Multi-Rate Fluid Scheduling of Mixed-Criticality Systems on Multiprocessors
NNoname manuscript No. (will be inserted by the editor)
Multi-Rate Fluid Scheduling of Mixed-CriticalitySystems on Multiprocessors
Saravanan Ramanathan · ArvindEaswaran · Hyeonjoong Cho the date of receipt and acceptance should be inserted later
Abstract
In this paper we consider the problem of mixed-criticality (MC)scheduling of implicit-deadline sporadic task systems on a homogenous multi-processor platform. Focusing on dual-criticality systems, algorithms based onthe fluid scheduling model have been proposed in the past. These algorithmsuse a dual-rate execution model for each high-criticality task depending onthe system mode. Once the system switches to the high-criticality mode, theexecution rates of such tasks are increased to meet their increased demand.Although these algorithms are speed-up optimal, they are unable to scheduleseveral feasible dual-criticality task systems. This is because a single fixed ex-ecution rate for each high-criticality task after the mode switch is not efficientto handle the high variability in demand during the transition period imme-diately following the mode switch. This demand variability exists as long asthe carry-over jobs of high-criticality tasks, that is jobs released before themode switch, have not completed. Addressing this shortcoming, we propose amulti-rate fluid execution model for dual-criticality task systems in this paper.Under this model, high-criticality tasks are allocated varying execution ratesin the transition period after the mode switch to efficiently handle the demandvariability. We derive a sufficient schedulability test for the proposed modeland show its dominance over the dual-rate fluid execution model. Further, wealso present a speed-up optimal rate assignment strategy for the multi-rate
This research was funded in part by the Ministry of Education, Singapore, Tier-1 grantRG21/13 and Tier-2 grant ARC9/14, and by the Start-Up-Grant from SCSE, NTU, Sin-gapore. This research was also partly supported by Basic Science Research Program of theNational Research Foundation of Korea (NRF- 2015R1D1A1A01057018).Saravanan Ramanathan · Arvind EaswaranSchool of Computer Science and Engineering, Nanyang Technological University, SingaporeE-mail: { saravana016,arvinde } @e.ntu.edu.sgHyeonjoong ChoDepartment of Computer and Information Science, Korea UniversityE-mail: [email protected] a r X i v : . [ c s . O S ] M a r Saravanan Ramanathan et al. model, and experimentally show that the proposed model outperforms all theexisting MC scheduling algorithms with known speed-up bounds.
Keywords
Mixed-Criticality · Implicit-deadline sporadic tasks · Multipro-cessors · Fluid scheduling
The mixed-criticality (MC) model proposed by Vestal [18] has received a lotof attention in the literature on real-time scheduling. Several studies exist onthe design of multiprocessor MC scheduling algorithms; see [5] for a review.To evaluate the schedulability performance of these algorithms two techniquesare generally used: 1) experimental evaluation in which the schedulability isassessed over a wide variety of task systems, and 2) analytical performancebounds such as the speed-up bound are derived. Scheduling algorithms withhigh schedulability in experimental evaluation as well as good (low) speed-upbound are highly desirable.Based on the above motivation, fluid scheduling algorithms have recentlybeen proposed for MC scheduling of task systems on homogenous multiproces-sor platforms [4,10]. For scheduling implicit-deadline MC task systems withtwo criticality levels (dual-criticality systems), Lee et al. [10] proposed a dual-rate fluid scheduling model in which each high-criticality task executes usingtwo rates and each low-criticality task executes using a single rate. The high-criticality task rates depend on the mode in which the system is operating;they execute using higher rates once the system switches to the high-criticalitymode. Note low-criticality tasks are suspended after the system switches to thehigh-criticality mode. A rate assignment strategy called MC-Fluid has alsobeen proposed, and it is shown to be dual-rate optimal ; if there is a feasi-ble dual-rate assignment for a dual-criticality task system, then MC-Fluid isguaranteed to find it. Subsequently, Baruah et al. [4] derived a simplified rateassignment strategy called MCF for dual-rate fluid scheduling, and showedthat both MC-Fluid and MCF have an optimal speed-up bound of 4 / / The speed-up bound of a scheduling algorithm is defined as the maximum additionalprocessor speed required to schedule any feasible task system using the algorithm [9].ulti-Rate Fluid Scheduling of Mixed-Criticality Systems on Multiprocessors 3 by low-criticality tasks, their left-over demand after mode switch may be dif-ferent (usually higher) than the demand of pure high-criticality jobs that arereleased after the mode switch. Therefore, in the transition period while suchcarry-over jobs are executing, the overall demand of high-criticality tasks isvarying depending on the number of pending carry-over jobs.To address the aforementioned limitation of the dual-rate fluid schedul-ing model, in this paper we propose a multi-rate fluid scheduling model fordual-criticality task systems scheduled on homogenous multiprocessor plat-forms. Under this model, each high-criticality task executes using a singleexecution rate in the low-criticality mode and a set of execution rates in thehigh-criticality mode. In particular, the task executes using different rates overtime until all carry-over jobs are guaranteed to be completed, and the purehigh-criticality jobs released subsequently execute using a single rate basedon task utilization. These rates and the time duration for which they are ap-plicable are all determined offline so as to ensure worst-case schedulability.Similar to the dual-rate model, each low-criticality task uses a single execu-tion rate based on task utilization before mode switch and is suspended afterthe mode switch. Thus, by using a fine-grained rate allocation in the transi-tion period, the multi-rate model is able to accommodate a higher left-overdemand for the carry-over jobs after the mode switch. This in turn enables itto accommodate a higher demand in the low-criticality mode, thus improvingschedulability over the dual-rate model. We now illustrate this benefit of themulti-rate model using a simple example.
Example 1
In Table 1 we show a dual-criticality task system that is schedulableunder the multi-rate model on 2 processors, but is not schedulable under thedual-rate model. Under the dual-rate model, both the carry-over job and thepure high-criticality jobs of a high-criticality task execute using a single rate.As can be seen, using the dual-rate model, the resulting execution rates in low-criticality mode are not feasible ( > w , while tasks τ and τ execute at themaximum rate of 1, τ does not execute at all. Later on in window w , when τ and τ are executing at a much lower rate equal to their utilization, τ is able toexecute at a rate of 0 .
5. This is even higher than the rate it was allocated in thedual-rate model (0 . τ across windows w , w and w in the multi-rate model is higher than the rate allocated to it Saravanan Ramanathan et al.
