A Schedulability Analysis of Deferrable Scheduling Using Patterns

Transcription

1 Euromicro Conference on Rea- Systems A Scheduabiity Anaysis of Deferrabe Scheduing Using Patterns Song Han The University of Texas at Austin shan@cs.utexas.edu Deji Chen Emerson Process Management Deji.Chen@Emerson.com Aoysius K. Mok The University of Texas at Austin mok@cs.utexas.edu Ming Xiong Be Labs, Acate-Lucent xiong@research.be-abs.com Abstract The scheduabiity testing for the deferrabe scheduing agorithm for fixed priority transactions (DS-FP) remains an open probem since its introduction. In this paper, we take the first step towards investigating necessary and sufficient conditions for the DS-FP scheduabiity. We propose a necessary and sufficient scheduabiity condition for the agorithm in discrete time systems, and prove its correctness. Based on this condition, we propose a scheduabiity test agorithm that is more accurate than the existing test that is ony based on a sufficient condition. Our agorithm expoits the fact that there is aways a repeating pattern in a DS-FP schedue in discrete time systems. We demonstrate through exampes that our scheduabiity test agorithm outperforms the existing agorithm in terms of accuracy. 1 Introduction The quaity and timeiness of data vaues samped from rea-word entities have been critica for rea-time data services. These samped data, usuay caed rea-time data, coud become invaid with the passage of time since the status of the corresponding entity in the rea-word may change continuousy. Sensor update transactions (or update transactions for short) need to constanty sampe rea-word data vaues and insta them into Rea- Databases (RTDBS). Otherwise, the system cannot detect and respond to environmenta changes in a timey manner. One efficient way to determine the correctness of reatimedataistodefineavaidity constraint [, 18, 11, 1, 13, 5,, 10, 14, 7, 8, 3, 3, 4] or age constraint [], which determines a vaidity interva ength for each reatime data object. A rea-time data vaue is ony vaid within the vaidity interva. In practice, for safety and reiabiity reasons, RTDBSs require continuous generation of update transactions to refresh rea-time data objects regardess of how much the status of the corresponding rea-word entities have been changed. To meet this constraint, it is important to produce a schedue for a update transactions such that for any consecutive updates of a rea-time data object, the next update is competed before the previous data version expires. It is crucia to schedue update transactions efficienty in the design of rea-time appications. Most of the work in update transaction scheduing assumes a periodic transaction mode. For exampe, a periodic update transaction mode is adopted in the more-ess (ML) scheme [3, 14, 3]. ML can guarantee the vaidity of rea-time data objects. Recenty, the deferrabe scheduing agorithm for fixed priority transactions (DS-FP) was proposed [4] to reduce the tota update transaction workoad. The intuition of DS-FP is to schedue an update transaction job as ate as possibe based on the samping time of its previous job. DS-FP reduces update workoad by sacrificing the periodicity of the update transactions. This poses great chaenges for its scheduabiity anaysis. One prominent method in cassica scheduabiity anaysis is based on the critica instant test. A critica instant makes sense for periodic tasks. Aso sporadic task sets are usuay reduced to periodic ones for their scheduabiity anaysis. For exampe, the minimum separation times are treated as periods in the scheduabiity anaysis of sporadic tasks. In a DS- FP schedue, however, the distance of the reease times of two consecutive jobs is not a constant. This distance may vary from a transaction s execution time to its vaidity interva minus the execution time. It is ony proved so far that DS-FP coud schedue any transaction set that is scheduabe by ML [5]. This sufficient condition has seriousy restricted the scheduabiity of a transaction set under DS- FP. Whie experimenta resuts have demonstrated that DS- FP outperforms ML significanty [4], the transaction sets used in the experiments are a scheduabe by ML. DS-FP is demonstrated with its ower processor untiization whie there exists no method to determine if a transaction set is scheduabe by DS-FP even if it is not scheduabe by ML. The open theoretic question is whether there is any necessary and sufficient condition to determine if a transaction /08 $ IEEE DOI /ECRTS Authorized icensed use imited to: University of Texas at Austin. Downoaded on Apri 7, 09 at :50 from IEEE Xpore. Restrictions appy.

