Geometric Programming Based Optimization of Multiple Periodic Resources in Hierarchical Scheduling

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1 Geometric Programming Baed Optimization of Multiple Periodic Reource in Hierarchical Scheduling Man-Ki Yoon, Jung-Eun Kim, Richard 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 optimally deign 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 report addree a holitic optimization of multiple periodic reource with regard to minimum ytem utilization. We extend the exiting analyi on the parameter bound 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. Partitioned reource cheduling can alo be ued to implement reource reervation to prevent aperiodic tak from being tarved [7], [8]. In uch a temporally partitioned hierarchical cheduling, one important quetion i how much of the computational reource need to be allocated to each partitioned reource in order for the ytem to be optimized for a certain metric. For intance, it i deirable in ytem deign proce 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, which can reduce the unit cot of production. 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 global optimal olution [4], [5]. Accordingly, each deign parameter cannot be choen independently. Thu, 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 report, 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

2 2 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) [9], [10]. GP i a non-linear optimization method that can olve a pecially formed non-convex 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. A will be hown later, 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]. The remaining ection of thi report are organized a follow: Section II ummarize the related work, and then Section III introduce the ytem model we conider and then formally decribe the parameter optimization problem of multiple periodic reource. In Section IV, we review the previou literature on the analyi of ingle reource bound, and then extend thee finding to multiple reource in Section V. In Section VI, we explain how to formulate and tranform the conidered optimization problem to geometric programming. The evaluation reult are given in Section VII. Finally, Section VIII conclude thi report. II. 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 erver (reource) 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 erver parameter optimization problem in a hierarchical cheduling ytem with a different approach of chedulability analyi. Thee three work all ued linear model of reource upply to repreent the reource upply (upply bound function, availability function, characteritic function, repectively) and conidered a ingle reource. 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. The author alo addreed the parameter election optimization of multiple erver and provided a greedy algorithm. Through an empirical invetigation, the author claimed that the optimal parameter election for multiple reource i a holitic problem. In Section VII, we compare the heuritic method in [5] with our GP-baed optimization method. Additionally, in [11], Saewong et al. developed a repone time analyi for real-time guarantee of tak under poradic erver and deferrable erver. In [12], Eawaran introduced a generalized periodic reource model called Explicit Deadline Periodic (EDP) reource model, 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. [14] and Fiher [15] 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 [9], [10] ha been widely applied to a broad range of non-linear, non-convex optimization problem uch a digital circuit gate izing [16], reource allocation in communication ytem [17], information theory [18], etc. An extenive dicuion of geometric programming can be found in [10]. A. Sytem Model III. PROBLEM DESCRIPTION We conider a uniproceor coniting of a et of independent periodic reourcer = {R i i = 1,...,N R }. Each reource R i i characterized by an unknown tuple of (, ), where and are the period and the execution length of the reource, repectively. In each reource R i, a et Γ i = {τ j j = 1,...,N Γ i } of tak run in a fixedpriority preemptive chedule uch a Rate Monotonic [19]. Each tak τ j i repreented by τ j := (e j,p j,d j ) 1, 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 1 More preciely, each tak hould be repreented a τ i,j if the tak belong to reource i. For the implicity of notation, however, we ue the abbreviation τ j.

