Telematics 2 & Performance Evaluation

Transcription

1 Teeacs & Pefoance Evauaon Chae Modeng and Anayss Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss Goas of hs chae Gves an ovevew on he hases & es used n heoeca efoance evauaon Dscusson on basc odeng echnues fo ava and sevce ocesses Cacuaon of sae obabes of Maov chans syboc and nuec Devaon of cosed fouas fo efoance ecs n fundaena ueung syses Geneaon of oe coex ueung syses Oen Cosed Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss

2 Ovevew Inoducon Fundaena es and noons Taffc chaacezaon ava & sevce ocesses Maov and Posson oees Le s aw Pefoance ecs of ueung nodes Kenda noaon Maov ocesses & Fundaena Queung Nodes Maov chans Te-hoogenous Maov chans Chaan Koogoov euaon Seady sae n Maov chans One-densona deah & bh ocesses Acaon o dffeen ueung syse M/M//, M/M/, M/G/,... Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss Ovevew Queueng newos Inoducon Foa defnon Pefoance ecs Seeced sovng ehods Nuec cacuaon of goba seady sae Condons fo goba and oca baance Syboc souon of oen newos Jacson Syboc souon of cosed newos Godon-Newe Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 4

3 Taffc Engneeng & Queung Theoy Too o ode and anayze daa and voce affc n councaon newos Teoa behavo of coue o newo syse s odeed by a sochasc ocess Modeed aaees ae sacay defned by ando vaabes, e.g., e beween avng ass T A o nube of ass n a syse o nube of ass n a ueue L Conseuence: esus of efoance evauaon foow aso a sasc dsbuon can be descbed by aveage, oens ec. Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 5 Taffc Engneeng & Queung Theoy a. Deands Rejeced ass Syse Acceed ass Sevced Tass b. Ava ae Syse Deaue ae Pobaby of success Loss obaby P V P E PV - P E Ine-ava e T A Syse Ine-deaue e T E c. Resonse e T V Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 6

4 Defnng Ccu-swched Taffc Usng a councaon esouce s caed aocaon Ieevan: aocaon eod, he eason fo he aocaon, whehe he councaon s successfu o no Channe bunde Obsevaon eod T Channe Channe Channe Exae: Teehony syse Su of a aocaon eods s caed affc voue Y, noazed voue by he nube of aocaons c s he aveage ca-hodng e Y Y Eh c The offeed o caed oad s denson-ess bu easued n Eang A. K. Eang, founde of ueueng heoy affc voue s hus easued n Eang hous Eh o Eh. Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss Defnng affc oad When oong a affc voue fo a defned obsevaon eod T, we oban he oad y, easued n Eang E o E: Y y c T T E E s defned by a fu aocaon of a snge channe fo a e un A snge channe ay anso a os E. A PCM- syse ay fowad u o E. A oca oo n a ubc newo ceaes abou, E of affc. The offeed oad o ncong affc s a affc voue ha s ansoed o eehony syse o a a of ndeenden of beng seved o no. When he nube of offeed aocaons whn a e wndow s caed C a, he offeed oad n a eehony syse s defned by: A C E a Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 8

5 Defnng affc oad The caed oad y s he oduc of he nube of successfu aocaons C y whn a gven e fae and he aveage hod e : y C E y The ao of offeed oad ha exceeds he caacy of he syse,.e. he axu caed oad, s ejeced and caed esdua affc R offeed oad - caed oad: R A - y C - C E a y The bocng obaby B s deved by eang he esdua affc o he offeed oad. I s defned by: B R Ca - C A C a y Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 9 Deay and oss syses Syses whou ueue ossbes ae caed oss syses. Syses wh ueues ae caed deay syses. Deendng on he enghs of he ueues deay syses ae cassfed n:. Pue deay syses ueues wh nfne enghs. Deay oss syses bound ueues Aso: esouce aocaon, ueueng dscnes and oy abuon ay a ajo oe n he oganzaon of he syses Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss

6 Ava ocesses and sevce es Sevce es b b b b4 b5 a a a a4 a45 Ineava es 4 5 Ava es Ineava & sevce es ae usuay assued o foow a sasca aen Obvous The bee he sasca oees ae caued, he hghe he sgnfcance of he evauaon Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss Ava ocesses Incong, ejeced and seved ass foow sochasc ocesses Incong ass defne he so-caed ava ocess. Is sasc oees us be deved by easueens The eod beween wo avng ass of an ava ocess s he neava e T A The execed vaue gven by he aveage n he easueen of he nube of avas e e un s caed ava ae: #avas E e E{T A } The ava ocess can be odeed by he dsbuon funcon o densy funcon of ehe he nube of ass e e un o neava es Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss

