7. Loss systems. Contents

Transcription

1 lect7.ppt S Itroducto to Teletrffc Theory - Fll 999 Cotets Refresher: Smple teletrffc system Posso model customers, servers Erlg model customers, < servers Boml model < customers, servers Egset model < customers, < servers 2

2 Smple teletrffc model Customers rrve t rte customers per tme ut / verge ter-rrvl tme Customers re served by prllel servers Whe busy, server serves t rte customers per tme ut / verge servce tme of customer There re m wtg plces It s ssumed tht bloced customers rrvg full system re lost m 3 Pure loss system No wtg plces m If the system s full wth ll servers occuped whe customer rrves, she s ot served t ll but lost Some customers re lost From the customer s pot of vew, t s terestg to ow e.g. the blocg probblty Note: I ddto to the cse where the rrvl rte s costt, we wll cosder the cse where t depeds o the stte of the system: x 4

3 Ifte system Ifte umber of servers No customers re lost or eve hve to wt before gettg served Note: Also here, ddto to the cse where the rrvl rte s costt, we wll cosder the cse where t depeds o the stte of the system: x 5 Blocg I loss system some clls re lost cll s lost f ll chels re occuped whe the cll rrves the term blocg refers to ths evet There re t lest two dfferet types of blocg quttes: Cll blocg B c probblty tht rrvg cll fds ll chels occuped the frcto of clls tht re lost Tme blocg B t probblty tht ll chels re occuped t rbtrry tme the frcto of tme tht ll chels re occuped The two blocg quttes re ot ecessrly equl If clls rrve ccordg to Posso process, the B c B t Cll blocg s better mesure for the qulty of servce expereced by the subscrbers but, typclly, tme blocg s eser to clculte 6

4 Cotets Refresher: Smple teletrffc system Posso model customers, servers Erlg model customers, < servers Boml model < customers, servers Egset model < customers, < servers 7 Posso model M/M/ Defto: Posso model s the followg smple teletrffc model: Ifte umber of depedet customers Iterrrvl tmes re IID d expoetlly dstrbuted wth me / > so, customers rrve ccordg to Posso process wth testy Ifte umber of servers Servce tmes re IID d expoetlly dstrbuted wth me / > No wtg plces m Posso model: Usg Kedll s otto, ths s M/M/ queue Ifte system, d, thus, lossless Notto: / trffc testy 8

5 Stte trsto dgrm LetXt deote the umber of customers the system t tme t Assume tht Xt t some tme t, d cosder wht hppes durg short tme tervl t, t+h]: wth prob. h + oh, ew customer rrves stte trsto + f >, the, wth prob. h + oh, customer leves the system stte trsto - Process Xt s clerly Mrov process wth stte trsto dgrm 2 2 Note tht process Xt s rreducble brth-deth process wth fte stte spce S {,,2,...} 3 9 Equlbrum dstrbuto Locl blce equtos LBE: !, Normlzg codto N:!!,,2, e e LBE N

6 Equlbrum dstrbuto 2 Thus, the equlbrum dstrbuto s Posso dstrbuto: X Posso P{ X } E[ X ], D 2! e [ X ],,,2, Remr sestvty: The result s sestve to the servce tme dstrbuto, tht s: t s vld for y servce tme dstrbuto wth me / So, sted of the M/M/ model, we c cosder, s well, the more geerl M/G/ model Cotets Refresher: Smple teletrffc system Posso model customers, servers Erlg model customers, < servers Boml model < customers, servers Egset model < customers, < servers 2

7 Erlg model M/M// Defto: Erlg model s the followg smple teletrffc model: Ifte umber of depedet customers Iterrrvl tmes re IID d expoetlly dstrbuted wth me / > so, customers rrve ccordg to Posso process wth testy Fte umber of servers < Servce tmes re IID d expoetlly dstrbuted wth me / > No wtg plces m Erlg model: Usg Kedll s otto, ths s M/M// queue Pure loss system, d, thus, lossy Notto: / trffc testy 3 Stte trsto dgrm LetXt deote the umber of customers the system t tme t Assume tht Xt t some tme t, d cosder wht hppes durg short tme tervl t, t+h]: wth prob. h + oh, ew customer rrves stte trsto + wth prob. h + oh, customer leves the system stte trsto - Process Xt s clerly Mrov process wth stte trsto dgrm Note tht process Xt s rreducble brth-deth process wth fte stte spce S {,,2,,} 4

8 Equlbrum dstrbuto Locl blce equtos LBE: Normlzg codto N: ,,!,!!, LBE N 5 Equlbrum dstrbuto 2 Thus, the equlbrum dstrbuto s tructed Posso dstrbuto: P{ X }!!,,,, Remr sestvty: The result s sestve to the servce tme dstrbuto, tht s: t s vld for y servce tme dstrbuto wth me / So, sted of the M/M// model, we c cosder, s well, the more geerl M/G// model 6

9 Tme blocg Tme blocg B t probblty tht ll chels re occuped t rbtrry tme the frcto of tme tht ll chels re occuped For sttory Mrov process, ths equls the probblty of the equlbrum dstrbuto. Thus, B t : P{ X }!! 7 Cll blocg Cll blocg B c probblty tht rrvg cll fds ll chels occuped the frcto of clls tht re lost However, due to Posso rrvls d PASTA property, the probblty tht rrvg cll fds ll chels occuped equls the probblty tht ll chels re occuped t rbtrry tme, I other words, cll blocg B c equls tme blocg B t : B c B t!! Ths s Erlg s blocg formul 8