2 Symbo Definition X i Rea-time data object i τ i Update transaction updating X i J i,j The jth job of τ i (j =0, 1,,..) R i,j Response time of J i,j C i Execution time of transaction τ i V i Vaidity (interva) ength of X i f i,j Finishing time of J i,j r i,j Reease (Samping) time of J i,j d i,j Absoute deadine of J i,j P i Period of transaction τ i in ML D i Reative deadine of transaction τ i in ML P A fixed pattern repeating in a DS-FP schedue P s Pattern P starting time P Length of pattern P S τ (t) State of transaction τ at time t S T (t) State of transaction set T at time t Θ i(a, b) Tota cumuative processor demands from higherpriority transactions received by τ i in interva [a, b) Tabe 1. Symbos and definitions. set is scheduabe by DS-FP (even if it is not scheduabe by ML). This paper takes the first step towards addressing this important probem. We prove that there aways exists a repeating pattern for a given DS-FP schedue in discrete time systems. This provides an important foundation to derive a necessary and sufficient condition for scheduabiity test of DS-FP. Based on this condition, a scheduabiity test agorithm is derived. This paper is organized as foows: Section briefy reviews the background and reated work in the area. Section 3 presents the major contributions of this paper. In particuar, we present the proof of our necessary and sufficient scheduabiity condition in discrete time systems, and our scheduabiity test agorithm based on this condition. We concude the paper in Section 4. Background and Reated Work This section briefy reviews the tempora vaidity of reatime data, and the More-Less (ML) [, 3] and deferrabe scheduing (DS-FP) [4] agorithms. Forma definitions of the frequenty used symbos are given in Tabe 1. Definition.1: A rea-time data object (X i ) at time t is temporay vaid (or absoutey consistent) if, for its j th update finished atest before t, the samping time (r i,j )pusthevaidity interva (V i ) of the data object is not ess than t, i.e., r i,j + V i t [18]. A data vaue for rea-time data object X i samped at any time t wi be vaid up to (t + V i ). The actua ength of the tempora vaidity interva of a rea-time data object is appication dependent [18, 19, 17]. One of the important design goas of RTDBS is to guarantee that rea-time data remain fresh, i.e., they are aways vaid. Given a set of transactions T = {τ i } m i=1,weassume without oss of generaity that τ k has higher priority than τ j for k < j. Pease note that both ML and DS-FP are preemptive scheduing agorithms. More-Less: ML adopts the periodic task mode [15] for sensor update transactions whose derived deadines are not arger than their periods. Consider synchronous transactions whose first jobs a start at time 0. A time instant after which a transaction job has the worst-case response time is caed a critica instant, e.g., time 0 is a critica instant for a the transactions with deadines no arger than their periods if those transactions are synchronous [15]. Note that we ony consider synchronous transactions. In ML, there are three constraints to foow for transactions τ i ( i, 1 i m) [3]: Vaidity constraint: the sum of the period and reative deadine of transaction τ i is aways ess than or equa to V i,i.e., P i + D i V i (1) Deadine constraint: the period of an update transaction is assigned to be more than or equa to haf of the vaidity ength of its updated object, whie its corresponding reative deadine is ess than or equa to haf of the vaidity ength of the same object. For τ i to be scheduabe, D i must be greater than or equa to C i, the worst-case execution time of τ i, i.e., C i D i P i. Scheduabiity constraint: for a given set of update transactions, the Deadine Monotonic scheduing agorithm is used to schedue the transactions. Consequenty, i Di j=1 ( P j C j ) D i (1 i m). ML assigns priorities to transactions based on Shortest Vaidity First (SVF), i.e., in the inverse order of vaidity ength, and ties are resoved in favor of transactions with arger execution time (C i ). It assigns deadines and periods to τ i as foows: D i = f m i,0 rm i,0, () P i = V i D i, (3) where f m i,0 and rm i,0 are finishing and samping times of the first job of τ i, respectivey. Note that in a synchronous system, r m i,0 =0and the first job s response time is the worstcase response time in ML. We use superscript m to distinguish the finishing and samping times in ML from those in DS-FP. DS-FP: ML is pessimistic on the deadine and period assignment. This is because it uses a periodic task mode that has a fixed period and reative deadine for each transaction, and the reative deadine D i is equa to the worst-case response time of the transaction. According to the vaidity constraint in ML, the arger the deadine D i, the smaer the period P i. In order to increase the separation of two consecutive jobs (and thus reduce the sensor update workoad), 48 Authorized icensed use imited to: University of Texas at Austin. Downoaded on Apri 7, 09 at :50 from IEEE Xpore. Restrictions appy.

3 r i, j J i, j d i, j V i r i, j+1 J i, j+1 r' i, j+1 J i, j+1 d i, j+1 V i V i J i, j J i, j+1 Higher-priority preemption to J i, j+1 d i, j+ d' i, j+ Figure 1. Iustration of DS-FP scheduing: r i,j+1 canbeshiftedtor i,j+1 without vioating the vaidity constraint. DS-FP adaptivey derives the reative deadine and separation of one job from its previous job and preemptions from higher-priority transactions. Given reease time r i,j of job J i,j and deadine d i,j+1 of job J i,j+1 (j 0), d i,j+1 = r i,j + V i (4) guarantees that Eq. 1 can be satisfied, as depicted in Figure 1. Correspondingy, the foowing equation foows directy from Eq. 4: (r i,j+1 r i,j )+(d i,j+1 r i,j+1 )=V i. (5) If r i,j+1 can be shifted onward to r i,j+1 aong the time ine in Figure 1, it does not vioate Eq. 5. After the shift, tempora vaidity can sti be guaranteed as ong as J i,j+1 is competed by its deadine d i,j+1. The idea of DS-FP is to defer the samping time (i.e., reease time), r i,j+1,ofj i,j s next job as ate as possibe whie sti guarantee Eq. 1. We now introduce a definition that is used in the description of DS-FP. Definition.: Let Θ i (a, b) denote the tota cumuative processor demands made by a jobs of higher-priority transaction τ j for j (1 j i 1) during the time interva [a, b) from a schedue S produced by a fixed priority scheduing agorithm. Then, i 1 Θ i (a, b) = θ j (a, b), j=1 where θ j (a, b) is the tota processor demands made by a jobs of singe transaction τ j during [a, b). According to the fixed priority scheduing theory, r i,j+1 in DS-FP can be derived backwards from its deadine d i,j+1 as