Hyperbolic utilization bounds for rate monotonic scheduling on homogeneous multiprocessors

Hyperbolic Utilization Bounds for Rate MonotonicScheduling on Homogeneous Multiprocessors

Hongya Wang, LihChyun Shu, Member, IEEE, Wei Yin, Yingyuan Xiao, and Jiao Cao

Abstract—The utilization bounds for partitioned multiprocessor scheduling are a function of task allocation algorithms andthe schedulability conditions selected for uniprocessor scheduling algorithms. In this paper, we use rate-monotonic schedulingon each processor and present the lower and upper limits of the utilization bounds for all reasonable task allocation heuristics.Unlike previous work, the hyperbolic bound due to Bini et al., instead of the Liu & Layland bound, is adopted to do the schedulabilitytest on uniprocessors. We also derive the utilization bounds with respect to the worst fit allocation algorithm and reasonableallocation decreasing heuristics, and the two bounds are found to coincide with the worst and best achievable multiprocessorutilization bounds, respectively. Analytical and experimental results show that the proposed utilization bound performs better than theexisting bound under quite a lot of parameter settings, and combining these two bounds together can significantly (up to 3 times)increase the number of schedulable task sets with little extra overhead.

Index Terms—Multiprocessor, real-time system, hyperbolic bound, rate-monotonic scheduling

Ç

1 INTRODUCTION

A real-time system distinguishes itself from otherhigh performance computing systems by its primary

requirement to guarantee the schedulability of its constit-uent tasks. The Rate Monotonic (RM) and Earliest DeadlineFirst (EDF) algorithms are two popular priority-basedscheduling algorithms to address this need. For RMscheduling, task priorities are statically assigned andinversely proportional to task periods, whereas the EDFalgorithm assigns high priorities to tasks with stringentabsolute deadlines, thus is a dynamic scheduling algorithm.For both RM and EDF scheduling, the task with the highestpriority is chosen to execute at each scheduling point. Bothalgorithms have their advantages and disadvantages, andwe refer readers to [1] for a comprehensive discussion ofthese algorithms. We focus on RM scheduling in this paperdue to its wide applications in real-life real-time systems.

Given a real-time task set, a fundamental problem isthat if all tasks in this set can meet their deadlines using RMscheduling, which is intrinsically a decision problem with

the answer of either ‘‘schedulable’’ or ‘‘unschedulable’’.Over the years, the uniprocessor version of this problemhas been nicely addressed by many researchers. Particu-larly, in 1973, Liu & Layland developed the first utilization-based schedulability test for RM scheduling in theirseminal paper [2]. One of the main results in the paper isthat any task set with its total utilization less than or equalto mð21=m � 1Þ, is schedulable under RM scheduling, wherem is the number of tasks.

To achieve higher utilization bounds for fast singleprocessor schedulability test, Bini et al. proposed a noveluniprocessor schedulability analysis technique, which iscalled the hyperbolic bound (HB), in [3]. Unlike LLB, HBuses the product of task utilizations to do the schedulabilitytest. As shown in (1), if the product of one plus ui, theutilization factor of task i, over all tasks is no greater thantwo, the task set will be schedulable under RM scheduling.HB offers better performance than LLB in that more tasks setscan be accepted by HB [3]. Put it another way, a task set thatis schedulable under LLB can always pass the schedulabilitytest with HB, but the converse of the statement may notnecessarily be true. HB has the same OðmÞ complexity asthe original Liu & Layland bound

Ymi¼1

ð1þ uiÞ � 2: (1)

Research into multiprocessor real-time scheduling hasbeen gaining new momentum in recent years due to therapid proliferation of multicore processors in computersystems. The problem is much more challenging than itsuniprocessor counterpart. The difficulty lies in the fact that,for a real-time task set, we not only have to decide wheneach task in the task set should be executed or suspended,but also on which processor a task would run. Broadlyspeaking, there are two main strategies, namely, thepartitioning and the global strategies, for scheduling real-time tasks on multiprocessors. With the partitioning

. H. Wang and J. Cao are with the School of Computer Science andTechnology, Donghua University, Songjiang District, Shanghai 201620,China. E-mail: [email protected].

. L. Shu is with the Department of Accountancy, National ChengKung University, Tainan 701, Taiwan, and is also with the College ofInformation and Engineering, Chang Jung Christian University, Taiwan.E-mail: [email protected].

. W. Yin is with the CASCO Signal Ltd., Shanghai, China. E-mail:[email protected].

. Y. Xiao is with the School of Computer and Communication Engineering,Tianjin University of Technology, Xiqing District, Tianjin 300384, China.E-mail: [email protected].

Manuscript received 14 Nov. 2012; revised 9 Aug. 2013; accepted 12 Aug.2013. Date of publication 22 Aug. 2013; date of current version 16 May 2014.Recommended for acceptance by J. Wang.For information on obtaining reprints of this article, please send e-mail to:[email protected], and reference the Digital Object Identifier below.Digital Object Identifier no. 10.1109/TPDS.2013.213

1045-9219 � 2013 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 25, NO. 6, JUNE 20141510

strategy, a task is assigned to a processor and then executeon this processor without any further migration. By contrast,task instances belonging to the same task might run ondifferent processors in global scheduling. The partitioningand global strategies both have their respective strengthsand weaknesses, and a comprehensive comparison betweenthem is beyond the scope of this paper. Interested readersare referred to [4] for more details. In this paper, we focuson the partitioning strategy because it seems much moreappealing in practice due to the simplicity and relativelylow scheduling overhead.

Given a specific partitioning strategy, which is oftencharacterized by the buniprocessor scheduling algorithm, taskallocation policyÀ pair, an important problem is how todetermine the schedulability of task sets on multiprocessors.While establishing general utilization bounds for multi-processor real-time scheduling is of great theoretical impor-tance in real-time systems [4], it was not until 1998 that thefirst general utilization bound, given in Equation (2), wasproposed by Oh and Baker [5] for RM scheduling usingLLB and the First Fit allocation algorithm. n is the numberof processors

Xmi¼1

ui � UFFLLB ¼ n 21=2 � 1

� �; n 9 1: (2)

By observing that the utilization bound for RM sched-uling on mulitprocessors is a function of the chosen taskallocation policy, Lopez et al. generalized the result in [5]by developing the minimum and maximum utilizationbounds for all (reasonable) task allocation policies undermultiprocessor RM scheduling. In their paper, the specificallocation policies which can provide the minimum and/ormaximum bounds are also identified. With the minimumand maximum utilization bounds, one can easily judge theschedulability of task sets under any given (reasonable)task allocation policies.