7 Sevce ocess Pobaby of sevng a as s caed obaby of success P E Pobaby ha a as s ejeced s caed oss obaby P V The ocess of sevng ass by one o oe seves s caed sevce ocess The sevce ocess can be odeed by he sevce e T B gven by s dsbuon o densy funcon The nube of seved ass e e un s caed sevce ae: #seved ass E e E{T B } The esonse e T V of as n he syse consss of ueueng e T W and sevce e T B. T V T W + T B Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss Dsbuon funcons of ava and sevce ocesses Exonena dsbuon Mos se & wdey used dsbuon Eang- dsbuon Su of ue exonena dsbuons Hyeexonena dsbuon Seecon of ue exonena dsbuons Hyoexonena dsbuon Cobnaon of Eang- & hyeexonena dsbuon Gaa dsbuon Eang- wh non-nega Cox ocess Deved fo Eang- dsbuon Douby sochasc,.e. execed vaue s aso a ando vaabe Fo ea-wod scenaos Deensc, Paeo, Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 4

8 Maov oey of exonena dsbuons The os oan fo exonena dsbuons: Maov oey aso caed eoyessness Sasca ocesses foowng an exonena dsbuon ae caed Maov ocesses Queson: How do evens ha haened n he as affec fuue? Pobaby of ava afe Sa of obsevaon P T > e - e Pobaby of ava afe +s P T > + st >??? e s No even Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 5 Maov oey of exonena dsbuons Poof of eoyessness: - P T > + s, T > - P T + s e P T > + s T > P T > P T > e + s - e -s P T > s The hsoy.. has no nfuence on he fuue neva..+s! Thus wang fo an even n a exonena ocess fo a eod whou success, does no have an nfuence on any fuue even. In Maov ocesses ony he cuen sae aes, no he hsoy of evens ha ed o he sae The exonena dsbuon s he ony connuous dsbuon, havng he Maov oey An exonena dsbuon occus aways when evens n he ocess ae coeey ndeenden of each ohe and soe ohe no axos Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 6

9 Posson oey of exonena dsbuons Cacuaon of he obaby ha ass ave whn e fae, ff neava es ae exonenay dsbued s... D D D D.... Cacuaon of obaby fo an ava whn neva Te neva dvded n e sos Sees exanson of he exonena dsbuon fo sa nevas D : T D - e T D» D D -[ - D +! D -! -D D nd d - s h ava +! ] Cacuaon of obaby fo no ava whn neva D T > D» - D D Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss Posson oey of exonena dsbuons æö ç è ø! -!! - - D - D Wh D! - D! -! - we oban: Fo >>:» -! Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 8

10 Posson oey of exonena dsbuons Usng he defnon of he exonena dsbuon he fo eads o he Posson dsbuon! - e The Posson dsbuon defnes he obaby ha exacy evens haen n whn a e fae, ff he neava es ae exonenay dsbued. If ehe neava o sevce es of a ocess foow an exonena dsbuon, he ando vaabes descbng he nube of avas o sevced ass esecvey ae Posson dsbued fo any gven e neva. Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 9 Cobnng and dvdng Posson ocesses Cobnng n Posson ocesses wh ava aes! eads o anohe Posson ocess P T - e - n wh n Subdvng a Posson ocess wh an ava ae! no n ocesses, such han each ocess obans ass wh a fxed obaby, hen each subocess s Posson dsbued n n n P T - e n - Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss

11 The hyeexonena dsbuon Aenave seecon of seves wh exonenay dsbued sevce es : nube of aae hases j j Iaon of age dsbuon by aae goung of exonena dsbued subocesses hases A as s assgned wh obaby o he seve hase A os a snge seve s acve a a es F P TB j j - e - j Used o aoxae non-exonena dsbuons wh a coeffcen of vaaon c > Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss Eang- dsbuon Sea aceen of denca hases wh exonena dsbuon of sevce es Ony a snge as ay be seved a once no ohe n ueue F P T B - e - - j j! j ³, Î{,,!} To aoxae non-exonena dsbuons wh coeffcen of vaaon c< Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss

12 Hyoexonena dsbuon Cobnng hyeexonena and Eang- dsbuon aows o ode abay coex dsbuons Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss Le s Law Mos oan and os ofen used eaon n ueung heoy Le s aw eaes he aveage nube of ass n a syse wh he aveage esonse e T V, assung he syse s sabe and non-eeve Queung syse Ava ae " ass Resonse e T V T V T W + T B T W Queung e T B Sevce e T V Le s aw aso aes o he ueue: L T W L avg. nube of wang ass Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 4

13 Le s Law A heusc oof The syse s sabe,.e. has on aveage a hghe sevng ae han he ava ae A as enes he syse, now conans ass beng ueued o beng seved A ass w be seved and afe T V he as as w eave he syse Whn hs eod " T V new ass ave n he syse on aveage Fo he syse beng sabe, hee us be agan ass n he syse Thus: T V Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 5 Le s Law A Gahca Ineeaon Nube of ass Avng and eavng ass shoud be n baance on aveage A Avas A T V T V T V T V4 aea T V5 F D Deaues Aveage esonse e T V T V τ Te Aτ T V Aτ Fτ Aτ Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 6

