Lectue 9 Multple Class Models Multclass MVA Appoxmate MVA 8.4.2002 Copyght Teemu Keola 2002 1
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 8.4.2002 Copyght Teemu Keola 2002 2
3 8.4.2002 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
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] 8.4.2002 Copyght Teemu Keola 2002 4
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 8.4.2002 Copyght Teemu Keola 2002 5
Aothe Smple Example Fg. 6.1 [Me 94] D, quey =1 update =2 cpu =1 0.105 0.375 Watch out fo dex odeg: D 1,2 = D 2, quey d1 =2 0.180 0.480 d3 =3 0 0.240 Populato book fgues ode (Update, Quey) 8.4.2002 Copyght Teemu Keola 2002 6
update quey Smple Example (5) (0,0) (1,0) ( 0) = 0= ( 1 ), A, X 01 10 1, (,) = ( 0105. + 0180. ) = 3509. 11, (,) 1 0 = 3509. * 0105. = 0. 368 (,) 10 = 3509. * 0180. = 0632. (,) 10 = 0 21, 31, ' R 12 (,) 10 =? (,) 10 = (,) 10 = (,) 10 = 0, [ ] ' ' ' R (,) 1 0 = D 1+ 0 = 0105. R (,) 1 0 = 0180. R (,) 1 0 = 0 11, 11, 21, 31, 12, 22, 32, (,) 10 = (,) 10 + (,) 10 = 0368. 1 11, 11, (,) 10 = 0632. (,) 10 = 0 2 3 8.4.2002 Copyght Teemu Keola 2002 Tbl 6.3 7
Smple Example (cotd) (6) (0,1): ( 01, ) = 0343. ( 01, ) = 0438. ( 01, ) = 0219. 1 2 3 devce (1,1): [ ] ' R (,) 11 = D 1+ A (,) 11 = D [ 1+ ( 01,)] 11, 11, 11, 11, 1 [ ] = 0105. 1+ 0. 343 = 0141. [ ] [ ] ' ' R (,) 11 = D 1+ ( 0,) 1 = 0180. 1+ 0. 438 = 0. 259 R 31 (,)= 11 0 21, 21, 2 X 01 11 1, (,) = 0141. + 0258. = 2. 500 dex ( ) class dex 11, 21, 31, (,) 11 = 2.* 5 0141. = 0. 352 (,) 11 = 2.* 5 0. 259 = 0. 648 (,) 11 = 0... 12, 22, 32, (,) 11 = 0. 334 (,) 11 = 0519. (,) 11 = 0156., total popul. (,) 11 = 0. 686 1 2 3 (,) 11 = 1158. (,) 11 = 0156. smlaly fo states (2,0), (3,0), (2,1), (3,1) Tbl 6.3 8.4.2002 Copyght Teemu Keola 2002 8
Smple Example (cotd) (slly ode, slly otato, soy!) Basele soluto Table 6.3 [Me 94] exteal: class update espose tme: 2.444 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 = 0.180 solve aga class update espose tme: 1.934 s Table 6.4 [Me 94] Table 6.4 8.4.2002 Copyght Teemu Keola 2002 9
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 8.4.2002 Copyght Teemu Keola 2002 10
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) 8.4.2002 Copyght Teemu Keola 2002 11
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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 3 2 2 54 5 20 1 100M 20 5 1 3200 20 5 9 10M 8.4.2002 Copyght Teemu Keola 2002 13
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 8.4.2002 Copyght Teemu Keola 2002 14
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 8.4.2002 Copyght Teemu Keola 2002 15
16 8.4.2002 Copyght Teemu Keola 2002 Sgle Class Schwetze Appoxmato ) ( 1 1) ( 1 1) ( ) ( costat ) ( = =
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 8.4.2002 Copyght Teemu Keola 2002 17
Example (4) D = (10, 10, 15), K=3, =2 [Fg. 5.1] step 0 step 1 step 2 ( 2) = 2 / 3 = 0. 667 R1 D1 1 2 1 ' ' ' = + 1 10*. 1333 13. 333 R2 R3 15*. 1333 20 2 = = = = = 2 X 0 = = 0. 04286 46. 67 1 = 0. 5714 = 2 3 = 0. 8572 = 2 ( ) R1 10 1 0 5714 12 857 R2 R3 15 1 0 8572 '. ' '. = +. * 2 = = = + 2 2 X 0 = = 0. 042424 47143. 1 = 05454. = 2 3 = 0. 9092 = 2 why stop hee? ( ) 8.4.2002 Copyght Teemu Keola 2002 tal guess: eve dstbuto = 0.5455 exact 0.9090 exact 21429. 18
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. 8.4.2002 Copyght Teemu Keola 2002 19
Questos (cotd) Q. What s good measue of covegece? A. E.g., max elatve chage must be less tha 1% ( ) 8.4.2002 Copyght Teemu Keola 2002 20
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. 8.4.2002 Copyght Teemu Keola 2002 21
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 8.4.2002 Copyght Teemu Keola 2002 22
How Good s Appoxmate MVA Petty good fo thoughput ad espose tme ot so good fo queue legths at heavy loads Fgs 34.1-34.3 [Ja 91] 8.4.2002 Copyght Teemu Keola 2002 23
sgle class: Multple Class Schwetze Appoxmato (1) ( ) costat ( ) ( 1) = 1 multple class: ( ) costat 1 ( 1) = ( ) ( ) ( 1 ) = 1 1 ( 1 ) = ( ) 8.4.2002 Copyght Teemu Keola 2002 24
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*... 8.4.2002 Copyght Teemu Keola 2002 25
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 8.4.2002 Copyght Teemu Keola 2002 (dstbute evely to evey ode vsted) (2,1) (3,1) (1,2) Fg. 6.5 [Me 94] 26
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% 8.4.2002 Copyght Teemu Keola 2002 27
Example (2) Model: Fg. 6.1 [Me 94] step 0 step 1 (,) 31 = 15. (,) 31 = 1/ 3= 0333. 11 12 (,) 31 = 15. (,) 31 = 0333. 21 22 (,) 31 = 0 (,) 31 = 0333. 31 32 [ ] ' R (,) 31 = D 1+ (,) 21 + (,) 30 Schwetze 0105 1 2 0 =. + 11( 31, ) + 12( 31, ) 0105. 1 2 15. 0 0210. 3 1 = + 3 + = R21 31 0180 1 2 ' 0 (,) =. + 21(,) 31 + 22(,) 3 1 0180. * 2 0. 360 3 1 = = R R 11 11 11 12 ' 31 ' 12 (,) 31 = 0 ' ' (,) 31 =... R (,) 31 =... R (,) 31 =... 22 32 Result: Tbl 6.6, Qsolve/1 output, PMVA output 8.4.2002 Copyght Teemu Keola 2002 =? 28
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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] 8.4.2002 Copyght Teemu Keola 2002 30
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... 8.4.2002 Copyght Teemu Keola 2002 31
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 8.4.2002 Copyght Teemu Keola 2002 32
Exact Soluto Methods fo Ope etwoks Queug etwoks Closed etwoks Covoluto MVA App. MVA 8.4.2002 Copyght Teemu Keola 2002 33
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