no had lmt on delays, although a use s pecepton of QoS does deteoate wth excessve delays. Howeve, the need fo epoducton of the specfed tmng maes t desable fo the ate of tansfe and vaaton n delay though the netwo to be elably nown. If such quanttes can be detemned, t allows the destnaton endpont to buffe a suffcent amount of data to begn playout befoe tansfe s complete, based on the pedcted tmng of the aval of futue pacets. Howeve, f delay vaaton cannot be lmted, the applcaton can stll functon by commencng play-out afte the ente fle has been tansfeed. Whle ths Chapte 1 3

17 1.1 Bacgound epesents seously compomsed QoS, the tansfe of the data though the netwo stll contbutes to the goals of the applcaton. Fnally, thee s a goup of applcatons that eques fathful epoducton of the souce s tmng and the fastest level of human esponse (of the ode of less than a second). These applcatons nclude telephony ove IP, vdeo ove IP and netwo gamng, and ae descbed as eal-tme applcatons. Fo these applcatons, any delay n data tansfe beyond a cetan pont completely negates the value of the tansfe. Such pacets cannot be used at the destnaton, and must be dscaded fo the applcaton to contnue functonng. The nvaldaton of late pacets esults fom a combnaton of the equements of tmng epoducton and mmedate nteacton. Ths thess s focused on how the QoS equements of eal-tme applcatons can be met n a netwo wth dvese taffc types. Netwo metcs elevant to quantfcaton of QoS fo applcatons patly depend on the foegong classfcaton. Howeve, two metcs elevant to all applcatons ae thoughput and loss. The thoughput can be loosely defned hee to be the aveage ate of tansfe though the netwo fo a flow. Ths s mpotant fo applcatons to mnmze the amount of tme needed to tansfe bul data. Any nteactvty equed upon completon of the tansfe nceases the need fo hghe thoughput, and any ncease n thoughput coesponds to an mpovement n QoS. On the othe hand, applcatons that eque tmng epoducton need to match some undelyng ate that would enable the tmng to be epoduced. Any thoughput acheved above ths equement s of no eal value to the applcaton. The othe metc elevant to all applcatons s loss pobablty. Ths s defned as the popoton of pacets offeed by any flow that do not each the destnaton n a usable fom. Such losses may be due to lac of elablty of the communcaton medum, oveflow of netwo esouces, delay of a pacet beyond ts useful lfe (fo eal-tme applcatons), o a host of othe factos. Fo flows that allow fo etansmsson, a hgh loss pobablty educes the thoughput of usable data (nown as the goodput). Fo flows that do not allow fo etansmsson, netwo loss esults n loss of data dectly, wth consequent QoS ssues. Chapte 1 4

18 1.1 Bacgound Fo applcatons that eque tmng epoducton, the delay expeenced by ndvdual pacets though the netwo s of equal mpotance to goodput. It s of patcula mpotance to eal-tme applcatons. Delay can be quantfed n vaous ways, but two measues of elevance to eal-tme applcatons ae the maxmum delay (o uppe bound on pacet delay), and the jtte (o maxmum vaaton n delay). Both of these can be expessed n absolute detemnstc tems, o n pobablstc tems. Fo eal-tme applcatons, the level of nteactvty between end uses mposes an acceptable maxmum delay (fo a gven QoS), whch s elated to the maxmum amount of tme a use wll wat to eceve a esponse. Also, the need fo tmng epoducton mposes a lmt on jtte, because thee ae lmts to the buffeng esouces avalable at the destnaton. If delay vaaton become too lage, the destnaton buffes can oveflow (esultng n data losses), o undeflow (esultng n a gap n the flow). It should be noted that, snce the mnmum delay possble s zeo, the jtte s bounded by the maxmum delay. Thus, the QoS constants fo each eal-tme applcaton flow can be specfed by just the equed goodput and maxmum delay. Howeve, t should be emphaszed that the maxmum acceptable delay fo a eal-tme applcaton s not elated to the goodput equed, but on the nteactvty nheent to the applcaton s functonalty. Fo example, a vdeo ove IP applcaton eques a much lage bandwdth than telephony ove IP, yet ts delay equements (on each and evey pacet) ae no moe stngent, because the level of nteactvty s exactly the same. An mpotant chaactestc of a netwo fo eal-tme applcatons s theefoe how well t s able to mae the delays expeenced by a flow ndependent of the natue of the othe taffc shang the netwo. Ths chaactestc s efeed to as solaton n ths thess. The netwo chaactestc moe elevant to othe types of data applcaton s how well t can shae the avalable thoughput between competng flows n popoton to the ageed ates. Ths s efeed to as faness n the maxmn sense [25]. Faness s only elevant to applcatons fo whch addtonal thoughput can be used to complete the tansfe of bul data to the destnaton faste. Fo eal-tme applcatons, whch have an undelyng tmng stuctue, any addtonal thoughput above that needed to meet the tmng equements s unnecessay. By the Chapte 1 5

