Approximate Bandwidth Allocation for Fixed-Priority-Scheduled Periodic Resources (WSU-CS Technical Report Version)

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1 Approximate Bandwidth Aocation for Fixed-Priority-Schedued Periodic Resources WSU-CS Technica Report Version) Farhana Dewan Nathan Fisher Abstract Recent research in compositiona rea-time systems has focused on determination of a component s rea-time interface parameters An important objective in interface-parameter determination is minimizing the bandwidth aocated to each component of the system whie simutaneousy guaranteeing component scheduabiity With this goa in mind, in this paper we deveop a fuy-poynomia-time approximation scheme FPTAS) for aocating bandwidth for sporadic task systems schedued by fixed priority eg, deadine monotonic, rate monotonic) upon an Expicit-Deadine Periodic EDP) resource Our parametric agorithm takes the task system and an accuracy parameter ɛ > 0 as input, and returns a bandwidth which is guaranteed to be at most a factor 1 + ɛ) times the optima minimum bandwidth required to successfuy schedue the task system By simuations over syntheticay generated task systems, we observe a significant decrease in runtime and a sma reative error when comparing our proposed agorithm with the exact agorithm and the sufficient agorithm 1 Introduction Recent research in rea-time systems has focused on designing frameworks for enabing component-based design in reatime systems State-of-the-art frameworks for compositiona rea-time systems incude [1, 9, 12, 25] Component-based design is highy desirabe due to its we-known benefits of reducing overa system compexity and enhancing system designers understanding of the system One of the major benefits of these systems is achieved by the goa of component abstraction, which hides the interna compexity and detais of one component from deveopers of other components and ony exposes information necessary to use the component via an interface In most compositiona frameworks, a component uses a rea-time interface to communicate with the other components of the system A component specifies its resource requirements to meet its rea-time constraints by the attribute interface bandwidth Thus, an important design issue of these compositiona frameworks is addressing the probem minimization of interface bandwidth MIB-RT) One simpe, yet fexibe, rea-time compositiona framework is the expicit-deadine periodic resource EDP) mode [25] An EDP resource Ω is characterized by a three-tupe Π, Θ, ) where Π is referred to as the period of repetition, Θ is the capacity, and is the reative deadine The interpretation of such a resource is that a component C executed upon Ω is guaranteed Θ units of processing resource suppy for successive Π-ength intervas given some initia starting time) Furthermore, the Θ units of resource suppy must be provided within Π) time units after the start of the Π-ength interva The interface bandwidth of C for this framework is Θ Π A system-eve scheduing agorithm aocates the processor time among the different periodic resources that share the same processor, such that each resource receives for every period) aggregate processor time equivaent to its capacity A component s tasks are then hierarchicay schedued by a componenteve scheduing agorithm upon the processing time suppied to resource Ω In this paper, we obtain soutions to MIB-RT for an EDP resource when components use fixed-priority as the componenteve scheduing agorithm The system-eve scheduing agorithm is not considered for this paper) Specificay, we consider the probem of determining the optima choice of capacity parameter ie, Θ) for an EDP resource Ω with a fixed period Π and deadine for component C Agorithms exist for determining Π and by searching over possibe vaues and using a This research has been supported by a Wayne State University Facuty Research Award F Dewan and N Fisher are with Department of Computer Science, Wayne State University in Detroit, MI, USA Corresponding Author s Emai:farhanad@wayneedu 1

2 capacity-determination agorithm as a subroutine eg, see [10, 13]); thus, since the search space may be quite arge, efficient capacity-determiniation agorithms are necessary The MIB-RT probem for fixed-priority periodic resource mode has been previousy studied An exact soution based on exact scheduabiity techniques for uniprocessor rea-time systems [6, 16]) has been proposed by Easwaran et a [11] There is aso a On)-time sufficient soution to MIB-RT for periodic resource Π equas ) by Shin and Lee [25] The exact resource aocation is computationay expensive pseudo-poynomia in this case) and thus might be impractica for agorithms that search for optima vaues of Π and On the other hand, though the sufficient resource aocation has ower inear) computationa compexity, these agorithms might provide over-estimated resource aocations and induce ower system utiization This might be impractica for deveoping rea-time systems in which resources are very scarce However, in many rea-time systems where tasks may be added or removed dynamicay, it is important to provision resources efficienty at run-time and an efficient aocation agorithm is desirabe Our goa is to design an agorithm which is computationay efficient on rea-time guarantee verification as we as to provide the system designer contro over accuracy of resource aocation In our prior work [15], we devised approximate bandwidth aocation agorithm for EDP resource with component eve scheduing agorithm EDF In this paper we extend those resuts for fixed priority schedued components However, the compositiona resuts for EDF does not directy appy for fixed priority, as we have to do maximum response time anaysis for each task; this fundamentay differs from the demand-based approach of [15] Our Contribution For EDP resource mode with sporadic tasks [20] as components, we deveop a parametric approximation agorithm that addresses the current gap between computationay-expensive, exact soutions and computationayinexpensive, sufficient soutions for MIB-RT probem We caim the foowing Given Π,, task system τ, and accuracy parameter ɛ > 0, et Θ Π,, τ) be the optima minimum capacity for τ to be fixed-priority-scheduabe upon EDP resource Ω = Π, Θ Π,, τ), ) Our agorithm returns Θ for the given parameters where Θ Π,, τ) Θ 1 + ɛ) Θ Π,, τ) Furthermore, the time compexity of our agorithm is poynomia in the number of tasks in τ and 1 ɛ In other words, our agorithm is a fuy-poynomia-time approximation scheme FPTAS) for the MIB-RT probem with the approximation ratio 1 + ɛ) This impies that the system designer can pre-specify an arbitrary eve of accuracy in obtaining soution to MIB-RT with the tunabe agorithm We aso vaidate our agorithm by means of simuation over randomy generated task systems Organization The remainder of the paper is organized as foows In Section 2, we briefy review the current iterature on compositiona rea-time frameworks and MIB-RT probem for fixed-priority In Section 3, we provide necessary notations required for the rest of the paper In Section 4, we present an approximate agorithm for bandwidth aocation, and prove its correctness Then, we give the approximation ratio resuts for the proposed agorithm in Section 5 Simuation resuts comparing our agorithm with both previousy-known exact and sufficient agorithms are given in Section 6 Finay, we concude with discussion and future direction of this research in Section 7 2 Reated Work In this section, we give a very high-eve overview of some of the prior work on MIB-RT for compositiona rea-time systems The concept of compositiona rea-time system was first introduced by Deng and Liu [9] in their work rea-time open environments and Rajkumar et a [21] in their work resource kernes Since then, researchers have proposed many different rea-time compositiona modes and studied the MIB-RT probem of the proposed modes Two of the we studied approaches in the iterature are the partition-based modes and the demand-based modes Feng and Mok [12] proposed the concept of tempora partitions to support hierarchica sharing of a processing resource Shin et a [23,25] proposed the reated periodic resource mode to characterize the suppy guaranteed to any component in compositiona system For the tempora partition modes where components are schedued by fixed-priority, Lipari and Bini [18] proposed exact, pseudo-poynomia time agorithm for MIB-RT, and Ameida and Pedreiras [3] proposed sufficient, poynomia-time bandwidth aocation techniques In the demand-based modes, components of the system are characterized by processor-demand curves, which describe the minimum amount of processing required by a component over any time interva Wandeer and Thiee [26] proposed the concept of interface-based design in which rea-time cacuus [8] is used to compute demand curves and service curves for each component in a compositiona rea-time system In another demand-based mode known as hierarchica event stream mode, Abers et a [1] have deveoped parametric agorithms for MIB-RT without known approximation ratios) Thus, for a variety of both partition-based modes, reativey efficient, sufficient agorithms for MIB-RT have been proposed; however, the existence of any work on obtaining poynomia-time agorithms with constant-factor approximation 2

