Holistic Design Parameter Optimization of Multiple Periodic Resources in Hierarchical Scheduling

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1 Holitic Deign Parameter Optimization of Multiple Periodic Reource in Hierarchical Scheduling Man-Ki Yoon, Jung-Eun Kim, chard Bradford, and Lui Sha Univerity of Illinoi at Urbana-Champaign, Urbana, IL {mkyoon, jekim314, Rockwell Collin, Cedar Rapid, IA Abtract Hierarchical cheduling of periodic reource ha been increaingly applied to a wide variety of real-time ytem due to it ability to accommodate variou application on a ingle ytem through trong temporal iolation. Thi lead to the quetion of how one can optimize over the reource parameter while atifying the timing requirement of real-time application. A great deal of reearch ha been devoted to deriving the analytic model for the bound on the deign parameter of a ingle reource a well a it optimization. The optimization for multiple periodic reource, however, require a holitic approach due to the conflicting requirement of the limited computational capacity of a ytem among reource. Thu, thi paper addree a holitic optimization of multiple periodic reource with regard to minimum ytem utilization. We extend the exiting analyi of a ingle reource in order for the variable interference among reource to be captured in the reource bound, and then olve the problem with Geometric Programming (GP). The experimental reult how that the propoed method can find a olution very cloe to the one optimized via an exhautive earch and that it can explore more olution than a known heuritic method. I. INTRODUCTION A the proceing power of proceor ha grown, there ha been an increaing trend toward integrating many real-time application on a ingle ytem and thu efficiently utilizing the ytem by allowing the application to hare common hardware device. In uch ytem, temporally partitioned hierarchical cheduling [1] [5] ha been widely adopted becaue of it trong iolation among et of real-time application, which either are independently developed or have different functionalitie or criticalitie. For example, in IMA (Integrated Modular Avionic) architecture [6], application are often grouped into different partition according to their deign-aurance level in order to protect high-criticality application from the faulty behavior of other application and guarantee their timing requirement. In uch a partitioned hierarchical cheduling, one important quetion i how much of the computational reource need to be allocated to each reource in order for the ytem to be optimized for a certain metric. For intance, it i deirable in ytem deign to minimize the ytem utilization while guaranteeing the timing requirement of both reource and their application. Thi i true ince a lower-utilized ytem can be more utilized by accommodating additional workload or, alternatively, the ame workload can be implemented by a lower-peed ytem. Thi work i upported in part by Rockwell Collin under RPS#6 4538, by Lockheed Martin under , by NSF under A17321, and by ONR under N Any opinion, finding, and concluion or recommendation expreed in thi publication are thoe of the author and do not necearily reflect the view of ponor /DATE13/ c 213 EDAA For a ingle reource cae, the optimized reource deign parameter, that i, period and execution length, can be obtained by a method baed either on an exact chedulability tet [4], [5] or on reource upply and demand function [1] [3]. However, it i often intractable to find the optimal et of reource parameter mainly becaue the local optimality of each reource doe not necearily lead to the globally optimal olution [4], [5]. Accordingly, each deign parameter cannot be choen independently; the optimal election require a brute-force earch, which i only practical when ome parameter are fixed and/or the number of reource i mall. Thu, in thi paper, we are intereted in finding a ub-optimal et of reource deign parameter that minimize the chedulable ytem utilization; both reource and their tak are chedulable. Specifically, we conider the periodic reource model introduced in [1] [5]; each reource R i periodically releaed at every T and upplie an execution amount of L to it tak. For global and local cheduling, we conider fixed-priority cheduling with the aumption that prioritie are pre-aigned. The reult on the reource parameter bound in previou work were derived by calculating the lower-bound on a reource upply that can atify the wort-cae demand of the workload. When other reource parameter are unknown, however, a peimitic aumption on the minimum upply need to be made; each reource uffer the maximum poible delay. We tackle thi problem by parameterizing the wort-cae reource upply with the unknown parameter of other reource that can be holitically optimized via Geometric Programming (GP) [7], [8]. GP i a non-linear optimization method that can olve a pecially formed nonconvex problem by tranforming it into a convex one through a logarithmic tranformation, thu finding the optimal olution efficiently. We preent a GP formulation a the olution to the deign parameter optimization of multiple periodic reource. We how that our method can find a olution that i cloe to the one that can be found by an exhautive earch, and it can explore more olution than a known heuritic method [5]. II. PROBLEM DESCRIPTION A. Sytem Model We conider a uniproceor coniting of a et of independent periodic reource R = {R i i = 1,...,N R }. Each R i i characterized by an unknown tuple of (,L i ),,L i R +, where and L i are the period and the execution length of the