14 Le s Law A Gahca Ineeaon Nube of ass A L nube of ass n he syse a on n e L Aea F Aveage nube of ass n he syse: ò L d F Te Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss Le s Law A Gahca Ineeaon By euang he aeas F we deve: Assuons fo he devaon The eads o F TV A A T V T V No assuons abou ava o sevng ocess egadng sasca dsbuons No assuons abou he ocessng ode of ass sevce saegy Ony eueen: eachng a sabe sae Concuson Le s aw s acabe fo ueueng syses n genea f hey ae n a sabe sae Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 8

15 Fundaena Queung Nodes Coex syses usuay conss of ue oe se ueung nodes ha ae neconneced Fundaena ueung nodes conss of:. Sevce nodes and/o. Queues Ava ae Tas Queung aea Sevce node Sevce ae Queue engh L Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 9 Pefoance of Fundaena Queung Syses Tage of odeng s aways he cacuaon of efoance ecs As ueung syse ae dynac by naue, he ecs deend on e Usuay we oo a he ecs n n saonay hase: Tansen henoena ae aenuaed Mecs ae ndeenden of e a eas he aveages Syse s a sasca eubu Ava and deaue ae ae eua Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss

16 Pefoance of Fundaena Queung Syses. Sae obaby The vaue efes o he obaby ha hee ae ass n he ueung syse ncudng he sevce nodes. Aveage uzaon The aon of he ovea e ha a sevce node s acve occued Fo any sabe syse,.e. hee s a sasca eubu he foowng condon us hod: avg. sevce e avg. ne ava e ava ae sevce ae Thee ay be no oe avng ass e e un han hee ay be seved on aveage,.e.: < Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss Pefoance of Fundaena Queung Syses. Thoughu The nube of ass seved on aveage e e un deaue ae. Fo sabe syses he foowng hods: 4. Aveage Queung Te T W The aveage eod of e a ass needs o say n he ueue befoe beng seved 5. Aveage Resonse Te T V Su of es ha a ass need o ass a subsyses of he ueung node I hods: Le! TV TW + TB T W+ avg. nube of ass n he syse Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss

17 Pefoance of Fundaena Queung Syses 6. Aveage nube of ass n he syse. Queue engh L The aveage nube of eeens n he ueue: L Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss Kenda Noaon fo Se Queung Nodes A/B/c/K//Z A Dsbuon of ne ava es B Dsbuon of sevce e c Nube of sevce nodes K Nube aces n ueung aea defau: Nube of avng ass defau: Z Queung dscne defau: FIFO Usuay abbevaed fo s used A/B/c e.g. M/G/ A/B/c/K e.g. M/M// Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 4

18 Kenda Noaon fo Se Queung Nodes Exaes fo dsbuons M Exonena dsbuon Maovan E Eang- dsbuon H Hye-exonena dsbuon wh aces D Degeneaed Dsbuon e.g. deensc es G Genec dsbuon no fuhe assuons Exaes fo ueung dscnes FCFS FIFO fs coe fs seved LCFS LIFO as coe fs seved SIRO seved n ando ode RR ound obn Sac o dynac oes Peeve saeges Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 5 Ovevew Inoducon Maov ocesses & Fundaena Queung Nodes Maov chans Te-hoogenous Maov chans Chaan Koogoov euaon Seady sae n Maov chans One-densona deah & bh ocesses Acaon o dffeen ueung syse M/M//, M/M/, M/G/,... Queueng newos Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 6

19 Maov ocesses Tye of sochasc ocesses Powefu oo fo odeng and efoance evauaon Queung nodes ay ofen be we descbed by a seca nd of Maov ocesses: Maov chans The anayss of Maov chans aows o cacuae he sae obaby of he ueung nodes and heeby he devaon of he efoance ecs A sochasc ocess s a Maov ocess, f ossesses he Maov oey,.e., evens ae ndeenden The fuue behavo of a ocess afe a on n e ony deends on he sae X and no evous saes. Exonena dsbuon of ne-ava and sevce es Posson dsbuon of ava and sevce evens Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss Maov chans Maov chans ae Maov ocesses havng a dscee sae sace and connuous e sace X P X x X x, X x, X x n + n +! n n Pas saes n- n n+ P X n+ xn+ X n xn Fuue? Fouaon of he Maov oey fo Maov chans The fuue of a ocesses ay be coeey descbed by s cuen sae Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 8

20 Seady Sae of Maov Chans Usua acaon of Maov chans: Loong a sae obabes afe eachng a saonay hase o seady hase Pocess an suffceny ong and seed, hus: Pobaby of beng n a ocess sae no onge deends on na sae Pobaby of beng n a ocess sae no onge deends on cuen e Such a saonay sae us exs, f Maov chans ae osve ecuen,.e. evey sae s eachabe n fne e educbe,.e. evey sae ay be eached fo evey ohe sae fne,.e. he nube of saes s bound Such saes as we as such chans heseves ae caed egodc The s of sae obabes of an egodc Maov chan e gong o ae caed seady-sae obabes Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 9 Te-Hoogeneous Maov Chans Te-Hoogenous Maov chans ae seca n a way ha he anson obaby beween a sae o a sae does no deend on he absoue e bu ony on %,.e. exonena dsbuon of eods T Z beween ocess ansons T + D T ³ T D Z Z Z!!! T Z D - e - D - [ - D + D! - D! aveage ae of ansons beween sae o sae +!] D Fo sa e nevas % he dsbuon funcon of anson obabes ay be aoxaed by: D» D Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 4