3 3 relative deadline. In thi report, 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 the releae of reource [5]. The reource R 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 report 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. Finally, there i no trict aumption on the mallet time unit of reource parameter, i.e.,, R + for all i. However, we alo conider cae when the parameter are contrained to integer, i.e., or N. A will be hown later, the integrality contraint make the parameter optimization much harder to olve. 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, {(, )} for i = 1,...,N R, which minimize the overall ytem utilization, U, while guaranteeing the chedulabilitie of the reource and the tak. Here, the overall ytem utilization can be repreented by U = N R i=1 c 1 δ +c 2, (1) where δ i the reource context-witch overhead, and c 1,c 2 are weight given by the ytem deigner [3]. In thi report, 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. In ummary, tak et and their temporal characteritic, i.e., {(e j,p j,d j ) τ j Γ i } for all R i R, are given a input, and we will find the optimal et of reource parameter {(, ) R i R} for a given c 1, c 2, and δ. IV. PARAMETER BOUNDS OF SINGLE RESOURCES In thi ection, we ummarize the previou literature on the analyi of ingle reource bound. The analyi preented in thi ection i primarily baed on the periodic reource model introduced in [1]. It hould be noted, however, that the periodic erver model in [2] can be imilarly ued without lo of generality. A. Sufficient Reource Bound for Tak Schedulability In a partitioned reource whoe period and length are unknown, we can derive the lower-bound of (or the upper-bound of ) with repect to (or ) that make τ j in R i chedulable by uing the periodic reource model introduced in [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) in [1] or A (t) in [2]) can match the maximum workload demand generated by τ j and it higher-priority tak during a time interval t. In fixed-priority global cheduling, the minimum upply of periodic reource i i delivered to tak when it (k 1) th execution ha jut been finihed at time 0 with minimum interference from higher-priority reource. Then, the ubequent execution from the k th releae i maximally delayed by higher-priority reource. Since no aumption i made on the period and length of other reource, we aume that the wort-cae occur when the reource uffer zero interference in the (k 1) th releae and thereafter, a depicted in Figure 1. For thi wort-cae minimum upply we can derive the linear lower-bound upply function lbf Ri (t) a in [1], which i defined a follow: lbf Ri (t) = (t 2 ( )). (2) Note that it i identical to A (t) with α = Li and = in [2]. Now, let u conider tak τ j in reource 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. Since we make no aumption on tak offet, the wort-cae repone time of τ j occur when τ j and the tak with higher priority than τ j are releaed imultaneouly at the end of R i execution and then uffer the wort-cae

4 4 ( k 1) th T th k i ( k + 1) th Ti Ri Ri Ri Reource Supply bf (t) R i 2( T L ) i i t lbf (t) R i Fig. 1: The wort-cae releae pattern of a periodic reource R i when the parameter of higher-priority reource are unknown. preemption from the higher-priority tak from k th releae and thereafter, which we define a the critical intant. 2 Now, let u denote I j a the wort-cae workload generated by τ j and the higher-priority tak 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) where hp(τ j ) i the et of tak with higher priority than τ j. In order to guarantee τ j chedulability, the minimum upply delivered by the reource ha to be greater than or equal to the wort-cae workload during the time interval d j. Thu, lbf Ri (d j ) = (d j 2 ( )) I j. (3) Accordingly, the minimum required reource length, i.e., L min i (τ j, ), for tak τ j with a given reource period can be obtained by olving the quadratic inequality in Eq. (3), which reult in L min i (τ j, ) = (d j 2 )+ (d j 2 ) 2 +8I j. (4) 4 Note that Eq. (4) i equivalent to Eq. (23) in [1] and to Eq. (12) with β = 1 in [2]. It i alo important to note that Eq. (3) 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 Ri (t) I j. In thi report, we ue the ufficient condition in Eq. (3) ince the preence of time in the neceary condition make the propoed optimization method not applicable to the problem under conideration. B. Optimization of Single Reource Bound A een in the previou ection, the required reource length that can guarantee the chedulability of τ j in R i i lower-bounded by Eq. (4). Figure 2 how L min i (τ j, ) for an example periodic reource coniting of {τ 1 = (5,20),τ 2 = (10,100),τ 3 = (15,150)}, where d j = p j for all j. In order to find the minimum required length of a reource i for a given period, 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, ). (5) τ j Γ i 2 In thi report, we do not conider tak jitter. However, without lo of generality, the preented analye in thi report can be imilarly applied to cae with jitter. For example, the wort-cae ituation of τ j i when all the higher-priority tak have experienced their maximum jitter and are releaed at the ame time with τ j.