10 Cotets Refresher: Smple teletrffc system Posso model customers, servers Erlg model customers, < servers Boml model < customers, servers Egset model < customers, < servers 9 Boml model M/M/// Defto: Boml model s the followg smple teletrffc model: Fte umber of depedet customers < o-off type customers ltertg betwee dleess d ctvty Idle tmes re IID d expoetlly dstrbuted wth me / > As my servers s customers Servce tmes re IID d expoetlly dstrbuted wth me / > No wtg plces m Boml model: Usg Kedll s otto, ths s M/M/// queue Although fte system, ths s clerly lossless O-off type customer ote: whe ctve, customer s servce: dleess servce 2

11 O-off type customer LetX t deote the stte of customer,2,, ttmet Stte dle, stte ctve servce Cosder wht hppes durg short tme tervl t, t+h]: f X t, the, wth prob. h + oh, the customer becomes ctve stte trsto f X t, the, wth prob. h + oh, the customer becomes dle stte trsto Process X t s clerly Mrov process wth stte trsto dgrm Note tht process X t s rreducble brth-deth process wth fte stte spce S {,} 2 O-off type customer 2 Locl blce equtos LBE: Normlzg codto N: + +, + + So, the equlbrum dstrbuto of sgle customer s the Beroull dstrbuto wth success probblty /+ From ths, we could deduce tht the equlbrum dstrbuto of the stte of the whole system tht s: the umber of ctve customers s the boml dstrbuto B, /+ 22

12 Stte trsto dgrm LetXt deote the umber of ctve customers Assume tht Xt t some tme t, d cosder wht hppes durg short tme tervl t, t+h]: f <, the, wth prob. -h + oh, dle customer becomes ctve stte trsto + f >, the, wth prob. h + oh, ctve customer becomes dle stte trsto - Process Xt s clerly Mrov process wth stte trsto dgrm Note tht process Xt s rreducble brth-deth process wth fte stte spce S {,,,} 23 Equlbrum dstrbuto Locl blce equtos LBE: !!!, LBE,,, Normlzg codto N: + N + 24

13 Equlbrum dstrbuto 2 Thus, the equlbrum dstrbuto s boml dstrbuto: X P{ X B, E[ X ] } + + Remr sestvty:, D 2 + [ X ] +,,, 2 + The result s sestve both to the servce d the dle tme dstrbuto, tht s: t s vld for y servce tme dstrbuto wth me / d y dle tme dstrbuto wth me / So, sted of the M/M/// model, we c cosder, s well, the more geerl G/G/// model, Cotets Refresher: Smple teletrffc system Posso model customers, servers Erlg model customers, < servers Boml model < customers, servers Egset model < customers, < servers 26

14 Egset model M/M/// Defto: Egset model s the followg smple teletrffc model: Fte umber of depedet customers < o-off type customers ltertg betwee dleess d ctvty Idle tmes re IID d expoetlly dstrbuted wth me / > Less servers th customers < Servce tmes re IID d expoetlly dstrbuted wth me / > No wtg plces m Note: If the system s Egset model: full whe dle cust. tres to become Usg Kedll s otto, ths s M/M/// queue ctve cust., ew dle Ths s pure loss system, d, thus, lossy perod strts. O-off type customer ote: whe ctve, customer s servce: dleess servce dle blocg! dle 27 Stte trsto dgrm LetXt deote the umber of ctve customers Assume tht Xt t some tme t, d cosder wht hppes durg short tme tervl t, t+h]: f <, the, wth prob. -h + oh, dle customer becomes ctve stte trsto + f >, the, wth prob. h + oh, ctve customer becomes dle stte trsto - Process Xt s clerly Mrov process wth stte trsto dgrm Note tht process Xt s rreducble brth-deth process wth fte stte spce S {,,,} 28

15 29 Equlbrum dstrbuto Locl blce equtos LBE: Normlzg codto N: LBE + + N 29,,,,!!! Equlbrum dstrbuto 2 Thus, the equlbrum dstrbuto s tructed boml dstrbuto: Remr sestvty: The result s sestve both to the servce d the dle tme dstrbuto, tht s: t s vld for y servce tme dstrbuto wth me / d y dlee tme dstrbuto wth me / So, sted of the M/M/// model, we c cosder, s well, the more geerl G/G/// model X P,,,, } {

16 Tme blocg Tme blocg B t probblty tht ll chels re occuped t rbtrry tme the frcto of tme tht ll chels re occuped For sttory Mrov process, ths equls the probblty of the equlbrum dstrbuto. Thus, B t : P{ X } 3 Cll blocg Cll blocg B c probblty tht rrvg cll fds ll chels occuped the frcto of clls tht re lost I the Egset model, however, the rrvls do ot follow Posso process. Thus, we cot utlze the PASTA property y more. I fct, the dstrbuto of the stte tht rrvg customer sees dffers from the equlbrum dstrbuto. Thus, cll blocg B c does ot equl tme blocg B t the Egset model. 32

17 Cll blocg 2 Let * deote the probblty tht there re ctve customers whe dle customer becomes ctve whch s clled rrvl Cosder log tme tervl,t: Durg ths tervl, the verge tme spet stte s T Durg ths tme, the verge umber of rrvg customers who ll see the system to be stte s- T Durg the whole tervl, the verge umber of rrvg customers s Σ - T Thus, * T T,,,, 33 Cll blocg 3 It c be show exercse! tht *,,,, If we wrte explctly the depedece of these probbltes o the totl umber of customers, we get the followg result: I other words, rrvg customer sees such system where there s oe customer less she herself! equlbrum *,,,, 34

18 Cll blocg 4 By choosg, we get the followg formul for the cll blocg probblty: B c t * B Thus, for the Egset model, the cll blocg system wth customers equls the tme blocg system wth - customers: B c B t Ths s Egset s blocg formul 35 THE END 36