Outline. Queuing Theory Framework. Delay Models. Fundamentals of Computer Networking: Introduction to Queuing Theory. Delay Models.

Transcription

1 Oule Fudaeals of Couer Neworg: Iroduco o ueug Theory eadg: Texboo chaer 3. Guevara Noubr CSG5, lecure 3 Delay Models Lle s Theore The M/M/ queug syse The M/G/ queug syse F3, CSG5 Fudaeals of Couer Neworg Lecure 3, F3, CSG5 Fudaeals of Couer Neworg Lecure 3, Delay Models ueug Theory Fraewor Delay or laecy of daa ace s a ora easure of he erforace of a ewor Delay ProagaoDelay TrasssoDelay ueugdelay ProagaoDelay Dsace/SeedOfLgh deede of essage sze TrasssoDelay MessageSze/Badwdh Badwdh daa-rae here ueugdelay delay due o e se wag queues os ora delay The queug delay deeds o several araeers: Arrval rocess Servce dscle Processg delay Ohers: badwdh of he l, buffer sze ueug syse: Servers oe or several: e.g., rouer, couer rocessor Cusoers: e.g., users, aces, rogra ass ueues: cusoers wa queues before geg servces Assuo: queues are ubouded. F3, CSG5 Fudaeals of Couer Neworg Lecure 3, 3 F3, CSG5 Fudaeals of Couer Neworg Lecure 3, 4

2 Lle Theore 96 Proof of Lle s Theore Measuree quaes of eres: T: average delay curred by a cusoer N: average uber of cusoers he syse Lle s Theore: N T where s rae of he arrval rocess Lle s Theore rovdes a geeral ad fudaeal relao bewee N, T, ad. I s deede of he aure of he arrval rocess or of he servce e dsrbuo. F3, CSG5 Fudaeals of Couer Neworg Lecure 3, 5 Noao: α: uber of users ha arrved before e β: uber of users ha deared before e T e se by user wh he syse N uber of users he syse a e Arrval rae: α s he l α Average e wh he syse: T T α Average uber of users a e : N N τ dτ F3, CSG5 Fudaeals of Couer Neworg Lecure 3, 6 Proof Co d The usage of he syse ca be bouded: β β T β N τ dτ Tag he l whe -> T β α α N τ dτ T N T α α F3, CSG5 Fudaeals of Couer Neworg Lecure 3, 7 T T Alcaos: Flow Corol Sldg wdow flow corol e.g., Go-Bac-N or Selecve eea wh wdow sze: The uber of aces he syse s always less ha : T N Cocluso: for a gve wdow sze, f T creases, he he arrval rae has o be decreased for a gve arrval rae, f T creases, he he wdow sze has o be creased for a gve T, f he arrval rae creases, he he wdow sze has o be creased F3, CSG5 Fudaeals of Couer Neworg Lecure 3, 8

3 Alcao: Throughu a Te Sharg Syse Assuos: N erals, oe rocessor, oe queue Terals are always occued Syse acvy: users log-o, refleco o average, sub as o he rocessor, ass are queued, ass execuo aes o average P us of e The delay for a user as s o average T s..: P T NP Usg Lle s Theore: N N NP P F3, CSG5 Fudaeals of Couer Neworg Lecure 3, 9 Te Sharg Co d The rocessor s also a queug syse where N I he seady sae ode: he arrval rae he syse s he sae as for he rocessor Usg Lle s Theore a secod e: P Cobg hese wo bouds we ge: N N {, } {, } P P P / P The salled er dcaes he boleec F3, CSG5 Fudaeals of Couer Neworg Lecure 3, The M/M/ queug syse Lle s Theore s a geeral ool ha allows us o calculae he seady-sae average delay of a queug syse Noao: M: eoryless, G: geeral, D: deersc, : uber of servers he syse M/M/: Arrval rae s Posso dsrbued Servce e s exoeally dsrbued These wo rocesses are deede Posso Process A Posso rocess wh arrval rae : The robably dsrbuo fuco df: e Pr arrvals erval[, τ ] τ The arrval dsrbuo of wo dsjo ervals s deede Proeres: execed uber of arrvals a legh-τ erval s: τ. τ F3, CSG5 Fudaeals of Couer Neworg Lecure 3, F3, CSG5 Fudaeals of Couer Neworg Lecure 3, 3