19 1.1 Bacgound same toen, a shotfall n povdng the thoughput equed fo epoducng the souce tmng can esult n the pemanent loss of data, and s unacceptable. Faness s theefoe elevant to eal-tme applcatons Delay Compason fo eal-tme Applcatons Fo all communcatons applcatons, the deal aangement fo tanspotaton of data s to have a dedcated ln connectng the souce and destnaton. Ths s patculaly tue fo a eal-tme applcaton, whose undelyng tmng can specfy a mnmum ln data speed (o ln capacty), and whose equements fo nteactvty specfy a maxmum delay. The pacet delay expeenced on an deal dedcated ln can be boen down nto thee components. These may be summazed as: () () () the spacng delay component; the tansmsson delay component; and the pocessng and popagaton delay component. The spacng delay component ases fom the fact that the souce can send data to the destnaton at a ate hghe than the ln capacty o ageed ate of tansfe. A pacet that aves at the ln befoe the pevous pacet can be tansmtted has to wat untl tansmsson s complete befoe t can be seved. Ths component s entely due to the chaactestcs of the souce. The tansmsson delay s the amount of tme equed to tansmt a dscete pacet at the ln tansmsson ate. Addtonal constant delays, such as those caused by pocessng at the tansmtte, and the tme equed to tavese the medum, mae up the last component. In the eal-wold stuaton of conveged netwos, tanspotaton of data taes place ove a common netwo sevcng many uses and applcatons. Ths means that many flows can be multplexed onto a sngle ln, and the ln s tansmsson capacty must be shaed between them. On a pacet-by-pacet bass, uses and applcatons must compete fo, and tae tuns n, the use of the esouces. The esultng need fo queueng of pacets, and a mechansm fo detemnng the ode n Chapte 1 6

20 1.2 Context of eseach whch those pacets ae to be seved (called a schedule) esults n an addtonal component of delay: (v) queueng delay component [8]. Ths last component s the delay expeenced by a pacet as t wats fo pacets fom dffeent flows to be seved by the common ln, albet at a faste ate than the dedcated ln. Anothe featue of shaed netwos s the fact that moe than one ln may need to be used to each the destnaton. In a stoe-and-fowad netwo (whch we assume fo ths wo), the delays at each hop accumulate addtvely; the tansmsson of a pacet on one ln must be completed befoe t can access the next ln. Thus, the total delay fom all hops befoe the last one becomes an addtonal component of the delay expeenced at the destnaton. When consdeng how well a netwo s able to meet the QoS equements of a eal-tme applcaton flow, an appopate measue s how much a pacet s sevce s delayed beyond the sevce epoch t would expeence at a dedcated ln popely povsoned fo that applcaton. 1.2 Context of eseach In a conveged netwo, the choce of pacet schedulng dscplne (o schedule) s cucal to whethe ndvdual flows eceve the equed QoS, because t affects the queueng delay component. The ode n whch competng pacets eceve sevce at a ln detemnes whethe each of them can be delveed to the destnaton wthn a tme fame appopate to ts espectve applcaton. Fo some data applcatons, a delay fo one of ts pacets, whle seveal othe pacets ae seved, does not mae a sgnfcant dffeence. Howeve, fo eal-tme applcatons thee ae stct delay equements that can be volated f one of ts pacets s caught behnd a lage numbe of othe pacets n a sevce queue. An appopate choce of schedule s theefoe patculaly mpotant to ensue that eal-tme applcatons can functon acoss a netwo. Chapte 1 7