Tasks Dual-rate/3-rate assignment Multi-rate assignment C Li C Hi T i u Li u Hi θ Li θ Hi (= θ Ci ) θ Pi (= u Hi ) θ Li θ Hi, w θ Hi, w θ Hi, w θ Hi (= u Hi ) τ τ τ τ (cid:80) Table 1 Example task system schedulable under the multi-rate fluid model, butnot under the dual-rate fluid model on processors . C Li denotes low-criticality exe-cution time estimate, C Hi denotes high-criticality execution time estimate, T i denotes taskperiod, u Li = C Li /T i and u Hi = C Hi /T i . For the dual-rate fluid model, θ Li denotes executionrate in the low-criticality mode and θ Hi denotes execution rate in the high-criticality mode.For the modified dual-rate fluid model (3-rate assignment), θ Li denotes execution rate inthe low-criticality mode, θ Ci denotes execution rate in the transition period and θ Pi denotesexecution rate in the pure high-criticality mode. For the multi-rate fluid model, θ Li denotesexecution rate in the low-criticality mode, θ Hi,j ( j ∈ { , , } ) and θ Hi denote several executionrates for the high-criticality mode, and w j ( j ∈ { , , } ) denotes duration of time for whichrate θ Hi,j will be used by task τ i . After w + w + w time units in the high-criticality mode,each high-criticality task τ i will use the rate θ Hi . in the dual-rate model. As a result its rate in low-criticality mode is reduced,and the overall task system becomes schedulable (schedulability is formallyverified in Sect. 4). Contributions.
The contributions of this paper can be summarized as follows. – We propose a new multi-rate fluid model for scheduling implicit-deadlineMC task systems on a homogenous multiprocessor platform (Sect. 3). – We derive a sufficient schedulability test for the multi-rate model (Sect. 4),and show that it dominates the schedulability test for the dual-rate model(Sect. 5). – We present a convex optimization based rate and window duration assign-ment strategy for the multi-rate model called SOMA (Speed-up OptimalMulti-rate Assignment), and prove that it is speed-up optimal with a speed-up bound of 4 / – We present results from extensive experimental evaluation and show thatSOMA outperforms all the other multiprocessor MC scheduling algorithmswith known speed-up bounds.
Related Work.
Several studies have been done on the design of multi-core MCscheduling algorithms in recent years ([1,3,4,8,10,11,13,16,17]). Of which,only a few provided both experimental evaluation and analytical performancebounds ([3,4,10,11]). Li and Baruah [11] proposed GLO-EDF VD, a globalscheduling algorithm combining the multiprocessor fixed priority algorithmfpEDF and uniprocessor virtual deadline based MC algorithm EDF-VD, andproved that the algorithm has a speed-up bound of √ / − / m . They also ulti-Rate Fluid Scheduling of Mixed-Criticality Systems on Multiprocessors 5 showed through experimental evaluation that the partitioned algorithm offersbetter schedulability than the global variant. Lee et al. [10] proposed the dual-rate fluid scheduling model and rate assignment algorithm called MC-Fluid,and showed that MC-Fluid has a speed-up bound of ( √ /
2. Recently,Baruah et al. [4] proposed a simplified rate assignment algorithm for the dual-rate fluid model called MCF, and proved that both MC-Fluid and MCF arespeed-up optimal with a speed-up bound of 4 / / Organization.
The remainder of the paper is as follows. We describe the systemmodel, notations and the dual-rate fluid scheduling algorithm in Sect. 2. Weintroduce the multi-rate fluid model in Sect. 3 and prove its correctness inSect. 4. We present the properties of the multi-rate fluid model in Sect. 5.In Sect. 6 we present the execution rates assignment strategy and a heuristicfor computing these execution rates. We describe the experiments conductedto evaluate the performance of our algorithm with the existing algorithms inSect. 7. Sect. 8 concludes the paper and presents the possible future work influid scheduling.
We consider an implicit-deadline sporadic MC task system with two crit-icality levels (LO and HI). Each MC task τ i is characterized by a tuple( T i , χ i , C Li , C Hi ), where – T i ∈ R + is the minimum release separation time; we assume an implicit-deadline task model, where deadline D i of a task is equal to T i . – χ i ∈ { LO, HI } denotes the criticality level of the task; we use the shortcutnotation LO-task and
HI-task to denote a LO-criticality and HI-criticalitytask respectively. – C Li ∈ R + is the LO-criticality worst case execution time (WCET) value(denoted as LO-WCET ). – C Hi ∈ R + is the HI-criticality WCET value (denoted as HI-WCET ); weassume C Li ≤ C Hi for all HI-tasks and C Li = C Hi for all LO-tasks.We consider a MC task system τ comprised of n sporadic tasks { τ , . . . , τ n } .Let n H denote the number of HI-tasks in the system. Let τ L def = { τ i ∈ τ | χ i = LO } and τ H def = { τ i ∈ τ | χ i = HI } denote the set of LO and HI-tasksrespectively.Task and system utilizations are denoted as follows. u Li def = C Li /T i , u Hi def = C Hi /T i , U LL def = (cid:80) τ i ∈ τ L u Li /m , U LH def = (cid:80) τ i ∈ τ H u Li /m and U HH def = (cid:80) τ i ∈ τ H u Hi /m .We assume that the tasks execute sequentially and are not allowed tosimultaneously execute on more than one processor at any given time (i.e., u Li ≤ u Hi ≤ m cores. Saravanan Ramanathan et al.
MC Modes.
The system starts in
LO-mode and remains in that mode as longas all the tasks signal completion before exceeding LO-WCET values. Thesystem switches to
HI-mode at the instant when any HI-task executes beyondits LO-WCET and does not signal completion.
Mode switch instant is definedas the time instant when this mode change occurs. The system can safelyreturn back to LO-mode at the time instant when all processors idle after themode switch. We assume that no job of LO-task τ i would exceed C Li and nojob of HI-task τ i would exceed C Hi . MC-Schedulable.
