Dynamic-Priority Scheduling. CSCE 990: Real-Time Systems. Steve Goddard. Dynamic-priority Scheduling

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1 CSCE 990: Real-Time Systems Dynamic-Priority Scheduling Steve Goddard htt:// Dynamic-riority Scheduling Real-Time Systems Dynamic-Priority Scheduling - The remaining scheduling discilines we consider are riority-based.» Each ob is assigned a riority, and the highest-riority ob executes at any time. We begin with dynamic-riority scheduling.» Under dynamic-riority scheduling, different obs of a task may be assigned different riorities.» Can have the following: ob J i,k of task T i has higher riority than ob J,m of task T, but ob J i, of T i has lower riority than ob J,n of T. We will consider static-riority scheduling later. Real-Time Systems Dynamic-Priority Scheduling - 2 2

2 Outline We consider both earliest-deadline-first (EDF) and least-laxity-first (LLF) (called least-slacktime-first by Liu) scheduling. Outline:» Otimality of EDF and LLF (Section 4.6 of Liu).» Utilization-based schedulability test for EDF (Section 6.3 of Liu).» Non-reemtive EDF from: Jeffay, Stanat, and Martel, On Otimal, Non-reemtive Scheduling of Periodic nad Soradic Tasks, 8th IEEE RTSS, 99, Otimality of EDF Theorem 4-: [Liu and Layland] When reemtion is allowed and obs do not contend for resources, the EDF algorithm can roduce a feasible schedule of a set J of indeendent obs with arbitrary release times and deadlines on a rocessor if and only if J has feasible schedules. Notes: Alies even if tasks are not eriodic. If eriodic, a task s relative deadline can be less than its eriod, equal to its eriod, or greater than its eriod. Real-Time Systems Dynamic-Priority Scheduling - 3 Real-Time Systems Dynamic-Priority Scheduling

3 Proof of Theorem 4- We show that any feasible schedule of J can be systematically transformed into an EDF schedule. Suose arts of two obs J i and are executed out of EDF order: If we inductively reeat this rocedure, we can eliminate all out-of-order violations. The resulting schedule may still fail to be an EDF schedule because it has idle intervals where some ob is ready: J i J i r k d k d i This situation can be corrected by erforming a swa : Such idle intervals can be eliminated by moving some obs forward: J i J i Real-Time Systems Dynamic-Priority Scheduling - 5 Real-Time Systems Dynamic-Priority Scheduling

4 LLF Scheduling Definition: At any time t, the slack (or laxity) of a ob with deadline d is equal to d t minus the time required to comlete the remaining ortion of the ob. laxity: deadline LLF Scheduling: The ob with the smallest laxity has highest riority at all times. Otimality of LLF Theorem 4-3: When reemtion is allowed and obs do not contend for resources, the LLF algorithm can roduce a feasible schedule of a set J of indeendent obs with arbitrary release times and deadlines on a rocessor if and only if J has feasible schedules. The roof is similar to that for EDF and is left as an exercise. Question: Which of EDF and LLF would be referable in ractice? Real-Time Systems Dynamic-Priority Scheduling - 7 Real-Time Systems Dynamic-Priority Scheduling

5 Preemtive vs. Nonreemtive EDF The rest of our discussion of dynamic-riority scheduling will focus reemtive and non-reemtive EDF. We first show the following: But under non-reemtive EDF, a deadline is missed! Theorem: Non-reemtive EDF is not otimal. Proof: Consider a system of three obs J, J 2, and J 3 such that (r, e, d ) = (0, 3, 0), (r 2, e 2, d 2 ) = (2, 6, 4), (r 3, e 3, d 3 ) = (4, 4, 2). Here s a schedule: J J 3 J Real-Time Systems Dynamic-Priority Scheduling - 9 J J 2 J r r 2 r 3 J 3 s deadline Question: Should we conclude from this result that reemtive EDF is always better than non-reemtive EDF in ractice? Note: The EDF otimality roof assumes there is no enalty due to reemtion. Are there other ractical issues we have ignored? Real-Time Systems Dynamic-Priority Scheduling

6 Utilization-based Schedulability Test for (Preemtive) EDF Note: Whenever we say EDF from now on, we mean reemtive EDF, unless secified otherwise. Theorem 6-: [Liu and Layland] A system T of indeendent, reemtable, eriodic tasks with relative deadlines equal to their eriods can be feasibly scheduled (under EDF) on one rocessor if and only if its total utilization U is at most one. Setting U the Proof We wish to show: U T is schedulable. We rove the contraositive, i.e., T is not schedulable U >. Assume T is not schedulable. Let J i,k be the first ob to miss its deadline. Proof: The only if art is obvious: If U >, then some task clearly must miss a deadline. So, we concentrate on the if art. T i t- ri,k r this is the last idle instant Real-Time Systems Dynamic-Priority Scheduling - Real-Time Systems Dynamic-Priority Scheduling - 2 2

7 Because J i,k missed its deadline the demand laced on the rocessor in [t -, r ) by obs with deadlines r is greater than the available rocessor time in [t -, r ]. Thus, r = = = = t r r = available rocessor time in [t,r < demand laced on the rocessor in [t,r N N (the number of N t t e e obs of T with deadlines r ] - ) - by obs with deadlines released in [t,r r - )) e Real-Time Systems Dynamic-Priority Scheduling - 3 r Thus, we have Cancelling r < i.e., t N = < U. e, < t t yields This comletes the roof. N = r e. Note: This roof is actually still valid if deadlines are larger than eriods. Real-Time Systems Dynamic-Priority Scheduling

