Schedulability Analysis of Deferrable Scheduling Algorithms for Maintaining Real-Time Data Freshness

Transcription

1 1 Scheduabiity Anaysis of Deferrabe Scheduing Agorithms for Maintaining Rea-Time Data Freshness Song Han, Deji Chen, Ming Xiong, Kam-yiu Lam, Aoysius K. Mok, Krithi Ramamritham UT Austin, Emerson Process Management, Googe, CityU of Hong Kong, IIT Bombay {shan, Abstract Athough the deferrabe scheduing agorithm for fixed priority transactions (DS-FP) has been shown to provide a better performance compared with the More-Less (ML) method, there is sti a ack of any comprehensive studies on the necessary and sufficient conditions for the scheduabiity of DS-FP. In this paper, we first anayze the necessary and sufficient scheduabiity conditions for DS-FP, and then propose a scheduabiity test agorithm for DS-FP by expoiting the fact that there aways exists a repeating pattern in a DS-FP schedue. To resove the imitation of fixed priority scheduing in DS-FP, we then extend the deferrabe scheduing to a dynamic priority scheduing agorithm caed DS- EDF by appying the eariest deadine first (EDF) poicy to schedue update jobs. We aso propose a scheduabiity test for DS-EDF and compare its performance with DS-FP and ML through extensive simuation experiments. The resuts show that the scheduabiity tests are effective. Athough the scheduabiity of DS-EDF is ower than DS-FP and the repeating patterns in DS-EDF schedues are onger than those in DS-FP due to the use of dynamic priority scheduing, the performance of DS-EDF is better than both DS-FP and ML in terms of CPU utiization and impact on ower priority appication transactions. Index Terms Rea-time Database, Rea-time Data, Scheduabiity, Tempora Vaidity, Rea-time Scheduing I. INTRODUCTION The timeiness of data vaues samped from rea-word entities is critica for many rea-time data services such as cyberphysica systems and inteigent robotic contro systems [2] [5]. For exampe in the cyber-physica avatar project [6], sensors are instaed in the humanoid robot for tracking the current status of dynamic entities either on the robot or in the physica operation environment. They pubish measurements through update transactions or update jobs to be instaed into a reatime database. These samped data, usuay caed rea-time data, wi become invaid with the passage of time since the entities in the operation environment may change continuousy. One of the common ways to determine the tempora vaidity of reatime data objects is to define a vaidity interva [7] or age constraint [8] for each data object such that a vaue for the rea-time data object is ony vaid within the vaidity interva of the data object. To maintain the data vaidity, the system has to continuousy generate update jobs to refresh the corresponding rea-time data object. Otherwise, the vaidity of the data object cannot be guaranteed and the system may not be abe to detect and respond to environmenta changes in a timey and effective manner as it cannot observe the atest status of entities in the operationa environment. 1 An earier version of this paper appeared in [1]. 2 In this paper, assuming that a transaction τ i is defined by {C i, V i } where C i is the worst-case execution time and V i is the vaidity interva, the scheduabiity of a scheduing poicy P is defined as the fraction of transaction sets that are scheduabe under P in the entire transaction set space. The typica method for maintaining the tempora vaidity of a set of rea-time data objects is to use periodic update transactions. One such method is ML [8], [9]. In ML, an update transaction is defined to generate an update job in every fixed period to refresh the vaidity of a rea-time data object regardess of how much the status of the corresponding rea-word entity has been changed. Unike the heuristic-based dynamic scheduing methods [10] [13] which are mosty proposed for soft rea-time systems where the arriva of update transactions is sporadic and unpredictabe, the goas of ML are to determine the update period, reative deadine and the scheduing priority for each update transaction such that a guarantee is provided on the vaidity of the set of rea-time data objects maintained in a reatime database. Athough ML has been shown to be effective in maintaining data vaidity, to further reduce the resuting update workoad in [14], an enhancement of ML, caed the deferrabe scheduing agorithm for fixed priority transactions (DS-FP) was proposed. By judiciousy deferring the generation time of an update job to be as ate as possibe without affecting the guarantee provided to the data vaidity, DS-FP has been shown to give a better performance than ML in terms of reduced update workoad. Athough DS-FP can reduce the update workoad, it sacrifices the periodicity of updates, which in turn, poses great chaenges for its scheduabiity anaysis. One prominent method in cassica scheduabiity anaysis is based on the critica instant test [9]. A critica instant makes sense for periodic tasks by assuming synchronous update tasks, i.e., the first jobs of a the update tasks are initiated at the same time. It has aso been adopted for sporadic task sets by converting the minimum separation times to be the periods in the scheduabiity anaysis. In a DS-FP schedue, however, the distance of the reease times of two consecutive jobs from the same update transaction is not fixed. It is ony proven so far that DS-FP coud schedue any transaction set that is scheduabe by ML [14]. Athough experimentay it has been demonstrated that DS-FP outperforms ML significanty [14], the open theoretic question of whether there is any necessary and sufficient conditions to determine if a transaction set is scheduabe by DS-FP (even if it is not scheduabe by ML) has remained unsoved. Furthermore, DS-FP is a fixed priority scheduing agorithm and its scheduing overhead in terms of deriving the reease time of an update job backwards from its deadine by considering the preemption from a higher priority jobs coud be very high [15]. This coud severey imit its appications for the systems which require to construct the schedue onine and the use of dynamic priorities for scheduing their jobs. Thus, another important question for DS-FP is whether it can be extended to support dynamic priority scheduing and incur ess scheduing overhead.

2 2 Contribution. Through this paper we address the aforementioned imitations of DS-FP and make the foowing contributions. To resove the imitation of fixed priority scheduing in DS- FP and reduce its onine scheduing overhead, we propose a new dynamic scheduing agorithm caed DS-EDF by adopting the eariest deadine first poicy in the deferrabe scheduing approach. We show that the compexity of DS- EDF is much ower than that of DS-FP. We aso address the probem of finding necessary and sufficient conditions for the scheduabiity of both DS-EDF and DS-FP. We prove that there aways exists a repeating pattern for a given DS- EDF (or DS-FP) schedue in discrete time systems. Based on this finding, we design pattern searching agorithms to find the eariest and shortest pattern in a DS-EDF (or DS- FP) schedue and propose corresponding scheduabiity test agorithms for checking their scheduabiity. The performance and scheduabiity of DS-EDF as compared with DS-FP and ML is investigated through extensive simuation studies. The resuts show that athough the scheduabiity of DS-EDF is ower than DS-FP and the repeating patterns in DS-EDF schedues are onger than those in DS-FP, DS-EDF outperforms both DS-FP and ML in terms of CPU utiization and impact on ower priority appication transactions. Organization. The remainder of the paper is organized as foows: Section II briefy reviews the background and reated work. In Section III, we study the necessary and sufficient scheduabiity conditions for DS-FP in discrete time systems, and propose a scheduabiity test agorithm. In Section IV, we introduce a new deferrabe scheduing agorithm using eariest deadine first scheduing, caed DS-EDF. The scheduabiity anaysis of DS-EDF is presented in Section V. Section VI presents the experimenta resuts on the performance and scheduabiity of both DS-FP and DS-EDF. We concude the paper and discuss future work in Section VII. II. BACKGROUND AND RELATED WORK In this section, we first briefy review the concept of tempora vaidity of rea-time data and the reated work. We then summarize the main principes of ML [8], [9] and deferrabe scheduing (DS-FP) [14] agorithms. Frequenty used symbos and their meanings are summarized in Tabe I. A. Rea-Time Data Vaidity One of the core goas in rea-time data management is to maintain the tempora vaidity of rea-time data objects so that they are vaid representations of the current status of reaword entities. In [7], an interva-based method is proposed to define the tempora vaidity constraint for a rea-time data object. Definition II.1: A rea-time data object (X i ) at time t is temporay vaid (or caed absoutey consistent) if, assuming its j th update is the atest to finish before t, the samping time (r i,j ) pus the vaidity interva (V i ) of the data object is not ess than t, i.e., r i,j + V i t [7]. A data vaue for rea-time data object X i samped at any time t wi be temporay vaid up to (t+v i ). Afterwards, it is invaid Symbo Definition X i Rea-time data object i (i = 1,.., m) τ i Update transaction updating X i J i,j The jth job of τ i (j = 0, 1, 2,..) R i,j Response time of J i,j C i Worst-case 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 P i Pattern of the first i highest priority transactions P i Estimated ength of P i 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) TABLE I Symbos and definitions. or caed stae. The actua ength of the vaidity interva of a rea-time data object is appication dependent and depends on the dynamic properties of the corresponding entity [7], [16], [17]. One of the important design goas of many rea-time database systems is to guarantee that the rea-time data remain fresh, i.e., they are aways vaid. Accessing stae data vaues coud seriousy affect the effectiveness of the rea-time functions provided by the systems, e.g., generate incorrect responses even though the responses may be timey [10], [18] [20]. There have been extensive research work on maintaining the vaidity and freshness of rea-time data [1], [9], [14], [15], [19], [21] [25]. Some of them use periodic update transactions whie the others assume the arriva of update transactions to be sporadic. The second type of methods, e.g., [10], [11], [13], [21], [26], are mainy designed for soft rea-time systems [18], and the main probem to be tacked is how to schedue update transactions at runtime to maximize the freshness of rea-time data objects whie minimizing their impact on the execution of rea-time transactions from appications. Athough these methods have been shown to be effective for achieving a better average performance, they cannot provide a guarantee on data vaidity for the execution of rea-time transactions. On the contrary, the prime performance goa of those methods using periodic update transactions is to provide a guarantee on data vaidity. The main probems studied in these methods are: (1) how to determine the period and deadine for each update transaction to maintain vaidity of each rea-time data object; and (2) how to define a schedue such that the deadines of a the update transactions can be guaranteed. In the foowing, we briefy review two of the most representative methods: ML and DS-FP. ML is the basic method using periodic update transactions whie DS-FP is an enhancement of ML with better performance. For the detais of the mechanisms of these two methods, readers may refer to [9], [14]. Note that in these two methods, the system is assumed to be synchronous. In addition, the transactions from appications (caed appication transactions) are assumed to be assigned ower priorities for scheduing compared with the update transactions so that the scheduing of the update transactions wi not be affected by

