Redes de Computadores
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1 Redes de Compuadoes Deay Modes i Compue Newoks Maue P. Ricado Facudade de Egehaia da Uivesidade do Poo
2 » Wha ae he commo muipexig saegies?» Wha is a Poisso pocess?» Wha is he Lie heoem?» Wha is a queue?» Wha is he meaig of sevice ime /m i a queue of packes?» Wha is he meaig of affic iesiy i a queue mode?» Wha is he pobabiiy of a M/M/ queue beig i a give sae?» Wha is he mea umbe of cies i a M/M/ queue? Wha is he mea waiig ime i a M/M/ queue? Wha is he eaioship bewee N ad i a M/M/ queue?» Wha ae he diffeeces bewee M/M/ ad M/G/ queues? How o esimae mea umbe of packes ad mea deay i a M/G/ queue?» How o mode a ewok of asmissio ies? How o cacuae he mea umbe of packes ad mea deay i his case?» Wha is a Jackso Newok? Why is i impoa? 2
3 Muipexig Taffic o a Lik Commuicaio ik» Bi pipe wih a give capaciy C bi/s» Lik capaciy ae a which bis ae asmied o he ik» Lik may aspo muipexed affic seams C Muipexig saegies» Saisica Muipexig» Fequecy Divisio Muipexig» Time Divisio Muipexig Muipexig saegy affecs affic deay 3
4 Saisica Muipexig Packes of a affic seams meged i a sige queue Packes asmied o a fis-come fis-seved basis Time equied o asmi a packe of egh L T fame =L/C C 4
5 FDM Fequecy Divisio Muipexig Lik capaciy C subdivided io m poios Chae badwidh W subdivided io m chaes of W/m Hz Capaciy of each chae C/m Time equied o asmi a packe of egh L T fame = Lm/C 5
6 TDM Time Divisio Muipexig Time axis divided io m sos of fixed egh usuay oe oce og Commuicaio m chaes wih capaciy C/m Time equied o asmi a packe of egh L T fame =Lm/C oe oce ime fame 6
7 Deay o Compue Newoks Deay» Impoa pefomace paamee i compue ewoks» Chaaceized usig queue modes Queue mode» Cusomes aive a adom imes o obai sevice» Cusome packe o be asmied hough a ik» Seve a packe = asmi a packe» Sevice ime packe asmissio ime =T pacfame = L/C Queue modes eabe he quaificaio of» Aveage umbe of cusomes/packes i he ewok» Aveage deay pe packe waiig pus sevice imes C 7
8 Compue Newoks Modeed as Queue Newoks Mobie ewok Goba ISP Home ewok Regioa ISP Isiuioa ewok 8
9 Poisso Disibuio ad Poisso Pocess Poisso disibuio wih paamee m Poisso pocess» T=m, e.g. aivas/s» P[ aivas i ieva T ] = 0,,,! ] [ m e p N P m m N Va N E ] [ ] [! e T p T p T T N Va N E ] [ ] [! e T T p T p T 9
10 Ie-Aiva Ieva A Saisica Chaaceizaio A Aiva of cies A e p A P A P F ] [ ] [ 0 A A e F pdf f A A e f [ ] A P ] [ A E 2 ] [ A Va A ime ieva bewee he aiva of cosecuive cies Expoeia disibuio 0
