7. Queueing and sharing systems. ELEC-C7210 Modeling and analysis of communication networks 1

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1 7. Queueg ad shag systes ELECC7 Modelg ad aalyss of coucato etwoks

2 7. Queueg ad shag systes Cotets Refeshe: Sle teletaffc odel Queueg dscle M/M/FIFO ( seve, watg laces, custoe laces M/M/PS ( seve, watg laces, custoe laces Alcato to acket level odellg of data taffc M/M/FIFO ( seves, watg laces, custoe laces M/M/PS ( seves, watg laces, custoe laces Alcato to flow level odellg of elastc data taffc M/M//k/kPS (k custoes, seve, k custoe laces

3 7. Queueg ad shag systes Sle teletaffc odel Custoes ave at ate l (custoes e te ut /l aveage teaval te Custoes ae seved by aallel seves he busy, a seve seves at ate (custoes e te ut / aveage sevce te of a custoe Thee ae + custoe laces the syste at least sevce laces ad at ost watg laces It s assued that blocked custoes (avg a full syste ae lost l + 3

4 7. Queueg ad shag systes Pue queueg syste Fte ube of seves ( <, sevce laces, fte ube of watg laces ( If all seves ae occued whe a custoe aves, t occues oe of the watg laces No custoes ae lost but soe of the have to wat befoe gettg seved Fo the custoe s ot of vew, t s teestg to kow e.g. what s the obablty that t has to wat too log? l 4

5 7. Queueg ad shag systes Delay I a queueg syste, soe ackets have to wat befoe gettg seved A avg acket s buffeed, f the lk s busy uo the aval Delay of a acket cossts of the watg te, whch deeds o the state of the syste uo the aval, ad the tassso te, whch deeds o the legth of the acket ad the caacty of the lk Exale: acket legth 5 bytes lk seed Gbs tassso te 5*8/,,,. s s 5

6 7. Queueg ad shag systes Pue shag syste Fte ube of seves ( <, fte ube of sevce laces ( +, o watg laces If thee ae at ost custoes the syste (x, each custoe has ts ow seve. Othewse (x >, the total sevce ate ( s shaed faly aog all custoes. Thus, the ate at whch a custoe s seved equals {,/x} No custoes ae lost, ad o oe eeds to wat befoe the sevce. But the delay s the geate, the oe thee ae custoes the syste. Thus, delay s a teesg easue fo the custoe s ot of vew. l 6

7 7. Queueg ad shag systes Thoughut I a shag syste the sevce caacty s shaed aog all actve flows. It follows that all flows get delayed (uless thee s oly a sgle actve flow By defto, the ato betwee the aveage flow sze S ad the aveage total delay D of a flow s called thoughut q, q S / D Exale: S Mbt D 5 s qs/d. Mbs 7

8 7. Queueg ad shag systes Cotets Refeshe: Sle teletaffc odel Queueg dscle M/M/FIFO ( seve, watg laces, custoe laces M/M/PS ( seve, watg laces, custoe laces Alcato to acket level odellg of data taffc M/M/FIFO ( seves, watg laces, custoe laces M/M/PS ( seves, watg laces, custoe laces Alcato to flow level odellg of elastc data taffc M/M//k/kPS (k custoes, seve, k custoe laces 8

9 7. Queueg ad shag systes Queueg dscle Cosde a sgle seve ( queueg syste Queueg dscle (also sevce dscle detees the way the seve seves the custoes It tells whethe the custoes ae seved oebyoe o sultaeously Futheoe, f the custoes ae seved oebyoe, t tells whch ode they ae take to the sevce Ad f the custoes ae seved sultaeously, t tells how the sevce caacty s shaed aog the custoes Note: I coute systes the coesodg cocet s schedulg A queueg dscle s called wokcosevg f custoes ae seved wth full sevce ate wheeve the syste s oety 9

10 7. Queueg ad shag systes okcosevg queueg dscles Fst I Fst Out (FIFO Fst Coe Fst Seved (FCFS oday queueg dscle ( queue aval ode sevce ode custoes seved oebyoe (wth full sevce ate always seve the custoe that has bee watg fo the logest te Last I Fst Out (LIFO Last Coe Fst Seved (LCFS evesed queug dscle ( stack custoes seved oebyoe (wth full sevce ate always seve the custoe that has bee watg fo the shotest te Pocesso Shag (PS fa queueg custoes seved sultaeously whe custoes the syste, each of the seved wth equal ate / also kow as oud ob (RR

