6. Loss systems. ELEC-C7210 Modeling and analysis of communication networks 1

Transcription

1 ELEC-C72 Modelg ad aalyss of commucato etwors

2 Cotets Refresher: Smple teletraffc model Posso model customers, servers Applcato to flow level modellg of streamg data traffc Erlag model customers, ; servers Applcato to telephoe traffc modellg tru etwor Bomal model ; customers, servers Egset model ; customers, ; servers Applcato to telephoe traffc modellg access etwor 2

3 Smple teletraffc model Customers arrve at rate κ customers per tme ut /κ = average ter-arrval tme Customers are served by parallel servers Whe busy, a server serves at rate customers per tme ut / = average servce tme of a customer There are m customer places the system at least servce places ad at most m watg places It s assumed that bloced customers arrvg a full system are lost κ m 3

4 Ifte system Ifte umber of servers, o watg places m No customers are lost or eve have to wat before gettg served Sometmes, ths hypothetcal model ca be used to get some approxmate results for a real system wth fte system capacty Always, t gves bouds for the performace of a real system wth fte system capacty t s much easer to aalyze tha the correspodg fte capacty models κ 4

5 Pure loss system Fte umber of servers ;, servce places, o watg places m If the system s full wth all servers occuped whe a customer arrves, t s ot served at all but lost Some customers may be lost From the customer s pot of vew, t s terestg to ow e.g. What s the probablty that the system s full whe t arrves? κ 5

6 Traffc testy The stregth of the offered traffc s descrbed by the traffc testy a By defto, the traffc testy a s the product of the arrval rate κ ad the mea holdg tme h: The traffc testy s a dmesoless quatty. Ayway, the ut of the traffc testy a s called erlag erl By Lttle s formula: traffc of oe erlag meas that oe chael s occuped o average Example: a κh O average, there are 8 ew calls a hour, ad the average holdg tme s 3 mutes. The the traffc testy s a 8 3/ 6 9 erlag 6

7 Blocg I a loss system some calls are lost a call s lost f all chaels are occuped whe the call arrves the term blocg refers to ths evet There are two dfferet types of blocg quattes: Call blocg B c = probablty that a arrvg call fds all chaels occuped = the fracto of calls that are lost Tme blocg B t = probablty that all chaels are occuped at a arbtrary tme = the fracto of tme that all chaels are occuped The two blocg quattes are ot ecessarly equal Example: your ow moble But f calls arrve accordg to a Posso process, the B c = B t Call blocg s a better measure for the qualty of servce expereced by the subscrbers but, typcally, tme blocg s easer to calculate 7

8 Call rates I a loss system each call s ether lost or carred. Thus, there are three types of call rates: κ offered = arrval rate of all call attempts κ carred = arrval rate of carred calls κ lost = arrval rate of lost calls κ offered κ carred κ lost κ κ κ offered carred lost κ κb c carred κ, B c κ lost κ 8

9 Traffc streams The three call rates lead to the followg three traffc cocepts: Traffc offered a offered = κ offered h Traffc carred a carred = κ carred h Traffc lost a lost = κ lost h a a a offered carred lost ab a c carred a, B Traffc offered ad traffc lost are hypothetcal quattes, but traffc carred s measurable, sce by Lttle s formula t correspods to the average umber of occuped chaels o the l c a lost a κ offered κ lost κ carred 9

10 Cotets Refresher: Smple teletraffc model Posso model customers, servers Applcato to flow level modellg of streamg data traffc Loss systems Erlag model customers, ; servers Applcato to telephoe traffc modellg tru etwor Bomal model ; customers, servers Egset model ; customers, ; servers Applcato to telephoe traffc modellg access etwor

11 Posso model M/M/ Defto: Posso model s the followg smple teletraffc model: Ifte umber of depedet customers Iterarrval tmes are IID ad expoetally dstrbuted wth mea /κ so, customers arrve accordg to a Posso process wth testy κ Ifte umber of servers Servce tmes are IID ad expoetally dstrbuted wth mea / No watg places m Posso model: Usg Kedall s otato, ths s a M/M/ queue Ifte system, ad, thus, lossless Notato: a κ.traffc testy, cf. slde 6

12 State trasto dagram Let Xt deote the umber of customers the system at tme t Assume that Xt at some tme t, ad cosder what happes durg a short tme terval t, t h]: wth prob. κh oh, a ew customer arrves state trasto f >, the, wth prob. h oh, a customer leaves the system state trasto, Process Xt s clearly a Marov process wth state trasto dagram κ κ 2 2 Note that process Xt s a rreducble brth-death process wth a fte state space S {,,2,...} κ 3 2