A task set τ is said to be MC-schedulable by a schedulingalgorithm if, – LO-mode guarantee: Every job of each task in τ is able to complete LO-WCET execution within its deadline, and – HI-mode guarantee: Every job of each task in τ H is able to complete HI-WCET execution within its deadline.Since no LO-task deadlines, including for jobs that are released before themode switch, are required to be met in HI-mode, it is possible to drop all theLO-task jobs immediately upon mode switch. The HI-task jobs in HI-modecan be classified into two types depending on their release time. A job of task τ i is said to be a carry-over job , if it is released before mode switch instantbut has not completed its execution until mode switch. All the remaining HI-mode jobs of τ i , those that are released after the mode switch, are called pureHI-mode jobs . The carry-over and pure HI-mode jobs of all the HI-tasksmust be guaranteed HI-WCET budgets by their deadlines. Similarly, the pureLO-mode jobs of all the tasks, those with deadlines before mode switch, mustbe guaranteed LO-WCET budgets by their deadlines.2.1 Dual-rate Fluid ModelThe dual-rate fluid model [10] was designed to schedule implicit deadline tasksystems with two criticality levels. It assigns different execution rates to tasksin each criticality level. The execution rate of a task is formally defined inDefinition 1. Definition 1 (Execution rate, from [10])
A task τ i is said to be executedwith execution rate θ i ∈ R + , s.t. 0 < θ i ≤
1, if every job of the task is executedon a fractional processor with a speed of θ i .The dual-rate model can be summarized as follows: – Each task τ i ∈ τ executes at a constant rate θ Li (where θ Li ∈ [ u Li , – All tasks in τ L are discarded immediately upon mode switch, and – Each task τ i ∈ τ H executes at a constant rate θ Hi (where θ Hi ∈ [ u Hi , ulti-Rate Fluid Scheduling of Mixed-Criticality Systems on Multiprocessors 7 An exact schedulability test for the dual-rate fluid model has also beenderived (Theorem 1 in [10]). Task set τ is said to be MC-schedulable underdual-rate fluid scheduling iff ∀ τ i ∈ τ, θ Li ≥ u Li (1) ∀ τ i ∈ τ H , u Li θ Li + u Hi − u Li θ Hi ≤ ∀ τ i ∈ τ H , θ Hi ≥ θ Li (3) (cid:88) τ i ∈ τ θ Li ≤ m (4) (cid:88) τ i ∈ τ H θ Hi ≤ m (5)Equations (4) and (5) ensure that the assigned rates are feasible in eachmode on the multiprocessor platform (denoted as platform feasibility tests ).Equation (1) ensures that each task is schedulable in the LO-mode, i.e., eachjob of the task is able to receive sufficient budget (proportional to u Li ) within itsdeadline (denoted as LO-mode task schedulability test ). Likewise, Equa-tions (2) and (3) ensure that each HI-task is schedulable in the HI-mode,including carry-over jobs (denoted as
HI-mode task schedulability test ).Note that although Equation (3) is not specified in Theorem 1 of [10], it isassumed in the derivation of the theorem. This HI-mode test is essentiallyderived by identifying a worst case mode switch instant for each HI-task. Asshown in [10], the worst case mode switch occurs at the same time instantwhen a carry-over job of the task completes its LO-WCET, i.e., the task itselftriggers the mode switch. This observation is intuitive, because in this case thecarry-over job executes using the higher HI-mode rate for the shortest possibleduration of time.
Under the multi-rate fluid scheduling model that we propose in this paper,each task τ i ∈ τ executes with a single rate in the LO-mode as in the dual-rate model. But, unlike the dual-rate model, each task τ i ∈ τ H executes withat most n H + 1 rates in the HI-mode, where n H = | τ H | is the number of HI-criticality tasks. The intuition behind these multiple rates in the HI-mode canbe explained as follows. When a mode switch occurs, there are potentially n H carry over jobs in the system, all of which may require an execution rate higherthan their HI-criticality utilization (i.e., u Hi ). By allowing them to adjust theirexecution rates several times in the transition period immediately after a modeswitch, it may be possible to accommodate more carry over demand in thesystem. Saravanan Ramanathan et al.
Multi-rate Fluid Scheduling Model : The multi-rate fluid model can be formallydefined as follows: – LO-mode execution rate : Each job of task τ i ∈ τ will start executing at arate θ Li in the LO-mode. If there is no mode switch while the job is active,then this is the only rate at which it will execute. – Transition execution rates : Upon a mode switch, all the LO-tasks will beimmediately dropped. The transition period after the mode switch is par-titioned into n H transition windows ( j : 1 ≤ j ≤ n H ), each of a fixedduration w j . The jobs of a HI-task τ i ∈ τ H execute at a constant rate of θ Hi,j in each transition window j . – HI-mode execution rate : After the completion of the transition period (after (cid:80) j w j time units from the mode switch), each job of HI-task τ i ∈ τ H executes at a constant rate of θ Hi .Thus, we denote the bounded period of time between the mode switch and (cid:80) j w j time units thereafter as the transition period . Note that the executionrates in the transition period can be used either by carry over jobs or pure HI-mode jobs that are released in the transition period. We denote the HI-modejobs released in the transition period as transition jobs and the HI-mode jobsreleased after the transition period as stable jobs . It is also worth noting thatin this multi-rate model the n H transition window durations are all determinedoffline, and remain fixed at runtime. As a consequence, similar to the dual-ratemodel, the runtime scheduling mechanism for this multi-rate model is also verysimple.The proposed multi-rate model is a generalization of the dual-rate modelthat was discussed in Sect. 2.1. If we set all the transition execution rates tobe equal to θ Hi , then in the resulting model, each task executes with a fixedrate in LO-mode and a fixed rate in HI-mode. This setting is identical to thedual-rate model. In the previous section we presented the multi-rate model for dual-criticalityimplicit-deadline sporadic task system and the worst-case mode switch patternfor which the execution rates need to be determined. In this section we derive asufficient schedulability test for the multi-rate fluid scheduling model defined inSect. 3. We also show that this test is equivalent to the dual-rate schedulabilitytest (Theorem 1 in [10]) if we set all the transition rates to θ Hi .This derivation is comprised of four steps:1. LO-mode task schedulability test : We derive a test using LO-modeexecution rates to ensure that each task is schedulable in the LO-mode.2.
LO-mode platform feasibility test : We derive a test using LO-modeexecution rates and the number of cores to ensure that the allocated LO-mode rates are feasible on the multiprocessor platform. ulti-Rate Fluid Scheduling of Mixed-Criticality Systems on Multiprocessors 9 HI-mode platform feasibility test : We derive a test using HI-modeexecution rates and the number of cores to ensure that the allocated HI-mode rates are feasible on the multiprocessor platform. Here we need toconsider all the n H + 1 rates assigned to each HI-task.4. HI-mode task schedulability test : We derive several conditions us-ing LO-mode and HI-mode execution rates to ensure that each HI-task isschedulable in the HI-mode. We need to consider carry over jobs, transitionjobs as well as stable jobs in this step.
LO-mode task schedulability test.