3 ratios where components are schedued by fixed priority is unknown In our preiminary work [15], we obtained such ratios for the periodic resource mode where components are schedued by dynamic-priority EDF) In this paper, our aim is to extend those resuts by deveoping an FPTAS for EDP framework where components are schedued by fixed-priority DM or RM) The agorithm of [13] may be used in conjunction with the resuts of this paper to find an optima period 3 Modes and Notation In this section, we present background and notation for the task mode, workoad functions, and periodic resource mode that we use throughout the paper Sporadic Task Mode A sporadic task τ i = e i, d i, p i ) is characterized by a worst-case execution requirement e i, a reative) deadine d i, and a minimum inter-arriva separation p i Such a sporadic task generates a potentiay infinite sequence of jobs, with successive job-arrivas separated by at east p i time units Each job has a worst-case execution requirement equa to e i and a deadine that occurs d i time units after its arriva time A sporadic task system τ = {τ 1,, τ n } is a coection of n such sporadic tasks We wi assume that each task τ i τ has d i p i ; such sporadic task systems are known as constrained-deadine sporadic task systems The task utiization for a sporadic task τ i is ined as u i = e i /p i The system utiization is denoted by Uτ) = τ u i τ i We wi assume that Uτ) 1; otherwise, task system τ cannot be schedued by any agorithm) to meet a deadines upon a dedicated preemptive uniprocessor We wi assume that each task has a fixed priority and tasks are indexed in non-increasing priority order That is, τ i has higher or equa) priority than τ j, if and ony if, i j As tasks generate jobs, each job inherits the priority of its generating task ie, a jobs generated by task τ i have the same priority as τ i ) For this paper, we assume each component uses fixed-priority scheduing as the component-eve scheduing agorithm Whenever component C is aocated the processor, C executes the highest-priority job with remaining execution; ties are broken in favor of the job generated from the ower task index One noteworthy fixed-priority scheduing agorithm is deadine monotonic DM) [17], which assigns each task a priority equa to the inverse of its reative deadine ie, tasks with shorter reative deadines have priority greater than tasks with onger reative deadines) DM is known to an optima fixed-priority uniprocessor scheduing agorithm for constraineddeadine sporadic tasks in the foowing sense; if a constrained-deadine sporadic task system is scheduabe upon a singe processor by a fixed-priority assignment, then it wi aso meet a deadines under the DM priority assignment Workoad Functions To determine the scheduabiity of a sporadic task system, it is often usefu to quantify the maximum amount of execution time requested by the task in its worst-case phasing over any given interva For sporadic task systems, it is known that the worst-case phasing is the synchronous arriva sequence The synchronous arriva sequence occurs when a tasks of a sporadic task system reease jobs at the same time instant and subsequent jobs as soon as permissibe Researchers [16] have derived the request-bound function, as ined beow Definition 1 Request-Bound Function) For any t > 0 and sporadic task τ i, the request-bound function RBF) quantifies the maximum cumuative execution requests that coud be generated by jobs of τ i arriving within a contiguous time-interva of ength t It has been shown that for sporadic tasks, RBF can be cacuated as foows [16] t RBFτ i, t) = e i 1) p i Figure 1 shows the request-bound function for a sporadic task τ i, which is a right continuous function with discontinuities at time points of the form t a p i where a N The cumuative request-bound function for task τ i is ined as foows: i 1 W i t) = e i + RBFτ j, t) 2) j=1 Audsey et a [5] have given a necessary and sufficient condition for sporadic task system τ to be fixed-priority-scheduabe upon a preemptive uniprocessor patform of unit speed: t 0, d i ] such that W i t) t, i Furthermore, it has aso been shown [4] that this condition needs to be verified at ony time points in the foowing ordered set: S i τ) = { t = b p a : a = 1,, i; b = 1,, di p a } {d i } 3) The above set is known as the testing set for sporadic task τ i The size of this set may be as arge as i d i j=1 p j which is dependent on the task periods, and thus requires pseudo-poynomia time feasibiity test Fisher and Baruah [14] proposed 3

4 5e i 4e i 3e i ta, D ta ) t a+1, D ta+1 ) RBFτ i, t) 2e i ta, D ta ) e i p i 2p i 3p i 4p i 5p i t Figure 1 The step function denotes a pot of RBFτ i, t) as a function of t The dashed ine represents the function δτ i, t), approximating RBFτ i, t) δτ i, t) is equa to RBFτ i, t) for a t k 1)p i The vaue of k equas three in the above graph the foowing approximation to RBF inspired by a simiar approximation for EDF due to Abers and Somka [2]) to reduce the number of points in the testing set { δτ i, t, k) = RBFτi, t), if t k 1)p i e i + t ei p i, otherwise 4) This function tracks RBF for exacty k 1 steps and after the k 1-th step, it uses inear interpoation of subsequent discontinuous points of RBF with sope equa to u i ) The steps in Figure 1 correspond to RBFτ i, t, k), and the thick steps and the soped-dashed ine correspond to δτ i, t, k) The approximate cumuative request bound function is ined as foows: i 1 Ŵ i t) = e i + δτ j, t, k) 5) For any fixed k N +, Fisher and Baruah [14] showed that if for a τ i τ there exists a t 0, d i ] such that Ŵit) t then the sporadic task system τ is static priority scheduabe upon a preemptive uniprocessor patform of unit speed The testing set for this condition is as foows: j=1 Ŝ i τ, k) = {t = b p a a = 1,, i 1; b = 1,, k 1; t 0, d i ]} {d i } {0} 6) Let t a, t a+1 denote any pair of consecutive vaues in the above ordered set Next, we give the reation between the request bound function RBF and the approximate request bound function δ Lemma 1 from [14]) Given a fixed integer k N +, RBFτ i, t) δτ i, t, k) ) k RBFτi, t) for a τ i τ and t R 0 We wi use this emma in our approximation agorithm Section 5) Next, we ine notation to represent the discontinuous ine segments of the cumuative request bound function Ŵi) Let ta, D ta ), t a+1, D ta+1), α be a ine segment in the Eucidean space, R 2, originating at point t a, D ta ) R 2 and ending at point t a+1, D ta+1 ) R 2 with sope α 0; more formay, ta, D ta ), t a+1, D ta+1), α = {x, y) R 2 x [t a, t a+1 ]) y = αx t a ) + D ta ))} 7) Pease note the term α is incuded in the notation for convenience ony; it is possibe to determine the sope from points t a, D ta ) and t a+1, D ta+1 ) aone We denote any point in the ine segment by t, D t ) t a, D ta ), t a+1, D ta+1 ), α 4