13 Equlbrum dstrbuto Local balace equatos LBE: Normalzg codto N: κ κ a a!,,,2,ϑ a! a,,! a, a e e LBE N 3

14 Equlbrum dstrbuto 2 Thus, the equlbrum dstrbuto s a Posso dstrbuto: X } Possoa P{ X } E[ X ] a, D 2 a! e, a [ X ] a,,,2, ϑ Remar: Isestvty wth respect to servce tme dstrbuto The result s sestve to the servce tme dstrbuto, that s: t s vald for ay servce tme dstrbuto wth mea / So, stead of the M/M/ model, we ca cosder, as well, the more geeral M/G/ model 4

15 Cotets Refresher: Smple teletraffc model Posso model customers, servers Applcato to flow level modellg of streamg data traffc Erlag model customers, ; servers Applcato to telephoe traffc modellg tru etwor Bomal model ; customers, servers Egset model ; customers, ; servers Applcato to telephoe traffc modellg access etwor 5

16 Flow level model for streamg CBR traffc Ifte system s sutable for descrbg streamg CBR traffc at flow level The trasmsso rate ad flow durato of a streamg flow are sestve to the etwor state Ths d of models were appled 9 s to the teletraffc aalyss of CBR traffc ATM etwors Cosder a l betwee two pacet routers traffc cossts of UDP flows carryg CBR traffc le VoIP ad loadg the l R R R R 6

17 Flow level model for streamg CBR traffc 2 Model: a fte system customer = UDP flow = CBR bt stream κ = flow arrval rate flows per tme ut servce tme = flow durato h = / = average flow durato tme uts Bufferless flow level model: whe the total trasmsso rate of the flows exceeds the l capacty, bts are lost uformly from all flows κ 7

18 Traffc process flow duratos flow arrval tmes total bt rate umber of flows lost traffc tme C carred traffc tme 8

19 Offered traffc Let r deote the bt rate of ay flow The volume of offered traffc s descrbed by average total bt rate R By Lttle s formula, the average umber of flows s Ths may be called traffc testy cf. slde 6 It follows that a κh R ar κhr 9

20 Loss rato Let N deote the umber of flows the system Whe the total trasmsso rate Nr exceeds the l capacty C, bts are lost wth rate The average loss rate s thus Nr, C E [ Nr, C ] E[max{ Nr, C,}] By defto, the loss rato p loss gves the rato betwee the traffc lost ad the traffc offered: p loss E[ Nr, C E[ Nr] ] ar E[ Nr, C ] 2

21 Teletraffc aalyss System capacty C = r = l speed bps Traffc load R = ar = offered traffc bps r = bt rate of a flow bps. Qualty of servce from the users pot of vew p loss = loss rato Assume a M/G/ fte system: flows arrve accordg to a Posso process wth rate κ flow duratos are depedet ad detcally dstrbuted accordg to ay dstrbuto wth mea h 2

22 Teletraffc aalyss 2 The the quattatve relato betwee the three factors system, traffc, ad the qualty of servce s gve by the followg formula Example: 2 a 4.36 p p loss. a, a a loss LR, : e a,! 22

23 Capacty vs. traffc Gve the qualty of servce requremet that p loss %, the requred capacty depeds o the traffc testy a as follows: a m{,2, ϑ LR, a ;.} 8 capacty traffc a 23

24 Qualty of servce vs. traffc Gve the capacty 2, the requred qualty of servce, p loss depeds o the traffc testy a as follows:, p loss a, LR2, a.8 qualty of servce, p loss traffc a 24

25 Qualty of servce vs. capacty Gve the traffc testy a 5. erlag, the requred qualty of servce, p loss depeds o the capacty as follows:, p loss, LR,5..8 qualty of servce, p loss capacty 25

26 Multplexg ga We determe traffc testy a so that loss rato p loss ; % Multplexg ga s descrbed by the traffc testy per capacty ut, a., as a fucto of capacty.8 ormalzed traffc a capacty 26

27 Summary of flow level modelg of streamg data Posso model may be appled to flow level modelg of streamg data traffc customer = UDP flow wth costat bt rate CBR κ = flow arrval rate flows per tme ut h = / = average flow durato tme uts a = κ/ = traffc testy r = bt rate of a flow data uts per tme ut N = r of actve flows obeyg Possoa dstrbuto Whe the total trasmsso rate Nr exceeds the l capacty C r, bts are lost loss rato p loss gves the rato betwee the traffc lost ad the traffc offered: p E[ Nr, C ] E[ N, ] loss, a E[ Nr] E[ N] a! e, a 27