In the LO-mode, jobs of each task τ i ∈ τ execute with a single rate θ Li . It is easy to see that θ Li ≥ u Li is a necessarycondition for task schedulability in LO-mode. Since our framework is basedon fluid scheduling, this is also a sufficient schedulability condition in the LO-mode. If the task receives an execution rate of at least u Li , then the totalallocated budget to each job of the task is at least u Li × T i , which is sufficientto meet deadlines. We record this test in the following proposition. Proposition 1
Task system τ is schedulable in the LO-mode iff ∀ τ i ∈ τ, θ Li ≥ u Li . LO-mode platform feasibility test.
In the LO-mode, a set of execution ratesis feasible on a multiprocessor platform comprising m cores if and only if thesum total of the rates is no more than m . If the total is more than m , thenclearly the rates cannot be assigned using m cores. Whereas if it is no morethan m , then it can be assigned because a single core can schedule multipletasks at the same time under the fluid scheduling model. Proposition 2
Execution rates assigned to task system τ are feasible in theLO-mode iff (cid:88) τ i ∈ τ θ Li ≤ m. HI-mode platform feasibility test.
Similar to the LO-mode case, feasibility ofthe assigned execution rates is ensured as long as the sum total of the ratesis no more than m . However, since we have multiple execution rates in theHI-mode, we need to ensure that rates are feasible in each of the n H + 1windows. Proposition 3
Execution rates assigned to task system τ are feasible in theHI-mode iff ∀ j (1 ≤ j ≤ n H ) , (cid:88) τ i ∈ τ H θ Hi,j ≤ m, and (6) (cid:88) τ i ∈ τ H θ Hi ≤ m. (7) k i θ Hi, θ Hi, θ Hi,k i u Hi θ Li θ Hi ≥ Mode Switch τ i w w w j w n H T i − C Li /θ Li θ Hi,n H ≥ u Hi C Li /θ Li ( T i − C Li /θ Li ) − P i ≤ j Worst case mode switch instant of a carry over job (cid:80) j :1 ≤ j ≤ n H w j time units from the mode switch), and stable jobs that are released after thetransition period. We first consider the stable jobs and then derive the testsfor the other jobs. Stable jobs. Since these jobs are released after the transition period, they al-ways execute using a single rate θ Hi . As long as θ Hi ≥ u Hi these jobs can meettheir deadlines because u Hi ∗ T i = C Hi units of execution would be guaranteed.It is easy to see that this is also a necessary condition for schedulability of thestable jobs; if θ Hi < u Hi then the stable jobs cannot meet their deadlines. Werecord this test in the following proposition. Proposition 4 A stable job of task τ i , that is a job released at or after (cid:80) j :1 ≤ j ≤ n H w j time units from the mode switch, is schedulable iff θ Hi ≥ u Hi Carry over jobs. Let k i denote the largest index (1 ≤ k i ≤ n H + 1) such thatthe earliest deadline of any carry over job of task τ i is strictly greater than (cid:80) j :1 ≤ j A carry over job that does not trigger a mode switch Note that no job of task τ i (carry over or otherwise) can have a deadlinewithin (cid:80) j :1 ≤ j A carry over job of task τ i ∈ τ H can meet its deadline in theHI-mode if the following three conditions are met. (cid:88) j :1 ≤ j We first show that Equation (8) is a necessary condition forthe schedulability of a carry over job that triggers the mode switch. But sinceEquations (9) and (10) are not necessary, the theorem itself only presents asufficient test. If Equation (8) does not hold then this carry over job will missits deadline. Consider the carry over job shown in Figure 1. Since it triggersthe mode switch, its remaining execution time at mode switch is C Hi − C Li ,and the time remaining to deadline is T i − C Li /θ Li . From the definition of k i we get (cid:80) j :1 ≤ j Assuming Equation (8) holds, the total execution a carry over job trig-gering a mode switch receives after the mode switch is ≥ C Hi − C Li . Since ithas already received C Li units of execution before mode switch, it can meet its p k i θ Hi, θ Hi, θ Hi,k i u Hi θ Li θ Hi ≥ Mode Switch τ i w w w j w n H u Hi θ Hi,n H ≥ tw p θ Hi,p task executionearly transition job (a) Early transition job k i θ Hi, θ Hi, θ Hi,k i u Hi θ Li θ Hi ≥ Mode Switch τ i w w w j w n H u Hi θ Hi,n H ≥ late transition job task execution (b) Late transition job Fig. 3 Transition jobs deadline. Let us consider the case of a carry over job that does not trigger themode switch as shown in Figure 2. In this case the carry over job has a dead-line greater than T i − C Li /θ Li ( > (cid:80) j :1 ≤ j Early transition jobs are released in the transition periodand use the execution rates in the transition windows before k i . To derive theschedulability conditions for this case, we first present a simple property ofnon-decreasing execution rates. Lemma 1 Suppose the execution rates in the transition windows prior to k i satisfy the following conditions. (cid:88) j :1 ≤ j The following theorem presents a sufficient schedulability condition for theearly transition jobs using the above lemma. Theorem 2 An early transition job of task τ i ∈ τ H can meet its deadline inthe HI-mode if the following four conditions are met. (cid:88) j :1 ≤ j Consider an early transition job released at some time instant t afterthe mode switch as shown in Figure 3(a). By definition, t < (cid:80) j :1 ≤ j The only remaining jobs to consider are late transitionjobs that are released in the transition period but no earlier than (cid:80) j :1 ≤ j Consider the task set τ with multi-rate assignments as shown inTable 1. We show that this rate and window assignment is schedulable. Wecan easily check that Propositions 1–3, Equations (9) and (10) of Theorem 1and all the equations of Theorem 2 are satisfied. Now consider Equation (8)of Theorem 1: for τ , 1 × . ≥ . − . 8; for τ , 1 × . × . ≥ − . 5; for τ , 0 × . . × . . × . ≥ . − . 5. Thus, Equation (8) is alsosatisfied, and hence the task set is schedulable. ulti-Rate Fluid Scheduling of Mixed-Criticality Systems on Multiprocessors 15 In this section we derive some important properties of the proposed multi-rate fluid scheduling model and its schedulability test. A rate and windowassignment algorithm for the multi-rate fluid scheduling model is an algorithmthat assigns values for all the execution rates in the model as well as for the n H transition window durations. We define the correctness criteria for such analgorithm as follows. Definition 3 (Multi-Rate MC-Correctness) A rate and window assign-ment algorithm is called Multi-Rate MC Correct iff the following holds:Whenever the algorithm returns a set of execution rates ( ∀ i : θ Li , θ Hi and ∀ i, j : θ Hi,j ) and transition window durations ( ∀ j : w j ), these rates and windowdurations satisfy Propositions 1–3 and Theorems 1 and 2. Dual-rate generalization and speed-up optimality. An interesting property ofthe proposed multi-rate model and schedulability test is that it generalizes thedual-rate model and schedulability test discussed in Sect. 2. We can obtain thedual-rate model by setting ∀ j : θ Hi,j = θ Hi for each HI-task τ i . For this case,the following lemma shows that the multi-rate schedulability test derived inSect. 