It is worth noting that all of the aforementionedmultiprocessor utilization bounds make use of the Liu &Layland bound as the uniprocessor schedulability condition[5], [6], which has been shown to be inferior to the HBbound on a uniprocessor as discussed earlier. While manyexperimental studies have confirmed that using HB as theuniprocessor schedualbility condition on multiprocessorscan increase the number of schedulable real-time task setssignificantly [7], [8], to the best of our knowledge, notheoretical result has been developed with respect toHB-based multiprocessor utilization bounds. With the increas-ing prevalence of multicore and multiprocessor systems, atheoretical treatment of the aforementioned problem not onlysatisfies our natural curiosity about whether HB outperformsLLB on multiprocessors as in the uniprocessor case, butalso has its practical significance. In this paper, using thehyperbolic bound as the uniprocessor schedulability condi-tion, we study the problem of establishing the minimumand maximum utilization bounds for partitioned multi-processor RM scheduling. The main contributions of thispaper include

. As shown in Equations (3) and (4), respectively, thelower and upper limits of the utilization bounds

associated with all reasonable task allocation (RA)policies, for partitioned multiprocessor RM schedulingunder HB are provided

MinRAHB ¼2n

ð1þ �Þn�1; n 9 1 (3)

MaxRAHB ¼ 2n�þ1�þ1 ; n 9 1: (4)

. We identify specific task allocation heuristics underwhich the utilization bounds coincide with thelower and upper limits. Specifically, the reasonableallocation decreasing (RAD) heuristic, a subclass oftask allocation policies, achieves the maximumutilization bound while the worst fit (WF) heuristicranks at the bottom.

. To show the gain (in terms of schedulability)achieved by MaxRAHB over MaxRALLB, we have con-ducted comprehensive simulation experiments,which indicates that, unlike the uniprocessor case,MaxRAHB and MaxRALLB are incomparable according tothe taxonomy in [4], i.e., neither of these two boundsdominates the other. Although MaxRAHB is not alwayssuperior to MaxRALLB, the practical significance of ourwork is that one can first use one of the bounds to dothe schedulability test for a given task set. If itsucceeds, we are done. Otherwise, we proceed touse the other bound. Experimental results demon-strate that, compared to the individual applicationof MaxRALLB, up to 3.4 times more feasible task setscan be obtained by the combined use of MaxRAHB andMaxRALLB. More importantly, this gain does not comewith an added cost in terms of the complexity ofschedulability test, which remains OðmÞ.

The rest of this paper is organized as follows. Section 2discusses preliminaries needed for our work. The minimumand maximum utilization bounds for multiprocessor RMscheduling under HB, denoted by MinRAHB and MaxRAHB, areprovided in Section 3. Sections 4 and 5 present the utilizationbounds for the worst fit allocation heuristic and thereasonable allocation decreasing policy, respectively. Anextensive experimental comparison of MinRAHB and MaxRAHBis reported in Section 6, and the related work is discussedin Section 7. Finally, Section 8 concludes the paper.

2 PRELIMINARIES

In this section, we shall formally define the system modelused throughout this paper and briefly discuss somerepresentative work that is closely related to our research.

We consider a homogeneous n processor platformfP1; P2; . . . ; Png, on which a set of m individual periodictasks f�1; �2; . . . ; �mg execute. Each task �i is characterizedby its computation time Ci, period Ti and relative deadlineDi. We assume that task deadlines are hard and equal to thetask periods ðDi ¼ TiÞ. The utilization ui of any task �i,defined as Ci=Ti, is assumed to be 0 G ui � � � 1, where �is the maximum utilization factor over all tasks.

Tasks are allocated to n processors fP1; P2; . . . ; Png usingthe reasonable allocation (RA) policies [9]. A task allocationpolicy is said to be reasonable if it fails to allocate a task

WANG ET AL.: HYPERBOLIC UTILIZATION BOUNDS FOR RATE MONOTONIC SCHEDULING 1511

only if the task does not fit into any processor in the system.Heuristics such as First Fit (FF), Worst Fit (WF), Best Fit(BF), etc. all fall into this category. We are also interested inthe reasonable allocation decreasing (RAD) algorithms, aspecial subset of RA policies, which first order tasks bydecreasing utilization factors before making the allocation,i.e., u1 � u2 � � � � � um. Typical members of this subclassinclude First Fit Decreasing (FFD), Best Fit Decreasing(BFD) and Worst Fit Decreasing (WFD) heuristics. Withineach processor, we assume that tasks are pre-emptivelyscheduled based on fixed priorities that are assignedaccording to the RM order.

The Liu & Layland bound, given by Inequality (5), is acommonly used condition to test the schedulability of atask set on uniprocessor systems

Xmi¼1

ui � m 21=m � 1� �

: (5)

LLB is only sufficient in the sense that a task set withutilization greater thanmð21=m � 1Þmight be still schedulableunder RM scheduling.

Exact schedulability tests yielding to necessary andsufficient conditions have been independently derived in[10], [11], [12]. Using the Response Time Analysis (RTA)proposed in [12], a periodic task set is schedulable with theRM algorithm if and only if the worst-case response time ofeach task is less than or equal to its deadline. In essence,RTA simulates the whole scheduling procedure for eachspecific task set by carrying out the schedulability test at thedeadlines of all task instances. While RTA outperforms LLBin schedulability, we do not consider this strategy in thispaper because 1) it is impossible to obtain a single (unified)utilization bound for all task sets with exact schedulabilitytests, and 2) compared with the OðmÞ complexity of LLB,the complexity of the exact test is pseudo-polynomial,which means that it is not suitable for online admissioncontrol in applications with large task sets [3].

Oh and Baker [5] provided the first general multiprocessorutilization bound for RM scheduling using LLB, which is,

however, overly pessimistic. By taking the maximum utili-zation factor over all tasks into consideration, Lopez et al.established the minimum and maximum utilization bounds,given in Equations (6) and (7) respectively, for multiprocessorRM scheduling and RA allocation

MinRALLB ¼naUa þ nbUb � ðn� 1Þ�; n 9 1 (6)

MaxRALLB ¼ðn� þ 1Þ 21=ð�þ1Þ � 1� �

; n 9 1: (7)

� is the maximum utilization factor over all tasks and �is defined as the maximum number of tasks of utilizationfactor � that can fit into one processor using LLB for RMscheduling. Please refer to [6] for more details about thedefinitions of na, nb, Ua, and Ub. � can be expressed as afunction of � as follows [9]:

� ¼ 1

log2ð1þ �Þ

� �: (8)

The hyperbolic bound, proposed by Bini et al. [3], is a novelschedulability analysis technique for RM scheduling, whichhas provable better performance than LLB on uniprocessorsystems and is also the best possible bound that can befound with the mere knowledge of task utilization factors.With HB, a set of m tasks is schedulable if Inequality (1) issatisfied. To simplify discussion, we refer to the product ofðui þ 1Þs of a set of tasks as HB-utilization in the rest of thispaper.

