Extended Empty-Slots Method for fixed Priority Feasibility Analysis

Li Jin The Missile Inst.

Air Force Engineering Univ. Sanyuan 713800, China

ljlxls@hotmail.com

Jian-Sheng Xing Embedded Systems Institute,

5600 MB Eindhoven, the Netherlands jiansheng03@gmail.com

Nasro Min-Allah COMSATS Institute of Information Technology,

Department of Computer Science, Islamabad 44000, Pakistan nasar.pathan@gmail.com

Xiu-Li WANG School of information ,Central University of Finance

and Economics, Beijing 100081,China;

Xiu_li_wang@hotmail.com

Abstract—For preemptive scheduling, the discrete scheduling has the same capability as the continuous scheduling. However, most work previously done has been focused on the continuous scheduling. In this paper, a discrete analysis approach Empty-Slots method is discussed and applied to fixed priority scheduling for constraint-deadline task model. Then a new schedulability test using the Empty-Slots method is presented. Compared with classical schedulability tests for continuous scheduling, this test is more convenient to be used and more efficient to tackle practical problems.

Keywordst; real-time system;discrete scheduling; fixed priority scheduling; Empty-Slots method; schedulability test

I. INTRODUCTION

A distinguishing characteristic of real-time system is the requirement that the system meets its temporal constraints. Temporal constraints are basically set be assigning deadlines to tasks. For hard real-time systems, all tasks must meet their deadlines and a missed deadline is treated as a fatal fault. Hard real-time systems are widely used in a number of application domains such as air-traffic control, process control, and numerous embedded systems. To ensure that all tasks meet their deadlines, appropriate scheduling algorithms must be employed in these hard real-time systems.

RM (rate-monotonic) algorithm[1], which assigns priorities to tasks in inverse proportion to their period parameters with ties broken arbitrarily, is an optimal scheduling algorithm for implicit-deadline synchronous periodic task sets. It was first introduced by Liu and Layland in 1973, together with a simple sufficient but not necessary schedulability test based on tasks’ processor utilization. This test is referred to as LL-Test and the least upper bound of total processor utilization LL-Bound. Burchard and Bini improved this bound respectively [3,4]. Recent research has also gained much progress by considering the relationship of task periods [5,6]. The bound-based tests have a low time complexity, but they are all based on sufficient but not necessary schedulability conditions and can reject many task sets actually schedulable. In [7], Han et al. derived two

polynomial-time schedulability tests and proved that they are better than LL-Bound and Burchard’s bound. But they are also founded upon sufficient but not necessary schedulability conditions and are pessimistic. Exact schedulability tests, which are based on sufficient and necessary schedulability conditions, were given in [2,8,9]. It is worth nothing that all these tests have a pseudo-polynomial time complexity. So they are not suited to be used for on-line admission control. Lots of work has been done to improve the performance of these tests [10,11]. However, for the RM approach, constraints imposed on task sets are severe: tasks must be periodic, independent and have deadline equal to period.

Many papers have largely weakened the constraints imposed by the RM approach, and provided associated schedulability tests. In [12-14], several methods were proposed to allow aperodic tasks to be included in the RM theory. In [9,15,16], non-independent tasks due to synchronization, mutual exclusion protection of a non-sharable resource and precedence relations were discussed.

As to the constraint that deadline equal to period, a model called constraint-deadline has been proposed, which weakens this constraint to allow deadline not larger than period. For constrained-deadline synchronous periodic task sets, DM (deadline-monotonic) algorithm, which assigns priorities to tasks in inverse proportion to their deadline parameters with ties broken arbitrarily, is an optimal scheduling algorithm [17]. DM algorithm was first introduced in 1982 [18]. Needless to say, any task set whose timing characteristics are suitable for RM scheduling would also be accepted by DM. Schedulability test for DM was first developed by Audsley in [19]. This test is founded upon the same concept “critical instant” introduced in the RM theory and only based on a sufficient but not necessary schedulability condition. Audsley et al. proposed a sufficient and necessary schedulability test in [20]. Like RM scheduling, similar work has been done to weaken the constraints as periodic, independent, etc imposed on task sets scheduled by DM [20].