2 reource, repectively. 1 In R i,aetγ i = {τ j j =1,...,N Γ i } of tak run in a fixed-priority preemptive chedule uch a Rate Monotonic [1]. Each τ j i repreented by τ j := (e j,p j,d j ), where e j i the wort-cae execution time, p j i the minimum inter arrival time between ucceive releae, and d j i the relative deadline. 2 In thi paper, we aume that d j = p j. We then further aume that there i no ynchronization or precedence contraint among tak, and tak releae are not bound to reource [5]. The reource are alo cheduled in a fixed-priority manner and we aume their deadline, D i, i equal to the period. In addition, we conider that reource prioritie are given, auming, for example, the prioritie are aigned according to criticalitie. We note that the optimization method in thi paper cannot be applied to cae when reource prioritie are not given. Additionally, a reource i idled if there i no tak ready to execute. We alo aume that there i no reource releae jitter. B. Problem Decription Given a et of reource {R i } and the correponding tak et {Γ i }, our problem i to find the et of the reource parameter, {(,L i )} for i = 1,...,N R, which minimize the overall ytem utilization, U, while guaranteeing the chedulabilitie of the reource and the tak. Here, U can be repreented by U = N R i=1 c 1 δ + c 2 L i, (1) where δ i the reource context-witch overhead, and c 1,c 2 are weight given by the ytem deigner [3]. In thi paper, we et both c 1 and c 2 to 1 and aume that each reource will conume a context-witch overhead of δ at each releae. III. PARAMETER BOUNDS OF MULTIPLE RESOURCES In thi ection, we preent the analyi of multiple reource bound by extending thoe of the ingle reource bound in the previou literature [1], [2], which i ummarized in [9]. In a partitioned reource whoe period and length L i are unknown, we can derive the lower-bound of L i (or the upper-bound of ) with repect to (or L i ) that make τ j in R i chedulable by uing the periodic reource model [1], [2]. Informally peaking, the key idea of previou work i that a tak can be chedulable if the minimum reource upply (bf Γ (t) [1] or A (t) [2]) can match the maximum workload demand generated by τ j and it higherpriority tak during a time interval t. In fixed-priority global cheduling, the minimum upply of a reource i delivered to it tak when it (k 1) th execution ha jut been finihed at time with minimum interference from higher-priority reource, and then the ubequent execution are maximally delayed by higher-priority reource. When the parameter of higher-priority reource are unknown, it i a afe aumption that the ubequent execution from the k th releae are delayed by L i.thii peimitic, ince, in reality, high priority reource would uffer no or only a few preemption. Thu, an exact method i required [4], [5], which, however, i not ueful for an optimization of 1 When,L i N, a branch-and-bound or a variable rounding heuritic i required. We note that the choice of uch method i orthogonal to the optimization preented in thi paper. In [9], we decribe the effect of uch contraint on the conidered optimization problem. 2 More preciely, each tak hould be repreented a τ i,j if it belong to R i. For the implicity of notation, however, we ue the abbreviation τ j. ( k 1) th Reource Supply L i ( T L) i i T th k i ( k + 1) th Δ L i lbf (t) R i Ti bf (t) R i Fig. 1: The wort-cae releae pattern of R i conidering the period and execution length of higher-priority reource. multiple reource parameter due to it high time complexity. Thi neceitate holitic optimization of multiple reource parameter, and thu in the following we preent the analyi of multiple reource bound via a parameterization of unknown reource parameter, which will be formulated and olved by Geometric Programming in Sec. IV. The optimization of a ingle reource ha been extenively tudied. Intereted reader can refer to [1] [5], [11] [15]. A. Lower-bound Supply Function Conidering Unknown Parameter of Higher-Priority Reource We conider a wort-cae releae pattern of R i occurring when it uffer zero delay in the (k 1) th releae and the maximum delay from higher-priority reource thereafter, i.e., Δ, a depicted in Fig. 1. The wort-cae buy period of R i, denoted a w, i the maximum time duration that R i can take to execute L i when it i releaed imultaneouly with it higher-priority reource, hp(r i ),atthek th releae, which can be obtained by the traditional exact analyi: w k w k+1 R i = L i + L h, (2) where wr i = L i and w = wr k i when it converge for ome k. Thu, the wort-cae delay at the k th releae and thereafter (called initial latency in [2]) can be repreented a w Δ = L h. (3) However, the iterative method of (2) can only be applicable to brute-force optimization. Thu, we approximate Δ.Duringa time interval of, the maximum workload generated by R i and hp(r i ) can be repreented by: w = L i + Ti L h. We can avoid iterative calculation by auming the number of invocation of higher-priority reource during, not during the exact buy period. Note that it i a afe bound a long a R i meet it deadline, i.e., D i =. Now, we linearize w a follow: ( Ti ) w = L i + +1 L h, becaue x x +1. Then, the wort-cae linear lower-bound upply function 3 of R i during a time interval t parameterized 3 It i identical to A (t) with Δ=Δ and α = L i in [2]. t

