On the Optimality of RM and EDF for Non-Preemptive Real-Time Harmonic Tasks

THE UNIVERSITY of TEHRAN

On the Optimality of RM and EDF for Non-Preemptive Real-Time

Harmonic Tasks

Mitra Nasri Sanjoy Baruah Gerhard Fohler Mehdi Kargahi

October 2014

2 of 29On the Optimality of RM and EDF

for Non-preemptive Harmonic Tasks 2 of 29

Benefits◦ No context switches◦ Cache and pipelines are not affected by other tasks

More precise estimation of WCET◦ Simpler mechanisms to protect critical sections

In some systems, preemption is either not allowed or too expensive

Non-preemptive Scheduling

Benefits from the application’s point of viewMinimum I/O delay

(the delays between sampling and actuation)


for Non-preemptive Harmonic Tasks

Non-Preemptive Scheduling is NP-Hard◦ For Periodic Tasks [Jeffay 1991]

◦ For Harmonic Tasks [Cai 1996]

◦ For Harmonic Tasks with Binary Period Ratio [Nawrocki 1998] ki = {1, 2, 4, 8, 16, …}

Existing Results

τi -1

Ti

Ti -1

τi Period Ratio



Schedulability test for npEDF, npRM, and Fixed Priority ◦ [Kim 1980, Jeffay 1991, George 1996, Park 2007, Andersson 2009, Marouf

2010,…]

Heuristic scheduling algorithms◦ Clairvoyant EDF [Ekelin 2006], Group-Based EDF [Li 2007]

Optimal scheduling algorithms for special cases ◦ [Deogun 1986, Cai 1996, Nasri 2014]

Related Works



[Deogun 1986]: An optimal algorithm if tasks have ◦ Constant integer period ratio K ≥ 3

[Cai 1996]: An optimal algorithm if tasks have ◦ Constant period ratio K = 2, or◦ Integer period ratio ki ≥ 3

Limitations Linear time in the number of jobs, exponential time in the number

of tasks! Constructs an offline time table with exponential number of entries

Optimal Algorithms for Harmonic Tasks

Period Ratio

An optimal scheduling algorithm is the one which guarantees all deadlines whenever a feasible schedule exists



Precautious-RM [Nasri 2014] is optimal if tasks have◦ Constant period ratio K = 2◦ Integer period ratio ki ≥ 3

◦ Arbitrary integer period ratio ki ≥ 1 and enough vacant intervals

Precautious-RM is online and has O(n) computational and memory complexity (in the number of tasks)

Optimal Algorithms, cont.


for Non-preemptive Harmonic Tasks 7 of 25A Framework to Construct Customized

Harmonic Periods for RTS

We study◦ The existence of a utilization-based test◦ The pessimism in the existing necessary and sufficient test◦ The efficiency of the recent processor speedup approach

Then for special cases of harmonic tasks we present◦ The schedulability conditions◦ A more efficient speedup factor

Contributions



Task set τ = {τ1, τ2, …, τn} Ti is the period of τi ci is the WCET of τi ki is the period ratio of τi to τi-1

Deadlines are implicit; Di = Ti

System Model

τi -1

Ti

Ti -1

τi Period Ratio



Is it possible to have a utilization-based test for non-preemptive scheduling of periodic tasks?

Existence of a Utilization-Based Test

[Liu 1973]EDF

[Bini 2003]RM

(Hyperbolic Bound)



Utilization of a non-preemptive task set which cannot be scheduled by any clairvoyant scheduling algorithm, can be arbitrarily close to zero.

Impossibility of Finding a Utilization-Based Test

2

1/ε

τ2:(2, 1/ε )

…εmissed

1

τ1:(ε, 1)

ε ~0 ⇒ U~0



It is impossible to find any relation between utilizations such that if it holds, schedulability of any scheduling algorithm is guaranteed.

