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DP-F AIR : A Simple Model for Understanding Optimal Multiprocessor Scheduling Greg Levin † Shelby Funk ‡ Caitlin Sadowski † Ian Pye † Scott Brandt † † University of California ‡ University of Georgia Santa Cruz Athens

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Backstory NQ-Fair by Shelby Funk (Rejected from RTSS) S N S by Greg, Caitlin, Ian, and Scott (Rejected from RTSS) DP-Fair

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Multiprocessors Most high-end computers today have multiple processors In a busy computational environment, multiple processes compete for processor time More processors means more scheduling complexity

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4 Real-Time Multiprocessor Scheduling Real-time tasks have workload deadlines Hard real-time = “Meet all deadlines!” Problem: Scheduling periodic, hard real- time tasks on multiprocessor systems. This is difficult.

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5 Real-Time Multiprocessor Scheduling easy!! Real-time tasks have workload deadlines Hard real-time = “Meet all deadlines!” Problem: Scheduling periodic, hard real- time tasks on multiprocessor systems. This is difficult.

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Taxonomy of Multiprocessor Scheduling Algorithms Uniprocessor Algorithms LLFEDF Partitioned Algorithms Globalized Uniprocessor Algorithms Global EDF Dedicated Global Algorithms Partitioned EDF Optimal Algorithms EKG LLREF pfair

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7 Partitioned Schedulers ≠ Optimal Example: 2 processors; 3 tasks, each with 2 units of work required every 3 time units

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8 Global Schedulers Succeed Example: 2 processors; 3 tasks, each with 2 units of work required every 3 time units Task 3 migrates between processors

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9 Problem Model n tasks running on m processors A periodic task T = (p, e) requires a workload e be completed within each period of length p T's utilization u = e / p is the fraction of each period that the task must execute job release time = 0 p 2p2p 3p3p job release period p workload e

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time work completed job release deadline Fluid Rate Curve period p workload e fluid rate curve utilization u = slope = e/p actual work curve slope = 0 or 1 10

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job release deadline Feasible Work Region period p workload e zero laxity work complete work completed time 11 CPU rate = 1

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12 Assumptions Processor Identity: All processors are equivalent Task Independence: Tasks are independent Task Unity: Tasks run on one processor at a time Task Migration: Tasks may run on different processors at different times No overhead: free context switches and migrations In practice: built into WCET estimates

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13 The Big Goal (Version 1) Design an optimal scheduling algorithm for periodic task sets on multiprocessors A task set is feasible if there exists a schedule that meets all deadlines A scheduling algorithm is optimal if it can always schedule any feasible task set

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14 Necessary and Sufficient Conditions Any set of tasks needing at most 1) 1 processor for each task ( for all i, u i ≤ 1 ), and 2) m processors for all tasks ( u i ≤ m) is feasible Proof: Small scheduling intervals can approximate a task's fluid rate curve Status: Solved pfair (1996) was the first optimal algorithm

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15 The Big Goal (Version 2) Design an optimal scheduling algorithm with fewer context switches and migrations ( Finding a feasible schedule with fewest migrations is NP-Complete )

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16 The Big Goal (Version 2) Design an optimal scheduling algorithm with fewer context switches and migrations Status: Solved BUT, the solutions are complex and confusing Our Contributions: A simple, unifying theory for optimal global multiprocessor scheduling and a simple optimal algorithm

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17 Greedy Algorithms Fail on Multiprocessors A Greedy Scheduling Algorithm will regularly select the m “best” jobs and run those Earliest Deadline First (EDF) Least Laxity First (LLF) EDF and LLF are optimal on a single processor ; neither is optimal on a multiprocessor Greedy approaches generally fail on multiprocessors

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18 Example (n = 3 tasks, m = 2 processors) : Why Greedy Algorithms Fail On Multiprocessors

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19 Why Greedy Algorithms Fail On Multiprocessors At time t = 0, Tasks 1 and 2 are the obvious greedy choice

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20 Why Greedy Algorithms Fail On Multiprocessors Even at t = 8, Tasks 1 and 2 are the only “reasonable” greedy choice

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21 Why Greedy Algorithms Fail On Multiprocessors Yet if Task 3 isn’t started by t = 8, the resultant idle time eventually causes a deadline miss

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22 Why Greedy Algorithms Fail On Multiprocessors How can we “see” this critical event at t = 8?

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23 Proportioned Algorithms Succeed On Multiprocessors Subdivide Task 3 with a period of 10

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24 Proportioned Algorithms Succeed On Multiprocessors Now Task 3 has a zero laxity event at time t = 8

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25 Proportional Fairness Insight: Scheduling is easier when all jobs have the same deadline Application: Apply all deadlines to all jobs Assign workloads proportional to utilization Work complete matches fluid rate curve at every system deadline

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26 Proportional Fairness Is the Key All known optimal algorithms enforce proportional fairness at all deadlines: pfair (1996) - Baruah, Cohen, Plaxton, and Varvel ( proportional fairness at all times ) BF (2003) - Zhu, Mossé, and Melhem LLREF (2006) - Cho, Ravindran, Jensen EKG (2006) - Andersson, Tovar Why do they all use proportional fairness?

