Lecture 9 Multiple Class Models

Transcription

1 Lectue 9 Multple Class Models Multclass MVA Appoxmate MVA Copyght Teemu Keola

2 Aval Theoem fo Multple Classes Wth jobs the system, a job class avg to ay seve sees the seve as equlbum wth oe job (hmself!) emoved fom the system A () = ( - 1 ) = () ( 1, 2,..., R ) A (0, 0,...,0, 1, 0,..., 0) R job classes th job class Copyght Teemu Keola

3 Copyght Teemu Keola 2002 Multple Class MVA (4), (0) = 0 statg pot: + = devce delay D devce queueg D R A, )] ( [1 ) (,,, ', esdece tme R Z X system Lttle + = ) ( ) ( ', 0, (espose tme law) R X R X Lttle devce = = = ) ( ) (, ) ( ) ( ) ( ) ( ) (, ', 0,,,, A () = ( - 1 ) = () A +1

4 Multple Class Mea Value Aalyss (MVA) (2) Compute solutos though class populato space =( 1, 2,..., R ), (0,0) statg fom empty system (1,0) (0,1) populato (state) space (2,0) (1,1) (0,2) ca stll be lage! (3,0) (2,1) (1,2) (0,3) two job classes taget populato (1,3)? taget populato (3,1)? taget populato (6, 15, 300)? (3,1) (1,3) Alg 7.2 [LZGS 84] Copyght Teemu Keola

5 Example (4) 2-class closed model, Fg. 7.2 [LZGS 84] Job classes A, B Tbl 7.3 [LZGS 84] (0,0) Q CPU, A = Q DISK, A = Q CPU, B = Q DISK, B = 0 (1,0) R CPU, A, R DISK, A, X 0, A, Q CPU, A, Q DISK, A (0,1) R CPU, B, R DISK, B, X 0, B, Q CPU, B, Q DISK, B (1,1) R CPU, A, R DISK, A, R CPU, B, R DISK, B, X 0, A, X 0, B, Q CPU, A, Q DISK, A, Q CPU, B, Q DISK, B Copyght Teemu Keola

6 Aothe Smple Example Fg. 6.1 [Me 94] D, quey =1 update =2 cpu = Watch out fo dex odeg: D 1,2 = D 2, quey d1 = d3 = Populato book fgues ode (Update, Quey) Copyght Teemu Keola

7 update quey Smple Example (5) (0,0) (1,0) ( 0) = 0= ( 1 ), A, X , (,) = ( ) = , (,) 1 0 = * = (,) 10 = * = (,) 10 = 0 21, 31, ' R 12 (,) 10 =? (,) 10 = (,) 10 = (,) 10 = 0, [ ] ' ' ' R (,) 1 0 = D 1+ 0 = R (,) 1 0 = R (,) 1 0 = 0 11, 11, 21, 31, 12, 22, 32, (,) 10 = (,) 10 + (,) 10 = , 11, (,) 10 = (,) 10 = Copyght Teemu Keola 2002 Tbl 6.3 7

8 Smple Example (cotd) (6) (0,1): ( 01, ) = ( 01, ) = ( 01, ) = devce (1,1): [ ] ' R (,) 11 = D 1+ A (,) 11 = D [ 1+ ( 01,)] 11, 11, 11, 11, 1 [ ] = = [ ] [ ] ' ' R (,) 11 = D 1+ ( 0,) 1 = = R 31 (,)= , 21, 2 X , (,) = = dex ( ) class dex 11, 21, 31, (,) 11 = 2.* = (,) 11 = 2.* = (,) 11 = , 22, 32, (,) 11 = (,) 11 = (,) 11 = 0156., total popul. (,) 11 = (,) 11 = (,) 11 = smlaly fo states (2,0), (3,0), (2,1), (3,1) Tbl Copyght Teemu Keola