4 is equivalent to the dual-rate schedulability test. Lemma 2 (Dual-rate Generalization) If ∀ i : 1 ≤ i ≤ n H , ∀ j : 1 ≤ j ≤ n H , θ Hi,j = θ Hi in the multi-rate fluid model, then Propositions 1–3 and Theo-rems 1 and 2 are satisfied if and only if Equations (1) – (5) hold.Proof The conditions for LO-mode platform feasibility and LO-task schedula-bility are identical for the dual-rate and multi-rate models (Proposition 1 andEquation (1), Proposition 2 and Equation (4)).We now consider the HI-task schedulability and HI-mode platform feasi-bility tests. (Dual-rate ⇒ Multi-rate): Suppose the tests are satisfied for the dual-ratemodel. That is, Equations (2), (3) and (5) hold.By substituting Equation (3) in Equation (2) we get,1 ≥ u Hi − u Li θ Hi + u Li θ Li Equation (2) ⇔ ≥ u Hi − u Li θ Hi + u Li θ Hi (Using θ Li ≤ θ Hi ) ⇔ θ Hi ≥ u Hi (17)First we consider Theorem 2. Equation (16) in this theorem is identical toEquation (17) above. Equations (13), (14) and (15) are all satisfied because θ Hi,j = θ Hi ≥ u Hi for all i and j based on the assumption of the lemma andEquation (17). Next we consider Proposition 3. Equation (7) in this proposition is identicalto Equation (5). Also, Equation (6) reduces to (cid:80) τ i ∈ τ H θ Hi ≤ m , which is alsosatisfied based on Equation (5). Thus the proposition is satisfied.Finally, we consider Theorem 1. Equation (8) in Theorem 1 is (cid:88) j :1 ≤ j This case is trivial because Proposition 3, and The-orems 1 and 2 subsume Equations (2), (3) and (5). (cid:117)(cid:116) Note that, as discussed in the introduction, dual-rate assignment strategyMC-Fluid has been shown to be speed-up optimal with a speed-up bound of4 / / .Since MC-Fluid has a speed-up bound of 4 / 3, this means any MC task sys-tem satisfying the condition max { U LL + U LH , U HH } ≤ / i { u Li , u Hi } ≤ / / / 3. Based on this intuition, the following lemma derives prop-erties for A that ensures A dominates MC-Fluid. Similar proof-technique has been used in [4], in which dominance of MC-Fluid over MCFhas been used to derive a speed-up bound for MC-Fluid.ulti-Rate Fluid Scheduling of Mixed-Criticality Systems on Multiprocessors 17 Lemma 3 (Dominance and Speed-up Bound) Suppose a rate and win-dow assignment algorithm A for the multi-rate fluid model is guaranteed toreturn some feasible assignment as long as there is at least one assignmentsatisfying, – ∀ i : 1 ≤ i ≤ n H , ∀ j : 1 ≤ j ≤ n H , θ Hi,j = θ Hi , – ∀ j : 1 ≤ j ≤ n H , w j = 0 , and – Propositions 1–3 and Theorems 1 and 2.Then Algorithm A dominates MC-Fluid and has a speed-up bound of / .Proof Suppose a MC task system is feasible under MC-Fluid. Then, let θ Li and θ Hi denote the execution rates for each task τ i ∈ τ H assigned by MC-Fluid thatsatisfies Equations (1)–(5). Now consider the multi-rate assignment ∀ i : 1 ≤ i ≤ n H , ∀ j : 1 ≤ j ≤ n H , θ Hi,j = θ Hi , ∀ j : 1 ≤ j ≤ n H , w j = 0 and θ Li identicalto the dual-rate assignment. From Lemma 2 we know that this rate assignmentsatisfies Propositions 1–3 and Theorems 1 and 2. Thus, there is at least onerate and window assignment that satisfies all the above three conditions ofthe lemma. From the assumption of the lemma, we then know that algorithmA is guaranteed to return a feasible multi-rate and window assignment. Thisshows that algorithm A dominates MC-Fluid in terms of schedulability. SinceMC-Fluid has a known speed-up bound of 4 / 3, by definition algorithm A alsohas a speed-up bound of 4 / (cid:117)(cid:116) It has been shown that no non-clairvoyant scheduling algorithm can have aspeed-up bound lower than 4 / / / Mapping to non-fluid platform. The multi-rate fluid scheduling model, simi-lar to other fluid models, assumes that a processing core can be fractionallyassigned to tasks. Since this is not possible on a real (non-fluid) platform, itis important to map the fluid execution rates to a non-fluid scheduling policy,ideally without any loss in schedulability. In [10], the dual-rate fluid modelhas been successfully mapped without any loss in schedulability to the DP-Fair non-fluid scheduling algorithm (short for Deadline-Partitions Fair) [7].We now show that the multi-rate fluid model can also be similarly mappedto DP-Fair without any loss in schedulability. In the classic non-MC multi-processor scheduling, DP-Fair scheduling algorithm has been used to map afluid execution model with single execution rate for each task to a non-fluidschedule without any loss in schedulability [7]. The main intuition behind thismapping is as follows. The entire scheduling window is partitioned based onjob deadlines. Between any two consecutive job deadlines, it is ensured thatthe total allocation to a task is equal to the length of the partition multipliedby its assigned execution rate. Thus, tasks are guaranteed processor allocations proportional to their execution rates at every job deadline , and hence schedula-bility is preserved. Between two consecutive job deadlines, the processor shareallocated to each task can be scheduled using any non-fluid optimal schedulingpolicy such as McNaughton’s algorithm [12]. To handle sporadic job releases,the allocations in each partition are scheduled using a work-conserving algo-rithm, and they are re-computed upon job releases. Note that this schedulingpolicy can be implemented fully online by computing allocations for pend-ing jobs between the current time instant (coinciding with a job release ordeadline) and the next earliest job deadline.In the multi-rate fluid model, apart from job deadlines, another importantevent is the mode switch instant. A mode switch is triggered when a HI-job ex-ecutes for its LO-WCET ( C Li ) and does not signal completion. For correctnessof the multi-rate schedulability test derived in Sect. 4, it is essential to ensurethat each HI-job of a task τ i ∈ τ H completes its LO-WCET execution C Li nolater than C Li /θ Li time units from its release. This is necessary to guaranteethe property that any job deadline of task τ i in HI-mode is no earlier than T i − C i /θ Li time units from the mode switch, which is used in Theorems 1and 2. Hence, when mapping the multi-rate fluid model to DP-Fair