11 7. Queueg ad shag systes Cotets Refeshe: Sle teletaffc odel Queueg dscle M/M/FIFO ( seve, watg laces, custoe laces M/M/PS ( seve, watg laces, custoe laces Alcato to acket level odellg of data taffc M/M/FIFO ( seves, watg laces, custoe laces M/M/PS ( seves, watg laces, custoe laces Alcato to flow level odellg of elastc data taffc M/M//k/kPS (k custoes, seve, k custoe laces

12 7. Queueg ad shag systes M/M/FIFO queue Cosde the followg sle teletaffc odel: Ifte ube of deedet custoes (k Iteaval tes ae IID ad exoetally dstbuted wth ea /l so, custoes ave accodg to a Posso ocess wth testy l Oe seve ( Sevce tes ae IID ad exoetally dstbuted wth ea / Ifte ube of watg laces ( Default queueg dscle: FIFO Usg Kedall s otato, ths s a M/M/ queue oe ecsely: M/M/FIFO queue Notato: l/taffc load

13 7. Queueg ad shag systes Related ado vaables X ube of custoes the syste at a abtay te queue legth equlbu X* ube of custoes the syste at a (tycal aval te queue legth see by a avg custoe watg te of a (tycal custoe S sevce te of a (tycal custoe D + S total te the syste of a (tycal custoe delay 3

14 7. Queueg ad shag systes State tasto daga Let X(t deote the ube of custoes the syste at te t Assue that X(t at soe te t, ad cosde what haes dug a shot te teval (t, t+h]: wth ob. lh + o(h, a ew custoe aves (state tasto f + f >, the, wth ob. h + o(h, a custoe leaves the syste (state tasto f Pocess X(t s clealy a Makov ocess wth state tasto daga l l Note that ocess X(t s a educble bthdeath ocess wth a fte state sace S {,,,...} l 4

15 7. Queueg ad shag systes 5 Equlbu dstbuto ( Local balace equatos (LBE: Noalzg codto (N:,,,K, (LBE + + l l (N ( f, < ł Ł

16 7. Queueg ad shag systes Equlbu dstbuto ( Thus, fo a stable syste ( <, the equlbu dstbuto exsts ad s a geoetc dstbuto: < X ~ Geo( P{ X E[ X ] }, (, D [ X ] (,,, K Reak: Ths esult s vald fo ay wokcosevg queueg dscle (FIFO, LIFO, PS,... Ths esult s ot sestve to the sevce te dstbuto fo FIFO eve the ea queue legth E[X] deeds o the dstbuto Howeve, fo ay syetc queueg dscle (such as LIFO o PS the esult s, deed, sestve to the sevce te dstbuto 6

17 7. Queueg ad shag systes Mea queue legth E[X] vs. taffc load E[X] Taffc load 7

18 7. Queueg ad shag systes Mea delay Let D deote the total te (delay the syste of a (tycal custoe cludg both the watg te ad the sevce te S: D + S Lttle s foula: E[X] le[d]. Thus, Reak: E[ D] E[ X ] l l l The ea delay s the sae fo all wokcosevg queueg dscles (FIFO, LIFO, PS, But the vaace ad othe oets ae dffeet! 8

19 7. Queueg ad shag systes Mea delay E[D] vs. taffc load E[D] Taffc load 9

20 7. Queueg ad shag systes Mea watg te Let deote the watg te of a (tycal custoe Sce D S, we have E[ ] E[ D] E[ S]

21 7. Queueg ad shag systes atg te dstbuto ( Let deote the watg te of a (tycal custoe Let X* deote the ube of custoes the syste at the aval te PASTA: P{X* } P{X }. Assue ow, fo a whle, that X* Sevce tes S,,S of the watg custoes ae IID ad ~ Ex( Due to the eoyless oety of the exoetal dstbuto, the eag sevce te S * of the custoe sevce also follows Ex(dstbuto (ad s deedet of eveythg else Due to the FIFO queueg dscle, S * + S + + S Costuct a Posso (ot ocess t by defg t S * ad t S * + S + + S,. Now (sce X* : > t t >t S * S S 3 t3 S S t t t t t