9 Smple Example (cotd) (slly ode, slly otato, soy!) Basele soluto Table 6.3 [Me 94] exteal: class update espose tme: s teal: Dsk1 utlzato too hgh Modfcato: move quees to Dsk2 move quey demad fom Dsk 1 to Dsk2 D 21 = D d1,quey = 0, D 31 = D d2,quey = solve aga class update espose tme: s Table 6.4 [Me 94] Table Copyght Teemu Keola

10 Poduct Fom Soluto Exsts: W s Aalytcally Solvable BCMP-etwoks Baskett, Chady, Mutz & Palacos (1975) Sevce dscple FCFS, PS, IS, LCFS-PR Job classes, class swtchg Sevce tme dstbutos, teaval tmes Expoetal teavals tmes fo FCFS seves o fo ope job class moe geec fo othes (atoal Laplace tasfomato exsts) Load-depedet seves (LD-seves) S = f( ) fo FCFS (I.e., same fo each class) S = f( ) fo othes Copyght Teemu Keola

11 Poduct Fom Soluto Exsts? Job Flow balace flow = flow out pe devce, pe system Oe step behavo Devce homogeety sgle esouce possesso o blockg depedet job behavo local fomato fa sevce outg homogeety A B µ = f ( ) µ = f ( ) p k costat (pob fo class chage s load depedet) Copyght Teemu Keola

12 Copyght Teemu Keola

13 How Useful ae Exact Soluto Methods? MVA & Covoluto based algothms Vey good fo sgle class cases Mght be too tme cosumg fo multple class cases f of classes o class populatos ae (vey) lage O( MVA ) = O( K R Π(1 + ) ) K R #ops M M Copyght Teemu Keola

14 Appoxmate MVA (1) Helps to solve MVA state space poblem Based o Schwetze-appoxmato: MVA empty system (0,0) (1,0) (0,1) (2,0) (1,1) (0,2) (2,1) (1,2) (3,1) stat hee () costat App. MVA (2,1) (3,1) (1,2) guess fst, teate Copyght Teemu Keola

15 MVA vs Appoxmate MVA MVA (0,0,0,0,0) (1,0,0,0,0) (0,1,0,0,0) (0,0,1,0,0) (0,0,0,1,0) (0,0,0,0,1)... (2,2,4,4,5) (3,1,4,4,5) (3,2,3,4,5) (3,2,4,3,5) (3,2,4,4,4) App MVA (3,2,4,4,5) quess tal state std MVA step Schwetze app Copyght Teemu Keola

16 Copyght Teemu Keola 2002 Sgle Class Schwetze Appoxmato ) ( 1 1) ( 1 1) ( ) ( costat ) ( = =

17 epeat utl covegece Sgle Class Appoxmate MVA guess tal : compute : stdmva step : R' X 0 ( 1) = ( ) = ( ) = ( ) = X ( ) D 0 1 [ 1+ ( 1) ] ( Z + R' ( ) ) ( )* R' ( ) ( ) schwetze pop=9 pop=10 mva Copyght Teemu Keola

18 Example (4) D = (10, 10, 15), K=3, =2 [Fg. 5.1] step 0 step 1 step 2 ( 2) = 2 / 3 = R1 D ' ' ' = * R2 R3 15* = = = = = 2 X 0 = = = = 2 3 = = 2 ( ) R R2 R '. ' '. = +. * 2 = = = X 0 = = = = 2 3 = = 2 why stop hee? ( ) Copyght Teemu Keola 2002 tal guess: eve dstbuto = exact exact

19 Questos (3) Q. How to quess tal job dstbuto A: ) = K ( (eve dstbuto o all devces) Q. Does t always covege? A. o. Almost always. o quaatee. Q. If t coveges, does t covege to the ght value? A. We hope so. It seems to do t Copyght Teemu Keola

20 Questos (cotd) Q. What s good measue of covegece? A. E.g., max elatve chage must be less tha 1% ( ) Copyght Teemu Keola

