Performance-Energy Trade-off in Multi-Server Queueing Systems with Setup Delay
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1 Performance-Energy Trade-off n Mlt-Server Qeeng Systems wth Setp Delay Saml Aalto Aalto Unversty, Fnland LCCC Clod Comptng Workshop 7 9 May 2014 Lnd, Sweden
2 Co-operaton wth Esa Hyytä, Pas Lassla, Mskr Gebrehwot (Aalto Unversty Rhonda Rghter (UC Berkeley 2
3 Qeeng models Sngle-server qee (M/G/1 Parallel qees l m homogeneos servers l m m Mlt-server qee (M/M/n l 1 m heterogeneos servers l m 1 n m m n Introdcton 3
4 Performance-energy trade-off Energy saved by swtchng the server off when dle However, performance mpared, f swtchng the server back on takes tme (setp delay BUSY / IDLE / OFF / SETUP? x x x x Introdcton 4
5 Cost model Performance: E[T] = mean delay per job (n seconds E[X] = mean nmber of jobs = le[t] Defnton: delay = response tme Energy: E[E] = mean energy per job (n joles E[P] = mean power consmed = le[e] Power consmpton levels: 0 P P P P off dle setp bsy Introdcton 5
6 Objectve fncton Energy-Response-tme- Weghted-Sm (ERWS: E[T] + E[E]/b e.g. Werman & al. (2009 Energy-Response-tme-Prodct (ERP: E[T]E[E] e.g. Gandh & al. (2010b General form: w 1 E[T] t1 E[E] e1 + w 2 E[T] t2 E[E] e2 by Macco & Down (2013 ERWS: w 1 = 1, t 1 = 1, e 1 = 0 w 2 = 1/b, t 2 = 0, e 2 = 1 ERP: w 1 = 1, t 1 = 1, e 1 = 1 w 2 = 0, t 2 = 0, e 2 = 0 Introdcton 6
7 Part I Sngle-server qee wth setp delays l m 7 Part I Sngle-server qee
8 Optmal swtchng on/off polcy Macco & Down (2013 M/G/1-FIFO Setp delay D generally dstrbted wth mean 1/g Control parameters: Delayed swtch-off for an exponental tme wth mean 1/a Server swtched on after k new job arrvals Objectve fncton: Gen. form [ ] 1 E[ ] 1 E[ ] 2 E[ ] 2 1 E t e t e T E w2 T E w + Polces: NEVEROFF: a 0 DELAYEDOFF: 0 a INSTANTOFF: a Theorem: For ERWS objectve fncton optmal polcy s ether NEVEROFF or INSTANTOFF Smlar reslt n Gandh & al. (2010b for ERP objectve fncton 8 Part I Sngle-server qee
9 Optmal swtchng on/off polcy Gebrehwot & al. (2014 M/G/1-FIFO Setp delay D generally dstrbted wth mean 1/g Control parameters: Delayed swtch-off for a gen. dstrbted tme wth mean 1/a Server swtched on after k new job arrvals Objectve fncton: Gen. form [ ] 1 E[ ] 1 E[ ] 2 E[ ] 2 1 E t e t e T E w2 T E w + Polces: NEVEROFF: a 0 DELAYEDOFF: 0 a INSTANTOFF: a Theorem: For gen. objectve fncton optmal polcy s ether NEVEROFF or INSTANTOFF NEVEROFF s better f P dle s sffcently small compared to P setp 9 Part I Sngle-server qee
10 Part II Mlt-server qee wth setp delays l 1 m n m 10 Part II Mlt-server qee
11 Analyss of server farms wth setp delays Gandh & al. (2010a M/M/n Setp delay D exponentally dstrbted Objectve fncton: Separately E[T] and E[P] Polces: ON/IDLE = NEVEROFF ON/OFF = INSTANTOFF ON/OFF/STAG = INSTANTOFF wth staggered bootp Mxed polcy: ON/IDLE(t swtchng dle server off only f nr of bsy and dle servers > t Conclsons: Under hgh loads, trnng servers off can reslt n hgher power consmpton and far hgher response tmes. As the sze of the server farm s ncreased, the advantages of trnng servers off ncrease. 11 Part II Mlt-server qee
12 Analyss of server farms wth setp delays Gandh & al. (2010a 12 Part II Mlt-server qee
13 Optmzaton of server farms wth setp delay Gandh & al. (2010b M/M/n Setp delay determnstc Addtonal sleep states S wth 0 P P P off and determnstc (setp delays 0 d d d dle sleep sleep Objectve fncton: ERP E[ T ] E[ P] dle off Polces: NEVEROFF INSTANTOFF SLEEP(S Probablstc and other Theorem: For n = 1, optmal statc control s ether NEVEROFF, INSTANTOFF or SLEEP(S Robst polcy: DELAYEDOFF wth MRB (Most Recent Bsy 13 Part II Mlt-server qee
14 Optmzaton of server farms wth setp delay Gandh & al. (2010b 14 Part II Mlt-server qee
15 Part III Parallel qees wth setp delays l m 1 m n 15 Part III Parallel qees