22 7. Queueg ad shag systes atg te dstbuto ( Sce X*, we have Deote by A(t the Posso (coute ocess coesodg to t It follows that: t > t A(t O the othe had, we kow that A(t ~ Posso (t. Thus, > > > > }( { } { } * { } * { } { } * { } { t P t P X P X t P t P X P P t t >! ( } ( { } { j t j t e t A P t P j t

23 7. Queueg ad shag systes 3 atg te dstbuto (3 By cobg the evous foulas, we get t t t j t j t j j j t j t j t j t e e e e e e t P t P j j j (! ( (! (! ( ( ( }( { } { t + + > >

24 7. Queueg ad shag systes 4 atg te dstbuto (4 atg te ca thus be eseted as a oduct JD of two deedet ado vaables J ~ Beoull( ad D ~ Ex((: ( ( ( ( ( ( ] [ ] [ ] [ ] [ } { ] [ ] [ ] [ ] [, }, { } { } { } { > > > E E D D E J P E D E J E E t e t D J P t P J P P t

25 7. Queueg ad shag systes Cotets Refeshe: Sle teletaffc odel Queueg dscle M/M/FIFO ( seve, watg laces, custoe laces M/M/PS ( seve, watg laces, custoe laces Alcato to acket level odellg of data taffc M/M/FIFO ( seves, watg laces, custoe laces M/M/PS ( seves, watg laces, custoe laces Alcato to flow level odellg of elastc data taffc M/M//k/kPS (k custoes, seve, k custoe laces 5

26 7. Queueg ad shag systes M/M/PS queue Cosde the followg sle teletaffc odel: Ifte ube of deedet custoes (k Iteaval tes ae IID ad exoetally dstbuted wth ea /l so, custoes ave accodg to a Posso ocess wth testy l Oe seve ( Sevce equeets ae IID ad exoetally dstbuted wth ea / Ifte ube of custoe laces ( Queueg dscle: PS. All custoes ae seved sultaeously a fa way wth equal shaes of the sevce caacty. Usg Kedall s otato, ths s a M/M/PS queue Notato: l/taffc load 6

27 7. Queueg ad shag systes State tasto daga Let X(t deote the ube of custoes the syste at te t Assue that X(t at soe te t, ad cosde what haes dug a shot te teval (t, t+h]: wth ob. lh + o(h, a ew custoe aves (state tasto f + f >, the, wth ob. (/h + o(h h+o(h, a custoe leaves the syste (state tasto f Pocess X(t s clealy a Makov ocess wth state tasto daga l l Note that ths s the sae educble bthdeath ocess wth a fte state sace S {,,,...} as fo the M/M/FIFO queue. l 7

28 7. Queueg ad shag systes Equlbu dstbuto Thus, fo a stable syste ( <, the equlbu dstbuto exsts ad s a geoetc dstbuto: < X ~ Geo( P{ X E[ X ] }, (, D ( Reak: Isestvty wth esect to sevce te dstbuto [ X ],,, K The esult fo the PS dscle s sestve to the sevce te dstbuto, that s: t s vald fo ay sevce te dstbuto wth ea / So, stead of the M/M/PS odel, we ca cosde, as well, the oe geeal M/G/PS odel 8

29 7. Queueg ad shag systes Mea delay Let D deote the total te (delay the syste of a (tycal custoe Sce the ea ube of custoes the syste, E[X], s the sae fo all wokcosevg queueg dscles, also the ea delay s the sae, by Lttle s esult. e ay aly the esult deved fo the FIFO dscle slde 8: E[D] 9

30 7. Queueg ad shag systes Mea delay E[D] vs. taffc load Note that the te ut s the aveage sevce equeet E[S] E[D] taffc load 3

31 7. Queueg ad shag systes Relatve thoughut A qualty of sevce easue s the elatve thoughut E[S]/E[D]: E[ S] E[ D] ( Note that above elatve thoughut easues the sze of the custoes uts of te,.e., E[S] /, ad slaly the ea delay E[D] s the sae uts. Hece, elatve thoughut easues how sall the sevce te equeet s elatve to the oveall ea delay Late o we wll defe thoughut a slghtly dffeet way the cotext of flowlevel odelg of elastc taffc, whee we take to accout seaately the sze bts (fle sze ad seve caacty exessed bt/s 3