3 with Δ i derived a follow: lbf (t) = L i (t ( L i ) Δ ), (4) where Δ = ( Ti ) +1 B. Sufficient Reource Bound for Tak Schedulability L h. (5) Let u now conider τ j in R i whoe period i fixed. Then, let u define L min i (τ j, ) a the minimum required length of R i that guarantee to chedule τ j. In order to derive L min i (τ j, ),we can conider the ituation in which τ j barely meet it deadline at time t = d j with the wort-cae interference from higher-priority tak hp(τ j ). Since we make no aumption on tak offet, the wort-cae repone time of τ j occur when it and hp(τ j ) are releaed imultaneouly at the end of R i (k 1) th execution and then uffer the wort-cae preemption from hp(τ j ) in k th releae and thereafter, which we define a the critical intant. 4 Now, let u denote I j a the wort-cae workload generated by τ j and hp(τ j ) from the critical intant to the deadline of τ j a follow: I j = e j + dj e h. p h τ h hp(τ j) τ j i guaranteed to be chedulable if the minimum upply delivered by the reource i greater than or equal to the wort-cae workload generated during the time interval d j. Thu, lbf (d j )= L i (d j ( L i ) Δ ) I j. (6) Accordingly, the minimum required reource length, L min i (τ j, ), for τ j with a given can be obtained by olving the quadratic inequality in (6), which reult in L min i (τ j, )= (d j Δ )+ (d j Δ ) 2 +4I j. 2 (7) Note that (7) i equivalent to Eq. (23) in [1] with Δ = L i and to Eq. (12) in [2] with β =1. In order to find the minimum required length of R i for a given, we take the maximum of the bound L min i (τ j, ) over all tak in Γ i, which therefore can be defined a follow: ( ) L min i ( )=max L min i (τ j, ). (8) τ j Γ i Thu, if we take L i from [L min i ( ), ], all τ j Γ i are guaranteed to meet their deadline. It i now important to note that (6) i only ufficient and not neceary condition; τ j can be chedulable if and only if there exit a time intant t d j uch that lbf (t) I j [1]. In thi paper, however, we ue the ufficient condition in (6) becaue the preence of time in the neceary condition make the propoed optimization method not applicable to the problem under conideration. Although the bound i not exact and may incur approximation error, it enable u to optimize multiple reource holitically with high efficiency, a will be decribed in Sec. IV. 4 In thi paper, we do not conider tak jitter. However, without lo of generality, the preented analye can be imilarly applied to cae with jitter. For example, the wort-cae ituation of τ j i when hp(τ j ) have experienced their maximum jitter and are releaed at the ame time with τ j. U (Sytem Utilization) U = U 1 + U 2 (Sytem Utilization) U 1 (Utilization of Reource 1) U 2 (Utilization of Reource 2) T=T 1 =T 2 (Reource Period) Fig. 2: Sytem utilization of two reource and it non-convexity. C. Non-convexity of Multiple Reource Optimization We preent an example of two reource in order to how the non-convexity of multiple reource optimization. Let u conider Fig. 2, which how the utilization function, U 1 and U 2, of two randomly generated reource, {R 1, R 2 }, and the ytem utilization function, U = U 1 + U 2, over the period in [1, 14]. In thi example, the reource have the ame period for implicity of repreentation, and δ i et to 1. R 1 ha a higher priority than R 2, thu Δ R1 =and Δ R2 =2 L 1. From the graph, we can firt ee that U i not convex, which i hown by the traight line drawn between T =6and 12. Furthermore, while the reource achieve minimum utilization at T 1 = 2.3 and T 2 =63.9, repectively, thee do not lead to the global optimality, which occur at T = T 1 = T 2 =23.1; thi iue i addreed alo in [4]. In thi example, the reource have the ame period. The optimization of multiple reource parameter will be harder to olve once we conider a higher number of reource and arbitrary reource period. IV. HOLISTIC OPTIMIZATION OF MULTIPLE RESOURCE PARAMETERS VIA GEOMETRIC PROGRAMMING In thi ection, we formulate the parameter optimization problem of multiple periodic reource with Geometric Programming (GP) [7], [8]. GP can olve a non-linear, non-convex optimization problem if it can be formulated in the following form: Minimize f (x) Subject to f i (x) 1, i =1,...,n p, g j (x) =1, j =1,...,n m, where f and g are poynomial and monomial function, repectively, and x are the optimization variable. A function g j (x) i monomial if it can be repreented a: n j g j (x) =c j x a k k, k=1 where c j R + and a k R. A poynomial function i a um of monomial, and thu can be expreed a: n i f i (x) = c k x a 1k 1 x a 2k 2 x a nk n, k=1 where c k R + and a jk R. Alo,f/g i a poynomial and f ax i alo a poynomial if a x R +. In ummary, only the objective function and the inequality contraint can be poynomial.

