New Slack-Monotonic Schedulability Analysis of Real-Time Tasks on Multiprocessors
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1 New Slack-Monotonic Schedulability Analysis of Real-Tie Tasks on Multiprocessors Risat Mahud Pathan and Jan Jonsson Chalers University of Technology SE-41 96, Göteborg, Sweden {risat, Abstract In this paper, ultiprocessor scheduling of a set of real-tie periodic tasks with iplicit deadlines is addressed. We propose two static priority-assignent policies, called policy P bound and policy P search. Coon for both policies is that a subset of a given task set is assigned the slack-onotonic priority while each of the other tasks is assigned the highest static-priority. The tasks are scheduled on processors using preeptive global ultiprocessor scheduling algorith. First, we show that the utilization bound of global ultiprocessor scheduling using our proposed policy P bound is higher than any other existing state-of-the-art static-priority ultiprocessor scheduling. Second, using the utilization inforation of individual task of a given task set, we propose our second priority assignent policy P search. We show that global ultiprocessor scheduling using policy P search doinates that of using P bound in the sense that any task set schedulable using P bound is also schedulable using P search and not conversely. I. INTRODUCTION Consider the following proble: Given a collection of n iplicit-deadline periodic tasks, is it possible to eet all the deadlines of the tasks scheduled on identical, unitcapacity processors? Real-tie task scheduling on ultiprocessors is priarily based on either global or partitioned approach. In global scheduling, a task is allowed to execute on any processor even when it is resued after preeption. In partitioned scheduling, the task set is grouped in different task partitions during design tie and each partition has a fixed processor on which only the tasks of that partition are allowed to execute. The ain design goal of any staticpriority global [3], [5], [6], [7], [1], [8] and partitioned [4], [1], [16] scheduling algoriths is to derive a schedulability condition that is when satisfied iplies that all the deadlines are et. One of the ost popular and expressive ways to derive a schedulability condition of a scheduling algorith is in ters of its utilization bound. The utilization bound of a scheduling algorith A is a nuber UB A such that all the tasks will eet their deadlines when scheduled by A on processors, if the total utilization of the task set is not greater than UB A. It has already been proved that neither global nor partitioned static-priority scheduling can have a utilization bound greater than 0.5 on processors [3], [16]. There exists a static-priority partitioned scheduling algorith, called R-BOUND-MP-NFR, having utilization bound of 0.5 [4]. However, the best state-of-the-art utilization bound of global ultiprocessor scheduling is 3 for 5 (RM-US scheduling [3]) and 3+ for > 5 (SM-US scheduling [1]). In this paper, 5 we propose two static priority-assignent policies, called P bound and P search, and prove that global ultiprocessor scheduling using either of these policies doinates both RM-US and SM-US scheduling. Both of our proposed priority assignent policies are based on slack-onotonic hybrid priority assignent schee that works as follows: if the utilization of a task is not greater than a threshold utilization, then the task is given slack-onotonic priority, otherwise, the task is given the highest static-priority. According to slack-onotonic priority assignent, the saller the difference between a task s period and worst-case execution tie (WCET), i.e., called slack, the higher is the assigned priority. For each of our proposed priority assignent policies, P bound and P search, we present the corresponding schedulability analysis of the Global ultiprocessor Scheduling algoriths, called GS bound and GS search, respectively. The threshold utilizations for policy P bound is deterined based on the schedulability analysis of a class of task sets, called special task sets (presented in Section V). The threshold utilization we use for policy P bound is where is the nuber of processors,. We prove that, the utilization bound of GS bound is in{ 1, }. It is easy to see that the utilization bound of GS bound is higher than (hence, doinates) that of both RM-US [3] and SM-US [1] scheduling. Note that, the threshold utilization used for P bound depends only on the nuber of processors and does not use any inforation (e.g., utilization) of the individual task of a given task set. Using the utilization inforation of individual task of a given task set in addition to the inforation about the nuber of processors, we propose our second slack-onotonic hybrid priority-assignent policy P search. In order to deterine the threshold utilization for policy P search, we borrowed the idea of deterining the joblevel priorities of priority-driven EDF (k) scheduling fro [13] and applied it to the static-priority setting at the tasklevel. In EDF (k) scheduling, jobs of the k highest utilization tasks are given the highest priority and the jobs of the reaining (n k) lowest utilization tasks are given the Earliest-Deadline-First (EDF) priority (in fact, one such k is searched). Based on policy P search, we show that
2 scheduling algorith GS search doinates GS bound in the sense that if a task set is schedulable using GS bound then it is also schedulable using GS search and not conversely. Consequently, GS search doinates both RM-US and SM-US since the utilization bound of GS bound is higher than that of RM-US and SM-US. To this end, we ake the following ajor contributions: 1) We propose priority-assignent policy P bound and show that the utilization bound of GS bound is in{ 1, }. This utilization bound is higher than that of the state-of-the-art algoriths RM-US and SM-US. Therefore, GS bound doinates both RM-US and SM-US. ) We propose priority assignent policy P search and show that GS search doinates GS bound scheduling algorith. We prove the doination of GS search over GS bound by showing that any task set schedulable using GS bound is also schedulable using GS search and not conversely. 