32 7. Queueg ad shag systes Relatve thoughut E[S]/E[D] vs. taffc load.8 E[S]/E[D] taffc load 3

33 7. Queueg ad shag systes Cotets Refeshe: Sle teletaffc odel Queueg dscle M/M/FIFO ( seve, watg laces, custoe laces M/M/PS ( seve, watg laces, custoe laces Alcato to acket level odellg of data taffc M/M/FIFO ( seves, watg laces, custoe laces M/M/PS ( seves, watg laces, custoe laces Alcato to flow level odellg of elastc data taffc M/M//k/kPS (k custoes, seve, k custoe laces 33

34 7. Queueg ad shag systes Packet level odel fo data taffc ( Queueg odels ae sutable fo descbg (acketswtched data taffc at acket level Poeeg wok ade by ay eole 6 s ad 7 s elated to ARPANET, atcula L. Kleock (htt:// Cosde a lk betwee two acket outes taffc cossts of data ackets tastted alog the lk R R R R 34

35 7. Queueg ad shag systes Packet level odel fo data taffc ( Ths ca be odelled as a ue queueg syste wth a sgle seve ( ad a fte buffe ( custoe acket l acket aval ate (ackets e te ut L aveage acket legth (data uts seve lk, watg laces buffe C lk seed (data uts e te ut sevce te acket tassso te / L/C aveage acket tassso te (te uts l 35

36 7. Queueg ad shag systes Taffc ocess acket status (watg/ tassso watg te tassso te 4 3 acket aval tes ube of ackets the syste lk occuato te te te 36

37 7. Queueg ad shag systes Taffc load The stegth of the offeed taffc s descbed by the taffc load By defto, the taffc load s the ato betwee the aval ate l ad the sevce ate C/L: l ll C The taffc load s a desoless quatty By Lttle s foula, t tells the utlzato facto of the seve, whch s the obablty that the seve s busy 37

38 7. Queueg ad shag systes Exale Cosde a lk betwee two acket outes. Assue that, o aveage, 5, ew ackets ave a secod, the ea acket legth s 5 bytes, ad the lk seed s Gbs. The the taffc load (as well as, the utlzato s 5,* 5* 8/,,,.6 6% 38

39 7. Queueg ad shag systes Teletaffc aalyss ( Syste caacty C lk seed kbs Taffc load l acket aval ate s (cosdeed hee as a vaable L aveage acket legth kbts (assued hee to be costat kbt Qualty of sevce (fo the uses ot of vew P z obablty that a acket has to wat too log,.e. loge tha a gve efeece value z (assued hee to be costat z. s s Assue a M/M/ queueg syste: ackets ave accodg to a Posso ocess (wth ate l acket legths ae deedet ad detcally dstbuted accodg to the exoetal dstbuto wth ea L queug dscle s FIFO 39

40 7. Queueg ad shag systes Teletaffc aalyss ( The the quattve elato betwee the thee factos (syste, taffc, ad qualty of sevce s gve by the followg foula: P z ll C at( C, l; L, z :, ex( ( C L l z ex( ( z, f f ll < C ll C ( < ( Note: The syste s stable oly the foe case ( <. Othewse the ube of ackets the buffe gows wthout lts. 4

41 7. Queueg ad shag systes Exale Assue that ackets ave at ate l6, s.6 ackets/s ad the lk seed s C. Gbs. kbt/s. The syste s stable sce ll C.6 < The obablty P z that a avg acket has to wat too log (.e. loge tha z s s P z at(.,.6;,.6ex( 4.» % 4

42 7. Queueg ad shag systes Caacty vs. aval ate Gve the qualty of sevce equeet that P z < %, the equed lk seed C deeds o the aval ate l as follows: C( l { c > ll at( c, l;, <.} lk seed C (Gbs aval ate l (ackets/s 4