21 Te-Hoogeneous Maov Chans The ovea obaby fo eavng sae : ¹ D - D» ¹ D The absoue anson obaby * fo an gven sae o a dffeen sae * D D Pobaby fo beng n sae Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 4 Chaan Koogoov Euaon Indeendeny dscoveed by Chaan and Koogoov The absoue obaby fo beng n sae a he on n e + %: + D D D * A ossbe saes Cacuaon of he obaby a + % fo he obaby a Vad fo e-hoogenous Maov chans Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 4

22 4 Chaan Koogoov Euaon Seaaon of e fo Subacng K Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss D + D D + ¹ D - - D - + D D ¹ ¹ ¹ D - D ¹ ¹ D - D» 44 Chaan Koogoov Euaon Dvson by % L Addonay hee s a consan fo he su ove a sae obabes of he sae sace Z Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss ¹ ¹ - D - + D D D D ¹ ¹ - d d ÎZ Lnea dffeena euaon syse Souon eues sang condon a

23 Hoogenous Maov Chans n Seady Sae Afe eavng ansen hase egodc Maov chans each a seady sae Sae obabes ae ndeenden fo e heeafe,.e.: d d Z ¹ Z ¹, Sevce and ava aes Consan Tansons eadng o a eave Sae Z Tansons eadng o ceaon ÎZ A sae obabes ae n sasca baance! Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 45 Exae: Cosed Queung Newo wh wo Tass Gven: a cosed no ass fo o o exena nodes ueung newo havng ueung nodes Two as ccuae n he newo Sevce es ae exonena dsbued wh sevce aes μ s -, μ s - Sevce node Sevce node Queue Queue We use he foowng noaon: Sae :, wh # of ass n node # of ass n node The esung sae sace: Z {,,,,,} Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 46

24 Exae: Cosed Queung Newo wh wo Tass Mode Sevce node Sevce node Queue Queue Coesondng sae cha,,, Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 4 Exae: Cosed Queung Newo wh wo Tass Koogoov s euaon syse o fnd he seady sae obabes Sae,: Sae,: Sae,:, -,, +, -, +, -, Lnea deenden euaon syse Su condon:, +, +, Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 48

25 Exae: Cosed Queung Newo wh wo Tass 4 æ ç ç- ç è - + æ öæ ç ç ç- ç ç ç è - øè " % " $ ' $ $ ' $ $ # 4 ' & $ # æ ç ç ç è öæ ç ç ç øè öæ ç ç ç øè,,,,,,,,,,,, ö æö ç ç ç ø èø ö æö ç ç ç ø èø % " ' % $ ' ' $ ' ' $ ' & # & ö æ ö ç ç ç ø èø Sae obabes:,, Aveage uzaon:, 4 Node wong n fs wo saes: 6, +, Þ 85,% Node wong n as wo saes, +, Þ 4,86% Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 49 One Densona Bh- and Deah Pocesses Maov ocesses ha ony aow ansons beween neghboed saes May be eesened by one-densona ando vaabe Rae fo anson fo sae o sae + s caed bh ae Rae fo anson fo sae + o sae s caed deah ae Raes beween any non-adjacen saes ae fo, + fo, -, +, - Adaed Chaan Koogoov euaon: d d fo > Tansons o sae Tansons fo sae Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 5

26 One Densona Bh- and Deah Pocesses Koogoov euaon fo sae d d - Noazaon condon ÎZ If a deah aes μ, he ocess s caed a ue bh ocess If a bh aes λ, he ocess s caed a ue deah ocess Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 5 Saonay One-densona Bh and Deah Pocesses A saes ae n seady sae In a one-densona bh and deah ocess goba baance exss fo sae f ansons o and fo a neghbong saes ae n baance fo > - fo ÎZ Condons fo goba baance + Noazaon condon λ λ λ λ λ Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 5

27 Saonay One-densona Bh and Deah Pocesses Suaon of goba condons fo baance Z Z Z Z Loca baance Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 5 Saonay One-densona Bh and Deah Pocesses Condons fo oca baance ony beween neghbong saes Afe ue ses of uggng n: usng and - - -! Ȭ + fo > Genea souon fo sae obaby n baance Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 54

28 Saonay One-densona Bh and Deah Pocesses ÎZ Due o noazaon condon : n- + Õ n + + +! + + +! + and + n- Õ + + n + +! Puggng n no - Õ + n- + Õ n + Genea souon fo sae obaby n baance fo > Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 55 The M/M/ Syse Queue has nfne engh Ava ae Queung aea Sevce node Sevce ae Tas Queue engh L A ansons o owe saes wh eua ae sevce ae, μ A ansons o hghe saes wh eua ae ava ae, λ Loca baance - + λ λ λ λ Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 56