By using HB as the uniprocessor schedulability condition, atask with utilization factor uk fits into processor Pj, whichalready has mj tasks allocated to it with total HB-utilization

Uj ¼Qmj

i¼1

ð1þ uiÞ, if the mj þ 1 tasks are schedulable under

HB, that is,Qmj

i¼1

ð1þ uiÞ � ð1þ ukÞ � 2. On multiprocessors of

size n, if the aforementioned inequality does not hold for all1 � j � n, the task set is said to be unschedulable with HBfor RM scheduling.

For ease of reference, we summarize the notations usedthroughout this paper in Table 1.

TABLE 1Notations


3 LOWER AND UPPER LIMITS OF HYPERBOLICUTILIZATION BOUNDS

The partitioned multiprocessor utilization bound is a functionof the btask allocation algorithm, uniprocessor schedulingalgorithmÀ pairs. Since HB has been shown to outperformLLB in uniprocessor RM scheduling, it is natural to ask if it isalso the case with respect to multiprocessor RM scheduling.To address this issue, we choose HB as the uniprocessorschedulability condition for RM scheduling in this paper, andthus the task allocation algorithms become the sole cause ofvariation in establishing the multiprocessor utilizationbounds. In this section, we will derive the expressions of theminimum and maximum values that the multiprocessorutilization bound can take in Theorem 1 and Theorem 2,respectively. Before presenting the lower limit, MinRAHB, andthe upper limit, MaxRAHB, of all possible utilization bounds forRA allocation, we introduce the parameter � first, which isdefined as the maximum number of tasks of utilization factor� that can fit into one processor using HB for RM scheduling.� can be expressed as a function of � as follows.

Lemma 1.

� ¼ 1

log2ð1þ �Þ

� �:

Proof. According to the definition of �, � tasks of utilizationfactor � must fit into one processor. Thus we haveQ�i¼1

ð1þ �Þ ¼ ð1þ �Þ� � 2, by using the hyperbolic bound

feasibility test. By taking the base-2 log of both sides of the

above inequality we obtain � � 1log2ð1þ�Þ

. Since � is an

integer we get

� � 1

log2ð1þ �Þ

� �: (9)

Recall that � is defined as the maximum number oftasks of utilization factor � that can fit into one processor.Hence, � þ 1 tasks of utilization factor � do not fit into

one processor using HB, viz.,Q�þ1

i¼1

ð1þ �Þ ¼ ð1þ �Þ�þ1 9 2.

By taking the base-2 log of both sides of this inequality weobtain � 9 1

log2ð1þ�Þ� 1. Since � is an integer we get

� � 1

log2ð1þ �Þ

� �: (10)

The lemma is proved directly from Inequalities (9)and (10). Ì

Lemma (1), together with Equation (8), indicates that � isalways equal to � for the same task set although they aredefined in different settings. With the definition of �, it is easyto see that any n processors can hold at least n� tasks, each ofwhich having its utilization factor less than or equal to �. So itis meaningless to discuss the notion of utilization boundswhen m � n�. In the rest of this paper, we assume m 9 n� forall theoretical analyses unless explicitly stated otherwise.

Next, in Theorem 1, the lower limit on the multiproces-sor utilization bound for RA allocation is given.

Theorem 1. For any reasonable allocation algorithm RA, it fol-lows that MinRAHB ¼ 2n

ð1þ�Þn�1 if m 9 n�.

Proof. Assume a set of tasks f�1; �2; . . . ; �mg cannot be allo-cated on the multiprocessor using HB as the uniprocessorschedulability condition. Without loss of generality, wereorder the task indices so that all the indices of allocatedtasks are less than the indices of tasks that do not fit intoany processor. Let �k be the first task that cannot beallocated to any processor, where k is the task index afterreordering. As RA is a reasonable allocation algorithm, wehave the following system of inequalities:

Uj � ð1þ ukÞ 9 2; 8 j ¼ 1; 2; . . . ; n (11)

where Uj is the total HB-utilization of the tasks thathave already been allocated to processor Pj and uk is theutilization factor of task �k. The total HB-utilization ofthe whole task set, denoted by U , satisfies

U ¼Ymi¼1

ð1þ uiÞ �Yki¼1

ð1þ uiÞ ¼Ynj¼1

Uj � ð1þ ukÞ: (12)

From (11), we have

Ynj¼1

Uj 9Ynj¼1

2

1þ uk

� �¼ 2n

ð1þ ukÞn:

Substituting this inequality into (12), we get

U 92n

ð1þ ukÞn� ð1þ ukÞ ¼

2n

ð1þ ukÞn�1:

Recall that � is the maximum task utilization factoramong all tasks, so uk � � and we have

U 92n

ð1þ �Þn�1: (13)

From all of the above analyses we know that any taskset that does not fit into the multiprocessor under HBmust satisfy inequality (13). Therefore, any task set withthe total HB-utilization no greater than 2n

ð1þ�Þn�1 can be

successfully allocated on the multiprocessor using HB for

RM scheduling and RA allocation, which indicates

MinRAHB ¼ 2n

ð1þ�Þn�1. Ì

In Theorem 2, we will provide the upper limit on themultiprocessor utilization bound for RA allocation.

Theorem 2. For any reasonable allocation algorithm RA, it fol-lows that MaxRAHB ¼ 2

n�þ1�þ1 if m 9 n�.

Proof. We need to show that, given "! 0þ, there must exist a

set of m tasks with total HB-utilizationQmi¼1

ð1þ uiÞ ¼2n�þ1�þ1 ð1þ "Þ, which does not fit into the multiprocessor.

There are two possible cases:Case 1 ðm ¼ n� þ 1Þ: We construct a task set composed

of n� þ 1 tasks, and each task has the same utilization


factor ui ¼ 21

�þ1ð1þ "Þ1

ðn�þ1Þ � 1. It is easy to see that the total

HB-utilization of this task set isQmi¼1

ð1þ uiÞ ¼ 2n�þ1�þ1 ð1þ "Þ.

Next, we will show the validity of our construction, i.e.,the utilization factors of all tasks are greater than 0 andless than or equal to �.

By definition of � and �, � þ 1 tasks of utilization � donot fit into one processor, therefore ð�þ 1Þ�þ1 9 2, and� 9 21=ð�þ1Þ � 1.