5 The connection between t a, D ta ), t a+1, D ta+1 ), α and Ŵi is as foows Consider a time t a Ŝiτ, k) Define D ta to be request bound function at time t a, that is Ŵit a ) Figure 1) At time t a, some set of tasks with priority greater τ i have job arrivas in the synchronous arriva sequence Let r i t) be the sum of the executions of these tasks Formay, r i t) = τ j τ:j<i) p j divides t) e j 8) At time t a there is a discontinuity in the function Ŵi in which Ŵi increases by r i t a ) and then is inear unti the next discontinuity in Ŵi ie, at time t a+1 Ŝiτ, k)) Thus, Ŵi is a ine segment from t a to t a+1 with sope equa to the tota utiization of a task τ j such that j < i and t a k 1)p j We denote D ta by the sum of request bound function at time point t a and job reease at time t a, that is, D ta = D ta + r i t a ) Finay, observe that α = u j 9) τ j τ:t a k 1)p j) j<i) From the above initions of t a, t a+1, D ta, Dta, D ta+1, and α, it is straightforward to verify that the ine segment ta, D ta ), t a+1, D ta+1 ), α is equivaent to t, Ŵit)) for a t t a, t a+1 ] The exception is at t a where D ta is not equa to Ŵit a ); this difference exists for the notationa and agebraic convenience throughout Section 4 From the initions of t a, t a+1, D ta, D ta, D ta+1, and α, the foowing emma is apparent Lemma 2 For any consecutive pairs of vaues t a, t a+1 ) Ŝiτ, k), Ŵit) D t for a t, D t ) t a, D ta ), t a+1, D ta+1 ), α Expicit-Deadine Periodic EDP) Resource Mode An EDP resource, denoted by Ω = Π, Θ, ), guarantees that a component C executed upon resource Ω wi receive at east Θ units of execution between successive time points in {t t 0 + Π N} where t 0 is some initia service start-time t 0 for the periodic resource Furthermore, the Θ units of service must occur units after each successive time point in the aforementioned set Obviousy, Θ ; for this paper, we wi make the simpifying assumption that Π, as we Furthermore, we wi assume in this paper that each component C is a sporadic task system τ schedued by fixed priority upon Ω From now on, we use τ in the context of component C) Definition 2 Suppy-Bound Function) For any t > 0, the suppy-bound function sbf) quantifies the minimum execution suppy that a component executed upon periodic resource Ω may receive over any interva of ength t It is ined as foows [11]: where y Ω = { sbfω, t) = yω Θ + max 0, t x Ω y Ω Π), if t Θ 0, otherwise t Θ) Π and x Ω = Π + 2Θ) 10) We denote the optima minimum capacity for Ω given task system τ by Θ Π,, τ) We use the concept of -feasibiity region of Ω simiar to [15] to ine the region under the -th step of sbf For our convenience, we reine -feasibiity region as foows: Definition 3 -Feasibiity Region of Ω) F Π, {, Θ) = t, D t) R 2 0 ) Θ D t t+π+ ) Θ D t } 11) Figure 2 shows graphica depiction of the suppy bound function sbf for EDP resource Ω The shaded region in the figure corresponds to the -feasibiity region for some step N of the sbf The foowing theorem states the exact scheduabiity condition for EDP resource Ω, where task system is schedued using fixed priority scheduing agorithm [11, 23, 24] It says that for the task system τ to be scheduabe with EDP resource, each task τ i in τ must have a fixed point t before its deadine at which the cumuative request bound function for τ i is ess than the suppy provided to the system at that point 5

6 sbf sbf 3Θ sbf 2Θ Θ ta, D ta ) t a+1, D ta+1 ) F 2 Π, Θ, ) Π + 2Θ + 2Θ 3Π + 2Θ Π + Θ + Θ 3Π + Θ t Figure 2 The soid ine step function is sbf for Ω The shaded region represents the -feasibiity region for = 2, containing a part of the ine segment t a, D ta ), t a+1, D ta+1 ), α Theorem 1 from [11]) A sporadic task system τ is fixed priority scheduabe upon an EDP resource Ω = Π, Θ, ), if and ony if, i, t 0, d i ] : W i t) sbfω, t)) Uτ) Θ ) 12) Π In the next section we present an approximate agorithm to obtain minimum capacity for EDP resource when the componenteve scheduing agorithm is fixed priority for the task system τ We consider fixed period Π) and deadine ) for the EDP resource Ω 4 An Agorithm for Determining Minimum Capacity In Figure 3, we present the pseudocode for our agorithm, FPMINIMUMCAPACITY Given task system τ and an EDP resource with Π and as input, the agorithm returns approximate minimum capacity to correcty schedue the task system with the resource The approximation parameter of the agorithm is the input k N + k = 1 ɛ ) For some fixed k input, the agorithm returns the approximate minimum capacity; if k is equa to, it returns exact minimum capacity If FPMINIMUMCAPACITY returns a vaue Θ min that does not exceed, then τ can be fixed-priority schedued to meet a deadines upon Ω = Π,, Θ min ) Note that the approximate capacity Θ min can be at most 1 + ɛ) times the exact capacity If FPMINIMUMCAPACITY returns a capacity greater than, then our agorithm cannot guarantee τ can be schedued on any Ω with parameters Π and Uness k =, the agorithm is an approximation, and, thus, a returned capacity greater than does not necessariy impy infeasibiity of τ) In our proposed agorithm, the objective is to compute minimum capacity Θ min for a task system τ such that τ is fixedpriority scheduabe under EDP resource mode For each task τ i τ, we find minimum capacity Θ min i such that there exists a fixed point t 0, d i ] at which the suppy bound function sbf exceeds the cumuative request bound function Ŵ i t) Theorem 1) To cacuate Θ min i, we determine, for each consecutive pair of vaues t a, t a+1 ) in the testing set Ŝ i τ, k), the minimum capacity Θ min t a required to guarantee that the ine segment t a, D ta ), t a+1, D ta+1 ), α is beneath sbfπ, Θ min t a, ), t) for some t t a, t a+1 ] Since t a, D ta ), t a+1, D ta+1 ), α is equivaent to Ŵi for a t t a, t a+1 ], this impies that there exist a t t a, t a+1 ] such that Ŵit) sbfπ, Θ min t a, ), t) To determine Θ min t a, we take specific steps of the sbf denote a seected step by ) and determine the minimum Θ such that some point of the ine segment is beow the -feasibiity region with capacity Θ Each Θ for t a, t a+1 ) is set in ines 9, 10, 11 and 12 The foowing functions are used to determine the vaues of Θ in our agorithm Φ 1 t a, D ta ), t a+1, D ta+1 ), α,, Π, ) Φ 2 t a, D ta ), t a+1, D ta+1 ), α,, Π, ) Φ 3 t a, D ta ), t a+1, D ta+1 ), α,, Π, ) = Dt a+1 ta+1+π+, = D ta, = D ta +απ+ t a) +α 13) We wi aso show that we ony need to consider the integer vaues of given by the foowing equations 6