3 3 the scheduing of the appication transactions. Athough this coud affect the performance of the appication transactions, it is assumed that the prime performance goa of the system is to guarantee the tempora vaidity of rea-time data by meeting the deadines of a the update transactions. After guaranteeing the data vaidity, the next performance goa is to minimize the impact on the performance of appication transactions by using a more efficient update scheduing agorithm. More-Less: is based on off-ine scheduabiity. To guarantee the vaidity of each data object X i, an update transaction τ i generates a job J i,j every fixed period P i after capturing the atest status of the entity ζ i. Given the vaidity interva of each data object, ML determines the periods and deadines for a set of update transactions and schedue their jobs using the Deadine Monotonic (DM) method [27]. In ML, the period P i pus the reative deadine D i for an update transaction τ i is set to be no arger than the vaidity interva V i of the corresponding data object X i, and P i needs to be no smaer than haf of V i whie D i needs to be no arger than haf of V i, i.e., P i + D i V i and D i P i. In cacuating the transactions deadines, it adopts a pessimistic approach by using the worst-case response time (WCRT) of each update transaction such that the vaidity of a data objects can be guaranteed by competing the jobs of the update transactions before their deadines. The cacuation order for each transaction foows the Shortest Vaidity First (SVF) poicy so that it starts from the update transaction with the shortest vaidity interva. DS-FP: ML is pessimistic on the deadine and period assignment as it uses the worst-case response time of τ i to derive the reative deadine D i for each update transaction τ i. According to the vaidity constraint in ML, if D i is arger, the period P i wi be smaer. Thus, the resuting update workoad wi be heavier. In order to increase the separation of two consecutive jobs from the same update transaction (and thus reduce the update workoad), DS-FP adaptivey derives the reative deadine and separation of one job from its previous job using the tota preemption time from higher-priority transactions instead of the worst-case response time of an update transaction. Simiar to ML, DS- FP determines the cacuation order for the update transactions according to SVF and schedues the jobs from the transactions using DM. The main steps of DS-FP are as foows: First, we set the reease time of the first job for each transaction r i,0 = 0, i, 1 i m. The highest priority job among the reeased jobs is aways schedued first. It is ony preempted when a new job with higher priority is reeased. 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. The foowing theorem shows that any update transaction set that can be schedued by ML is aso scheduabe by DS-FP [14]. Theorem II.1: (Coroary 3.2 in [14]) Given a synchronous update transaction set T with known C i and V i (1 i m), if T can be schedued by ML, then it can aso be schedued by DS-FP. Unfortunatey, up to now, it is sti uncear what the scheduabiity conditions of DS-FP are if a set of update transactions cannot be schedued by ML. This question wi be answered in the next section by expoiting the patterns that aways exist in the DS-FP schedues. Fig. 1. T 1 : {C 1 =2, V 1 =6} J 2, 0 competes after V 2 /2 T 2 : {C 2 =3, V 2 =12} (a) ML is unscheduabe J 2, 0 competes after V 2 / (b) DS-FP is scheduabe Time Repeating pattern Time Two transactions that can be schedued by DS-FP but not by ML III. DS-FP SCHEDULABILITY ANALYSIS In this section, we first present the pattern anaysis for DS-FP, and then we introduce an agorithm for testing the scheduabiity of DS-FP. From here on, it is assumed that transactions are studied in a discrete time system uness it is specified otherwise. The probem for deaing with continuous time wi be discussed in Section III-F. Given a set of update transactions T = {τ i } m i=1, we assume without oss of generaity that τ k has higher priority than τ j for k < j. A. DS-FP Patterns Theorem II.1 states that DS-FP is at east as good as 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 III.1: Consider a set of two transactions {τ 1, τ 2 } with computation times 2 and 3, and vaidity intervas 6 and 12 respectivey. Figure 1 (a) depicts a schedue of the transactions by ML. The first job of τ 2, J 2,0, competes at time 7, which is greater than V2 2 = 6. Thus, the set of transactions is not scheduabe by ML. Figure 1 (b) depicts a schedue of the transactions by DS-FP. The same set of transactions is scheduabe by DS-FP because the schedue pattern between time points 12 and 24 repeats itsef forever. DS-FP is better in Exampe III.1 because DS-FP aows J 2,0 to be competed ater than V2 2. There are aso transaction sets in which for every transaction τ i, J i,0 is competed no ater than Vi 2 in DS-FP, and furthermore these transaction sets can be schedued by DS-FP but not by ML. The next exampe iustrates this point. Exampe III.2: Consider a set of three transactions {τ 1, τ 2, τ 3 } with computation times 2, 3, 3, and vaidity intervas 6, 15, 47, respectivey. Figure 2 (a) depicts a schedue of the transactions by ML. The ML period and deadine for τ 2 are 8 and 7, respectivey. The first job of τ 3, J 3,0, competes at time 24, which is greater than V 3 2 = Thus the set of transactions is not scheduabe by ML. Figure 2 (b) depicts a schedue of the transactions by DS-FP. The same transaction set is scheduabe by DS-FP because the schedue pattern between time 26 and 50 repeats itsef forever. In this schedue J 3,0 competes at time