Since we always can find one real number between tworeal numbers, a positive value " must exist such that

ui ¼ 21

�þ1ð1þ "Þ1

ðn�þ1Þ � 1 � �, i.e., ui is valid.Equation (14) tells us that no processor can hold more

than � tasks that we construct. However, given nprocessors and n� þ 1 such tasks, at least one processorhas to accommodate � þ 1 or more of these tasks, whichindicates the infeasibility of allocating these n� þ 1 tasks

with total HB-utilizationQmi¼1

ð1þ uiÞ ¼ 2n�þ1�þ1 ð1þ "Þ on n

processors using HB for RM scheduling given "! 0þ.In other words, a task set is schedulable only if the

HB-utilization of this set is less than or equal to 2n�þ1�þ1

ðui þ 1Þ�þ1 ¼ 2ð1þ "Þ�þ1ð1þn�Þ 9 2: (14)

Case 2 ðm 9 n� þ 1Þ: We construct a task set consistingof two subsets as follows. The first task set has n� þ 1tasks, each of which having the identical utilization factor

ui ¼ 21

�þ1ð1þ "Þ1

2ðn�þ1Þ � 1. The size of the second task set is

m� n� � 1, which is composed of tasks with the same

utilization factor ui ¼ ð1þ "Þ1

2ðm�n��1Þ � 1.

One can readily check that the total HB-utilization ofthe whole task set is

Qmi¼1

ð1þ uiÞ ¼ 2n�þ1�þ1 ð1þ "Þ. Also, the

validity of task utilizations in the first set can be proved in

a similar way as in Case 1.

For the second task set, by choosing a proper smallvalue for ", we have ui ¼ ð1þ "Þ

12ðm�n��1Þ � 1 G 2

1�þ1 � 1 G �

ðui þ 1Þ�þ1 ¼ 2ð1þ "Þ�þ1

2ð1þn�Þ 9 2: (15)

Similar to the analysis with Equation (14), we know,from Equation (15), that the n� þ 1 tasks in the firstsubset can not be allocated on the n processorssuccessfully, and the whole task set with HB-utilizationQmi¼1

ð1þ uiÞ ¼ 2n�þ1�þ1 ð1þ "Þ does not fit into the n processors

either. Thus, the necessary condition for a task set to be

schedulable is that the total HB-utilization must be less

than or equal to 2n�þ1�þ1 in this case.

As a conclusion, by combining the results in both cases,

we proved that MaxRAHB ¼ 2n�þ1�þ1 . Ì

So far we have established the upper and lower limits of

HB bounds for partitioned multiprocessor RM scheduling.

In the next two sections, we will demonstrate that the RAD

and WF heuristics coincide with the upper and lower

limits, respectively. We consider these two heuristics because: 1)

the bin packing problem and multiprocessor scheduling are

closely related; 2) in general, RAD provides reasonably good

performance and WF tends to be one of the worst heuristics for

the bin packing problem. We surmise that might be the same

case for our problem, which turns out to be true as we

demonstrate in Sections 4 and 5.

4 HYPERBOLIC UTILIZATION BOUND FOR WORSTFIT ALLOCATION

In this section, we will show that the WF heuristic provides theminimum utilization bound among all reasonable task allocationalgorithms. The WF heuristic works in the following way:

1. Tasks are assigned to a processor one at a time in theorder that they appear in the task set, i.e., �1 is assignedfirst, followed by �2, then �3, �4, and so on and so forth.

2. For a task to be assigned, the processor with the mostremaining capacity, among all processors having enoughcapacity to hold this task, is chosen to accommodate it.

3. If two (or more than two) processors have the sameremaining capacity, the tie is broken by choosing theprocessor with the lowest index among all eligiblecandidates.

Theorem 3 gives an upper limit on the multiprocessorbound for WF allocation and HB.

Theorem 3. For the worst fit allocation algorithm, it follows thatUWFHB � 2n


Proof. The proof strategy here is to construct a set of tasks,f�1; �2; . . . ; �mg, with the utilization factor of each individ-ual task no greater than � and the total HB-utilizationequal to 2n

ð1þ�Þn�1 � ð1þ "Þ, and then show that this task setcannot fit into the multiprocessor using HB and WFwhen "! 0þ.

Firstly, the set of tasks are constructed as follows and alltasks are indexed strictly in the order they are generated.

1. bm�1n c task subsets are constructed one by one, each

of which consists of n ¼ n1 þ n2 tasks, where

n1 ¼m� 1� m� 1

n

� �� n

n2 ¼n� n1:

2. For each subset of tasks, the utilization factors of thefirst n1 tasks are identical and equal to

u1 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2

1þ � � ð1þ "Þ1

2n1

m�1nd er

� 1 (16)

and the utilization factors of the remaining n2 tasks are

u2 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2

1þ � � ð1þ "Þ1

2n2

m�1nb cr

� 1: (17)

3. After the construction of bm�1n c task subsets, another

n1 tasks, each with utilization factor u1, are added in,followed by the final task �m of utilization factor �.

Next we will show the validity of our construction. Bythe construction, we know that there are ðbm�1

n c þ 1Þ � n1

tasks with utilization factor u1, ðbm�1n cÞ � n2 tasks with


utilization factor u2 and one task of utilization factor �. Itis easy to see that the total number of tasks is equal tom according to the definition of n1 and n2. By multiplyingall task utilization factors together, the total HB-utilizationof the generated task set is

U ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2

1þ � � ð1þ "Þ1

2n1

m�1nd er" # m�1

nb cþ1ð Þn1

�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2

1þ � � ð1þ "Þ1

2n2

m�1nb cr" # m�1

nb cð Þn2

� ð1þ �Þ ¼ 2n

ð1þ �Þn�1� ð1þ "Þ

which indicates that our construction in terms of thetotal HB-utilization is correct.

Besides the total HB-utilization, it is also necessary toprove that the utilization factor of each task is no greaterthan �, viz., 0 G ui � �, 8 i ¼ 1; 2; . . . ;m.

It is straitforward to prove u1 9 0 and u2 9 0 because �is less than or equal to 1. To show u1 � �, we rewrite u1

as follows:

u1 ¼ffiffiffiffiffiffiffiffiffiffiffiffi

2

1þ �m�1nd er

� ð1þ "Þm�1nd e� 1

2n1 � 1:

Bearing in mind that ð1þ "Þðm�1Þnd e�ð 1

2n1Þ

approaches 1þ as

"! 0þ. Hence, ifffiffiffiffiffiffiffi

21þ�

ðm�1Þnd eq

� 1 G �, then one can always

find an approximate " which makes u1 � � because of

the continuity of real numbers. We will showffiffiffiffiffiffiffi

21þ�

ðm�1Þnd eq

�1 G � below.