21 1.2 Context of eseach Schedules Most of the avalable schedulng dscplnes fo output buffeed swtches use dstbuted abtaton methods; that s abtaton between pacets fo sevce at each ln s pefomed ndependently. A common and smple schedule used n data netwos s Fst-Come-Fst-Seved (FCFS), also nown as Fst-In-Fst-Out (FIFO). Ths dscplne seves pacets meely n the ode of the aval, and maes no decson on how to allocate tansmsson esouces between ndvdual flows. Flows eceve sevce n accodance wth the demand. If any flows send lage amounts of data nto the netwo wthn a shot peod of tme, othe competng flows can expeence longe delays as they wat fo these pacets to be seved fst. Ths schedule theefoe has poo solaton chaactestcs and, wthout addtonal contols beng placed on flows, cannot guaantee any QoS to eal-tme applcatons (o ndeed to any applcaton). Howeve, f each flow s spaced at an ageed ate comng nto the netwo, and the total amount of such taffc enteng the netwo s lmted to pevent congeston (admsson contol), FCFS s sutable fo all flow types. FCFS seves each flow accodng to ts demand, but the demand fom each flow wthn the netwo s lmted to an exact level. The advance of convegence n the last two decades has seen the poposal of a lage numbe of altenatve schedules that ae ntended to enable all types of flows to eceve an appopate level of QoS (see [3] and [1] fo a suvey of some basc types). The ound obn schedule s one that mantans a sepaate queue fo evey flow wth pacets awatng sevce, and t seves one pacet fom each queue n tun [4]. Evey flow wth watng pacets theeby eceves sevce n each ound. Ths concept s extended to allow fo dffeent amounts of sevce n a ound fo each nonempty queue wth Weghted ound obn (W) [2]. These schedules ensue that no flow wth watng pacets s staved of sevce fo any peod longe than a ound, and ntoduce the concept of faness. Any ncease n avalable bandwdth s accompaned by a decease n ound sze, so all queues n the ound shae n t popotonately. Howeve, delays ae dependent on the numbe of competng flows, as wth FCFS. Chapte 1 8

22 1.2 Context of eseach One dawbac of these schedules s that, wth vaable pacet lengths, the amount of sevce that each flow eceves n a ound s andom. Anothe poblem s that, although W can gve dffeent flows vayng amounts of sevce n a ound, that vaable allocaton of esouces s lmted by the need fo ntegal weghts. These poblems ae solved by the Weghted Fa Queueng schedule (WFQ) [3], whch genealzes the concept of the ound nto a contnuous functon, an example of the vtual tme functon. WFQ s a pacet mplementaton of the flud concept of Genealzed Pocesso Shang (GPS) [2] that allocates the avalable bandwdth esouces contnuously among competng flows accodng to ageed weghts assocated wth those flows. Ths concept, whch evolved fom unfom pocesso shang [7], povdes an deal fo faness, n popoton to those weghts, that WFQ attempts to emulate as closely as possble. Abtaton between pacets fo sevce n WFQ can be mplemented by means of a set of equatons that attbutes a paamete to each pacet. Ths paamete eflects the fnsh tme of that pacet f t wee seved by an magnay flud GPS seve (such sevce s not possble n eal netwos). A WFQ seve then seves watng pacets n nceasng ode of the fnsh-tme paamete. Ths mode of opeaton ensues that WFQ shaes, to wthn cetan bounds, the faness chaactestcs of GPS. Howeve, fo eal-tme applcatons the delay chaactestcs ae moe mpotant. It s establshed n [5] and [6] that WFQ, also nown as Pacet-based Genealzed Pocesso Shang (PGPS), delves a delay bound to a flow that deceases wth an ncease n the ageed weght of the flow. The weght attbuted to a flow s dectly popotonal to the ate of sevce t eceves, snce t epesents a popoton of bandwdth esouces. The delay bound equed fo a flow s not necessaly nvesely elated to ts thoughput equements, so ths means that meetng the delay equements could necesstate allocatng moe bandwdth to the flow than t eques. Ths s a seous poblem wth PGPS, because t would cause undeutlzaton of avalable bandwdth esouces. The eason fo the ate dependency of the delay bound n PGPS s eadly undestood. Snce PGPS s geaed towads attemptng to meet the fnsh tme fo sevce n the magnay GPS seve, flows wth lowe allocated ates automatcally have the pacets seved late than pacets fom hghe-ate flows that ave at aound the same tme. Chapte 1 9

