Chap 5: Product-Form Queuing Networks (QN)

Transcription

1 CPU disk Chap 5: Poduct-Fom Queuig Netwoks (QN) Etities: ) sevice cetes with diffeet sevice disciplies ) customes (obs) sigle class multiple classes (each/w a diffeet wokload) ) liks coectig sevice cetes queue CPU CPU disk exit 9

2 Sevice disciplies ) FCFS ) Pioity ca be peemptive o o-peemptive ) Roud Robi (RR) time-slot based ) Pocesso Shaig (PS) the seve s capability is equally divided amog all obs 5) Last-Come-Fist-Seve Peemptive Resume (LCFSPR) stack push-pop style Ope vs. Closed QNM Ope: customes aive fom a exteal souce, sped time i the system & fially depat. Closed: # of customes ciculatig amog the sevice cetes is a costat, i.e., o exteal souce of obs & o depatue. 0

3 What is a poduct-fom solutio fo a QNM? - The oit pobability of the queue sizes i the etwok is a poduct of the pobabilities of queue sizes i idividual sevice cetes. e.g., a tadem queuig etwok with seves a Makov chai, - -, +, - a ob aives at seve, +, a ob moves fom seve to seve -, +, + a ob leaves seve

4 By solvig the steady-state global balace equatios (oe fo each state), it ca be show that: P(, ) Pob. that the system has at seve & at seve ( P( ) ) P( ( ) Fo the st M/M/ queue ) Fo the d M/M/ queue whee & The oit pobability that thee ae obs at seve & obs at seve is the poduct of the pobabilities fo two M/M/ queues.

5 Poduct-Fom Queuig Netwoks A QN is said to have a poduct-fom solutio if P(,,... ) J J P ( This is tue whe the followig chaacteistics hold (p. 9, text):. The outig of customes fom oe sevice cete to the ext must be histoy idepedet, i.e., memoy less (o Makovia).. The queuig disciplies may be FCFS, PS (Pocesso Shaig), IS (Ifiite Seve) o LCFSPR (Last Come Fist Seve with Peemptive-Resume). Fo a FCFS cete, the sevice time distibutio must be expoetial; fo othe seves, the sevice time distibutio does ot have to be expoetial but must be diffeetiable (w.. t. time). A poduct-fom etwok may have multiple chais (multiple classes) of obs ad may be ope with espect to some chais of obs ad closed with espect to othes. Exteal aivals fo all ope chais must be Poisso. ) P ( ) is a fuctio oly of the -th cete

6 Ope poduct-fom QNs: All obs aive fom a exteal souce & depat to a sik. Aival ate to Fom the exteal souce J k k q k Fom othe cetes i the etwok q k : pob {a custome goes whe it depats to k} Ex: = = = = A geeal method to solve a ope system QNM with a PF solutio: ) get iput aival ate fo each cete. ) aalyze each cete sepaately. ) get aggegate measues, e.g., k k

7 5 e.g. = 0.5, = & = , Evaluate each cete idepedetly 0.5 R ; 0.5 R ; time spet i the system by a custome: uits time 8 6 Waitig time at cete, R w Waitig time at cete, 6 R w

8 Ex: ope poduct-fom QNM with feedback P P P P P o P P P P P Q: x,, R? Thoughput x = P = ( /P ) P = 6

9 Closed Poduct-Fom Netwoks A etwok with a set of obs ciculatig idefiitely o a etwok i which a ob leavig the etwok will be eplaced istatly by a statistically idetical ew ob. e.g., N = I geeal, coside a etwok with J sevice cetes sevig N obs: Visit cout to cete v J k v k q k pobability that a ob leavig cete k moves to cete A paticula cete s visit cout is set to oe based o the model s physical meaig.

10 Solutio Techique: Mea Value Aalysis Algoithm it yields the aveage values of pefomace measues. I a closed system with N obs, whe a ob aives, it actually sees oly (N-) obs distibuted i the system. Notatio: :sevice ate at cete t k k k T k :aveage# of :aveage espose timeof whe theeae k obs i thesystem :system thoughput : thoughput obs at cete whe k obs at cete a ob at cete ae i the system 8

11 Fomulas: t k k T k k R v k k by Little's k v Tk k t k k by Little's Law J (k) is estimated ust like i M/M/ except that populatio is oe less Law Recusio: Give Fo v all, J k=0 k= 0 0 k fo all s A paticula cete s visit cout is set to oe based o the model s physical meaig. J R v k Tk t k k k=k+ util k=n 9

12 0 e.g., = = N= Set v to the v = ( elative visit cout is the same fo both cetes i this example) P.00, text k=0 Statig with 0 0 0; 0 & & 0 ; 0 t t T R k= (Little s Law) ( v = v = ) & 6 6 & ; t t T R k=

13 t t T R k= t t T R k= (last iteatio)