foows: r i,j+1 = d i,j+1 R i,j+1 (r i,j+1, d i,j+1 ); (6) R i,j+1 (r i,j+1, d i,j+1 )=Θ i (r i,j+1, d i,j+1 )+C i. (7) where R i,j+1 (r i,j+1, d i,j+1 ) (or R i,j+1 for simpicity in DS-FP) denotes the response time of J i,j+1 deriving backwards from its deadine d i,j+1. Note that the schedue of a higher-priority jobs that are reeased prior to d i,j+1 needs to be computed before computing Θ i (r i,j+1, d i,j+1 ). Simiar to ML, DS-FP aso assigns priorities to transactions according to SVF. Readers are referred to [5] for the detais of the DS-FP agorithm. Now we summarize the agorithm as foows. First we set r i,0 =0, i, 1 i m. The highest priority job among the outstanding jobs is aways schedued first. It is ony preempted when a new job with higher priority is ready. As soon as a job J i,j is competed, we derive the r i,j+1 of its next job according to above cacuations. The agorithm fais when a job misses its deadine. Otherwise it keeps running. It is proved that any task set that is schedued by ML is aso schedued by DS-FP [5]. Theorem.1: (Coroary 3. in [5]) Given a synchronous update transaction set T with known C i and V i (1 i m),ift can be schedued by ML, then it can aso be schedued by DS-FP. 3 DS-FP Scheduabiity Anaysis It is uncear how to perform a scheduabiity test for DS- FP if a set of transactions cannot be schedued by ML. This section presents a method based on DS-FP pattern anaysis that can be used for DS-FP scheduabiity testing. In Section 3.1, our exampes demonstrate that transaction sets scheduabe by DS-FP are not necessariy scheduabe by ML. Section 3. proves the existence of repeating patterns in a vaid DS-FP schedue. Section 3.3 presents an agorithm that searches for the repeating pattern in a DS-FP schedue. The search agorithm eads to a scheduabiity test agorithm in Section 3.4. Finay Section 3.5 discusses DS-FP in Continuous Systems. From here on, it is assumed that transactions are studied in a discrete time system uness it is specified otherwise. It is aso assumed that the DS-FP scheduer ony considers the worst-case execution time of a transaction in its schedue. In other words, if the execution time of a transaction job is ess than the worst-case execution time of the transaction, the job can be ided in its assigned time sots after its execution is finished. This assumption simpifies our scheduabiity anaysis. 3.1 ML vs. DS-FP Theorem.1 states that DS-FP dominates ML in terms of scheduabiity. That is, if T can be schedued by ML, then it can aso be schedued by DS-FP. However, the converse statement is not true. This can be demonstrated in the foowing exampes. Exampe 3.1: Consider a set of two transactions {τ 1, τ } with execution times and 3, and vaidity intervas 6 and 49 Authorized icensed use imited to: University of Texas at Austin. Downoaded on Apri 7, 09 at :50 from IEEE Xpore. Restrictions appy.

4 T 1 : {C 1 =, V 1 =6} T : {C =3, V =1} T 1 : {C 1 =, V 1 =6} T : {C =3, V =15} T 3 : {C 3 =3, V 3 =47} J, 0 competes after V / (a) ML is unscheduabe J 3, 0 competes after V 3 / (a) ML is unscheduabe J 3, 0 competes before V 3 / J, 0 competes after V / (b) DS-FP is scheduabe Figure. Two transactions that can be schedued by DS-FP but not by ML (b) DS-FP is scheduabe Figure 3. Three transactions that can be schedued by DS-FP but not by ML 1 respectivey. Figure (a) depicts a schedue of the transactions under ML.Thefirstjobofτ, J,0, competes at time 7, which is greater than V =6. Thus the set of transactions is not scheduabe by ML. Figure (b) depicts a schedue of the transactions under DS-FP. The same transaction set is scheduabe by DS-FP because the schedue pattern between time 7 and 19 repeats itsef forever. DS-FP is better in Exampe 3.1 because DS-FP aows J,0 to be competed ater than V. There are aso transaction sets in which for every transaction τ i, J i,0 is competed no ater than Vi in DS-FP, and further more these transaction sets can be schedued by DS-FP but not by ML. This is because DS-FP eaves extra processor time eary on to be used by J i,0 s of ower priority transactions. The next exampe iustrates this point. Exampe 3.: Consider a set of three transactions {τ 1, τ, τ 3 } with execution times, 3, 3, and vaidity intervas 6, 15, 47, respectivey. Figure 3 (a) depicts a schedue of the transactions under ML. TheML period and deadine for τ are 8 and 7, respectivey. The first job of τ 3, J 3,0, competes at time 4, which is greater than V3 =3.5. Thus the set of transactions is not scheduabe by ML. Figure 3 (b) depicts a schedue of the transactions under DS-FP. The same transaction set is scheduabe by DS-FP because the schedue pattern between time 6 and 50 repeats itsef forever. In this schedue J 3,0 competes at time 19, which is smaer than V3 (i.e., 3.5). This is beacuse J, is schedued ater than that of ML. Note that DS-FP fuy utiizes the processor in both exampes. We coud easiy derive exampes in which the pro- cessor ides once in a whie. For exampe, in Figure we coud change C to.5 and in Figure 3 we coud change C 3 to.75. After both changes the transaction sets sti cannot be schedued by ML but can by DS-FP. Furthermore,we coud then scae up the numbers to make them a integers again. Note that we prove that a transaction set is scheduabe by DS-FP by demonstrating that a repeating pattern occurs for the transaction set. The remaining questions are whether a repeating pattern aways exists in a DS-FP schedueif the transaction set is scheduabe, and if so, how to find it. We answer those questions next. 3. DS-FP Pattern Anaysis As mentioned before, we study the DS-FP scheduabiity probem in a discrete time system. In order to prove that there aways exists a repeating pattern if a transaction set is scheduabe by DS-FP, we sha first review the Pigeonhoe Principe. The Pigeonhoe Principe [6]: If m pigeons occupy n pigeonhoes and m>n, then at east one pigeonhoe has two or more pigeons roosting in it. In a discrete time system, a repeating pattern aways exists for any successfu DS-FP schedue because we know the fact that the execution times, vaidity intervas, and the number of transactions are a finite integers, and so an execution state can be defined that characterizes the progress of an execution in meeting the timing constraints for any particuar time. Given infinite time, there must be a pattern repeating itsef in the DS-FP schedue inasmuch as the number of distinct execution states is finite. The foowing theorem states that there must exists a fixed pattern repeat- 50 Authorized icensed use imited to: University of Texas at Austin. Downoaded on Apri 7, 09 at :50 from IEEE Xpore. Restrictions appy.