23 1.2 Context of eseach An altenatve way of calculatng the fnsh-tme paamete fo PGPS s to use equatons that epesent sevce by the magnay GPS seve n vtual tme, whch s an altenatve tmelne that evolves at a ate that can vay fom eal tme. The defnton of vtual tme fo PGPS ases fom equatng the weght fo a flow wth ts actual equed thoughput (.e. the ate of sevce t would eque of a dedcated ln). The equatons epesent the sevce epochs whee each flow eceves the equed ate of sevce fom the GPS seve, but n vtual tme. If the total ate equed by all competng flows s less than the ln capacty, then the vtual tme fo PGPS evolves popotonately faste. The total sevce ate of the magnay seve n eal tme s theeby ept at exactly the same ate as the ln capacty, even though n vtual tme the sum of the sevce ate fo all competng flows s less than the ln capacty. Othe schedules have been poposed that use smla equatons, epesentng sevce n vtual tme by an magnay seve, fo abtatng between competng pacets fo sevce. The dffeence between these schedules s n the defnton of the vtual tme functon, the mathematcal functon of tme that defnes the evoluton of vtual tme. We call these schedules ate-based schedules. Two such schedules that we consde n ths thess ae VtualCloc Sevce (VCS) and Self-Cloced Fa Queueng (SCFQ). VCS s poposed n [9], and attempts to emulate Tme Dvson Multplexng (TDM) on a pacet netwo by usng a vtual tme defnton conguent wth eal tme. SCFQ, poposed n [1], s an attempt to appoxmate PGPS wth a smple algothm. The vtual tme functon of PGPS s dependent on the aval sequence of pacets, and can only be obtaned by the schedule by smulatng the magnay GPS seve. Ths complexty s emoved n SCFQ, whch esets the vtual tme functon to zeo when the ln s dle and specfes the vtual tme functon to be the vtual fnsh tme n the magnay seve of the pacet beng seved at the ln when t s busy. The vtual tme functon then becomes a step functon that s easly detemned wheneve a pacet aves at the ln. All of the ate-based schedules mentoned thus fa use abtaton based on the fnsh tme fo pacet sevce n an magnay seve. Consequently, they all shae the poblem of the couplng between the ageed sevce ate and delay bound fo a flow; hghe-ate flows automatcally have lowe delay bounds. A ate-based schedule that Chapte 1 1

24 1.2 Context of eseach uses stat tmes n the magnay seve to detemne the ode of pacet sevce s Stat-tme Fa Queueng, poposed n [11]. Ths schedule s n fact a stat-tme veson of SCFQ, usng a stat-tme paamete to abtate between pacets. The vtual tme functon s defned by the vtual stat tme n the magnay seve of the pacet beng seved at the ln. In ths thess we evaluate the sutablty of ate-based schedules to ensue appopate QoS specfcally fo eal-tme applcaton flows. Based on the concept of SFQ, we popose stat-tme vesons of both PGPS and VCS, whch we call Vtual Space Genealzed Pocesso Shang (VS-GPS) and Vtual Space VtualCloc Sevce (VS-VCS) espectvely. In elaton to delay, we examne the chaactestcs of both stat- and fnsh-tme ate-based schedules, and compae these to what can be acheved wth FCFS, usng admsson contol and spacng of flows at the netwo ngess ponts Chaactezaton of ate-based Schedules In ths thess, we popose that the chaactezaton of ate-based schedules fo ealtme applcatons must focus on the delays (assocated wth a gven ate) fo an abtay flow, and on solaton. We suggest that the concept of faness s moe elevant to othe types of data applcaton, and s of seconday mpotance to ealtme flows. In fact, the faness popety of a schedule can only assst a eal-tme flow when ts equed ate has been mschaactezed, and then the extent and tmng of those benefts cannot be pedcted. Faness benefts that accue to a flow ae entely dependent on the natue of the bacgound taffc, and ths s antthetcal to the vew that eal-tme flows should be well solated fom the vagaes of bacgound taffc. Much of the lteatue snce the poposal of the ognal ate-based schedules descbed n the pevous sub-secton loos at both the delay bounds and faness chaactestcs. Fom ths analyss, a wdespead assumpton appeas to be that these schedules ae appopate fo all types of taffc, ncludng eal-tme. In ths wo, we test that assumpton wth espect to eal-tme taffc. Chapte 1 11