3) We show through siulation experients that ore than 9% of the one illion randoly-generated GS search schedulable task sets are not SM-US schedulable. This percentage increases up to 99.99% when we increase the nuber of processors fro 4 to 3 in our experients. Therefore, GS search scheduling scales very well in coparison to SM-US with increasing nuber of processors. The rest of the paper is organized as follows. First, other works related to ultiprocessor scheduling are discussed in Section II. The syste odel we use in this work is presented in Section III. Useful definitions and soe iportant prior results that we use are presented in Section IV. The schedulability analysis of special task sets and the two priority assignent policies are presented in Section V and Section VI, respectively. In Section VII, we prove the doination of GS search over GS bound and doination of GS bound over both RM-US and SM-US scheduling, and present our siulation results. Finally, we conclude the paper in Section VIII. II. RELATED WORK The well-known Rate-Monotonic (RM) static-priority assignent is optial for uniprocessor [15] and is not optial for global ultiprocessor scheduling due to so called Dhall s effect [11]. Andersson, Jonsson and Baruah proposed a RM hybrid static-priority assignent policy for RM-US[/(3 )] algorith that has the utilization bound of /(3 ) on processors [3]. In RM-US[/(3 )] algorith, the Dhall s effect is avoided by assigning the highest priority to the tasks having utilization greater than /(3 ) while the rest of the tasks are given the traditional RM priority. Analysis of static-priority ultiprocessor global scheduling is also addressed by Baker [5] based on the iniu aount of interference in an interval that can cause a tasks deadline to be issed. Baker s analysis is general for any fixed-priority scheduling and arbitrary deadline task systes. Baker [5] showed that, for iplicit-deadline task sets the utilization bound of RM scheduling is (1 uax) + u in, where u ax and u in are the axiu and iniu utilization of any task in the task set, respectively. RM scheduling is studied for unifor ultiprocessors in [6], and it is shown that the utilization bound is 3 for iplicit-deadline tasks on unit-capacity processors. Using an analysis of the worst-case workload in an interval, siilar to that of the Baker s work in [5], Bertogna et al. [7] showed that, the utilization bound for deadline-onotonic scheduling of iplicit deadline tasks is (1 uax) +u ax. The work in [7] is further iproved in [8], where an iterative algorith for separately testing the schedulability of each task is proposed. Based on slack-onotonic hybrid priority assignent schee, a static priority-assignent policy is proposed by Andersson for SM-US scheduling algorith [1]. In SM-US scheduling, each task having utilization greater than /(3 + 5) is given the highest static-priority and the rest of the tasks are given slack-onotonic priorities. The utilization bound of the SM-US scheduling algorith is /(3 + 5) [1]. Recently, static-priority assignent schee for global scheduling is proposed by Davis and Burns [10] which has been shown (using siulation) to have better perforance than any other fixed-priority scheduling algoriths. In suary, the current state-of-the-art utilization bound for global ultiprocessor scheduling is 3 for 5 (RM-US scheduling) and 3+ for > 5 (SM-US scheduling). In this paper, we show that our proposed schedul- 5 ing algorith GS bound has a higher utilization bound than that of both RM-US and SM-US. Moreover, we show that GS search doinates GS bound which also iplies GS search doinates both RM-US and SM-US. III. SYSTEM MODEL In this paper, we consider global preeptive scheduling of n independent periodic tasks in set Γ ={τ 1,τ,...τ n } on identical, unit-capacity processors. Each task τ i Γ is characterized by a pair (C i, ), where C i represents the WCET and is the period of task τ i. An instance of each task, called job, is released at each period and requires at ost C i units of execution tie before the next period. Without loss of generality we assue that tasks are sorted based on decreasing priority order, that is, task τ k has lower priority than task τ i for i k. Since the execution of a task τ k can be interfered only by the higher-priority tasks, whether task τ k eets its deadline or not depends on the tasks in {τ 1,τ,...τ k } but are copletely unaffected by the tasks in {τ k+1,τ k+,...τ n }. We find it useful to define the task set Γ k def = {τ 1,τ,...τ k } for k = 1,...n. We
3 soeties, hereafter, also use the notation Γ k to denote the collection of all the jobs of the tasks in Γ k. We define the utilization of a task τ i as u i = C i / and the total utilization of the task set Γ k as U k = k u i for k = 1,...n. Note that, U n is the total utilization of Γ n =Γ. We define the axiu utilization u k ax and the iniu utilization u k in of a periodic task syste Γk to be the largest and sallest utilization of any task in Γ k, respectively. IV. USEFUL DEFINITIONS AND PRIOR RESULTS The schedulability analysis presented in this paper is based on the schedulability analysis of a class of task sets called special task sets. A task set is said to be special on processors based on two particular properties (defined shortly in Definition ). We prove that a task set that is special on processors is schedulable using global slackonotonic scheduling, denoted by GS SM, on processors (proved in Theore 6). It will be evident later that the schedulability analysis of special task sets is based on the aount of work done by GS SM within a particular tie interval. To forally characterize the aount of work done by any scheduling algorith A over a tie interval of length t on processors each having speed s, we use the following well-known definition of work (also used in [3], [6], [13]). Definition 1 (W(A,,s,Γ k,t)). Let the jobs of the task set Γ k are to be executed using any algorith A on an identical ultiprocessor platfor where each processor has speed s. For tie instant t 0, let W(A,,s,Γ k,t) denote the aount of work done by algorith A on jobs of the task set Γ k over the interval [0,t), while executing on processors each of speed s. Scheduling algorith A on a ultiprocessor platfor is called work-conserving algorith, if it never idles a processor when there are jobs awaiting execution. Note that, global static-priority scheduling is work-conserving by definition. The following Theore 1 provides a lower bound on the aount of work copleted by a work-conserving scheduling algorith A on an identical ultiprocessor platfor that has each of the processors ( 1 ) ties faster copared to that of copleted by any algorith A on a platfor with identical processors. Theore 1 is based on the work in [17] that exploits resource augentation for on-line scheduling of real-tie jobs. Theore 1 (Based on [17]). For the set of all jobs of task systes Γ k, any tie-instant t 0, any work-conserving algorith A, and any algorith A, it is the case that W(A,, s ( 1 ), Γk, t) W(A,, s, Γ k, t) (1) Theore 1 will be useful later to find the lower bound on the aount of work copleted by tie instant t using global slack-onotonic scheduling algorith GS SM. We will use the following function in Eq. () in rest of this paper: (1 x) F (x) = + x () x where Z + and 0 x 1. Two iportant features of the function in Eq. () are given in Lea. Lea. Consider a, b, x, c and d such that 0 a b x c d 1 for any integer > 0. The following two inequalities hold: in{f (b),f (c)} F (x) (3) in{f (a),f (d)} in{f (b),f (c)} (4) Proof: Proof is given in Appendix. V. SPECIAL TASK SET AND ITS SCHEDULABILITY In this section, first we forally define the two properties of a task syste that is special on processors. Then, we present an upper and lower bound on the aount of work within a given interval for GS SM scheduling (subsections V-A and V-A). Finally, based on the difference between these two bounds and using the two properties of the special task syste, we prove that all the deadlines of the special task syste are et using GS SM scheduling on unit-capacity processors (subsection V-C). Then based on the schedulability analysis of special task syste of this section, we propose the two slack-onotonic hybrid priority-assignent policies P bound and P search and the corresponding schedulability analysis of algoriths GS bound and GS search, respectively, in Section VI. Definition (Special Task Syste). A periodic task syste Γ k is special on processor if it satisfies the following two properties: Property 1: u k ax 1 Property : U k in{f (u k in),f (u k ax)} According to Property 1, the axiu utilization of any task in set Γ k, that is special on processors, is not greater than 1. According to Property, the total utilization of the special task syste Γ k is not greater than the iniu of F (u k in ) and F (u k ax). A. Upper Bound on Work of GS SM Over the interval [0,t), exactly t coplete jobs of task τ i are scheduled by GS SM and the ( t + 1) th job of τ i ay be scheduled for at ost in(t t,c i ) tie units. Thus, the axiu aount of work to be copleted by the jobs of the tasks {τ 1,...