Energy-Efficient Thermal-Aware Scheduling for RT Tasks Using T CP N

Transcription

1 Energy-Effcent Thermal-Aware Schedulng for RT Tasks Usng T CP N L. Rubo-Anguano G. Desrena-López A. Ramírez-Trevño J.L. Brz CINVESTAV-IPN Undad Guadalajara, Av. del Bosque 11, CP 19, Zapopan, Jalsco, Mexco DIIS/I3A Unv. de Zaragoza, María de Luna Zaragoza, España. Abstract: Ths work leverages TCPNs to desgn an energy-effcent, thermal-aware real-tme scheduler for a multprocessor system that normally runs n a low state energy at maxmum system utlzaton but ts capable of ncreasng the clock frequency to serve aperodc tasks, optmzng energy, and honorng temporal and thermal constrants. An off-lne stage computes the mnmum frequency requred to run the perodc tasks at maxmum CPU utlzaton, the proporton of each task s job to be run on each CPU, the maxmum clock frequency that keeps temperature under a lmt, and the avalable cycles (slack) wth respect to the system wth mnmum frequency. Then, a Zero-Laxty onlne scheduler dspatches the perodc tasks accordng to the offlne calculaton. Upon the arrval of aperodc tasks, t ncreases clock frequency n such a way that all perodc and aperodc tasks are properly executed. Thermal and temporal requrements are always guaranteed, and energy consumpton s mnmzed. Keywords: Real-Tme schedulng, Tmed Contnuous Petr Nets, Modellng. 1. INTRODUCTION There s a great nterest n usng multcore processors n embedded real-tme systems. Multcores reduce the Sze, Weght and Power (SWaP) requrements n avoncs, have evdent advantages n automotve electroncs and satellte systems, and are already common n consumer devces. Lmted power, battery lfespan and thermal restrctons requre careful energy management, for whch real-tme schedulers can leverage power management mechansms provded by current MPSoCs, such as dynamc voltage and frequency scalng (DVFS). Many DVFS schedulers, thermal-aware schedulers and schedulers that mnmze energy consumpton have been proposed over the past few years (Kong et al. (214), Schor et al. (212), Ahmed et al. (216), Hettarachch et al. (214)). In ths work we present a mnmal energy, thermal-aware real-tme scheduler for a multprocessor system. We consder a set of perodc ndependent tasks subject to temporal, thermal and energy restrctons, modeled by means of a Tmed Contnuos Petr Net (TCPN). Aperodc tasks arrve asynchronously, and should only be served when every perodc task s guaranteed to meet ts deadlne. An off-lne stage frst computes the mnmum frequency requred to run the perodc task set at maxmum CPU utlzaton. By means of a Lnear Programmng Problem (LPP) t fnds a maxmum frequency (F + ) at whch the system can run subject to temporal and thermal constrants. The fnal offlne step leverages a deadlne parttonng approach (Funk et al. (211)) to compute a schedule by calculatng the proporton of each task that must be executed on each CPU durng the nterval. In ths way, the executon of ths schedule consumes mnmum energy and meets temporal and thermal constran. In a second on-lne stage, a Zero Laxty scheduler (Davs and Burns (211b)) performs the schedule computed off-lne for the perodc task set. