The Value of Even Distribution for Temporal Resource Partitions

Transcription

1 The Value of Even Distribution for Temoral Resource Partitions Yu Li, Albert M. K. Cheng Deartment of Comuter Science University of Houston Houston, TX, 7704, USA htt:// Technical Reort Number UH-CS February 5, 016 Keywords: Hierarchical Real-time Scheduling, Temoral Resource Partition, Real-time Task Scheduling, Resource Utilization, Schedulability Rate Abstract Most Hierarchical Real-time Scheduling (HiRTS) techniues have focused on temoral resource artitions which time units are eriodically distributed. Although such eriodic artitions could rovide great flexibility for resource artitioning, engineers are stuck when trying to determine the schedulability of real-time tasks running on them. The main reason is that eriodic artitions fail to effectively bound the difference between the ideal and the actual resource allocation. To solve this roblem, some researchers introduced the Regular Partition, a tye of temoral resource artition which is almost evenly distributed. Recent research has shown that it achieves maximal transarency for task scheduling. Some classical real-time scheduling roblems on a regular artition can be easily transformed into euivalent roblems on a dedicated single resource. However, the resource artitioning roblem for regular artitions is much more comlicated than the one for eriodic artitions. Based on a ractical -layer HiRTS latform, this aer first introduces new resource artitioning techniues for regular artitions. After that, it comares the overall erformance of the eriodic artition and the regular artition. We conclude that the regular artition is a better choice for the integration of real-time alications.

2 The Value of Even Distribution for Temoral Resource Partitions Yu Li, Albert M. K. Cheng Abstract Most Hierarchical Real-time Scheduling (HiRTS) techniues have focused on temoral resource artitions which time units are eriodically distributed. Although such eriodic artitions could rovide great flexibility for resource artitioning, engineers are stuck when trying to determine the schedulability of real-time tasks running on them. The main reason is that eriodic artitions fail to effectively bound the difference between the ideal and the actual resource allocation. To solve this roblem, some researchers introduced the Regular Partition, a tye of temoral resource artition which is almost evenly distributed. Recent research has shown that it achieves maximal transarency for task scheduling. Some classical real-time scheduling roblems on a regular artition can be easily transformed into euivalent roblems on a dedicated single resource. However, the resource artitioning roblem for regular artitions is much more comlicated than the one for eriodic artitions. Based on a ractical -layer HiRTS latform, this aer first introduces new resource artitioning techniues for regular artitions. After that, it comares the overall erformance of the eriodic artition and the regular artition. We conclude that the regular artition is a better choice for the integration of real-time alications. Index Terms Hierarchical Real-time Scheduling, Temoral Resource Partition, Real-time Task Scheduling, Resource Utilization, Schedulability Rate I. INTRODUCTION Aiming to integrate multile real-time alications onto one single hysical latform, hierarchical realtime scheduling (HiRTS) allows different real-time alications to share sace or time on one comutation resource. This roblem is increasingly imortant as oen systems [18] become more oular. Oen systems make it easy to add and remove software alications as well as to increase resource utilization and reduce imlementation cost when comared to systems which hysically assign distinct comutation resources to run different alications. In this aer, we focus on how to share time on one comutation resource based on a ractical -layer HiRTS system shown in Figure 1. A comutation resource could be a single resource or an identical multiresource, and it is temorally divided into a grou of Resource Partitions [], which are managed by a global resource-level scheduler. On each resource artition, the real-time tasks belonging to a real-time alication is scheduled by its own task-level scheduler. Fig. 1. A -Layer HiRTS System The resource artition is an intermediate layer in this -layer system, and each resource artition only uses a fraction of the time on the comutation resource. There are several HiRTS resource models. Tyically, each HiRTS model has its own definition of resource artition containing some arameters. For examle, a resource artition in the Periodic Model [10] exactly obtains c comutation time units in each eriod. A time unit is a 1

3 system-defined unit of time for the urose of scheduling, and there is no (either artition or task) reemtion in it. Therefore, we may assume that all time arameters in this aer are integers without loss of generality. The arameter values of a resource artition is called its signature. For examle, eriodic artition (10, 4) reresents a artition which obtains 4 time units in each eriod of 10 time units. The signature of a resource artition is the only sharing information between the global resource-level scheduler and its own task-level scheduler. The resource-level scheduler gathers all the artition signatures in the system and decides how to assign time units to the artitions. We call it the resource artitioning roblem in HiRTS. Meanwhile, on each resource artition, real-time tasks are usually migrated from a non-hierarchical real-time system where they were directly running on a dedicated comutation resource under a secific scheduling olicy, such as Earliest Deadline First (EDF) or Rate Monotonic (RM) [1]. In most cases, the original schedulability tests for a dedicated resource do not work for a resource artition which only reemts a fraction of the comutation time. Therefore, new task scheduling techniues have to be develoed. We call it the task scheduling roblem in HiRTS. Then, we start to address the roblems we shall discuss in this aer. At the resource level, a basic roblem is how to schedule resource artitions on a single resource. We use (P1) Res-Single to reresent this roblem. In the multiresource scenario, there are two dominating categories of scheduling algorithms: global scheduling and artitioned scheduling. Global scheduling allows resource artitions to migrate between different comutation resource units, while artitioned scheduling does not. They may dominate each other deending on different characteristics of comutation resources. Generally, if the migration overhead is relatively small, global scheduling could rovide higher efficiency; Otherwise, artitioned scheduling could erform better. Therefore, when considering the resource artitioning roblem, we need to investigate both global scheduling and artitioned scheduling. We call them (P) Res-Global and (P3) Res-Partitioned roblems resectively. For task scheduling, we shall discuss the schedulability roblems for several oular task models. One task model we consider is the Periodic Task Model, in which two successive reuests of each task are searated by exactly the same time interval, called its eriod. In this model, a task t i is denoted by (c i, i, d i, o i ), where c i, i, d i and o i are its execution time, eriod, deadline and offset, resectively. Most current HiRTS Models have investigated a simle case of the Periodic Task Model, in which each task s deadline is same as its eriod and offset is 0. We call (P4) Task-Periodic-Simle as the schedulability roblem for this simle Periodic Task Model, and (P5) Task-Periodic-Generic as the one for the general Periodic Task Model. Another considered task model is the Soradic Task Model, in which two successive reuests of a task are searated by at least a time interval, called its minimum searation time. Similarly, we use (P6) Task-Soradic to reresent the schedulability roblem for the Soradic Task Model. Related Work: There are several HiRTS resource models, such as the Regularity-based Model [1, 3], the Bounded-Delay Model [, 3], the Periodic Model [10] and the EDP Model [11]. The Periodic Model (or Constant Bandwidth Server [16]) is the most oular one due to its simlicity for resource artitioning. Since a resource artition in this model is defined similarly to a eriodic real-time task, the existing eriodic task scheduling techniues can be used for eriodic resource artitioning without changes. However, schedulability tests have been found only in the simlest case (P4) Task-Periodic-Simle for task scheduling in this model. Due to the blacked-out intervals (Figure ) without any comutation time available, researchers have been stuck in more comlicated roblems, such as (P5) Task-Periodic-Generic and (P6) Task-Soradic. Fig.. Blacked-out Intervals on a Periodic Partition (, c) Besides the Periodic Model, other resource models had not received enough attention for a long time because they lacked effective resource artitioning algorithms. Recently, the Regularity-based Model has been greatly exlored. Other than the Periodic Model, this model tends to evenly distribute the time units on a resource artition, and a arameter regularity is used to restrain the time-unit distribution. A resource artition is