Impossibility of Finding a Utilization-Based Test

We build an infeasible task set with those utilizations

τ1 to τn-1 :T1 = T1 = … = Tn-1 = an arbitrary valuec1 = u1T1, c2 = u2T2, … , cn-1 = un-1Tn-1

…

c1 + c2 + … + cn-1

T1-cr T1-cr +ε

τn: cn = 2(T1-cr)+ εTn = cn/un cr = c1 + c2 + … + cn-1

cn = 2(T1-cr) +ε

τn

…

c1 + c2 + … + cn-1

τ1 to τn-1

At least one deadline miss



Yes! (for npEDF)

[Jeffay 1991]: Necessary and sufficient conditions for the schedulability of periodic tasks with unknown release offsets (with npEDF):

Is There Any Necessary & Sufficient Test?

Does it provide necessaryconditions for task sets with known release offset?



For task sets with no or known release offset, Jeffay’s conditions are only sufficient.

Pessimism in the existing schedulability test

This task set is feasible by npEDF, but it is rejected in Jeffay’s test



Processor Speedup Approach

ci …τiProcessor

with speed 1

Processor with speed S

ci/S …Ti

τi

[Thekkilakattil 2013]


for Non-preemptive Harmonic Tasks15 of 29

The speed S that guarantees the feasibility of a non-preemptive execution of a harmonic task set is upper bounded by

The proof can be done by finding the bound on the maximum possible execution time of a non-preemptive task!

Applying the Existing Speedup Factor (in Harmonic Task Sets)

S ≤ 8It may reduce U=1 to U’=0.125



Can we find cases where npEDF and npRM are optimal?

Can we find better speedup factor?

We Focus on Harmonic Tasks



Schedulable and Non-schedulable Tasks

What if the execution times are limited to ci ≤ T1 – c1?

Schedulable!

Non-Schedulable!



If we have U ≤ 1 and ci ≤T1 – c1 can we guarantee schedulability?

Intuition: maximum blocking will be bounded to T1 – c1

Is it enough?

Is a Task Set with Limited Execution Times, Schedulable with npEDF?



npRM and npEDF are not optimal for harmonic tasks with U ≤ 1 and ci ≤T1 – c1

Limiting the Execution Times Is Not Enough

This task set is infeasible

The relation between periods cannot be ignored easily!



We can count the vacant intervalsto make sure each task has its own place to be scheduled!

Next Step: Finding Feasible Cases for npEDF



A vacant interval is constructed by the slack of τ1

The number of vacant intervals is defined as

Definition of Vacant Intervals

τ3

c3 = T1 – c1

τ1

c1

τ2

c2 = T1 – c1 V2 = 3

V3 = 2



ci = T1 – c1

npRM and npEDF have no deadline miss if in the harmonic task set we have

U ≤ 1 ci ≤T1 – c1 Vi ≥ 1, 1 < i < n; and Vn ≥ 0

Sufficient Schedulability Conditions for npRM and npEDF

τ3

c1

τ1

τ2

c2 = T1 – c1

c3 = T1 – c1

τi……

ki > 1



The New Speedup Factor

npEDF and npRM guarantee schedulability if ci ≤T1 – c1 and ki > 1, or

ci ≤T1 – c1 and Vi ≥ 1

[Deogun 1986] ci ≤ 2(T1 – c1) and K ≥ 3

[Cai 1996] ci ≤ 2(T1 – c1) and K = 2 or ki ≥

3

S is bounded to 2



npEDF and npRM with speedup factor 2 are optimal

for task sets with enough vacant interval orinteger period ratio greater than 1

In ECRTS 2014, we have introduced a framework to construct customized harmonic periods.

It can be used to increase the applicability of our results.



Experimental Results



Non-preemptive RM (npRM) Precautious-RM (pRM) [Nasri 2014] Cai’s Algorithm (GSSP) [Cai 1996] Group-Based EDF (gEDF) [Li 2007]

npEDF + Speedup Factor of [Thekkilakattil 2013] (TSP-EDF)

npEDF + Our Speedup Factor (OSP-EDF)

Scheduling Algorithms



Task Sets with ci ≤ 2(T1 – c1)

Task setsci ≤ 2(T1 – c1)Vi ≥ 1ki {1, 2, …, 6}∊Parameter: u1 from 0.1 to 0.9

OSP-EDF, TSP-EDF, and Precautious-RM have no misses.

gEDF has the highest amount of miss ratio. In this case, it is worse than npRM.

The goal is to show the efficiency of the speedup of TSP- and OSP-EDF.