4 g i (L i ) and approximate value L i g i (L i )=L i +3 g i (1) g i (5) g i (1) Fig. 3: g i (L i )=L i +3 and g i (L i ) at x =1, 5, and 1. g i (x ) i tangent to g i (L i ) at L i = x. We now formulate the optimization problem of the multiple reource parameter in GP form. A tated in Sec. II-B, we are given a et of periodic reource {R i } with unknown parameter, and L i ; their tak et {(e j,p j,d j ) τ j Γ i }, and the reource context-witch overhead δ are known. Optimization variable are T =(T 1,...,T N R) and L =(L 1,...,L N R). Objective Function The objective function (1) in Sec. II-B i already in a poynomial form, thu it can be repreented a follow: f o (T, L) = (c 1 δ + c 2 L i ) T 1 i, (9) N R i=1 where c 1,c 2,δ. Reource Bound Contraint The reource bound for each R i i contrained by (6) for each τ j Γ i, which can be reexpreed a follow: (L i + I j )+Δ L i 1, (1) L i (L i + d j ) where d j and I j are contant for a given input. However, the inequality above doe not conform to a poynomial form becaue of the poynomial term in the denominator, i.e., L i (L i + d j )= L 2 i + L i d j (Recall that a denominator mut be monomial). Oberve, however, that L i + d j can be approximated with a monomial by the following geometric mean approximation [16]. Let u firt denote it a g i (L i ) = u 1 (L i )+u 2 (L i ) where u 1 (L i )=L i and u 2 (L i )=d j. Then, we now approximate g i (L i ) with ) γ1 ( ) γ2, g i (L i )= (u 1 (L i )/γ 1 u 2 (L i )/γ 2 (11) where γ 1 = u1(x) g and γ i(x ) 2 = u2(x) g where x i(x ) R + i a contant that atifie g i (x ) = g i (x ). The approximated monomial g i (L i ) then can be rewritten a: ) γ1 ( ) γ2, g i (L i )= (L i /γ 1 d j /γ 2 with γ 1 = x x +d j and γ 2 = dj x +d j. Finally, (1) can be formulated a the following poynomial contraint: ( (L i + I j )+Δ L i ) (L i g i (L i )) 1 1, (12) where Δ i ( R h hp(r (Ti i) + ) T 1 ) h L h. Note that the approximation quality of g i (L i ) depend on the choice of x, a hown in Fig. 3. Thu, in the optimization procedure, we iteratively approximate g i (L i ) by updating γ 1 and γ 2 according to the intermediate olution of L i ; until the objective value converge, we ue L i at k th tep a x at (k +1) th tep. The objective value converge normally in 1 or 2 iteration. Reource Schedulability Contraint Each reource mut be chedulable, that i, L i +Δ, which correpond to the following poynomial contraint: ( L i + ( (Ti + ) T 1 h V. EVALUATION L h ) ) T 1 i 1. (13) A. Evaluation Method Parameter Value Number of reource, N R {2, 3, 4, 5} Number of tak per reource, N Γ i [2, 8] Tak execution time, e j [1, 3] Tak period, p j [5, 2] Context-witch overhead, δ 1 The table above ummarize the experimental parameter ued for the evaluation. We conider the cae with 2, 3, 4, and 5 reource, and for each cae, we generated 1 random input et with the parameter. The number of tak per reource, the tak execution time and period are uniformly randomly choen in the given range. For the implicity of evaluation, the context-witch overhead wa fixed to 1. With thee parameter, we compare the following method: Exhautive Search: From the highet priority reource to the lowet one, we recurively aign each reource period from 1 to T max with a tep ize of. For each period, L min i i determined by (8) and (7) with Δ calculated by (3). Recall that the ytem utilization obtained with thi exhautive earch i till not the exact globally optimal olution a explained in Sec. III. GP with the upper-bound on T max : The GP optimization method with additional contraint on the upper-bound on reource period, that i, Tmax 1 1. GP without the upper-bound on T max : Identical to the above except that there i no upper-bound on T max. Note that in our GP-baed optimization, the upper-bound on reource period i unneceary. For the evaluation purpoe only, the priority of each reource i aigned according to the utilization um of tak in each reource; the higher the utilization i, the higher it priority. Reader intereted in priority optimization can refer to [4], [5]. Tak prioritie in each reource are aigned by RM priority aignment [1]. The GP were olved uing GGPLAB [17]. B. Evaluation Metric We compare the method above in term of the minimum ytem utilization, i.e., Eq (1). We denote the olution of each method a U Exh, U GP1, and U GP2, repectively. For each input, we calculate the difference of U Exh from U GP1 and U GP2, that i, U GP1 U Exh and U GP2 U Exh, repectively, and then take the average of 1 random input et for each etting. It hould be noted that we do not compare the olving time of each method becaue while GP can olve a problem within a few econd, the exhautive earch normally take 1 6 minute or more depending on the problem ize and the choice of T max and.