29 The M/M/ Syse Fo we oban fo unde he condon of eua ansons aes and an nfne nube of ueung aces nfne nube of saes ossbe + n- Õ n fo +! + / < T A T B T A o eed oad T B - - Pobaby ha hee s no as n he syse / oad n ueung node Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 5 The M/M/ Syse Fo we oban - Genea souon fo sae obaby n baance fo > Pobaby ha hee ae ass n he syse...5 ρ Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 58

30 The M/M/ Syse Aveage nube execed vaue of ass n he syse - fo - < - - Aveage nube L execed vaue of ass n he ueue L L - fo < Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 59 The M/M/ Syse Lρ Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 6

31 The M/M/ Syse Reang o e: The aveage esonse e T V usng Le s aw T V - - The aveage ueung e T W T W T V -T B T V - - Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 6 The M/M/ Syse Ovevew - Sae obaby T V - T V L - TW - - Pobaby of ey syse Aveage nube of syse ass Aveage nube of ueued ass L Resonse e T V Queung e T W Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 6

32 The M/M/ Syse Exae Tass ave sascay ndeenden of each ohe wh a ae of λ 4s - n a syse Posson ocess. Thee s he aenave of usng wo ocessos wh a sevce ae of μ s - each, o a snge ocesso wh a sevce ae of μ 5s -. Whch aenave woud you efe? 5 Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 6 The M/M/ Syse Exae T V. s T V. s Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 64

33 The M/M// oss syse The syse conans aae sevce nodes wh eua sevce ae μ and no ueue Node λ Leads o dffeen sevce aes Node - λ λ λ λ λ S fee sevce nodes avaabe A sevce nodes ae busy Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 65 The M/M// oss syse Pobabes fo saes nube of ass n he syse Y 8,..., Due o he noazaon condon +! X Z ! + + +! +!!!! - [ ]! Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 66

34 The M/M// oss syse Pobabes fo saes! P! Pobaby fo oss o bocng obaby! P Eang oss foua o Eang-B foua! Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 6 The M/M// oss syse.8 Loss obaby Offeed oad E / Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 68

35 The M/M// oss syse - Ovevew! Sae obaby - [ ]! Pobaby fo an ey syse! P! Bocng obaby Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 69 The M/M// oss syse - Exae In a vae banch exchange ee wh nes o he ubc hone syse oubound cas ha exceed he caacy ae boced. Ava and sevce ocesses can be assued o be Maovan. Ava ae Lne Lne Lne Sevce ae A. Daw he sae cha of he coesondng Maov chan. B. Sech he sae obaby as a funcon of he offeed oad n a ange fo o 8 Eang. How ay he obaby be neeed? C. Sech he sae obaby A fo A.5 Eang; A.5 Eang; A Eang. Whch vaues of yed n axa? Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss

36 The M/M// oss syse - Exae A. Daw he sae cha of he coesondng Maov chan. B. Sech he sae obaby as a funcon of he offeed oad n a ange fo o 8 Eang. How ay he obaby be neeed? Bocng obaby! A! A A Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss The M/M// oss syse - Exae C. Sech he sae obaby A fo A.5 Eang; A.5 Eang; A Eang. Whch vaues of yed n axa?.! A! A A.5 A.5 A Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss

37 The M/M// Deay Syse The ueung syse has aae sevng nodes wh eua sevce ae μ and a shaed ueue wh an nfne nube of aces: Node λ Leads o dffeen sevce aes: Node - + λ λ λ λ λ λ S fee sevce nodes avaabe A sevce nodes ae busy Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss The M/M// Deay Syse Cacuaon of sae obabes: fo fo ae fo Y + fo By defnng: fo,...,! fo!! fo,...,! fo Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 4

38 5 The M/M// Deay Syse Cacuaon of fo he noazaon consan: Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss X Z !!!! - - ú û ù ê ë é Pobaby of an ey syse 6 The M/M// Deay Syse Pobaby w ha a as s ueued fs and no edaey seved Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss +! + + ³ + + W P !!! W - W! - Eang C foua

39 The M/M// Deay Syse Queung obaby Noazed oad E / Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss The M/M// Deay Syse Cacuaon of he aveage ueue engh E{L}: E { L} - +! - - E { L}! - W - Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 8

40 The M/M// Deay Syse Cacuaon of he aveage esonse e E{T V }: { T } E{ T } E{ T } E + V W B Usng Le s aw E { } { L} W E T E{ T B } W - Aveage ueung e n he syse E{ T V } W λ ρ ρ + Aveage sevce e n he syse Aveage esonse e E W - { } E{ L} + E{ B} + Aveage nube of ass n he syse Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 9 The M/M// Deay Syse - Ovevew! fo,...,! fo é ê + ë -! ù +! - ú û - Pobaby of an ey syse W! - Pobaby of beng ueued E { L} { } E T W! - W { L} E W - E{ T V } W λ ρ ρ + - E - { } E{ L} + E{ B} W + TV Aveage nube of ass n ueue Aveage ueung e Aveage esonse e Aveage nube of ass n syse Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 8

