Metamodeling : a study of approximations in queueing models

273 Pages·1984·37.9 MB·English

by  Subhash Chandra Agrawal.

METAMODELING
MIT PreSsesr iiens Computer Systems
Herb Schwetemdaint,o r
Ah-JUIIlUddSitflgu:do yfA ApproximinaQ tuieoufeMliosnd ge lsS,ub hbaysh Chandra
Agrawa1l9,8 5
METAMODELING
A Studoyf A pproximatiinoQ nuse ueMiondge ls
SubhaCshha ndArgar awal
The M PIrTe ss
Cambridge, Massachusetts
LondonE,n gland
If) 1985 by The MaI�nssatciohtfuu steet ts
Technology
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direcftrloytm h aeu thorc'asm era-recaodpy y.
LibraoryfC ongreCastsa logiinnP gu blication
Data
AgrawaSlu.b haCshha ndra.
Metamodela isntgu:do yfa pproximaitni ons
queueimnogd els.
(ThMeI Ts eries in computer systems)
Bibliograpp.h y:
Incluidnedse x.
I.D igictoamlp uter simuEllaetcitornon.i c2 .
digitcaolm puters-Evalu3at.Qi uoenu.e itngh e­
ory.4 . ApproximatioTni ttleh.e ory. I.
11. Series.
QA76.9.C16958A503 081 .6844 ·19456
ISBN0 ·262I· 00 80-1
ISBN: 0-262(-p51a1p4e4r-4b ack)
To my parents
and
tot heme mory ofm y grandmother
CONTENTS
Figures IX
Boxes xi
Tables xiii
SeriFeosr eword xv
Preef ac XVII
1 Introduction
1.1Q ueueiNnegt woMrokd els 3
1.1.S1i ngRlees ouMrocdeel s 3
1.1.2Q ueueNientgwo rMko dels 4
1.1M.u3l tiWpolrek loads 5
1.10P4r oducFto rmS olution 5
1.1.F5a sCto mputatioAnlaglo rithms 8
1.1.R6o bustnaensdsA ccuracy 8
1.1.7 History 10
1.2A pproximMaotdee lofsC omplCeoxm putSeyrs tems 12
1.2.G1e neSrearlv iTciem eD istribuattai noF nC FS 13
1.2.2M emory Queueing 16
1.2.S3i multaneRoeusso urPcoes sessiino nI S/uOb systems 18
1.20P4r oiritSyc heduling 19
1.2.S5e rialization 19
1.2.I6n ternal PrCoognrcaumr rency 22
1.2.B7l OCking 23
1.2A.p8p roximaAtnea lysoifLs a rge Product Networks 25
13. Plaonft heM onograph 29
14. RelatWeodr k 33
2 TheS tructouftr heeM odeling Process 37
2.1D asiMco deliPnrgo cess 37
2.2S yntax ofC omDpaosniecon fta s T ransformation 40
2.3E xamp1l:eS ingSleer vQeur eue 43
2.4E xampleQ u2e:u eNientwgo Mrokd el 45
25. Examp3l:eS evcik's CSPhUa doMwo del 47
CONTENTS vii
2.6E xampl4e: D ecompositwiiotonhu tI teration 49
2.7E xampl5e: D ecompositiwoint Iht eration 50
2.8 Examp6l:eM odelUss inRge cursion 55
2.9H ierarcohfMy o delasn dI tSso lution 57
2.10C omparison withE xampleosfH ierairccahl
Modelinign L iterature 61
2.1E1q uatiOorni entAepdp roximations 64
2.11C.o1ur toiMse'tsh oodf A ggregation 65
2.11.D2i ffusiAopnp roximation 67
2.12C onclusion 69
3 Beha\S'eiqoure nTcrea nsformatiMoondse lwsai ntSdhh adoSwe ners 73
3.1 Background 74
3.2 ExamplIe: P reemptiPvrei orSicthye duling 76
3.2C.l1a sLs C ustomerBe'hsa voriS equence 77
3.2C.l2a sIsI C ustomer's BehaviSoerq uence 79
3.2T.r3a nsformMeodd el 81
3.2C.o4m parisonw ithS evcikS'hsa dowCP U Algorithm 81