25 1.2 Context of eseach To acheve ths, we undetae a two-pat analyss. We fst genealze fo the detemnstc delay chaactestcs of all ate-based schedules n tems of the defnng vtual tme functon. Based on ths, we mae obsevatons on how the vtual tme functon affects the behavou of ate-based schedules. We then pefom a compehensve smulaton study on the delay dstbutons obtaned fo a test flow fom the ognal ate-based schedules. Fom ths, we can mae assetons about the sutablty of stat-tme and fnsh-tme ate-based schedules fo eal-tme flows n compason to FCFS when thee s admsson contol (smla to that equed fo ate-based schedules anyway) and spacng of flows. The genealzatons fo delays bounds ae based on the delay bounds obtaned fo specfc schedules n [5] and [6] fo PGPS, n [12] fo VCS, n [13] fo SCFQ and n [11] fo SFQ. The concepts developed n [5] enable sevce tmes at a seve employng PGPS to be elated to the sevce tmes n the magnay GPS seve by mang use of ctcal tmes at whch sevce n the two seves can be elated, such as when a seve becomes busy, o when thee s a dvegence n the ode of sevce between the two seves. Much of the late lteatue uses these concepts to develop delay bounds, and we use them n obtanng ou genealzatons. In [5], [12] and [13], the absolute delay bounds ae deved n elaton to a leay bucet constant. In ou wo, we choose nstead to mae use of the concept of latency developed n [14] and [15]. The latency fo a schedule of a specfed class (to whch ate-based schedules belong) s defned as the maxmum lag n the tme taen fo the schedule to allow sevce fo a patcula quantum of data compaed to the tme when a dedcated ln sevng at an allocated ate fo the flow would seve that data. Data s consdeed to be seved only when ts ente pacet has been completely seved, so the cuve fo sevce s a step functon. Howeve, sevce n the dedcated seve s defned to be contnuous. We use ths concept because t elates delays to sevce at a dedcated ln, an appopate compason fo a eal-tme flow. Howeve, latency as defned specfes the maxmum lag fo the complete sevce of a pacet afte that pacet begns sevce on the dedcated ln. We modfy ths to specfy the maxmum lag between a pacet s abtated stat tme and ts stat tme on the dedcated ln. Ths defnton Chapte 1 12

26 1.2 Context of eseach allows us to solate the queueng delay component, whch s the only component that the schedule can affect. It excludes both the spacng delay component, due only to the egulaty of the souce, and the tansmsson delay component, dependent only on the ln capacty and pacet lengths. We note that ths analyss easly leads to delay bounds fo leay-bucet constaned flows. Fundamental wo on the quantfcaton of netwo delay bounds s found n the semnal papes [16] and [17], ntoducng the feld of netwo calculus. The feld s futhe developed n [19], [23] and [24], and appled to quantfyng delay bounds n [18], [21] and [22]. Netwo calculus ntoduces the concepts of the aval and sevce cuves, and delay beng bounded by the maxmum devaton between them. Ths motvates some of the technques used n ths thess to genealze delay bounds fo ate-based schedules. Also, the delay beng bounded by the maxmum devaton between two cuves occus n ou wo wth espect to the vtual tme functon. A method smla to that n [16] of ntegatng aval ates s used extensvely n ou devatons. The second method of chaactezaton of the delay popetes of ate-based schedules undetaen n ths thess s by smulaton study to obtan delay dstbutons. The goal of ths wo s to study the mpact of bacgound taffc chaactestcs (patculaly bustness and ate composton) on the delay dstbuton of a flow fo the ognal fnsh-tme ate-based schedules (PGPS, VCS and SCFQ), and the stat-tme countepats (VS-GPS, VS-VCS and SFQ). Fom these esults, we deduce the solaton chaactestcs of these schedules. The ssue of how to model the bacgound taffc, patculaly the bustness, needs to be addessed. One opton fo modellng bustness s as n the statstcal analyss of GPS pesented n [26], whch uses an exponentally bounded bustness (EBB) taffc model suggested n [27], nstead of a fxed bound on bustness. The model s used n that pape to statstcally obtan delay dstbutons fo a smple topology. Fo ou puposes we do not use ths to model taffc bustness because we wsh to detemne how changes n bustness affect the dstbuton. We theefoe mantan a fxed bustness fo each smulaton, so that the effect of dffeent levels of bustness s appaent. We employ the same appoach fo bacgound ate composton. Chapte 1 13

