8. Queueing systems. Contents. Simple teletraffic model. Pure queueing system

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1 8. Quug sysms Cos 8. Quug sysms Rfrshr: Sml lraffc modl Quug dscl M/M/ srvr wag lacs Alcao o ack lvl modllg of daa raffc M/M/ srvrs wag lacs lc8. S Iroduco o Tlraffc Thory Srg 5 8. Quug sysms 8. Quug sysms Sml lraffc modl ur quug sysm Cusomrs arrv a ra cusomrs r m u / avrag r-arrval m Cusomrs ar srvd by aralll srvrs h busy a srvr srvs a ra cusomrs r m u / avrag srvc m of a cusomr Thr ar m cusomr lacs h sysm a las srvc lacs ad a mos m wag lacs I s assumd ha blockd cusomrs arrvg a full sysm ar los F umbr of srvrs < srvc lacs f umbr of wag lacs m If all srvrs ar occud wh a cusomr arrvs occus o of h wag lacs No cusomrs ar los bu som of hm hav o wa bfor gg srvd From h cusomr s o of vw s rsg o kow.g. wha s h robably ha has o wa oo log? m 3 4

2 8. Quug sysms 8. Quug sysms Cos Quug dscl Rfrshr: Sml lraffc modl Quug dscl M/M/ srvr wag lacs Alcao o ack lvl modllg of daa raffc M/M/ srvrs wag lacs Cosdr a sgl srvr quug sysm Quug dscl drms h way h srvr srvs h cusomrs I lls whhr h cusomrs ar srvd o-by-o or smulaously Furhrmor f h cusomrs ar srvd o-by-o lls whch ordr hy ar ak o h srvc Ad f h cusomrs ar srvd smulaously lls how h srvc caacy s shard amog hm No: I comur sysms h corrsodg coc s schdulg A quug dscl s calld work-cosrvg f cusomrs ar srvd wh full srvc ra whvr h sysm s o-my Quug sysms 8. Quug sysms ork-cosrvg quug dscls Cos Frs I Frs Ou FIFO Frs Com Frs Srvd FCFS ordary quug dscl quu arrval ordr srvc ordr cusomrs srvd o-by-o wh full srvc ra always srv h cusomr ha has b wag for h logs m dfaul quug dscl hs lcur Las I Frs Ou LIFO Las Com Frs Srvd LCFS rvrsd quug dscl sack cusomrs srvd o-by-o wh full srvc ra always srv h cusomr ha has b wag for h shors m rocssor Sharg S far quug cusomrs srvd smulaously wh cusomrs h sysm ach of hm srvd wh qual ra / s Lcur 9. Sharg sysms 7 Rfrshr: Sml lraffc modl Quug dscl M/M/ srvr wag lacs Alcao o ack lvl modllg of daa raffc M/M/ srvrs wag lacs 8

3 8. Quug sysms 8. Quug sysms M/M/ quu Rlad radom varabls Cosdr h followg sml lraffc modl: If umbr of dd cusomrs k Irarrval ms ar II ad xoally dsrbud wh ma / so cusomrs arrv accordg o a osso rocss wh sy O srvr Srvc ms ar II ad xoally dsrbud wh ma / If umbr of wag lacs m faul quug dscl: FIFO Usg Kdall s oao hs s a M/M/ quu mor rcsly: M/M/-FIFO quu Noao: /raffc load umbr of cusomrs h sysm a a arbrary m quu lgh qulbrum umbr of cusomrs h sysm a a ycal arrval m quu lgh s by a arrvg cusomr wag m of a ycal cusomr S srvc m of a ycal cusomr S oal m h sysm of a ycal cusomr dlay 9 8. Quug sysms 8. Quug sysms Sa raso dagram qulbrum dsrbuo L do h umbr of cusomrs h sysm a m Assum ha a som m ad cosdr wha has durg a shor m rval h: wh rob. h oh a w cusomr arrvs sa raso f h wh rob. h oh a cusomr lavs h sysm sa raso rocss s clarly a Markov rocss wh sa raso dagram No ha rocss s a rrducbl brh-dah rocss wh a f sa sac S... Local balac quaos LB: LB K N f < Normalzg codo N:

4 8. Quug sysms 8. Quug sysms qulbrum dsrbuo Ma quu lgh vs. raffc load Thus for a sabl sysm < h qulbrum dsrbuo xss ad s a gomrc dsrbuo: < Gom K Rmark: Ths rsul s vald for ay work-cosrvg quug dscl FIFO LIFO S... Ths rsul s o ssv o h srvc m dsrbuo for FIFO v h ma quu lgh dds o h dsrbuo Howvr for ay symmrc quug dscl such as LIFO or S h rsul s dd ssv o h srvc m dsrbuo Traffc load 4 8. Quug sysms 8. Quug sysms Ma dlay Ma dlay vs. raffc load L do h oal m dlay h sysm of a ycal cusomr cludg boh h wag m ad h srvc m S: S Ll s formula:. Thus Rmark: Th ma dlay s h sam for all work-cosrvg quug dscls FIFO LIFO S Bu h varac ad ohr moms ar dffr Traffc load 5 6