4 4 Fig. 2. T 1 : {C 1 =2, V 1 =6} T 2 : {C 2 =3, V 2 =15} T 3 : {C 3 =3, V 3 =47} J 3, 0 competes after V 3 / (a) ML is unscheduabe J 3, 0 competes beforev 3 /2 (b) DS-FP is scheduabe Repeating pattern Time Time Time Three transactions that can be schedued by DS-FP but not by ML 19, which is smaer than V 3 2 (i.e., 23.5). This is because J 2,2 is reeased ater than its reease time in the schedue obtained from ML. Note that DS-FP fuy utiizes the processor in both exampes. We coud easiy derive exampes in which the processor ides once in a whie. For exampe, in Figure 1 we coud change C 2 to 2.5 and in Figure 2 we coud change C 3 to After both changes the transaction sets sti cannot be schedued by ML but can be schedued by DS-FP. Furthermore, we can scae up the numbers to make them a integers again. We denote by a tupe P = (P s, P ) the DS-FP schedue of ength P starting from time P s. Let tupe S τ (t) = (d, e) denote the state of transaction τ at time t, where d is the distance to τ s ast job reeased before time t, and e is the remaining execution time of τ at time t. In particuar, e = 0 if τ s ast job before t has aready been finished at time t. We denote by S T (t) the aggregated states of a transactions in T at time t, i.e., S T (t) = {S τ (t) τ T }. Note once S T (t) is known, the DS-FP schedue from t onward can be determined. Definition III.1: Given a transaction set T, if (1) a DS-FP schedue repeats pattern P = (P s, P ) forever in time interva [P s + np, P s + (n + 1)P ) (n = 0, 1, 2,..); and (2) S T (P s + np + t) = S T (P s + (n + 1)P + t) (t = 0, 1, 2,.., P 1), then P is a repeating pattern of T s DS-FP schedue. Coroary III.1: If S T (t + s) = S T (t), then (P s = t, P = s) is a repeating pattern. Proof. By definition of S T (t), for any t, S T (t + 1) is fuy determined based on S T (t). So if S T (t + s) = S T (t), we have S T (t+s+1) = S T (t+1) and the same transaction is schedued at time t+s+1 and t+1 if time t is not ide. Foowing the same argument, we have, for a k > 0, S T (t + s + k) = S T (t + k) and thus S T (t + s) = S T (t + s + s). This impies that (P s = t, P = s) is a repeating pattern. Now we proved 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 schedue if the transaction set is scheduabe, and if so, how to find it. We answer those questions in the next subsection. B. DS-FP Pattern Anaysis Using Pigeonhoe Principe In this subsection, we prove that for each transaction set which is scheduabe by DS-FP, there aways exists a repeating pattern. Note that DS-FP is not necessariy ide immediatey before P s for a pattern P. Exampe III.3: Consider the same transaction set as in Exampe III.1. P = (12, 24) is a repeating pattern because the schedue between time points 12 and 24 repeats itsef forever. Aso S T (12) = S T (24) because S τ1 (12) = S τ1 (24) = (4, 0) and S τ2 (12) = S τ2 (24) = (5, 0). However, athough the schedue between time 6 and 10 does repeat itsef, P = (6, 4) is not a repeating pattern because S T (6) S T (10). As mentioned before, we study the DS-FP scheduabiity probem in a discrete time system. In order to prove that there exists a repeating pattern if a transaction set is scheduabe by DS-FP, we first review the Pigeonhoe Principe. The Pigeonhoe Principe [28]: 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 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 as the number of distinct execution states is finite. The foowing theorem states that a DS-FP schedue has a repeating pattern that must occur at east once in a bounded time interva. Theorem III.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 in the bounded time interva [0, (V m C m ) + Π m i=1 (V i C i + 1) 1], then the DS-FP schedue has a repeating pattern that must occur at east once in the bounded time interva [V m C m, (V m C m ) + Π m i=1 (V i C i + 1) 1]. Proof. The proof is incuded in Appendix I due to space imit. According to the proof of Theorem III.1, 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 repeating pattern appearing in the interva repeats itsef forever. Thus we have the foowing coroary. Coroary III.2: 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]. C. DS-FP Pattern Properties Theorem III.1 proves the existence of a repeating pattern for a given DS-FP schedue. This subsection further studies the properties of the DS-FP pattern. Given a repeating pattern P = (P s, P ), the foowing 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 III.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.

5 5 We now prove the next emma. Lemma III.1: Given a the patterns P, P,... of a DS-FP schedue for transaction set T, et P 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 2, 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 Coroary III.1 that the pattern (t 1+P, s) repeats itsef from time (t 1 +P ) with ength s. As 0 < 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 of Lemma III.1, since P is a mutipe of P, (P s, P ) must aso be a pattern. Coroary III.4: 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 III.1 and Coroary III.4 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 III.2: 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 eariest and shortest pattern. According to Lemma III.1, P is a mutipe of P. According to Coroary III.4, (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. Given a transaction set T 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 III.3: If P i is the eariest and shortest pattern of the first i (1 i < m) highest priority transactions, and P i+1 is the eariest and shortest pattern of the first i + 1 highest priority transactions, then 1) Ps i Ps i+1. 2) 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 III.1, P i+1 is a mutipe of P i. DS-FP patterns in the genera case. Pease note that we assume the worst-case execution times for a jobs in our DS-FP pattern anaysis. However, this assumption does not aways hod. Lemma III.4 states that in the genera case where a job s actua execution time can be ess than its worst-case execution time, the DS-FP pattern can sti be kept if the DS-FP scheduer sti assigns the worst-case execution time to the job. In such cases, the processor may ide after the job s competion unti the job s assigned time sots expire. Lemma III.4: Given an update transaction set T with known C i and V i (1 i m), if it can be schedued by DS-FP with worst-case execution times, then it can aso be schedued by DS-FP in the genera case and each job J i,j in both schedues can have the same reease time r i,j and deadine d i,j. Proof. Let c i,j be the actua execution time of J i,j in the genera case and we have c i,j C i. Let S be a feasibe DS-FP schedue with the worst-case execution times of a the tasks. Let us keep J i,j s reease time r i,j and deadine d i,j unchanged and repace J i,j s worst-case execution time C i in S with its corresponding actua execution time c i,j. Since c i,j C i, J i,j must be scheduabe after the repacement. If we repace every J i,j s worst-case execution time C i with its actua execution time c i,j, we have a feasibe schedue S for the genera case execution times. D. DS-FP Pattern Search Agorithm Coroary III.2 forms a basis for the scheduabiity test of DS-FP. However, the ength of the interva in Coroary III.2 is O(Π m i=1 V i), and it does not take into consideration the time sots occupied by C i (1 i m). We now present an improved upper bound estimation of the pattern ength by restricting the possibe pigeonhoes ony to the ide sots in the DS-FP schedues. Given a set of transactions T = {τ i } m i=1, we denote by Pi the upper bound ength of the pattern P i formed by transactions τ 1, τ 2,..., τ i, and I i 1 the number of ide sots in P i 1. Consider I i consecutive jobs of τ i foowing Ps i 1, there are two notabe facts about those τ i jobs: 1) there exist two jobs starting at the same offset within their corresponding pattern P i 1 instances according to the pigeonhoe Principe; and 2) the separation between any two consecutive jobs of τ i can not exceed V i C i. The schedue between the request time points of the two jobs in the first fact forms a repeating pattern. Thus, P i (2 i m) can be estimated as foows. P i = (I i 1 + 1) (V i C i ) (1) Foowing Eq. 1, the initia conditions P 1 = V 1 C 1 and I 1 = V 1 2 C 1, the upper bound of the pattern ength P i for transactions τ i (1 i m) can be estimated iterativey from high to ow priority transactions, which heps to improve the efficiency of our pattern search Agorithms 1 and 2. Note that the computation of upper bound P i ony takes into account the ide sots in P i 1 (i.e., I i 1 ) whie not P i 1. This significanty reduces the ength of the time interva for finding its pattern P i. Our pattern search agorithm foows the idea in Theorem III.1. It searches for the pattern of the first i (2 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. Ag. 1 invokes Ag. 2 whose input is the pattern of the first i 1 (1 < i m) highest priority transactions, and output