7 FPMINIMUMCAPACITYΠ,, τ, k) 1 Θ min Uτ) Π 2 for each τ i τ 3 Θ min i 4 for each t a, t a+1 ) Ŝiτ, k) In order) 5 Dta Ŵit a ) + r i t a ) 6 D ta+1 Ŵit a+1 ) 7 α τ i τ:t d i+k 1)p i u i Set ine segment variabe 8 AB t a, D ta ), t a+1, D ta+1 ), α 9 Θ 1 +1 Φ 1 AB, 1 + 1, Π, ) 10 Θ 2 1 Φ 2 AB, 2 1, Π, ) 11 Θ 1 Φ 3 AB, 1, Π, ) 12 Θ 2 Φ 3 AB, 2, Π, ) 13 Θ min t a min{θ 1 +1, Θ 2 1, Θ 1, Θ 2 } 14 Θ min i min{θ min i, Θ min t a } 15 end of inner oop) 16 Θ min max{θ min, Θ min i } 17 end of outer oop) 18 return Θ min Figure 3 Pseudo-code for determining minimum capacity for a periodic resource given Π,, and τ using fixed-priority scheduing agorithm Note the agorithm is exact when k equas See the description of Section 4 for inition of 1, 2, and the Φ functions 1 = t a+1 ) + t a+1 ) 2 + 4ΠD ta+1, 14) 2 = t a ) + t a ) 2 + 4Π D ta 15) That is, we consider 1, 1 + 1, 2 1 and 2 to evauate Θ The ogic behind the choice of Φ functions and our inition of 1 and 2 wi be more apparent in the proof of correctness section beow Since we are ooking for ony one point in t 0, d i ] for task τ i where Ŵit) sbfω, t), we ony need a singe ine segment of Ŵit) that intersects with sbfω, t) and gives minimum capacity Thus, we set Θ min i Θ min t a vaues for each of the ine segment of Ŵi Finay, we set Θ min to be the maximum of a Θ min i to be the minimum of a vaues This ensures that for each task τ i τ, we find a t d i such that Ŵit) sbfπ, Θ min, ), t) Since Ŵit) W i t) for a t, this impies Theorem 1; thus τ is fixed priority scheduabe upon EDP resource Ω = Π, Θ min, ) Agorithm Compexity The compexity of FPMINIMUMCAPACITY depends on the number of tasks n in the task set τ and the cardinaity of testing set Ŝiτ, k) for each task τ i The outer oop of the agorithm Lines 2 to 17) iterates for each task, thus n times in tota The inner oop Lines 4 to 15) scans every pair of testing set points in Ŝiτ, k) in non-decreasing order) for task τ i, and this can take at most 1+i 1)k 1) times for a singe task Using a heap-of-heaps described by Mok [19], the time compexity to obtain an eement of the testing set is Oog n) Setting D ta, D ta+1 and α Lines 5, 6 and 7) is done in constant time on each iteration of the inner oop Again, setting vaues and evauating Θ vaues using these Line 9 to 12) takes constant time Therefore, the runtime compexity of FPMINIMUMCAPACITY is Oog n n i=1 Ŝiτ, k) ) If k =, the compexity for exacty determining the minimum capacity is the same compexity as the test of Theorem 1 on a fixed Ω, which may be pseudo-poynomia depending on the period of tasks Otherwise, if k is a fixed integer, the compexity is at most O og n n i=1 1 + i 1)k 1))) times, which is Okn2 og n) Agorithm Correctness To prove the correctness of FPMINIMUMCAPACITY, we prove the foowing theorem which states that the vaue returned by the agorithm ie, Θ min ) is at east the optima minimum capacity vaue Θ Π,, τ) Furthermore, if the input k equas, then the returned capacity is optima 7

8 Theorem 2 For a k N + { }, FPMINIMUMCAPACITY returns Θ min Θ Π,, τ) Furthermore, if k =, Θ min = Θ Π,, τ) We require some additiona initions simiar to [15] for notationa convenience The next inition quantifies the minimum capacity Θ ) that is required for sbf to exceed the ine segment t a, D ta ), t a+1, D ta+1 ), α at some point t, D t ) We wi use the convention that inf returns on an empty set Definition 4 Minimum Capacity for t a, D ta ), t a+1, D ta+1 ), α ) Θ Π,, t a, D ta ), t a+1, D ta+1 ), α ) = inf { Θ R + Θ ) t, D t) t a, D ta ), t a+1, D ta+1 ), α : D t sbfπ, Θ, ), t) ) } 16) The next function determines the minimum capacity for any given ine segment t a, D ta ), t a+1, D ta+1 ), α to have a point in the -feasibiity region Definition 5 -Minimum Capacity for t a, D ta ), t a+1, D ta+1 ), α ) Θ Π,, ta, D ta ), t a+1, D ta+1 ), α ) = inf {Θ ) R + t, D t) t a, D ta ), t a+1, D ta+1 ), α : t, D t) F Π, Θ, ) } 17) Note the two above initions use infimum, since they are ined over infinite sets; however, we wi ater see Coroary 4) that the infimum corresponds to the minimum ie, the vaue returned by inf exists in the set specified in the right-hand side of Equations 16 and 17) In order to prove Theorem 2, we must prove some additiona emmas We start by presenting the three conditions on the vaue of Θ that are necessary and sufficient condition for a ine segment t a, D ta ), t a+1, D ta+1 ), α to have a point in the -feasibiity region Lemma 3 For any two consecutive pair of vaues t a, t a+1 ) Ŝiτ, k), there exists t, D t ) t a, D ta ), t a+1, D ta+1 ), α such that t, D t ) F Π,, Θ) for some N +, if and ony if, the foowing conditions hod: Θ Φ 1 t a, D ta ), t a+1, D ta+1 ), α,, Π, ) 18a) Θ Φ 2 t a, D ta ), t a+1, D ta+1 ), α,, Π, ) 18b) Θ Φ 3 t a, D ta ), t a+1, D ta+1 ), α,, Π, ) 18c) 18) Proof: For the ony if direction, we must show if some point of the ine segment is in the -feasibiity region for any given N + then the three conditions of Equation 18) hod We wi show this by contrapositive; that is, if any of the three conditions is vioated, the ine segment wi not be in F Π,, Θ) for that We now consider the negation of the conditions of Equation 18) By negation, at east one of the Equations 18a), 18b), or 18c) must be vioated We wi show that if any of the conditions is vioated, then for a t, D t ) t a, D ta ), t a+1, D ta+1 ), α, t, D t ) F Π,, Θ) Case 1: Equation 18a) is fase That is, Θ < Dt a+1 ta+1+π+ Θ < D ta +αt a+1 t a) t a+1+π+ The second inequaity foows from the fact that D ta+1 = D ta + αt a+1 t a ) Consider any t, D t ) t a, D ta ), t a+1, D ta+1 ), α Let x = t t a where 0 x t a+1 t a ; thus, t = t a + x and D t = D ta + αx Consider the expression D ta + αx) t a + x) + Π Obviousy, the above expression is non-increasing in x, since Uτ) 1 and α is at most the utiization of tasks D with higher priority than τ i Therefore, ta +αt a+1 t a) t a+1+π+ D ta +αx) t a+x)+π+ Dt t+π+ for a t, D t ) t a, D ta ), t a+1, D ta+1 ), α This impies that the first condition of -feasibiity is vioated for a t, D t ) 8

