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fakultät für informatikinformatik 12
technische universität dortmund
Classical scheduling algorithms
for periodic systems
Peter MarwedelTU Dortmund, Informatik 12
Germany
Gra
phic
s: ©
Ale
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ra N
olte
, Ges
ine
Mar
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003
These slides use Microsoft clip arts. Microsoft copyright restrictions apply. 2010 年 12 月 15 日
- 2 -technische universitätdortmund
fakultät für informatik
p. marwedel, informatik 12, 2010
Structure of this course
2:Specification
3: ES-hardware
4: system software (RTOS, middleware, …)
8:Test
5: Evaluation & validation & (energy, cost, performance, …)
7: Optimization
6: Application mapping
App
licat
ion
Kno
wle
dge Design
repositoryDesign
Numbers denote sequence of chapters
- 3 -technische universitätdortmund
fakultät für informatik
p. marwedel, informatik 12, 2010
Classes of mapping algorithmsconsidered in this course
Classical scheduling algorithmsMostly for independent tasks & ignoring communication, mostly for mono- and homogeneous multiprocessors
Dependent tasks as considered in architectural synthesisInitially designed in different context, but applicable
Hardware/software partitioningDependent tasks, heterogeneous systems,focus on resource assignment
Design space exploration using evolutionary algorithms; Heterogeneous systems, incl. communication modeling
- 4 -technische universitätdortmund
fakultät für informatik
p. marwedel, informatik 12, 2010
Periodic scheduling
Each execution instance of a task is called a job.
Notion of optimality for aperiodic scheduling does not make sense for periodic scheduling.
For periodic scheduling, the best that we can do is to design an algorithm which will always find a schedule if one exists. A scheduler is defined to be optimal iff it will find a schedule if one exists.
T1
T2
- 5 -technische universitätdortmund
fakultät für informatik
p. marwedel, informatik 12, 2010
Periodic scheduling- Scheduling with no precedence constraints -
Let {Ti } be a set of tasks. Let: pi be the period of task Ti, ci be the execution time of Ti , di be the deadline interval, that is,
the time between Ti becoming available and the time until which Ti has to finish execution.
li be the laxity or slack, defined as li = di - ci
fi be the finishing time.
li
di
cit
i
pi
- 6 -technische universitätdortmund
fakultät für informatik
p. marwedel, informatik 12, 2010
Average utilization, a very important characterization of scheduling problems:
Average utilization:
n
i i
i
p
c
1
Necessary condition for schedulability(with m=number of processors):
m
- 7 -technische universitätdortmund
fakultät für informatik
p. marwedel, informatik 12, 2010
Independent tasks:Rate monotonic (RM) scheduling
Most well-known technique for scheduling independentperiodic tasks [Liu, 1973].Assumptions: All tasks that have hard deadlines are periodic.
All tasks are independent.
di =pi, for all tasks.
ci is constant and is known for all tasks.
The time required for context switching is negligible.
For a single processor and for n tasks, the following equation holds for the average utilization µ:
)12( /1
1
nn
i i
i np
c
- 8 -technische universitätdortmund
fakultät für informatik
p. marwedel, informatik 12, 2010
Rate monotonic (RM) scheduling- The policy -
RM policy: The priority of a task is a monotonically decreasing function of its period.
At any time, a highest priority task among all those that are ready for execution is allocated.
Theorem: If all RM assumptions are met, schedulability is guaranteed.
- 9 -technische universitätdortmund
fakultät für informatik
p. marwedel, informatik 12, 2010
Maximum utilization for guaranteed schedulability
Maximum utilization as a function of the number of tasks:
)2ln()12((lim
)12(
/1
/1
1
n
n
nn
i i
i
n
np
c
- 10 -technische universitätdortmund
fakultät für informatik
p. marwedel, informatik 12, 2010
Example of RM-generated schedule
T1 preempts T2 and T3.T2 and T3 do not preempt each other.
t
- 11 -technische universitätdortmund
fakultät für informatik
p. marwedel, informatik 12, 2010
Failing RMS
Task 1: period 5, execution time 3Task 2: period 8, execution time 3µ=3/5+3/8=24/40+15/40=39/40 0.975 2(21/2-1) 0.828
- 12 -technische universitätdortmund
fakultät für informatik
p. marwedel, informatik 12, 2010
Case of failing RM scheduling
Task 1: period 5, execution time 2Task 2: period 7, execution time 4µ=2/5+4/7=34/35 0.97 2(21/2-1) 0.828
Missed deadline
Missing computations scheduled in the next period
leviRTS animation
t
- 13 -technische universitätdortmund
fakultät für informatik
p. marwedel, informatik 12, 2010
Intuitively: Why does RM fail ?
No problem if p2 = m p1, mℕ :
T1
T2
t fits
T1
T2
t
should be completed
Switching to T1 too early, despite early deadline for T2
t
t
- 14 -technische universitätdortmund
fakultät für informatik
p. marwedel, informatik 12, 2010
Critical instants
Definition: A critical instant of a task is the time at which the release of a task will produce the largest response time.