However, so far most of the work has been focused on continuous scheduling. Little work has been done to investigate discrete scheduling. In [21], it has been pointed

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out that the discrete scheduling is preferred as opposed to the continuous scheduling when considering preemptive scheduling problems. For any periodic task set with integer-valued parameters, there is a valid discrete schedule if there is a valid continuous schedule for it on one processor. So it’s reasonable to traverse from continuous scheduling to discrete scheduling.

The Empty-Slots method is for the discrete scheduling. It was first introduced in [22] to deal with priority assignment in local LANs operating in hard real-time environments. In[23], Santos et al. extended it to be used in RM scheduling. In this paper, we thoroughly analyze the Empty-Slots method and apply it to fixed priority schedulability analysis for constraint-deadline task model. Based on this method, we propose a novel sufficient and necessary schedulability test and demonstrate that it is more efficient than classical ones to tackle practical problems.

II. EMPTY-SLOTS METHOD

A. Definition

Simply to say, the main idea of Empty-Slots method is as follows: Time is considered to be slotted and the duration of one slot taken as unit of time. Slots are notated t and numbered 1, 2 … .The expressions instant t and at the beginning of slot t are equivalent. All tasks are assumed to be released at the beginning of slots. Parameters such as execution time, period and deadline are assumed to be positive integers. Initially, all slots are empty. Each task acquires its slots according to the scheduling algorithm.

B. Task Model and Previous Results

Let S(n) denote the set of tasks 1 2, ,... nτ τ τ . It is

completely specified as ({ ),D,T,C)n(S 111=

2 2 2( , , ),..., ( , , )n n nC T D C T D , where , ,i i iC T D denote

the execution time, period, deadline of task i and each task

is characterized by the relationship i i iC D T≤ ≤ . Slot 1 is

assumed to be the first slot and all tasks are released at slot 1.

If , i ii D T∀ = , then the above task model is implicit-

deadline synchronous. It has been proved that for any implicit-deadline synchronous task set S(n) , it is fixed priority schedulable if and only if

1

1{2,..., }, 1

i h

hh

Ci n

T

−

=∀ ∈ <∑ and ( 1)ii C iT e

−≥ , where

( 1)iC ie−

denotes the iC th empty slot of the higher priority

1i − tasks. As RM approach is optimal for implicit-deadline

synchronous task model, the above schedulability test can also be used in RM scheduling. Because this schedulability test has several good properties [23], we have a reasonable assumption that if a similar result can also be obtained when

we weaken the timing constraint , i ii D T∀ = such

that , i ii D T∀ ≤ . In the following sections, we investigate

the schedulability of constraint-deadline task model using the empty-slots method. For later notation convenience,

N denotes natural number, and N + denotes natural number except 0.

C. General Properties of Empty-Slots Method

In this paper, we only consider the Empty-Slots method for preemptive fixed priority scheduling. We know that

/ it T⎡ ⎤⎢ ⎥ gives the maximum number of execution times of iτ

in interval [1, ]t and ⎡ ⎤ii T/tC denotes the workload of iτ

in the same interval. Let us define the function:

1( ) /

n

i iif t C t T

== ⎡ ⎤⎢ ⎥∑ N t ∈ (1)

Then ( )f t denotes the maximum workload of all tasks

in interval [1, ]t . Some conclusions can be immediately obtained from this definition:

(A) 2 1( ) ( )f t f t− is the workload of all tasks in

interval 1 2[ 1, ]t t+ .

(B) ( 1) ( )f t f t+ − is the workload generated at the

beginning of slot 1t + . If the system is schedulable and ( 1) ( )f t f t+ > , then the interval

[ 1, ( ( 1) ( ))]t t f t f t+ + + − is full. If ( 1) ( )f t f t+ = ,

then slot 1t + can be full or empty. (C) If ( )f t t≥ , slot t is full. If ( )f t t< , slot t can be

full or empty. It is obvious that ( )f t is non-decreasing monotonically.

