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

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1 1 Scheduabiity Anaysis of Deferrabe Scheduing Agorithms for Maintaining Rea- 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 comparing with the More-Less (ML) method, it is sti 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. Instead of using fixed priority scheduing as adopted in DS-FP, we extend the deferrabe scheduing to be a dynamic priority scheduing agorithm caed DS-EDF by appying the eariest deadine first (EDF) poicy to schedue update jobs. We propose a scheduabiity test agorithm for DS-EDF and compare its performance with DS-FP and ML through extensive simuation experiments. The resuts show that the scheduabiity test agorithms are effective. Athough the scheduabiity of DS-EDF is ower than DS-FP and the ength of 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 impacts to 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 rea-time data services. These samped data, usuay caed rea-time data, wi become invaid with the passage of time since the status of the corresponding entities in the rea-word may change continuousy. One of the commonest way to determine the tempora vaidity of rea-time data objects is to define a vaidity interva [2] or age constraint [3] 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. For this case, 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 the entity in the operation environment. The typica method for maintaining the tempora vaidity of a set of rea-time data objects is to use periodic update transactions. One of such basic methods is the More-Less (ML) [3], [4]. 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 [5] [8] which are 1 An earier version of this paper appeared in [1]. mosty proposed for soft rea-time systems where the arrivas of update transactions are 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 deterministic guarantee is provided on the vaidity of the set of rea-time data objects maintained in a rea-time database. Athough ML has been shown to be effective in maintaining data vaidity, the resuting update workoad coud be heavy. In [9], 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 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, sacrificing the periodicity in update generation poses great chaenges for its scheduabiity anaysis. One prominent method in cassica scheduabiity anaysis is based on the critica instant test [4]. A critica instant makes sense for periodic tasks by assuming synchronous update tasks to generate their first jobs 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 sets that are scheduabe by ML [9]. However, the truth of the converse is sti an open probem. This sufficient condition has seriousy restricted the scheduabiity of a transaction set by DS-FP. Athough experimentay it has been demonstrated that DS-FP outperforms ML significanty [9], 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) is sti unsoved. Furthermore, DS-FP is a fixed priority scheduing agorithm. This coud imit its appications for the systems which require to use dynamic priorities for scheduing their jobs. Thus, another important performance question on DS-FP is what the scheduabiity wi be if it is modified to use dynamic priority scheduing instead of fixed priority scheduing. A study on efficient methods for testing the scheduabiity of DS-FP is highy important for the design of further efficient scheduing methods for maintaining tempora vaidity of rea-time data objects [10]. Contribution. In this paper, we address the probem of finding the necessary and sufficient conditions for the scheduabiity test of DS-FP. We first prove that there aways exists a repeating pattern for a given DS-FP schedue in discrete time systems. Based on this finding, we propose a scheduabiity test agorithm for checking the scheduabiity of DS-FP. To

2 2 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. resove the imitation of fixed priority scheduing in DS-FP, we propose a new dynamic scheduing agorithm caed DS-EDF by adopting the eariest deadine first poicy in the deferrabe scheduing approach. By modifying the proposed scheduabiity test designed for DS-FP, we deveop another scheduabiity test agorithm for DS-EDF. We have shown that the compexity of DS-EDF is much ower than that of DS-FP. The performance and scheduabiity of DS-EDF as compared with DS-FP and ML is aso investigated through extensive simuation studies. Organization. The remainder of the paper is organized as foows. Section II briefy reviews the background and reated works. In Section III, we study the necessary and sufficient scheduabiity conditions of DS-FP in discrete time systems, and prove its correctness with the proposa of a scheduabiity test agorithm. In Section IV, we introduce a new deferrabe scheduing agorithm with dynamic priority assignment, caed DS-EDF. The scheduabiity anaysis of DS-EDF is presented in Section V. Section VI summarizes the experiment resuts on the performance and scheduabiity of both DS-FP and DS-EDF. We concude the paper and discuss the future works in Section VII. II. BACKGROUND AND RELATED WORKS In this section, we first briefy review the concept of tempora vaidity of rea-time data and the reated works. We then summarize the main principes of the More-Less (ML) [3], [4] and deferrabe scheduing (DS-FP) [9] agorithms. Forma definitions of the frequenty used symbos are summarized in Tabe I. A. Rea- Data Vaidity One of the core studies in rea-time data management is to maintain the tempora vaidity of rea-time data objects so that they are vaid representation of the current status of rea-word entities. In [2], 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, for its j th update finished atest 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 [2]. 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 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 [2], [11], [12]. 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, i.e., generating incorrect responses even though the responses can be produced timey [5], [13] [15]. There have been extensive research works on maintaining vaidity and freshness of rea-time data [1], [4], [9], [14], [16] [21]. Some of them use periodic update transactions whie the others assume the arrivas of update transactions are sporadic. The second types of methods, e.g., [5], [6], [8], [16], [22], are mainy designed for soft rea-time systems [13], and the main probem to be tacked is how to schedue update transactions in runtime to maximize the freshness of rea-time data objects whie minimizing their impacts to 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 deterministic guarantee on data vaidity for the execution of rea-time transactions. On the contrary, the performance goa of those methods using periodic update transactions is to provide a deterministic guarantee in data vaidity. The main probems to be 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 foowings, we briefy review two of the most representative methods: the More-Less (ML) and DS-FP. ML is the basic method using periodic update transactions whie DS-FP is an enhancement of ML showing a better performance. For the detais of the mechanisms of these two methods, readers may refer to [4], [9]. Note that in these two methods, the transactions from appications are assumed to be assigned to ower priorities in scheduing comparing with the update transactions so that the scheduing of the update transactions wi not be affected by the scheduing of the appication transactions. Athough this coud affect the performance of the appication transactions, it is important to provide a deterministic guarantee in data quaity for their execution. More-Less: More-Less (ML) [4] is an off-ine deterministic method. 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. With given vaidity interva of each data object, ML determines the periods and deadines for a set of update transactions and schedue them using the Deadine Monotonic (DM) method [23]. 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