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27 Scheduling Multiple Tasks is Complicated time work completed

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28 Scheduling Multiple Tasks with Same Deadline is Easy work completed time

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Actual Feasible Regions 29 work completed time job deadlines

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Restricted Feasible Regions Under Deadline Partitioning 30 work completed time all system deadlines

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31 The DP-Fair Scheduling Policy Partition time into slices based on all system deadlines Allocate each job a per-slice workload equal to its utilization times the length of the slice Schedule jobs within each slice in any way that obeys the following three rules: 1. Always run a job with zero local laxity 2. Never run a job with no workload remaining in the slice 3. Do not voluntarily allow more idle processor time than ( m - u i ) x ( length of slice )

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time DP-Fair Work Allocation time slice Allocated workload slope = utilization 32 work completed

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DP-Fair Scheduling Rule #1 zero local laxity line When job hits zero local laxity, run to completion 33 time work completed time slice

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DP-Fair Scheduling Rule #2 time slice local work complete line When job finishes local workload, stop running 34 time work completed

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DP-Fair Scheduling Rule #3 time slice Don’t voluntarily allow idle time in excess of this limit slope = m - u i slope = m Allowable idle time 35 time idle time

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36 DF-Fair Guarantees Optimality We say that a scheduling algorithm is DP-Fair if it follows these three rules Theorem: Any DP-Fair scheduling algorithm for periodic tasks is optimal

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37 Proof of Theorem

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38 DP-Fair Implications ( Partition time into slices ) + ( Assign proportional workloads ) Optimal scheduling is almost trivial Minimally restrictive rules allow great latitude for algorithm design and adaptability What is the simplest possible algorithm?

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All time slices are equivalent (up to linear scaling), so normalize time slice length to 1 Jobs have workload = utilization Schedule by “stacking and slicing” Line up job utilization blocks Slice at integer boundaries 39 The DP-Wrap Algorithm

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40 The DP-Wrap Algorithm Example: 3 processors, 7 tasks

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41 The DP-Wrap Algorithm Example: 3 processors, 7 tasks

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42 The DP-Wrap Algorithm Example: 3 processors, 7 tasks

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43 The DP-Wrap Algorithm

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44 DP-Wrap Performance n−1 context switches and m−1 migrations per time slice This is about 1/3 as many as LLREF, similar to BF, and somewhat more than EKG DP-Wrap has very low (O(n)) computational complexity

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45 Extending DP-Fair: Sporadic Arrivals, Arbitrary Deadlines Sporadic arrivals: jobs from a task arrive at least p time units apart Arbitrary deadlines: A job from T = (p, e, D) has a deadline D time units after arrival; D may be smaller or larger than p DP-Fair and DP-Wrap are easily modified to handle these cases

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46 DP-Fair Easily Extended Extending optimal algorithms to sporadic tasks and arbitrary deadlines has previously been difficult pfair : 10 years LLREF : 3 years DP-Fair & DP-Wrap: 3 weeks

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Ongoing Work Extend to uniform multiprocessors (CPUs running at different speeds) Extend to random, unrelated job model Heuristic improvements to DP-Wrap Work-conserving scheduling Slice merging

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48 Summary DP-Fair : Simple rules to make optimal real-time multiprocessor scheduling easy!! DP-Wrap : Simplest known optimal multiprocessor scheduling algorithm Extensible : DP-Fair and DP-Wrap easily handle sporadic tasks with arbitrary deadlines We plan to extend DP-Fair and DP-Wrap to more general problem domains

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49 Thanks for Listening Questions?

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50 === EXTRA SLIDES === 1. Sporadic Tasks, Arbitrary Deadlines

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51 Sporadic Arrivals, Arbitrary Deadlines We extend our task definition to T i = (p i, e i, D i ), where each job of T i has a deadline D i time units after its arrival; D i may be smaller or larger than p i. These are referred to as arbitrary deadlines. Sporadic arrivals means that p i is now the minimum inter-arrival time for T i 's jobs, so that a new job will arrive at least p i time units after the previous arrival. Scheduling is on-line, i.e. we don't know ahead of time when new jobs will arrive

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52 Task Density Our expression for utilization must be updated to the density of a task: i = e i / min{ p i, D i } and are still sufficient conditions for the feasibility of a task set, but they are no longer necessary. Example: T 1 = (2, 1, 1), T 2 = (2, 1, 2) on m = 1 is feasible, although = 1.5.

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53 Optimality Our updated DP-Fair rules for sporadic tasks with arbitrary deadlines will allow an algorithm to schedule any task set with and ; in fact, we require that these conditions hold Since feasible task sets exist where these conditions do not hold, DP-Fair algorithms are not optimal in this problem domain This is an acceptable limitation, since it has recently been proven that no optimal scheduler can exist for sporadic tasks

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54 Updated DP-Fair Scheduling The rules for scheduling within time slices is nearly the same with sporadic tasks and arbitrary deadlines The third rule is now: "Do not voluntarily allow more than units of idle time to occur during the time slice before time t" The F j (t) term represents unused capacity between tasks' deadlines and next arrivals

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55 Updated DP-Fair Allocation The rules for allocating workloads in time slices become more complicated than simple proportionality A task is active between its arrival and its deadline. When a task becomes active, it is allocated a workload proportional to its density and the length of its active segment in the slice If a job arrives during a slice that will have a deadline within the slice, the remainder of the slice is partitioned into two new subslices, with a subslice boundary at the deadline, and remaining work for all tasks allocated proportionally to subslice lengths

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Handling a Sporadic Arrival time slice Workload allocated upon arrival Sporadic arrival in the middle of a slice 56 time work completed

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Managing Idle Time time slice Available idle time grows by utilization of unarrived sporadic job… slope = m - u i slope = m Original idle time pool …until that job finally arrives Additional idle time from late arrival 57 time idle time

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58 Theorem 3 Any DP-Fair scheduling algorithm for sporadic tasks with arbitrary deadlines will successfully schedule any task set where and

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59 Modifying DP-Wrap The DP-Wrap algorithm is easily modified to accommodate the DP-Fair scheduling rules for sporadic tasks with arbitrary deadlines If a job arrives during a time slice, wrap it onto the end of the stack of blocks If the arriving job has a deadline within the current slice, partition the remainder of the slice into two subslices, as per our previous rule, and apply the usual scheduling rules to both of these subslices

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