We can prove that it has the following property: Lemma 1.

1 2 1 2( ) ( ) ( )f t t f t f t j+ = + −1

{0,1,... }n

iij C

=∈ ∑

(2)Proof.

As/ /

( ) / , , ,/ / 1

{ a c b ca b c a b c N

a c b c++⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥+ = ∀ ∈⎡ ⎤⎢ ⎥ + −⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥

( ) / / /a b c a c b c j+ = + −⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥ , {0,1}j ∈

So 1 2 1 21( ) ( ) /

n

i iif t t t t T C

=+ = +⎡ ⎤⎢ ⎥∑

1 21{ / / }

n

i i i i i iit T C t T C j C

== + −⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥∑

1 21 1 1/ /

n n n

i i i i i ii i it T C t T C j C

= = == + −⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥∑ ∑ ∑

1 2( ) ( )f t f t j= + −

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Where1

n

i iij j C

==∑ , as {0,1}ij ∈ ,

1{0,1,... }

n

iij C

=∈ ∑ .

It has been pointed out that Lemma 2. Let τ be a synchronous periodic task set with

constrained deadlines, when scheduled using a fixed-priority scheduler under fixed priority assignment, the response time

of the first job of task iτ is the largest among all the jobs of

task iτ .

As each task is periodic, task set is also periodic. Task set

( )S n repeats its behavior every 1 2( , ,... )nM LCM T T T=

slots, which is the least common multiple of all tasks’ periods. Therefore, the system is periodic, its period is M and to study its behavior it suffices to analyze it in the slot interval [1, ]M . If task set ( )S n is schedulable in the

interval [1, ]M , it is schedulable forever.

We know that if task set ( )S n is schedulable, then the

total workload generated by ( )S n in the interval [1, ]Mmust execute to completion, namely ( )f M M≤ . So

( )f M M≤ is a necessary condition for ( )S n to be

schedulable. As / /i iM T M T=⎡ ⎤⎢ ⎥ , it easily follows

that ( )f M M≤ if and only if 1

/ 1n

i iiC T

=≤∑ . If

( ) 0M f M− > , then it gives the number of empty slots in

the interval [1, ]M and the system is said to be non-

saturated. If ( ) 0M f M− = , the system has no empty slots

in the interval [1, ]M and the system is called saturated. It

can be easily proved that ( ) 0M f M− > is equivalent to

1/ 1

n

i iiC T

=<∑ , and ( ) 0M f M− = is equivalent to

1/ 1

n

i iiC T

==∑

For later notation convenience, we denote ie as the thiempty slot in a non-saturated system.

Theorem 1. For a non-saturated system under any fixed priority scheduling algorithm, the ith empty slot is the minimum value such that ( )t i f t= + , [1, ]t ∈ +∞

i N +∀ ∈

Proof. As ie is the thi empty slot in the system, there

are 1i − empty slots in the interval [1, 1]ie − . Since ie is

empty, all workload generated by all tasks in the interval

[1, 1]ie − can execute to completion. Therefore,

( 1) ( 1) ( 1)i i if e e i e i− = − − − = −

Since no workload is generated at the instant ie :

( ) ( 1)i i if e f e e i= − = − , then ( )i ie i f e= + .

It remains to prove that ie is the minimum

| ( )t t i f t= + .

Since it e∀ < , there are at most 1i − empty slots in the

interval[1, ]t . Thus we have

( ) ( 1)f t t i t i≥ − − > − ⇒ ( )t i f t< + ,

so ie is the minimum | ( )t t i f t= + .