3 3 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. In here, we briefy summarize the main steps of DS- FP 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 [9]. Theorem II.1: (Coroary 3.2 in [9]) 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 that patterns aways exist in the DS-FP schedues. 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 priority higher 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 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 Repeating pattern Two transactions that can be schedued by DS-FP but not by ML 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 V 2 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 19, which is smaer than V3 2 (i.e., 23.5). This is because J 2,2 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 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.

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 Three transactions that can be schedued by DS-FP but not by ML 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 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 [24]: 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 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 theorem can be proved by the foowing two caims using induction: 1) There is a pattern for τ 1 in the interva [V m C m, V m C m +V 1 C 1 ] which repeats itsef in the DS-FP schedue. 2) For any k, 1 k < m, if there is a pattern 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] which repeats itsef in the DS-FP schedue, then there is a pattern 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] which repeats itsef in the same DS-FP schedue. The first caim is obvious because there is a repeating pattern of ength (V 1 C 1 ) repeating itsef from time 0. As a matter of fact, any schedue of ength (V 1 C 1 ) is a repeating 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 repeating 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 instances of the recurring pattern 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; and 2) 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 repeating 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 repeating 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 repeating 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)

5 5 recurring patterns since time t. It foows from the Pigeonhoe Principe that there must be two jobs reease times at the same offset within their corresponding pattern instances. Denote t 1 and t 2 to be the two jobs reease times, and t 1 < t 2. We then have a repeating pattern that must occur at east once in the interva [t 1, t 2 ] repeating itsef thereafter for transactions τ 1,.., τ k, τ k+1. Since t 1 t (V m C m ) and t 2 < (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 some instance (not necessariy the same instance) 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 2 to be the two jobs reease times in DS-FP, and t 1 < t 2. We then have a repeating pattern that must occur at east once in the interva [t 1, t 2 ] repeating itsef for transactions τ 1,.., τ k, τ k+1. Since t 1 t (V m C m ) and t 2 < (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 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. 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 shortest and eariest 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 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 + 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

6 6 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 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 SearchNextT ask(i, P i 1 ); 9: end for 10: return P m ; Ag. 1 invokes Ag. 2 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. 2 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: maxl 1 + I i 1 ; 7: whie (k < maxl) 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; 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 shortest and eariest pattern. Proof. We sha prove that if the input to Ag. 2 is the shortest and eariest 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 ex-

7 7 amine 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 shortest and eariest pattern, so is the output. We aso know that Line 5 in Ag. 1 assigns the shortest and eariest pattern for τ 1. By induction, P m returned by Ag. 1 is the shortest and eariest 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 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, 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 is finished by 8, which is more than V3 2 = 7. Now we test whether it can be 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 shortest and eariest 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 = (8, 18) can be returned from Ag. 1. 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 (T estnextt ask(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: maxl 1 + I i 1 ; 11: whie (k < maxl) 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 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. F. DS-FP in Continuous Systems So far we assume a discrete time system 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.