9 Case 2: Equation 18b) is fase That is, Θ < D ta / Again, consider any t, D t ) t a, D ta ), t a+1, D ta+1 ), α Observe that D t = D ta + αt t a ) D ta, since t t a and α 0 Thus, Θ < D ta / impies Θ < D t / for a t, D t ); this impies that the second condition of F Π,, Θ) is vioated Case 3: Equation 18c) is fase That is, Θ < D ta + απ + t a ) 19) + α Consider any t, D t ) t a, D ta ), t a+1, D ta+1 ), α We consider two further subcases based on the vaue of t We wi show in both subcases, t, D t ) F Π,, Θ) Subcase 3a: t < D ta αt a+π+ )Θ 1 α By soving for Θ, we obtain Θ < D ta 1 α)t αt a+π+ Θ < Dt t+π+ The impication foows from D t = D ta + αt t a ) The above inequaity impies that the first condition of -feasibiity is vioated Subcase 3b: t D ta αt a+π+ )Θ 1 α Again, soving for Θ, Θ D ta 1 α)t αt a+π+ Θ Dt t+π+ Now consider the vaue of the first partia derivative of Φ 3 with respect to α; ie, Φ3 α to Π + t a ) D ta + α) 2 20) which is equa Note the sign of the above partia derivative is independent of the vaue of α; therefore, either Φ3 α 0, or Φ3 Φ3 α > 0 for any α : 0 α 1; in other words, the sign remains constant for a α If α > 0, then Φ 3 is maximized when α is as arge as possibe ie, α equas one) By Equation 19), this impies that Θ < D ta +Π+ t a which is impossibe due to Equation 20) Thus, Φ3 α 0 must be true If the partia derivative is non-positive, then Φ 3 is maximized when α is as sma as possibe ie, α equas zero) By Equation 19), Θ < Dt which vioates the second condition of -feasibiity Thus, we have proved that if the ine segment has a point in the -feasibiity region, then the conditions in Equation 18) hod For the if direction, we need to show, if the conditions hod then there exists a point on the ine segment that is incuded in the -feasibiity region Again, we wi show by contrapositive; that is, if the ine segment is competey outside the -feasibiity region, then there is a condition of Equation 18) that is not satisfied Assume that for a t, D t ) t a, D ta ), t a+1, D ta+1 ), α that t, D t ) F Π,, Θ) The previous statement impies that the first or the second condition of -feasibiity must be vioated for each t, D t ) We now consider two cases based on the ocation of the eft end point of the ine segment t a, D ta ) Case 1: The second condition of -feasibiity is vioated for t a, D ta ) In this case, Θ < condition of Equation 18b) D ta Indeed, this vioates the Case 2: The second condition of -feasibiity is not vioated for t a, D ta ) In this case, Θ D ta We now consider two further subcases regarding the ocation of t a+1, D ta+1 ) Subcase 2a: The first condition of -feasibiity is vioated for the right end point of ine segment t a+1, D ta+1 ) In this case, Θ < D t a+1 t a+1+π+ This ceary vioates the condition of Equation 18a) 9

10 Subcase 2b: The first condition of -feasibiity is not vioated for t a+1, D ta+1 ) In this subcase, D t a+1 t a+1+π+ Θ Consider the function θt) = D ta +αt t a) t+π+ for t [t a, t a+1 ] Thus, by this subcase and = D ta + αt a+1 t a ) we obtain the foowing equation, D ta+1 θt a+1 ) = D ) ta + αt a+1 t a ) t a+1 + Π + Θ 21) + 1 By Case 2, the second condition of -feasibiity is not vioated for t a, D ta ) Thus, the first condition must be; ie, θt a ) = D ) ta t a + Π + > Θ 22) + 1 Therefore, Θ [θt a+1 ), θt a )) Observe that θt) is continuous for a t [t a, t a+1 ] Therefore, the Intermediate Vaue Theorem impies that there exists a t [t a, t a+1 ] such that θt ) equas Θ That is, D ta + αt t a ) t + Π + = Θ 23) + 1 By the above equaity, the first condition of -feasibiity is not vioated for t, D t ); therefore, the second condition must be fase: D ta + αt t a ) > Θ 24) Soving Equation 23) for t, we obtain t = D ta αt a + Π + + 1)Θ 1 α Substituting the above soution to t into Equation 24) and soving for Θ, we obtain Θ < D ta απ + t a ) + α which indeed vioates the condition of Equation 18c) Thus, if the ine segment is stricty above the -feasibiity region, at east one of the three conditions is vioated The foowing emma formaizes the equivaence between the concept of a ine segment t a, D ta ), t a+1, D ta+1 ), α being incuded in some -feasibiity region and the concept of a cumuative request-bound function Ŵi faing beow a suppy-bound function sbf Lemma 4 For consecutive pair of vaues t a, t a+1 ) Ŝiτ, k) and t, D t ) t a, D ta ), t a+1, D ta+1 ), α such that t a < t t a+1, the inequaity Ŵit) sbfπ, Θ, ), t) hods, if and ony if, there exists N + such that t, D t ) F Π,, Θ) Proof Sketch: For the if direction, we must show that if the point t, D t ) t a, D ta ), t a+1, D ta+1 ), α satisfies t, D t ) F Π,, Θ), then there is sufficient suppy over an interva of ength t to satisfy the execution of a job of τ i and the approximated execution times of a higher-priority tasks formay, Ŵit) sbfπ, Θ, ), t)) Observe that every point in F Π,, Θ) is beow the sbf function see Figure 2) 1 Thus, if t, D t ) F Π,, Θ), then D t sbfπ, Θ, ), t) Finay, Lemma 2 states that Ŵit) D t impying the if direction For the ony if direction, observe that t a, D ta ), t a+1, D ta+1 ), α and Ŵit) are equivaent for t t a, t a+1 ] Thus, we must show that if ine segment t a, D ta ), t a+1, D ta+1 ), α has point t, D t ) contained beow the sbf function for Ω, then there exists an N + such that t, D t ), α F Π, Θ, ) Consider = D t Θ The second condition of -feasibiity Equation 11)) is triviay satisfied for this It aso must be true that D t > 1)Θ Thus, t, D t ) must be beow of the ine ined by y = x Π + + 1)Θ) otherwise, t, D t ) woud be above the sbf function at t) This ast constraint is equivaent to the first condition of -feasibiity region Therefore, for = D t Θ we have satisfied the two conditions of Equation 11), impying that t, D t ) F D t Θ Π, Θ, ) 1 A fu agebraic proof of this is rather invoved and wi be incuded in an extended version of this paper 10