τ } within [0,t) is at ost t C i + in(t t,c i ) and this quantity is upper bounded by the right hand side of Eq. (5) given in Lea 3 (proved in [1], [9]). Lea 3 (Fro [1], [9]). The axiu aount of work t C i + in(t t,c i ) that need to be copleted by tasks {τ 1,τ...τ } over an interval [0,t) is upper bounded by the following inequality in Eq. (5)
4 [ t C i + in(t t, C i) ] [ C i + (t C i)u i] (5) We will use Eq. (5) as the upper bound of the aount of work that need to be copleted by the higher priority jobs of task τ j within a particular tie interval [0,t]. B. Lower Bound on Work of GS SM In this subsection, first we show in Lea 4 that there exists an algorith, called OPT, that can feasibly schedule special task set Γ k on processors each of speed 1. Then, we will show that the aount of work copleted by GS SM on unit-capacity processors within an interval is no less than that of copleted by OPT on processors each of speed 1 (i.e., we derive a lower bound on the aount of work actually copleted by GS SM ). Lea 4. Consider the task set Γ k that is special on processors. Task set Γ k is feasible on processors each with speed 1. Proof: By Property 1 of special task syste we have u k ax 1. Therefore, we have u i 1 since u i u k ax for all i = 1,,...k. To prove this lea, we need to show that the following inequality holds U k (6) 1 This is because, if u i 1 and Uk 1, then the task syste Γ k can be scheduled to eet all the deadlines on processors each of speed 1 using the algorith, called OPT, as follows The processor sharing schedule, called OPT, assigns a fraction u i of a processor to τ i at each tie-instant, and thus ensures that each instance of task τ i copletes C i units of execution within its deadline. The algorith OPs also used in the schedulability analysis of RM-US in [3] where the above inequality in Eq. (6) was straightforward to prove for so called light task syste. However, showing that the inequality in Eq. (6) also holds for special task syste is not as straightforward as for RM-US. To show that inequality in Eq. (6) holds we, in fact, show that the inequality in Eq. (7) holds. in{f (u k in),f (u k ax)} (7) 1 The proof that Eq. (7) holds is given in Lea 10 in the Appendix. Since Γ k is special on processors, using Property of special task syste we have U k in{f (u k in),f (u k ax)} (8) Fro Eq. (7) and Eq. (8), the inequality in Eq. (6) follows iediately. The outcoe of Lea 4 is that there exists a scheduling algorith, called OPT, through the use of which a task syste Γ k that is special on unit-capacity processors is feasible on identical processors each of speed 1. Now, since 1 ( 1 ) = 1, using Theore 1 together with the existence of an scheduling algorith OPT plus the fact that GS SM is work-conserving, we get W(GS SM,, 1, Γ k, t) W(OPT,, 1, Γk, t) (9) for all t 0. Note that according to algorith OPT, W(OPT,, 1,Γk,t) = t k u i = t U k for any tie instant t 0. Thus, the lower bound on the aount of work actually copleted by GS SM within [0,t) for executing the jobs of the task set Γ k is given in Eq. (10) as follows. k W(GS SM,, 1, Γ k, t) t u i = t U k (10) We will use Eq. (10) as the lower bound of the actual aount of work copleted by GS SM scheduling within a particular tie interval [0, t]. C. Slack-Monotonic Schedulability Analysis In this subsection, we prove that a task syste Γ k that is special on processors is feasible using GS SM on unitcapacity processors. First, we prove in Lea 5 that all the jobs of the lowest priority task τ j of task set Γ j, which is special on processors, coplete by their deadlines using GS SM scheduling. Then, it follows using induction on j = 1,,...k that the special task syste Γ k is feasible on processors using GS SM. Lea 5. Consider task set Γ j that is special on processors. All the jobs of task τ j eet their deadlines when Γ j is scheduled using GS SM on unit-capacity processors. Proof: We will prove this Lea using induction. Lets assue that all the (l 1) jobs of τ j have et their deadlines using GS SM scheduling algorith. We will prove that the l th job of τ j also eets the deadline. Using induction on l 1, the correctness of Lea 5 then iediately follows. Since each job of task τ j is released at each period T j, the l th job arrives at tie (l 1)T j and requires C j units of execution tie before its deadline lt j. Reeber fro Lea 4 that task set Γ j is schedulable using algorith OPT on processors each having speed 1. Thus, fro Eq. (10) we have, j W(GS SM,, 1, Γ j, (l 1)T j) (l 1)T j = (l 1)T j u i + (l 1)T ju j (11) According to Eq. (11), the iniu aount of work copleted by GS SM before the l th job of τ j arrives at tie (l 1)T j is (l 1)T j u i + (l 1)T j u j. Note that, prior to tie instant (l 1)T j, the aount of work u i
5 generated for task τ j is exactly (l 1)T j u j. Since we assue that all the (l 1) jobs of task τ j have et their deadlines (inductive hypothesis), the total work executed by GS SM for the higher-priority tasks τ 1, τ,... τ is at least (l 1)T j u i prior to the tie instant (l 1)T j. Lea 3 ensures that the axiu aount of work that can be copleted by all the higher-priority tasks τ 1, τ,...τ over the interval [0,lT j ) is bounded fro above by [C i + (lt j C i )u i ]. In the previous paragraph, we saw that at least (l 1)T j u i of this work is copleted prior to tie instant (l 1)T j. Therefore, at ost [(C i + (lt j C i )u i)] (l 1)T j u i = (C i + (T j C i )u i) aount of work reains to be executed after tie instant (l 1)T j for all the higher priority tasks τ 1,τ,...