16 Dspatchng problem Dspatchng = Task assgnment = Rotng Random job arrvals wth random servce reqrements Dspatchng decson made pon the arrval l m 1 m n Or settng: M/G/. Posson arrvals generally dstrbted job szes heterogeneos servers wth FIFO qeeng dscplne (NEVEROFF or INSTANTOFF 16 Part III Parallel qees
17 Statc dspatchng polces RND = Bernoll splttng choose the qee pre randomly no sze nor state nformaton needed SITA = Sze Interval Task Assgnment choose the qee wth smlar jobs based on the sze of the arrvng job, bt no state nformaton needed Harchol-Balter et al. ( Part III Parallel qees
18 MDP approach Any statc polcy (RND, SITA reslts n parallel M/G/1 qees Fx the statc polcy and determne relatve vales for all these parallel M/G/1 qees Dspatch the arrvng job to the qee that mnmzes the mean addtonal costs As the reslt, yo get a better dynamc dspatchng polcy Ths s called Frst Polcy Iteraton (FPI n the MDP theory Applcable for the ERWS objectve fncton 18 Part III Parallel qees
19 Relatve vales Defnton: For a fxed polcy resltng n a stable system, the vale fncton v(x gves the expected dfference n the nfnte horzon cmlatve costs between the system ntally n state x, and the system ntally n eqlbrm Defnton: For a fxed polcy resltng n a stable system, the relatve vale v(x - v(0 gves the expected dfference n the nfnte horzon cmlatve costs between the system ntally n state x, and the system ntally n state 0 19 Part III Parallel qees
20 Sze-aware M/G/1 qee wthot setp delays Hyytä et al. (2012 State descrpton: D1 + + D n D = remanng servce tme of job = backlog = nfnshed work Mean vales: Reslt: Sze-aware relatve vales v v T P E[ T ] E[ P] ( - v ( - v E[ S] + (1 - P T P (0 (0 le[ S 2 ] 2(1- dle l 2 2(1- ( P + P bsy - bsy P dle 20 Part III Parallel qees
21 Sze-aware M/G/1 qee wth setp delays Hyytä et al. (2014a State descrpton: D0 + D1 + + D n D = remanng servce tme of job D 0 = remanng setp delay = vrtal backlog Assme: Determnstc setp delay d and Psetp P bsy Mean vales: Reslt: Sze-aware relatve vales v v T P E[ T ] E[ P] ( - v ( - v E[ S] + + ld 1+ ld T P (0 (0 le[ S 2 ] 2(1- P bsy l 2 2(1-1+ ld - P + d (2+ ld 2(1+ ld ld (2+ ld 2(1- (1+ ld bsy 21 Part III Parallel qees
22 FPI polcy Hyytä et al. (2012, 2014a For NEVEROFF servers: Dspatch the job wth servce tme x to qee mnmzng the mean addtonal costs: For INSTANTOFF servers: Dspatch the job wth servce tme x to qee mnmzng the mean addtonal costs: 22 Part III Parallel qees, ( 0, 1( (,, (, ( 0, 1( ( 0 1(,, ( v d x v x a v d x v d x x a P P P T T T , (, (,, (, (, (,, ( v x v x a v x v x x a P P P T T T
23 Nmercal reslts Hyytä et al. (2014a 23 Part III Parallel qees
24 Nmercal reslts Hyytä et al. (2014a 24 Part III Parallel qees
25 Nmercal reslts Hyytä et al. (2014a 25 Part III Parallel qees
26 Other qeeng dscplnes Hyytä et al. (2014b LIFO n the M/G/. settng wth setp delays PS n the M/D/. settng wth setp delays Bt t s another story 26 Part III Parallel qees
27 References Harchol-Balter, Crovella & Mrta (1999 On Choosng a Task Assgnment Polcy for a Dstrbted Server System, JPDC Werman, Andrew & Tang (2009 Power-aware speed scalng n processor sharng systems, n IEEE INFOCOM Gandh, Harchol-Balter & Adan (2010a Server farms wth setp costs, PEVA Gandh, Gpta, Harchol-Balter & Kozch (2010b Optmalty analyss of energy-performance trade-off for server farm management, PEVA Hyytä, Penttnen & Aalto (2012 Sze- and state-aware dspatchng problem wth qee-specfc job szes, EJOR Macco & Down (2013 On optmal polces for energy-aware servers, n MASCOTS Hyytä, Rghter & Aalto (2014a Task assgnment n a heterogeneos server farm wth swtchng delays and general energy-aware cost strctre, to appear n PEVA Hyytä, Rghter & Aalto (2014b Energy-aware job assgnment n server farms wth setp delays nder LCFS and PS, accepted to ITC Gebrehwot, Lassla & Aalto (2014 Energy-aware qeeng models and controls for server farms, ongong work 27 References
28 The End 28
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