41 The M/M// Deay Syse - Exae Thee ae wo ndeenden ava ocesses, boh havng he Posson oey and a ae of λ λ s -. They ae beng ocessed n a syse wh wo nodes, havng an eua sevce ae of μ s -. Evauae he foowng ocessng saeges by coang he aveage esonse es:. Boh ocessos have dsnc ueues.. Boh ocessos shae a snge ueue. Whch saegy s o efe? Node λ λ + λ Node Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 8 The M/M// Deay Syse - Exae Vaan : λ s -, λ s - E{ TV } E{ TV }, s E{ TV } + E{ T } E{ T } +, s V V Vaan : M/M/, WS, λ λ + λ 4 s é ù - - é ù ê 4 4 ú ê + +!! ú ê + + ú ë - û ê - ú ë û W! ,5-5 Vaan s oe effcen E V W { T } + +, s Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 8

42 The M/G/ Syse Snge sevce node wh abay genea bu nown dsbuon of of sevce es Does no need o be ndeenden of evous sae! Infne nube of ueung aces Ava ocess s Maovan Dsbuon of sevces es chaacezed by: E { } T B s Aveage sevce e Vaance Ava ae By defnon: Tas Queung aea { B }- E{ TB} E{ T } s - E T B Thus aso he foowng s gven: Sevce node Sevce ae Queue engh L { } s + E T B Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 8 The M/G/ Syse Consdeng he ava of he h as a : Le he esdua sevce e R be he eod of e s eued o ocess he cuen as ay deend on he eod of e ha he as s aeady n he sevce un, no exonena dsbuon! One has: ³ R Gven he engh of he ueue L a The ueung e of he as w be: T W R + L j If we hn of he vaues as ando vaabes, we deve he foowng exec vaues: E{ T } E{ R} + E{ L} E{ T } W T Bj E{ R} + E{ L} B Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 84

43 The M/G/ Syse Usng Le s aw:.e.: E { L } E { T W } E } E{ R} + E{ T } { TW W hus: E{ R} E{ T } W - Fo he esonse e: E { TV } E{ TB} + E{ TW } We deve: E{ R} The esonse e deends E{ T V } + - on he execed vaue of he esdua sevce e! Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 85 The M/G/ Syse Execed vaue of he esdua sevce e R F aea unde he cuve R T B Cacuaon of he aea T B T B T B T B- ò E{ R} R d F F T Bj - j Usng he foua fo he execed vaue of he esdua sevce e E{ R}, - j T Bj E { R} E{ T B } Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 86

44 The M/G/ Syse Puggng E{R} no he euaon of he esonse e E{ T E{ T } + - Puggng E{T B } no he euaon,.e. E T B s + E{ T V } + - V } B { } s + E { R} Gven he eads o he euaons of Poacze-Khnchne usng Le s aw { } [ E T - - s V ] E{ } E{ TV } [ Resonse e Nube of ass n he syse Seca cases: s Exonena dsbuon eads o M/M/ syse s Deensc dsbuon esus n M/D/ syse s E{ ] T B } Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 8 Queung Newos Queung odes conssng of oe han one fundaena ueung nodes ae caed ueung newos o ueue newos Two an nds: Oen ueueng newos Tass ene he newo fo he ousde and ex he newo agan afe beng ocessed Cosed ueueng newos Nube of ass s consan Tass ccuae Fuheoe: Newos wh dffeen nds of ass May have dffeen sevce es May be ueued o ocessed dffeeny n he nodes Mxed ueueng newos Oen fo soe nds of ass, cosed fo ohes Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 88

45 Queung Newos Noaon Nube of nodes n a newo Nube of ass n a cosed newo Nube of ass n node Obvous fo cosed newos: Sae of he newo Nube of as seved n aae n node Aveage sevce ae a node Aveage sevce e a node N K N K,,!, N T B Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 89 Queung Newos Noaon Pobaby ha a as ocessed a node changes o node j Pobaby ha a new ass aves a node befoe any ohe Pobaby ha a ass eaves he syse afe beng ocessed a node I hods j N - j j The aveage ae of ass fowng fo he ousde o a node The aveage oa ae of ass avng a node In sasca baance: N + j j,!, N j Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 9

46 Queung Newos Pefoance Mecs Sae obaby,,!, N Pobaby ha newo s a cean sae sasca baance! Su ove a obabes:,,! Z Magna sae obaby, N Su ove a sae obabes such ha node conans exacy ass: Thoughu λ n node Rae of ass eneng and eavng a node unde sasca baance Sevce ae ay deend on oad Z &,,!,,!, N Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 9 Queung Newos Pefoance Mecs Cuuave houghu Rae a whch ass eave he newo; n baance aso ae a whch hey ene N N Aveage uzaon & n node Pobaby ha a node conans a ass o ha new as woud have o wa - Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 9

47 Queung Newos Pefoance Mecs Aveage nube of ass n a node : In sasca baance he aveage nube of ass n a node can be deved by he execed vaue of he weghed sae obabes o he aveage esonse e. T V Aveage ueue engh L n node L - T W Reave vsng feuency o eave ava ae e n a node Oen ueung newos Cosed ueung newos e e e + e N j e j j N e e j j j Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 9 Queung Newos Goba Baance Cacuaon of newos ossbe n heoy by usng and sovng Koogoov euaons fo a! ossbe saes Î Z ZX X Z X 6 6 Reachng Leavng Max noaon: Z Noazaon consan Q~ ~ havng ~ T,,..., Z Geneao ax Q P 6 P 6 P 6 C A Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 94