τ. The aount of processors capacity left unused by tasks τ 1, τ,... τ during the interval [(l 1)T j,lt j ) on the ultiprocessor platfor is therefore at least T j (C i + (T j C i )u i) (1) Not all of this capacity is available to the l th job of τ j if several processors are available at the sae tie. In the worst case (i.e., all the processors are available at the sae tie) at least 1 of this available capacity can be used by τ j. Consequently, the aount of processing capacity available to the l th job of τ j during the interval [(l 1)T j,lt j ) on the ultiprocessor platfor is at least 1 [ ] T j (C i + (T j C i )u i) To guarantee that the l th job of τ j eets its deadline, we need this capacity to be at least as large as the execution tie of τ j ; that is, we ust have, C j 1 [ ] T j (C i + (T j C i )u i) (13) In the reaining part of this proof, we show that Eq. (13) holds; which guarantees that the l th job of τ j eets its deadline. Since task set Γ j is special on processors, according to Property of special task set we have U j in{f (u j in ),F (u j ax)} (14) For task τ j Γ j, we have u j in u j u j ax. Thus, according to Property 1 of special task syste Γ j, we have 0 u j in u j u j ax 1. And using Eq. (3) of Lea, we have in{f (u j in ),F (u j ax)} F (u j ) (15) Fro Eq. (14) and Eq. (15), we have U j F (u j ) which is equivalent to j u i (1 uj) u j + u j [fro Eq. ()] u i( u j) (1 u j) u j 1 1 u i( u j) u j 1 [ [ ui + u i(1 u ]] j) Cj 1 [ [C i + Ci Tj Cj ( ) ] ] T j T j (According to slack-onotonic priority assignent, we have C i T j C j) Cj 1 [ [C i + Ci Ti Ci ( ) ] ] T j T j C j 1 [ ] [C it j T j + C i C ] i C j 1 [ [ ] ] T j Ci + (T j C i)u i Eq. (13) Since the inequality in Eq. (13) is true, we can conclude that the l th job of task τ j eets its deadline using GS SM. Based on Lea 5 we prove in Theore 6 that the task set Γ k that is special on processors is feasible using GS SM. Theore 6. Task syste Γ k that is special on processors is feasible using GS SM on processors. Proof: Using induction on j and applying Lea 5 for task set Γ j for j = 1,,...k, it is easy to see that the special task syste Γ k is feasible on processors using GS SM. Based on the GS SM feasibility of special task systes, we propose two slack-onotonic hybrid priority-assignent policies P bound and P search for an arbitrary task set Γ. VI. HYBRID PRIORITY ASSIGNMENT In this section, we propose the two hybrid priority assignent policies P bound and P search. Coon for both policies is that the priorities to the tasks are assigned based on soe threshold utilization u ts such that all the tasks having utilization not greater than u ts are given the slackonotonic priorities and each task having utilization greater than u ts is given the highest priority. Using such hybrid policy, the task set Γ is visualized as the union of two sets Γ = Γ L Γ H such that the tasks in set Γ L have the slackonotonic priorities and each task in set Γ H has the highest priority. It will be evident shortly that the value of u ts for policy P bound only depends on the nuber of processors. However, the value of u ts for policy P search depends not only on the nuber of processors but also on the utilization inforation of the individual task of a given task set. Before the thresholds for the two priority assignent policies are presented, we present two general conditions,
6 denoted as C1 and C, in Lea 7 that can iply the feasibility of a task set based on priority assignent policy P bound (P search ). The proof technique in Lea 7 is based on the notion of predictable scheduling algorith that is proposed by Ha and Liu in [14] and used in [1] as follows. Predictability (fro [14]): A job is characterized by its arrival tie, its deadline, its iniu execution tie and its axiu execution tie. The execution tie of a job is unknown but it is no less than and greater than its iniu and axiu execution tie, respectively. A scheduling algorith A is predictable if for every set J of jobs, the following fact scheduling all jobs in J by A with execution ties equal to their axiu execution ties causes all the deadlines to be et iplies that scheduling all jobs in J by A with execution ties being at least their iniu execution ties and at ost their axiu execution ties causes all the deadlines to be et. This notion of predictable scheduling algorith iplies that we only need to analyze the schedulability of the jobs considering the WCET of the jobs. Since a periodic task set generates a set of jobs, the notion of predictability can be extended in a straightforward anner to algoriths for scheduling periodic task systes. Ha and Liu s work also iplies that global static-priority scheduling of periodic tasks on ultiprocessors is predictable [3], [1]. Lea 7. Let u ts be the threshold utilization that is used to deterine the sets Γ L and Γ H such that Γ = Γ L Γ H assuing the priority assignent policy P bound (P search ). The task set Γ is schedulable using GS bound (GS search ) if the following two conditions C1 and C are satisfied (C1) (C) Γ H < Γ L is special on ( Γ H ) processors Proof: We will show that if C1 and C are true for priority assignent policy P bound (P search ) that uses u ts as the threshold utilization, then the task set Γ is schedulable using the GS bound (GS search ) scheduling. Consider the following task set Γ H such that Γ H = {τ i τ i Γ H and T i = C i = } For each task τ i Γ H, there is a corresponding task τ i in set Γ H such that the period and the WCET of τ i are equal to. Therefore, each task τ i Γ H has utilization 1. Note that, Γ H = Γ H and we let k = Γ H = Γ H. Now consider the task set Γ = Γ L Γ H that is to be scheduled on processors using GS bound (GS search ). According to policy P bound (P search ) that uses the threshold utilization u ts, each of the tasks in Γ H is given the highest priority and the tasks in Γ L are given the slack-onotonic priorities. When scheduling the task set Γ using GS bound (GS search ), then k = Γ H processors are devoted to tasks in set Γ H since these are the highest priority tasks each with utilization 1. All these tasks in Γ H are schedulable on k processors (one task is executed in one dedicated processor) since Γ H = Γ H = k < according to C1. According to C, the tasks in set Γ L are special on ( Γ H ) = ( k) processors. Reeber that according to Theore 6, task set Γ L that is special on ( k) processors is schedulable using global slack-onotonic scheduling GS SM on ( k) processors. Consequently, the task set Γ is schedulable on processors using GS bound (GS search ) scheduling. However, since global static-priority algorith GS bound (GS search ) is predictable, jobs of the tasks in set Γ H, in fact, coplete earlier than the jobs of the tasks in set Γ H. In other words, no jobs in task syste Γ finishes later than the corresponding job in Γ for predictable scheduling algorith GS bound (GS search ). Therefore, all the deadlines of the tasks in Γ are et using GS bound (GS search ) scheduling where the tasks are given the priorities based on policy P bound (P search ) whenever the conditions C1 and C are satisfied. Based on these two general conditions (C1 and C) of Lea 7, the schedulability analysis of GS bound and GS search, assuing the priority assignent policies P bound and P search, are presented in subsection VI-A and subsection VI-B, respectively. A. Priority Assignent Policy P bound In this section, we present the threshold utilization used for slack-onotonic hybrid priority assignent policy P bound and derive the corresponding utilization bound for GS bound scheduling. The value of u ts is defined based on the solution of the equation F (u ts )= u ts where is soe constant, > 1. One of the solutions of F (u ts )= u ts is u ts = for > 1. The value of u ts for policy P bound is u ts = B() for > 0 where B() is defined as follows: { 1 if = 1 B() = (16) if > 1 The two following inequalities in Eq. (17) and Eq. (18) hold for B() and B( ) where 1 B() 1 1 (17) B() B( ) (18) The proof that Eq. (17) and Eq. (18) hold are given in Lea 11 and Lea 1 in the Appendix. Now the schedulability condition in ters of utilization bound of GS bound scheduling is given in Theore 8.