11 Wha is he diffeece bewee deemiisic ad Poisso aivas?
12 Makov Pocess - Popeies Megig Popey» A, A 2, A k ae idepede Poisso Pocesses wih aes λ, λ 2, λ k» A=S A i si is a Poisso pocess, wih ae λ=sλ i Spiig popey» Packes aive o a oue accodig o a Poisso Pocess A,λ» They ae oued adomy o wo oupu ies wih pobabiiies p ad -p» Packes eavig he oue si ae Poisso Pocesses, chaaceized by A,pλ ad A,-pλ λ λ 2 λ k λ pλ -pλ λ=sλ i 2
13 Queue Mode Queue mode used fo» Cusomes waiig i ie» Packes i a ewok Cie aiva Aiva ae = cie/s Used o deemie» Aveage umbe of cies i he sysem N» Aveage deay expeieced by a cie T λ Queue m Cie depaue Sevice ae= m cie/s Sevice ime= / m Queue chaaceized i ems of» - aiva ae of cie aveage umbe of cies pe ime ui» m - sevice ae aveage umbe of cies he seve pocesses pe ime ui» /m affic iesiy occupaio of he seve Keda oaio A/S/s/K» A aiva saisica pocess» S sevice saisica pocess» s umbe of seves» K capaciy of he sysem i buffes 3
14 Lie s Theoem N=T» N- aveage umbe of cies i a sysem» T aveage amou of ime a cie speds i he sysem» aiva ae of cies o he sysem T=T w +T s» T w ime a cie wais i he queue fo beig seved» T s sevice ime N=N w +N s» N w umbe of cies waiig i he queue fo beig seved» N s umbe of cies beig seved N w =T w 4
15 N w =T w T w =N w / The mea ime a cie has o wai befoe beig seved T w depeds o he umbe of cies waiig N w ad o he aiva ae of cies No depedece o he sevice ae?! Ca you expai i? 5
16 Lie s Theoem Ca be appied o a sige Queue λ m N,T Ca be appied o a compex sysem» Fo each seam i N i = i T i» Fo he sysem: λ =S i N=S N i T=S N i / S i T=N/ λ k λ k 6
17 M/M/ Queue M/M/» Poisso aiva, expoeia sevice ime λ m Modeed by a Makov Chai» Sae k - k cies i he queue» pi,j pobabiiy of asiio fom sae i o sae j» Whe d 0 pi, i+= d pi, i-= md pi, i= - dmd p0, 0= - d pi, j=0 fo ohe vaues i, j d d N,T d d d 0 2 k md md md md» Bih-deah chai Tasiios bewee adjace saes d ad md become fow aes bewee saes p i, i p d d e p 0,0 p0 d e d d d d 7
18 M/M/ Queue Equiibium Aaysis Pj pobabiiy of he Makov chai be i sae j Makov Chai - goba baace equaios d P j p j, i P i p i, j d i0 i0 i j i j I he case of M/M/ P 0 d P md P P0 P2 P P P0 2 P 0 i0 i0 P i d d d 0 2 k md md md md i P0 P0 P0 P 8
19 M/M/ Queue Aveage Queue size N Aveage amou of ime he cie speds i he sysem, T» Lie s fomua, T=N/ Aveage waiig ime T w Aveage umbe of cies waiig i he queue, N w 0 0 P N m m m 0 P N m T m m m s w T T T m m N T N w w 9
20 M/M/ Queue N=f
21 M/M/: 0.9 N=9 Why have cies o wai if he seve is busy oy 90% of his ime? Wha woud happe fo D/D/, 0.9? 2
22 Packe Legh, Sevice Time, Speed» 00 packe/s ae equied o be asmied hough a ik» Packes aive accodig o a Poisso pocess» Packe eghs ae expoeiay disibued E[L]=0 4 bi/packe» Lik has capaciy C=0 Mbi/s The» Aiva ae: =00 packe/s» Sevice ae: m=c/e[l]= 0 7 /0 4 = 0 3 packe/s» =/m0., N//9, TN//900 s Assume ow: 0 ad C =0C m 0C/E[L]0m» The ad N =N bu T =N / =T/0 The speed of he sysem iceases! 22
23 M/M//B Queue M/M/ queue has imied capaciy B buffes» Packes ca be os» Pobabiiy of packe beig os = PB Queue is fu Aaysis simia o M/M/ P B i0 P i P P0 0 B P B B B Paicua cases, P B B, P B m 23