27 1.2 Context of eseach We use Monte Calo smulatons to develop an ensemble fo bacgound taffc, based on the deas n [28] and [29]. In those papes, t s establshed fo peodc bacgound steams n a non-congested netwo, that the aval patten fo one pecedng flow peod s all that s elevant to the delay fo a test flow. Thus, by samplng dffeent aval pattens wth each new peod, an ensemble can be developed. Ths appoach s modfed fo ou puposes to allow fo bustness by samplng the delays ove a cycle of seveal peods befoe changng the aval patten. Ths enables the bacgound bustness to affect the test flow befoe the aval patten s changed. Thus, each sample path s followed fo a one cycle befoe a new aval patten s mposed Impotance of ths Wo The mpotance of ths evaluaton of ate-based schedules s clea n lght of the amount of publshed wo consdeng these schedules. The ate-based schedules n ths thess ae also nown n the lteatue as fa queueng algothms [1, 13], and the fnsh-tme vesons ae ncluded n the class of ate-popotonal Seves [32-33] and the class of guaanteed ate schedulng algothms [35, 44]. A consdeable amount of effot has been devoted n the lteatue to analyss of these goups of schedules. Thee s also contnung analyss [45, 46] and smulaton study [47] of PGPS n patcula. Futhemoe, thee have been many poposed modfcatons to basc ate-based schedules, such as [39], [42], [48] and [49] modfyng WFQ, [5] and [51] modfyng SCFQ, and [52] modfyng VCS. The focus of much of ths lteatue s on quantfyng faness and bounds on delay. Ths tends to ceate the assumpton that ate-based schedules ae nheently sutable fo dealng wth both eal-tme and othe data taffc types. We evst ths assumpton by tunng the focus away fom faness, whch s lagely a dstacton when consdeng eal-tme flows. The man focus of ths thess s on delay. The statng pont fo ths wo s wth delay bounds, based on pevous wo n the aea. Howeve, because these bounds tend to be extemely lage compaed to commonly expeenced pacet delays, the study has to poceed futhe to tae nto account the delay dstbutons. Much of the po wo nvolvng smulatons have used measues such as aveage delays and aveage bandwdth [43], whch ae clealy not Chapte 1 14

28 1.3 Summay of Contbutons suffcent fo eal-tme flows, o have attempted to fnd wost-case delays based on the smulatons [42, 48, 47]. Pobablstc analyses have tended to mae assumptons about aval dstbutons [26, 46]. A majo contbuton of ths thess s to undetae a compehensve smulaton analyss of some of the ognal ate-based schedules poposed, as well as some newly poposed ones. Thee does not appea to be such a complete nvestgaton of the mpact of bustness and bacgound ate composton on the delay dstbutons poduced by dffeent ate-based schedules. We use deas fom [28] and [29] to develop Monte Calo smulatons that do not assume an aval dstbuton. Bustness and ate-composton ae ept fxed wthn each smulaton to allow detaled obsevaton of the mpact of dffeent levels of bustness on each schedule, allowng compasons of how each schedule behaves n dffeent ccumstances. An solaton measue based on delay dstbuton s developed n ths thess. Ths s poposed as a moe appopate measue than one based on delay bounds fo evaluatng the sutablty of a schedule fo eal-tme applcatons. The solaton esults fo ate-based schedules lead to an altenatve achtectue. Ths s based on the demonstated unsutablty of ate-based schedules fo eal-tme flows. Thee has been po wo suggestng modfcatons to WFQ to decouple allocated ate fom delay bounds. Some of the technques poposed, such as poty classes [42] and polcng [48] ae smla to those suggested by us. Howeve, those poposals stll use a ate-based schedule fo eal-tme flows, whch ou wo would ndcate may not be deal. 1.3 Summay of Contbutons The ey contbutons of the eseach pesented n ths thess ae as follows: I. Ognal geneal delay bounds ae deved fo stat-tme and fnsh-tme atebased schedule n tems of the vtual tme functons. These bounds ae pesented n tems of the queueng latency, a concept developed to solely eflect the queueng component of delay. Ths s a useful measue fo evaluatng the pefomance of schedules fo eal-tme taffc. Chapte 1 15