6 6 Ag 1 SearchPattern 1: Input: A successfu DS-FP schedue. 2: Output: The eariest and shortest pattern P m. 3: 4: // Pattern of the first transaction. 5: P 1 (0, V 1 C 1); 6: for i = 2 to m do 7: // Find the pattern when adding the next transaction. 8: P i SearchNextTask(i, P i 1 ); 9: end for 10: return P m ; Ag 2 SearchNextTask 1: Input: Pattern P i 1 of transactions τ 1,.., τ i 1. 2: Output: Pattern P i of transactions τ 1,.., τ i 1, τ i. 3: 4: k 1; 5: r i,j first τ i reease time after Ps i 1 ; 6: L max 1 + I i 1 ; 7: whie (k < L max ) do 8: k k + 1; 9: for r = r i,j to r i,j+k 1 do 10: if ((r i,j+k r) % P i 1 = 0) then 11: // Find the shortest pattern 12: P i (r, r i,j+k r); 13: // The next oop finds the eariest pattern 14: whie S T (Ps i 1) = S T (Ps i 1 + P i ) do 15: P i (Ps i 1, P i ); 16: end whie 17: return P i ; 18: end if 19: end for 20: end whie 21: return No pattern found; is the pattern of the first i highest priority transactions. Ag. 2 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. Note that in Ag. 2, τ i can ony be schedued in the ide sots of the input pattern P i 1. According to the Pigeonhoe Principe, Ag. 2 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 Line 7 in Ag. 2 does not need to oop more than the number of ide sots in P i 1 pus 1. Thus, the condition at Line 10 can be true at east once before the whie oop beginning from Line 7 ends. Line 12 in Ag. 2 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 Line 10, the whie oop at Line 14 cannot run for more than V i C i times. Otherwise the end of the found pattern must have hit the time point r i,j+k 1 and the beginning of the pattern must aso be a τ i s request time because the beginning and the end of a pattern have the same state. However, this pattern must have aready satisfied the condition on ine 7 during the previous whie oop of ine 4 and must have been returned by the agorithm. Aso note that the whie oop cannot move back to τ i s first job J i,0 (i 2) 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. Theorem III.2: P m returned by Ag. 1 is the eariest and shortest pattern. Proof. We sha prove that if the input to Ag. 2 is the eariest and shortest pattern, so is the output. We first prove that Ag. 2 returns a pattern. Ag. 2 returns ony when the condition at Line 10 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 Line 12 is a pattern for transactions τ 1,.., τ i 1, τ i. Furthermore, the condition of Line 14 guarantees that P i remains to be a pattern when it is shifted aong the time ine. We then prove that the returned P i is the shortest. Let us examine P i produced at Line 12. Assume that the shortest pattern is of ength L and L P i, then according to Coroary III.4 (Ps, i L) must be a pattern. The agorithm indicates that τ i must have a job J reeased at time Ps i + L, which is earier than or equa to Ps i + P i. According to Lemma III.3, L is a mutipe of P i 1. This means that J satisfies the condition at Line 10. Since (Ps, i L) is the shortest, J shoud be the first examined job that satisfies the condition. In other words, L = P i. Finay, since P i at Line 12 is the first pattern that starts with a τ i s reease time, the whie oop at Line 14 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. 2 is the eariest and shortest pattern, so is the output. We aso know that Line 5 in Ag. 1 assigns the eariest and shortest pattern for τ 1. By induction, P m returned by Ag. 1 is the eariest and shortest pattern of the transaction set. Ag. 1 has time compexity O(m(Π m i=1 V i) 2 ). However, it can be further improved to O(mΠ m i=1 V i) if an array of size O(P i 1 ) can be used when searching for pattern P i. We simpy wak through the job requests of τ i. For each job, we save its index number in the array entry where the entry index is equa to this job s reative offset in its corresponding P i 1 instance. If the array entry aready has saved a job index, then these two jobs form a pattern. Thus, the compexity is O(P i + V i P i) = O(V i P i) = O(Πm i=1 V i). If Ag. 2 is impemented in this way, the compexity of Ag. 1 wi be O(mΠ m i=1 V i). Note that the whie oop at Line 14 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. E. DS-FP Scheduabiity Test Agorithm Ag. 1 aso impies a scheduabiity test agorithm. The agorithm begins the scheduabiity test with τ 1. Given a subset of transactions τ 1,.., τ i 1 (1 < i m) that has been tested, the agorithm tests 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, the agorithm schedues 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 and Ag. 4 are modified versions of Ag. 1 and Ag. 2 for the scheduabiity test, respectivey. If Ag. 3 returns TRUE, it aso produces the shortest pattern and the DS-FP schedue. The foowing exampe iustrates how the agorithm works. Exampe III.4: Consider a set of three transactions {τ 1, τ 2, τ 3 } with computation times 1, 1, 2, and vaidity intervas 3, 7, 14, respectivey. It is not scheduabe by ML because τ 3 finishes by 8, which is more than V 3 2 = 7. Now we test whether it can be

7 7 schedued by DS-FP or not. Figure 3 (a) corresponds to Line 5 of Ag. 3. It shows the pattern of τ 1. Figure 3 (b) depicts the resut of invoking Ag. 4 for τ 2. There is ony one ide time sot in {τ 1 } s pattern, so the reease times of two consecutive jobs J 2,1 and J 2,2 after Ps 1 = 0 forms a pattern P 2 = (5, 6). Figure 3 (c) depicts the resut of invoking Ag. 4 for τ 3. There are two ide time sots in {τ 1, τ 2 } s pattern P 2 = (5, 6), and the agorithm examines three consecutive jobs J 3,1, J 3,2, and J 3,3 after Ps 2 = 5 to find an output pattern P 3 = (9, 18). Note that r 3,1 has an offset 4 within the pattern P 2 = (5, 6), whie r 3,2 has an offset 2 within its corresponding pattern P 2 = (17, 6), and r 3,3 has an offset 4 within its corresponding pattern P 2 = (23, 6). The offset of r 3,3 matches that of r 3,1. So Ag. 4 goes to Line 20, and Ag. 3 returns that the transaction set is scheduabe. The eariest and shortest pattern for P 3 is (8, 18), one time unit earier than the starting time of P 3 returned from Ag. 3. The eariest pattern P 3 can be returned from Ag. 1. F. DS-FP in Continuous Time Systems So far we assume a discrete time system for DS-FP. To make the discussion more compete, now we move on to the scheduabiity discussions of DS-FP in continuous time systems where update transactions execution times and vaidity intervas can be rea numbers. 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 integer 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 III.5: Consider a set of two transactions {τ 1, τ 2 } with computation times 1 and 1 + d, and vaidity intervas 5 and 9 respectivey. Suppose that d is an infinitey sma rea number. Figure 4 (a) depicts a schedue of the transaction set by DS-FP. Let i to be the argest integer such that 3 i d > 1, i.e., i = 2 d. r 2,1, r 2,2,..., r 2,i occur in every other repeating pattern of τ 1. In addition, k, (1 k i), the offset of r 2,k within τ 1 s pattern P is 3 k d. There exists no pattern for τ 2 s first i jobs. Hence there exists no pattern from time 0 to t = 2P i = 8 2 d. Time t can be arbitrariy arge if d is infinitey sma. In other words, if execution time C 2 of τ 2 is a rea number infinitey cose to 1, there exists no repeating pattern for the DS-FP schedue. Note that the transaction set has finite 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 2,0 and J 2,1 are scheduabe. We can easiy prove that if J 2,i, i > 0 is scheduabe, so is J 2,i+1. Another proof foows from Theorem II.1 because the transaction set is obviousy scheduabe by ML. Our observation from Exampe III.5 is the foowing: given an arbitrariy arge time t (t + ), there aways exists Ag 3 ScheduabiityTest 1: Input: A transaction set T. 2: Output: Whether T is scheduabe. 3: 4: // Pattern of the first transaction 5: P 1 (0, V 1 C 1); 6: for i = 2 to m do 7: if (TestNextTask(i, P i 1 ) = FALSE) then 8: return T is unscheduabe; 9: end if 10: end for 11: return T is scheduabe; Ag 4 TestNextTask 1: Input: Pattern P i 1 of transactions τ 1,.., τ i 1. 2: 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. 3: 4: Schedue up to, incuding τ i s first request after Ps i 1 ; 5: if (Line 4 fais) then 6: return FALSE; 7: end if 8: r i,j τ i s first reease time since Ps i 1 ; 9: k 1; 10: L max 1 + I i 1 ; 11: whie (k < L max ) do 12: k k + 1; 13: Schedue r i,j+k ; 14: if (Line 13 fais) then 15: return FALSE; 16: end if 17: for r = r i,j to r i,j+k 1 do 18: if ((r i,j+k r) % P i 1 = 0) then 19: // Found the shortest pattern. 20: P i (r, r i,j+k r); 21: return TRUE; 22: end if 23: end for 24: end whie T 1 : {C 1 =1, V 1 =3} T 2 : {C 2 =1, V 2 =7} T 3 : {C 3 =2, V 3 =14} Time (a){t 1 }'s pattern Time (b) {T 1, T 2 } spattern (c) {T 1, T 2, T 3 }'s pattern Repeating pattern Fig. 3. r 2,1 r 2,2 Repeating pattern Repeating pattern r 3,1 r 3,2 r 3,3 Iustration of the scheduabiity test agorithm Time