11 In the above emma, we did not incude t a in the interva of time vaues where ine segment incusion in the -feasibiity region impies that the approximate request-bound function is beow the suppy-bound function The excusion of t a from the above emma is due to the fact that Ŵi is discontinuous at t a However, notice that t a is the right end point of the predecessor ine segment immediatey to the eft of t a, D ta ), t a+1, D ta+1 ), α Lemma 3 equates the concept of finding t such that Ŵit) is beow the sbf for a given Θ and the concept of point t, D t ) of a ine segment t a, D ta ), t a+1, D ta+1 ), α being contained in some -feasibiity region for Θ The next emma uses Definitions 4 and 5 to show that if we can compute Θ Π,, t a, D ta ), t a+1, D ta+1 ), α ) for any N +, then we can aso compute Θ Π,, t a, D ta ), t a+1, D ta+1 ), α ) Lemma 5 Θ Π,, t a, D ta ), t a+1, D ta+1 ), α { ) = inf Θ Π,, t a, D ta ), t a+1, D ), α ta+1 ) } 25) >0 Proof: Let Θ RHS denote the right-hand side of Equation 25) We wi show that both Θ RHS Θ Π,, t a, D ta ), t a+1, D ta+1 ), α ) and Θ RHS Θ Π,, t a, D ta ), t a+1, D ta+1 ), α ) which wi impy the emma First, we show Θ RHS Θ Π,, t a, D ta ), t a+1, D ta+1 ), α ) By inition of infimum, for any δ > 0, there exists N + such that Θ Π,, t a, D ta ), t a+1, D ta+1 ), α ) Θ RHS + δ Definition 5 states that there exists t, D t ) ta, D ta ), t a+1, D ta+1), α such that t, D t ) F Π, Θ RHS +δ, ) for this Therefore, for a δ > 0, Θ RHS +δ must be in the set considered in the inf on the right-hand side of Equation 16) by Definition 4 Thus, Θ RHS Θ Π,, t a, D ta ), t a+1, D ta+1 ), α ) Next, we wi show Θ RHS Θ Π,, t a, D ta ), t a+1, D ta+1 ), α ) By Definition 4 and appication of Lemma 2, there exist t, D t ) t a, D ta ), t a+1, D ta+1 ), α such that Ŵ i t) sbfπ,, Θ Π,, t a, D ta ), t a+1, D ta+1 ), α )), t) Lemma 4 impies that there exists N + such that t, D t ) F Π, Θ Π,, t a, D ta ), t a+1, D ta+1 ), α ), ) By Definition 5, this impies that Θ Π,, t a, D ta ), t a+1, D ta+1 ), α ) is in the set considered in the right-hand side of Equation 17) which impies the inequaity In the next few emmas, we derive the vaues 1 and 2 Equations 14) and 15)), and prove that we ony need to evauate the Φ functions at these vaues to obtain minimum capacity Consider the three conditions given in Equation 18) of Lemma 3 There are three possibe cases We invite the reader to verify that these cases are compete and mutuay excusive In the cases et AB denote the ine segment t a, D ta ), t a+1, D ta+1 ), α Case I: Φ 1 AB,, Π, ) > Φ 2 AB,, Π, ) ) Φ 1 AB,, Π, ) > Φ 3 AB,, Π, ) ) ; Case II: Φ 2 AB,, Π, ) > Φ 3 AB,, Π, ) ) Φ 2 AB,, Π, ) Φ 1 AB,, Π, ) ) ; Case III: Φ 3 AB,, Π, ) Φ 1 AB,, Π, ) ) Φ 3 AB,, Π, ) Φ 2 AB,, Π, ) ) For each of the above cases, we sove for the vaue of and obtain bounds for the vaue of Lemma 6 For any AB = t a, D ta ), t a+1, D ta+1 ), α, N +, Π,, Case I hods, if and ony if, t a+1 ) + t a+1 ) 2 + 4ΠD ta ) Proof: Let us consider the ony if direction of the emma; that is, Case I hods From Case I, we have that both Φ 1 AB,, Π, ) > Φ 2 AB,, Π, ) and Φ 1 AB,, Π, ) > Φ 3 AB,, Π, ) For Φ 1 AB,, Π, ) > Φ 2 AB,, Π, ), soving for, D ta+1 t a+1+π+ > D ta > [1 α)ta+1+αta ]+ 1 α)t a+1+αt a ) 2 +4Π D ta 27) 11

12 The bidirectiona impication foows since Inequaity 27) is a quadratic inequaity with respect to, ining a convex paraboa Π 2 1 α)t a+1 + αt a ) D ta The zeros of the paraboa are [1 α)t a+1 + αt a ] ± 1 α)t a+1 + αt a ) 2 + 4Π D ta Since the square-root term in the numerator is aways greater than the term preceding the ±, one root is positive and the other is negative Inequaity 27) impies that we are interested in vaues of N + such that the paraboa stricty exceeds zero Since the paraboa is convex, a vaues of stricty greater than the positive root satisfy this inequaity For Φ 1 AB,, Π, ) > Φ 3 AB,, Π, ), soving for, D ta+1 t a+1+π+ > D ta +απ+ t a) +α > ta+1 )+ t a+1 ) 2 +4ΠD ta+1 The bidirectiona impication foows since Inequaity 28) is a quadratic inequaity with respect to, ining a convex paraboa Π 2 t a+1 ) Dta + αt a+1 t a ) ) By simiar reasoning done for Inequaity 27), a vaues of stricty greater than the positive root satisfy this inequaity Combining Equations 27) and 28), we obtain > max [1 α)t a+1+αt a ]+ 1 α)t a+1+αt a ) 2 +4Π D ta, t a+1 )+ t a+1 ) 2 +4ΠD ta+1 28) 29) Observe that 1 α)t a+1 + αt a equas t a+1 αt a+1 t a ) which is at most t a+1, since t a+1 > t a and 0 α 1 Thus, we concude that the second vaue of Equation 29) is the maximum of the two bounds obtained in this case The emma foows by observing that is an integer The if direction foows by simpy reversing the direction of each impication in the proof Lemma 7 For any AB = t a, D ta ), t a+1, D ta+1 ), α, N +, Π,, Case II hods, if and ony if, t a ) + t a ) 2 + 4Π D ta 1 30) Proof: Let us consider the ony if direction of the emma; that is, Case II hods From Case II, we have that both Φ 2 AB,, Π, ) > Φ 3 AB,, Π, ) and Φ 2 AB,, Π, ) Φ 1 AB,, Π, ) For Φ 2 AB,, Π, ) > Φ 3 AB,, Π, ), soving for, D ta > D ta +απ+ t a) +α < ta )+ t a ) 2 +4Π D ta The bidirectiona impication foows since Inequaity 31) is a quadratic inequaity with respect to, ining a convex paraboa Π 2 t a ) D ta The zeros of the paraboa are t a ) ± t a ) 2 + 4Π D ta Since the square-root term in the numerator is aways greater than the term preceding the ±, one root is positive and the other is negative Inequaity 31) impies that we are interested in vaues of N + such that the paraboa is stricty beow zero Since the paraboa is convex, a positive integer vaues of stricty ess than the positive root satisfy this inequaity For Φ 2 AB,, Π, ) Φ 1 AB,, Π, ), soving for, 31) D ta D t a+1 t a+1+π+ [1 α)ta+1+αta ] α)t a+1+αt a ) 2 +4Π D ta 32)