Conclusion

npEDF

npRM

Negative Results

Non-existence of any utilization-based test

Pessimism in the existing testFor tasks with known release time

Inefficiency of the recent processor speedup approachfor many cases of harmonic tasks



Conclusion

npEDF

npRM

New Results

Schedulability conditionswith limited execution time

Extending those conditions to task sets with ki > 1

Deriving more efficient speedup factor when the execution time is not limited




Questions

Thank you



The speedup factor that guarantees feasibility of npRM and npEDF for task sets with U ≤ 1 and ci ≤ 2(T1 – c1) and

Vi ≥ 1 (for 1 < i < n), and Vn ≥ 0 is bounded to

An Efficient Speedup Factor

S is bounded to 2



U ≤ 1 ci ≤ 2(T1 – c1)

Necessary Conditions of Schedulability of Non-preemptive Task Sets

We call it the Slack Rule




We can use a sort of packing in the tasks so that in the formula, each vacant interval can be occupied by different subset of tasks.

We might be able to show that the problem of finding the minimum number of Vis is NP-Complete because it can reduces to subset sum problem.

Future Works

T1- c1

τ1

c1 …

…

c2 + c5 + … + cj

τ2 , τ5 , … , τj …

T2



[Kim 1980]: Exact schedulability analysis for npEDF [Jeffay 1991]:

◦ Necessary and sufficient conditions for schedulability of npEDF for periodic tasks with unknown release phase

◦ npEDF is optimal among non-work conserving algorithms

[George 1996, Park 2007, Andersson 2009]: Sufficient conditions for RM and FP algorithms

[Marouf 2010]: Schedulability analysis for strictly periodic tasks

Schedulability Analysis for Periodic Tasks



Clairvoyant EDF [Ekelin 2006]◦ It looks ahead in the schedule and tries to …◦ Not optimal

Group-Based EDF [Li 2007]◦ Creates groups of tasks with close deadlines◦ Selects a task with the shortest execution time from a group with the

earliest deadline◦ Efficient for soft real-time tasks◦ Not optimal

Heuristic Algorithms



General task sets with limited execution time◦ ci ≤ 2(T1 – c1 )

◦ ki {1, 2, …, 6}∊

◦ Parameter: U from 0.1 to 0.9

Experiment Setup



Task Sets with Limited Execution Times

TSP-EDF is only optimal algorithm (it uses S ≤ 8).

OSP-EDF has in average only 1% miss ratio, however, it cannot guarantee schedulability because of not having Vi ≥ 1 condition.

Precautious-RM is very efficient among other algorithms, yet it is not optimal.

Note: some of those task sets are infeasible.



General task sets◦ ki {1, 2, …, 6}∊

◦ Parameter: U from 0.1 to 0.9 (with uUniFast)

Experiment Setup



General Task Sets

TSP-EDF is only optimal algorithm (it uses speedup).

OSP-EDF has in average only 0.02 miss ratio, however, it cannot guarantee schedulability

Precautious-RM is very efficient among other algorithms, yet it is not optimal.

Note: many of those task sets are infeasible.



Feasible task sets◦ ci ≤ T1 – c1

◦ Vi ≥ 1

◦ ki {1, 2, …, 6}∊

◦ Parameter: u1 from 0.1 to 0.9

Experiment Setup



Feasible Task Sets

gEDF has a lot of misses.

GSSP cannot handle ki =1

Others have no misses

Before u1=0.5, Cmax is usually from other tasks, after that c1 becomes larger than others

[Thekkilakattil 1013]



For the proof we use necessary condition ci ≤ 2(T1 – c1), thus:◦ Cmax is either 2(T1 – c1) or c1

◦ Dmin is T1

The speed S that guarantees the feasibility of a non-preemptive execution of a harmonic task set is upper bounded by

S ≤ 8It may reduce U=1 to U’=0.125



npRM and npEDF are identical if the period is the tie breaker (for npEDF)

In Harmonic Task Sets

From now on, any result for npRM is applicable on npEDF as well

RM and EDF are also identical

On the Optimality of RM and EDF for Non-Preemptive Real-Time Harmonic Tasks

Documents

Transcript of On the Optimality of RM and EDF for Non-Preemptive Real-Time Harmonic Tasks