Theorem 2. For a non-saturated system under any fixed priority scheduling, there is

i a i ae e e+

− ≤ ,i N a N∀ ∈ ∈ (3)

Proof. To prove theorem 2 it suffices to prove

i∀ , [ 1, ]i a i i ae e e e+

∈ + + . There are ae slots in the

interval [ 1, ]i i ae e e+ + . Since ie is empty, all workload

generated by all tasks in the interval [1, 1]ie − has been

dealt with. Therefore, according to lemma 1 and conclusion A the workload generated in the interval is:

( ) ( )i a iW f e e f e= + − ( ) ( ) ( )i a if e f e j f e= + − −

( )af e j= −

Form theorem 1, there is ( )a af e e a= − , so

a aW e a j e= − − < . We can see that the difference between

the workload and the length of the interval is larger than a , so there are at least a empty slots in the interval. Therefore

[ 1, ]i a i i ae e e e+

∈ + +

III.A NEW FIXED PRIORITY FEASIBILITY ANALYSIS

In the above section we have presented several important properties of Empty-Slots method for fixed priority scheduling. Now we investigate the schedulability of fixed priority scheduling using this method. For later convenience,

( )i ne denotes the ith empty slot of task set ( )S n .

Theorem 3. If task set ( 1)S n − is fixed priority

schedulable, when added a new low priority task nτ , then

( )S n is fixed priority schedulable if and only if ( 1)S n − is

non-saturated and ( 1)nn c nD e−

≥ .

Proof. First we prove the sufficient condition: if ( 1)S n − is fixed priority schedulable, non-saturated and

suffices condition ( 1)nn c nD e−

≥ , then ( )S n is schedulable.

As task n ’s period is nT , deadline is nD , it generates

workload at instant 1, 1, 2 1,...,n nT T+ + . To check the

schedulability of the task set it suffices to check if there are enough empty slots left to execute task n in the interval

[ 1, ]n n nkT kT D+ + . The length of the interval is nD , from

Theorem 2 we know that ( 1) ( 1) ( 1)n ni c n i n c ne e e+ − − −

− ≤ . So

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there are at least nC empty slots in the interval whose length

is ( 1)nc ne−

. As ( 1)nn c nD e−

≥ , therefore there are enough

empty slots in the interval [1, ]nD to execute task nτ to

completion, so ( )S n is schedulable.

Next we prove the necessary condition. If ( )S n is fixed

priority schedulable, then obviously ( 1)S n − is non-

saturated and schedulable. As task nτ ’s priority is the

lowest, there must be ( 1)nn c nD e−

≥ .

From the theorem above, we can easily get the following theorem.

Theorem 4. A task set ( )S n is fixed priority schedulable

if and only if for 2,..., ,i n= 1

1/ 1

i

i ijC T

−

=<∑ and ( 1)ii c iD e

−≥ .

This schedulability test includes two conditions. The first one guarantees that ( 1)S i − is non-saturated and

consequently it can incorporate another task iτ . The second

one guarantees that there are sufficient slots to execute task

iτ to completion before its deadline. We refer this test to

ESMFP (Empty –Slots Method fixed Priority) schedulability test. For each task, we should calculate and check if it’s the

iC th empty slot of the prior 1i − tasks. The maximum

number of times to calculate is nD . So the time complexity

of this test is ( * )nO n D .

As DM is an optimal scheduling algorithm for constrained-deadline task model, we can also get the following theorem.

Theorem 5. A task set ( )S n is DM schedulable if and

only if for 2,..., ,i n= 1

1/ 1

i

i ijC T

−

=<∑ and ( 1)ii c iD e

−≥ .

We must keep in mind that in the above theorem, the task set must be sorted in DM ordering.

IV.COMPARISON WITH CLASSICAL FEASIBILITY ANALYSIS

One classical schedulability test is [20]:

:1 : i ii i n WR D∀ ≤ ≤ ≤ (4)

Where iWR is the worst-case response time of task iτ .

iWR is given by the smallest x N +∈ that satisfies the

follow recursive equation. 0

1

i

i jjWR C

==∑ (5)

11

1/

il li i i j jj

WR C WR T C−+

=⎡ ⎤= + ⎢ ⎥∑ (6)

The above procedure stops when the same value is found for two successive iterations of l , or when the deadline

iD is exceeded. In the former case, the smallest solution of

the recursive equation is iWR . In the latter case, the task set

is not schedulable. In [20], it has been pointed that this test’s time complexity is pseudo-polynomial.