Lemma: For any task, the critical instant occurs if that task is simultaneously released with all higher priority tasks.
Proof: Let T={T1, …,Tn}: periodic tasks with i: pi ≦ pi +1.
Source: G. Buttazzo, Hard Real-time Computing Systems, Kluwer, 2002
- 15 -technische universitätdortmund
fakultät für informatik
p. marwedel, informatik 12, 2010
Critical instances (1)
Response time of Tn is delayed by tasks Ti of higher priority:
cn+2ci
Tn
Ti
t
Maximum delay achieved if Tn and Ti start simultaneously.
Maximum delay achieved if Tn and Ti start simultaneously.
cn+3ci
Tn
Ti
Delay may increase if Ti starts earlierDelay may increase if Ti starts earlier
t
- 16 -technische universitätdortmund
fakultät für informatik
p. marwedel, informatik 12, 2010
Critical instants (2)
Repeating the argument for all i = 1, … n-1:
The worst case response time of a task occurs when it is released simultaneously with all higher-priority tasks. q.e.d.
Schedulability is checked at the critical instants.
If all tasks of a task set are schedulable at their critical instants, they are schedulable at all release times.
Observation helps designing examples
- 17 -technische universitätdortmund
fakultät für informatik
p. marwedel, informatik 12, 2010
The case i: pi+1 = mi pi
Lemma*: If each task period is a multiple of the period of the next higher priority task, then schedulability is also guaranteed if µ 1.
Proof: Assume schedule of Ti is given. Incorporate Ti+1:
Ti+1 fills idle times of Ti; Ti+1 completes in time, if µ 1.
Ti
Ti+1 t
T’i+1
Used as the higher priority task at the next iteration.
* wrong in the book of 2003
- 18 -technische universitätdortmund
fakultät für informatik
p. marwedel, informatik 12, 2010
Proof of the RM theorem
Let T={T1, T2} with p1 < p2.
Assume RM is not used prio(T2) is highest:
T1
T2 t
Schedule is feasible if c1+c2 ≦ p1 (1)Schedule is feasible if c1+c2 ≦ p1 (1)
c1
c2
p1
Define F= p2/p1: # of periods of T1 fully contained in T2Define F= p2/p1: # of periods of T1 fully contained in T2
Morein-depth:
- 19 -technische universitätdortmund
fakultät für informatik
p. marwedel, informatik 12, 2010
Case 1: c1 p2 – Fp1
Assume RM is used prio(T1) is highest:
T1
T2 t
Case 1*: c1 p2 – F p1
(c1 small enough to be finished before 2nd instance of T2)
Case 1*: c1 p2 – F p1
(c1 small enough to be finished before 2nd instance of T2)
p2Fp1
Schedulable if (F +1) c1 + c2 p2 (2)Schedulable if (F +1) c1 + c2 p2 (2)
* Typos in [Buttazzo 2002]: < and mixed up]
- 20 -technische universitätdortmund
fakultät für informatik
p. marwedel, informatik 12, 2010
Proof of the RM theorem (3)
Not RM: schedule is feasible if c1+c2 p1 (1)
RM: schedulable if (F+1)c1 + c2 p2 (2)
From (1): Fc1+Fc2 Fp1
Since F 1: Fc1+c2 Fc1+Fc2 Fp1
Adding c1: (F+1) c1+c2 Fp1 +c1
Since c1 p2 – Fp1: (F+1) c1+c2 Fp1 +c1 p2
Hence: if (1) holds, (2) holds as well
For case 1: Given tasks T1 and T2 with p1 < p2, then if the schedule is feasible by an arbitrary (but fixed) priority assignment, it is also feasible by RM.
- 21 -technische universitätdortmund
fakultät für informatik
p. marwedel, informatik 12, 2010
Case 2: c1 > p2 – Fp1
Case 2: c1 > p2 – Fp1
(c1 large enough not to finish before 2nd instance of T2)
T1
T2 tp2Fp1
Schedulable if F c1 + c2 F p1 (3)
c1+c2 p1 (1)
Multiplying (1) by F yields F c1+ F c2 F p1
Since F 1: F c1+ c2 F c1+ Fc2 F p1
Same statement as for case 1.
- 22 -technische universitätdortmund
fakultät für informatik
p. marwedel, informatik 12, 2010
Calculation of the least upper utilization bound
Let T={T1, T2} with p1 < p2.
Proof procedure: compute least upper bound Ulup as follows
Assign priorities according to RM
Compute upper bound Uup by setting computation times to fully utilize processor
Minimize upper bound with respect to other task parameters
As before: F= p2/p1
c2 adjusted to fully utilize processor.