5 ing itsef in a bounded time interva in the DS-FP schedue. Theorem 3.1: Given an update transaction set T with known C i and V i (1 i m), if it can be schedued by DS-FP, then the DS-FP schedue has a fixed repeating pattern that occurs at east once in the interva [V m C m, V m C m +Π m i=1 (V i C i +1) 1]. Proof. The theorem can be proved by the foowing two caims using induction: 1. There is a fixed pattern repeating itsef for τ 1 in the interva [V m C m, V m C m + V 1 C 1 ].. For any k, 1 k<m,ifthereisafixed pattern repeating itsef for the schedue of τ 1,..,τ k in the interva [V m C m, V m C m +Π k i=1 (V i C i +1) 1], then there is a fixed pattern repeating itsef for τ 1,.., τ k, τ k+1 in the interva [V m C m, V m C m +Π k+1 i=1 (V i C i + 1) 1]. The first caim is obvious because there is a fixed pattern of ength (V 1 C 1 ) repeating itsef from time 0. Asa matter of fact, any schedue of ength (V 1 C 1 ) is a fixed pattern, thus the schedue of ength (V 1 C 1 ) from time V m C m must repeat itsef. Note that the theorem does not require that the first instance of the fixed pattern in the interva [V m C m, V m C m +Π m i=1 (V i C i +1) 1] starts exacty on time V m C m. Now et us prove the second caim. We sha rey on the Pigeonhoe Principe to identify two time points in two recurring patterns for transactions τ 1,.., τ k such that the foowing two conditions are satisfied: 1) the two time points are τ k+1 s reease times; ) the two time points have the same offsets within their patterns. If such two time points are identified, then the schedue of τ 1,.., τ k, τ k+1 between those two time points is a fixed pattern repeating itsef thereafter. This is because the schedues after those two time points are identica for transactions τ 1,.., τ k, thus it is aso identica for τ k+1. Suppose that the fixed pattern for τ 1,.., τ k starting from time t has ength L. We have t (V m C m ) and L Π k i=1 (V i C i +1) 1. There are two cases, L (V k+1 C k+1 ) and L<(V k+1 C k+1 ). Case I: Suppose L (V k+1 C k+1 ). In every recurring instance of the fixed pattern of ength L starting from time t, there is at east one job of τ k+1 since L (V k+1 C k+1 ). Let us examine the ast τ k+1 job in each recurring instance of the pattern. Denote d to be the distance from its reease time to the end of the pattern. The ength of d cannot exceed (V k+1 C k+1 ), otherwise there must be another job afterwards in the pattern in order to satisfy τ k+1 s vaidity constraint. Since d>0, it can be one of (V k+1 C k+1 ) possibe vaues (pigeonhoes). Let us ook at τ k+1 s ast job (pigeon) in each of the (V k+1 C k+1 +1)recurring patterns since time t. It foows from the Pigeonhoe Principe that there must be two jobs reease time at the same offset within their corresponding pattern instances. Denote t 1 and t to be the two job reease times, and t 1 <t. We then have a fixed pattern in the interva [t 1,t ] repeating itsef thereafter for transactions τ 1,.., τ k, τ k+1. Since t 1 t (V m C m ) and t < (V m C m )+L(V k+1 C k+1 +1) < (V m C m )+Π k+1 i=1 (V i C i +1) 1, we have proved the first case. Case II: Suppose L < (V k+1 C k+1 ). Denote J k+1,w to be the first τ k+1 s job that executes after time t. J k+1,w and a its subsequent jobs appear in different instances of the pattern. There are ony L possibe offsets (pigeonhoes) within a pattern for τ k+1 s jobs to start. Let us ook at the first (L +1)jobs (pigeon) of τ k+1, i.e., J k+1,w through J k+1,w+l. It foows from the Pigeonhoe Principe that there must be two jobs starting at the same offset within their corresponding pattern instances. Denote t 1 and t to be the two job reease times in DS-FP,andt 1 <t.wethen have a fixed pattern in the interva [t 1,t ] repeating itsef for transactions τ 1,.., τ k, τ k+1.sincet 1 t (V m C m ) and t < (V m C m )+(L+1)(V k+1 C k+1 ) < (V m C m )+ Π k+1 i=1 (V i C i +1) 1, we have proved the second case. Based on the above two caims, the theorem is proved. According to the theorem, if a transaction set can be schedued by DS-FP in the interva [0, (V m C m )+ Π m i=1 (V i C i +1) 1], then it is scheduabe by DS-FP because a fixed pattern appearing in the interva repeats itsef forever. Thus we have the foowing coroary. Coroary 3.1: An update transaction set T can be schedued by DS-FP if and ony if it can be schedued by DS-FP in the interva [0, (V m C m )+Π m i=1 (V i C i +1) 1]. Athough the ength of the interva in Coroary 3.1 is O(Π m i=1 V i), Coroary 3.1 provides a foundation for the scheduabiity test of DS-FP. 3.3 DS-FP Pattern Search Agorithm Theorem 3.1 proves the existence of a repeating pattern for a given DS-FP schedue. This subsection studies the properties of the pattern and provides a search agorithm that finds the shortest and eariest pattern. Denote tupe P =(P s, P ) to be the DS-FP schedue of ength P starting from time P s. If a DS-FP schedue repeats P forever after time P s +P,thenP is a fixed pattern of the DS-FP schedue. Given a pattern P, DS-FP may not necessariy be ide immediatey before P s. Denote tupe S τ (t) =(d, e) to be the state of transaction τ at time t, whered is the distance to τ s ast job reease time before time t, ande is the remaining outstanding execution time of τ at time t. e =0if τ s ast job before t is aready finished. Let S T (t) be the combined S τ (t) states of a the transactions in the set T at time t. Note once S T (t) is known, the DS-FP schedue from t onward is determined. Given a repeating pattern P =(P s, P ),thefo- 51 Authorized icensed use imited to: University of Texas at Austin. Downoaded on Apri 7, 09 at :50 from IEEE Xpore. Restrictions appy.