5 min (τ1, ) min (τ2, ) min (τ3, ) L = T (Reource Length) Feaible Region τ 1 τ 3 τ (Reource Period) Fig. 2: Minimum required reource length L min i (τ j, ) for {τ 1 = (5,20),τ 2 = (10,100),τ 3 = (15,150)}. Thu, if we take from the feaible region, that i, L min i ( ), all τ j Γ i are guaranteed to meet their deadline. For example, we can ee from Figure 2 that L min i ( ) i lower-bounded by τ 3 until around = (τ 1, ) become the new bound. If = 15, the reource length ha to be longer than approximately 9.12 in order to guarantee that the tak meet their deadline. Although the main conideration of thi report i the optimization of multiple reource, we briefly addre the effect of variou contraint for the cae of a ingle reource. In Figure 3, we drew the reource utilization function For > 11.6, L min i U i ( ) = δ + Lmin i ( ) for [1,20] of the example ued in Figure 2: (a) no context-witch overhead (δ = 0), (b) δ = 1, (c) δ = 4, and (d) δ = 1 and integrality contraint on L min i. Firt of all, if context-witch overhead i not a conideration, the minimum reource utilization i achieved at the minimum poible period becaue ( L min i (τ j, ) ) 0 if I j d j. Note that if I j = d j, L min i (τ j, ) become, which mean that a dedicated proceor need to be allocated to the reource in order to make the tak chedulable. When a context-witch overhead i conidered, in contrat, the reource utilization function i no longer monotonically increaing with, and the optimal period δ appear at a longer period due to the hyperbolic nature of. If an integrality contraint on reource length i enforced, the graph become a awtooth function and the optimal period in uch a cae i not necearily identical to the one obtained with real-valued. The optimization of a ingle reource ha been extenively tudied in previou literature. Intereted reader can refer to [1] [5], [12] [15], [20]. V. PARAMETER BOUNDS OF MULTIPLE RESOURCES In thi ection, we extend the analyi of the ingle reource bound in the previou ection to the cae of multiple reource via a parameterization of unknown reource parameter. A. Lower-bound Supply Function Conidering Unknown Parameter of Higher-Priority Reource The bound for ingle reource explained in Section IV wa derived with the aumption that each reource experience no interference in the (k 1) th releae and then uffer the delay of from the k th releae. Thi i a peimitic aumption, ince, in reality, high priority reource would uffer no (if they are the highet one)

6 (c) U i ( ) = min (Ti ) / + 4 / U i ( ) (Reource Utilization) U i ( ) = min (Ti ) / + 1 / (b) min (d) U (T )= L (Ti ) / T + 1 / i i i i 0.55 (a)u i ( ) = min (Ti ) / (Reource Period) Fig. 3: Reource Utilization U i ( ) for [1,20] with (a) δ = 0 (no context-witch overhead), (b) δ = 1, (c) δ = 4, and (d) δ = 1 and integrality contraint on L min i ( ). Each circle repreent the minimum U i ( ) for each cae. or only a few preemption. Thu, an exact method i required [4], [5], which i not ueful for an optimization of multiple reource parameter due to it high time complexity. Thi neceitate holitic optimization of multiple reource parameter. Thu, we now parameterize the linear lower-bound upply function lbf Ri (t) (Eq. (2)) with the period and execution length of the higher-priority reource, hp(r i ). The wort-cae releae pattern of R i occur when R i and hp(r i ) are releaed imultaneouly. The wort-cae buy period of R i, denoted a w Ri, i the maximum time duration that R i can take to execute when it i releaed imultaneouly with the higher-priority reource at the k th releae, which can be obtained by the traditional exact analyi: w k+1 R i = + R h hp(r i) w k Ri L h, T h where wr 0 i = and the wort-cae buy period of R i i w Ri when it converge, i.e., w Ri = wr k i = w k+1 R i for ome k. Thu, the wort-cae delay at the k th releae and alo thereafter (called the initial latency in [2]) can be repreented a wri Ri = L h. (6) T h R h hp(r i) However, thi iterative method can only be applicable to brute-force optimization. Thu, we take a different approach; we approximate Ri. During a time interval of, the maximum workload generated by R i and hp(r i ) can be repreented by: Ti w Ri = + L h. T h R h hp(r i) Note that with thi equation, we can avoid iterative calculation by auming the number of invocation of higherpriority reource during, not during the exact buy period of R i. Alo note that it i a afe bound a long a R i meet it deadline, i.e., D i =. Now, we remove the ceiling in order to linearize w Ri, which reult in ( Ti ) w Ri = + +1 L h, T h R h hp(r i)