8 8 T 1 : {C 1 =1, V 1 =5} T 2 : {C 2 =1+d, V 2 =9} Ag 5 Deferrabe Scheduing with Eariest Deadine First Fig d 3-2d p 3-3d (a) A schedue without a repeating pattern T 1 : {C 1 =p, V 1 =3p} 4p 6p 8p T 2 : {C 2 =1, V 2 =5e} Repeating pattern (b) A schedue with a repeating pattern 25 Time Time DS-FP schedues for transaction sets with rea number parameters a transaction set with finite number of transactions and finite rea number parameters that has a successfu DS-FP schedue without any repeating pattern that must occur at east once 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 III.6: Define two rea numbers p = π = and e = Consider a set of two transactions {τ 1, τ 2 } with computation times p, 1, and vaidity intervas 3p, 5e, respectivey. Figure 4 (b) depicts the DS-FP schedue of the transaction set with a repeating pattern. IV. DEFERRABLE SCHEDULING WITH DYNAMIC PRIORITY ASSIGNMENT DS-FP is a fixed priority scheduing agorithm and it imits its appications for the systems which require to use dynamic priorities for scheduing their jobs. To overcome this shortcoming, in this section we present a dynamic scheduing agorithm using deferrabe scheduing. The new agorithm is caed Deferrabe Scheduing with Eariest Deadine First (DS-EDF) in which the Eariest Deadine First (EDF) scheduing poicy is adopted for assigning priorities to the update jobs [27]. Simiar to DS-FP, in DS-EDF, the reease time of a job is cacuated backwards from its deadine such that the period for generating a job is in genera onger than ML which uses the worst-case execution to determine the period for generating a job from an update transaction. Therefore, it is expected that the tota update workoad resuted from DS-EDF is aso ower than ML but it is more adaptive in scheduing as it uses the deadines of the jobs to determine how to schedue the set of jobs. The performance comparison of DS-EDF with DS-FP and ML is reported in Section VI. In the foowings, we concentrate on the scheduabiity anaysis of DS-EDF using the pattern anaysis technique introduced in the previous section. Simiar to the discussion on DS-FP, it is assumed that transactions are studied in a discrete time system to simpify the discussions. Ag. 5 presents the detais of DS-EDF in which update jobs to be schedued are queued in an EDF queue, i.e., Q EDF, in the ascending order of the deadines of the jobs. There is aways one job per each transaction avaiabe for scheduing in the Q EDF. Input: A set of update transactions T = {τ i} m i=1 with known {C i} m i=1 and {V i } m i=1 Output: Construct a partia schedue S EDF if T is feasibe; otherwise, reject. 1: Enqueue a first jobs J i,0 of τ i (i = 1,..., m) to Q EDF in the ascending order of V i ; 2: whie TRUE do 3: Dequeue the first job J i,k from Q EDF ; 4: if k == 0 then 5: t = 0; 6: ese 7: t = d i,k 1 ; 8: end if 9: r i,k = CacReeaseTime(i, k, t, d i,k ); 10: if r i,k < t then 11: return FAILURE; 12: end if 13: d i,k+1 = r i,k + V i; 14: Enqueue J i,k+1 to Q EDF in the ascending order of deadines; 15: end whie Ag 6 CacReeaseTime(i, k, t s, t e ) Input: J i,k and time interva [t s, t e ). Output: Reease time of J i,k, r i,k. 1: C R = C i; // C R is the remaining execution time of J i,k. 2: r i,k = t e ; 3: whie r i,k t s do 4: if Time sot r i,k is not schedued then 5: Schedue time sot r i,k for J i,k 6: C R ; 7: end if 8: if C R == 0 then 9: return r i,k ; 10: end if 11: end whie 12: return FAILURE; Simiar to DS-FP, the reease time r i,k of job J i,k is cacuated in CacReeaseTime(i, k, t, d i,k ), where t is the deadine of job J i,k 1 or time 0 if k = 0, and d i,k is the deadine of the current job. Once r i,k is computed from Ag. 6 (Line 9), the deadine of its next job J i,k+1 is computed (Line 13), and J i,k+1 is enqueued in Q EDF (Line 14). In Ag. 6, CacReeaseTime(i, k, t s, t e ) computes the time sots taken by job J i,k in time interva [t s, t e ), which returns the reease time r i,k of J i,k, i.e., the eariest time sot that can be taken by J i,k as r i,k is computed backwards from d i,k. The worst-case time compexity of CacReeaseTime is O(V max ) where V max = max i {V i }. Aso, the time compexity of enqueue and dequeue operations can be O(n m) if a priority queue is used. Thus, the time compexity of the whie oop in DS-EDF (Line 2 to Line 15) is O(V max + n m). Note that the space compexity of DS-EDF can be maintained O(V max ) because schedue information that is not usefu for the cacuation of job reease times can be discarded. Overa, the worst-case time compexity of DS-EDF is much ower than that of DS- FP, which is O(m V 2 max) [14] for cacuating reease time r i,k aone. Therefore, DS-EDF has ess scheduing overhead than DS-FP.

9 9 V. DS-EDF SCHEDULABILITY ANALYSIS In this section, we first prove that for each update transaction set which is scheduabe by DS-EDF, there aways exists a repeating pattern. We then introduce a search agorithm to find the eariest and shortest pattern in DS-EDF schedue. An agorithm for testing the scheduabiity of DS-EDF is aso presented. The foowing emma provides an upper bound on the distance between the finish times of any two consecutive jobs in DS-EDF. Lemma V.1: The distance between the finish times of any two consecutive jobs in DS-EDF is no arger than d min = min i {V i C i }. Proof. Assume that d min = min i {V i C i } = V k C k and two consecutivey finished jobs in DS-EDF, J i,p and J j,q have the smaest distance between their finish times which are f i,p and f j,q respectivey. We prove the emma by contradiction. Suppose that f j,q f i,p > V k C k. Since J i,p and J j,q are consecutivey finished jobs, there is no job from τ k which finishes in the interva (f i,p, f j,q ). Let J k,s denote the ast job of τ k finished before f i,p and J k,s+1 denote the first job of τ k finished after f j,q. According to DS-EDF, we have r k,s f k,s C k and f k,s+1 r k,s f k,s+1 f k,s + C k f j,q f i,p + C k > V k C k + C k = V k. This vioates the vaidity constraint of τ k. Therefore it must not be true that f j,q f i,p > V k C k, i.e., the distance between the finish times of any two consecutive jobs in DS-EDF is no arger than d min = min i {V i C i }. Theorem V.1: Given an update transaction set T with known C i and V i (1 i m), if it can be schedued by DS-EDF, then the DS-EDF schedue has a fixed repeating pattern that must occur at east once in the bounded time interva m [V m, V m + (2 (V i C i ) + 1) (V i C i + 2) d min ] i=1 Proof. The proof is incuded in Appendix II due to space imit. According to Theorem V.1, if a transaction set can be schedued by DS-EDF in the interva [V m, V m + m i=1 (2 (V i C i ) + 1) (V i C i + 2) d min ], then it is scheduabe by DS-EDF because a fixed pattern appearing in the interva repeats itsef forever. Thus we have the foowing coroary. Coroary V.1: An update transaction set can be schedued by DS-EDF if and ony if it can be schedued by DS-EDF in the interva [V m, V m + m i=1 (2 (V i C i ) + 1) (V i C i + 2) d min ]. A. DS-EDF Pattern Search Agorithm Theorem V.1 proves the existence of a repeating pattern for a given DS-EDF schedue. Now, we present the pattern search agorithm to find the eariest and shortest pattern in the schedue. The agorithm is summarized in Ag. 7 and it foows the idea in Theorem V.1 that if the transaction set T has the same state at two time points which are the deadines of two different jobs, the interva between them forms a pattern and wi repeat it forever. Starting from time V m, we compare the state of the transaction set T at each time point t which is a certain job s deadine backward to those time points in [V m, t) which are aso deadines of certain jobs. Theorem V.1 guarantees that either we can find a repeating pattern by checking at m most i=1 (2 (V i C i ) + 1) (V i C i + 2) time points or the transaction set T is unscheduabe by DS-EDF. Ag 7 Pattern Searching Agorithm for DS-EDF 1: Input: A successfu DS-EDF schedue S. 2: Output: The eariest and shortest DS-EDF pattern P. 3: 4: k 0; 5: Sort the job deadines at and after V m in S in ascending order, {t 0, t 1, t 2,..., t k,...} where t 0 = V m 6: N max 1 + m i=1 (2 (V i C i ) + 1) (V i C i + 2) 7: 8: whie k N max do 9: k k + 1; 10: for h = k 1 to 1 do 11: // Suppose that t k is the deadine of job d i,j 12: if t h d i,j 1 and S Q T (t h) = S Q T (t k) then 13: // Find the shortest pattern in S 14: P = (P s, P ) [t h, t k ) 15: whie S Q T (P s 1) = S Q T (P s 1 + P ) do 16: P (P s 1, P ) 17: end whie 18: // Find the eariest pattern in S 19: return P; 20: ese 21: continue; 22: end if 23: end for 24: end whie 25: return No pattern found; Given a repeating pattern P = (P s, P ), foowing the fact that a transactions have the same states at times P s + t and P s + t + P for t > 0, Coroary III.3 sti hods for DS-EDF patterns. Theorem V.2: pattern. P returned by Ag. 7 is the eariest and shortest Proof: We prove the theorem by contradiction. Suppose the pattern returned by Ag. 7 is P = (P s, P ) and there exists another pattern P = (P s, P ) with P < P. Assuming P s + (k 1) P < P s P s +k P (k 1), foowing Coroary III.3 we have S Q T (P s + k P ) = S Q T (P s + k P + P ). This further derives S Q T (P s) = S Q T (P s + P ) and thus [P s, P s + P ) is aso a pattern. However, this pattern shoud be found in Line 12 in Ag. 7 before pattern P is found. So P must be the shortest pattern. Simiary, suppose there exists another pattern P = (P s, P ) with P P and P s +P < P s +P. We can derive [P s, P s + P ) is aso a pattern which shoud be found before pattern P. So P must aso be the eariest pattern. This finishes the proof. B. DS-EDF Scheduabiity Test Agorithm The scheduabiity test agorithm for DS-EDF is more compicated than its pattern search agorithm. The input of the pattern search agorithm (Ag. 7) is a successfu schedue and at each time point t k, the state of the transaction set T is avaiabe for comparison. However, when we test the scheduabiity of T in the run time, the state information may not be compete. This is because at the job execution time, the reease times of the jobs in its Q EDF may have not been derived yet. For this reason, the scheduabiity test at time t k (k 1) has to be deayed unti