4 called a Regular Partition when its regularity is minimal, which causes its time-unit distribution to be almost even. This idea was originally introduced in [7], and develoed in [1, 3] for the single-resource scenario with some rimitive results. Li and Cheng [4, 5] alied it onto a multiresource latform with global scheduling. Furthermore, they also comrehensively exlored task scheduling on regular artitions in [15]. They found that some classical real-time scheduling roblems on a regular artition can be easily transformed into euivalent roblems on a dedicated single resource. Therefore, regular artitions are able to rovide maximal transarency for task scheduling. In Table I, we summarize the current state of the art of the Regularity-based and the Periodic Models. Problem the Periodic Model the Regularity-based Model (P1) Res-Single Converted to a Periodic Task Scheduling Problem Primitive Results [1, 3] (P) Res-Global Same As Above Otimal Aroximation Alogrithm (Magic7 [5]) (P3) Res-Partitioned Same As Above No Result (P4) Task-Periodic-Simle New Schedulability Tests for Periodic Partitions [10] Converted to a Task Scheduling Problem on a Dedicated Resource [15] (P5) Task-Periodic-Generic No General Result Same As Above (P6) Task-Soradic No General Result Same As Above TABLE I THE CURRENT STATE OF THE ART: THE PERIODIC MODEL AND THE REGULARITY-BASED MODEL Contributions: One major contribution of this aer is to alleviate the weaknesses of the Regularity-based Model listed in Table I. The first weakness is that the current results on (P1) Res-Single are very rimitive. Since the scheduling roblem of regular artitions is announced NP-hard, the current solutions fall into the category of aroximation algorithms. Mok and Feng [1, 3] gave an initial solution to schedule regular artitions on a single resource, where the weight of each regular artition is aroximated by the values in an infinite seuence 1, 1, 1 4, 1 8,.... However, they did not consider other aroximation seuences, and their results have not been roved otimal. This aer is the first to comrehensively study (P1) Res-Single for the Regularity-based Model. It derives its schedulability bound for all feasible aroximation seuences, and finds a grou of sub-otimal aroximation seuences which achieve higher resource utilization. Based on the new results on (P1) Res-Single, this aer also studies (P3) Res-Partitioned in the Regularitybased Model for the first time, which is imortant because the migration overhead cannot be neglected in most real-time systems [13, 14]. It first derives the schedulability bound when a single aroximation seuence is used. Then it introduces MulZ, a novel algorithm for (P3) Res-Partitioned, which drastically imroves the overall resource utilization by using multile aroximation seuences. Even without considering the migration overhead due to global scheduling, MulZ outerforms the otimal global scheduling algorithm, esecially on middle-to-large multiresource latforms. Moreover, we conclude that MulZ does not affect the original schedulability bound. Another imortant contribution of this aer is to comare the overall erformance of the Periodic and the Regularity-based Models. Since there has been no general result on (P5) Task-Periodic-Generic and (P6) Task-Soradic in the Periodic Model, we can only comare the erformance of (P4) Task-Periodic-Simle in both models with three different resource artitioning scenarios, (P1) Res-Single, (P) Res-Global and (P3) Res-Partitioned, resectively. Our exerimental result shows that the Regularity-based Model outerforms the Periodic Model in each scenario. Furthermore, considering the Regularity-based Model is able to handle comrehensive task models at the task level, it should be a better choice than the Partitioned Model when alying real-time alication integrations, esecially after we make u its weaknesses on resource artitioning in this aer. Organization: The rest of this aer is organized as follows. We review the Regularity-based Model and its current aroximation algorithm for (P) Res-Global in Section II. We introduce our solutions for (P1) 3

5 Res-Single and (P3) Res-Partitioned in Sections III and IV, resectively. Section V resents our exerimental results. Finally, we draw the conclusion in Section VI. II. THE REGULAR PARTITION AND ITS APPROXIMATION GLOBAL SCHEDULING ALGORITHMS In this section, we review some basic definitions and imortant results in the Regularity-based Model introduced in [, 5]. We make some changes or imrovements to them for consistency and brevity. These rereuisite knowledge falls into two categories: one is about the roerties of regular artitions; the other is about the current aroximation algorithms based on global scheduling. A. Regular Partition The theoretical definition of a resource artition shows how time units are eriodically assigned to it. As shown in Def..1, it is secified by a eriod and a time-unit seuence with length. W P = denotes the weight of P. In the Regularity-based Model, the weight of a resource artition is always a rational number between 0 and 1. Definition.1 A Resource Partition P is a tule (S, ), where S = s 1, s,..., s : 0 s 1 < s <... < s < is the time-unit seuence; is the eriod and, are co-rime. As shown in Figure 3, S P (t) denotes the Suly Function of P, eual to the total number of time units that are available in P from time 0 to t; I P (t) = S P (t) t W P denotes the Instant Regularity of P, which shows the difference between the actual and ideal resource allocation on P at time t. Lemma.1 Suose P = (S, ) is resource artition where is the length of S, then t [0, ), { 1 if t S; I A (t + 1) I A (t) = otherwise. Proof: Immediately follows from the definition of Instant Regularity. Definition. A resource artition P is a Regular Partition if and only if t 1, t, I P (t 1 ) I P (t ) < 1. A regular artition minimizes the deviation range of its instant regularity. As shown in Figure 3, the size of this range is limited to less than 1. We want to oint out that this bounded range concet is very similar to the lag of a P-fair task [8, 9], but the lag range in P-fair is bounded by. Therefore, we cannot use the P-fair algorithm to schedule regular artitions because a regular artition has a much tighter restriction on the deviation range than a P-fair task. Fig. 3. A Regular Partition of Weight 4 7 For a given regular artition P, let β P denote the minimum value of P s instant regularities. Since I P (0) = 0, by Def.., it is obvious that β P ( 1, 0]. Suose is P s eriod and is its weight where, are co-rime, then by Lemma.1, the ossible values of β P are {0, 1, 1,..., }. Li and Cheng [15] rove that each value determines a -weight regular artition. Therefore, there are exactly different -weight regular artitions. Definition.3 A regular artition P is a Standard Regular Partition when β P = 0. 4