21 Geeal Statoay Iteatve Method (1) (fom umecal Lea Algeba) = B+ c Fxed pot equato coveges fom abtay tal ( 0) f ρ( B) = max λ( B) <1 spectal adus of B th egevalue of B Q. Do we check ths befoe usg appoxmate MVA? A. o Copyght Teemu Keola

22 Geeal Theoy of Iteato (1) Thm: = f ( ) If = f ( ) has oot α { } ad f ( ) exsts close to α,.e., J = : α < ρ ad f ' ( ) < 1 J The (a) J each teato (b) coveges to α (c) α s the oly oot Q. Do we check ths fo appoxmate MVA? A. o. J Copyght Teemu Keola

23 How Good s Appoxmate MVA Petty good fo thoughput ad espose tme ot so good fo queue legths at heavy loads Fgs [Ja 91] Copyght Teemu Keola

24 sgle class: Multple Class Schwetze Appoxmato (1) ( ) costat ( ) ( 1) = 1 multple class: ( ) costat 1 ( 1) = ( ) ( ) ( 1 ) = 1 1 ( 1 ) = ( ) Copyght Teemu Keola

25 Multple Class Appoxmate MVA 1 ( 1 ) = ( ) (2,2,4,4,5) (3,1,4,4,5) (3,2,3,4,5) (3,2,4,3,5) (3,2,4,4,4) 2/3*... (3,2,4,4,5) 4/5* Copyght Teemu Keola

26 Multple Class Appoxmate MVA (1) guess tal () = /K at taget state (2) compute back (-1 ) = ( -1)/ () (3) use stadad MVA step to compute ew estmate of () teate steps (2) ad (3) utl covegece usually 4-6 teatos eough std output fom last teato Copyght Teemu Keola 2002 (dstbute evely to evey ode vsted) (2,1) (3,1) (1,2) Fg. 6.5 [Me 94] 26

27 Multple Class Appoxmate MVA Does t covege? Almost always. o quaatee. If t coveges, does t covege to the ght value? It seems to do good wok... What s good measue of covegece? e.g., max elatve chage () < 1% Copyght Teemu Keola

28 Example (2) Model: Fg. 6.1 [Me 94] step 0 step 1 (,) 31 = 15. (,) 31 = 1/ 3= (,) 31 = 15. (,) 31 = (,) 31 = 0 (,) 31 = [ ] ' R (,) 31 = D 1+ (,) 21 + (,) 30 Schwetze =. + 11( 31, ) + 12( 31, ) = = R ' 0 (,) =. + 21(,) (,) * = = R R ' 31 ' 12 (,) 31 = 0 ' ' (,) 31 =... R (,) 31 =... R (,) 31 = Result: Tbl 6.6, Qsolve/1 output, PMVA output Copyght Teemu Keola 2002 =? 28

29 Copyght Teemu Keola

30 Poblems wth Aalytcal Solutos Load depedet seves Memoy Some pat of system ot poduct fom howeve, soluto method s obust Computatoal poblems accuacy, oveflow, udeflow Fgs 8.1, 8.3 [Me 94] Copyght Teemu Keola

31 Commecal Soluto Packages (Softwae) BGS, Bosto, MA Best/1 AT&T, Holmdale, J Q+ SES, Aust, TX PAWS Uvesty Pojects Jeff Bumfeld, U of Aust, TX PMVA Copyght Teemu Keola

32 Pet ets vs. Queueg etwoks Petets good fo cocuecy Stochastc Pet ets (SP) Tmed Pet ets smulatos Queug etwoks good fo queug smple bouds Bottleeck Bouds (ABA) Balaced Job Bouds (BJB) exact soluto methods appoxmate soluto methods smulatos Copyght Teemu Keola

33 Exact Soluto Methods fo Ope etwoks Queug etwoks Closed etwoks Covoluto MVA App. MVA Copyght Teemu Keola

34 Copyght Teemu Keola