28 Cotets Refresher: Smple teletraffc model Posso model customers, servers Applcato to flow level modellg of streamg data traffc Erlag model customers, ; servers Applcato to telephoe traffc modellg tru etwor Bomal model ; customers, servers Egset model ; customers, ; servers Applcato to telephoe traffc modellg access etwor 28

29 Erlag model M/M// Defto: Erlag model s the followg smple teletraffc model: Ifte umber of depedet customers Iterarrval tmes are IID ad expoetally dstrbuted wth mea /κ so, customers arrve accordg to a Posso process wth testy κ Fte umber of servers ; Servce tmes are IID ad expoetally dstrbuted wth mea / No watg places m Erlag model: Usg Kedall s otato, ths s a M/M// queue Pure loss system, ad, thus, lossy Notato: a κ.traffc testy, cf. slde 25 29

30 State trasto dagram Let Xt deote the umber of customers the system at tme t Assume that Xt at some tme t, ad cosder what happes durg a short tme terval t, t h]: wth prob. κh oh, a ew customer arrves state trasto wth prob. h oh, a customer leaves the system state trasto, Process Xt s clearly a Marov process wth state trasto dagram κ κ κ 2,, Note that process Xt s a rreducble brth-death process wth a fte state space S {,,2,,} κ 3

31 3 Equlbrum dstrbuto Local balace equatos LBE: Normalzg codto N:!! N, a a 32 LBE κ a κ a,,,,! ϑ

32 Equlbrum dstrbuto 2 Thus, the equlbrum dstrbuto s a trucated Posso dstrbuto: P{ X } a!,,, ϑ, j a j j! Remar: Isestvty wth respect to the servce tme dstrbuto The result s sestve to the servce tme dstrbuto, that s: t s vald for ay servce tme dstrbuto wth mea / So, stead of the M/M// model, we ca cosder, as well, the more geeral M/G// model 32

33 Tme blocg Tme blocg B t = probablty that all servers are occuped at a arbtrary tme = the fracto of tme that all servers are occuped For a statoary Marov process, ths equals the probablty of the equlbrum dstrbuto. Thus, B t : P{ X } a! j a j j! 33

34 Call blocg Call blocg B c = probablty that a arrvg customer fds all servers occuped = the fracto of arrvg customers that are lost However, due to Posso arrvals ad PASTA property, the probablty that a arrvg customer fds all servers occuped equals the probablty that all servers are occuped at a arbtrary tme, I other words, call blocg B c equals tme blocg B t : B c B t a! j a j j! Ths s Erlag s blocg formula 34

35 Cotets Refresher: Smple teletraffc model Posso model customers, servers Applcato to flow level modellg of streamg data traffc Erlag model customers, ; servers Applcato to telephoe traffc modellg tru etwor Bomal model ; customers, servers Egset model ; customers, ; servers Applcato to telephoe traffc modellg access etwor 35

36 Classcal model for telephoe traffc Loss models have tradtoally bee used to descrbe crcutswtched telephoe etwors Poeerg wor made by Dash mathematca A.K. Erlag Cosder a l betwee two telephoe exchages traffc cossts of the ogog telephoe calls o the l 36

37 Classcal model for telephoe traffc 2 Erlag modelled ths as a pure loss system m = customer = call κ = call arrval rate calls per tme ut servce tme = call holdg tme h = / = average holdg tme tme uts server = chael o the l = r of chaels o the l κ 37

38 Traffc process chael-by-chael occupato call holdg tme chaels r of chaels call arrval tmes r of chaels occuped bloced call traffc volume tme tme 38

39 Teletraffc aalyss System capacty = umber of chaels o the l Traffc load a = offered traffc testy Qualty of servce from the subscrbers pot of vew B c = call blocg = probablty that a arrvg call fds all chaels occuped Assume a M/G// loss system: calls arrve accordg to a Posso process wth rate κ call holdg tmes are depedetly ad detcally dstrbuted accordg to ay dstrbuto wth mea h 39

40 Teletraffc aalyss 2 The the quattve relato betwee the three factors system, traffc, ad qualty of servce s gve by Erlag s formula: Bc Erl, a : a! a! Also called: Erlag s B-formula Erlag s blocg formula Erlag s loss formula Erlag s frst formula!, ϑ 2,! 4

41 Example Assume that there are = 4 chaels o a l ad the offered traffc s a = 2. erlag. The the call blocg probablty B c s B 2 4 Erl 4,2 4! c ! 3! 4! % If the l capacty s rased to = 6 chaels, the B c reduces to B 2 6 Erl 6,2 6! c ! 3! 4! 5! 6!.2% 4