6 owing coroary foows directy from the fact that a transactions have the same states at times P s + t and P s + t + P for t>0,i.e.,s T (P s + t) =S T (P s + t + P ). Coroary 3.: If P =(P s, P ) is a pattern repeating itsef from time P s,then(p s + t, P ) (t >0) is aso a pattern repeating itsef from time P s + t. We now prove the next emma. Lemma 3.1: Given a the patterns P, P,... of a DS-FP schedue for transaction set T,etP be a pattern with the minimum ength among a patterns, i.e., P P for any other pattern P. Then P is a mutipe of P, i.e., P = NP where N is a positive integer. Proof. Let t 1 = P s + P n 1 (n 1 > 0 is an integer) such that t 1 > P s.both(t 1, P ) and (t 1, P ) are patterns. We prove the emma by contradiction. Suppose P is not a mutipe of P and P = P r s, r, and 0 < s < P. We have the state S T ((t 1 + P )+s) = S T (t 1 + P r) =S T (t 1 )=S T (t 1 + P ). It foows that the pattern (t 1 + P,s) repeats itsef from time (t 1 + P ) with ength s. As0 <s<p, this contradicts the fact that P is the minimum ength among a repeating patterns. So P must be a mutipe of P. In the proof above, since P is a mutipe of P, (P s, P ) must aso be a pattern. Coroary 3.3: Given a the patterns of a DS-FP schedue, et P be a pattern with the minimum P. For any other pattern P, (P s, P ) is aso a repeating pattern. Lemma 3.1 and Coroary 3.3 impy that there exists a shortest pattern P that is aso the eariest. Any other pattern P coud be derived from P. P coud be of the same ength but with some offset from a P s repeat; P coud be a mutipe of P s repeats; or P coud be a mutipe of P s repeats with some offset. Lemma 3.: If P and P are two different repeating patterns of a DS-FP schedue, then (P s, P ) and (P s, P ) are aso repeating patterns. Proof. Let P be the shortest and eariest pattern. AccordingtoLemma3.1,P is a mutipe of P. According to Coroary 3.3, (P s, P ) is aso a repeating pattern. Because (P s, P ) is a pattern and P is a mutipe of P, (P s, P ) is aso a repeating pattern. By the same argument, (P s, P ) is a repeating pattern. GiventransactionsetT of size m, we ca P i a pattern of the first i (1 i m) highest priority transactions (τ 1,.., τ i ) by ignoring a other ower priority transactions τ i+1,...,τ m in the schedue. In other words, P i is a pattern of the transaction set consisting of ony the first i highest priority transactions. Lemma 3.3: If P i is the shortest and eariest pattern of the first i (1 i<m) highest priority transactions, and P i+1 is the shortest and eariest pattern of the first i +1highest priority transactions, then 1.Ps i Pi+1 s..p i+1 is a mutipe of P i. Proof. By ignoring the schedue of τ i+1 in P i+1, P i+1 is aso a repeating pattern of the first i highest priority transactions. By definition, Ps i Ps i+1. By Lemma 3.1, P i+1 is a mutipe of P i. We now present the pattern search agorithm. The agorithm foows the idea in the proof of Theorem 3.1. It searches for the pattern of the first i ( i m) highest priority transactions based on the repeating pattern of the first i 1 highest priority transactions. After the agorithm competes for the owest priority transaction, it returns the pattern for the transaction set. Agorithm 3.1 SearchPattern: Input: A successfu DS-FP schedue. Output: The eariest and shortest pattern P m. 1 P 1 (0, V 1 C 1 ); // Pattern of the first transaction. for i =to m do 3 // Find the pattern when adding the next transaction. 4 P i SearchNextTask(i, P i 1 ); 5 od 6 return P m ; Ag. 3.1 invokes Ag. 3. whose input is the pattern of the first i 1(1<i m) highest priority transactions, and output is the pattern of the first i highest priority transactions. Ag. 3. scans the DS-FP schedue for the jobs of the i th highest priority transaction to find the first two jobs such that each starts at the same offset within its corresponding input pattern P i 1. The schedue between these two reease times forms the output pattern for the first i highest priority transactions. Agorithm 3. SearchNextTask: Input: Pattern P i 1 of transactions τ 1,.., τ i 1. Output: Pattern P i of transactions τ 1,.., τ i 1, τ i. 1 k 1; r i,j first τ i reease time after Ps i 1 ; 3 maxl 1+ No. of ide sots in P i 1 ; 4 whie (k maxl) do 5 for r = r i,j to r i,j+k 1 do 6 if ((r i,j+k r)%p i 1 =0) 7 then 8 P i (r, r i,j+k r); //Find the shortest pattern. 9 // The next oop finds the eariest pattern. 10 whie (S T (Ps i 1) = S T (Ps i 1+Pi )) do 5 Authorized icensed use imited to: University of Texas at Austin. Downoaded on Apri 7, 09 at :50 from IEEE Xpore. Restrictions appy.