7 Theore 8. A task set Γ is schedulable using GS bound if U n in{1/,b()} for. Proof: Given the task set Γ and the nuber of processors, we can deterine the two subsets Γ L and Γ H such that Γ = Γ L Γ H based on the threshold utilization u ts = B(). Note that, based on policy P bound, the tasks in set Γ L and Γ H are given the slack-onotonic and the highest static priorities, respectively. We will show that if the total utilization U n in{1/,b()}, then the two general conditions C1 and C of Lea 7 hold; which guarantees the schedulability of Γ using GS bound. (C1 holds) It is easy to see that B() in{1/,b()}. Then it follows that each task in Γ H has utilization greater than in{1/, B()} since each task in Γ H has utilization greater than u ts = B() using policy P bound. If the total utilization (i.e., U n ) of the task set Γ is not greater than in{1/,b()}, then the nuber of tasks that are given the highest priority is less than. In other words, we have Γ H <, and this iplies that condition C1 of Lea 7 holds. (C holds) To show that C of Lea 7 holds, we have to show that Γ L is special on processors where = ( Γ H ). Let UL be the total utilization of all the tasks in Γ L. Also let u axl and u inl be the axiu utilization and iniu utilization of any task in set Γ L, respectively. To show that Γ L is special on processors, we show that Property 1 and Property (given in Definition ) of special task set are satisfied. In other words, we have to show that the following two inequalities hold. Property 1 u axl 1 Property UL in{f (u inl),f (u axl)} (Property 1 holds for Γ L ) Using policy P bound, no task in Γ L has utilization greater than the threshold utilization u ts = B(). So, we have u axl B(). Moreover, fro Eq. (17), we have B() 1. Consequently, u axl 1 and thus Property 1 is satisfied for Γ L. (Property holds for Γ L ) The total utilization of the tasks in Γ H is greater than ( Γ H in{1/,b()}) because each task in Γ H has utilization greater than u ts = B() and B() in{1/, B()}. Since the total utilization (i.e., U n ) of the task set Γ is not greater than in{1/,b()}, the total utilization of the tasks in Γ L is at ost in{1/,b()} where = ( Γ H ). Therefore, Eq. (19) holds. UL in{1/, B()} (19) Based on the threshold utilization u th = B() of policy P bound, we have u axl B(). Moreover, fro Eq. (18), we have B() B( ). Thus, u axl B( ). Furtherore, fro Eq. (17), we have B( ) 1 (by replacing by in the left-hand side inequality in Eq. (17)). Therefore, u axl B( ) 1. Because 0 u inl u axl, the following inequality in Eq. (0) holds. 0 u inl u axl B( ) (0) 1 Based on Eq. (0) and fro Eq. (4) of Lea we have in{f (0), F (B( ))} in{f (u inl),f (u axl)} (1) Fro the function definition given in Eq. () we have F (0) = (1 0) = / () Fro Eq. (16), for = 1 and setting x = B( ) we have F (B( ))= 1 =. And for > 1 we have F (B( )) = B( ) because one solution for x of F (x) = x is x = B( ). Thus, for any F (B( )) in{1, B( )} (3) It follows fro Eq. () and Eq. (3) that in{f (0), F (B( ))} in{1/,b( )} (4) Now fro Eq. (4) and using the fact that B() B( ) fro Eq. (18), we have in{1/, B()} in{f (0), F (B( ))} (5) Thus it now follows fro Eq. (19) and Eq. (5) that UL in{f (0), F (B( )) } (6) Finally, fro Eq. (1) and Eq. (6), we have UL in{f (u inl), F (u axl) } (7) Therefore, Property is satisfied for task set Γ L. Consequently, Γ L is special on processors (i.e., C holds). B. Priority Assignent Policy P search Inspired by the priority assignent schee for EDF (k) scheduling in [13], the hybrid priority assignent policy P search is defined as follows. 1) Each of the k highest utilization tasks is given the highest priority, and ) the reaining (n k) lowest utilization tasks are given the slack-onotonic priorities for soe k such that 0 k <. The schedulability condition of GS search scheduling assuing policy P search is given in Theore 9. Theore 9. A task set Γ is schedulable using GS search if the set of (n k) lowest utilization tasks of Γ is special on ( k) processors for soe k and k <. Proof: Using policy P search the (n k) th highest utilization of the task in set Γ is used as the threshold utilization u ts, for soe k, 0 k <. This threshold utilization decides the tasks in set Γ L and Γ H that are given the slack-onotonic and the highest priorities such that Γ = Γ L Γ H. Note that, using policy P search, the nuber of tasks having the highest priority is Γ H = k for soe k and k <. Consequently, condition C1 of Lea 7 is satisfied
8 for policy P search. According to Lea 7, to guarantee the feasibility of task set Γ using GS search scheduling, the value of k has to be chosen such that the condition C of Lea 7 holds as well. In other words, task set Γ is feasible using GS search scheduling whenever Γ L is special on ( k) processors such that Γ L contains all the (n k) lowest utilization tasks. Deriving a k, if one exists, that satisfies Theore 9 is straightforward. One such exaple algorith, called P search (Γ), that finds (if exists) the value of k is presented in Figure 1. The algorith P search (Γ)returns True if it can find soe k such that the set of (n k) lowest utilization tasks of Γ is special on ( k) processors such that k <, otherwise, it returns False. Algorith P search (Γ) 1. Γ H =. Γ L = Γ 3. For k = 0 to ( 1) 4. If Γ L is special on ( k) processors Then 5. Print All tasks in Γ L are assigned SM priority 6. Print All tasks in Γ H are assigned the highest priority 7. Return True 8. End If 9. Find τ ts such that u ts is the largest utilization in set Γ L 10. Γ H = Γ H {τ ts} 11. Γ L = Γ Γ H 1. End For 13. Print Priority Assignent Fails 14. Return False Figure 1. Priority Assignent Algorith In line 1, the algorith P search (Γ)in Figure 1 initializes local variables Γ L and Γ H as Γ L = Γ and Γ H = to consider first whether all the tasks in Γ are special on processors (this condition is checked during the first iteration of the For loop in line 3 1). The For loop in line 3 1 iterates at ost ties for the iterative variable k that iterates for 0 to ( 1). In each iteration of the For loop, we check whether the (n k) lowest utilization tasks in set Γ L are special on ( k) processors. Note that in order to deterine whether Γ L is special on ( k) processors, we need to verify that both Property 1 and Property (Definition ) of special task syste are satisfied. If the task set Γ L is special on ( k) processors (condition at line 4 is true), then slack-onotonic priorities are assigned to the tasks in Γ L (line 5), each of the tasks in Γ H is assigned the highest static-priority (line 6) and the algorith returns True (line 7). During a particular iteration of the For loop, if the task set Γ L is not special on ( k) processors (condition at line 4 is false), then the highest utilization task