48 Queung Newos Goba Baance Reca Exae! Cosed newo, nodes, ass N, K, μ., μ.4 Sevce node Sevce node Q Queue Queue! T,,,,,,,,,,, + + C A C A Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 95 Queung Newos Nueca Anayss Q~ ~ Euaons syse and sovabe wh sandad nueca aoaches Resus n sae obabes ÎZ Pefoance ecs easy o deve Pobes: Reues fne sae sace,.e., no acabe o oen ueung newos Pacca acaon ony o newos wh ow node and as coun hgh nube of euaons! Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 96

49 Queung Newos Nueca Anayss. Gaussan enaon Exchange one ow wh he noazaon condon Q ~,,,..., T P 6 P Q 6 C..... A Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 9 Queung Newos Nueca Anayss Exae Q ~ C B, A, B A,.5,.66,.,.66 Souon of sae obabes Devaon of agna sae obabes,66,,66,5 Cacuang he uzaon of sevce nodes Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 98

50 Queung Newos Nueca Anayss Exae Cacuang he houghu of node Sasca baance!.86 Aveage nube of ass n node E } + + { E } + + {.66. Aveage esonse e of node TV / TV / T / V Le s aw Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 99 Queung Newos Nueca Anayss. Ieave Nuec Aoach Aoxaon wh uch fase une ~ Q~ ~ Q~s ~ Q~s + ~ s Scaa > ~ n+ Qs + E~ n Se of eaon Scaa s chosen s.. ages eeen of Qs s sae han condon fo convegence s /ax E beng a of he ax assues ha hee s an egenvaue e Advanage: ax s no changed dung eaon, aaezabe Dsadvanage: S consdeaby sow fo age nsances Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss

51 Goba & Loca Baance n Queung Newos When a ueung nodes n a newo fuf cean condons,.e. egadng Dsbuon of ava and sevce es Queueng dscne s ossbe o deve condons fo a oca baance of newo saes Loca baance n hs conex eans: The ae a whch a sae Z changes o Z- euas he ae a whch Z- changes o Z,.e. aes of avng and eavng ass ae eua n he node. Condons fo goba baance ay be eesened by addng condons fo oca baance. Euaons fo oca baance ae uch ease o hande, as seaae euaons exs fo each ueung node! Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss Loca Baance n Queung Newos Loca baance has been shown o exs fo he ueung newos conssng of he foowng nodes M/M/-FCFS M/G/-PSRR M/G/ Pobaby ha node conans ass Sevce ae n node, f hee ae ass. Pobaby ha node conans - ass Ava ae n node, f hee ae - ass. I/O devces CPUs Infne Seve, Tenas - Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss

52 Loca Baance n Queung Newos - Exae Cosed newo, nodes, ass N, K, μ., μ.4 Sevce node Sevce node Queue Queue,,,,,,,,,..4, é, ê + + ë ù + ú û,,,, Noazaon consan, +, +, +, é, ê + ë + ù + 4 8ú û 4 8,.5 5 Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss Loca Baance n Queung Newos - Exae,,,,,,,,,, + - Euaons fo goba baance ay be deved by addng o subacng of euaons fo oca baance., - -,,, Is hee a souon ayng oca baance euaons, euas he goba baance souon., +,, + Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 4

53 Poduc-fo Souons If a nodes of a ueung newo fuf he condons fo oca baance hee s a oduc fo souon,.e., he newo sae s decy ooona o a coesondng obabes of he nodes,,, N Pobaby of ovea newo sae GK N N [ ] Magna obabes fo snge nodes GK Noazaon consan, such ha sae obabes of newo add o I s ossbe o seaaey cacuae efoance ecs fo a nodes and cobne he o an ovea ec seaabe newo Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 5 Poduc-fo Souons - Exae Cosed newo, nodes, ass N, K, μ., μ.4 Sevce node Sevce node Queue Queue Ayng he euaons fo M/M/ syse o he sevce nodes: Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 6

54 Poduc-fo Souons - Exae The obabes of he ovea newo ae heefoe:, GK GK, GK GK The consan GK us be chosen such ha he obabes add o :, GK GK, GK GK, +, +, +, Thus: GK Leadng o we nown esus: - é ù é ù, ê ú ê ú.5 ë û ë û - Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss Poduc-fo Souons fo Oen and Cosed Newos A ajo beahough n he anayss of ueung newos wee he fndngs Jacson 5s & 6s n he anayss of oen newos The based on he esus Godon and Newe ae ubshed anayss echnues fo cosed newos noazng obabes Genea consans fo he acaon of he aoaches o ueung newos: A snge nd of ass s used n he newo The nube of ass Is nfne Jacson, oen newos Is fne and consan Godon/Newe, cosed newos The avas fo he ousde have exonenay dsbued neava es n he case of an oen newo Fo a nodes sevce es ae exonenay dsbued Fo a nodes ueung dscne s FCFS Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 8