6 Definition.4 T 0 (, ) reresents the time-unit seuence of a standard regular artition with weight. For examle, Figure 3 shows T 0 (7, 4) = 0, 1, 3, 5. The motivation to oint out such a secial regular artition is that any other regular artition with the same weight can be easily obtained from it by alying a right-shift oeration on its time-unit seuence. Therefore, we usually only need to check the roerties of regular artitions on the standard ones. Figure 4 shows an examle containing such kind of right shifts. Definition.5 T (,, δ) reresents the time-unit seuence right shifted from T 0 (, ) by δ time units, where a modulus oeration is alied when a time unit is out of [0, ). Secially, T (,, 0) = T 0 (, ). Fig. 4. Right Shifting Regular Partition (T 0(3, ), 3) The following discussion answers the uestion of whether a regular artition can be a art of another one in some secial cases. These results are very imortant for our aroximation method of regular artition scheduling. We always use it to determine counter examles when we check the feasibility of an aroximation seuence. In this aer, we formally define a artial order r to indicate the containment relation between regular artitions. Definition r if and only if δ, T (, 1 1, δ) T 0 (, ), where = LCM( 1, ). For convenience, in this aer, we use RP w to reresent a regular artition whose weight is w. Then, we can also describe r like this: w 1 r w if and only if RP w1 could be contained by RP w. This containment relation is checked freuently while scheduling regular artitions. The following two lemmas investigate this containment relation in some secial cases. Lemma. If w 1 r w where w 1 + w < 1 and w < w 1, then n > 0, w 1 = n+1 4n+3, w = n+1 4n+3. Proof: Lemma 4.3 in [5] shows the same roerty. The roof is also resented Aendix B. Lemma.3 If w 1 w 3 and < w 1 < w, w 1 r w. Proof: Immediately follows from Lemma.. For examle, by Lemma.3, we easily conclude that a 1 3 -weight regular artition does not contain any 1 4 -weight one; by Lemma., a 5 9 -weight regular artition does not contain any 1 3-weight one. B. Aroximation Algorithms for Regular Partitions Global Scheduling The roblem is: given {w i : 1 i n} as the weights of n regular artitions, how to schedule them on an identical multiresource with global scheduling? An aroximation method [5] adjusts each w i to the closest greater or eual value in an Aroximation Boundary Seuence (ABS) or an Extended Aroximation Boundary Seuence (E-ABS) that are defined as follows: Definition.7 An Aroximating Boundary Seuence (ABS) is an infinite number seuence b 1, b, b 3,..., where i, 0 < b i+1 < b i < 1; lim b n = 0 when n. Definition.8 An Extended Aroximating Boundary Seuence (E-ABS) is a tule (B, B ), where ABSes B = b 1, b,..., B = b 1, b,..., and b 1 + b 1 < 1. E-ABS is introduced to achieve lower aroximation overhead than ABS because it could contain more and denser elements used for aroximation. For a given E-ABS E = (B, B ), we say b E if and only if b B or 1 b B. Sometimes we use..., 1 b, 1 b 1, b 1, b,... to reresent an E-ABS for convenience, though it is not a formal reresentation for seuences. The following are some imortant functions for a given ABS/E-ABS B originally defined in [5]. Secially, if B is an ABS, let B include 1 for boundary control. 5

7 Definition.9 Aroximation Function R B (w): R B (w) = min{b : b B; b w}. R B (w) aroximates a weight w at the closest greater or eual value in B. Definition.10 Schedulability Bound Υ B : Υ B = min{ b b : b, b B; b < b ; b B, b < b < b }. Υ B euals the minimal uotient of any two consecutive elements in B, which indicates the low-bound resource utilization due to the aroximation strategy by B. Definition.11 Aroximation Overhead O B : O B determines the average resource utilization of B. We do not resent the euation of O B here because this aer does not use this euation. In Figure 5, the stairs shae indicates R B (w), and O B euals the area of the slash-filled triangles. Fig. 5. Aroximation Function and Overhead of an ABS Let us make a naming convention. In this aer, we use m-resource to reresent a multiresource with m identical resource-units. Secially, 1-resource reresents a single-resource. Then, we can define the feasibility of an ABS/E-ABS in Def..1, where {n i w i : i = 1,,..., k} r denotes a artition grou containing n i times RP wi, and 1 w i can be simlified to w i. Definition.1 An ABS/E-ABS B is globally-feasible if and only if b i B (i = 1,,..., n) where n i=1 b i m, {b i : i = 1,,..., n} r is schedulable on an m-resource via global scheduling. The thought behind B is globally-feasible is: For a grou of regular artitions {w i : i = 1,,..., n} r, we first aroximate the weight of each artition at its closest boundary in B using R B (w). If the sum of these aroximated weights does not exceed m, they are always schedulable on an m-resource via global scheduling. On the contrary, if we are able to find a artition grou as a counter examle that is unschedulable after being aroximated by B via global scheduling, we can claim that B is not globally-feasible. Name Definition Globally-Feasible G n,m 1, 1 1,,... n m n m n m 3 H n,m n 1, n,..., 1, 1, 1 1,,... n n n n m n m n m 3 yes iff n RMN Z n,m (H n,m, G n,m) iff n RMN TABLE II TYPICAL ABSES AND E-ABSES Table II describes some tyes of ABSes/E-ABSes defined in [5] and their feasibility for global scheduling, and we also list some secific ABSes/E-ABSes as follows, which are widely used in our later discussion. Secially, Z 7, is the otimal ABS/E-ABS for global scheduling found in [5]. RMN (Regularity Magic Numbers) is an integer set {,3,4,5,7}. 6