24 M/G/ Queue Poisso aivas λ m Geea idepede Sevice imes Poisso aivas a ae λ Sevice ime has abiay disibuio wih give E[X] ad E[X 2 ]» Sevice imes Idepede ad Ideicay Disibued IID» Idepede of aiva imes» E[sevice ime] =E[X]= /μ» Sige Seve queue 24
25 M/G/ Queue Poaczek-Khichi P-K Fomua T w 2 E[ X ] 2 whee ρ = λ/μ = λe[x] = ie uiizaio Fom Lie s Theoem» N w = λt w» T = T w + E[X] = T w + /m» N = λt= λt w + /m =N w + ρ 25
26 M/G/ Queue Poof of P-K Fomua T w 2 E[ X ] 2 Le T w i - waiig ime i queue of i h aiva Ri esidua sevice ime see by he i h aiva N w i umbe of cies foud i queue by he i h aiva Xi sevice ime of he i h aiva» Usig Lie s fomua T T w w E i Tw m i jin w E X j R i i w T i T EN i EX i ER i ER i w w Ri w T w E Ri N m λ i h aiva [i-n w i] h aiva Ri N w i,t w i m 26
27 M/G/ Queue Poof of P-K Fomua M umbe of cies seved by ime M i i M i i M X M X R i R E M, = aiva ae= depaue ae X E M X i R E M i i Ri E T w 2 ] [ 2 X E T w 27
28 M/G/ Exampes Case M/M/» E[X]= /μ ; E[X 2 ]= 2/μ 2 T w 2 m m Case M/D/» Deemiisic, cosa sevice ime /μ» E[X]= /μ ; E[X 2 ]= /μ 2 T w 2m 2 2m 28
29 Poisso aivas λ 2 Assume Queue is M/D/. Ca he aiva of packes o Queue 2 be descibed as a Poisso pocess? 29
30 Newoks of Tasmissio Lies - Pobems Case» Aiva o Q Poisso, λ Poisso aivas» Assume coa packe egh Q = M/D/ λ 2» Aiva o Q 2 is o Poisso; λ 2 <m 2 /λ 2 >/m 2 o waiig a Q 2 Case 2» Q =M/M/» aiva o Q 2 sogy eaed o packe egh» og packes equie og sevice a each ode» shoe packes wi cach up og packes ieaiva imes chage Q 2 cao be modeed as M/M/ 30
31 Keiock Idepedece Appoximaio Megig sevea packe seams o a asmissio ie esoes idepedece of ieaiva imes ad packe eghs M/M/ ca be used o mode each commuicaio ik Appoximaio good fo» sysems ivovig Poisso seam aivas a he ey pois» packe eghs eay expoeiay disibued» desey coeced ewoks» Modeae o heavy affic oads 3
32 Keiock Idepedece Appoximaio Le» x p = aiva ae of packes aog pah p» λ ij = aiva ae of packes o ik i,j» μ ij = sevice ae o ik i,j N Lik queues idepede M/M/ queues ij x p a p avesig iki,j N ij Ad» N= Aveage umbe of packes i ewok» T Aveage packe deay i ewok ij m ij ij ij N ij xp oaexea aiva ae T i, j a pahsp ij N 32
33 Jackso Newoks Aiva ae a ode j K j j ipij, j,2,..., K i Node j j λ j λ j m j K i j P ji λ P j λ 2 P 2j λ K P Kj λ j P j λ P j2 λ P jk Idepede ouig of packes» Whe a packe eaves ode i i comes o ode j wih pobabiiy P ij» Packes ca oop iside ewok» Packe eaves he sysem a ode j wih pobabiiy 2 P K i P ji λ λ m 2 m 2 /2 2/3 /2 /3 33
34 Jackso Newoks Le he sae of he sysem be defied by j umbe of cies i Q j, 2,, K Jackso s heoem:» Sae of Q j j is idepede K of sae of ohe queues» Simia o idepede M/M/ queues!» Simia o Keiock s idepedece Agai, by Lie s heoem K j N j N j P K K j j P j j j j, whee j m j j j j K N j N j T j j 34
35 Newoks of asmissio ies / Jackso ewoks Give exampes of ewoks ha coud be descibed by hese modes 35
36 Jackso Newok - Exampe 6 i i 9 s - 2 T N N 6 i N i s i N i 36
37 Homewok. Review sides 2. Read Besekas&Gaage» Secios 3., 3.2, 3.3, 3.5, 3.6, Aswe quesios a moode 37
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