Online Scheduling with QoS Constraints

Transcription

1 Online cheduling with Qo Constraints Uwe chwiegelshohn cheduling for large-scale systems Knoville, TN, UA May 4, 9

2 Problem Description Online scheduling r j,online Jobs are submitted over time. At its submission r j, job J j is immediately allocated to an eligible machine. Parallel identical machines P m Each job J j has the same processing time p j on each eligible machine. Ordered machine eligibility There is a fied order of the machines:,,,m Using this order, the first machine eligible to eecute job J j is machine k j. Every machine i with i k j is also eligible to eecute job J j : M j ={i i k j } Makespan C ma It is the goal to minimize the makespan of the schedule. P m r j,online,m j C ma

3 Previous Results P m M j C ma with no restrictions on M j. The problem is NP-hard as P m C ma is already NP-hard. P m p j =,M j C ma with nested machine eligibility constraints. M j = M k, M j M k,m j M k, or M j M k = ø. The Least Fleible Job First (LFJ) rule optimally solves this problem. M. Pinedo: cheduling: Theory, Algorithms, and ystems, Prentice Hall,. P m M j C ma with ordered eligibility. Least eligibility longest processing time order guarantees the approimation factor -/(m-). H-C. Hwang,.Y. Chang, K. Lee. Parallel machine scheduling under a grade of service provision, Computer & Operations Research 3, 55-6 (4). P m r j,online,m j C ma with no restrictions on M j. Competitive ratio log n for deterministic and randomized cases. Y. Azar, J. Naor, R. Ron. The Competitiveness of On-Line Assignments, Journal of Algorithms 8, -37 (995). 3

4 Relevance of the Problem hall I allocate my precious resources to somebody not paying enough for them or run the risk that these resources are not used at all? In practice, this is a fied capacity problem with customer rejection. This problem is different from the utilization of a fied number of machines. There is a close connection with utilization if there ins no rejection. The makespan can represent this objective. 4

5 Application Eamples Packaging in lattice boes The granularity of the material determines the required size of the lattice. ervers with different amount of main memory The storage requirement of a job determines the eligibility of a server. Class of transportation The ticket determines the class of transportation. 5

6 Continuous Model for Large-scale ystems We allow fractional machines and normalize the machine space. The machine space is represented by the interval [,] of real numbers. Job J j can only be allocated to the interval [k j,] with k j. Each job has a very short processing time. Job allocation does not need to consider individual processing times. Machine eligibility is represented by the job density function p() p(): [,] R : Total processing time of jobs with k j =. Release dates are not considered within the job density function. There is a completion time function that determines the makespan. c (): [,] R : Completion time function of schedule C ma ()=ma{c () }: Makespan of schedule Idle times are included. 6

7 Long and hort Processing times time machines machines 7

8 Job Density and Completion Time Functions idle areas in the schedule p() C ma () c s () Job density function Completion time function p()d c () d if there are no intermediate idle areas in the schedule. 8

9 imple Approaches with Bad Results Constant interval approach: A new job J j is allocated such that the maimum of the function c s () is increased the least in the interval [k j, ma{k j +ε,}). The competitive factor is ε -. Constant interval schedule Optimal schedule ε Greedy approach: A new job J j is allocated such that the maimum of the function c s () is increased the least in the interval [k j,). p()=, jobs are submitted in quick succession in order of k j. The competitive factor is not constant. 9

10 Greedy Approach c (t)dt ( ) c () dc() c() c() ( ) d dc() c d () Cln c ()d C ( ) ln( ) C ln Cln C lim c( )

11 Interval Approach A job J j is only eecuted in the interval [k j, k j +(-k j )/α) with α>. In this approach, additional jobs cannot decrease the makespan of a schedule. Eample: Assume that a group of jobs with k = is released at time and immediately followed by another group of jobs with k =δ δ α+δ (-α)

12 Making a chedule Worse Jobs are added until c opt ()=const for all and the schedule contains no idle areas. The ratio ma {c ()} to ma {c opt ()} cannot decrease. We normalize the job density function such that c opt ()=. C ma () c opt ()

13 Worst Case: Job ubmission Order The jobs are submitted in quick succession in increasing order of k j. The difference between the starting values of two intervals k -k is larger than the difference between the ending values of these intervals (-/α) (k -k ). An increasing order of k j produces larger C ma values than a decreasing order. Increasing order of k j Decreasing order of k j Eample with α= 3

14 Worse Case Input Data Assume y=arg{ma {c ()}}. y: Every job with that eecutes in the optimal schedule on a machine with a number greater than y is changed to a job with k j =y. The makespan of the optimal schedule remains unchanged. C ma () cannot decrease as jobs with k j >y do not contribute to c s (y). <y: If the eligibility bound k j of a job J j is less than the machine number at which it is eecuted in the optimal schedule then k j is increased to. The makespan of the optimal schedule remains unchanged. C ma () cannot decrease as this transformation can only increase the machine number on which a job is eecuted in schedule. The job density function of worst case input data is for <y and a Dirac pulse for =y such that the area of the pulse is (-y). 4

15 Job Density Function of Worst Case Input Data y 5

16 Differential Equation of the Interval Approach c (t)dt ( ) c () c s () (-) c s ()/α 6

17 Differential Equation of the Interval Approach c (t)dt ( ) c () ( ) dc() c() c() d dc() d c () dc() d c () c () 7

18 olution of the Differential Equation h - c () ep d ep( ) ln( ) ( ) p c () ( ) d ( ) ( ) - ( ) c () Cc h () c p () C(- ) - c () C c () ( ) - 8

19 Interval chedule for α= and a Given y C ma ()=c s (y) c s () y 9

20 dortmund university Determination of the Optimal Value for α y) ( dy df(y) (y) c f(y) f(y) y ) ( ) ( ) ( d d 4 f(y) y

21 Transformation to the Discrete Case The interval allocation is approimated by allowing a job to be allocated to a machine if the continuous interval of the job would contain at least of fraction of this machine and afterwards applying list scheduling. ome machines may receive more total processing time than in the continuous case. The additional processing time is upper bounded by the processing time of the longest job. Due to the different processing times of the individual jobs, not all machines of an interval may achieve the same makespan although this might have happen in the continuous case. The difference between the makespan of such two machines is at most the processing time of the longest job (list scheduling). Altogether the approimation cannot increase the competitive factor by more than.

22 Consequence of the Approimation time machines of an interval

23 Conclusion For very large systems, online job scheduling on parallel identical machines with ordered machine eligibility achieves a competitive factors of 5. For the proof, we used a generalization to a continuous case and derived and solved a differential equation. We approimated the continuous case by simply applying list scheduling. For systems with few machines, the competitive factor is smaller as the value of y in the continuous case is bounded by m-. 3