schedul-ing, in addition to ensuring task allocation fairness at job deadlines, we alsoneed to ensure this fairness at worst case mode switch instants for each HI-job.That is, we need to ensure fairness at C Li /θ Li time units from each job releaseinstant for each HI-task τ i .After the mode switch, one simple way to map the multi-rate model toDP-Fair scheduling is to ensure allocation fairness at the boundary of eachtransition window j (1 ≤ j ≤ n H ), in addition to job deadlines. Of course, thisis not necessary, and schedulability would be guaranteed even if fairness is onlyensured at job deadlines. Suppose a partition (interval between two consecutivejob deadlines or between the mode switch and the earliest job deadline) spansmore than one transition window. Let [ t , t ] denote this partition, j ( ≤ n H )denote a transition window that contains t , and j ( ≤ n H ) denote anothertransition window that contains t . Then, for each HI-task τ i it is sufficientto ensure that its total allocation in this partition is equal to ( (cid:80) ≤ j ≤ j w j − t ) θ Hi,j + ( (cid:80) j Given a rate and window assignment inthe multi-rate fluid model, consider the following task allocations. – LO-mode mapping: Partition the scheduling window based on job dead-lines and worst case mode switch instants ( C Li /θ Li time units from jobrelease for each HI-task τ i ). In each partition p having length L p , allocatean execution of θ Li × L p time units for each task τ i that has an unfinishedjob at the beginning of the partition. – Transition period mapping: After mode switch drop all the jobs ofLO-tasks. Partition the transition period (within (cid:80) ≤ j ≤ n H w j time unitsfrom the mode switch) based on HI-job deadlines. Consider a partition ulti-Rate Fluid Scheduling of Mixed-Criticality Systems on Multiprocessors 19 p spanning the interval [ t , t ], such that j ( ≤ n H ) denotes a transitionwindow that contains t , and j ( ≤ n H ) denotes another transition win-dow that contains t . Allocate an execution of ( (cid:80) ≤ j ≤ j w j − t ) θ Hi,j +( (cid:80) j Consider a set of execution rates and window durations that satisfyPropositions 1–3 and Theorems 1 and 2. If these rates are mapped to the DP-Fair scheduling algorithm using Definition 4, then in the resulting schedule alljob deadlines are met.Proof In the LO-mode, each job of a task τ i is guaranteed to receive executionproportional to its assigned rate θ Li within its deadline. This follows fromProposition 2 and the fact that allocations in each partition are scheduledusing a work-conserving optimal algorithm. Then, using Proposition 1 we canconclude that all the job deadlines in the LO-mode are met.Each job of a HI-task τ i is guaranteed to receive at least C Li time units ofexecution within C Li /θ Li time units from its release. Therefore, the deadline ofany job of τ i in the HI-mode is no earlier than T i − C Li /θ Li time units fromthe mode switch instant.In the HI-mode, each job is guaranteed to receive execution proportionalto the assigned rates within its deadline (using Proposition 3 and the factthat allocations are scheduled using a work-conserving optimal algorithm).Therefore, using Theorems 1 and 2 we can conclude that all job deadlines inthe HI-mode will be met. (cid:117)(cid:116) In this section, we present an algorithm called SOMA (short for Speed-up Op-timal Multi-rate Assignment) to determine the execution rates and windowdurations for the multi-rate model. A convex optimization based solution, sim-ilar to the dual-rate algorithm MC-Fluid, is desirable because it can optimizethe assignments. However, one major hurdle towards using an optimizationframework for the multi-rate model is that the earliest completion windowparameters, k i s in Definition 2, control the number of terms involved in thesummations used in the schedulability test. Since k i s are dependent on the execution rates and window durations, this would imply that the constraintsof the optimization problem are no longer fixed. To address this issue, we firstuse a simple heuristic to fix these k i s and then formulate the optimizationproblem.Algorithm 1 presents SOMA. Since it is sufficient for all jobs of LO-tasksto execute at their minimum required rate, we assign θ Li = u Li for all τ i ∈ τ L .The algorithm first sorts all the HI-tasks in increasing order of the parameter T i − C Li /u Hi . Note that u Hi denotes the maximum possible rate that a HI-task τ i can be assigned in the LO-mode (easily seen by combining equations inTheorems 1 and 2). Then T i − C Li /u Hi denotes the earliest completion timefor any carry over job of this task, assuming the maximum possible rate inthe LO-mode. Hence, in any feasible assignment, a HI-task τ i with a smaller T i − C Li /u Hi , will most likely end up with a smaller k i when compared to aHI-task τ j with a larger T j − C Lj /u Hj . We use this intuition to fix the values for k i s, and thus address the challenge discussed above. Inspite of this restriction,we show that SOMA has a speed-up bound of 4 / n H !possible combinations for the k i s (Sect. 7). Algorithm 1 SOMA rate and window assignment strategy Input: τ , m 1: For each τ i ∈ τ L assign θ Li = u Li .2: Sort τ H in increasing order of the parameter T i − ( C Li /u Hi ).3: Solve the optimization problem in Definition 5, assuming task indices are sorted basedon the above sorting order.4: if Optimization returns an assignment then if (cid:80) τ i ∈ τ θ Li ≤ m then 6: Declare Success7: else 8: Declare Failure9: end if else 11: Declare Failure12: end if SOMA then solves the convex optimization problem given in Definition 5.Using a fixed value for all the k i s ( k i = i ), where we assume task indices aresorted based on increasing value of the parameter T i − ( C Li /u Hi ), this optimiza-tion encodes Propositions 1 and 3 and Theorems 1 and 2 of the schedulabilitytest in its constraints. Its objective is to minimize the total LO-mode executionrates. If it returns a feasible assignment for the rates and window durations,we check whether this returned assignment satisfies the remaining conditionfor schedulability, i.e., Proposition 2. ulti-Rate Fluid Scheduling of Mixed-Criticality Systems on Multiprocessors 21 Definition 5 (Multi-rate convex optimization) Suppose k i = i for eachHI-task τ i ∈ τ H . minimize (cid:88) τ i ∈ τ H θ Li subject to,P ropositions − T heorems and ∀ i : 1 ≤ i ≤ n H , ∀ j : 1 ≤ j ≤ n H , θ Hi,j ≤ ∀ i : 1 ≤ i ≤ n H , θ Hi ≤ ∀ j : 1 ≤ j ≤ n H , w j ≥ SOMA is a Multi-Rate MC-Correct assignment strat-egy. Any rate and window assignment returned by SOMA satisfies Proposi-tions 1-3 and Theorems 