Recall that we assume m 9 n� to eliminate the trivialcases where each processor only holds at most � tasks.As m is a natural number, we have m � n� þ 1. For

m � n� þ 1,ffiffiffiffiffiffiffi

21þ�

ðm�1Þnd eq

is a non-increasing function of m.

Therefore, this function is maximized atffiffiffiffiffiffiffi

21þ�

�

q� 1 when

m ¼ n� þ 1. Consequently, to proveffiffiffiffiffiffiffi

21þ�

ðm�1Þnd eq

� 1 G �, we

only need to show thatffiffiffiffiffiffiffi

21þ�

�

q� 1 G �.

By the definition of �, � þ 1 tasks of utilization factor� cannot fit into one processor under HB, which meansQ�þ1

i¼1

ð1þ �Þ ¼ ð1þ �Þ�þ1 9 2. Dividing both sides of the

inequality by ð1þ �Þ� , we get

2

ð1þ �Þ G ð1þ �Þ�:

By rearranging this inequality we obtainffiffiffiffiffiffiffiffiffiffiffiffi2

1þ ��

r� 1 G �

which proves the validity of u1. The correctness of u2 canbe validated in a similar way, which is omitted in thispaper for the sake of brevity.

We are now ready to prove that the constructed taskset does not fit into the multiprocessor. Please note thatall tasks are allocated sequentially (according to the orderof indices) by the WF heuristic, which assigns a task to

the processor with the largest remaining capacity. As aresult, the first m� 1 tasks in the task set are assigned toprocessors in the following way:

1. Each of the first n1 processors holds dðm� 1Þ=netasks of utilization factor u1.

2. Each of the remaining n2 processors holds bðm� 1Þ=nctasks of utilization factor u2.

For the first n1 processors, any of them can only hold

one more task with utilization factor2

ðu1þ1Þdðm�1Þ=ne � 1 ¼ð1þ �Þ

12n1 � 1. Therefore, the last task of utilization factor�

can not be allocated to any of these processors successful-ly. Likewise, none of the remaining n2 processors can holdthe last task because the capacity left for each of these

processor is 2

ðu2þ1Þbðm�1Þ=nc � 1 ¼ ð1þ �Þ1

2n2 � 1.

As a conclusion, the constructed task set, with total

HB-utilization of 2n

ð1þ�Þn�1 � ð1þ "Þ, does not fit into n

processors under the WF heuristic and HB given "! 0þ.

Hence, we proved UWFHB � 2n

ð1þ�Þn�1. Ì

Corollary 1 presents the multiprocessor utilizationbound associated with the WF heuristic and HB.

Corollary 1. For the worst fit allocation algorithm, it follows thatUWFHB ¼ 2n


Proof. The Corollary can be proved directly from Theorem 1and Theorem 3. Ì

5 HYPERBOLIC UTILIZATION BOUND FORRAD ALLOCATION

In this section, the utilization bound for RAD allocation,denoted by URAD

HB , is presented. We also prove that URADHB

coincides with MaxRAHB. Recall that reasonable allocationdecreasing (RAD) algorithms are a subclass of reasonableallocation algorithms with the following characteristics:

1. Tasks are ordered by decreasing utilizations factorbefore allocation.

2. Tasks are allocated sequentially and the allocationalgorithm fails only if there is no processor havingsufficient capacity to hold the pending task.

Typical representatives of the RAD heuristics includeFFD, BFD and WFD mentioned in Section 2.

Theorem 4 provides a lower limit on the multiprocessorutilization bound for RAD allocation.

Theorem 4. For any reasonable allocation decreasing algorithm,it follows that URAD

HB � 2n�þ1�þ1 if m 9 n�.

Proof. To prove the theorem, we need to show that if a taskset f�1; �2; . . . ; �mg does not fit into the multiprocessor,the following inequality must hold

Ymi¼1

ð1þ uiÞ 9 2n�þ1�þ1 :

Let �k, with utilization uk, be the first task that does notfit into any of the n processors. According to the definition


of RAD, we have the following equation:

Uj � ð1þ ukÞ 9 2; 8 j ¼ 1; 2; . . . ; n (18)

where Uj ¼Qmj

i¼1

ð1þ uiÞ is the total HB-utilization of the

mj tasks allocated to Pj. The system of inequalities fromEquation (18) gives

ð1þ ukÞnYnj¼1

Uj 9 2n: (19)

The total HB-utilization of the first k� 1 tasks satisfiesQnj¼1

Uj¼Qk�1

i¼1

ð1þuiÞ. Substituting this equation into Equation

(19), we get

ð1þ ukÞn�1Yki¼1

ð1þ uiÞ 9 2n: (20)

Since tasks are sorted in the decreasing order of the uti-

lization factor before allocation, we know ðQki¼1

ð1þ uiÞÞ1=k �

ð1þ ukÞ.Substituting this inequality into Inequality (20) and

finding

Yki¼1

ð1þuiÞ !n�1=k

�Yki¼1

ð1þuiÞ ¼Yki¼1

ð1þuiÞ !nþk�1=k

9 2n

which indicates

Yki¼1

ð1þ uiÞ 9 2nk

nþk�1: (21)

It is easy to check 2nk

nþk�1 increases monotonically as k

increases. Recall that k is constrained to be k � n� þ 1,

otherwise the task set will be schedulable trivially. Underthis constraint, we know that 2

nknþk�1 is minimized when

k ¼ n� þ 1, i.e., 2nk

nþk�1 � 2n2�þnnþn� ¼ 2

n�þ1�þ1 . By substituting the

inequality into (21) we have

Yki¼1

ð1þ uiÞ 9 2nk

nþk�1 � 2n�þ1�þ1 : (22)

Since we knowQmi¼1

ð1þ uiÞ �Qki¼1

ð1þ uiÞ, from (22) we have

Ymi¼1

ð1þ uiÞ 9 2n�þ1�þ1 : (23)

Inequality (23) indicates that the total HB-utilizationof any unschedulable task set on a homogeneous nprocessor platform using RAD and HB for RM schedulingmust be greater than 2

n�þ1�þ1 . Thus, any task set with the total

HB-utilization less than or equal to 2n�þ1�þ1 fits into the

multiprocessor using HB. As a conclusion, we have

URADHB � 2

n�þ1�þ1 . Ì

Corollary 2 presents the multiprocessor utilization boundassociated with any RAD heuristic under HB.

Corollary 2. For any reasonable allocation decreasing algorithm,it follows that URAD

HB ¼ 2n�þ1�þ1 if m 9 n�.