5 7 8. Quug sysms Ma wag m L do h wag m of a ycal cusomr Sc S w hav S 8 8. Quug sysms ag m dsrbuo L do h wag m of a ycal cusomr L do h umbr of cusomrs h sysm a h arrval m ASTA:. Assum ow for a whl ha Srvc ms S S of h wag cusomrs ar II ad x u o h mmorylss rory of h xoal dsrbuo h rmag srvc m S of h cusomr srvc also follows x-dsrbuo ad s dd of vryhg ls u o h FIFO quug dscl S S S Cosruc a osso o rocss τ by dfg τ S ad τ S S S. Now sc : τ τ τ τ τ S S S 3 τ3 S S 9 8. Quug sysms ag m dsrbuo Sc w hav o by A h osso cour rocss corrsodg o τ I follows ha: τ A O h ohr had w kow ha A osso. Thus τ τ A τ 9 8. Quug sysms ag m dsrbuo 3 By combg h rvous formulas w g τ

6 8. Quug sysms 8. Quug sysms ag m dsrbuo 4 Cos ag m ca hus b rsd as a roduc J of wo dd radom varabls J Broull ad x: J J J J Rfrshr: Sml lraffc modl Quug dscl M/M/ srvr wag lacs Alcao o ack lvl modllg of daa raffc M/M/ srvrs wag lacs 8. Quug sysms 8. Quug sysms Alcao o ack lvl modllg of daa raffc Mullxg ga M/M/ modl may b ald o som x o ack lvl modllg of daa raffc cusomr I ack ack arrval ra acks r m u / avrag ack rasmsso m akayks. / raffc load Qualy of srvc s masurd.g. by h ack dlay z robably ha a ack has o wa oo log.. logr ha a gv rfrc valu z z z z drm load so ha rob. z < % for z m us Mullxg ga s dscrbd by h raffc load as a fuco of h srvc ra load srvc ra 4

7 8. Quug sysms 8. Quug sysms Cos M/M/ quu Rfrshr: Sml lraffc modl Quug dscl M/M/ srvr wag lacs Alcao o ack lvl modllg of daa raffc M/M/ srvrs wag lacs Cosdr h followg sml lraffc modl: If umbr of dd cusomrs k Irarrval ms ar II ad xoally dsrbud wh ma / so cusomrs arrv accordg o a osso rocss wh sy F umbr of srvrs < Srvc ms ar II ad xoally dsrbud wh ma / If umbr of wag lacs m faul quug dscl: FCFS Usg Kdall s oao hs s a M/M/ quu mor rcsly: M/M/-FCFS quu Noao: / raffc load Quug sysms 8. Quug sysms Sa raso dagram qulbrum dsrbuo L do h umbr of cusomrs h sysm a m Assum ha a som m ad cosdr wha has durg a shor m rval h: wh rob. h oh a w cusomr arrvs sa raso f h wh rob. mh oh a cusomr lavs h sysm sa raso rocss s clarly a Markov rocss wh sa raso dagram No ha rocss s a rrducbl brh-dah rocss wh a f sa sac S... 7 Local balac quaos LB for < : K Local balac quaos LB for : LB LB K 8

8 9 8. Quug sysms qulbrum dsrbuo Normalzg codo N: Noao : f N < Quug sysms qulbrum dsrbuo 3 Thus for a sabl sysm < ha s: < h qulbrum dsrbuo xss ad s as follows: < : : K K 3 8. Quug sysms robably of wag L do h robably ha a arrvg cusomr has o wa L do h umbr of cusomrs h sysm a a arrval m A arrvg cusomr has o wa whvr all h srvrs ar occud a hr arrval m. Thus ASTA:. Thus : : Quug sysms Ma umbr of wag cusomrs L do h umbr of wag cusomrs qulbrum Th 3 : :

9 33 8. Quug sysms Ma wag m L do h wag m of a ycal cusomr Ll s formula:. Thus : : Quug sysms Ma dlay L do h oal m dlay h sysm of a ycal cusomr cludg boh h wag m ad h srvc m S: S Th S : : Quug sysms Ma quu lgh L do h umbr of cusomrs h sysm quu lgh qulbrum Ll s formula:. Thus : : Quug sysms ag m dsrbuo L do h wag m of a ycal cusomr L do h umbr of cusomrs h sysm a h arrval m Th cusomr has o wa oly f. Ths has wh rob.. Udr h assumo ha h sysm howvr looks lk a ordary M/M/ quu wh arrval ra ad srvc ra. L do h wag m of a ycal cusomr hs M/M/ quu L do h umbr of cusomrs h sysm a h arrval m I follows ha ' '

10 37 8. Quug sysms ag m dsrbuo ag m ca hus b rsd as a roduc J of wo d. radom varabls J Broull ad x: ' ' ' J J J J Quug sysms xaml rr roblm Cosdr h followg wo dffr cofguraos: O rad rr II rg ms x Two slowr aralll rrs II rg ms x Crro: mmz ma dlay O rad rr M/M/ modl wh /: Two slowr rrs M/M/ modl wh /: Quug sysms xaml Traffc load /