τ ts Γ L is extracted fro Γ L (line 9) and is included in set Γ H (line 10). Note that during the k th iteration of the For loop, the largest utilization of the tasks in Γ L is the (n k) th largest utilization of the tasks in Γ. It is easy to see that, during each iteration of the For loop, total k largest utilization tasks are in set Γ H and the rest (n k) lowest utilization tasks are in set Γ L. If the task set Γ L is not special on ( k) processors for soe k such that 0 k <, then P search fails to assign the static-priorities to the tasks in Γ (line 13) and the algorith returns False (line 14). VII. PERFORMANCE EVALUATION In this section, first we show that GS bound doinates both the best state-of-the-art scheduling algoriths RM-US and SM-US (note that, if 5, RM-US is the best one, otherwise, SM-US is the best one). To show the doinance of GS bound over RM-US and SM-US, we show that the utilization bound of GS bound is higher than that of both RM-US and SM-US. Second, we show that GS search doinates GS bound. To show the doinance of GS search over GS bound, we show that any task set schedulable using GS bound is also schedulable using GS search but the converse is not true. Doinance of GS bound over RM-US and SM-US: The utilization bound of GS bound is in{1/,b()} where B() = for (Theore 8). The utilization bound of RM-US is 3 [1]. It is easy to see that, in{1/,b()} > 3 for any > (they are equal only for = ). The utilization bound of SM-US is [1]. It is also easy to see that, in{1/,b()} > for any. Consequently, the utilization bound of GS bound doinates that of both RM-US and SM-US. Doinance of GS search over GS bound : We will show that any task set Γ that is schedulable using GS bound is also schedulable using GS search. Assue a contradiction where task set Γ is not schedulable using GS search but schedulable using GS bound. If Γ is not schedulable using GS search, then there exist no k such that k < and the set of (n k) lowest utilization tasks is special on ( k) processors (contrapositive of Theore 9). When Γ is schedulable using GS bound scheduling, the proof of the schedulability condition of GS bound in Theore 8 guarantees that there exists a task set Γ L that is special on ( Γ H ) processors and Γ H <. So, there exists soe k such that k < and the set of (n k) lowest utilization tasks is special on ( k) processors (contradiction!). Therefore, any task set schedulable using GS bound is also schedulable using GS search. We will now show, using an exaple, that the converse is not true; that is, there is a task set that is schedulable using GS search but not schedulable using GS bound. Exaple: Consider n = 11 tasks in set Γ = {τ 1,...τ 11 } such that u 1 =... = u 10 = 0.40 and u 11 = Thus, the total utilization of Γ is U n = The task set Γ is to be scheduled on = 10 processors. The entire task set Γ is special on = 10 processors (i.e., Property 1 and Property of special task syste are
9 satisfied) and hence schedulable using GS search. Notice that Property 1 of special task syste is satisfied for Γ because u n ax = 0.4 < /( 1) for = 10. And Property of special task syste is satisfied for Γ as well. This is because for = 10, u n ax=0.40 and u n in =0.15, we have F (u n in ) and F (u n ax) = Consequently, in{f (u n in ),F (u n ax)} = which iplies U n in{f (u n in ),F (u n ax)} (i.e., Property ). Since Property 1 and Property are satisfied, Γ is special on = 10 processors. Thus, according to Theore 9, the set of (n k) lowest utilization tasks (i.e., entire task set Γ ) is special on ( k) processors for the choice of k = 0. However, the schedulability condition of GS bound in Theore 8 is not satisfied for task set Γ (i.e., utilization bound in{1/,b()} = < U n ). Consequently, we can not guarantee the schedulability of Γ using GS bound. In suary, GS search doinates GS bound which doinates both RM-US and SM-US scheduling. A. Siulation Results To quantitatively estiate the degree of doinance of GS search over the state-of-the-art algorith, we conducted experients using randoly generated task sets for = 4, 8, 16 and 3 processors. For siulation purpose, we considered SM-US as the copetitive state-of-the-art algorith and estiated the doinance of GS search over SM-US scheduling. The conclusions fro our experients is that GS search provides an order-of-agnitude perforance iproveents in ters of nuber of scheduled task sets. 1) Siulation Setup: We run a nuber of 1 experients. Each experient has three siulation paraeters:, inu, axu. Each experient is characterized by the value of and the range (inu,axu]. The value of denotes the nuber of processors we consider for an experient. Each experient is carried out for a nuber of task sets. The utilization u i of a randoly generated task τ i of a task set is uniforly distributed within (inu, axu]. The nuber of processors we consider in our experients are = 4, 8, 16, and 3. This paraeter is used to easure the ipact of increasing nuber of processors (scalability) on nuber of schedulable task sets. We consider three different utilization ranges (0, 0.5], (0.5, 0.75] and (0, 1] for (inu, axu]. The utilization ranges (0, 0.5], (0.5, 0.75] and (0, 1] are used to experient with light, ediu and ixed tasks, respectively. The four values of and the three utilization ranges for (inu, axu] constitute our 1 different experients. For each a total of task sets, that are all schedulable using GS search on processors, are randolygenerated using a siilar approach as in [8]. The task sets are generated according to the following procedure: 1) Initially, we generate + 1 tasks. ) Then we verify if the generated task set is schedulable using GS search (i.e., Theore 9 is used for verification). 3) If this task set is GS search schedulable, then a) it is counted as one of the task sets, and b) by adding one new task, we extend this (old) task set to a new task set and return to Step. 4) If this task set is not GS search schedulable, then we discard this task set and go to Step 1. For each experient, we calculated the doinance of GS search over SM-US as the percentage of the 1 illion task sets that are not schedulable using SM-US scheduling. First, the total nuber of task sets (of the GS search schedulable task sets) that are also schedulable by SM-US scheduling is counted in variable P searchcount. Second, the doinance of the GS search over SM-US scheduling, denoted by DOM Psearch, is given as follows: DOM Psearch = P searchcount 100% The higher the value of DOM Psearch, the higher is the degree of doinance of GS search over SM-US scheduling. For exaple, if DOM Psearch = 0%, then 0% (0000 task sets) of the task sets are not schedulable using SM-US scheduling. ) Siulation Result: The result of the 1 experients are given in Table I. Each