4 Posso Process Co d Oher Proeres Probables for sall ervals: Pr arrval e -δ / δ οδ Pr arrval δ e -δ / δ οδ Pr arrvals δ e -δ / οδ If δ eds o, he we have Pr arrvals δ, ad Pr arrval δ. Ier-arrval es: Le be he arrval e of he h cusoer ad τ - The: Prτ > s e -s > exoeal dsrbuo Posso rocesses are used o odel he raffc of a large uber of slar ad deede users If deedely ad decally dsrbued ace arrval rocesses rae / occur a he head of a l he he aggregaed rocess ca be show o be well aroxaed by a Posso rocess of rae. s cosdered o be a large value. The aggregao of deede Posso rocesses of raes,,, yelds a Posso rocess of rae: F3, CSG5 Fudaeals of Couer Neworg Lecure 3, 3 F3, CSG5 Fudaeals of Couer Neworg Lecure 3, 4 Exoeal Servce Te Aalyss of he M/M/ ueug Syse Le s deoe he servce e for he h cusoer. The servce e dsrbuo s exoeal wh araeer µ f: Pr[s s] e- µs The execed servce e for a job s: /µ The exoeal servce e s eoryless he sese ha: Prs > r s > Prs > r Posso rocesses are closely relaed o exoeal dsrbuos: er-arrval es of a Posso rocess wh rae have a exoeal dsrbuo wh araeer. The sae of he syse s caured by he uber of cusoers he syse a e e cosder a dscree verso of he rocess evoluo: Te:, δ, δ, 3δ, δ, N : uber of cusoers a e δ, Proeres: Pr[N l N l] Pr[ arrvals ad dearures δ erval] Pr[N N ] - δ δµδ - δ F3, CSG5 Fudaeals of Couer Neworg Lecure 3, 5 F3, CSG5 Fudaeals of Couer Neworg Lecure 3, 6 4

5 M/M/ Aalyss Co d Le: P,j Pr[N j N ] P, δ οδ δ P, δ µδ οδ δ µδ for P, δ οδ δ for P,- µδ οδ µδ for P,j οδ οδ for j,, - The sae rasos rerese a Marov cha Saoary Dsrbuo of a Syse Afer a log erod of e he syse reaches a seady sae Le: l Pr[ N ] Fro he Marov cha dagra we have: - δ - δ µδ µδ Hece: - -, where /µ Le, he - for > e also have: - F3, CSG5 Fudaeals of Couer Neworg Lecure 3, 7 F3, CSG5 Fudaeals of Couer Neworg Lecure 3, 8 Sce: Saoary Dsrbuo of a Syse j j, he, ad F3, CSG5 Fudaeals of Couer Neworg Lecure 3, 9 Seady Sae Averages Seady sae average uber of cusoers: Average delay T usg Lle s Th.: T µ Average wag e : delay-servce e µ µ F3, CSG5 Fudaeals of Couer Neworg Lecure 3, 5

6 Alcaos A : Blocg Probably Scalg u he arrval rae ad servce rae If we crease he arrval ad servce raes by he sae facor he average uber of cusoers he syse says he sae, whle he average delay goes dow Mullexg several coecos o oe l Beef of sascal ullexg Cosder a queug syse wh: K servers N K syse cusoers servce wag Dearg cusoers are edaely relaced by ew cusoers s he average cusoer servce e Average cusoer e he syse T? T N/ ad K Thus: T N / K F3, CSG5 Fudaeals of Couer Neworg Lecure 3, F3, CSG5 Fudaeals of Couer Neworg Lecure 3, A : Blocg Probably Co d Assue ha cusoers are bloced ad los f he syse s full: β s he rooro of cusoers ha are bloced The syse ay go hrough oes where less ha K servers are acve The: K β K K β F3, CSG5 Fudaeals of Couer Neworg Lecure 3, 3 A : Teral Coceraor Cosder a eral coceraor: 4 u les, each le of 64 bs ouu le of 8 bs Mea ace sze s 8 bs Each of he four u les delvers Posso raffc wh s/s Mea delay of a ace wh he coceraor: 8 s/s, µ s/s, T /µ - 5 s Average uber of aces wh he coceraor: N /- 4 F3, CSG5 Fudaeals of Couer Neworg Lecure 3, 4 6

7 A : Teral Coceraor Co d ears: The ouu le s caable of hadlg he geeraed raffc 8Kbs > 8 * 8, bu a subsaal u queue bulds u. The reaso s he radoess of he arrvals Usefuless of odelg ad aalyss: Delay esao Buffers desog F3, CSG5 Fudaeals of Couer Neworg Lecure 3, 5 A 3: Sascal Mullexg vs. Dedcaed Chaels Le a syse coss of: Two couers coeced usg a 64Kbs le 8 arallel sessos Each sesso geeraes Posso raffc wh s/s Paces legh s exoeally dsrbued wh ea bs. Two ossble sraeges: Gve each sesso a dedcaed oro of he chael e.g. TDM or FDM Have all he aces coee for he shared chael F3, CSG5 Fudaeals of Couer Neworg Lecure 3, 6 A 3: Sascal Mullexg vs. Dedcaed Chaels Co d Dedcaed chaels 8*8Kbs: s/s, µ 4 s/s, T /µ - 5 s Sascal ullexg: 6 s/s, µ 3 s/s, T /µ s Exlaao: because of he radoess of he arrval rae, soe of he dedcaed chael ay be uused because he corresodg sesso s dle whle aces are queued for oher sessos The M/G/ Syse M/G/ syse: Arrval rae s Posso Servce e has a geeral dsrbuo I s o ossble o derve a closed-for saoary dsrbuo as M/M/ bu we ca derve oher resuls Assue ha: Cusoers are served o a FCFS bass servce e of h arrval decally dsrbued, uually deede, ad deede of he er-arrval es F3, CSG5 Fudaeals of Couer Neworg Lecure 3, 7 F3, CSG5 Fudaeals of Couer Neworg Lecure 3, 8 7