7 7 ( k 1) th T th k i ( k + 1) th Ti Ri Ri Ri Reource Supply bf (t) R i ( T L ) i i Ri lbf (t) R i t Fig. 4: The wort-cae releae pattern of R i conidering the period and execution length of higher-priority reource. becaue x x+1. Then, the new linear lower-bound upply function of R i during a time interval t parameterized with hp(r i ) can be preented a follow: where lbf Ri (t) = (t ( ) Ri ), (7) Ri = R h hp(r i) ( Ti ) +1 L h. (8) T h If the reource period are harmonic with each other, we can ue Ri = R h hp(r i)( Ti T h ) L h intead. Now, the minimum upply of R i i delivered to tak when it experience no interference in the (k 1) th execution and the maximum preemption delay from it higher-priority reource thereafter, that i, Ri (Figure 4). Now, the ufficient reource bound contraint for tak chedulability, i.e., Eq. (3), i refined a follow: lbf Ri (d j ) = (d j ( ) Ri ) I j. (9) Accordingly, the minimum required reource length for tak τ j with a given reource period, i.e., L min i (τ j, ) in Eq. (4), become Again, L min i (τ j, ) = (d j Ri )+ (d j Ri ) 2 +4I j. (10) 2 ( ) L min i ( ) = max L min i (τ j, ). (11) τ j Γ i Although the bound preented here i not exact and may incur approximation error, it enable u to optimize multiple reource holitically with high efficiency, a will be decribed in Section VI. B. Non-convexity of Multiple Reource Optimization We preent a imple example of two reource in order to how the non-convexity of the optimization problem of multiple reource. Let u conider Figure 5, 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 [1,140]. In thi example, the reource have the ame period, i.e., T 1 = T 2, for implicity of repreentation, and δ i et to 1. R 1 ha a higher priority than R 2, thu R1 = 0 and R2 = 2 L 1. From the graph, we can firt ee that the ytem utilization function U i not convex (and neither i U 1 ), which i hown by the traight line drawn between

8 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. 5: Sytem utilization of two reource, {R 1,R 2 }, and it non-convexity. The awtooth-haped graph repreent the ytem utilization when the reource execution length are contrained to integer. Circle repreent the minimum utilization of the ytem and the reource. T = 60 and T = 120: U (90) U (60)+U (120). 2 Furthermore, while the reource achieve minimum utilization at T 1 = 20.3 and T 2 = 63.9, repectively, thee do not necearily lead to the global optimality which occur at T = T 1 = T 2 = 23.1; thi iue i addreed alo in [4]. Additionally, a in the ingle reource cae in Figure 3, if are to be dicrete, the optimal olution occur at a different point (T = 30.3). In fact, when we conider uch integrality contraint, determining the optimal olution require extenive branch and bound earche. 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. VI. HOLISTIC OPTIMIZATION OF RESOURCE PARAMETERS VIA GEOMETRIC PROGRAMMING In thi ection, we formulate the parameter optimization problem of multiple periodic reource with Geometric Programming (GP) [9], [10]. A. Geometric Programming Formulation A non-linear, non-convex optimization problem can be olved by geometric programming if the problem can be formulated in a pecial form a follow [10]: Minimize f 0 (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 k=1 x a k k,

