Mathematical models for computer systems behaviour

Transcription

1 Mthemtcl models for comuter systems ehvour Gols : redct comuter system ehvours - erformces mesuremets, - comrso of systems, - dmesog, Methodology : - modellg evromet (stochstc rocess) - modellg system (utomto) - ehvour recto of utomto o stochstc stmul

2 Orgsto utomt + rolstc trstos : Dscrete tme Mrkov chs utomt + rolstc trstos + tme: osso rocesses, cotuous tme Mrkov chs Stte sce structure : Smle ueues, roduct form ueug etworks Stochstc utomt etworks Smulto of Mrkov chs : Drect smulto/erfect smulto

3 lctos Oertg systems Networks rotocols Mufcturg systems roducto les Mddlewres...

4 Scetfc doms led mthemtcs : stochstc rocesses, ergodc theory,... Mrkov rocesses Refereces : R. Nelso rolty theory wth... R. J The rt of comuter systems erformce lyss

5 frst exmle : Fl-flo B - - # % & Trsto mtrx tme stte B ( ) ( ) ( ) ( ) B B B B

6 Log ru ehvour ( ) ( ) ( ) ) ( Ler recurrece euto Cse : -- < Cse 2 : -- > Cse 3 : -- - > B B ( ) # % & ( (

7 Results Covergece B lm lm ( ) ( B) + + ; Geometrc ( ) symtotc stsfes Lck of memory B ( ) + B + B ( )

8 Ergodc covergece ( ) ( ) ( ) ( ) B B lm lm ; lm lm erformces dexes : lk utlsto commucto delys

9 { } N Dscrete tme Mrkov chs Trjectory of the system Dscrete stte sce Ste y ste evoluto ( j, ) ( j ) +,..., + Lck of memory Homogeety ( tme) ( ) ( ) + j j, j Trsto mtrx (stochstc) ( ),., j, j, j j

10 Chm-Kolmogorov eutos ( ) ( ) ( ) k j k k k k k j j,, 2 2 Iterto ---> roduct of mtrces symtotc ehvour?? lm

11 Clssfcto of sttes sorg stte erodc recurret sttes Trset sttes erodc recurret sttes : rreducle clss

12 { } N Covergece theorem Homogeeous, erodc, rreducle Mrkov ch (fte stte sce) lm ( ) j j ( ), 2,..., Stedy-stte vector N Uue soluto of the ler system Geometrc covergece (module of the secod egevlue)

13 { } N Ergodc theorem Homogeeous, erodc, rreducle Mrkov ch (fte stte sce) For y fucto f (cost fucto) lm ( ) ( ) k # f f k Estmto of the stedy-stte dstruto y -> smulto lm # k k

14 Eulrum eutos, j j, k k Iterretto : j, j j j, k k j

15 Exmle : cche mgemet Cche sce Memory sce M N Otmsto : Tme to ccess cche << Tme to ccess memory Cche relcemet olcy M cche sze N glol memory sze

16 LRU olcy Cche ht M N Cche mss M N

17 Move hed olcy Cche ht M N Cche mss M N

18 Evromet model rogrm seuece of memory ccesses hyothess : deedet seuece sme dstruto Stte sce : ermuttos of (,,N) sze : N

19 Stte sce reducto Hyothess 2 : oe referece s more freuet others re eully dstruted (uform) + ( N ( referece ), ( referece other th ), ).

20 { } N Mrkov model osto of referece t ste Mrkov ch : homogeeous, erodc, rreducle > covergece d ergodcty > comutto of the stedy-stte

21 LRU N 2 (N-) (N-2) (N-3) # % & N N N N N LRU ) ( 2) ( 3) ( 2 2) ( ) ( ( ) ( ) ( ) N N.. + +

22 Move hed # % & N M ) ( ( ) # # # # # 2 3 N C # % &

23 Numercl exmle Toy exmle N8.3. Stedy-stte rolty vectors LRU [ ] M [ ] Self otmsg lgorthm Rd decresg

24 Cche mss evluto ge M LRU M Glol M LRU M

25 Exmle: cocluso Self dtg lgorthm, Mmse cche mss Move-hed etter th LRU Seed of covergece? 2d module of egevlue LRU :.7 M :.96 LRU reches sttory regme more uckly th M

26 Exmle: cocluso Hyothess 2 : relxg uformty (ex Zf lw) > sme ehvour (smulto) Hyothess : model of evoluto of rolty of refereces > deeds o the vrlty of the rocess comromse etwee rte of covergece d seed of evoluto

27 Geerlztos Ifte stte sce Sme ehvour (f o recurret ull) Trset lyss study of owers of Geerlzed Mrkov rocesses Tmed sttes comutto of stedy-stte weghted roltes

28 Solvg Mrkov chs N< 5 Forml methods Mle N<5 Clsscl umercl methods (Guss elmto, ) Mthemtc, Lck N< Itertve methods, recodtog, N< Secfc umercl lgorthms (srse mtrces, ) Mrc, es, N> Smulto Over roxmtos d lytcl techues

29 Lks comuter scece Commo formlsms wth verfcto tools Queug etworks etr ets rocess lger utomt etworks > stte sce costructo, ehvours secfctos