- 23 -technische universitätdortmund
fakultät für informatik
p. marwedel, informatik 12, 2010
Case 1: c1 p2 – Fp1
Largest possible value of c2 is c2= p2 – c1 (F+1)Corresponding upper bound is
T1
T2 tp2Fp1
)1(11
11
1
2
2
1
2
1
1
1
2
12
1
1
2
2
1
1 Fp
p
p
c
p
F c
p
c
p
F – cp
p
c
p
c
p
cUub
)(
)(
{ } is <0 Uub monotonically decreasing in c1
Minimum occurs for c1 = p2 – Fp1
- 24 -technische universitätdortmund
fakultät für informatik
p. marwedel, informatik 12, 2010
Case 2: c1 p2 – Fp1
Largest possible value of c2 is c2= (p1-c1)FCorresponding upper bound is:
T1
T2 tp2Fp1
Fp
p
p
cF
p
pF
p
c
p
cF
p
p
p
F – cp
p
c
p
c
p
cUub
1
2
2
1
2
1
2
1
1
1
2
1
2
11
1
1
2
2
1
1 )(
{ } is 0 Uub monotonically increasing in c1 (independent of c1 if {}=0)
Minimum occurs for c1 = p2 – Fp1, as before.
- 25 -technische universitätdortmund
fakultät für informatik
p. marwedel, informatik 12, 2010
Utilization as a function of G=p2/p1-F
For minimum value of c1:
F
p
pF
p
pF
p
pF
p
p
p
FppF
p
pF
p
p
p
cF
p
pUub
1
2
1
2
2
1
1
2
2
12
2
1
1
2
2
1
2
1
Let ;1
2 Fp
pG
GF
GG
GF
GGGF
GF
GF
FFpp
GF
pp
GFGF
p
pUub
11
)(
//
22
12
2
12
22
2
1
Since 0 G< 1: G(1-G) 0 Uub increasing in F
Minimum of Uub for min(F): F=1
G
GUub
1
1 2
- 26 -technische universitätdortmund
fakultät für informatik
p. marwedel, informatik 12, 2010
Proving the RM theorem for n=2
This proves the RM theorem for the special case of n=2
83.0)11222
224
121
121
:10
;21;21
0)1(
12
)1(
)1()1(2
:1
1
2
2
21
2
2
2
2
2
2
1
2(2)()(
)(
since ,only gConsiderin
of minimum find to derivative Using
lub
ub
ub
ub
U
GG
GG
G
GG
G
GGG
dG
dU
UG
GU
end
- 27 -technische universitätdortmund
fakultät für informatik
p. marwedel, informatik 12, 2010
Properties of RM scheduling
RM scheduling is based on static priorities. This allows RM scheduling to be used in an OS with static priorities, such as Windows NT.
No idle capacity is needed if i: pi+1=F pi: i.e. if the period of each task is a multiple of the period of the next higher priority task, schedulability is then also guaranteed if µ 1.
A huge number of variations of RM scheduling exists.
In the context of RM scheduling, many formal proofs exist.
- 28 -technische universitätdortmund
fakultät für informatik
p. marwedel, informatik 12, 2010
EDF
EDF can also be applied to periodic scheduling.
EDF optimal for every hyper-period(= least common multiple of all periods)
Optimal for periodic scheduling
EDF must be able to schedule the example in which RMS failed.
- 29 -technische universitätdortmund
fakultät für informatik
p. marwedel, informatik 12, 2010
Comparison EDF/RMS
T2 not preempted, due to its earlier deadline.
EDF-animation
RMS:
EDF:
- 30 -technische universitätdortmund
fakultät für informatik
p. marwedel, informatik 12, 2010
EDF: Properties
EDF requires dynamic priorities
EDF cannot be used with an operating system just providing static priorities.
However, a recent paper (by Margull and Slomka) at DATE 2008 demonstrates how an OS with static priorities can be extended with a plug-in providing EDF scheduling(key idea: delay tasks becoming ready if they shouldn’t be executed under EDF scheduling.
- 31 -technische universitätdortmund
fakultät für informatik
p. marwedel, informatik 12, 2010
Comparison RMS/EDF
RMS EDF
Priorities Static Dynamic
Works with OS with fixed priorities
Yes No*
Uses full computational power of processor
No,just up till µ=n(21/n-1)
Yes
Possible to exploit full computational power of processor without provisioning for slack
No Yes
* Unless the plug-in by Slomka et al. is added.
- 32 -technische universitätdortmund
fakultät für informatik
p. marwedel, informatik 12, 2010
Sporadic tasks
If sporadic tasks were connected to interrupts, the execution
time of other tasks would become very unpredictable.
Introduction of a sporadic task server,periodically checking for ready sporadic tasks;
Sporadic tasks are essentially turned into periodic tasks.
- 33 -technische universitätdortmund
fakultät für informatik
p. marwedel, informatik 12, 2010
Dependent tasks
The problem of deciding whether or not a schedule existsfor a set of dependent tasks and a given deadlineis NP-complete in general [Garey/Johnson].
Strategies:
1. Add resources, so that scheduling becomes easier
2. Split problem into static and dynamic part so that only a minimum of decisions need to be taken at run-time.
3. Use scheduling algorithms from high-level synthesis
- 34 -technische universitätdortmund
fakultät für informatik
p. marwedel, informatik 12, 2010
Summary
Periodic scheduling
Rate monotonic scheduling
EDF
Dependent and sporadic tasks (briefly)