8 G 1, = 1, 1 4, 1 8, 1 16, 1 3,... Z, = Z 4, =..., 31 3, 15 16, 7 8, 3 4, 1, 1 4, 1 8, 1 16, 1 3,... Z 3, =..., 3 4, 11 1, 5 6, 3, 1 3, 1 6, 1 1, 1 4,... Z 5, =..., 39 40, 19 0, 9 10, 4 5, 3 5, 5, 1 5, 1 10, 1 0, 1 40,... Z 7, =..., 55 56, 7 8, 13 14, 6 7, 5 7, 4 7, 3 7, 7, 1 7, 1 14, 1 8, 1 56,... At the end of this review section, we list the symbols used in this aer in Table III. B, B, A ABS or E-ABS b, b, b, b i item in ABS or E-ABS w, w i weight of regular artition R B(w) Υ B O B T 0(, ) T (,, δ) RP w r aroximation function of B schedulability bound of B aroximation overhead of B time-unit seuence of standard regular artition time-unit seuence of regular artition regular artition of weight w containment relation between regular artitions G, {...} r regular artition grou TABLE III SYMBOL TABLE III. SINGLE-RESOURCE SCHEDULING FOR REGULAR PARTITIONS Although there already have been some aroximation algorithms for single-resource scheduling of regular artitions, such as AAF [] and Magic7-Single [5], these algorithms have not been roved otimal. We will deely study this roblem by also robing into aroximation algorithms in this section. First, we define the ABS/E-ABS feasibility on a single resource as follows. For convenience, we say a regular artition grou is on-1-schedulable if it is schedulable on a single resource. Definition 3.1 An ABS/E-ABS B is on-1-feasible if and only if b i B (i = 1,,..., n) where n i=1 b i 1, {b i : i = 1,,..., n} r is on-1-schedulable. This definition is very similar to Def..1 for globally-feasible, but on-1-feasible only reuires that an ABS/E-ABS always works on a 1-resource. Therefore, B is globally-feasible is sufficient but not necessary for B is on-1-feasible. There are some ABSes/E-ABSes that are on-1-feasible but not globally-feasible. 3, 1, 1 6, 1 1, 1 4,... is such an examle. We formally examine it as follows. Observation 1 3, 1, 1 6, 1 1, 1 4,... is on-1-feasible. Proof: Let B denote the given ABS. Following Def. 3.1, we only need to rove that b i B (b 1 b... b n ) where n i=1 b i 1, G = {b i : i = 1,,..., k} r is on-1-schedulable. CASE 1: b In Table II, we know G 3, = 1 6, 1 1, 1 4,... is globally-feasible, then it is also on-1-feasible. It follows that G is on-1- schedulable in this case. CASE : b 1 = 1. If b = 1, it is easy to schedule G; otherwise, we need to examine whether G = {b i : i =,..., k} r is schedulable on a 1 -weight regular artition, which is euivalent to that G = {b i : i =,..., k} r is on-1-schedulable, where all items in G belong to 1 3, 1 6, 1 1, 1 4,.... From Table II, H 3, = 3, 1 3, 1 6, 1 1, 1 4,... is on-1-feasible, and it follows that G is on-1-schedulable. CASE 3: b 1 = 3. It is obvious that G does not contain 1. Since H 3, = 3, 1 3, 1 6, 1 1, 1 4,... is on-1-feasible, G is on-1-schedulable. Observation 3, 1, 1 6, 1 1, 1 4,... is not globally-feasible. 7

9 Proof: We only need to show that artition grou G = { 3, 1 } r is unschedulable on a -resource. Figure 4 resents the three forms that 3 -weight regular artitions have. CASE 1: If the two 3-weight regular artitions in G is same, without loss of generality, suose their time-unit seuences are both T 0 (3, ). Obviously, the remaining time units (a coule of, 5, 8,... ) cannot roduce a 1 -weight regular artition, whose time-unit seuence should be 0,, 4,... or 1, 3, 5,.... CASE : Otherwise, without loss of generality, suose G contains the first two forms of artitions in Figure 4, and the remaining time units 0,, 3, 5, 6, 8,... also cannot roduce a 1 -weight regular artition. A. Schedulability Bound on a Single Resource We figure out the uer limit of the schedulability bound of an on-1-feasible aroximation seuence in Theorem 3.1 by roving that it is not on-1-feasible if its schedulability bound exceeds 0.5. Lemma 3.1 on-1-feasible ABS/E-ABS B where Υ B > 0.5, 3, 1 B. Proof: resented in Aendix C. Lemma 3. > 1, after scheduling ( 1) times 1 -weight regular artitions on a 1-resource, the remaining time units comose another 1 -weight regular artition. Proof: It is true because T (, 1, δ) = δ for δ [0, ). Theorem 3.1 on-1-feasible ABS/E-ABS B, Υ B Proof: If Υ B > 0.5, by Lemma 3.1, 3, B. And by Def..10, w B, w (, 1 ). Then {( 1) 1, w} 1 r is not on-1-schedulable because w r (Lemmas 3.,.3). This is a counter examle against that B is on-1-feasible. It follows that Υ B cannot be great than 0.5. B. Sub-otimal Aroximation on Single Resources Next, we start to consider the aroximation overhead. Based on the shaes shown in Figure 5, intuitively, the aroximation overhead will decrease if we add more elements into the seuence. So we first define the containment relation between aroximation seuences. Definition 3. Suose A,B are ABSes/E-ABSes, A B if b A, b B; A B if A B but B A. The next conclusion is obvious from Figure 5: Lemma 3.3 Suose A,B are ABSes/E-ABSes, O A < O B if A B. Lemma 3.3 shows that a feasible aroximation seuence can reach its minimum aroximation overhead if we cannot add any element into it without violating its feasibility. This conclusion leads to the idea of Saturated. Definition 3.3 An ABS/E-ABS A is saturated if A is on-1-feasible, and b / A, A b is not on-1-feasible. Furthermore, we say a saturated aroximation seuence is sub-otimal if it also reaches the maximal schedulability bound. Definition 3.4 An ABS/E-ABS A is sub-otimal if A is saturated and Υ A = 0.5. In this aer, we find a grou of sub-otimal E-ABSes based on the idea of Regularity Magic Numbers [5]. Reminder that Z n, has been defined in Table II. Theorem 3. Z n, is sub-otimal if n {3, 4, 5, 7}. Proof: It is obvious that Υ Zn, = 0.5 and Z n, is on-1-feasible. We only need to rove that Z n, is saturated. Since Z n, =..., 4n 1 4n, n 1 n, n 1 n, n n,..., 1 n, 1 n, 1 4n,..., b Z n,, there are four cases: CASE 1: b ( 1 k n, 1 k 1 n ) where k > 0. {( k 1 n 1) 1 k 1 n, b} 1 r is not on-1-schedulable because b r k 1 n (Lemmas 3.,.3). CASE : b (1 1 k 1 n, 1 1 k n ) where k > 0. 1 {b, k n } 1 r is not on-1-schedulable because k n r (1 b) (Lemmas 3.,.3). 8