10 10 Ag 8 Scheduabiity Test Agorithm for DS-EDF 1: Input: A transaction set T. 2: Output: Whether T is scheduabe by DS-EDF. 3: 4: k 0; 5: Sort the job deadines at and after V m in S in ascending order, {t 0, t 1, t 2,..., t k,...} where t 0 = V m 6: N max 1 + m i=1 (2 (Vi Ci) + 1) (Vi Ci + 2) 7: N ex i dmax V i C i 8: I c 0; 9: 10: whie I c N max do 11: I o I c ; 12: if k < N max + N ex then 13: k k + 1; 14: // Suppose that t k is the deadine of τ i,j and t c is the eariest time point arger than t k (V i C i ) 15: Schedue τ i,j and derive its reease time r i,j 16: if r i,j < d i,j 1 then 17: // Job τ i,j is not scheduabe 18: return FALSE; 19: ese 20: for h = c to k do 21: Adding r i,j into S Q T (t h) 22: if S Q T (t h) is compete then 23: I c h; 24: end if 25: end for 26: end if 27: if I c I o then 28: for r = I o to I c do 29: // Suppose that t r is the deadine of τ p,q 30: for s = r 1 to 0 do 31: if t s d p,q 1 and S Q T (t s) = S Q T (t r) then 32: // Find the shortest pattern in S 33: P = (P s, P ) [t s, t r ) 34: return TRUE; 35: end if 36: end for 37: end for 38: end if 39: end if 40: end whie a the state information at time t i (1 i k) is avaiabe. To make sure that the job that finishes at t Nmax has compete state information, we need to execute more jobs after time t Nmax. Let us use N ex to denote this number and d max = max i {V i C i }, we have N ex i dmax V i C i. Ag. 8 presents the framework of the DS-EDF scheduabiity test agorithm. The agorithm records the state information of the transaction set T at the finish time of each executed job and keeps an index k for the current job to be executed. It aso maintains an index I c to record the atest state whose information is compete. In Ag. 8, every time it schedues a job, it derives its reease time backwards from its deadine (Line 15). If the job is not scheduabe, the transaction set T does not pass the scheduabiity test and a faiure wi be reported (Line 18). Otherwise, it wi insta the derived reease time to previous states whose information is incompete and update the index I c accordingy (Line 20 to Line 25). If a new state with compete information is identified (Line 27), it wi be compared with previous states for pattern searching (Line 28 to Line 37). This process continues unti either a faiure is reported or a pattern is found. Parameter Cass System Update Transactions Appication Transactions Parameter Meaning Vaue N CP U No. of CPU 1 N T No. of rea-time data objects [1, 18] V i Vaidity interva of X i [20, 200] C i CPU time for updating X i [1, 10] Length No. of data to update 1 CPU Time CPU time per data access [1, 5] Arriva Rate Transaction arriva rate [20, 50] Sack Factor Transaction sack factor 8 TABLE II Experiment Parameters VI. PERFORMANCE EVALUATION AND DISCUSSIONS In this section, we summarize the important resuts from our simuation studies on the performance and scheduabiity of DS-FP and DS-EDF. In the experiments, we compare their performance with ML as a baseine for reference. A. Simuation Mode and Parameters The simuation mode is deveoped foowing the mode used in [14] which is for typica rea-time database systems such as in the avatar cyber-physica project [6]. It is a singe CPU system with a main memory rea-time database. Tabe II summaries the mode parameters and defaut settings for the experiments. To verify the mode, we test it with different settings. The baseine vaues of the parameters are scaed down from those used in [14] for the effective demonstration of the scheduabiity of DS-FP and DS-EDF. If a arge number of update transactions is used, the pattern ength coud be very ong. Since the main goa of the experiments is to study how the pattern changes in DS- FP and DS-EDF and effectiveness of the scheduabiity testing agorithms, a sma set of update transactions wi be sufficient provided that the workoad is heavy enough. As shown in Tabe II, the number of rea-time data objects is varied from 1 to 18 and the vaidity interva of a data object is uniformy distributed between 20 and 200ms. Simiar to the assumption used in previous work [9], [14], each update transaction updates one data object. The worst-case execution time for each update job is uniformy distributed between 1ms and 10ms, and its actua execution time is uniformy distributed between 1ms and its worst-case execution time. For the appication transactions, the worst-case execution time is 5ms and its actua execution time is aso assumed to be uniformy distributed between 1 and 5ms. Foowing the definition in [14], the sack factor determines the sack of an appication transaction before its deadine and it is fixed at 8. An appication transaction wi be aborted if it misses its deadine. Let AT (τ i ), ET (τ i ) and Deadine(τ i ) be the arriva time, execution time and deadine of an appication transaction τ i, respectivey. The deadine of τ i is then cacuated as: Deadine(τ i ) = AT (τ i ) + (ET (τ i ) SackF actor) (2) Foowing the definition in [29], we define the density factor for a set of update transactions T as m C i i=1 V i. The primary performance metrics used in the experiments are the CPU utiization, the success ratio of scheduabiity, the average pattern ength, and the success rate and average response time of the appication transactions. The success ratio of scheduabiity is the number of update transaction sets which can be successfuy schedued over the tota number of randomy generated transaction sets. This is used to compare the average scheduabiity among

