Report

SELFISHMIGRATE: A Scalable Algorithm for Non-clairvoyantly Scheduling Heterogeneous Processors Janardhan Kulkarni, Duke University Sungjin Im (UC Merced ), Kamesh Munagala (Duke), Kirk Pruhs (UPitt) ONLINE UNRELATED MACHINE MODEL 1 2 3 rj p j - processing length ℓij - rate of processing M w j - weight ONLINE UNRELATED MACHINE 1 2 (rate = 1, length=100) (rate = 0, length = infinity) rj 100 3 (rate = 0.5, length 200) t Time p j - processing length (rate = 2, length =50) M ℓij - rate of processing NON-CLAIRVOYANCE 1 (rate = 1, length=100) rj 2 (rate = 0, infinity) 3 t (rate = 0.5, length = 200) p j - processing length ℓij - rate of processing M (rate = 2, length =50) w j - weight Time FLOW-TIME (DELAY OR RESPONSE TIME) ANOTHER FORM SPEED-AUGMENTATION: - Online scheduler runs at higher speed PREMPTION: - Stop a job in middle of processing and continue later. MIGRATION: - Change the assignment of jobs to machines at no cost. NO PARALLEL PROCESSING - No job is simultaneously processed on multiple machines. 1 OK 2 NO PARALLEL PROCESSING: - No job is simultaneously processed on multiple machines. 1 NOT ALLOWED 2 How to schedule jobs to minimize sum of weighted flow-time for unrelated machines in non-clairvoyant model? Motivation It was very difficult Motivation • Heterogeneous architectures are ubiquitous • Non-clairvoyance is very important in practice THE MAIN THEOREM SELFISHMIGRATE is -speed -competitive for minimizing weighted flow-time on unrelated machines. Result also extends to energy + weighted flow-time. CLAIRVOYANT ALGORITHM Anand-Garg-Kumar, SODA ‘12 Chadha-Garg-Kumar-Muralidhara, STOC ‘09 ALGORITHM 20 SJF 1 10 100 Increase A = 31 + 20 NEW 20 20 SJF 3 5 70 Increase B = 28 + 20 xijt å p ³1 ij i,t åx ijt j £1 For all jobs æ pj ö çç pij = ÷÷ ℓij ø è j For all machines i, t THE DUAL For all jobs j machines i and time instants t ³ rj The whole frame work breaks down for non-clairvoyant settings… 1. We cannot measure the increase in the objective 2. Migrating jobs is necessary 3. Migration of jobs makes calculating the flowtimes difficult and complicates how jobs interfere with each other SELFISHMIGRATE KEY IDEA I Focus on speed at which jobs are getting processed and monotonicity of speed Speed of a job = fraction of the job processed at time If is monotone, then If is monotone, then Dual Constraint: If is monotone, then Dual Constraint: Round Robin or Proportional Sharing 1 Fraction of processor allocated = N(i, t) If is monotone, then Dual Constraints: If is monotone, then But which machine we should assign the job Dual Constraints: ? If is monotone, then But which machine we should assign the job ? ANS IS Dual Constraints: MINIMUM Dual constraints need us… For all jobs j machines i and time instants t 1. BEST MACHINE- which maximizes speed. 2. The speed MONOTONICALLY increases. KEY IDEA II: VIRTUAL QUEUES, VIRTUAL SPEED j RR ℓij f ( j, t) = * N ( j, t) +1 VIRTUAL QUEUES, VIRTUAL SPEEDS RR ℓij f ( j, t) = N( j, t) +1 Inverse of instantaneous delay the job induces on the other jobs ahead of its virtual queue. KEY IDEA II: VIRTUAL QUEUES, VIRTUAL SPEED RR Total delay induced by job if it stays on the machine till its completion [t,C j ] is proportional to j p (t) f ( j, t) KEY IDEA III: SELFISHMIGRATE RULE 1: NO JUMPING (not even virtual ones!) RULE 1: NO JUMPING Virtual speed of other jobs does not decrease if a job enters the machine! RULE 2: BE SELFISH! ℓij f ( j, t) = N *( j, t) +1 Increases virtual speed… migrate! Any time! RULE 2: BE SELFISH! Remains Same Increases! Remains Same Jobs migrate whenever they can increase their virtual speed... Jobs migrate whenever they can increase their virtual speed….. Jobs migrate whenever they can increase their virtual speed… Jobs migrate whenever they can increase their virtual speed….. This cannot go on forever… as virtual speeds are finite We get …MONOTONICITY Virtual speeds are monotone non-decreasing! BEST MACHINE ℓij f ( j, t) = N( j, t) +1 each job chooses the best machine on arrival BEST MACHINE (Nash Equilibrium on Virtual Speed) SELFISHMIGRATE ensures this is true at all times by design! BEST MACHINE Entire process can be thought as Coordination ‘game’ on virtual speeds DUAL SETTINGS aj The total delay job causes on the jobs ahead of its virtual queue over its entire sojourn DUAL SETTINGS aj åa j The total delay job j causes on the jobs ahead of its virtual queue over its entire sojourn + its total processing time. j The flow-time of all jobs (Cost of the algorithm) Dual setting b 1 = it (2 + e ) å i,t 1 bit = (2 + e ) The total number of jobs in the queue at time t. The total flow-time of all jobs (Cost of the algorithm) DUAL OBJECTIVE Maxåa j - å bit j i,t DUAL = O(e )*COSTALGO What About Constraints? For all jobs j machines i and time instants t DELAY LEMMAS Total delay a job induced on jobs ahead of it over the interval Total delay a job induced on jobs ahead of it in t rj £ (t - r ) j Cj pj £ 2* f ( j, t) (Follows from a property of Round Robin) DELAY LEMMA DELAY LEMMA MINIMUM (BEST MACHINE) CONCLUSIONS and OPEN PROBLEMS We gave a new framework to analyze non-clairvoyant scheduling problems - norms of weighted flow-time THANKS