7 11 P i (Ps i 1, Pi ); 1 od 13 return P i ; 14 fi 15 od 16 k k +1; 17 od 18 return No pattern found; Note that in Ag. 3., τ i can ony be schedued in the ide sots of the input pattern P i 1. According to the Pigeonhoe Principe, Ag. 3. does not need to examine more jobs than the number of ide sots pus 1 in P i 1. In other words, the whie oop of ine 4 in Ag. 3. does not need to oop more than the number of ide sots in P i 1 pus 1. Thus the condition at ine 6 can be true at east once before the whie oop beginning from ine 4 ends. Line 8 in Ag. 3. produces the shortest pattern starting from the reease time of one of τ i s jobs. Given a job τ i,j+k that satisfies condition ((r i,j+k r) %P i 1 =0) at ine 6, the whie oop at ine 10 cannot not run for more than V i C i times, otherwise the agorithm must have returned P i based on τ i,j+k s previous job because ((r i,j+k 1 r) %P i 1 = 0) impies that the pattern (r, r i,j+k 1 r) can be returned. This whie oop shifts the identified pattern to its eariest possibe starting time. Aso note that the whie oop cannot move back to τ i s first job J i,0 (i ) because the reease time of J i,0 (i.e., time 0) is not the beginning time of J i,0 s execution in the DS-FP agorithm. As an exampe, in Figure. (b), P =(7, 1) is the shortest pattern. At time 6, S τ (6) = (6, 1) and S τ (18) = (4, 1). This impies S T (P s 1) S T (P s 1+P ) thus P is the eariest pattern and can not be moved back further. Here S τ (6) S τ (18) because the reease time of J,0 (time 0) is not the beginning time of J,0 s execution. Theorem 3.: P m returned by Ag. 3.1 is the shortest and eariest pattern if such a pattern exists. Proof. We sha prove that if the input to Ag. 3. is the shortest and eariest pattern, so is the output. We first prove that Ag. 3. returns a pattern. Ag. 3. returns ony when the condition at ine 6 is true. The condition impies that r and r i,j+k are of the same offsets within their respective input patterns. So P i derived at ine 8 is a pattern for transactions τ 1,.., τ i 1, τ i. Furthermore, the condition of ine 10 guarantees that P i remains to be a pattern when it is shifted aong the timeine. We then prove that the returned P i is the shortest. Let us examine P i produced at ine 8. Assume that the shortest pattern is of ength L and L P i, then according to Coroary 3.3 (Ps,L) i must be a pattern. The agorithm indicates that τ i must have a job J reeased at time Ps i + L, whichis earier than or equa to Ps i + Pi. According to Lemma 3.3, L is a mutipe of P i 1. This means that J satisfies the condition at ine 6. Since (Ps,L) i is the shortest, J shoud be the first examined job that satisfies the condition. In other words, L = P i. Finay, since Pi at ine 8 is the first pattern that starts with a τ i s reease time, the whie oop at ine 10 guarantees that the returned P i is the eariest pattern for the first i highest priority transactions. We have now proved that if the input to Ag. 3. is the shortest and eariest pattern, so is the output. We aso know that ine 1 in Ag. 3.1 assigns the shortest and eariest pattern for τ 1. By induction, P m returned by Ag. 3.1 is the shortest and eariest pattern of the transaction set. Ag. 3.1 runs in pseudo-poynomia time and has time compexity O(m(Π m i=1 V i) ). However, it can be improved to O(mΠ m i=1 V i) if an array of size O(P i 1 ) can be used to keep reative offsets of τ i s jobs within their corresponding patterns (i.e., P i 1 ). Note that the whie oop at ine 10 is ony executed once athough it is within the two outer oops. It oops at most V i C i times. Thus it is ignored in the compexity cacuation. 3.4 DS-FP Scheduabiity Test Agorithm Ag. 3.1 aso impies a scheduabiity test agorithm. We begin the scheduabiity test with τ 1. Given a subset of transactions τ 1,.., τ i 1 (1 <i m) that has been tested, we test transaction τ i by adding the transaction to the subset unti an added transaction is not scheduabe or a pattern for a transactions is found. Given transaction τ i, we schedue it aong with the schedue of the higher priority transactions τ 1,.., τ i 1, for which a pattern has aready been found. Ag. 3.3 and Ag. 3.4 are modified versions of Ag. 3.1 and Ag. 3. for the scheduabiity test, respectivey. Agorithm 3.3 ScheduabiityTest: Input: A transaction set T. Output: Whether T is scheduabe. 1 P 1 (0, V 1 C 1 ); // Pattern of the first transaction. for i =to m do 3 if (TestNextTask(i, P i 1 )= FALSE) 4 then return T is unscheduabe; fi 5 od 6 return T is scheduabe; Agorithm 3.4 TestNextTask: Input: Pattern P i 1 of transactions τ 1,.., τ i 1. Output: Returns TRUE and pattern P i of transactions τ 1,.., τ i 1, τ i if a pattern of those transactions exists. Otherwise, returns FALSE. 1 Schedue up to, incuding τ i s first request after P i 1 s ; if (Line 1 fais) 3 then return FALSE; fi 4 r i,j τ i s first reease time since P i 1 s ; 53 Authorized icensed use imited to: University of Texas at Austin. Downoaded on Apri 7, 09 at :50 from IEEE Xpore. Restrictions appy.