42 Capacty vs. traffc Gve the qualty of servce requremet that B c %, the requred capacty depeds o the traffc testy a as follows: a m{,2, ϑ Erl, a ;.} 5 4 capacty traffc a 42

43 Qualty of servce vs. traffc Gve the capacty 2 chaels, the requred qualty of servce, B c depeds o the traffc testy a as follows:, B c a, Erl2, a.8 qualty of servce, B c traffc a 43

44 Qualty of servce vs. capacty Gve the traffc testy a 5. erlag, the requred qualty of servce, B c depeds o the capacty as follows:, B c, Erl,5..8 qualty of servce, B c capacty 44

45 Multplexg ga We determe traffc testy a so that call blocg B c ; % Multplexg ga s descrbed by the traffc testy per capacty ut, a., as a fucto of capacty.8 ormalzed traffc a capacty 45

46 Summary of traffc modellg tru etwor Erlag model may be appled to modellg of telephoe traffc tru etwor where the umber of potetal users of a l s large customer = call κ = call arrval rate calls per tme ut h = / = average call holdg tme tme uts a = κ/ = traffc testy = l capacty chaels A call s lost f all chaels are occuped whe the call arrves call blocg Erlag s blocg formula B c gves the probablty of such a evet B c a! a j j j! 46

47 Cotets Refresher: Smple teletraffc model Posso model customers, servers Applcato to flow level modellg of streamg data traffc Erlag model customers, ; servers Applcato to telephoe traffc modellg tru etwor Bomal model ; customers, servers Egset model ; customers, ; servers Applcato to telephoe traffc modellg access etwor 47

48 Bomal model M/M/// Defto: Bomal model s the followg smple teletraffc model: Fte umber of depedet customers ; o-off type customers alteratg betwee dleess ad actvty Idle tmes are IID ad expoetally dstrbuted wth mea / As may servers as customers Servce tmes are IID ad expoetally dstrbuted wth mea / No watg places m Bomal model: Usg Kedall s otato, ths s a M/M/// queue Although a fte system, ths s clearly lossless O-off type customer: dleess servce 48

49 O-off type customer Let X j t deote the state of customer j j,2,, at tme t State = dle, state = actve = servce Cosder what happes durg a short tme terval t, t+h]: f X j t, the, wth prob. h oh, the customer becomes actve state trasto f X j t, the, wth prob. h oh, the customer becomes dle state trasto Process X j t s clearly a Marov process wth state trasto dagram Note that process X j t s a rreducble brth-death process wth a fte state space S = {,} 49

50 5 O-off type customer 2 Local balace equatos LBE: Normalzg codto N: So, the equlbrum dstrbuto of a sgle customer s the Beroull dstrbuto wth success probablty /+ offered traffc s / From ths, we could deduce that the equlbrum dstrbuto of the state of the whole system that s: the umber of actve customers s the bomal dstrbuto B, /+ j j j j, j j j j j

51 State trasto dagram Let Xt deote the umber of actve customers Assume that Xt at some tme t, ad cosder what happes durg a short tme terval t, t h]: f ;, the, wth prob.,h oh, a dle customer becomes actve state trasto f =, the, wth prob. h oh, a actve customer becomes dle state trasto, Process Xt s clearly a Marov process wth state trasto dagram, 2 2,, Note that process Xt s a rreducble brth-death process wth a fte state space S {,,,} 5

52 52 Equlbrum dstrbuto Local balace equatos LBE: Normalzg codto N: LBE, N,, 53,,,,!!! ϑ,,

53 53 Equlbrum dstrbuto 2 Thus, the equlbrum dstrbuto s a bomal dstrbuto: Remar: Isestvty w.r.t. servce tme ad dle tme dstrbuto The result s sestve both to the servce ad the dle tme dstrbuto, that s: t s vald for ay servce tme dstrbuto wth mea / ad ay dle tme dstrbuto wth mea / So, stead of the M/M/// model, we ca cosder, as well, the more geeral G/G/// model 2 2 ] [, ] [,,,, } {, B, } X D X E X P X ϑ

54 Cotets Refresher: Smple teletraffc model Posso model customers, servers Applcato to flow level modellg of streamg data traffc Erlag model customers, ; servers Applcato to telephoe traffc modellg tru etwor Bomal model ; customers, servers Egset model ; customers, ; servers Applcato to telephoe traffc modellg access etwor 54