29 1.4 Wo fom ths Thess Publshed by Autho II. We genealze fo a class of ate-based schedules whose vtual tme functons satsfy confomance condtons that ensue they exhbt the mnmum latency possble. Ths s acheved usng an ognal appoach nvolvng the use of an magnay seve assocated wth the vtual tme functon. Useful nsghts nto the effect of the vtual tme functon on schedule behavou ae theeby obtaned. Also, the mpact of volaton of the confomance condtons by the vtual tme functon s dscussed. III. Usng a new and useful measue of solaton n tems of a FCFS benchma, we pesent a compehensve smulaton study of the solaton chaactestcs of a ange of ate-based schedules. We demonstate that, despte ndcatons to the contay fom delay bounds, the delay dstbutons of stat-tme schedules can possess solaton chaactestcs that fnsh-tme schedules do not have. Insght s also developed nto how the vtual tme functon mpacts on solaton. IV. It s demonstated that all ate-based schedules can poduce solaton poblems ove multple hops of a netwo, whch ae nheent to the dstbuted natue. We suggest an altenatve achtectue nvolvng FCFS wth admsson contol and spacng that elmnates these and othe poblems caused by ate-based schedules. 1.4 Wo fom ths Thess Publshed by Autho The ealy wo on genealzatons fo ate-based schedules n ths thess has been publshed n the followng pape: [31] T. N. Devadason and G. Mecanos, Obsevatons on a Class of atebased Schedules, 11th IEEE Intenatonal Confeence on Netwos, Sydney, Austala, pp , Oganzaton of Thess Chapte 2 pesents genealzatons fo a class of stat-tme and fnsh-tme ate-based schedules (confomng schedules). Stat-tme schedules of ths class possess a common queueng latency, and fnsh-tme schedules of ths class shae a sepaate common queueng latency. The popetes of ths class ae abstacted fom the poofs Chapte 1 16

30 1.5 Oganzaton of Thess of delay bounds fo specfc schedules. Sevce epochs n an magnay hypothetcal seve opeatng n vtual tme fom the bass of abtaton fo these schedules, and actual sevce epochs ae elated to sevce epochs n the hypothetcal seve. Sevce epochs n the hypothetcal seve ae then elated to sevce epochs n a seve dedcated to the elevant flow wth an ageed ate. The queueng latency s then obtaned. The deas developed n Chapte 2 lead logcally to nsghts nto how the vtual tme functon mpacts on the behavou of the schedule. These obsevatons ae pesented n Chapte 3. The concepts of steady and tansent states ae developed, and the mpact on faness s befly consdeed. Ths ssue s then elated to the specfc atebased schedules consdeed n ths thess. In Chapte 4, delay bounds ae deved fo ate-based schedules wth geneal vtual tme functons. Ths wo uses some of the deas fom netwo calculus. Ognal expessons fo queueng latences ae deved n tems of the vtual tme functon fo stat- and fnsh-tme schedules. Ths wo s dstngushed fom pevous wo because the bounds can easly be appled to exstng and new schedules to obtan queueng latences. Also, the ctcal events that cause maxmal delays can be dentfed fom the expessons. Confomng schedules ae shown to have the mnmum queueng latences possble. Chapte 5 pesents a compehensve smulaton analyss of delay dstbutons expeenced by a test flow unde vaous ate-based schedules. Isolaton fom bacgound taffc chaactestcs s the focus of ths study, wth a benchma based on FCFS beng poposed. The elatve mets of stat-tme and fnsh-tme schedules wth espect to delay dstbutons ae examned. The mpact of bacgound bustness and ate composton on test flow delay dstbutons ae consdeed. The effect of the patcula vtual tme functon on solaton chaactestcs s also obseved. Isolaton poblems that ase fo all ate-based schedules consdeed ove multple hops become evdent fom ths wo. The thess s concluded wth Chapte 6, whch evaluates the assumpton that atebased schedules ae appopate fo delveng QoS to eal-tme flows n a conveged netwo. A compason s made wth an achtectue nvolvng FCFS Chapte 1 17