It can be seen that the new ESMFP schedulability test takes a similar form to the classical schedulability test, though they are obtained from different methods. The most important distinction is that the classical schedulability test is for continuous scheduling and the new ESMFP schedulability test is for discrete scheduling.

Compared with the classical schedulability test, the new ESMFP schedulability test has many advantages. First, as natural number is easier to operate in real-time computer systems than real number, ESMFP is more convenient to be implemented on real-time computer systems. Second, ESMFP can be used to easily solve many practical problems:given ( )S n , non-saturated and DM (or fixed priority) schedulable.

(1) If ( 1)S n + , obtained by added a new low priority task, is also schedulable?

(2) How much the execution time of a task in ( )S n can be expanded while keeping the system

schedulable? (3) How long is the execution time of the task that

saturates the system? Third, although ESMFP also has a pseudo-polynomial

time complexity, we can largely reduce it to be capable for on-line admission control. We can do this by the following methods:

(1) If ( 1)S n + is non-saturated, then

( ), 1,..., 2S i i n= − is also non-saturated. So we only need

to check ( 1)S n − .

(2) For ( 1)ii c iD e−

≥ to hold, since

1

( 1) 1 1min | /

i

i i

c i i j j jj je t t C C t T C

−

− = =⎡ ⎤= = + ≥⎢ ⎥∑ ∑ ,

we can check this inequality from the slot 1

i

jjC

=∑ instead

of 1.

(3) If we have 1

1/ '

i

i j jjC C t T t t

−

=⎡ ⎤+ = >⎢ ⎥∑ , then

the next slot to be checked is not t but 't . (4) For each task, the number of ceiling calculations is

not always nD , but increases from 1

i

jjC

=∑ up to iD .

So the number of effective operations needed for

schedulability test is much less than * nn D .

From above, we can see that the new schedulability test is more convenient to be used and more efficient to tackle practical issues.

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V. CONCLUSIONS

This paper thoroughly investigates the Empty-Slots method with respect to a different model, constraint-deadline synchronous task model, and presents several important properties of Empty-Slots method. Based on the results, this paper proposes a new schedulability test ESMFP. Compared with classical schedulability test, ESMFP is more convenient to be used and it can easily solve many practical problems.

However, there still exists a constraint needs to be weakened for the task model studied above, i.e. the constraint that tasks have deadlines not larger than their periods. Many applications have the property that they have tasks whose deadlines are larger than periods. A new model called arbitrary-deadline is proposed to deal with these systems. MRM (Modified Rate-Monotonic) scheduling can partially solve the scheduling problem of these systems. As the convenience and good properties of ESMFP, it will be beneficial to extend this approach to be used in these systems.

REFERENCES

[1] C. L. Liu and J. W. Layland, Scheduling Algorithms for Multiprogramming in a Hard Real-Time Environment, Journal of the ACM, 20(1): 46-61,1973.

[2] J. P. Lehoczky, L. Sha, and Y. Ding. The Rate Monotonic Scheduling Algorithm: Exact Characterization and Average Case Behavior. In Proceedings of 10th IEEE Real-Time Systems Symposium, pages 166-171, 1989.

[3] A. Burchard, J. Liebeherr, Y. Oh, and S. H. Son. New Strategies for Assigning Real-Time Tasks to Multiprocessor Systems. IEEE Transactions on Computers, 44(12): 1429-1442, 1995.

[4] E. Bini, G. C. Buttazzo and G. Buttazzo. A Hyperbolic Bound for the Rate Monotonic Algorithm. In IEEE Proceedings of the 13th Euromicro Conference on Real-Time Systems, pages 59-68, 2001.