8 8 T 1 : {C 1 =1, V 1 =3} T 2 : {C 2 =1, V 2 =7} T 3 : {C 3 =2, V 3 =14} T 1 : {C 1 =1, V 1 =5} T 2 : {C 2 =1+d, V 2 =9} Repeating pattern 3-d 3-2d 3-3d (a){t 1 }'s pattern (a) A schedue without a repeating pattern 25 r 2,1 r 2,2 Repeating pattern T 1 : {C 1 =p, V 1 =3p} T 2 : {C 2 =1, V 2 =5e} (b) {T 1, T 2 } spattern Repeating pattern Repeating pattern (c) {T 1, T 2, T 3 }'s pattern Fig. 3. r 3,1 r 3,2 r 3,3 Iustration of the scheduabiity test agorithm 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. 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 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 = Fig p 4p 6p 8p (b) A schedue with a repeating pattern DS-FP schedues for transaction sets with rea number parameters 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 probem, 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 [23]. Simiar to DS-FP, in DS-EDF, the reease time of a job is cacuated backwards from its deadine. 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 Deferrabe Scheduing with Eariest Deadine First 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 = CacReease(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 Agorithm 5 presents the detais of DS-EDF in which update jobs to be schedued are queued in an EDF queue, i.e., Q EDF,

9 9 Ag 6 CacReease(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 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; 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. Simiar to DS-FP, the reease time r i,k of job J i,k is cacuated in CacReease(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 Agorithm 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 Agorithm 6, CacReease(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 CacReease 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 in 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 Vmax) 2 [9] for cacuating reease time r i,k aone. Therefore, DS-EDF has ess scheduing overhead than DS-FP. 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. In DS-EDF, there is aways one job per update transaction avaiabe for scheduing in Q EDF. For the sake of simpicity, we denote tupe Sτ Q (t) = (rτ Q (t), fτ Q (t)) to be the state of transaction τ in Q EDF at time t, where rτ Q (t) and fτ Q (t) are the offsets of τ s job s reease time and finish time to t respectivey. Notice that rτ Q (t) is positive when the job s reease time is arger than t. Otherwise rτ Q (t) is zero or negative. fτ Q (t) is aways non-negative. We denote S Q T (t) = (RQ T (t), F Q T (t)) to be the aggregated states of a the transactions in T at time t. The theorem can be proved by the foowing two caims. There exist two time points, t 1 and t 2 in interva [V m, V m + m i=1 (2 (V i C i ) + 1) (V i C i + 2) d min ] so that t 1 and t 2 are two jobs finish times (not necessariy from the same transaction), and S Q T (t 1) = S Q T (t 2). The schedue in the interva [t 1, t 2 ] is a fixed pattern repeating itsef thereafter. The proof of the first caim reies on the Pigeonhoe Principe. At time t 1, assume that the job of update transaction τ i in Q EDF is τ i,j and its state is Sτ Q i (t 1 ) = (rτ Q i (t 1 ), fτ Q i (t 1 )). Since τ i,j is the first job of τ i to be finished after t 1 and τ i,j 1 is finished before t 1, fτ Q i (t 1 ) can ony be one of (V i C i + 1) possibe vaues (pigeonhoes) in [0, V i C i ]. Simiary, rτ Q i (t 1 ) can ony be one of 2 (V i C i ) possibe vaues (pigeonhoes) in [ V i + C i, V i C i ). As V m is the finish time of τ m,1, et us take V m as t 1 and ook at the finish times of m i=1 (2 (V i C i ) + 1) (V i C i + 2) consecutivey executed jobs from t 1. It foows from the Pigeonhoe Principe that there must exist a job s finish time t 2 such that S Q T (t 1) = S Q T (t 2). According to Lemma V.1, the distance between t 1 and t 2 is no arger than m i=1 (2 (V i C i ) + 1) (V i C i + 2) d min. Now et us prove the second caim. Assume that the sequence of executed jobs from t 1 and t 2 are S 1 = {J1 0, J1 1, J1 2,...} and S 2 = {J2 0, J2 1, J2 2,...}, and the finish times of the corresponding jobs are F 1 = {t 0 1, t 1 1, t 2 1,...} and F 2 = {t 0 2, t 1 2, t 2 2,...}. Here J1 0 and J2 0 are the currenty finished job at t 1 and t 2 and we have t 0 1 = t 1 and t 0 2 = t 2. Since t 2 aso beongs to F 1, there must exist a job J1 x in S 1 and J1 x = J2 0. We wi prove that at any time t k 1 and t k 2 (k 0), we have F Q T (tk 1) = F Q T (tk 2) and the schedues in [t 1, t k 1] and [t 2, t k 2] are the same after the execution of J1 k and J2 k. Proof for case k = 0 is obvious because Sτ Q i (t 0 1) = Sτ Q i (t 0 2) F Q T (t0 1) = F Q T (t0 2) and the intervas [t 1, t 0 1] and [t 2, t 0 2] are empty.