8 P-K Forula Average servce e: Secod oe of servce e: Pollacze-Khch P-K forula: The: T Usg Lle s Theore: N E{ } E{ } F3, CSG5 Fudaeals of Couer Neworg Lecure 3, 9 µ ; N Verfcao of P-K Forula for Exoeally Dsrbued Servce Te he servce es are exoeally dsrbued as he M/M/ syse: / µ ; / µ ; T µ µ he he servce e s decal for all cusoers: M/D/: / µ ; / µ µ M/D/ rovdes lower bouds for, T, N, ad N F3, CSG5 Fudaeals of Couer Neworg Lecure 3, 3 Proof of he P-K Forula e use he coce of ea resdual servce e Noao: : wag e of cusoer : resdual e o coleo of he curre cusoer a sa whe arrves, f o cusoer s beg servced : uber of cusoers wag queue whe arrves Sce cusoers are servced order, we have: N j F3, CSG5 Fudaeals of Couer Neworg Lecure 3, 3 j Proof of he P-K Forula Co d Fro Lle s Theore: N, he: /- τ dτ β Thus he P-K forula F3, CSG5 Fudaeals of Couer Neworg Lecure 3, 3 8

9 hy Posso Assuo? here dd we use Posso Process arrvals assuo? A he oe whe a ace arrves he queue s ycal l P{N a arrval occurred jus afer } l {N } Seco 3.3. If arrval o Posso: Ier-arrval: uforly dsrbued bewee ad 4 secods Cusoer servce e s: secod > A arrvg cusoer fds he queue ey > bu a exeral cusoer sees a average queue legh of /3 F3, CSG5 Fudaeals of Couer Neworg Lecure 3, 33 Usable M/G/ Syses For several robably dsrbuo fucos he secod oe s fe rooroal o he square of he ea: e.g., exoeal, cosa, ufor. However, s o geeral o all dsrbuos. Le be he rado varable rereseg he servce e for a cusoer s..: Pr[] /3; Pr[ ]/4 for > The ea of s fe, bu he secod oe s fe I hs d of syses we ay have a accuulaos of arrvals ha exceeds he servce caably F3, CSG5 Fudaeals of Couer Neworg Lecure 3, 34 Alcaos of M/G/: GBN A Alcaos of M/G/: GBN A Slfed aalyss of Go-Bac- A: No-odulus, all acowledgees are receved If he lowes uber he wdow s o aced by he ed of he wdow he seder assues ha he error occurred ad sars rerasg Errors are deede fro oe o aoher All fraes ae a u of e o be rased The servce e dsrbuo s: Pr[] - F3, CSG5 Fudaeals of Couer Neworg Lecure 3, 35 If he aces are geeraed a he seder by a Posso rocess, he we have a M/G/ syse: Forulas good o ow: T 3,, F3, CSG5 Fudaeals of Couer Neworg Lecure 3, 36 9

10 F3, CSG5 Fudaeals of Couer Neworg Lecure 3, 37 M/G/ wh Prores Syse wh rory: Cusoers are dvded o classes: Cusoers class are gve rory over cusoers of class j for ay j> No-reeve: Cusoers are served her order of arrval Noao: Arrval rocess for class : Posso wh rae Servce e of cusoers of class : average wag e for a cusoer class I average resdual e average uber of cusoers of class wag queue F3, CSG5 Fudaeals of Couer Neworg Lecure 3, 38 M/G/ wh Prores Co d Class Class Class F3, CSG5 Fudaeals of Couer Neworg Lecure 3, 39 M/G/ wh Prores Co d As for he P-K forula: Thus he average wag e for a cusoer class : > f f F3, CSG5 Fudaeals of Couer Neworg Lecure 3, 4 M/M/ Marov Syse Seady sae robables: > < µ } { P ueug P Erlag C Forula: robably of havg o wa for servce P

11 F3, CSG5 Fudaeals of Couer Neworg Lecure 3, 4 M/M// Marov Syse Seady sae robables: ; µ µ Blocg robably: Erlag B forula / / / / µ µ