3.3 Exampl2e: S imultnaeousR esourPcoes session 83
3.3O.v1e rvieowft hel acobson-LazMoewtshkoad 84
3.3B.eh2a viorS equenAcpep roach 87
3.3O.r3i ginBaelh aviSoerq uence 90
3.3T.r4a nsformatLeiaodninsg t oM odel. H, 93
3.3T.r5a nsformatLieoandsi ntgoM odelM 2 96
3.3.G6e neralization 101
3.4E xample3 :S erialization 102
3.4.O1v ervieowft heA ggregaSetrev eMre thod 102
3.4.T2r ansformations toLt ehaeAd gignrge gaSteer veMro del 106
3.5C onclusion 114
4 StatSep aceT ransformations 115
4.1S tatSep acTer anosrfmations 116
4.2 ExamplDee:c ompoisointof M emoryQ ueue 120
4.3E xamplSeh:a dowC PU Modeflo r
PreemptiPvrei orSicthye duling 122
4.4E xamplLeo:a d-DependSehnatd owC PUM odelf or
PreemptiPvrei orSicthye duling 125
viii CONTENTS
4.5A CataloogfG eneraTlr ansformations 128
4.5.L1o ad ConcealTmreannts formation 129
4.5.S2t aAtgegr egatTiroann sformation 132
4.5.S3e rveArgg regaTtriaonsnf ormation 135
4.5.R4e sponTsei mPer eservaTtriaonns formation 140
4.5.L5o adS eparatTirona nsformation 140
4.5C.l6a sAsg gregatTiroan nsformation 143
4.5.L7o adS caliTnrga nsformation 145
4.5.R8e sponTsei mMeo deliTnrga nsformation 146
4.5.D9e laSye rver IntroTdruacntsifoonr mation 152
5 ConsistSenotl utIitoenr,a tainodCn oO\',e rgence 155
5.1C onsistSeonltu tion 156
5.2C onvergeTnhceeo remfso rI teraAtligvoeri thms 161
5.3M onotonicRietsyu lftosrQ ueueiNnge tworks 165
5.3.M1o notoniciinCt uys tomePro pulation 166
5.3.M2o notoniciinPt ayr ameters 167
5.4C onvergenPcreo offo rB ard-SchwAeligtozrietrh m 169
5.5C onvergenPcreo offo rt heJ acobson-LazoMwestkhao d 172
5.6C onvergenPcreoo ffo rS evcikS'hsa dowCP U Algorithm 175
5.7A nomalies 180
5.7.D1i vergence 180
5.7.M2u ltipSltea blFei xePdo ints 180
6 Conclusion 187
References 189
Appendixes
AppendAi:Cx o mparisoofnM odelosf P reemptPirvieo rSiysttye ms 201
AppendiBx:T heA ggregaSteer veMre thodf orM ulti-ClNaestsw orks 207
AppendCi:Rx e sponTsiem eP reservaTtriaonns formation 221
Appendi0x: ImplicatoifoB niss tability iPnr iPorrSeiyetsmytp etmisv2 e5 9
261
Index
FIGURES
11. A hyperexponesnetrviearl 15
2.1T heb asipca tteirnnt hemo delipnrgo cess 39
2.2M ultipmloed els chemuas edt oc onstraunce tq uivalseenrtv feorr
thes ystem modelbeydA 10 49
2.3A schemfa otrh el imitdeedg reoefm ultiprogramemxianmgp le 51
2.4R ecursiinSo OnL VE 55
25. Modelt ransformaitnit ohneCs o nvolutiMoVnA,, andP BH
algorithms 58
2.6A n examploef h ierarocfhm yo dels 59
2.7F ulelxyp andetdr efeo rt heh ierarichnFy i gur2e. 6 60
2.8H ierarchsitcraulc toufrh ey brids imulation 64
3.1B ehavisoerq uenocfea clasLs c ustomer 78
3.2T rasnformebde havisoerq uenocfea clasLs c ustomer 78
3.3 Behavisoerq uenocfea clas/Is c ustomer 80
3.4T ransformed besheaqvuieonroc fea clasIsIc ustomer 80.