10 CASE 3: b ( k 1 n, k n ) and b + k n < 1, where 1 < k < n. {1 k n, b} r is on-1-schedulable b r k n m > 0, b = m+1 4m+3 and k n = m+1 4m+3 m = 1, n = 7, k = 3, b = 7. This result contradicts b > k 1 n. (Lemma.) CASE 4: b ( k 1 n, k n ) and b + k n {b, 1 k n } r is on-1-schedulable (1 k n ) r (1 b) > 1, where 1 < k < n. m > 0, k n = 3m+ m+ 4m+3 and b = 4m+3 (Lemma.) m = 1, n = 7, k = 5, b = 4 7. This result also contradicts b > k 1 n. IV. PARTITIONED MULTIRESOURCE SCHEDULING FOR REGULAR PARTITIONS We have deely studied the single-resource scheduling roblem for regular artitions. A maximum schedulability bound and some sub-otimal E-ABSes have been found. Next, we start to investigate the artitioned multiresource scheduling roblem for regular artitions. There are two stes in a common artitioned scheduling algorithm: (1) allocate resource artitions (or realtime tasks) to resource-units; and () schedule them on each resource-unit. Ste 1 has two concerns when allocating a resource artition: one is which resource-units can contain it; the other is that if multile resourceunits can do it, which one should be chosen. These two issues corresond to the single-resource scheduling algorithm and the allocation algorithm, resectively. We adot the naming convention in [6], where SA-RA denotes the artitioned scheduling algorithm combining a reasonable allocation algorithm RA and a single-resource scheduling algorithm SA. For examle, Z 3, -WF reresents the combination of the Worst Fit First resource allocation algorithm and the aroximation singleresource scheduling algorithm using Z 3,. Also, Υ SA-RA denotes the Schedulability Bound of SA-RA. A. Using a Single Aroximation Seuence There are some articularity when scheduling regular artitions. Since we use an aroximation methodology for single-resource scheduling, we can aroximate the regular artitions before the allocation ste. If the aroximation algorithm is based-on a single aroximation seuence, we only need to guarantee that the total weight allocated to each resource-unit does not exceed 1. We easily conclude that the schedulability bound of such a artitioned algorithm cannot exceed 0.5. Theorem 4.1 ABS/E-ABS B, Υ B-RA Υ B. Proof: Since the weight of each regular artition has to be aroximated by B, no matter how these artitions are allocated, Υ B-RA cannot exceed Υ B. Corollary 4.1 ABS/E-ABS B,Υ B-RA 0.5. We assume that the weight of each regular artition is static, such that we can sort the regular artitions by their weights before the allocation ste. Some allocation algorithms are based on this assumtion, such as First Fit Decreasing (FFD) and Best Fit Increasing (BFI). We first notice that G 1, -FFD and Z, -FFD reach the maximal schedulability bound shown in Corollary 4.1. Lemma 4.1 Υ G1,-FFD = 0.5. Proof: From Table II, G 1, = 1, 1 4, 1 8,..., and Υ G 1, = 0.5. Suose {n i 1 : i = 1,, 3,...} i r is the aroximated artition set. Obviously, these artitions can be successfully allocated to an m-resource by FFD when i>0 ni m. Therefore, G i 1, -FFD always works when the total weight of the original artitions is not greater than 0.5. It follows Υ G1,-FFD 0.5. By Corollary 4.1, Υ G1,-FFD = 0.5. Lemma 4. Υ Z4,-FFD = 0.5. Proof: Z 4, =..., 7 8, 3 4, 1, 1 4, 1 8,.... Let s comare the scheduling of Z 4,-FFD and G 1, -FFD. Suose {N i (1 1 i ) : i =, 3,...; n i 1 i : i = 1,, 3,...} r is the aroximated artition set by Z 4,, then {( i>1 N i) 1; n i 1 i : i = 1,, 3,...} r is the one by G 1,. Figure 6 shows the artition allocations of these two algorithms. 9

11 Fig. 6. Z 4,-FFD and G 1,-FFD Their allocations of heavy artitions (weight 1 ) are exactly the same. The difference is between the lightartition allocations. Since Z 4, -FFD leaves some blanks in the heavy art, some light artitions could be assigned into these blanks. Nevertheless, these changes do not negatively imact the schedulability. Then we rove that each aroximation seuence with the maximal schedulability bound has to contain 1. Lemma 4.3 If 1 B, then RA, Υ B-RA max{0, b : b B, b < 1 }. Proof: Let b 1 = max{0, b : b B, b < 1 } and b = min{1, b : b B, b > 1 }. Schedule {(m+1) (b 1 +ɛ)} r on an m-resource with B, where ɛ < b b 1. This set is aroximated to {(m+1) b } r. It is unschedulable for any allocation algorithm because b > 1. It follows Υ B-RA (m+1) (b1+ɛ) m b 1 when m and ɛ 0. Therefore, Υ B-RA b 1. Corollary 4. If 1 B, then RA, Υ B-RA < 0.5. Theorem 4. shows that Z 4, is the only sub-otimal seuence reaching the maximal schedulability bound. Lemma 4.4 Given an on-1-feasible ABS/E-ABS B where Υ B = 0.5, if 1 1 B, then k > 0, B and k w ( 1, 1 k+1 ), w / B. k Proof: Use an inductive method. When k = 1, 1 B. Assume w ( 1 4, 1 ), w B. Check the schedulability of { 1, w} r on a 1-resource. Suose RP 1 reemts all the time units at even numbers. No matter in which case among 1 w (, 3), 1 w (3, 4) and 1 w = 3, there exist two consecutive time units on RP w whose distance is 3. It follows that { 1, w} r is not on-1-schedulable because all the time units at even numbers are already reemted by RP 1. Thus, w ( 1 4, 1 ), w B. 1 When k > 1, by the inductive assumtion, B and k 1 w ( 1 1, k ), w / B. Then 1 k 1 B (otherwise, Υ k B < 0.5 by Def..10). Assume w ( 1, 1 k+1 ), k w B. Let = k 4, then {( 1) 1, w} 1 r is not on-1-schedulable because w r (Lemmas 3.,.3). Therefore, w ( 1, 1 k+1 ), w / B. k Theorem 4. If B is a sub-otimal ABS/E-ABS and Υ B-FFD = 0.5, then B = Z 4,. Proof: By Lemma 4. and Corollary 4., we only need to show that Z 4, is the only sub-otimal ABS/E-ABS containing 1 1. Suose B is such a seuence. By Lemma 4.4, k > 0, B and w ( 1 k, 1 k+1 ), w / B. k On the other hand, k > 0, w (1 1, k ), w B; otherwise, {w, k+1 } k+1 r is on-1-schedulable 1 k+1 r (1 w), which contradicts Lemma.. It follows B Z 4,. Since B is saturated, B = Z 4,. B. Using Multile Aroximation Seuences The erformance of a artitioned scheduling algorithm strongly deends on its aroximation overhead. For a given weight, the aroximation overhead is not the same on different aroximation seuences. For examle, R Z4, (0.45) = 0.5 and R Z7, (0.45) = This fact insires us to use multile aroximation seuences in a artitioned scheduling algorithm. We call it MulZ when we use Z 3,, Z 4,, Z 5,, Z 7, simultaneously, and resent its seudocode as follows. (0) resource-units R := {R j {factor = 0, rest = 1} : j [1, m]}; 10