8 EDF with Deadlines < Periods If deadlines are less than eriods than U is no longer a sufficient schedulability condition. This is easy to see. Consider two tasks such that, for both, e i = and i = 2. If both have deadlines at.9, then the system is not schedulable, even though U =. For these kinds of systems, we work with densities instead of utilizations. Definition: The density of task T k is defined to be δ k = e k /min(d k, k ). The density of the system is defined to be = k=.,,,.,n δ k. Deadlines < Periods (Continued) Theorem 6-2: A system T of indeendent, reemtable, eriodic tasks can be feasibly scheduled on one rocessor if its density is at most one. The roof is similar to that for Theorem 6- and is left as an exercise. Note: This theorem only gives sufficient condition. We refer to the following as the schedulability condition for EDF: n k= ek min(d, ) k k Real-Time Systems Dynamic-Priority Scheduling - 5 Real-Time Systems Dynamic-Priority Scheduling

9 Proof of Non-tightness To see that > doesn t imly non-schedulability, consider the following examle. Examle: We have two tasks T = (2, 0.6, ) and T 2 = (5, 2.3). = 0.6/ + 2.3/5 =.06. Nonetheless, we can schedule this task set under EDF: T T reeats Non-reemtive EDF (Jeffay et al.) Theorem : Let T = {T, T 2,, T n } be a system of indeendent, eriodic tasks with relative deadlines equal to their eriods such that the tasks in T are indexed in non-decreasing order by eriod (i.e., if i <, then i ). T can be scheduled by the non-reemtive EDF algorithm if: n ei ) i= i 2) i : i n :: L : < L < :: L e + i i i- = L e Note: This condition is actually necessary and sufficient for real-world soradic tasks. Real-Time Systems Dynamic-Priority Scheduling - 7 Real-Time Systems Dynamic-Priority Scheduling

10 Exlanation The first condition is ust a constraint on utilization. In the second condition, the term i- L L ei + e = gives an uer bound on rocessor demand in an interval [t, t+l]. Intuition: The worst-case attern of ob releases occurs when a ob of some T i begins executing (non-reemtively!) one time unit before some tasks with smaller eriods begin releasing some obs. These other obs are blocked by the ob of T i. Real-Time Systems Dynamic-Priority Scheduling - 9 Second Condition (Continued) Here s an illustration: T i T T 2 T 3 For any L over the range < L < i, the total demand on the rocessor in [t, t + L] due to obs with deadlines at or before t + L is: i- L ei + e = For the system to be schedulable, this demand must not exceed the length of the interval (which is L). Real-Time Systems Dynamic-Priority Scheduling

11 Proof of Theorem Suose conditions () and (2) hold for T but a deadline is missed. Let t d be the earliest oint in time at which a deadline is missed. There are two cases. Case : No ob with a deadline after time t d is scheduled rior to time t d. The analysis is ust like with reemtive EDF. Case 2: Some ob with a deadline after time t d is scheduled rior to time t d. Let T i be the task with the last ob with deadline after t d that is scheduled rior to t d. Then, we have the following: As before, let t - be the last idle instant. As before, because a deadline is missed at t d, demand over [t -, t d ] exceeds t d t -. In addition, this demand is at most =,..,n (t d t - )/ e. Thus, we have t d t - < =,..,n (t d t - )/ e =,..,n [(t d t - )/ ] e. This imlies utilization exceeds one, which contradicts condition (). Time T i t i Let us bound the rocessor demand in [t i, t d ] t d Real-Time Systems Dynamic-Priority Scheduling - 2 Real-Time Systems Dynamic-Priority Scheduling

12 Observe the following:» i > t d t i. This follows from the fact that the ob of task T i scheduled at time t i had a deadline after t d.» No task with index greater than i is scheduled in the interval [t i, t d ].» Other than a ob of task T i, no ob scheduled in [t i, t d ] could have been released at time t i.» There is no idle time in the interval [t i, t d ].» There is a least one ob that is released in [t i, t d ] with a deadline at or before time t d. From these facts, we conclude that demand over [t i, t d ] is less than or equal to i- t d (ti + ) ei + e. = Let L = t d t i. Then, L < e + This contradicts condition (2). i i- = L e. Real-Time Systems Dynamic-Priority Scheduling - 23 Real-Time Systems Dynamic-Priority Scheduling

13 Notes Note that this scheduling condition requires seudoolynomial time to evaluate. (Why?) Using real-world terminology, this condition is necessary and sufficient for soradic and non-concrete eriodic task systems. (Why?) Concrete = fixed release times (though maybe not all 0). For a non-concrete task system to be feasible, it must be schedulable for any initial hasing. In the rest of the aer, it is shown that the feasibility roblem for non-reemtive concrete eriodic task systems is NP-hard in the strong sense. Imlies that a seudo-olynomial-time feasibility test is unlikely for such systems. (We cover this result later when we consider intractability.) Real-Time Systems Dynamic-Priority Scheduling