1 and 2 because of the constraints of the optimizationproblem as well as the check in Step 5 of Algorithm 1. The following lemmashows that SOMA has a speed-up bound of 4 / Lemma 5 Algorithm SOMA has a speed-up bound of / .Proof From Lemma 3 we know that any algorithm that always return somefeasible assignment if there exists an assignment satisfying the following threeconditions has a speed-up bound of 4 / ∀ i : 1 ≤ i ≤ n H , ∀ j : 1 ≤ j ≤ n H , θ Hi,j = θ Hi ,2. ∀ j : 1 ≤ j ≤ n H , w j = 0, and3. Propositions 1–3 and Theorems 1 and 2.To prove this lemma, it is then sufficient to show that such an assignmentis considered by SOMA. Observe that the boundary constraints on executionrates and window durations in the optimization problem allow an assignmentthat satisfies the above three conditions.The constraints of the optimization problem exactly encode Propositions 1–3 and Theorems 1 and 2 of the schedulability test.Now we show that for an assignment satisfying the above conditions, theearliest completion window parameter k i is irrelevant.In Theorem 1, Equation (9) is subsumed by Equation (10). That is, whencondition 1 holds ∀ j : k i ≤ j ≤ n H , θ Li ≤ θ Hi,j is subsumed by θ Li ≤ θ Hi .Also, Equation (8) reduces to θ Hi ( T i − C i /θ Li ) ≥ C Hi − C Li . Substituting R i = θ Hi (by condition 1) and condition 2 in Equation (8) we get, (cid:88) j :1 ≤ j SOMA can determine the values for k i s in linearith-mic time ( O ( n H log ( n H ))). The optimization has n H + 3 n H real variables, n H integer variables and 9 × n H constraints. Hence, the number of variables andconstraints is polynomially bounded in the number of HI-tasks in the system.The complexity of SOMA is therefore bounded by the complexity of solvingthe convex optimization problem. By replacing C Li /θ Li in the optimizationproblem with a variable x i , it is easy to see that the problem reduces to aconvex optimization problem with objective of the form (cid:80) i /x i and all linearconstraints. This is one of the simplest convex optimization problems, and weplan to investigate a polynomial time algorithm for solving it in future work. Theoretical performance and experimental evaluation. Scheduling algorithmsare evaluated based on either analytical performance bound or experimentalevaluation. Scheduling algorithms with good theoretical performance such aslow speed-up bound and good performance in experimental evaluation areusually preferred. Speed-up bound is a good metric to compare the worst-caseperformance of different scheduling algorithms. For scheduling algorithms suchas partitioned scheduling algorithms that are based on heuristics, it is hardto determine these speed-up bounds. The performance of these algorithms isoften evaluated based on experimental evaluation.The dual-rate fluid model is shown to be speed-up optimal [4] and has goodperformance in experimental evaluation for multiprocessor task systems [10].Though the dual-rate model is optimal in terms of speed-up bound, it is notoptimal in terms of schedulability [14]. There are several feasible task sys-tems that are deemed unschedulable by MC-Fluid [10] - the optimal dual-rateassignment algorithm. Thus, we propose SOMA the multi-rate and window as-signment algorithm which has an optimal speed-up bound and performs betterthan MC-Fluid in experimental evaluation. In fact, our experimental evalua-tion together with Lemma 3 also shows that SOMA strictly dominates MC ulti-Rate Fluid Scheduling of Mixed-Criticality Systems on Multiprocessors 23 Fluid in terms of schedulability. That is, all task systems deemed schedulableby MC Fluid are also deemed so by SOMA, and there exist task systems thatare not schedulable by MC Fluid but are deemed so by SOMA.To provide some insights on the complexity of these two rate assignmentalgorithms - SOMA and MC-Fluid - we compare their offline and runtimecomplexity. The rate assignment in both these algorithms is done offline whendetermining the execution rates. The offline complexity of MC-Fluid rate as-signment algorithm is O ( n H ). Whereas, the offline complexity of SOMA islarger compared to MC-Fluid because the complexity of SOMA is boundedby the complexity of solving the optimization problem defined in Definition 5.For details on the complexity of SOMA refer to the complexity discussionabove. During runtime, the scheduler executes the task with the assigned ex-ecution rates. The runtime scheduling mechanism of SOMA is quite similarto MC-Fluid because the n H window durations that are determined offlineremain fixed during runtime. The runtime complexity of SOMA is not signifi-cantly high compared to dual-rate model because the only additional overheadincurred is in the transition period ( n H windows) when the mode switch is trig-gered. Essentially, upon mode switch in the dual-rate model, HI-tasks switchtheir execution rate at most once. Whereas, in the case of multi-rate model,each HI-task switches its execution rate at most n H + 1 times. In this section we evaluate the schedulability performance of SOMA and com-pare it with other scheduling algorithms that have known speed-up bounds.These include GLO-EDF VD [11], PAR-EDF VD [3] and MC-Fluid [10]. Wecompare with only these algorithms because our focus is on designing an algo-rithm that has a worst-case performance guarantee. We do not compare withthe dual-rate strategy MCF [4] because MC-Fluid is known to be dual-rateoptimal. Task set generation. Our experiments are carried out using the task set gen-erator proposed in [15]. The task set parameters used in our generator aredescribed as follows:1. m ∈{ , , } denotes the number of cores.2. u min (= 0 . u max (= 1 . 0) denote the minimum and maximumindividual task utilization respectively.3. U B = max ( U HH , U LH + U LL ) /m denotes the normalized system utilization inboth LO- and HI-modes. We consider U B ∈ [0 . , . , . . . , . U HH /m ∈ [0 . , . , . . . , . 0] denotes the normalized system utilization inHI-mode.5. U LH /m ∈ [0 . , . , . . . , U HH /m ] denotes the normalized system utilizationHI-tasks in LO-mode.6. U LL /m ∈ [0 . , . , . . . , − U LH /m ] denotes the normalized system utiliza-tion of LO-tasks in LO-mode. 7. Total number of tasks is lower bounded by m + 1 and upper bounded by10 ∗ m .8. Total number of HI-tasks in the system is lower bounded by m + 1 andupper bounded by 3 ∗ m . Note that the performance of SOMA is identicalto MC-Fluid for values of n H ≤ m . This is expected because if such a taskset is dual-rate infeasible, it means that even allocating the maximum rateof 1 to each task in the HI-mode is not sufficient.9. P H ∈ [0 . , . , . . . , . 