Proof. The Corollary follows directly from Theorem 2 andTheorem 4. Ì

6 EXPERIMENTS AND ANALYSIS

The hyperbolic multiprocessor bounds we obtain arederived for the worst-case task sets as shown in Theorem 1and Theorem 2. Considering the rare occurrences of theseworst cases in practice, these bounds are relatively pessimisticcompared to the multiprocessor utilization bounds (thatmay be obtained by) using exact schedulability analysisas the uniprocessor schedulability condition. We, however,will not address this issue here because detailed evaluationof such pessimism has been reported in [6].

In this paper, we concentrate on the comparison of themultiprocessor utilization bounds under LLB and thoseunder HB. Since the lower limits on the multipocessorutilization bound under LLB and HB are too pessimistic,and only of importance from a theoretical perspective, wefurther narrow our focus on evaluating the performance ofMaxRALLB and MaxRAHB in this section.

To evaluate the performance of utilization bounds, anideal method is to analytically compute the acceptanceratio for a test, i.e., the fraction of task sets that pass the testwith respect to the total number of feasible ones.

Fig. 1. Visual comparison of LLB-based and HB-based multiprocessorbounds in terms of their respective acceptance ratios represented by twoplanes in a 3-dimensional space.

Fig. 2. Uniform utilization (n ¼ 16, � ¼ 1).


To illustrate, Fig. 1 visualizes the schedulability spacesunder HB and LLB for a simple example, where three tasksare assigned to two processors. In Fig. 1, the x, y and z axesrepresent the utilization factors of these three tasks, respec-tively. In this 3-dimensional space, the upper limit of LLB isrepresented by a red 3-dimensional plane, while the maxi-mum HB bound is represented by a black 3-dimensionalhyperbolic surface. It is easy to see that, in this example, thespace beneath the HB hyperbolic surface is much largerthan that under the LLB plane, which suggests that the HBcan provide much better schedulability than LLB. (Actually,there are a few task sets that are feasible under LLB but notfeasible under HB, but it is hard to see them in this figurebecause the volume is too small.)

Unfortunately, as pointed out in [13], computing thegeneral acceptance ratio using a rigorous approach isnot always possible, except for very simple cases. As analternative, we experimentally evaluate the performanceof MaxRAHB, in comparison with MaxRALLB, under a largevariety of parameter settings.

To be specific, we enumerate as many task sets aspossible in the task space, and then collect acceptanceratios experimentally so as to see under which bound thenumber of schedulable task sets is significantly greater. Thecollected statistics could be a good estimate of the relativetheoretical performance for HB and LLB only if a largenumber of task sets are tested. As a result, we choose anexperimental setup similar to the one adopted in [14],which is the de facto standard for generating workload inmultiprocessor real-time scheduling. Task utilizations arechosen according to one of following distributions:

1. uniform distribution between (0, 21=� � 1), � ¼1; 2; . . . ; 20;

2. bimodal distribution: heavy tasks with uniformdistribution between (0.5, 1) and light tasks withuniform distribution between (0, 0.5), probabilityof being heavy ¼ 0:25, 0.33, 0.67 and 0.75;

3. exponential distribution with mean 0.1, 0.25, 0.5.

We do not simulate task periods and computation timesexplicitly because the two feasibility tests are only affectedby task utilization. Datasets are generated for 4, 8, 12, and 16processor systems as follows. Initially, 1,000,000 task sets,each of which consisting of n tasks, are generated according

to the chosen utilization distribution. Then for each taskset, one new task is generated and added in, and bothfeasibility tests are carried out for the new task set. Fora single task set, the procedure of adding new tasks isrepeated until the total utilization exceeds n. The wholeprocedure ends if the utilization of every of the 1,000,000task sets is greater than n.

The main performance measure is the Success Ratio,which is defined as the proportion of generated task setsthat can pass the LLB-based test (HB-based test). To gain adeeper insight into the performance of both bounds, wedivide the full range of possible utilizations into intervalsof length 0.1. The statistics are collected for each interval.Besides, the total number of feasible task sets using HB(denoted by Num(HB-M)), the total number of feasible tasksets using LLB (denoted by Num(LLB-M)), the number oftask sets that are feasible using HB and infeasible usingLLB, and the number of task sets that are infeasible usingHB and feasible using LLB are also collected.

6.1 Experimental ResultsWe have observed similar performance patterns across allparameter settings in our experiments. As a result, we onlyshow distinguishing results in this section. Figs. 2, 3, 4, 5and Figs. 14, 15, 16, 17 (which is available in the ComputerSociety Digital Library at http://doi.ieeecomputersociety.org/10.1109/TPDS.2013.213) show the experimental resultsfor the uniform utilization distribution, where HB-M and





LLB-M are shorthand notations for MaxRAHB and MaxRALLB,respectively. As shown in these figures, HB-M is alwayssuperior to LLB-M in the case of small values of �ð¼ 1; 2Þ.As � increases, the performance of HB-M degradesgradually, and eventually becomes worse than that ofLLB-M in terms of the total number of feasible task sets. Thereason can be explained with the help of Observation 1.

6.1.1 Observation 1Given

Pmi¼1

ui ¼ C, where ui 2 ½0; 1�, i ¼ 1; 2; . . . ; m and C is a

constant,Qmi¼1

ðui þ 1Þ is maximized when the variance of uis

is zero, i.e., u1 ¼ u2 ¼ � � � ¼ um, and the value ofQmi¼1

ðui þ 1Þdecreases as the variance of uis increases.

When � takes small valuesð¼ 1; 2Þ, the maximum achiev-able task utilizations are 1 and

ffiffiffi2p� 1, respectively.

Since task utilizations follow the uniform distribution, thevariance of uis is relatively large, which leads to small

values ofQmi¼1

ðui þ 1Þ. As a result, many infeasible task sets

using LLB-M become schedulable using HB-M. As �increases, the maximum achievable task utilizationsbecome smaller and smaller, and finally the values of uis

will converge to ðPmi¼1

uiÞ=m. By Observation 1 we know, for

task sets falling into the same utilization interval,Qmi¼1

ðui þ 1Þ

increases as the variance of uis decreases, which makesHB-M lose its advantage over LLB-M in accepting more tasksets in the end. Fortunately, Observation 2 indicates that theperformance of HB-M will not deteriorate continuously as �increases.

6.1.2 Observation 2As � tends to infinity, MaxHBRA ¼ 2

n�þ1�þ1 and MaxLLBRA ¼

ðn� þ 1Þð21=ð�þ1Þ � 1Þ converge to 2n and nln2, respectively.In other words, as the task utilizations become smaller andsmaller, and eventually almost identical to each other, themultiprocessor utilization bound is equal to the sum of

single processor utilization bounds in the case of LLB, orthe product of single processor HB-utilization bounds inthe case of HB.