43 7. Queueg ad shag systes Qualty of sevce vs. aval ate Gve the lk seed C. Gbs. kbt/s, the qualty of sevce P z deeds o the aval ate l as follows: P ( l at(., l;, z.8 qualty of sevce P z aval ate l (ackets/s 43

44 7. Queueg ad shag systes Qualty of sevce vs. caacty Gve the aval ate l6, s.6 ackets/s, the qualty of sevce P z deeds o the lk seed C as follows: P z ( R at( C,.6;,.8 qualty of sevce P z lk seed C (Gbs 44

45 7. Queueg ad shag systes Suay o alcato to acket level odellg of data taffc M/M/ odel ay be aled (to soe extet to acket level odellg of data taffc custoe IP acket l acket aval ate (ackets e te ut / aveage acket tassso te (te uts l/ taffc load Qualty of sevce s easued e.g. by the acket delay P z obablty that a acket has to wat too log,.e. loge tha a gve efeece value z P z P{ > z} e ( z 45

46 7. Queueg ad shag systes Multlexg ga e detee load so that ob. P z < % fo z (te uts Multlexg ga s descbed by the taffc load as a fucto of the sevce ate.8 load sevce ate 46

47 7. Queueg ad shag systes Cotets Refeshe: Sle teletaffc odel Queueg dscle M/M/FIFO ( seve, watg laces, custoe laces M/M/PS ( seve, watg laces, custoe laces Alcato to acket level odellg of data taffc M/M/FIFO ( seves, watg laces, custoe laces M/M/PS ( seves, watg laces, custoe laces Alcato to flow level odellg of elastc data taffc M/M//k/kPS (k custoes, seve, k custoe laces Deo: M/M//FIFO 47

48 7. Queueg ad shag systes M/M/FIFO queue Cosde the followg sle teletaffc odel: Ifte ube of deedet custoes (k Iteaval tes ae IID ad exoetally dstbuted wth ea /l so, custoes ave accodg to a Posso ocess wth testy l Fte ube of seves ( < Sevce tes ae IID ad exoetally dstbuted wth ea / Ifte ube of watg laces ( Default queueg dscle: FIFO Usg Kedall s otato, ths s a M/M/ queue oe ecsely: M/M/FIFO queue Notato: l/(taffc load 48

49 7. Queueg ad shag systes State tasto daga Let X(t deote the ube of custoes the syste at te t Assue that X(t at soe te t, ad cosde what haes dug a shot te teval (t, t+h]: wth ob. lh + o(h, a ew custoe aves (state tasto f + f >, the, wth ob. {,} h + o(h, a custoe leaves the syste (state tasto f Pocess X(t s clealy a Makov ocess wth state tasto daga l l Note that ocess X(t s a educble bthdeath ocess wth a fte state sace S {,,,...} l l + l 49

50 7. Queueg ad shag systes 5 Equlbu dstbuto ( Local balace equatos (LBE fo < : Local balace equatos (LBE fo : (LBE ( l + +,K,, (LBE!! ( l l,,,,! ( ( K l

51 7. Queueg ad shag systes 5 Equlbu dstbuto ( Noalzg codto (N:!( (! (!( (! (! (! (!! (, Notato : f, (N b a b a < + ł Ł + ł Ł + ł Ł +

52 7. Queueg ad shag systes 5 Equlbu dstbuto (3 Thus, fo a stable syste ( <, that s: l<, the equlbu dstbuto exsts ad s as follows: < b a b a b a b a ,, :,, : K K,,,,,,, } {!! ( X P b a b a

53 7. Queueg ad shag systes 53 Pobablty of watg Let deote the obablty that a avg custoe has to wat Let X* deote the ube of custoes the syste at a aval te A avg custoe has to wat wheeve all the seves ae occued at he aval te. Thus, PASTA: P{X* } P{X }. Thus, } * { X P b a b +!( (! } * { X P + : :

54 7. Queueg ad shag systes 54 Mea ube of watg custoes Let X deote the ube of watg custoes equlbu The!( ( ( ( ( ] [ X E 3 ] [ : ] [ : + X E X E

55 7. Queueg ad shag systes 55 Mea watg te Let deote the watg te of a (tycal custoe Lttle s foula: E[X ] le[]. Thus, l l l X E E ( ] [ ] [ ( ] [ : ] [ : E E