12 (1) artitions P := {P j {weight, res-unit = 0} : j [1, s]}; () bool MulZ FFD() (3) sort P in non-increasing order; (4) for j = 1 to s do (5) P j.resource := MulZ FFD Alloc(P j.weight); (6) if P j.resource = 0 return false; (7) od (8) return true; (9) int MulZ FFD Alloc(w) (10) for i = 0; n {3, 4, 5, 7}; i++ do (11) A i = R Zn, (w); (1) od (13) for k = 1 to 4 do (14) r := the k-th minimum item in array A; (15) f := n, where w is aroximated at r by Z n, ; (16) for j = 1 to m do (17) if R j.factor = f and R j.rest r do (18) R j.rest := R j.rest r; (19) return j; (0) od (1) else if R j.factor = 0 do () R j.factor := f; (3) R j.rest := 1 r; (4) return j; (5) od (6) od (7) od (8) return 0; - The first two lines define and initialize the data structures. In line (0), a ositive value of factor, n, indicates that a resource-unit is not emty and the artitions on it are aroximated by Z n,. In line (1), a ositive value of res-unit, m, indicates that a artition is assigned to the m-th resource-unit. Function MulZ FFD first sorts the resource artitions in non-increasing order, and then calls MulZ FFD Alloc within a loo to allocate resource for each artition. Lines 10-1 comute the aroximated weights of the sub-otimal E-ABSes, and store them as an array. The loo of lines 16 6 checks the availability of each resource-unit one by one for the current artition, where line 14 determines which E-ABS is chosen for aroximation in the current iteration; lines 17 0 search available non-emty resource-units using the chosen E-ABS; lines 1 5 assign the current artition to an emty resource-unit if the condition in line 17 fails for all non-emty resource-units (always having lower indexes in R). If there are emty resource-units remaining when starting function MulZ FFD Alloc, the loo of lines 16 6 must terminate at either line 19 or line 4 when k = 1. Therefore, the branch of lines 1 5 is only executed when k = 1. It follows that an emty resource-unit always chooses its working E-ABS such that the minimum aroximation overhead is achieved for its first assigned artition. A Partitioning Examle of MulZ-FFD: Partition G = {0.65, 0.6, 0.55, 0.5, 0.35, 0.3, 3 0.5} r on a 4-resource. Let U(G, n) = w G R Z n, (w). It is easy to check that n {3, 4, 5, 7}, U(G, n) > 4. Therefore, any single Z n, does not work in this scenario, even if we use a global scheduling strategy without considering migration overhead. However, MulZ-FFD is able to do that. Let s see how the first artition 0.65 is assigned. When n = 3, 4, 5, 7, its aroximated weight euals 3, 3 4, 4 5, 5 7, resectively. We choose the minimum one 3, and the working E-ABS is Z 3,. Then we assign this artition to the No. 0 resource-unit because currently all the resource-units are emty. Meanwhile, the working E-ABS on the No. 0 resource-unit is set to Z 3,. The rest can be done in the same manner, and the final result is shown in Table IV. The overall resource utilization is 9.5%. Notice that although Z 7, cannot achieve the minimum aroximated overhead for the last remaining 0.5-weight artition, this artition is still assigned to the No. resource-unit because the other three resource-units cannot accommodate it at that moment. MulZ-FFD is an intuitive algorithm, which is unable to romise otimal erformance theoretically. Nevertheless, our exerimental results in the next section show that it has better erformance than the current 11