55 Poduc-fo Souons fo Oen and Cosed Newos Howeve hee s aso a o of feedo: Tass ay ave a any node oen newos Tass ay eave a any node oen newos Evey ueueng node ay have ue dffeen sevng nodes The sevce ae ay deend on he oad of a node The ava ae ay deend on he oad of he node Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 9 Jacson Theoe fo Oen Newos If a nodes,, N n an oen newo fuf he saby consan: < wh + NX j j j Then he obaby fo each newo sae s:,,..., N NY GK! Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss

56 Agoh o Sove Jacson Newos Se : Cacuae fo a nodes of an oen newo he ava ae λ nea euaons syse wh N unnowns Se : Chec fo a nodes of he newo he fufen of he saby consan Se : Cacuae he agna sae obabes fo each node. Se 4: Use hese obabes o cacuae he sae obabes fo he ovea newo Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss Agoh o Sove Jacson Newos - Exae Oen newo, N4, FCFS λ λ λ4 4 4 λ Pne IO devce CPU Soage λ λ Sevce es: μ -.4s, μ -.s, μ -.6s, μ 4 -.5s Ava ae: λ λ 4 4 ass/s Tanson oees:.5, 4,.6,.4 Deve:. Aveage nube of ass n each node. Aveage esonse e of each node & he ovea syse. Aveage ueue engh and ueung es n each node 4. The obaby,,4, Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss

57 Agoh o Sove Jacson Newos - Exae Se : Cacuang he ava aes n each node + NX j j j s s - s - s - Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss Agoh o Sove Jacson Newos - Exae Se : Checng he saby consan fo each node 4 < Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 4

58 Agoh o Sove Jacson Newos - Exae Se.5: Cacuae esonse es T v fo each node and he ovea syse T v T V - T V. s T T 4 4 V V T V. 4s T V. 5 s T. V 4 65 s T V. 545 s Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 5 Agoh o Sove Jacson Newos - Exae Se.5: Cacuae ueue engh L & es T w fo each node T W T V - - T W. 6 s L T W - L. s T W 9. L.9 T W. s L.9 T. W 4 5 s L 4.5 Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 6

59 Agoh o Sove Jacson Newos - Exae Se : Cacuae he sae obabes fo each node - Souon fo M/M/ syse Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss Agoh o Sove Jacson Newos - Exae Se 4: Cacuae newo sae obaby,,4,,,, N N N,,4, Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 8

60 Godon/Newe Theoe fo Cosed Newos '*+ '*+ dffeen newo saes ossbe Saes of a cosed ueung newo ae no ndeenden Pobaby fo cean newo saes ay be exessed by he foowng oduc-fo souon,,, N GK,. s agan a noazaon consan By defnon: N G K F N Õ K N F coeaes o he feuency a nube of ass can be found n one of he sevce nodes Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 9 Godon/Newe Theoe fo Cosed Newos defned by: Wh: æ e ö F ç è ø b e N j e j Reave vsng feuency o eave ava ae j e j j 5! fo ; ;! ; < *> fo ; fo ; Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss

61 Agoh o Use Godon/Newes Aoach Se : Cacuae he eave vsng feuency A B fo a nodes C E usng he nea euaon syse Se : Cacuae F B G B nodes C E usng he eave vsng feuences and C Se : Cacuae he noazaon consan HI usng he vaues Buzen s agoh s fase: JE., bu no coveed hee Se 4: Cacuae he a eevan sae obabes of he ovea newo Se 5: Su eevan sae obabes o deve agna obabes of he saes of snge nodes z,,!,,!, zîz & o cacuae fuhe efoance ecs Noe: Mus no be deved ndvduay as no a saes can be eached, e.g. an M/M/ syse can neve ueue oe han. ass N Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss Godon/Newe Theoe Exae Reca! Gven: a cosed no ass fo o o exena nodes ueung newo havng ueung nodes Two as ccuae n he newo Sevce es ae exonena dsbued wh sevce aes μ s -, μ s - Sevce node Sevce node Queue Queue We use he foowng noaon: Sae :, wh # of ass n node # of ass n node The esung sae sace: Z {,,,,,} Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss

62 Godon/Newe Theoe Exae Se : Cacuae he eave vsng feuency A B e e Se : Cacuae F B G B æ e ö F ç b è ø b F F F F F F Se : Cacuae he noazaon consan HI G K N Õ N K F Ony one sevce node! 4 G K F F + F F + F F G K Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss Godon/Newe Theoe Exae Se 4: Cacuae he a sae obabes of he ovea newo N,,, N Õ F G K! 4, F F G K 4, F F G K 4, F F 4 G K Se 5: Deve agna obabes K 4 4 Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 4

63 Godon/Newe Theoe Exae Se 5.: Cacuae efoance ecs Uzaon of sevce nodes Aveage nube of ass n each node Thoughu of he nodes Aveage esonse e n he nodes 4 T V 6 6 T V 6 5 Teeacs / Pefoance Evauaon WS /8: Modeng & Anayss 5