9 9 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 3 if a x R +. In ummary, the objective function and the inequality contraint mut be in poynomial form, and the equality contraint can only be in monomial form. We now formulate the optimization problem of the multiple reource parameter in a GP form. A previouly tated in Section III-B, we are given a et of periodic reource {R i } with unknown parameter, and, their tak et {(e j,p j,d j ) τ j Γ i }, and the reource context-witch overhead are knownδ. Thu, the optimization variable are T = (T 1,...,T N R) and L = (L 1,...,L N R). Objective Function The objective function (1) in Section III-B i already in a poynomial form, thu it can be repreented a follow: where c 1,c 2,δ 0. f o (T,L) = (c 1 δ +c 2 ) T 1 i, (12) N R i=1 Reource Bound Contraint The reource bound for each reource R i i contrained by Eq. (9) for each τ j Γ i, which can be reexpreed a follow: ( +I j )+ Ri 1, (13) ( +d j ) where Ri = ) Ti R h hp(r i)( T h +1 L h (i.e., Eq. (8)), d j i the relative deadline of τ j, and I j i the wort-cae workload generated by the tak itelf and the higher-priority tak during the time interval of d j, both of which are contant for a given input. However, the above inequality doe not conform to a poynomial form becaue of the poynomial term in the denominator, i.e., ( + d j ) = L 2 i + d j (recall that a denominator mut be monomial). Oberve, however, that +d j can be approximated with a monomial by the following geometric mean approximation [17]. Let u firt denote it a g i ( ) = u 1 ( )+u 2 ( ), where u 1 ( ) = and u 2 ( ) = d j. Then, we now approximate g i ( ) with ( ) γ1 ) γ2 u 1 ( ) u 2 ( ) g i ( ) = (, (14) γ 1 γ 2 where γ 1 = u 1(x 0 ) g i (x 0 ) and γ 2 = u 2(x 0 ) g i (x 0 ) where x 0 R + i a contant that atifie g i (x 0 ) = g i (x 0 ). The approximated monomial g i ( ) then can be rewritten a: ( ) γ1 ) γ2 d j g i ( ) = (, γ 1 γ 2 with γ 1 = x 0 and γ 2 = d j. x 0 +d j x 0 +d j 3 If α x i allowed to be a non-integer, the form i called Generalized Geometric Program (GGP), which can be tranformed to GP.

10 10 g i ( ) and approximate value g i ( ) = + 30 g i (10) g i (5) g i (1) Fig. 6: g i ( ) = +30 and it approximated monomial g i ( ) at x 0 = 1,5, and 10. g i (x 0 ) i tangent to g i ( ) at = x 0. Finally, Eq. (13) can be formulated a the following poynomial contraint: where ( ( +I j )+ Ri ) ( g i ( )) 1 1, (15) Ri = R h hp(r i) ( (Ti +T h ) T 1 h L h ). Note that the approximation quality of g i ( ) depend on the choice of x 0, a hown in Figure 6. Thu, in the optimization procedure, we iteratively approximate g i ( ) by updating γ 1 and γ 2 according to the intermediate olution of. That i, until the objective value converge, we ue at k th tep a x 0 at (k +1) th tep. In our experiment, the initial value of x 0 wa choen a 1, and the objective value converged within one or two iteration. Reource Schedulability Contraint Each reource mut be chedulable, that i, + Ri, which can be expreed a the following poynomial contraint: ( + R h hp(r i) B. Mixed-Integer Geometric Programming for Integrality Contraint ( (Ti +T h ) T 1 ) 1 h L h ) Ti 1. (16) In a real ytem, the reource period and execution length are multiple of the mallet time unit becaue of cheduling granularity. In thi cae, we can think of integrality contraint on and value, however thi make the optimization problem much harder to olve a illutrated in Section IV-B. A GP i called Mixed-Integer Geometric Programming (MIGP) [10] if one or more variable are contrained to be integer. A branch and bound method [21] can be ued and often find a global optimal olution; however, it cannot be calable with problem ize. In thi report, we ue a heuritic for rounding fractional variable. The heuritic i not a globally optimal method but can efficiently find a near-optimal olution. We note that the choice of a branching or rounding method i orthogonal to the optimization preented in the previou ubection. The key idea of the rounding heuritic i that we firt find the optimal olution without any integrality contraint, which i called GP Relaxation. Then, we round each variable up or down to it nearet integer value at each tep. Here we aume the integrality contraint on reource execution length, L; however, thi can be imilarly applied to T. Now, let u denote {T,L } a the optimal olution found with the relaxed GP. 4 For each L i L, we 4 The problem itelf i infeaible if no olution exit for the relaxed GP.