14 Usig shape to solve closed QNMs Sigle-chai Multiple-chai Thee ae 6 types of sevice cetes i a closed QNM which ca be specified i a shape pogam:. FCFS Sytax: statio-ame fcfs ate Q: how to calculate (k) fo a IS cete?. IS Sytax: statio-ame is ate thee ae ifiite # of seves i the cete. MS Sytax: statio-ame ms #seves ate thee ae multiple seves i the cete, each with the idetical sevice ate. LCFSPR Sytax: statio-ame lcfsp ate 5. PS Sytax: statio-ame ps ate all obs peset at the cete shae oe seve with each ob seeig the seve speed educed by a facto of 6. LDS (Load-depedet seve) Sytax: statio-ame lds ate, ate, all obs at the cete agai shae oe seve but the sevice ate of the seve is load depedet (i.e., depedig o the # of obs peset i the cete) Must be specified i the shape code Automatically computed by shape

15 B..6. P.6 text Pogam stuctue fo sigle-chai (o sigle class) poduct-fom queuig etwoks (fo a closed system) pfq {(paa-list)} * sectio : statio-to-statio pobabilities < statio-ame statio-ame expessio> ed * sectio : statio types & paametes < statio-ame statio-type expessio, > ed * sectio : umbe of customes pe chai (o pe class) < chai-ame expessio> ed

16 e.g., p. & p.0 Sevice time pe visit 5 s 0 ms CPU P 0 = 0. P = 0.66 P = 0. 0 ms disk.9 ms disk These ae ate paametes M Temials Sectio * cetal seve model bid P 0.66 P 0. ed pfq csm CPU disk P CPU disk P CPU temials -P-P temials CPU disk CPU disk CPU ed Sectio Sectio * statio types & paametes temials IS /5 CPU fcfs 000/0 disk fcfs 000/0 disk fcfs 000/.9 ed * umbe of obs chai NoCustomes ed Othe sevice disciplie see p.6, text

17 qlegth etus pe-cete populatio * (cotiued) * epotig pe cete (CPU) measues loop i,, 0, bid NoCustomes i exp tput (csm, CPU) exp util (csm, CPU) exp qlegth (csm, CPU) exp time (csm, CPU) ed * calculate the system espose time * by applyig Little s Law * R * : * x : / x, whee populatio thoughput i the cetal system of the cetal system fuc x() \ tput (csm, CPU) * (-P -P ) fuc ba() \ qlegth (csm, CUP) + \ qlegth (csm, disk) + \ qlegth (csm, disk) bid NoCustomes 0 * calculate the aveage espose time pe * temial use oce it etes the cetal * system fo N=0 exp ba()/x() ed /* ed the etie pogam */ Apply MVA to this closed system ad set the visit cout to fo the temials cete. You should get the same output. Note: set temials = 5s because it is a IS cete. 5

18 Multiple-chai poduct-fom queuig etwoks (fo a closed system) mpfq {(paamete-list)} * sectio : statio to statio pobabilities fo each chai. <chai chai-ame <statio-ame statio-ame expessio> : ed> ed * sectio : statio types & paametes <<statio-ame statio-type expessio, > <chai-ame expessio, > : ed> ed * sectio : umbe of obs pe chai <chai-ame expessio> ed 6

19 Example: CPU disk Two classes: chai A: obs chai B: ob Sevice time (sevice demad) D A,CPU = ; D A,disk = D B,CPU = ; D B,disk = Pefomace measues of iteest? Respose time of a ob (system) Respose time of a class A ob (pe class) Respose time of a class B ob i the CPU cete (pe cete pe class) Thoughput of the CPU cete fo class A obs (pe cete pe class) Utilizatio of the CPU cete fo class B obs (pe cete pe class)

20 * A example of usig shape fo solvig a multiple-class poduct fom queuig etwok * Two classes: A ad B; assume visit cout is fo each cete * umbe of obs: (A, B) * umbe of statios: -- cpu ad disk * D A,cpu = * D A,disk = * D B,cpu = * D B,disk = *wat to kow R A, X A,( A,cpu ),(U A,cpu ) mpfq simple * sectio: statio to statio tasitio pobabilities chai A cpu disk disk cpu ed chai B cpu disk disk cpu ed ed CPU disk D A,CPU = ; D A,disk = D B,CPU = ; D B,disk = 8

21 *sectio : statio types ad paametes cpu ps A / B / ed disk ps A / B / ed ed *sectio : umbe of customes i each * chai A B ed *I geeal eed to calculate R A = A /X A. *But fo the simple system hee, we *ca calculate R A = R A,cpu + R A,disk exp mtime(simple,cpu,a) +mtime(simple,disk,a) *X A = X A, cpu fo this simple system exp mtput(simple,cpu,a) *populatio of class A at CPU: A,cpu exp mqlegth(simple,cpu,a) *utilizatio of class A at CPU: U A,cpu exp mutil(simple,cpu,a) ed Output: mtime(simple,cpu,a) +mtime(simple,disk,a): 6.8e mtput(simple,cpu,a):.50e mqlegth(simple,cpu,a):.95e mutil(simple,cpu,a): 6.0e-0 9