11 11 CPU Utiization ML (Genera) ML (WCET) DS-FP (Genera) DS-FP (WCET) DS-EDF (Genera) DS-EDF (WCET) Number of Transactions Average Response Time ML (Genera) ML (WCET) DS-FP (Genera) DS-FP (WCET) DS-EDF (Genera) DS-EDF (WCET) Number of Transactions Success Ratio ML (Genera) ML (WCET) 0.60 DS-FP (Genera) DS-FP (WCET) 0.55 DS-EDF (Genera) DS-EDF (WCET) Number of Transactions Fig. 5. CPU utiization vs. No. of update trans. Fig. 6. Avg. response time of appication trans. Fig. 7. Success rate of appication trans. different scheduing agorithms experimentay. The success rate of appication transactions is the number of appication transactions competed before their deadines over the tota number of appication transactions generated. B. Expt. 1: Comparison of CPU Utiization In the first set of experiments, we quantitativey compare the CPU utiization of scheduing the same set of update transactions using DS-FP, DS-EDF and ML in both the worst-case and genera case scenarios. In the worst-case scenario, c i,j, the execution time of J i,j, is aways set to C i. In the genera case, c i,j is chosen using a uniform distribution between 1 and C i. In the experiments, we increase the number of update transactions in steps of one. For each given transaction set size, we randomy generate 10,000 transaction sets according to the parameter settings summarized in Tabe II. Ony those transaction sets that are scheduabe under a the three approaches (ML, DS-FP and DS-EDF) are taken into account to cacuate the average CPU utiization incurred by the update transactions. Since under the current parameter settings, if the number of transactions goes over 18, it is very difficut to find transaction sets that can be schedued by a the three approaches especiay for ML, we ony show the experiment resuts when there is no arger than 18 update transactions in the system. As shown in Figure 5, the CPU utiization of both DS-FP and DS-EDF is consistenty ower than that of ML in both the worsecase and genera case scenarios especiay when the number of update transactions is arger. The improvement over ML reaches about 19.5% in the worse-case and 23.5% in the genera case when there are more than 15 update transactions in the system. In addition, consistent with our expectation, the CPU utiization of the three methods in the genera case is around haf of that in the worst-case. This is because, in the genera case, athough ML, DS-FP and DS-EDF sti assign the worst-case execution time to a job J i,j to keep the pattern, J i,j s actua execution time, c i,j, foows the uniform distribution between 1 and C i. Thus, the CPU wi be ide after competing a job unti its assigned time sots expire. Another important observation from Figure 5 is that the CPU utiization of DS-EDF is consistenty ower than that of DS-FP and ML when the system workoad is heavy. When there are 18 update transactions in the system (the CPU workoad incurred by the ML approach in the worstcase scenario reaches 0.77%), the improvement of DS-EDF over DS-FP and ML in the worst-case scenario is 3.27% and 19.5%; whie in the genera case scenario, the improvement further reaches 8.18% and 23.5% respectivey. This indicates that the performance of DS-EDF is better than DS-FP as we as ML in terms of CPU utiization as it adaptivey schedues the jobs based on their urgencies according to their deadines. The number of update transactions in the experiments is reativey sma under the current parameter settings. This is for the ease of setting up the scheduabiity comparison experiments which require the pattern anaysis to be presented in Section VI-C and VI-D. Since an exact pattern anaysis is computationay intensive and we ony need it for the simuation comparison, we keep the number of distinct parameter sets sma to save time. Notice that this does not restrict the number of update transactions in a rea appication since different update transactions can have the same vaidity interva parameter; what matters for the scheduabiity anaysis is the variation in the vaidity interva which we seect from a uniform distribution. We conducted additiona experiments by enarging the vaidity intervas and varying the worst-case execution time of the update transactions, thus increase the number of update transactions in the experiments. Our resuts show that the effectiveness of the DS-EDF agorithm in reducing the CPU utiization over DS-FP and ML is consistent and reativey insensitive to the parameter setting. For this reason and due to the page imit consideration, we do not incude them in the paper. To further compare the performance of DS-EDF with ML and DS-FP, we repeat the experiments with the introduction of appication transactions into the system. The appication transactions are assigned ower priorities for execution compared with the update transactions. Note that they are firm rea-time transactions. Thus, if they miss their deadines, they wi be aborted immediatey. So, the average response time is cacuated using those appication transactions which are competed before their deadines. Consistent with the resuts shown in Figure 5, as shown in Figure 6 and Figure 7 respectivey, the appication transactions in the genera case have much ower average response times and higher success rates compared with that in the worst-case scenario. This is because the system workoad in the genera case scenarios is roughy haf of that in the worstcase scenarios. In the genera case scenario, the success rates of the appication transactions under DS-EDF and DS-FP are amost the same, and compared with DS-FP, ower average response time of appication transactions under DS-EDF can be observed in Figure 6 because DS-EDF incurs ower CPU workoad. When the update workoad is heavy, i.e., the number of update trans-

12 12 actions is more than 10, DS-FP and DS-EDF show significant improvement over ML in both average response time and success rate of the appication transactions. For instance, when there are 18 update transactions in the system, the success rate of the appication transactions under DS-EDF and DS-FP is around 6.84% higher than that under ML, and the average response time of the appication transactions under DS-EDF is 22.5% shorter than that of ML. It is because the CPU workoad of the update transactions of DS-FP and DS-EDF is much ower than that of ML. Therefore, more CPU times can be aocated to the appication transactions and improve their performance. In the worst-case scenario, the improvement in performance of DS-FP and DS-EDF over ML are more significant when the number of update transactions is arge. When there are 18 update transactions in the system, the improvement on appication transaction success rate from DS-EDF and DS-FP over ML are 40.36% and 15.18% respectivey. The improvement on the average response time of the appication transactions under DS-EDF and DS-FP aso reaches 19.14% and 24.1% respectivey. It is aso interesting to see in Figure 6 that when the system workoad is heavy, for instance, when there are more than 12 update transactions in the system, the average response time of appication transactions under DS-FP is ower than that of DS-EDF. This is because in this scenario, the success rate of appication transactions under DS-EDF is significanty better than that of DS-FP as shown in Figure 7. Thus more appication transactions are abe to meet their deadines in DS- EDF compared with DS-FP due to the use of the eariest deadine first scheduing. Therefore, the tota system workoad is heavier and makes their average response time arger. C. Expt. 2: Comparison of Scheduabiity In the second set of experiments, we compare the success ratio of scheduabiity of DS-FP, DS-EDF and ML under various CPU utiizations by changing the density factor. Figure 8 depicts the success rate of scheduabiity of DS-EDF, DS-FP and ML when the density factor is varied from 0.35 to 0.65 for a system consisting of 5 update transactions with execution time aways set to the worst-case vaue. The increase in density factor is achieved by fixing C i and decreasing vaue for V i. We have conducted 1,000 runs for each setting and present the average vaues from them in the figure. As shown in Figure 8, DS-FP consistenty outperforms ML in terms of success ratio of scheduabiity. The resuts are consistent with the caim in Theorem II.1 which states that a transaction sets that are successfuy schedued by ML are aso scheduabe by DS-FP. The success ratio of ML drops beow 0.85 when the density factor is This happens to DS-FP ony when the density factor is about 0.6. Aso, when the density factor is 0.63, most of the transaction sets cannot be schedued by ML whie the success ratio of DS-FP is sti around Athough the performance of DS-EDF is better in terms of CPU utiization as shown in the first set of experiments, it is a bit surprising to see that the success ratio of scheduabiity of DS-EDF is quite varied for different vaues of density factor as shown in Figure 8. For ow density factor vaues, its scheduabiity is simiar to both ML and DS-FP as the workoad is ight. However, when the density factor is more than 0.45, its scheduabiity varies a ot and is even ower than that of ML. Success Ratio Fig. 8. ML DS-FP DS-EDF Density Factor Success ratio of scheduabiity vs. Density factor On the contrary, when the density factor is more than 0.56, its scheduabiity becomes significanty higher than that of ML but is sti ower than that of DS-FP. The resuts indicate that athough the resuting CPU utiization of DS-EDF is ower than both DS-FP and ML, its scheduabiity is worse than DS-FP and ony better than ML when the update workoad is heavy. The main reason for the poor and unstabe scheduabiity of DS- EDF as compared with DS-FP is that introducing a dynamic scheduing mechanism to schedue update jobs using eariest deadine first may make some jobs miss the deadines after being served by the CPU for a ong time. This is consistent with the genera performance of the eariest deadine first scheduing [27]. D. Expt. 3: Pattern Anaysis of DS-FP and DS-EDF In the fina set of experiments, we compare the pattern engths (caed practica pattern engths) of the DS-FP and DS- EDF obtained from the scheduabiity test agorithms (Ag. 1 and Ag. 7) and the pattern engths (caed the theoretica upper bound) obtained from the theoretica anaysis (Coroary III.2 and Coroary V.1). A shorter pattern ength indicates a more effective scheduabiity test agorithm and a faster pattern searching agorithm to find the eariest and shortest pattern in the constructed schedue. The parameter settings of the experiments are the same as those in Expt. 2. Figure 9 compares the theoretica upper bounds and the corresponding practica pattern engths of DS-FP and DS-EDF. As shown in Figure 9, the theoretica upper bounds of both DS- FP and DS-EDF are very arge compared with the corresponding practica pattern engths. When the density factor is 0.5, the ratio between the theoretica upper bound ( ) and the practica pattern ength ( ) of DS-FP is about The ratio for DS-EDF is even arger as can be cacuated from the resuts. The reason for the great difference ies in the fundamenta principes of DS-FP and DS-EDF. In DS-FP as we as in DS-EDF, each job J i,j cacuates the reease time r i,j backwards from its deadine d i,j. This mechanism can easiy generate bocks. Each bock is a chunk of continuousy occupied time sots in a DS-FP or DS-EDF schedue. For instance, in DS-FP, given a detected pattern, P i 1, from transaction set τ 1, τ 2,..., τ i 1, if there are two jobs of τ i, J i,j and J i,k, whose deadines have different offsets but ie in the same bock in different occurrences of pattern P i 1, their reease times shoud