of the shaded cells in Table I represents the value of DOM Psearch for the siulation paraeters nuber of processors and utilization range given in the corresponding second row and first colun of Table I. (inu,axu] (0, 0.5] (0.5, 0.75] (0, 1] = % % 9.06 % = % % % = % % 99.1 % = % % % Table I VALUE OF DOM PS E A R C H FOR THE 1 EXPERIMENTS The ipact of utilization ranges (inu,axu] on the doinance of GS search over SM-US is significant. The value of doinance DOM Psearch for ediu (i.e. u i is in [0.5, 0.75]) and ixed (i.e. u i is in [0, 1]) tasks is significantly higher (see the third and forth coluns of Table I) than that of for the light tasks (i.e. u i is in [0, 0.5]). The SM-US scheduling fails to schedule ore than 9% of the randoly generated one illion task sets (having ediu/ixed tasks) that are schedulable using GS search. This significant perforance gain of GS search scheduling in ters of nuber of schedulable ediu/ixed task sets results fro our schedulability analysis of the special task systes. Reeber that the threshold utilization used for SM-US is constant (i.e., /(3 + 5)) while that of for
10 GS search scheduling is searched as one of the utilizations of the tasks of a given task set. Thus, a task τ i with utilization greater than /(3 + 5) is always given the highest priority in SM-US scheduling while this task ay be assigned slack-onotonic priority in GS search scheduling. Such exploitation of the tasks utilization to deterine the threshold utilization for GS search results in fewer nuber of tasks that are given the highest priority in order to avoid the Dhall s effect. In other words, GS search is ore effective than SM-US in dealing with tasks having large utilization. Thus, GS search scheduling significantly outperfors SM-US scheduling for ediu/ixed tasks in ters of nuber of schedulable task sets. The nuber of processors used in an experient has an ipact on the value of doinance as well. As the nuber of processors increases, the scalability of GS search scheduling in ters of doinance increases for experients with ediu/ixed tasks (notice the increasing trend of doinance value in third and forth coluns of Table I). This is because as increases, the nuber of tasks in a task set also increases due to the way tasks are generated for our experients. As the nuber of tasks increases for the experients with ediu and large tasks having larger nuber of processors, it is ore likely that the nuber of tasks having relatively larger individual task utilization in a task set also increases. The ability of GS search to deal with relatively larger nuber of high utilization tasks results GS search to schedule large nuber of ediu/ixed task sets in coparison to that of SM-US. Consequently, the value of doinance for GS search increases as increases. However, the value of doinance decreases for experients with light tasks as we increase the nuber of processors (notice the decreasing trend of doinance in second colun of Table I). This is because when tasks are light (i.e., u i 0.5), ost of the individual task s utilization of a task set are relatively sall. As a result ost of the tasks of such task sets are given the slack-onotonic priority for both SM-US and GS search scheduling. In other words, the difference between the assigned priorities to the tasks in SM-US and GS search scheduling decreases as we increase the nuber of processors for experients with light tasks. Consequently, the value of doinance decreases. Nevertheless, the value of doinance DOM Psearch for the light tasks is still greater than 3.5% for our experients with at ost 3 processors. In suary, GS search scheduling of ediu/ixed tasks scales very well with the increase in nuber of processors, and the doinance of GS search over SM-US for light tasks is also significant. In general, if the utilization of the tasks are within [0,1], then ore than 9% of the 1 illion randoly-generated GS search schedulable task sets are not schedulable using SM-US scheduling. This percentage increases up to 99.99% as we increase the nuber of processors in our experients fro 4 to 3. The conclusion fro the experients is thus that GS search provides an order-of-agnitude perforance iproveent in ters of nuber of scheduled task sets, and scales very well as the nuber of processors increases. B. Uniprocessor Slack-Monotonic Scheduling The schedulability analysis of special task syste on ultiprocessors (Section V) enables the derivation of a higher utilization bound for uniprocessor slack-onotonic scheduling copared to that of the state-of-the-art result in []. Our proposed utilization bound for uniprocessor slack-onotonic scheduling is F 1 (u n in ). We now show how we derive this bound F 1 (u n in ) and also show that it is higher than that of the state-of-the-art result (i.e., 50%) proposed in []. Consider a task syste Γ that is special on uniprocessor (i.e. = 1). According to Property 1 of special task syste Γ, we have u n ax 1 because /( 1) = 1 for = 1. Therefore, special task syste Γ is in fact an arbitrary task syste for uniprocessor. Note that we have 0 < u n in u n ax 1 where u n in and un ax are the iniu and axiu utilization of any task in Γ, respectively. For = 1, the function F 1 (x) is increasing within [0,1] since F 1(x) 1 = 1 > 0 within (0,1). Consequently, in{f 1 (u n in ),F 1(u n ax)} = F 1 (u n in ) since u n in un ax. It is obvious fro Property of special task syste Γ that we have U n in{f 1 (u n in),f 1 (u n ax)} = F 1 (u n in) (8) Using Theore 6, the special task set Γ is schedulable using GS SM (uniprocessor slack-onotonic scheduling) on = 1 processor. Thus, based on Eq (8) our proposed utilization bound for uniprocessor slack-onotonic scheduling is F 1 (u n in ). The current state-of-the-art utilization bound for slack-onotonic uniprocessor scheduling is 50% which is proposed in []. We now show that F 1 (u n in ) > 50%. Since the function F 1 (x) is increasing within [0,1], we have F 1 (u n in ) > F 1 (0) since u n in > 0. Note that F 1(0) = 1(1 0) = 1/ = 50%. Therefore, F 1 (u n in ) > 50%. Our proposed utilization bound F 1 (u n in ) for the uniprocessor slack-onotonic scheduling is higher than that of the stateof-the-art result in []. VIII. CONCLUSION In this paper, we have proposed two static-priority assignent policies and proved their superior schedulability copared to the state-of-the-art global ultiprocessor scheduling algoriths. We have also derived a higher utilization bound for uniprocessor slack-onotonic scheduling than that of the state-of-the-art result. Our experiental results showed that around ore than 9% of the one illion GS search schedulable task sets are not schedulable using SM-US. In addition, the GS search scheduling scales very well with the increase in nuber of processors.