[5] W. Lu, H.W. Wei, K.J. Lin, Rate Monotonic Schedulability Conditions Using Relative Period Ratios, pp.3-9, 12th IEEE International Conference on Embedded and Real-Time Computing Systems and Applications (RTCSA'06), pages 3-9, 2006

[6] H.-W. Wei, K.-J. Lin, W.-C. Lu, and W.-K. Shih, Generalized rate monotonic schedulability bounds using relative period ratios, Information Processing Letters, vol. 107, no. 5, pp. 142 – 148, 2008.

[7] C. C. Han and H. Y. Tyan. A Better Polynomial-Time Schedulability Test for Real-Time Static-Priority Scheduling Algorithm. In Proceedings of the 18th IEEE Real-Time Systems Symposium, pages 36-45, 1997.

[8] M. Joseph and P. Pandya. Finding Response Times in a Real-Time System. The computer Journal, 29(5): 309-395, 1986.

[9] N.C. Audsley, A. Burns, M. Richardson, K.Tindell and A. Wellings. Applying New Scheduling Theory to Static Priority Preemptive Scheduling. Software Engineering Journal 8(5): 284-292, September 1993.

[10] E. Bini, G. C. Buttazzo. The Space of Rate Monotonic Schedulability. In Proceedings of 23rd IEEE Real-Time Systems Symposium, pages 169-178, 2002.

[11] J. Liu, Y. Wang, M. Cartmell. An Improved Rate Monotonic Schedulability Test Algorithm. Journal of Software 16(1): 89-100, 2005.

[12] J.P. Lehoczky, L. Sha and J.K. Stronsnider. Enhancing Aperiodic Responsiveness in Hard Real-Time Enviroment. In Proceedings of 8th IEEE Real-Time Systems Symposium, 1987.

[13] L. Sha, J.B. Goodenough, and T. Ralya. An Analytical Approach to Real-Time Software Engineering. Software Engineering Institute Draft Report, 1988.

[14] B. Sprunt, L. Sha and J.P.Lehoczky. Aperiodic Task Scheduling for Hard Real-Time Systems. The journal of Real-Time Systems 1: 27-69, 1989.

[15] A. Burns, R. Davis, and S. Punnekkat. Feasibility Analysis of Fault-Tolerant Real-Time Task Sets. In IEEE Proceedings of the Euromicro Workshop on Real-Time Systems, pages 29-33, 1996.

[16] L. Sha, R. Rajkumar, and J.P. Lehoczky. Priority Inheritance Protocols: An Approach to Real-Time Synchronization. IEEE Transactions on Computers 39(9): 1175-1185, 1990.

[17] J. Goossens and P. Richard. Overview of Real-Time Scheduling Problem. In Proceedings of the ninth international conference on project management and scheduling, 2004.

[18] J.Y.T. Leung and J. Whitehead. On the Complexity of Static-Priority Scheduling of Periodic, Real-Time Tasks. Performance Evaluation 2: 237-250, 1982.

[19] N.C. Audsley. Deadline Monotonic Scheduling. YCS 146. Department of Computer Sciences, University of York, 1990.

[20] N.C. Audsley, A.Burns, M.F. Richardson, A.J. Wellings. Hard real-time scheduling: the deadline monotonic approach. In Proceedings of 8th IEEE Workshop on Real-Time Operating Systems and Software, pages 133-137, 1991.

[21] K. B. Sanjoy, R. H. Rodney and E. R. Louis. Algorithms and Complexity Concerning the Preemptive Scheduling of Periodic, Real-Time Tasks on One Processor. Real-Time Systems 2:301-324, 1990.

[22] J. Santos, M.L. Gastaminza, J. Orozco, D. Picardi and O.Alimenti. Priorities and Protocols in Hard Real-Time LANs. Computer and Communications 14(9): 507-514, 1991.

[23] J. Santos and J. Orozco. Rate monotonic scheduling in hard real-time systems. Information Processing Letters, 48: 39-45, 1993.

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