5 Average of U GP 1 U Exh and U GP 2 U Exh GP with T max Number of reource Fig. 4: The average difference of U GP1 different number of reource. GP without T max and U GP2 to U Exh with C. Evaluation Reult Fig. 4 compare the minimum ytem utilization found by the exhautive earch and our GP method increaing the number of reource from 2 to 5. 5 T max and were et to 1 and.5, repectively. A we can ee from the reult, the average difference of U GP1 and U Exh increae with the number of reource. Thi i mainly becaue of the approximation error in Δ (ee Sec. III). Recall that while the exhautive earch calculate the minimum upply of each reource by uing the iterative equation (Eq. (3)), our GP method approximately calculate the interference from higher-priority reource during the interval of it period (Eq. (5)). With two reource, the error of the approximation i mall; however a the number of reource increae, the error accumulate from higher-priority reource to lower-priority one. Neverthele, we can ee that the difference between the two method are quite mall, indicating that our method can find a olution that i cloe to the one that can be found by the exhautive earch;.7.27 average difference compared to the exhautive earch. Another intereting obervation i that GP 2 can find better olution than GP 1.Thi reult might be expected becaue of the limited earch pace of GP 1 : when the workload of a reource i ignificantly lower than the other reource, it optimal period may appear beyond T max. In fact, for ome input et, U GP2 were lower than U Exh.One can find better olution with the exhautive earch by etting T max higher, however, thi can be limited by the input ize. With the ame input, we evaluated the performance of our GP method (GP 2 ) with variou bae utilization, i.e., the um of all tak utilization. From Fig. 5, we can ee that the difference increae with the bae utilization. A imilar argument a above can be ued to explain thi correlation. That i, a higher bae utilization implie that there exit reource with higher utilization, and thu thoe tend to have horter period and longer execution length. We attribute thi, again, to the approximation error of Δ in (5). Latly, we compared our GP method with the heuritic propoed in [5]. The method find the optimal parameter for each reource in turn from the highet to the lowet reource; for each reource, it iterate over a range of period and for each 5 The exhautive earch take a longer time to olve cae with ix or more reource, and thu we evaluated cae with 2, 3, 4, and 5 reource. Difference of U GP 2 U Exh Fig. 5: U GP2 - U Exh Bae utilization with variou bae utilization. period, it find the optimal reource length by a binary earch. The ame proce i applied to the next priority reource. For the comparion, we ued the ame input a above. In the heuritic, each i aigned from 1 to 1 with of.1, and each L i wa found at the granularity of.1. Fig. 6 how i) the number of olution found by each method and ii) the average difference of the minimum ytem utilization between the method. A can be een from the bar, our method find more olution than the heuritic, and in the experiment, all input et for which a olution wa found by the heuritic were alo olved by our method. We can alo ee that a the number of reource increae, the gap alo increae; with 5 reource, our method found 49 olution among 1 input et, but only 7 olution were found by the heuritic. 6 Thi follow from the greedy nature of the heuritic; the parameter for a high-priority reource were locally optimized without conidering the feaibilitie of lower-priority reource. In contrat, although our method i not a global optimal method either, it can explore more olution due to it ability to take into account the variable interference among reource imultaneouly in the GP optimization proce. However, the qualitie of the olution found by our method are wore than thoe found by the heuritic, a the line plot in Fig. 6 how. Each marker on the line i the average of the difference of U GP2 and U Heu, for the input et the heuritic found; with 2, 3 and 4 reource, the average difference are between.55 and.65, and with 5 reource, the difference i.18. The pike at 5 reource could be explained by the low number of olution found. Although our method achieved lower ytem utilization for ome input et, the heuritic could find better olution in mot