13 13 Pattern Length DS-FP Practica Pattern DS-FP Theoretica Bound DS-EDF Practica Pattern DS-EDF Theoretica Bound Density Factor Pattern Length Density Factor DS-FP DS-EDF Pattern Length DS-FP Practica Pattern DS-FP Theoretica Bound DS-EDF Practica Pattern DS-EDF Theoretica Bound Number of Transactions Fig. 9. Theoretica bound vs. Practica pattern ength Fig. 10. Practica pattern ength vs. Density factor Fig. 11. Pattern ength vs. No. of update tran. have the same offset in the pattern and a new pattern P i is detected. In this manner, a pattern that is much shorter than the theoretica anaysis can be detected. Simiar to the theoretica upper bound shown in Figure 9, we observe that the practica pattern engths of DS-FP and DS-EDF decrease graduay with an increase in density factor as shown in Figure 10. This is because an increase in density factor is achieved by decreasing V i, which decreases the ength of the patterns for DS-FP and DS-EDF. In addition, the pattern engths of both theoretica and practica patterns of DS-EDF are both arger than that of DS-FP. As shown in Figure 10, the practica pattern ength of DS-EDF varies a ot with an increase in density factor athough the genera trend is decreasing. This is because the dynamic priority scheduing in DS-EDF introduces addition variations into the patterns making them to have onger and ess stabe pattern engths. We aso evauate the pattern engths of DS-FP and DS-EDF by varying the number of update transactions. Figure 11 shows the comparison of the practica pattern ength and theoretica upper bound when the number of update transactions is varied and the density factor is fixed at 0.6. As shown in Figure 11, the theoretica upper bounds of both DS-FP and DS-EDF increase dramaticay with an increase in number of update transactions and are greaty arger than their corresponding practica pattern engths. Aso, the practica pattern ength of DS-EDF is consistenty arger than the corresponding practica pattern ength of DS-FP confirms the beief that introducing dynamic scheduing makes the pattern ength onger. Summary of Experiment Resuts. It is demonstrated in the experiment resuts that both DS-EDF and DS-FP give a better performance in terms of CPU utiization compared with ML. With the scheduabiity test agorithms, it is shown that the scheduabiity of DS-FP is much better than that of ML [14]. In addition, the practica pattern engths of both DS-FP and DS-EDF obtained from the experiments are significanty s- maer than the corresponding theoretica upper bounds. This observation demonstrates that the agorithms are effective for the scheduabiity tests of DS-FP and DS-EDF. Athough the scheduabiity of DS-EDF is ower than that of DS-FP and its patterns are onger than DS-FP due to the use of dynamic scheduing, its scheduabiity is sti significanty better than that of ML in most cases especiay when the update workoad is heavy. Furthermore, the performance of DS-EDF is better than DS-FP and ML in terms of CPU utiization and its impact on the performance of appications is smaer. VII. CONCLUSIONS AND FUTURE WORK Athough DS-FP gives a better performance in maintaining tempora vaidity of rea-time data by adopting the aperiodic update mode in update job generation, this poses a great chaenge for its scheduabiity test. In this paper, we address the scheduabiity issue by deriving the necessary and sufficient scheduabiity conditions for DS-FP. We prove the existence of a repeating pattern for any successfu DS-FP schedue, and derive a scheduabiity test agorithm for DS-FP. Since DS-FP is a fixed priority scheduing method, we extend it to dynamic scheduing with the proposa of DS-EDF which uses the eariest deadine first poicy to schedue update jobs. We have shown that the compexity of DS-EDF is much ower than that of DS- FP. Based on the pattern anaysis method used for checking the scheduabiity of DS-FP, we design the scheduabiity test agorithm for DS-EDF. In the performance evauation studies, it is demonstrated that the scheduabiity testing agorithms are effective to check the scheduabiity of DS-FP and DS-EDF. In the experiments, we aso observe that the performance of DS- EDF is better than both DS-FP and ML in terms of utiization and impact on ower priority appication transactions. However, DS-FP outperforms DS-EDF in terms of scheduabiity and the pattern ength of DS-EDF is onger compared with DS-FP due to the use of dynamic scheduing. Note that in comparison with ML, both DS-FP and DS-EDF are stricty better in the genera case where the actua execution time is ess than the worst-case. In our case, the scheduer must wait for the sack time to expire before continuing with the pre-computed schedue. This does not mean that ML is better inasmuch as the scheduabiity anaysis requires the worst-case execution time in order to guarantee that no deadine wi be missed for ML, DS-FP and DS-EDF. So there is no advantage for ML to guarantee hard deadines. On the other hand, if the comparison is made for soft rea-time systems, then the resuts in this paper show that our agorithm aso outperforms in terms of better scheduabiity and ower CPU workoad. It wi be usefu to study the scheduabiity probem of tempora consistency on mutiprocessor patforms. To our knowedge, very few studies have been conducted in this area. Recenty, Li et. a. [30] investigated the issue of workoad-aware partitioning for maintaining tempora consistency under EDF. The scheduabiity and performance of tempora consistency maintenance agorithms on mutiprocessor patforms is eft as our future work.

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Grimadi, Discrete and combinatoria mathematics: An appied introduction, Addison-Wesey, [29] Ming Xiong, Qiong Wang, and Krithi Ramamritham, On eariest deadine first scheduing for tempora consistency maintenance, Rea- Time Systems, vo. 40, no. 2, pp , [30] Jianjun Li, Jian-Jia Chen, Ming Xiong, and Guohui Li, Workoadaware partitioning for maintaining tempora consistency on mutiprocessor patforms, in Proc. of Rea-Time Systems Symposium, Song Han received the BS degree in computer science from Nanjing University, Peopes Repubic of China in 2003 and the M.Phi degree in computer science from City University of Hong Kong in He is currenty a PhD candidate in the Department of Computer Science at the University of Texas at Austin. His research interests incude cyber-physica systems, rea-time and embedded systems, database systems and wireess networks. He is a student member of the IEEE. Deji Chen received his PhD degree in Computer Science from the University of Texas at Austin in He is currenty a senior principa software engineer at Emerson Process Management. His research interests incude rea-time systems and wireess process contro. He co-authored the book WireessHART - Rea-time Mesh Network for Industria Automation. He is a member of the IEEE and the IEEE Computer Society. Ming Xiong is currenty a Member of Technica Staff at Googe Inc. He hods the BS degree in computer science and engineering from Xian Jiaotong University, the MS degree in computer science from Sichuan University, and the PhD degree in computer science from University of Massachusetts, Amherst. From 2000 to 2009, he is a Member of Technica Staff at Be Laboratories Research, Lucent Technoogies. His research interests incude rea-time systems, database systems and mobie computing. Kam-Yiu Lam received the BSc (Hons; with distinction) degree in computer studies and the PhD degree from City University of Hong Kong in 1990 and 1994, respectivey. He is currenty an associate professor in the Department of Computer Science at City University of Hong Kong. His research interests incude rea-time database systems, rea-time active database systems, mobie computing, and distributed mutimedia systems. Aoysius K. Mok received the BS degree in Eectrica Engineering, the MS degree in Eectrica Engineering and Computer science, and the PhD degree in Computer Science, a from the Massachusetts Institute of Technoogy. He is the Quincy Lee Centennia Professor in Computer Science at the University of Texas at Austin. His current interests incude rea-time and embedded systems, robust and secure network centric computing, and rea-time knowedge-based systems. He is a member of the IEEE. Krithi Ramamritham received the PhD degree in computer science from the University of Utah and then joined the University of Massachusetts. He is currenty at IIT Bombay as a professor in the Department of Computer Science. He has served on numerous program committees of conferences and workshops. His editoria board contributions incude IEEE Transactions, the Rea Time Systems Journa, and the VLDB Journa. He is a feow of the IEEE and a feow of ACM.