13 The bidirectiona impication foows since Inequaity 32) is a quadratic inequaity with respect to, ining a convex paraboa Π 2 1 α)t a+1 + αt a ) D ta By simiar reasoning done for Inequaity 31), a positive integer vaues of at most the positive root satisfy this inequaity Now consider the foowing term: 1 α)t a+1 +αt a which equas 1 α)t a+1 t a )+t a which is at east t a since t a+1 > t a and 0 α 1 Thus, we concude that the vaue on the right-hand-side of Equation 31) is the minimum of the two vaues obtained in this case The emma foows by observing that must be an integer The if direction of the emma foows by simpy reversing the impications of the proof Lemma 8 For any AB = t a, D ta ), t a+1, D ta+1 ), α, N +, Π,, Case III hods, if and ony if, t a ) + t a ) 2 + 4Π D ta t a+1 ) + t a+1 ) 2 + 4ΠD ta+1 33) Proof: Let us consider the ony if direction of the emma; that is, Case III hods From Case III, we have that both Φ 3 AB,, Π, ) Φ 1 AB,, Π, ) and Φ 3 AB,, Π, ) Φ 2 AB,, Π, ) For Φ 3 AB,, Π, ) Φ 1 AB,, Π, ), soving for, D ta +απ+ t a) +α ta+1 )+ D t a+1 t a+1+π+ t a+1 ) 2 +4ΠD ta+1 The bidirectiona impication foows since Inequaity 34) is a quadratic inequaity with respect to, ining a convex paraboa Π 2 t a+1 ) Dta + αt a+1 t a ) ) The zeros of the paraboa are t a+1 ) ± t a+1 ) 2 + 4ΠD ta+1 Since the square-root term in the numerator is aways greater than the term preceding the ±, one root is positive and the other is negative Inequaity 34) impies that we are interested in vaues of N + such that the paraboa is at most zero Since the paraboa is convex, a positive integer vaues of at most the positive root satisfy this inequaity For Φ 3 AB,, Π, ) Φ 2 AB,, Π, ), soving for, 34) D ta +απ+ t a) +α ta )+ D ta t a ) 2 +4Π D ta 35) The bidirectiona impication foows since Inequaity 35) is a quadratic inequaity with respect to, ining a convex paraboa Π 2 t a ) D ta The zeros of the paraboa are t a ) ± t a ) 2 + 4Π D ta Since the square-root term in the numerator is aways greater than the term preceding the ±, one root is positive and the other is negative Inequaity 35) impies that we are interested in vaues of N + such that the paraboa is at east zero Since the paraboa is convex, a positive integer vaues of at east the positive root satisfy this inequaity The emma foows by observing that must be an integer The if direction of the emma foows by simpy reversing the impications of the proof We now prove three emmas and coroaries which show that for a N + not equa to the vaues 1, 1 + 1, 2 or 2 1 wi resut in a arger minimum Θ The first emma, towards this goa, shows that if a point on the ine segment is in an -feasibiity region and is at east 1 + 1, then the point is aso in the feasibiity region Lemma 9 For any t a, t a+1 Ŝiτ, k), t, D t ) t a, D ta ), t a+1, D ta+1 ), α, N +, Π,, and Θ, if and Θ then [t, D t ) F Π,, Θ)] [ t, D t ) F 1 +1Π,, Θ) ] 13

14 Proof: By Lemma 6 and 1 + 1, Case I must hod for a such Combining Case I and Lemma 3, we have that if t, D t ) F Π,, Θ), then Θ Φ 1 t a, D ta ), t a+1, D ta+1 ), α,, Π, ) Now consider the first partia derivative of Φ 1 with respect to ; ie, Φ 1 = D t a+1 +t a+1 Π +Π) ) 2 = [ta+1 D t a+1 ]+[Π ] ) 2 Since Π, the second term in the numerator is positive Consider the first term, t a+1 D ta+1 By t, D t ) F Π,, Θ) and the first condition of -feasibiity, t D t + Π + + 1)Θ t + t a+1 t) D t + αt a+1 t) + Π + + 1)Θ since α < 1) t a+1 D ta+1 + Π + + 1)Θ t a+1 D ta+1 The second to ast impication is due to D t = D ta + αt t a ) and D ta+1 = D ta + αt a+1 t a ) The ast impication is due to Θ Therefore, the first term in the numerator of Φ1 is aso positive Thus, Φ1 is non-decreasing for a Thus, the Φ 1 evauated at is a ower bound; ie, for a 1 + 1, Φ 1 t a, D ta ), t a+1, D ta+1 ), α,, Π, ) Φ 1 t a, D ta ), t a+1, D ta+1 ), α, 1 + 1, Π, ) The above inequaity impies that Θ Φ 1 t a, D ta ), t a+1, D ta+1 ), α, 1 + 1, Π, ), satisfying Equation 18a) of Lemma 3 For 1 + 1, Case I hods, impying that Equations 18b) and 18c) must aso hod Thus, by Lemma 3, t, D t ) F 1 +1Π,, Θ) The next coroary foows from the above emma and the inition of Θ Definition 5) Coroary 1 For any t a, t a+1 Ŝiτ, k), N +, Π, and, if 1 + 1) then Θ Π,, t a, D ta ), t a+1, D ta+1 ), α ) Θ 1 +1 Π,, t a, D ta ), t a+1, D ta+1 ), α ) The next emma shows that if a point on the ine segment is in an -feasibiity region and is at most 2 1, then the point is aso in the 2 1-feasibiity region Lemma 10 For any t a, t a+1 Ŝiτ, k), t, D t ) t a, D ta ), t a+1, D ta+1 ), α, N +, Π,, and Θ, if 2 1 and Θ then [t, D t ) F Π,, Θ)] [ t, D t ) F 2 1Π,, Θ) ] Proof: By Lemma 7 and 2 1, Case II must hod for a such Combining Case II and Lemma 3, we have that if t, D t ) F Π,, Θ), then Θ Φ 2 t a, D ta ), t a+1, D ta+1 ), α,, Π, ) Now consider the first partia derivative of Φ 2 with respect to ; Φ 2 = D t 2 Therefore, Φ 2 is a decreasing function for a N + such that 2 1 Thus, the Φ 2 evauated at 2 1 is an upper bound for a such ; ie, for a 2 1, Φ 2 t a, D ta ), t a+1, D ta+1 ), α,, Π, ) Φ 2 t a, D ta ), t a+1, D ta+1 ), α, 2 1, Π, ) The above inequaity impies that Θ Φ 2 t a, D ta ), t a+1, D ta+1 ), α, 2 1, Π, ), satisfying Equation 18b) of Lemma 3 For 2 1, Case II hods, impying that Equations 18a) and 18c) must aso hod Thus, by Lemma 3, t, D t ) F 2 1Π,, Θ) The next coroary foows from the above emma and the inition of Θ Definition 5) 14