13 otimal global scheduling algorithm. Meanwhile, Theorem 4.3 shows that the maximal schedulabiltiy bound 0.5 is still ket. Lemma 4.5 Suose weights w 1, w,..., w s (w 1 w... w s > 0) are already assigned to a resource-unit by MulZ-FFD, where Z n, (n {3, 4, 5, 7}) is its working E-ABS, and s i=1 w i < 0.5. w s+1 (0, w s ], this resource-unit is still able to accommodate w s+1. Proof: We only check Z 7, here. The others can be checked similarly. Let r i = R Z7, (w i ) be the aroximated weight of w i for i = 1,,..., s + 1. Notice w i r i < w i. Since Z 7, is sub-otimal, we need to show U = s+1 i=1 r i 1. CASE 1: w 1 ( 3 7, 1 ), r 1 = 4 7. This case is imossible. When MulZ-FFD assigns w 1 to an emty resource-unit, it should choose Z 4, as the working E-ABS because R Z4, (w 1 ) = 1 < r 1. CASE : w 1 ( 7, 3 7 ], r 1 = 3 7. Similarly, to choose Z 7, as the working E-ABS when assigning w 1, w 1 must be in ( 5, 3 7 ]. If s = 1, then U r 1 < 1; otherwise, s > 1 s i= w i < 0.5 w 1 < 0.1 U < r 1 + s i= w i + w s s i= w i < 1. CASE 3: w 1 ( 1 7, 7 ], r 1 = 7. To choose Z 7,, w 1 must be in ( 1 4, 7 ]. If s, then U 3r 1 < 1; otherwise, s > s i= w i < 0.5 w 1 < 1 4 : CASE 3.1: w ( 1 7, w 1], then s i=3 w i < = 3 U < r 1 + r + s i=3 w i + w s s CASE 3.: w i=3 w i < 1. CASE 3..1: w s 1 14, then U < r 1 + s i= w i + w s+1 < < 1. CASE 3..: w s > 1 14, then 1 7 w w 3... w s > Since s i= w i < 1 4, s 4. U = r 1 + s i= r i + r s+1 r 1 + 4r = < 1. CASE 4: w 1 1 7, then (i) for i = 1,,..., s + 1, r i { 1 7, 1 14, 1 8, 1 56,...}; (ii) r 1 r... r s r s+1. Since s i=1 w i < 0.5 and r i < w i, (1 s i=1 r i) > 0. By (i) and (ii), i [1, s], r i is divisible by r s+1. It follows (1 s i=1 r i) is divisible by r s+1. Therefore, r s+1 1 s i=1 r i U 1. Theorem 4.3 Υ MulZ-FFD = 0.5. Proof: (i) Since {(m + 1) (0.5 + ɛ)} r is unschedulable on an m-resource by MulZ-FFD, Υ MulZ-FFD (m+1) (0.5+ɛ) m 0.5 when m and ɛ 0. It follows Υ MulZ-FFD 0.5. (ii) Suose {w 1, w,..., w n : 1 w 1 w... w n > 0} r is unschedulable on an m-resource by MulZ-FFD. Find the roer t where {w i : i = 1,,..., t} r is schedulable and {w i : i = 1,,..., t+1} r is unschedulable. If t i=1 w i < 0.5, there is a resource-unit whose utilization is less than 0.5. By Lemma 4.5, it is able to accommodate a regular artition of weight w t+1. This contradicts that {w i : i = 1,,..., t + 1} r is unschedulable. Therefore, t i=1 w i 0.5. It follows that the total weight of any unschedulable artition set is greater than 0.5 and Υ MulZ-FFD 0.5. A. Regular Partition Scheduling on Mulitresources V. EXPERIMENTAL RESULTS Let us briefly exlain this art of the simulation exeriments. For each weight ercentile, we generate random artition sets. In these sets, the weight of each regular artition is randomly chosen in the interval Resource-Unit E-ABS Partitions Arox. Weights No. 0 Z 3, 0.65, 0.3 No. 1 Z 5, 0.6, 0.35 No. Z 7, 0.55, 0.5 No. 3 Z 4, 0.5, 0.5, 0.5 3, , 5 4 7, 7 1, 1 4, 1 4 TABLE IV A PARTITIONING EXAMPLE OF MulZ-FFD 1

14 Θ = (low, high). Then we simulate different artitioned scheduling algorithms and count their schedulability rates. When a single aroximation seuence is alied, Figure 7 shows that Z 7, -FFD has the highest overall schedulability rate due to its lowest aroximation overhead. Fig. 7. Schedulability % of Z n,-ffd on a 64-resource Fig. 8. MulZ-FFD Greatly Imroves Schedulability % Figure 8 comares the schedulability rate among MulZ-FFD, Z 7, -FFD and Z 7, -Global. We have shown that Z 7, -FFD achieves the otimal overall resource utilization with a single E-ABS. Meanwhile, Z 7, -Global (Magic7 [5]) is the current otimal global scheduling algorithm for regular artitions. We ignore its migration overhead because it is hard to estimate a roer value for it. This kind of overhead deends on the hysical latform architecture, which is beyond the scoe of this aer. Meanwhile, this absence will not imact our conclusions. The simulation results indicate that MulZ-FFD has better erformance than the others. First, each chart shows that MulZ-FFD outerforms Z 7, -FFD, which means MulZ-FFD drastically imroves the overall resource utilization by using multile aroximation seuences to reduce the aroximation overhead. Second, even without considering the migration overhead due to global scheduling, MulZ-FFD erforms better or no worse than Z 7, -Global when Θ = (0, 1), and significantly outerforms Z 7, -Global for light-weight artitions where Θ = (0, 0.5). Moreover, MulZ-FFD erforms even better on middle-to-large multiresource latforms. B. Comare the Periodic and the Regularity-based Models Then we comare the overall resource utilization between the Periodic and the Regularity-based Models. As shown in Table I, for task-level scheduling in the Periodic Model, we can only find solutions for the simle eriodic task model. Therefore, our comarison only focuses on this task model, where a eriodic task t i is defined as (c i, i ). c i and i are t i s execution time and eriod, resectively. At the beginning of the simulation, a size of the comutation resource, m, is selected. The main body is a run loo. In each run, we generate a grou of eriodic task sets {T 0, T 1,...T n : T i = {t i0, t i1,...}} for each weight ercentile r, where the total weight of these task sets is exactly m r/100 and the weight of each task set is randomly chosen in the interval Θ = (low, high). It follows two major hases. Phase I simulates task scheduling. For each task set T i, we determine a resource artition P i which has the exact size to accommodate it with the EDF olicy. And Phase II simulates resource artitioning. We determine the 13

15 schedulability of resource artitions P 0, P 1,..., P n on an m-resource. After the run main loo, we count the schedulability rate. We imlement our simulation in three scenarios: on a single resource, on a multiresource with global resource scheduling and on a multiresource with artitioned resource scheduling. In all of them, we aly schedulability tests in [10] and [15] for task scheduling in the Periodic and the Regularity-based Models, resectively. Figure 9 shows our exerimental results on a single resource. We aly EDF and Z 7, for resource scheduling in the two models resectively. The results show that the Regularity-based Model has higher schedulability rate most of the time. Figure 10 shows the exerimental results on a 64-resource with global resource scheduling. P-fair and Magic7 are alied for resource scheduling. We find that the Regularitybased Model also outerforms the Periodic Model esecially when task sets are light (Θ = (0, 0.5)). Figure 11 shows the exerimental results on a 64-resource with artitioned resource scheduling. EDF-FFD and MulZ- FFD are alied for resource scheduling. The results show that the Regularity-based Model outerforms the Periodic Model in both scenarios, no matter the task sets are heavy or light. In general, the Regularity-based Model achieves higher schedulability rate than the Periodic Model, which shows that it also rovides higher overall resource utilization. Fig. 9. Schedulability % on a Single Resource Fig. 10. Schedulability % on a 64-Resource with Global Resource Scheduling Fig. 11. Schedulability % on a 64-Resource with Partitioned Resource Scheduling VI. CONCLUSION The Periodic Model is the most oular resource model for HiRTS due to its simlicity in resource artitioning. However, it has not solved most classical task scheduling roblems because of the significant 14