11 11 TABLE I: Evaluation parameter. 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 [1,30] Tak period, p j [50,2000] Context-witch overhead, δ 1 calculate the ditance between L i and it nearet integer a follow: ε(l i ) = min( L i, L i ), where x = x x and x = x x. Then, we find the leat fractional L i L i = arg min L i L (ε(l i)). uch that Once we have found uch L i, we then olve the GP by adding the following monomial contraint to the original GP: L i 1 = 1, where L i i L i +L i or L i L i depending on L i and L i. Now let u conider the cae when both T and L are to be integer. The rounding proce i imilar to that decribed above; however, in each iteration we round one and one L j up or down, where i i not necearily equal to j. The heuritic work a follow. We firt find the relaxed optimal olution {T,L }. Then, we round the leat fractional Ti up or down to it nearet integer,. We then again olve the modified GP and then find the leat fractional L j. If T j i an integer we round L j up. Otherwie, we find it nearet integer a decribed above. In the next iteration, we find the leat fractional Ti among the remaining fractional T. However, thi time we check if L i i already an integer. If thi i the cae, we round Ti down. In ummary, the heuritic iterate N R time, and during each iteration we update one pair of fractional Ti and L j ; however, T i (L j ) can only be rounded down (up) if it correponding L i (T j ) i already an integer. The reaon i that for each T i found by GP, it correponding L i i the lower-bound of. Thu, if L i i already an integer, T i cannot be rounded up even if it i cloer to Ti becaue thi will make the problem infeaible by violating the reource bound contraint (15). A imilar argument i applied to L j. Note that we need to update g i( ) (Eq. (14)) a well in each iteration. We note that thi rounding heuritic i baed on the hope that a better olution of parameter pair could be found by independently rounding each parameter of a reource; updating both Ti and L i of a reource at the ame time may increae the poibility of local optima by moving radically in the feaible region. A previouly mentioned, however, the preented rounding heuritic doe not guarantee the optimality of the olution. VII. EVALUATION In thi ection, we evaluate the propoed optimization method preented in Section V and VI. A. Evaluation Method Table I 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 100 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 Eq. (11) and (10) with Ri calculated by Eq. (6). Recall that the ytem utilization obtained with thi exhautive earch i till not the exact globally optimal olution a explained in Section V. GP with the upper-bound on T max : The GP optimization method preented in Section VI with an additional et of 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 of the reource period i unneceary.

12 12 Average of U GP 1 U Exh and U GP 2 U Exh Fig. 7: The average difference of U GP1 0 GP with T max Number of reource and U GP2 GP without T max to U Exh with different number of reource. 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. Thi aumption i only for the evaluation purpoe. Reader intereted in priority optimization can refer to [4], [5]. The priority of each tak in each reource i aigned baed on Rate Monotonic priority aignment [19]. The GP were olved uing GGPLAB [22]. 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 and U GP2, that i, U GP1 U GP1 U Exh and U GP2 U Exh, repectively, and then take the average of 100 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 minute or more depending on the problem ize and the choice of T max and. C. Evaluation Reult Figure 7 compare the minimum ytem utilization found by the exhautive earch and our GP method increaing the number of reource from 2 to 5. It hould be noted that the exhautive earch take a longer time to olve cae with ix or more reource. Thu, we evaluated cae with 2, 3, 4, and 5 reource. It took about 30 minute 1 hour to olve one input coniting of 5 reource with a tep ize of 0.5. T max and the tep ize were et to 100 and 0.5, repectively, and no integrality contraint wa poed. A we can ee from the reult, the average difference of U GP1 and U Exh, i.e., U GP1 U Exh, increae with the number of reource. Thi i mainly becaue of the approximation error in Ri decribed in Section V. Recall that while the exhautive earch calculate the minimum upply of each reource by uing the iterative equation (Eq. (6)), our GP method take the ceiling off and calculate the interference from higher-priority reource during the interval of it period (Eq. (8)). When there are only 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; average difference compared to the olution of the exhautive earch. Another intereting obervation i that GP 2, a GP without the upper-bound on T max, can find better olution than GP 1 ; the error of GP 2 compared to the exhautive earch i only on average.