15 Coroary 2 For any t a, t a+1 Ŝiτ, k), N +, Π, and, if 2 1) then Θ Π,, t a, D ta ), t a+1, D ta+1 ), α ) Θ 2 1 Π,, t a, D ta ), t a+1, D ta+1 ), α ) Lemma 11 For any t a, t a+1 Ŝiτ, k), t, D t ) t a, D ta ), t a+1, D ta+1 ), α, N +, Π,, and Θ, if 2 1 then [t, D t ) F Π,, Θ)] [ t, D t ) F 1 Π,, Θ) ] [ t, D t ) F 2 Π,, Θ) ] Proof: By Lemma 8 and 2 1, Case III must hod for a such Combining Case III and Lemma 3, we have that if t, D t ) F Π,, Θ), then Θ Φ 3 t a, D ta ), t a+1, D ta+1 ), α,, Π, ) Now consider the first partia derivative of Φ 3 with respect to ; Φ 3 = α2 Π D ta α + αt a + α) 2 Note the sign of the above partia derivative is independent of the vaue of ; therefore, either Φ3 any N + ; in other words, the sign remains constant for a If Φ3 0, or Φ3 > 0 for > 0, then Φ 3 is minimized when is as sma as possibe; ie, equas 2 In this case, the Φ 3 evauated at 2 is a ower bound for a such ; ie, for a such that 2 1, Φ 3 t a, D ta ), t a+1, D ta+1 ), α,, Π, ) Φ 3 t a, D ta ), t a+1, D ta+1 ), α, 2, Π, ) The above inequaity impies that Θ Φ 3 t a, D ta ), t a+1, D ta+1 ), α, 2, Π, ), satisfying Equation 18c) of Lemma 3 For 2, Case III hods, impying that Equations 18a) and 18b) must aso hod Thus, by Lemma 3, t, D t ) F 2 Π,, Θ) when Φ3 > 0 If Φ3 0, then Φ 3 is minimized when is as arge as possibe; ie, equas 1 In this case, the Φ 3 evauated at 1 is an upper bound for a such ; ie, for a such that 2 1, Φ 3 t a, D ta ), t a+1, D ta+1 ), α,, Π, ) Φ 3 t a, D ta ), t a+1, D ta+1 ), α, 1, Π, ) By the same argument for Φ3 > 0, t, D t ) F 1 Π,, Θ) when Φ3 0 The next coroary foows from the above emma and the inition of Θ Definition 5) Coroary 3 For any t a, t a+1 Ŝiτ, k), N +, Π, and, if 2 1 ) then Θ Π,, t a, D ta ), t a+1, D ta+1 ), α ) min{θ 1 Π,, t a, D ta ), t a+1, D ta+1 ), α ), Θ 2 Π,, t a, D ta ), t a+1, D ta+1 ), α )} Combining Coroaries 1, 2, 3, and using Definitions 4 and 5, we obtain the foowing coroary Coroary 4 Θ Π,, t a, D ta ), t a+1, D ta+1 ), α ) = min { {Θ Π,, t a, D ta ), t a+1, D ta+1 ), α )} 1, 1 +1, 2, 2 1} Proof: By Lemma 5, we may determine Θ Π,, t a, D ta ), t a+1, D ta+1 ), α ) by evauating Θ Π,, t a, D ta ), t a+1, D ta+1 ), α ) for a possibe N + The coroary foows by appying Coroaries 1, 2, and 3, respectivey, for the foowing regions of : [1, 2 1], [ 2, 1 ], and [ 1 + 1, ) By the above coroary, we now know how to compute Θ ) efficienty from Θ ) The next emma shows that we may use the Φ functions to efficienty compute Θ ) Lemma 12 Let AB represent t a, D ta ), t a+1, D ta+1 ), α For any N +, Θ Π,, t a, D ta ), t a+1, D ta+1 ), α ) Φ 1 AB,, Π, ), if 1 + 1; = Φ 2 AB,, Π, ), if 2 1; Φ 3 AB,, Π, ), otherwise 36) 15

16 Proof: From Definition 5, Θ Π,, AB) is the minimum Θ such that there exists t, D t) AB where t, D t ) F Π, Θ, ) By Lemma 3, such a Θ is aso the minimum vaue that satisfied a three conditions of Equation 18) Since each of the conditions is a ower bound on Θ with equaity permitted), Θ must satisfy equaity of at east one of the three conditions of Equation 18) and must exceed or equa the other two conditions Notice that, if 1 +1, then by Lemma 6, Θ equas Φ 1 AB,, Π, ) We can show an identica proof for intervas 0, 2 1] and[ 2, 1 ], by appying Lemmas 7 and 8, respectivey The fina emma that we prove before providing a proof for Theorem 2 shows that a choice of Θ based on the computation of Θ ) is a safe choice in the sense that a tasks in τ i wi compete by their deadine under an EDP resource Ω = Π, Θ, ) Lemma 13 For a τ i τ, t 0, d i ] such that Ŵit) sbfπ, Θ, ), t) and Uτ) Θ Π, if and ony if, { { maxτi τ minta,t Θ max a+1 Ŝi Θ Π,, t a, D ta ), t a+1, D ), α )}} ta+1, Uτ) Π ) 37) Proof: We wi prove this emma by contrapositive For the if direction, we must prove if either Uτ) > Θ Π or t 0, d i ] : Ŵit) > sbfπ, Θ, ), t), then the negation of the inequaity of Equation 37) is true If we consider Uτ) > Θ Π, the inequaity of Equation 37) is triviay vioated due to the second expression in the outer max of Equation 37) Now, consider the case when there exists a τ i τ such that Ŵit) > sbfπ, Θ, ), t) for a t in 0, d i ] By Lemma 4, this impies for a N +, t a, t a+1 Ŝiτ, k), and t, D t ) t a, D ta ), t a+1, D ta+1 ), α that t, D t ) F Π,, Θ) By Definition 5, it must be for a N + that Θ < Θ Π,, t a, D ta ), t a+1, D ta+1 ), α ) By Lemma 5, this impies that Θ < Θ Π,, t a, D ta ), t a+1, D ta+1 ), α ) for any t a, t a+1 Ŝiτ, k), which vioates the inequaity of Equation 37) due to the first term in the outer max For the ony if direction of the emma, we wi aso consider the contrapositive The contrapositive wi foow by simpy reversing the impications of the proof for the if direction After proving the above conditions, we are ready to prove Theorem 2 which states that FPMINIMUMCAPACITY returns a vaid vaue for finite k and an exact vaue for k = Proof of Theorem 2 We wi show that Θ min returned from FPMINIMUMCAPACITY corresponds to the vaue on the righthand side of Equation 37) of Lemma 13 The oop from Line 4 to 15 iterates through each consecutive pair of vaues t a and t a+1 in Ŝiτ, k) to find optima capacity for each ine segment ined by the endpoints t a, D ta ) and t a+1, D ta+1 ) It sets variabes corresponding to Ŵit a ) and Ŵit a+1 ) in Lines 5 and 6 respectivey Then, in the next few ines it sets four different vaues to based on 1 and 2, ined in Equations 14) and 15)) and evauates Φ j ) according to Lemma 12 to compute Θ ) for each of the four integer vaues of Therefore, Θmin t a, set in Line 13, equas Θ Π,, t a, D ta ), t a+1, D ta+1 ), α ) by Lemma 5 At the end of this oop it sets Θ min i to be the minimum of Θ min t a and Θ min i Line 14) Thus, once the inner oop is executed for a t a, t a+1 Ŝiτ, k), Θ min i contains the minimum of a Θ min t a vaues The outer oop from Line 2 to Line 17 finds Θ min i for a task τ i in τ Finay, in Line 16, Θ min is set to the maximum of Uτ) Π and Θ min i over a vaues τ i in τ By Lemma 13, Ŵit) sbfπ, Θ min, ), t) for some t 0, d i ] and Uτ) Θ Π, by Lemma 1, W it) Ŵit) These impies W i t) sbfπ, Θ min, ), t) for a t 0 which is the scheduabiity condition given by Theorem 1 Therefore, τ wi aways meet a deadines when schedued by fixed-priority scheduing upon Ω = Π, Θ min, ) When k =, Ŵit) equas W i t) for a t 0; in this case, Θ min equas Θ Π,, τ) ie, Θ min is exact capacity) 5 An Approximation Scheme In the previous section, we have shown that FPMINIMUMCAPACITY gives a vaid answer when k is finite and an exact answer when k is infinite In this section, we show that as k increases, the guaranteed accuracy of FPMINIMUMCAPACITY increases aong with its running time Theorem 3 presents the tradeoff between accuracy and computationa compexity, in terms of k Theorem 3 Given Π,, τ, and k N +, the procedure FPMINIMUMCAPACITY returns Θ min such that ) k + 1 Θ Π,, τ) Θ min Θ Π,, τ) k Furthermore, FPMINIMUMCAPACITY Π,, τ, k) has time compexity Okn 2 og n) 16