It has been proved in [3] that LLB and HB are equivalenton uniprocessors when the utilizations of all tasks areequal. Together with Observation 2, we know that the ratiobetween the number of feasible task sets using HB-M andthe number of feasible task sets using LLB-M converges to 1as � approaches infinity, which is borne out by resultsshown in Fig. 6.

Figs. 7 and 8, together with Figs. 18 and 19 (in Appendix I)show the performance of LLB-M and HB-M for task setsgenerated according to the bimodal distribution of taskutilizations. As shown in these figures, when the probabilityof tasks being light gets low, the gain of HB-M over LLB-Mdecreases gradually, which indicates that HB-M is morefavored large-utilization tasks dominate the task popula-tion. Once again, these results confirm our observation thatHB-M performs better when the variance of utilizationfactors is large.

The experimental results for the exponential distribution aredepicted in Figs. 9, 10, and 20 (in Appendix I). As we can see,HB-M outperforms LLB-M in most cases when the mean valueis equal to 0.5. As the mean value decreases, the performance ofHB-M begins to degrade, and eventually HB-M is over-shadowed by LLB-M. Two reasons account for this phenom-enon. Firstly, the utilizations of most tasks are small in thecase of small mean values, which leads to the increase inQmi¼1

ðui þ 1Þ. Secondly, HB-M ð2n�þ1�þ1 Þ has much chance to be

minimized with respect to � because large utilization tasksmay still be generated even if the mean values are small.Please note that it is impossible for these two cases to occursimultaneously in the uniform distribution.

6.2 Compatibility of Both BoundsThe experimental results discussed in Section 4.1 indicatethat neither HB-M nor LLB-M outperforms the other across

Fig. 7. Bimodal utilization, Probability of being heavy ¼ 0:75 ðn ¼ 16Þ. Fig. 8. Bimodal utilization, Probability of being heavy ¼ 0:25 ðn ¼ 16Þ.

Fig. 6. Success Ratio vs. � value ðn ¼ 16Þ.


all parameter settings. According to the taxonomy in [4],HB-M and LLB-M are incomparable in that none of themis superior to the other in every case. Fortunately, bothfeasibility tests (HB-M and LLB-M) are based on RMscheduling and RA, and therefore compatible with eachother. To be specific, as long as a task set passes eitherHB-M or LLB-M test, it can be scheduled successfully usingRM and RA on multiprocessors. More importantly, sucha combined feasibility test does not incur much extraoverhead because it keeps the OðmÞ complexity of schedul-ability test unchanged.

Figs. 11, 12, and 13 show that the combined use of HB-Mand LLB-M improves the overall performance significantlycompared with the separate use of LLB-M.

7 RELATED WORK

Multiprocessor real-time scheduling with its origin tracedback to the late 1960s and early 1970s has gone through arenaissance in recent years due to the advent of multi-corearchitectures. One interesting problem in multiprocessorreal-time scheduling is to find out utilization boundsfor given scheduling algorithms, so that one can quicklyevaluate the feasibility of a specific task set. For our study,we only focus on partitioned fixed-priority multiprocessorscheduling. We refer interested readers to [4] for a morecomplete discussion of other research topics on multi-processor real-time scheduling.

Most of the early research into partitioning-based static-priority multiprocessor scheduling centered on the designand analysis of novel allocation algorithms, by which tasksare partitioned into groups and then assigned to differentprocessors [15], [16], [17]. In 1998, Oh and Baker derivedthe utilization bound based on RM scheduling and the firstfit heuristic. It was proved that RM scheduling, togetherwith the FF heuristic, meets all task deadlines if less than41 percent of the processor capacity is requested [5].

In 2001, B. Andersson et al. showed that with respect to thepartitioning strategy, the utilization bound for any allocationalgorithm is less than ðnþ 1Þ=2 [18]. Two years later, theyproposed an allocation algorithm, ROUND-MP-NFR, thatrenders a utilization bound of n=2. This result demonstratesthat a task allocation algorithm, which can almost achievemaximum possible utilization bound, does exist.

Lopez et al. generalized the result in [5] by takingmaximum task utilization and number of tasks to bescheduled into consideration. A series of utilization boundshad been provided for RM scheduling under FF, BFD,FFD and WF, which have better performance than theaforementioned bounds when the maximum task utiliza-tion is relatively small.

8 CONCLUSION

In this paper, the problem of establishing utilization boundsfor partitioned multiprocessor RM scheduling is investigated.

Fig. 9. Exponential utilization with mean ¼ 0:5 ðn ¼ 16Þ.

Fig. 10. Exponential utilization with mean ¼ 0:1 ðn ¼ 16Þ.

Fig. 11. Ratio of feasible task sets w.r.t. feasible task sets using LLB-M,uniform utilization ðn ¼ 16Þ.

Fig. 12. Ratio of feasible task sets w.r.t. feasible task sets using LLB-M,bimodal utilization ðn ¼ 16Þ.


Different from most existing work, we use the hyperbolicbound as the uniprocessor schedulability condition andprovide the lower and upper limits on the multiprocessorutilization bound for all reasonable allocation algorithms.Also, we show that the utilization bounds for the WF heuristicand RAD allocation coincide with the minimum and maxi-mum achievable multiprocessor utilization bounds underHB. Comprehensive experimental results demonstrate thatneither the proposed hyperbolic multiprocessor utilizationbound nor the traditional bound based on LLB dominatesin all circumstances. Thanks to the compatibility of thesetwo bounds, we can achieve up to 3.4 times more feasibletask sets by combining them together compared with theseparate use of either one.

ACKNOWLEDGMENT

The authors would like to thank the anonymous reviewersfor their valuable comments and suggestions that improvedthe quality of this paper. The work reported in this paperis partially supported by NSFC under grant numbers61370205 and 61170174, NSF of Shanghai under grantnumber 13ZR1400800, the Fundamental Research Funds forthe Central Universities and NSC grant 101-2221-E-006-219.A preliminary version of this paper appeared in theproceedings of 2012 IEEE International Conference onEmbedded and Real-Time Computing Systems and Appli-cations (RTCSA 2012) Seoul, Korea, August 19-22, 2012.

REFERENCES

[1] G.C. Buttazzo, ‘‘Rate Monotonic vs. EDF: Judgment Day,’’ Real-Time Syst., vol. 29, no. 1, pp. 5-26, Jan. 2005.