8 5 k 1; 6 whie (k P i 1 ) do 7 Schedue r i,j+k ; 8 if (Line 7 fais) 9 then return FALSE; fi 10 for r = r i,j to r i,j+k 1 do 11 if ((r i,j+k r) %P i 1 =0) 1 then // Found the shortest pattern. 13 P i (r, r i,j+k r); 14 return TRUE; 15 fi 16 od 17 k k +1; 18 od If Ag. 3.3 returns TRUE, it aso produces the shortest pattern and the DS-FP schedue. The foowing exampe iustrates how the agorithm works. Exampe 3.3: Consider a set of three transactions {τ 1, τ, τ 3 } with execution times 1, 1,, and vaidity intervas 3, 7, 14, respectivey. It is not scheduabe by ML because τ 3 is finished by 8, which is more than V3 =7. Now we test whether it can be schedued by DS-FP or not. Figure 4 (a) corresponds to ine 1 of Ag It shows the pattern of τ 1. Figure 4 (b) depicts the resut of invoking Ag. 3.4 for τ. There is ony one ide time sot in {τ 1 } s pattern, so the reease times of two consecutive jobs J,1 and J, after Ps 1 =0forms a pattern P =(5, 6). Figure 4 (c) depicts the resut of invoking Ag. 3.4 for τ 3. There are two ide time sots in {τ 1,τ } s pattern P = (5, 6), and the agorithm examines three consecutive jobs J 3,1, J 3,,andJ 3,3 after Ps =5tofindanoutput pattern P 3 = (9, 18). Note that r 3,1 has an offset 4 within the pattern P =(5, 6), whie r 3, has an offset within its corresponding pattern P =(17, 6), andr 3,3 has an offset 4 within its corresponding pattern P =(3, 6). The offset of r 3,3 matches that of r 3,1. So Ag. 3.4 goes to ine 13, and Ag. 3.3 returns that the transaction set is scheduabe. Note that the pattern returned from Ag. 3.3 is the shortest but not the eariest. The shortest and eariest pattern for P 3 is (8, 18), one time unit earier than the starting time of P 3 returned from Ag This eariest pattern can be returned by Ag. 3.1, which is actuay cacuated at ines 10 and11inag.3.. Given a DS-FP schedue, there aways exists a repeating pattern and our scheduabiity test agorithm can be appied. However, the space and time compexity of the agorithm is high. The question remains if there is a better scheduabiity test that is more efficient. On the other hand, a repeating pattern in the DS-FP schedue can be found off-ine and used repeatedy to construct the DS-FP schedue. This can significanty reduce the on-ine scheduing overhead of the DS-FP agorithm. T 1 : {C 1 =1, V 1 =3} T : {C =1, V =7} T 3 : {C 3 =, V 3 =14} (a) {T 1 }'s pattern (b) {T 1, T } r,1 r, pattern (c) {T 1, T, T 3 }'s pattern r 3,1 r 3, r 3, Figure 4. Iustration of the scheduabiity test agorithm 3.5 DS-FP in Continuous Systems So far we assume discrete time systems for DS-FP.Now we move on to the scheduabiity discussions of DS-FP in continuous time systems. Given a DS-FP schedue, it can be proved that a repeating pattern sti exists if ony rationa numbers are considered for transaction parameters (i.e., vaidity intervas and execution times). Denote to be the east common mutipe of a the denominators of a those rationa numbers. If we measure time in the unit of 1,then we again have an interger probem which has a pattern for a successfu DS-FP schedue. This schedue is the same as the one that ony has transaction parameters with origina rationa numbers athough their granuarities are different. However, if execution times or vaidity intervas can be rea numbers, it may not be possibe to identify such a repeating pattern in a DS-FP schedue. We sha iustrate this with the foowing exampe. Exampe 3.4: Consider a set of two transactions {τ 1, τ } with execution times 1 and 1 + d, and vaidity intervas 5 and 9 respectivey. Suppose that d is an infinitesimay sma rea number. Figure 5 (a) depicts a schedue of the transaction set under DS-FP. Leti to be the argest integer such that 3 i d>1, i.e.,i = d. r,1,r,,..., r,i occur in every other repeating pattern of τ 1. In addition, k, (1 k i), the offset of r,k within τ 1 s pattern P is 3 k d. There exists no pattern for τ s first i jobs. Hence there exists no pattern from time 0 to t =P i =8 d. t can be arbitrariy arge if d is infinitesimay sma. In other words, if execution time C of τ is a rea number infinitesimay cose to 1, there exists no repeating pattern for the DS-FP schedue. Note that the transaction set has fi- 54 Authorized icensed use imited to: University of Texas at Austin. Downoaded on Apri 7, 09 at :50 from IEEE Xpore. Restrictions appy.

9 T 1 : {C 1 =1, V 1 =5} 3-d T : {C =1+d, V =9} 3- d 3-3 d (a) A schedue without a repeating pattern T 1 : {C 1 =p, V 1 =3p } 0 p 4p 6p 8p T : {C =1, V =5e} (b) A schedue with a repeating pattern 5 Figure 5. DS-FP schedues for transaction sets with rea number parameters nite number of transactions, and finite vaues for execution times and vaidity intervas. We can aso prove that the transaction set is scheduabe by DS-FP using induction. We know J,0 and J,1 are scheduabe. We can easiy prove that if J,i,i > 0 is scheduabe, so is J,i+1. Another proof foows from Theorem.1 because the transaction set is obviousy scheduabe by ML. Our observation from Exampe 3.4 is the foowing: given an arbitrariy arge time t (t + ), there aways exists a transaction set with finite number of transactions and finite rea number parameters that has a successfu DS- FP schedue without any repeating pattern in the interva [0,t]. However, there aso exist transaction sets with finite number of transactions and finite rea number parameters that have successfu DS-FP schedues with repeating patterns, which is iustrated by the foowing exampe. Exampe 3.5: Define two rea numbers p = π = and e = Consider a set of two transactions {τ 1, τ } with execution times p, 1, and vaidity intervas 3p, 5e, respectivey. Figure 5 (b) depicts the DS-FP schedue of the transaction set with a repeating pattern. 