cae. Such difference mainly arie from the optimality of the analyi ued by the heuritic. That i, when the parameter of higher-priority reource and the period of the reource under analyi are fixed, the (local-)optimal reource length i found by the binary earch which i baed on the exact analyi [5]. On the other hand, a explained Sec. III, our analyi conider the wort-cae cenario that are ufficient but not neceary, and it i alo baed on the approximation of Δ, both of which 6 The exact number of feaible olution i unknown a thi require a true optimal method. Among 1 input et for each cae, ome input et might be infeaible in the firt place. Alo, the main reaon that each method find fewer number of olution a the number of reource increae i becaue the bae ytem utilization alo increae. For example, the average bae utilization of 1 input et with 2 and 5 reource are.281 and.663, repectively.

6 Number of olution found Number of reource Our GP method Heuritic in [R. Davi, RTNS 28] Fig. 6: The number of olution found by the propoed GP method and the heuritic in [5] (Bar), and the average difference of U GP2 and U Heu (Line). lead to chedulability lo. From thi evaluation, we can conclude that there i a trade-off between the olution feaibility (our GP method) and the olution quality (the heuritic of [5]). VI. RELATED WORK Shin et al. [1] propoed the periodic reource model in a hierarchical cheduling that facilitate the chedulability analyi of the workload of tak (child) under a periodic reource upply (parent). The author preented the exact chedulability analyi of a workload et in a periodic reource under RM and EDF cheduling and derived the utilization bound. In [2], Almeida et al. analyzed a imilar periodic erver model by introducing the erver availability function. They alo developed a heuritic algorithm for a erver parameter optimization for the minimum ytem utilization, in which the earch pace i reduced to a et of deadline point. Lipari et al. [3] alo conidered the ame problem with a different approach of chedulability analyi uing a notion of characteritic function. Thee three work all ued linear reource upply model. In contrat, Davi et al. [4], [5] preented the exact wort-cae repone time analyi of tak under deferrable erver, periodic erver, and poradic erver. Through an empirical invetigation, the author claimed that the optimal parameter election for multiple reource i a holitic problem and provided a greedy algorithm, which we compare with our GP-baed method in Sec. V. Additionally, in [18], Saewong et al. developed a repone time analyi for real-time guarantee of tak under poradic erver and deferrable erver. In [11], Eawaran introduced a generalized periodic reource model called Explicit Deadline Periodic (EDP) reource, and propoed an exact algorithm for determining the optimal reource parameter that minimize the ratio of length to period of an EDP reource. The ame problem for periodic reource model wa addreed by Shin et al. [13], in which the author preented a polynomial-time ufficient algorithm. Both problem were addreed by Dewan et al. [15] and Fiher [14] by propoing fully-polynomial-time approximation algorithm that improve both the optimality and time complexity. None of thee paper, however, conider the problem of optimizing the parameter of multiple reource. Geometric Programming [7], [8] ha been widely applied to a broad range of non-linear, non-convex optimization problem uch a digital circuit gate izing [19], reource allocation in Average of U GP 2 U Heu communication ytem [16], information theory [2], etc. An extenive dicuion of GP can be found in [8]. VII. CONCLUSION In thi paper we addreed the problem of deign parameter optimization of multiple periodic reource in hierarchical cheduling. We extended the exiting analyi on a ingle reource in order for our reource upply model to be able to capture the variable parameter of higher-priority reource. The preented analyi on the reource bound and the GP-baed optimization i not a globally optimal method due to the approximation error of wort-cae reource interference and tak chedulability. However, we believe that one can benefit from the preented optimization method in deigning a hierarchical ytem with a large number of partitioned reource due to it ability to yield a high-quality olution with a high calability. For future work, we plan to apply the preented method to a hierarchical ytem under non-preemptive global cheduling uch IMA cheduling. REFERENCES [1] I. Shin and I. 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