56 7. Queueg ad shag systes 56 Mea delay Let D deote the total te (delay the syste of a (tycal custoe cludg both the watg te ad the sevce te S: D + S The, l ( ] [ ] [ ] [ + ł Ł + + S E E D E ( ( ] [ : ] [ : ł Ł + + ł Ł + D E D E

57 7. Queueg ad shag systes 57 Mea queue legth Let X deote the ube of custoes the syste (queue legth equlbu Lttle s foula: E[X] le[d]. Thus, l l l l D E X E + + ] [ ] [ ] [ : ] [ : X E X E

58 7. Queueg ad shag systes 58 atg te dstbuto ( Let deote the watg te of a (tycal custoe Let X* deote the ube of custoes the syste at the aval te The custoe has to wat oly f X*. Ths haes wth ob.. Ude the assuto that X*, the syste, howeve, looks lke a oday M/M/ queue wth aval ate l ad sevce ate. Let deote the watg te of a (tycal custoe ths M/M/ queue Let X* deote the ube of custoes the syste at the aval te It follows that, } *' ' { } * { } * { } { } { ( > > > > t e X t P X t P X P t P P t

59 7. Queueg ad shag systes 59 atg te dstbuto ( atg te ca thus be eseted as a oduct JD of two de. ado vaables J ~ Beoull( ad D ~ Ex((: ( ( ( ( ( ( ( ] [ ] [ ] [ ] ' [ } { ] [ '] [ ] [ ] [, } ', { } { } { } { > > > t E E D D E J P E D E J E E t e t D J P t P J P P

60 7. Queueg ad shag systes Exale ( Pte oble Cosde the followg two dffeet cofguatos: Oe ad te (IID tg tes ~ Ex( Two slowe aallel tes (IID tg tes ~ Ex( Cteo: ze ea delay E[D] Oe ad te (M/M/ odel wth l/(: E[D ] Two slowe tes (M/M/ odel wth l/(: ( (+ + E [ D ] E[ D ] > E[ D ] 6

61 7. Queueg ad shag systes Exale ( E[D ]/E[D ] Taffc load 6

62 7. Queueg ad shag systes Cotets Refeshe: Sle teletaffc odel Queueg dscle M/M/FIFO ( seve, watg laces, custoe laces M/M/PS ( seve, watg laces, custoe laces Alcato to acket level odellg of data taffc M/M/FIFO ( seves, watg laces, custoe laces M/M/PS ( seves, watg laces, custoe laces Alcato to flow level odellg of elastc data taffc M/M//k/kPS (k custoes, seve, k custoe laces 6

63 7. Queueg ad shag systes M/M/PS queue Cosde the followg sle teletaffc odel: Ifte ube of deedet custoes (k Iteaval tes ae IID ad exoetally dstbuted wth ea /l so, custoes ave accodg to a Posso ocess wth testy l Fte ube of seves ( < Sevce equeets ae IID ad exoetally dstbuted wth ea / Ifte ube of custoe laces ( Queueg dscle: PS. If thee ae at ost custoes the syste (, each custoe has ts ow seve. Othewse ( >, the total sevce ate ( s shaed faly aog all custoes. Usg Kedall s otato, ths s a M/M/PS queue Notato: l/(taffc load 63

64 7. Queueg ad shag systes State tasto daga Let X(t deote the ube of custoes the syste at te t Assue that X(t at soe te t, ad cosde what haes dug a shot te teval (t, t+h]: wth ob. lh + o(h, a ew custoe aves (state tasto f + f >, the, wth ob. {,/} h + o(h {,} h + o(h, a custoe leaves the syste (state tasto f Pocess X(t s clealy a Makov ocess wth state tasto daga l l Note that ths s the sae educble bthdeath ocess wth a fte state sace S {,,,...} as fo the M/M/FIFO queue. l l + l 64

65 7. Queueg ad shag systes Equlbu dstbuto (3 Thus, fo a stable syste ( <, that s: l<, the equlbu dstbuto exsts ad s as follows: < P{ X } (!! a + b a + b,,,, K,, +, K Reak: Isestvty wth esect to sevce te dstbuto The esult fo the PS dscle s sestve to the sevce te dstbuto, that s: t s vald fo ay sevce te dstbuto wth ea / So, stead of the M/M/PS odel, we ca cosde, as well, the oe geeal M/G/PS odel 65