4 Concusions and Future Work Distinct from past studies of maintaining the freshness (or tempora vaidity) of rea-time data that adopt the periodic task mode, DS-FP adopts the aperiodic task mode. In DS-FP, the deadines of jobs and the separation of two consecutive jobs of an update transaction are adjusted judiciousy so that the farthest distance of the samping times of two consecutive jobs can be achieved. However, the aperiodic mode in DS-FP poses a great chaenge for its scheduabiity test. This paper addresses this issue in discrete time system by providing a necessary and sufficient scheduabiity condition for DS-FP. We prove the existence of a repeating pattern for any successfu DS-FP schedue, and derive the corresponding scheduabiity test agorithm. Whie this provides a scheduabiity anaysis of DS-FP in discrete time systems, there are sti unanswered questions for DS-FP scheduabiity in continuous time systems. Furthermore, the achievabe time compexity of our scheduabiity test agorithm is O(mΠ m i=1 V i), thus it is highy desirabe to improve the efficiency of the agorithm. References [1] S. K. Baruah, A. K. Mok, L. E. Rosier, Preemptivey Scheduing Hard-Rea- Sporadic Tasks on One Processor, IEEE Rea- Systems Symposium, December [] A. Burns and R. Davis, Choosing task periods to minimise system utiisation in time triggered systems, in Information Processing Letters, 58 (1996), pp [3] D. Chen and A. K. Mok, Scheduing Simiarity-Constrained Rea- Tasks, pp. 15-1, ESA/VLSI, 04. [4] H. Chetto and M. Chetto, Some Resuts of the Eariest Deadine Scheduing Agorithm, IEEE Transactions on Software Engineering, Vo. 15, No. 10, pp , October [5] R. Gerber, S. Hong and M. Saksena, Guaranteeing Endto-End Timing Constraints by Caibrating Intermediate Processes, IEEE Rea- Systems Symposium, December [6] R. P. Grimadi, Discrete and Combinatoria Mathematics: An Appied Introduction, Addison-Wesey, 4th Edition, pp , [7] T. Gustafsson, J. Hansson, Data Management in Rea- Systems: a Case of On-Demand Updates in Vehice Contro Systems, IEEE Rea- and Embedded Technoogy and Appications Symposium, pp , 04. [8] T. Gustafsson and J. Hansson, Dynamic on-demand updating of data in rea-time database systems, ACM SAC, 04. [9] C. C. Han, K. J. Lin and J. W.-S. Liu, Scheduing Jobs with Tempora Distance Constraints, Siam Journa of Computing, Vo. 4, No. 5, pp , October [10] K. D. Kang, S. Son, J. A. Stankovic, and T. Abdezaher, A QoS-Sensitive Approach for iness and Freshness Guarantees in Rea- Databases, EuroMicro Rea- Systems Conference, June 0. [11] T. Kuo and A. K. Mok, Rea- Data Semantics and Simiarity-Based Concurrency Contro, IEEE Rea- Systems Symposium, December 199. [1] T. Kuo and A. K. Mok, SSP: a Semantics-Based Protoco for Rea- Data Access, IEEE Rea- Systems Symposium, December [13] S. Ho, T. Kuo, and A. K. Mok, Simiarity-Based Load Adjustment for Rea- Data-Intensive Appications, IEEE Rea- Systems Symposium, [14] K.Y. Lam, M. Xiong, B. Liang and Y. Guo, Statistica Quaity of Service Guarantee for Tempora Consistency of Reatime Data Objects, IEEE Rea- Systems Symposium, 04. [15] J. Leung and J. Whitehead, On the Compexity of Fixed- Priority Scheduing of Periodic Rea- Tasks, Performance Evauation, (198), [16] C. L. Liu, and J. Layand, Scheduing Agorithms for Mutiprogramming in a Hard Rea- Environment, Journa of the ACM, (1), Authorized icensed use imited to: University of Texas at Austin. Downoaded on Apri 7, 09 at :50 from IEEE Xpore. Restrictions appy.

10 [17] D. Locke, Rea- Databases: Rea-Word Requirements, in Rea- Database Systems: Issues and Appications, edited by Azer Bestavros, Kwei-Jay Lin and Sang H. Son, Kuwer Academic Pubishers, pp , [18] K. Ramamritham, Rea- Databases, Distributed and Parae Databases 1(1993), pp , [19] K. Ramamritham, Where Do Constraints Come From and Where Do They Go? Internationa Journa of Database Management, Vo. 7, No., Spring 1996, pp [] X. Song and J. W. S. Liu, Maintaining Tempora Consistency: Pessimistic vs. Optimistic Concurrency Contro, IEEE Transactions on Knowedge and Data Engineering, Vo. 7, No. 5, pp , October [1] J. A. Stankovic, S. Son, and J. Hansson, Misconceptions About Rea- Databases, IEEE Computer, Vo. 3, No. 6, pp. 9-36, June [] M. Xiong, K. Ramamritham, J. A. Stankovic, D. Towsey, and R. M. Sivasankaran, Scheduing Transactions with Tempora Constraints: Expoiting Data Semantics, IEEE Transactions on Knowedge and Data Engineering, 14(5), , 0. [3] M. Xiong and K. Ramamritham, Deriving Deadines and Periods for Rea- Update Transactions, IEEE Transactions on Computers, 53(5), pp , 04. [4] M. Xiong, S. Han, and K.Y. Lam, A Deferrabe Scheduing Agorithm for Rea- Transactions Maintaining Data Freshness, IEEE Rea- Systems Symposium, 05. [5] M. Xiong, S. Han, K. Y. Lam and D. Chen. Deferrabe Scheduing for Maintaining Rea- Data Freshness: Agorithms, Anaysis and Resuts, IEEE Transactions on Computers, to appear. A onger version is avaiabe as Technica Report TR-07-44, Dept. of Computer Sciences, Univ. of Texas at Austin, Sept Authorized icensed use imited to: University of Texas at Austin. Downoaded on Apri 7, 09 at :50 from IEEE Xpore. Restrictions appy.