16 blacked-out intervals on its resource artitions. The Regularity-based Model achieves maximal transarency for task scheduling, but its resource artitioning roblem is comlicated due to the very strict timing constraint on regular artitions. To alleviate the weaknesses of the Regularity-based Model, this aer introduces new resource artitioning techniues for it. After alying these new techniues, our simulation results show that the Regularity-based Model achieves higher overall resource utilization than the Periodic Model. Since the Regularity-based Model can also handle much more task models at the task level, we conclude that the Regularity-based Model is a better choice than the Partitioned Model for the integration of real-time alications. ACKNOWLEDGMENT This aer is based uon work suorted in art by the National Science Foundation under Awards No and No REFERENCES [1] A. K. Mok and X. Feng. Towards comositionality in real-time resource artitioning based on regularity bounds. RTSS, 001. [] A. K. Mok, X. Feng, and D. Chen. Resource artition for real-time systems. RTAS, 001. [3] X. Feng. Design of real-time virtual resource architecture for largescale embedded systems. Ph.D. dissertation, Deartment of Comuter Science, The University of Texas at Austin, 004. [4] Y. Li, A. M. K. Cheng, A. K. Mok. Regularity-based artitioning of uniform resources in real-time systems. RTCSA, 01. [5] Y. Li, and A. M. K. Cheng. Static aroximation algorithms for regularity-based resource artitioning. RTSS, 01. [6] J.M. Loez, J.L. Diaz, and D.F. Garcia. Utilization bound for EDF scheduling on real-time multirocessor systems. Real-Time Systems, 8(1):39C68, 004. [7] S. Shigero, M. Takashi, and H. Kei. On the schedulability conditions on artial time slots. RTCSA, [8] S. Baruah, N. Cohen, G. Plaxton, and D. Varvel. Proortionate rogress: A notion of fairness in resource allocation. Algorithmica, [9] S. Baruah, J. Gehrke, and G. Plaxton. Fast scheduling of eriodic tasks on multile resources. IPPS, [10] I. Shin and I. Lee. Periodic resource model for comositional real-time guarantees. RTSS, 003. [11] A. Easwaran, M. Anand, and I. Lee. Comositional analysis framework using EDP resource models. RTSS, 007. [1] C. L. Liu and J. W. Layland. Scheduling algorithms for multirogramming environment in a hard real-time environment. J. ACM, 0(1):46-61, [13] A. Bastoni, B.B. Brandenburg and J.H. Anderson. An emirical comarison of global, artitioned, and clustered multirocessor EDF schedulers. RTSS, 010. [14] A. Bastoni, B. Brandenburg, and J. Anderson. Is semi-artitioned scheduling ractical? ECRTS, 011. [15] Y. Li and A. M. K. Cheng. Transarent Real-Time Task Scheduling on Temoral Resource Partitions. IEEE Transaction on Comuters (TC), 015. [16] L. Abeni and G. Buttazzo. Integrating multimedia alications in hard real-time systems. RTSS, [17] Y. Li, and A. M. K. Cheng. The Value of Even Distribution for Temoral Resource Partitions. Technical Reort, University of Houston, 016. [18] Z. Deng and J. Liu. Scheduling real-time alications in an oen environment. RTSS,

17 APPENDIX A Lemma A1 shows that the time slices on a regular artition are always evenly distributed. For convenience, a san denotes two neighboring time slices on a regular artition. Meanwhile, the size of a san is the distance between the two time slices it reresents. Lemma A1 The size of any san on a regular artition is either 1 w or 1 w, where w is the artition weight. Both tyes of sans coexist if 1 w is a fraction. Proof: Suose w = where, are co-rime. We only need to check the roerty on the standard regular artition P with time-slice seuence T 0 (, ). By Def..3, the first time slice reemted on P must be 0; otherwise, the instant regularity at time 1 is less than 0. Notice that n (0, ), n n n; ( +1) > n n (n, n]. Consider the second time slice on P. By Lemma.1, a reemted time slice increases the instant regularity of P by (1 ), and a non-reemted time slice decreases it by. To ensure the value of every instant regularity in [0,1) (Def.s. and.3), the second reemted time slice must be because (1, 1]. Similarly, the third time slice must be ( 1) ;... the -th time slice must be. Therefore, T 0 (, ) euals 0, ( 1),,...,. k (0, ], let san(k) denote the distance between the k-th and (k+1)-th time slices in T 0 (, ), then san(k) = k (k 1) [, + 1]. When is a fraction, k, san(k) = + 1; otherwise, the total size of all sans in T 0(, ) is <. APPENDIX B Proof of Lemma.: We only need to show that if T (, r, δ) T 0 (, ) where + r < and r < < r, then n > 0, = n+1 4n+3 ; r = n+1 4n+3. Let S (res. S ) denote the infinite seuence including all time slices on the regular artition T 0 (, ) (res. T (, r, δ)), then S is a subseuence of S. Let d = and d = r. By Lemma A1, any san size in S (res. S ) is either d or d + 1 (res. d or d + 1). Meanwhile, since + r < and r < < r, we have d d d + 1. CASE 1: d 3. Since d d d + 1 and d > (d + 1) + 1, the size of any san in S can only be chosen from either {d, d+1} or {d, d+1}. The first case is imossible because there must be a air of neighboring time slices in S searated by a time slice in S S, the distance of which is not less than d. In the second case, the time slices in S must be assigned to S and S S alternately. It follows = r, which contradicts < r. CASE : d = 1. Then = d = 1; > + r; r < < r 1 < < 3 ; 1 4 < 1 < r < 1 < 1 1 < < 3 ; < r < 4. From 1 < < 3, there are no neighboring 1-size sans in S ; from < r < 4, d = r = or 3. CASE.1: d =. Suose there are x (res. y) times 1-size (res. -size) sans in the first eriod of S. Then x + y =, x + y = y =. As shown in the figure, since there are no neighboring 1-size sans in S, each -distance san in S corresonds to a -distance one in S, and each 3-distance san in S corresonds to a 1-distance one and a -distance one in S. Suose there are x (res. y ) times 3-size (res. -size) sans in the first eriod of S, then x = x, y = y x r = x + y = y r =, which contradicts + r <. Suose the second reemted time slice is t. If t <, then IP (t + 1) 1. If t >, then IP (t) < 0. Both cases contradict that t, I P (t ) [0, 1). Three consecutive reemtive time slices increase the instant regularity on S by 3 (1 ) > 1, which contradicts Def... 16