13 Difference of U GP 2 U Exh Fig. 8: The difference of U GP2 Bae utilization and U Exh with variou bae utilization Average of U GP 2 U Exh Number of reource Fig. 9: The average difference of U GP2 and U Exh when integrality contraint on reource parameter are poed. 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 optimal olution with the exhautive earch by etting T max higher, however, thi can be limited by the input ize. Hereafter we compare U GP2 with U Exh. With the ame input, we evaluated the difference of U GP2 to U Exh with variou bae utilization, i.e., the um of all tak utilization, a hown in Figure 8. From the graph, 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 Ri in Eq. (8). Next, we evaluate our method when both reource period and execution length are be integer. For thi

14 Our GP method Heuritic in [R. Davi, RTNS 2008] Number of olution found Average of U GP 2 U Heu Number of reource 0 Fig. 10: 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). evaluation, we ued the ame input a above. In the exhautive earch, the reource period are recurively aigned from 1 to 100 with a tep ize of 1. Then, the ceiling of the minimum execution length for each reource period, L min i ( ) (Eq. (11)), wa ued in Eq. (6). For our GP, the rounding heuritic explained in Section VI-B wa ued. A can be een from Figure 9, the difference between the two method increae with the number of reource for the ame reaon a above. However, thi time the difference became bigger with the contraint compared to the reult in Figure 7. One can eaily expect that the increaed difference arie from the implicity of the rounding heuritic ued. A more accurate method could be combining an etimation-baed election of rounding variable. Note that reource utilization function are in different hape depending on the prioritie and workload of tak, a can be een in Figure 5. In the example, R 2 i more flexible in chooing it parameter ince it utilization function, Lmin i (), i almot flat during a wide range of it period. In uch a cae, it would be better to round off R 2 firt and then proceed to higher utilization reource in order to avoid local optima. However, it i till difficult to find the global optimal olution with uch a rounding-baed approach due to the awtooth nature of the curve. Thu, one may want to conider uing a cla of branch-and-bound method to enhance the quality of the olution. Latly, we compare 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 period, it find the optimal reource length by a binary earch. When the optimal pair of period and length i found, the ame proce i applied to the next priority reource. For the comparion, we ued the ame input a above. In the heuritic, each reource period i aigned from 1 to 1000 with a tep ize of 0.1, and each reource length wa found at the granularity of 0.1. For our GP method, both T max and integrality were not aumed. Figure 10 how i) the number of olution found by each method and ii) the average difference of the minimum utilization between the two method. A can be een from the bar, our GP 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 100 input et, but only 7 olution were found by the heuritic. 5 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 globally optimal method either, it can explore more olution due to it ability to take into account the 5 It hould be noted that the exact number of feaible olution i unknown a thi require a true optimal method. Among 100 input et for each cae, ome input et may not be feaible 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 100 input et with 2 and 5 reource are and 0.663, repectively.

15 15 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 Figure 10 how. Each marker on the line i the average of the difference of the minimum ytem utilization found by our method, i.e., U GP2, to that found by the heuritic, i.e., U Heu, for the input et that the heuritic found; with 2,3 and 4 reource, U GP2 U Heu are between and in average, and with 5 reource, the difference i 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 Section V, our analyi conider the wort-cae cenario that are ufficient but not neceary, and it i alo baed on the approximation of Ri, both of which 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]). VIII. CONCLUSION In thi report 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. In order to olve the problem, we formulated it via Geometric Programming and provided a heuritic method for integrality contraint. The preented analyi on the reource upply model and it optimization i not a globally optimal method due to the approximation error of wort-cae reource interference. 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 will invetigate the poibility of applying the preented analyi and optimization method to a hierarchical ytem under non-preemptive global cheduling uch IMA (Integrated Modular Avionic) cheduling. REFERENCES [1] I. Shin and I. Lee, Periodic reource model for compoitional real-time guarantee, in Proceeding of the 24th IEEE Real-Time Sytem Sympoium, 2003, pp [2] L. Almeida and P. 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