9] denotes the percentage of HI-tasks in the system.10. T i , the period of task τ i is drawn uniformly at random from [5 , u Li and u Hi are determined using techinques MRandFixed-Sum [6] and BoundedUniform [15]12. The execution requirements C Li and C Hi are defined as u Li × T i and u Hi × T i respectively.Using the above procedure we generate atleast 1000 task sets for each valueof U B and m . For each such successfully generated task set we evaluate theperformance of the SOMA with other multi-core MC scheduling algorithmswith known speed-up bound. Results. Figure 4 shows the overall performance of the algorithms where weplot the acceptance ratio, i.e., fraction of generated task sets that are deemedschedulable, versus normalized utilization U B for varying m ∈ [2 , , m at high U B values mainly because the number of tasks in the task sets arealso higher.In the remainder of this section we take a closer look at the performanceof SOMA in comparison to MC-Fluid. Since an exact feasible test for MCtask systems is not known ( U B ≤ dual-rate infeasible task sets that are deemed tobe schedulable under SOMA in Figure 5. Due to speed-up optimality of MC-Fluid (speed-up of 4 / U B ≤ . U B ≤ . 75, because the two algorithmshave optimal performance for U B ≤ . 75. As expected, we can see that SOMAperforms better for U B ≥ . 80. For m = 2, as much as 35 . 8% more tasksets are deemed to be schedulable under SOMA. Thus, there are significantnumber of task sets that are schedulable under SOMA but not under MC-Fluid (about 358 out of 1000 when U B = 0 . ulti-Rate Fluid Scheduling of Mixed-Criticality Systems on Multiprocessors 25(a) m = 2(b) m = 4(c) m = 8 Fig. 4 Comparison of acceptance ratio for varying m Fig. 5 Percentage of dual-rate infeasible task sets that are schedulable under multi-rate consider the dual-rate infeasible task sets. For m = 2, when U B = 0 . 80 theacceptance ratio of SOMA and MC-Fluid in Figure 4 is 97 . 1% and 95 . . 6% (97 . − . . 6% out of the 4 . 5% (100% − . . 5% (1 . / . × m and utilization. For m = 4 and m = 8,the performance gap between SOMA and MC-Fluid increases gradually as U B increases. This is expected because the number of HI-tasks in the task sets arealso higher for same value of U B .Next, we present the weighted acceptance ratios for varying values of taskset parameters P H and u max . Weighted Acceptance Ratio for an algorithmis defined as W AR ( S ) = (cid:80) UB ∈ S ( AR ( U B ) XU B ) (cid:80) UB ∈ S U B , where S is the set of U B valuesand AR( U B ) is the acceptance ratio of that algorithm for a specific value of U B . This metric essentially gives more importance to task sets with higher U B values, consistent with the fact that such task sets are in general harder toschedule. Task sets are generated using the procedure described in Sect. 7. Theresults of these task sets are combined based on their normalized utilization U B .An important factor in MC fluid scheduling is the number of HI-tasks( n H ). Therefore, in Figure 6(a) we compare the weighted acceptance ratios ofSOMA and MC-Fluid for varying n H and m . As n H increases the performancegap between SOMA and MC-Fluid also increases. This is expected because forlarger n H at the same U B value, there is more flexibility in assigning multipleexecution rates in the HI-mode. ulti-Rate Fluid Scheduling of Mixed-Criticality Systems on Multiprocessors 27(a) m = 2(b) m = 4(c) m = 8 Fig. 6 Weighted Acceptance ratio with varying n H In Figure 7(a) we compare the weighted acceptance ratios of SOMA andMC-Fluid for varying P H values. The performance gap widens as the number P H )(b) Varying maximum HI-utilization of HI-tasks ( max { u Hi } ) Fig. 7 Comparison of weighted acceptance ratio of HI-tasks in a task set increases, indicating that SOMA performs better whenthere are more number of HI-tasks in the system. This is as expected becausethe main benefit of assigning multiple rates to HI-tasks in the HI-mode is thatthe total rate for these tasks in the LO-mode can be reduced. This reductionis useful only when there are LO-tasks that can benefit from it. At high P H values, there are very few LO-tasks and as a result, all these tasks will bebenefitted.In Figure 7(b) we compare the weighted acceptance ratios of SOMA andMC-Fluid for varying max { u Hi } values. The performance of both algorithmsdecrease with increasing max { u Hi } , this is reasonable because typically thenumber of tasks in each task set reduces with increasing max { u Hi } and asshown in Sect. 7, SOMA tends to perform better with increasing n H . When max { u Hi } is large, there is a significant performance gap between SOMA andMC-Fluid. This is because in such task systems there are typically more num-ber of heavy utilization tasks. As a result, the minimum required HI-modeexecution rate under MC-Fluid is high and hence it does not have much flex- ulti-Rate Fluid Scheduling of Mixed-Criticality Systems on Multiprocessors 29 Fig. 8 Varying normalized system utilization of LO-tasks ( U LL /m ) ibility. However, under SOMA, some execution rates in the HI-mode can bemuch lower than u Hi as long as the average rate over all the transition win-dows is reasonably high. This flexibility in rate assignment is a key propertyof SOMA and its benefit is clearly visible for task sets with high max { u Hi } values.In Figure 8 we compare the weighted acceptance ratios of SOMA andMC-Fluid for varying U LL /m values with U B = 0 . 95. The performance gapwidens as U LL /m increases, indicating that SOMA performs better when thesystem utilization of LO-tasks is high. This is reasonable because SOMA triesto minimize the total LO-mode execution rate of HI-tasks and as a result, thisreduction becomes more useful as U LL /m increases.The presented experiments show that SOMA outperforms all the existingalgorithms with known speed-up bounds. The overall performance gap betweenSOMA and MC-Fluid increases with utilization. Further, we have also shownthat this gap is significant in terms of the percentage of dual-rate infeasibletask sets that SOMA can schedule, and that the gap widens with increasingnumber of HI-tasks in the system. Scheduling algorithms with good speed-up bounds as well as good performancein schedulability experiments are highly desirable. In this paper, we addressedthe shortcoming in the schedulability performance of the speed-up optimaldual-rate fluid model by proposing the multi-rate fluid model. We derived asufficient schedulability test for the multi-rate model and also showed thatit dominates the dual-rate model. 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