55 Egset model M/M/// Defto: Egset model s the followg smple teletraffc model: Fte umber of depedet customers ; o-off type customers alteratg betwee dleess ad actvty Idle tmes are IID ad expoetally dstrbuted wth mea / Less servers tha customers ; Servce tmes are IID ad expoetally dstrbuted wth mea / No watg places m Egset model: Usg Kedall s otato, ths s a M/M/// queue Ths s a pure loss system, ad, thus, lossy O-off type customer: Note: If the system s full whe a dle cust. tres to become a actve cust., a ew dle perod starts. dleess servce dle blocg! dle 55

56 State trasto dagram Let Xt deote the umber of actve customers Assume that Xt at some tme t, ad cosder what happes durg a short tme terval t, t h]: f ;, the, wth prob.,h oh, a dle customer becomes actve state trasto f =, the, wth prob. h oh, a actve customer becomes dle state trasto, Process Xt s clearly a Marov process wth state trasto dagram,, 2, 2,, Note that process Xt s a rreducble brth-death process wth a fte state space S = {,,,} 56

57 57 Equlbrum dstrbuto Local balace equatos LBE: Normalzg codto N: LBE, N, 58,,,,!!! ϑ,,

58 Equlbrum dstrbuto 2 Thus, the equlbrum dstrbuto s a trucated bomal dstrbuto: P{ X } j j j j j j,, j,, ϑ, Offered traffc s / Remar: Isestvty w.r.t. servce tme ad dle tme dstrbuto The result s sestve both to the servce ad the dle tme dstrbuto, that s: t s vald for ay servce tme dstrbuto wth mea / ad ay dle tme dstrbuto wth mea / So, stead of the M/M/// model, we ca cosder, as well, the more geeral G/G/// model 58

59 Tme blocg Tme blocg B t = probablty that all servers are occuped at a arbtrary tme = the fracto of tme that all servers are occuped For a statoary Marov process, ths equals the probablty of the equlbrum dstrbuto. Thus, B t : P{ X } j j j 59

60 Call blocg Call blocg B c = probablty that a arrvg customer fds all servers occuped = the fracto of arrvg customers that are lost I the Egset model, however, the arrvals do ot follow a Posso process. Thus, we caot utlze the PASTA property ay more. I fact, the dstrbuto of the state that a arrvg customer sees dffers from the equlbrum dstrbuto. Thus, call blocg B c does ot equal tme blocg B t the Egset model. 6

61 6 Call blocg 2 Let * deote the probablty that there are actve customers whe a dle customer becomes actve whch s called a arrval Cosder a log tme terval,t: Durg ths terval, the average tme spet state s T Durg ths tme, the average umber of arrvg customers who all see the system to be state s, T Durg the whole terval, the average umber of arrvg customers s Ρ j,j j T Thus, j T j T j j j j,,,, * ϑ,,,,

62 Call blocg 3 It ca be show exercse! that *,,, ϑ,, j j, j If we wrte explctly the depedece of these probabltes o the total umber of customers, we get the followg result: I other words, a arrvg customer sees such a system where there s oe customer less tself! equlbrum *,,,, ϑ, 62

63 Call blocg 4 By choosg, we get the followg formula for the call blocg probablty: Bc *, Bt, Thus, for the Egset model, the call blocg a system wth customers equals the tme blocg a system wth, customers: B c B t, j,, j j Ths s Egset s blocg formula 63

64 Cotets Refresher: Smple teletraffc model Posso model customers, servers Applcato to flow level modellg of streamg data traffc Erlag model customers, ; servers Applcato to telephoe traffc modellg tru etwor Bomal model ; customers, servers Egset model ; customers, ; servers Applcato to telephoe traffc modellg access etwor 64

65 Applcato to telephoe traffc modellg access etwor Egset model may be appled to modellg of telephoe traffc access etwor where the r of potetal users of a l s moderate customer = call = call arrval rate per dle user calls per tme ut / = average call holdg tme tme uts = umber of potetal users = l capacty chaels A call s lost f all chaels are occuped whe the call arrves call blocg B c gves the probablty of such a evet B c j,, j j 66 65

66 Multplexg ga We assume that a access l s loaded by potetal users We determe traffc testy / so that call blocg B c ; % Multplexg ga s descrbed by the traffc testy per capacty ut, /, as a fucto of capacty.8 ormalzed traffc capacty 66

67 THE END What you should uderstad/remember: example of a fte system what d of systems you model as loss systems examples of such systems cocept of traffc testy tme blocg, call blocg ad the dfferece betwee these what s meat by sestvty of some systems to terarrval ad/or servce tme dstrbuto what s Erlag s blocg formula 67