66 7. Queueg ad shag systes Mea delay Let D deote the total te (delay the syste of a (tycal custoe Sce the ea ube of custoes the syste, E[X], s the sae fo all wokcosevg queueg dscles, also the ea delay s the sae, by Lttle s esult. e ay aly the esult deved fo the FIFO dscle slde 55: E[ D] ( + ( whee w efes to the obablty P{ X * }! (!( b a + b 66

67 7. Queueg ad shag systes Mea delay E[D] vs. taffc load Note that the te ut s the aveage sevce equeet E[S] 6 5 E[D] taffc load 67

68 7. Queueg ad shag systes 68 Relatve thoughut A qualty of sevce easue s the elatve thoughut E[S]/E[D]: ( ( ( ( ( ( ] [ ] [ + + D E S E ( ( ( ] [ ] [ ( ] [ ] [ : : + + D E S E D E S E

69 7. Queueg ad shag systes Relatve thoughut E[S]/E[D] vs. taffc load E[S]/E[D] taffc load 69

70 7. Queueg ad shag systes Cotets Refeshe: Sle teletaffc odel Queueg dscle M/M/FIFO ( seve, watg laces, custoe laces M/M/PS ( seve, watg laces, custoe laces Alcato to acket level odellg of data taffc M/M/FIFO ( seves, watg laces, custoe laces M/M/PS ( seves, watg laces, custoe laces Alcato to flow level odellg of elastc data taffc M/M//k/kPS (k custoes, seve, k custoe laces 7

71 7. Queueg ad shag systes Flow level odel fo elastc data taffc ( Shag odels ae sutable fo descbg elastc data taffc at flow level Elastcty efes to the adatve sedg ate of TCP flows Ths kd of odels have bee oosed, e.g., by J. Robets ad hs eseaches (htt://eso.d.faceteleco.f/obets/ Cosde a lk betwee two acket outes taffc cossts of TCP flows loadg the lk R R R R 7

72 7. Queueg ad shag systes Flow level odel fo elastc data taffc ( The slest odel s a sgle seve ( ue shag syste wth a fxed total sevce ate of custoe TCP flow fle to be tasfeed l flow aval ate (flows e te ut S aveage flow sze aveage fle sze (data uts seve lk C lk seed (data uts e te ut sevce te fle tasfe te wth full lk seed / S/C aveage fle tasfe te wth full lk seed (te uts l 7

73 7. Queueg ad shag systes Taffc ocess flow duato tasfe te wth full lk ate exta delay flow aval tes te 4 3 ube of flows the syste elatve tassso ate fo sgle flows te / /4 /3 te 73

74 7. Queueg ad shag systes Teletaffc aalyss ( Syste caacty C lk seed Mbs Taffc load l flow aval ate flows e secod (cosdeed hee as a vaable S aveage flow sze kbts (assued hee to be costat Mbt the load / / Qualty of sevce (fo the uses ot of vew q thoughut E[S] / E[D] Assue a M/G/PS shag syste: flows ave accodg to a Posso ocess (wth ate l flow szes ae deedet ad detcally dstbuted accodg to ay dstbuto wth ea S q E[S] / E[D] l E[S] / E[N] C ( 74

75 7. Queueg ad shag systes Teletaffc aalyss ( The the quattatve elato betwee the thee factos (syste, taffc, ad qualty of sevce s gve by the followg foula: q Xut( C, l; S : C ls, C(, f f ls ls < C C ( < ( Note: Iteetato: The thoughut that a gve flow obtas equals the eag (o excess caacty C(. Cf. the esult slde 68 (M/G/PS wth,, dffeece s that elatve thoughut ad thoughut ae defed slghtly dffeetly. I elatve thoughut E[S] easues the sevce te whle thoughut E[S] s the ea fle sze bts. The syste s stable oly the foe case ( <. Othewse the ube of flows as well as the aveage delay gows wthout lts. I othe wods, the thoughut of a flow goes to zeo. 75