3.5G enerarle sourscter uctfuorrte h eJ acobson-Lazoawpspkroaa fcho r
analyzsiinmgu ltaneroeusso urpcoes sessio.n 85
3.6M odel!l- ll-primaryr esourccoen gesmtoidoenl 86
3.7M odelM 2-secondraesryo urccoen gesmtoidoenl 88
3.8I nteracbteiotnw eperinm ary (MaI)n ds ecnodary (M2)r esource
congestmioodne ls 88
3.9O vervioefbw e havisoerq uencfeosrm odelisn t he
Jacobson-Lazoawpspkrao ach 89
3.1O0r igibneahla vsieoqru enfcoert hsei multaneroeussoc uerp ossession
example 91
3.1F1i gur3.e 10be havisoerq uenacfet reera rrangemoenfpt r imary
resouracceq uisidteiloany( sq l'anl'd q />q) 94
3.1F2i gur3e.1 1b ehavisoerq uenacftee rt ransforma1t:si eoapnr atioofn
queueainndgs ervice 94
3.1F3i gur3e. 1b2e havisoerq uenacftee rd elaseyr veQrD 'si ntroducti9on5
3.1M4o del1 behavisoerq uence 95
x FIGURES
3.1F5i gur3e. 1b0e havisoerq uencaeft etrr ansforma0t:ri eoanr range-
mento fd elay(so rdoefrq ""a,n dq ,(,!,,d ifferefnrto m
Figur3e.1 1) 97
3.1F6ig ure3 .1b5e havisoerq uencaeft etrr ansforma1t:Pi Do n
introduced 97
3.17F igur3e. 1b6e havisoerq uenacftee rt ransforma2t:si eornv ice
overlbaept weePn,a ndQ ! removed 99
3.81 Figur3e. 1b7e haviors equeanftceret ransforma3t:fli oown-
equivalseenrtv QeSr introduced 99
3.19 Behavisoeqru enceo fa customreerq uestai cnrgi tisceaclt ion 107
3.2B0e hvaiors equenacftee rl oacdo ncealment 108
3.2B1e havisoeqru ence afterse rveagrgr egation (decomposition) 110
32.2M odelh ierarfcohrayg gregsaetrev emre thod III
4.1S tatter ansidtiiaognr afmo rp rioriyt servers 117
4.2M emory examplset atter ansitsitornu cture 121
4.3 Flow-equivsaelrevneftro rce ntrals ubsystaenmd m emory queue 138
4.4M odelh ierarfcohryM VA respontsiem meo delitnrga nsformati1o4n8
51. Decisitorne feo ri nconsisstoelnutt ions 157
5.2A n iteratfiuonnc tiwoint hco ntracatnidve ex pansiivnet ervals 163
5.3 Alternasttianbgal ned u nstabfilxee dp oints 164
5.4 A mUltipfilxee dp oint 166
55. Meanq ueulee ngetsht imasteeq uencfeosrt heB ard-Schweitzer
approximation 170
5.6C onvergeinnct eh Jea cobs-Loanzowskmae thod 174
57. ShadoCwP Ua lgoritihtme ratfiuonnc tion-sotnaeb filxee dp oint 178
5.8S hadowC PUa lgoritihtme ratfiuonnc tion-mulfitxiepdpl oei nts 179
5.9S hadowC PUa lgorititherma tion functions-tnaob filxee dp oint 181
5.1C0l as1s/ C PU utilizattriaocfneo rb istasbylset em 183
5.11S tatper obabidliisttyr ibuftoirbo ins tasbylset em 186
Appendixes
A.IS tretcoheudtC PU-Ls ervtiicmeev s.cl ass L population 205
A.2S tretcohuetdC PU-Lse rvifcuen ction 206
8.1H ierarcfhoyrt hea ggregsaetrev emro delf orm ixde networks 220
C.lR TPAf ora nF CFSs ervewri tghe nerals ervitciem e 225