31 1.5 Oganzaton of Thess schedulng, and some poblems wth ate-based schedules ae hghlghted. Fnally, aeas fo futue wo ae befly mentoned. Chapte 1 18

32 2.1 Intoducton 2. Confomng Schedules and Confomance Condtons 2.1 Intoducton As dscussed n Chapte 1, a numbe of ate-based schedules have been poposed to facltate tansfe of eal-tme flows, as well as tadtonal data flows, ove conveged communcatons netwos. They ae ntended to facltate the povson of an ageed mnmum ate of sevce to each connecton, when equed, subject to a bounded ncemental delay at each node along ts data path. It s thus poposed that the specal equements of eal-tme applcatons can be met, whle smultaneously sevcng data fom connectons that ae not delay senstve. The goal of ths chapte s to unfy esults obtaned egadng delay bounds fo a numbe of schedules. We fst consde the VtualCloc Sevce (VCS) [9], [12] and the Pacet-by-pacet Genealzed Pocesso Shang (PGPS) [5] schedules. These schedules have smla delay bounds (see summay n [15]), although they may ode pacets dffeently. The ctcal popetes that detemne the delay bounds ae dentfed and specfed. We elmnate schedule-specfc popetes used n [12] and [5] that ae extaneous, and theeby genealze fo an ente class of atebased schedules that shae the same delay bounds (whch ae shown n Chapte 4 to be mnmal). In addton, two new schedules nown as Vtual Space based on VCS (VS-VCS) and Vtual Space based on GPS (VS-GPS) ae descbed and ncluded n the dentfed class. These schedules ae smla to VCS and PGPS espectvely, but use a dffeent quantty fo abtaton amongst pacets and theefoe have dffeent delay bounds, whch ae obtaned n ths chapte. We defne the concepts of stat-tme and fnsh-tme schedules based on the quantty used fo abtaton, to futhe categoze ate-based schedules. The possble benefts of these poposed stat-tme schedules ae dscussed and nvestgated n late chaptes. Chapte 2 19

33 2.1 Intoducton Pevous wo genealzng fo a class of schedules that nclude PGPS and VCS has been publshed n [32-33]. The condtons defnng confomng schedules n ths chapte ae equvalent to those defnng ate-popotonal seves (PSs) n [32], as the system potental n that pape coesponds to the vtual tme functon fo atebased schedules. Howeve, the man focus of that wo s faness, although a delay bound s obtaned n the elated thess [34]. In ths wo, the objectve s an evaluaton of schedules elatng to eal-tme flows, and so delay s the focus. The esults fo PSs ae only applcable to fnsh-tme schedules, whle ou genealzatons ae extended to stat-tme schedules. Also, the genealzed methods fo quantfyng bounds usng an ntemedate magnay seve that ae developed n ths chapte ae ognal. These ae developed futhe n chapte 4 to cove both statand fnsh-tme schedules that do not fall wthn the puvew of PSs. Fo the puposes of ths analyss, we use a quantty based on the latency concept fom [14-15] as a measue of delay at a sngle node, to compae dffeent schedules. Howeve, the quantty used s modfed to ntentonally exclude the pacet tansmsson tme and focus on the queueng delay component. The queueng delay component s a functon of competton between sessons, but pacet tansmsson tme s dependent only on the pacet length and outgong ln ate. Once a detemnaton s made to tansmt a pacet, the tansmsson tme s ndependent of the schedule, and does not contbute to ts chaactezaton. The defnng popetes dentfed n ths chapte elate specfcally to the vtual tme functon, and we theefoe consde ts mpact on the behavou of a schedule. By mang use of and extendng some of the technques ntoduced n [5] and followed n [12], delay bounds fo all membes of the class ae deved. The concept of a hypothetcal seve s ntoduced to elucdate the way n whch the vtual tme functon affects delay bounds, sevce ate and faness. It s found that the ate of sevce delveed by a schedule depends on the gadent of the vtual tme functon. We also demonstate that the latency fo a schedule at a node s affected by the length of tme and amount by whch cetan vtual tme popetes ae not satsfed by the schedule. It s obseved that schedules wth vey dffeent faness popetes (n tems of allocaton of excess bandwdth) can have smla delay bounds. Chapte 2 2