[2] C.L. Liu and J.W. Layland, ‘‘Scheduling Algorithms for Multi-programming in a Hard-Real-Time Environment,’’ J. ACM,vol. 20, no. 1, pp. 46-61, Jan. 1973.

[3] E. Bini, G.C. Buttazzo, and G. Buttazzo, ‘‘Rate MonotonicAnalysis: The Hyperbolic Bound,’’ IEEE Trans. Comput., vol. 52,no. 7, pp. 933-942, July 2003.

[4] R.I. Davis and A. Burns, ‘‘A Survey of Hard Real-TimeScheduling for Multiprocessor Systems,’’ ACM Comput. Surv.,vol. 43, no. 4, p. 35, Oct. 2011.

[5] D.-I. Oh and T.P. Baker, ‘‘Utilization Bounds for n-Processor RateMonotone Scheduling with Static Processor Assignment,’’ Real-Time Syst., vol. 15, no. 2, pp. 183-192, Sept. 1998.

[6] J.M. Lopez, J.L. Dıaz, and D.F. Garcıa, ‘‘Minimum and MaximumUtilization Bounds for Multiprocessor Rate Monotonic Scheduling,’’IEEE Trans. Parallel Distrib. Syst., vol. 15, no. 7, pp. 642-653, July 2004.

[7] I.I. Lupu, P. Courbin, L. George, and J. Goossens, ‘‘Multi-CriteriaEvaluation of Partitioning Schemes for Real-Time Systems,’’ inProc. ETFA, 2010, pp. 1-8.

[8] T.A. AlEnawy and H. Aydin, ‘‘Energy-Aware Task Allocation forRate Monotonic Scheduling,’’ in Proc. IEEE Real-Time Embed.Technol. Appl. Symp., 2005, pp. 213-223.

[9] J.M. Lopez, M. Garcıa, J.L. Dıaz, and D.F. Garcia, ‘‘UtilizationBounds for Multiprocessor Rate-Monotonic Scheduling,’’ Real-Time Syst., vol. 24, no. 1, pp. 5-28, Jan. 2003.

[10] M. Joseph and P.K. Pandya, ‘‘Finding Response Times in a Real-Time System,’’ Comput. J., vol. 29, no. 5, pp. 390-395, 1986.

[11] J.P. Lehoczky, L. Sha, and Y. Ding, ‘‘The Rate MonotonicScheduling Algorithm: Exact Characterization and AverageCase Behavior,’’ in Proc. IEEE Real-Time Syst. Symp., 1989,pp. 166-171.

[12] N. Audsley, A. Burns, M. Richardson, K. Tindell, and A.J. Wellings,‘‘Applying New Scheduling Theory to Static Priority Pre-EmptiveScheduling,’’ Softw. Eng. J., vol. 8, no. 5, pp. 284-292, Sept. 1993.

[13] E. Bini and G.C. Buttazzo, ‘‘Measuring the Performanceof Schedulability Tests,’’ Real-Time Systems, vol. 30, no. 1/2,pp. 129-154, May 2005.

[14] T.P. Baker, ‘‘A Comparison of Global and Partitioned EDFSchedulability Tests for Multiprocessors,’’ in Proc. RTSN, 2005,pp. 119-127.

[15] S. Lauzac, R.G. Melhem, and D. Mosse, ‘‘An Improved Rate-Monotonic Admission Control and Its Applications,’’ IEEETrans. Comput., vol. 52, no. 3, pp. 337-350, Mar. 2003.

[16] A. Burchard, J. Liebeherr, Y. Oh, and S.H. Son, ‘‘New Strategiesfor Assigning Real-Time Tasks to Multiprocessor Systems,’’ IEEETrans. Comput., vol. 44, no. 12, pp. 1429-1442, Dec. 1995.

[17] Y. Oh and S.H. Son, ‘‘Allocating Fixed-Priority Periodic Tasks onMultiprocessor Systems,’’ Real-Time Syst., vol. 9, no. 3, pp. 207-239,Nov. 1995.

[18] B. Andersson, S.K. Baruah, and J. Jonsson, ‘‘Static-PriorityScheduling on Multiprocessors,’’ in Proc. IEEE Real-Time Syst.Symp., 2001, pp. 193-202.

[19] H. Wang, L. Shu, W. Yin, J. Jin, and J. Cao, ‘‘The HyperbolicSchedulability Bound for Multiprocessor RM Scheduling,’’ inProc. RTCSA, 2012, pp. 124-133.

Hongya Wang received the BS and MS degreesin electrical engineering from Central ChinaNormal University, Wuhan, Hubei, China, in1998 and 2001, respectively, and the PhDdegree in computer Science from HuazhongUniversity of Science and Technology, Wuhan,Hubei, China, in 2005. He is currently anassociate professor in the School of ComputerScience and Technology at Donghua University,Shanghai, China. His research interests includereal-time computing, mobile computing and

database systems.

LihChyun Shu received the PhD degree incomputer science from Purdue University, WestLafayette, IN, USA, in 1994. He is a professorin the Department of Accountancy, NationalCheng Kung University, Tainan, Taiwan. FromAugust 2011, he also serves as Dean of collegeof information and engineering of Chang JungChristian University, Tainan, Taiwan. He acts asthe corresponding author of this article. Hisresearch interests include real-time systems,data stream mining and location-based query

processing, and software transactional memory. He is a member of theIEEE and IEEE Computer Society.

Wei Yin received the Master’s degree in computerscience from theDonghuaUniversity, in 2012. He iscurrently working as a software engineer in CASCOSignal Ltd. Hismain research interests include real-time scheduling theory and real-time systems.

Fig. 13. Ratio of feasible task sets w.r.t. feasible task sets using LLB-M,exponential utilization ðn ¼ 16Þ.


Yingyuan Xiao received the PhD degree incomputer science from Huazhong Universityof Science and Technology, Wuhan, Hubei,China, in 2005. He is currently an AssociateProfessor in the School of Computer Scienceand Technology, Tianjin University of Technology,Tianjin, China. His research interests includeadvanced databases, real-time systems andmobile computing. He has published more than30 journal and conference papers in these areas.He has served as a publicity chair and program

committee member for a number of international conferences, includingFSKD2009, APWeb2011, and APSCC2011. He was a visiting scholar inthe School of Computing at the National University of Singapore from2009 to 2010.

Jiao Cao is a graduate student at the school ofcomputer science and technology in DonghuaUniversity. Her main research interests includereal-time computing and database theory.

. For more information on this or any other computing topic,please visit our Digital Library at www.computer.org/publications/dlib.


Hyperbolic utilization bounds for rate monotonic scheduling on homogeneous multiprocessors

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