11 REFERENCES [1] B. Andersson. Global Static-Priority Preeptive Multiprocessor Scheduling with Utilization Bound 38%. In Proc.of OPODIS, pages 73 88, 008. [] B. Andersson. The Utilization Bound of Uniprocessor Preeptive Slack-Monotonic Scheduling is 50%. In Proc. of ACM Syp. On Applied Coputing, pages 81 83, 008. [3] B. Andersson, S. Baruah, and J. Jonsson. Static-Priority Scheduling on Multiprocessors. In Proc. of RTSS, pages 193 0, 001. [4] B. Andersson and J. Jonsson. The utilization bounds of partitioned and pfair static-priority scheduling on ultiprocessors are 50%. In Proc. of ECRTS, pages 33 40, 003. [5] T. P. Baker. An Analysis of Fixed-Priority Schedulability on a Multiprocessor. Real-Tie Systes, 3(1-):49 71, 006. [6] S. Baruah and J. Goossens. Rate-onotonic scheduling on unifor ultiprocessors. IEEE Transactions on Coputers, 5(7): , 003. [7] M. Bertogna, M. Cirinei, and G. Lipari. New Schedulability Tests for Real-Tie Task Sets Scheduled by Deadline Monotonic on Multiprocessors. In Proc. of OPODIS, pages , 005. [8] M. Bertogna, M. Cirinei, and G. Lipari. Schedulability Analysis of Global Scheduling Algoriths on Multiprocessor Platfors. IEEE Transactions on Parallel and Distributed Systes, 0(4): , 009. [9] E. Bini, T. H. C. Nguyen, P. Richard, and S. Baruah. A Response-Tie Bound in Fixed-Priority Scheduling with Arbitrary Deadlines. IEEE Trans. Coput., 58():79 86, 009. [10] R. I. Davis and A. Burns. Priority Assignent for Global Fixed Priority Pre-Eptive Scheduling in Multiprocessor Real-Tie Systes. In Proc. of RTSS, pages , 009. [11] S. K. Dhall and C. L. Liu. On a Real-Tie Scheduling Proble. Operations Research, 6(1):17 140, [1] N. Fisher, S. Baruah, and T. P. Baker. The Partitioned Scheduling of Sporadic Tasks According to Static-Priorities. In Proc. of ECRTS, pages , 006. [13] J. Goossens, S. Funk, and S. Baruah. Priority-driven scheduling of periodic task systes on ultiprocessors. Real-Tie Syst., 5(-3):187 05, 003. [14] R. Ha and J. Liu. Validating tiing constraints in ultiprocessor and distributed real-tie systes. In Proc. of ICDCS, pages , [15] C. L. Liu and J. W. Layland. Scheduling Algoriths for Multiprograing in a Hard-Real-Tie Environent. Journal of the ACM, 0(1):46 61, [16] D.-I. Oh and T. P. Baker. Utilization Bounds for N-Processor Rate Monotone Scheduling with Static Processor Assignent. Real-Tie Systes, 15():183 19, [17] C. A. Phillips, C. Stein, E. Torng, and J. Wein. Optial tie-critical scheduling via resource augentation (extended abstract). In Proc. of the ACM syposiu on Theory of coputing, pages , APPENDIX Lea. Consider a, b, x, c and d such that 0 a b x c d 1 for any integer > 0. The following two inequalities hold: in{f (b),f (c)} F (x) (3) in{f (a),f (d)} in{f (b),f (c)} (4) Proof: To show that Eq. (3) holds we will show that, within [b,c] where b x c, the function F (x) = (1 x) x +x achieves its absolute iniu at one of the endpoints of [b,c] for any given. Thus, the iniu between F (b) and F (c) is the absolute iniu of F (x), and consequently Eq. (3) holds. The first derivative of F (x) = (1 x) x + x with respect of x is F (x) = 1. By setting F (x) = 0, we have x = ( ± ). For any value of > 0, the point x = ( + ) is outside of (b,c) since c 1 1 for > 0. Moreover, the point x = ( ) is outside of (0, 1 ) for both = 1 and 4. Consequently, x = ( ) is also outside of (b,c) because (b,c) is entirely contained within (0, 1 ) for = 1 and 4. So, the only possible x values where x = ( ) and F (x) = 0 are x = ( ) and x = ( 3) for = and = 3, respectively (stationary points). Since there is no stationary point of F (x) within (b,c) for = 1 or 4, the absolute iniu of F (x) occurs at one of the end points of [b,c] for = 1 and 4. Now for =, if the point x = ( ) is outside of (b,c), then the absolute iniu of F (x) occurs at one of the endpoints of [b,c]. Otherwise, if the point x = ( ) is within (b,c), then the absolute iniu occurs at one of the points x = a, x = ( ), or x = b. The function F (x) is increasing within [b, ) since F (x) =1 > 0 within (b, ) and F (x) is decreasing within (,c] since F (x) =1 < 0 within (,c). Therefore, the function F (x) has its absolute axiu at x = ( ). Thus, the absolute iniu of F (x) occurs at one of the end points of [b,c]. Siilarly for = 3, if the point x = ( 3) is outside of (b,c), then the absolute iniu of F 3 (x) occurs at one of the endpoints of [b, c]. Otherwise, if the point x = ( 3) is within (b,c), then the absolute iniu occurs at one of the points x = a, x = ( 3), or x = b. The function F 3 (x) is increasing within [b, 3) since F 3(x) =1 3 > 0 within (b, 3) and F 3 (x) is decreasing within ( 3,c] since F 3(x) =1 3 < 0 within ( 3,c). Therefore, the function F 3 (x) has its absolute axiu at x = ( 3). Consequently, the absolute iniu of F 3 (x) occurs at one of the end points of [b,c]. Since the function F (x) has its iniu at one of the end points of [b,c] for any, we can conclude that if x is within [b,c] then F (x) is not less than the iniu between F (b) and F (c). Therefore, Eq. (3) holds. Since a b d and a c d, it follows fro Eq. (3) that in{f (a),f (d)} F (b) and in{f (a),f (d)} F (c). Consequently, in{f (a),f (d)} in{f (b),f (c)} which proves that Eq. (4) also holds.
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