76 7. Queueg ad shag systes Exale Assue that flows ave at ate l6 flows e secod ad the lk seed s C Mbs. Gbs. The syste s stable sce Thoughut s C S l 6.6 < q Xut(,6; 6 4 Mbs.4 Gbs 76

77 7. Queueg ad shag systes Caacty vs. aval ate Gve the qualty of sevce equeet that q 4 Mbs, the equed lk seed C deeds o the aval ate l as follows: C( l { c > ls Xut( c, l; 4} ls lk seed C (Mbs aval ate l (flows e secod 77

78 7. Queueg ad shag systes Qualty of sevce vs. aval ate Gve the lk seed C Mbs, the qualty of sevce q deeds o the aval ate l as follows: q ( l Xut(, l; ls, l < /S 8 thoughut q (Mbs aval ate l (flows e secod 78

79 7. Queueg ad shag systes Qualty of sevce vs. caacty Gve the aval ate l6 flows e secod, the qualty of sevce q deeds o the lk seed C as follows: q ( C Xut( C,6; C 6S, C > 6S thoughut q (Mbs lk seed C (Mbs 79

80 7. Queueg ad shag systes Suay of alcato to flow level odellg of elastc data taffc M/G/PS odel s alcable to flow level odellg of elastc data taffc custoe TCP flow l flow aval ate (flows e te ut access lk seed fo a flow (data uts e te ut C seed of the shaed lk (data uts e te ut E[L] aveage flow sze (data uts E[S] / E[L]/ aveage flow tasfe te wth access lk ate l/( taffc load A qualty of sevce easue s the thoughut q E[ L] E[ D] E[ S] E[ D] ( ( + ( C ( ( + ( 8

81 7. Queueg ad shag systes Thoughut q vs. taffc load Note that the ate ut s the lk ate C.8 C thoughut q.6.4. C/ C/3 C/ C/ taffc load 8

82 7. Queueg ad shag systes Cotets Refeshe: Sle teletaffc odel Queueg dscle M/M/FIFO ( seve, watg laces, custoe laces M/M/PS ( seve, watg laces, custoe laces Alcato to acket level odellg of data taffc M/M/FIFO ( seves, watg laces, custoe laces M/M/PS ( seves, watg laces, custoe laces Alcato to flow level odellg of elastc data taffc M/M//k/kPS (k custoes, seve, k custoe laces 8

83 7. Queueg ad shag systes M/M//k/kPS queue Cosde the followg sle teletaffc odel: Fte ube of deedet custoes (k < ooff tye custoes (alteatg betwee dleess ad actvty Idle tes ae IID ad exoetally dstbuted wth ea / Oe seve ( Sevce equeets ae IID ad exoetally dstbuted wth ea / As ay custoe laces as custoes ( k Queueg dscle: PS. Usg Kedall s otato, ths s a M/M//k/kPS queue Ooff tye custoe: dleess sevce 83

84 7. Queueg ad shag systes State tasto daga Let X(t deote the ube of custoes the syste at te t Assue that X(t at soe te t, ad cosde what haes dug a shot te teval (t, t+h]: f < k, the, wth ob. (kh + o(h, a dle custoe becoes actve (state tasto f + f >, the, wth ob. (/h + o(h +o(h, a actve custoe becoes dle (state tasto f Pocess X(t s clealy a Makov ocess wth state tasto daga k (k k k Note that ocess X(t s a educble bthdeath ocess wth a fte state sace S {,,,k} 84

85 7. Queueg ad shag systes Equlbu dstbuto ( Local balace equatos (LBE: + ( k ( k + (LBE ( k! ( k k,,, K, k 85

86 7. Queueg ad shag systes 86 Equlbu dstbuto ( Noalzg codto (N: ł Ł k k k k k k k k k k k k k k k ' ' '! (! (! (! (! ( ( ( ( ( (N (

87 7. Queueg ad shag systes THE END hat you should udestad/eebe: exales of queueg ad shag systes what s eat by queueg dscle ad what kd of dscles ae FIFO, LIFO ad PS cocets of delay, taffc load ad thoughut what s eat by sestvty of soe systes to teaval ad/o sevce te dstbuto stablty codtos fo the systes (.e., whe a syste s stable? how ea delay ad watg tes ae calculated, dea behd devato of the watg te dstbutos 87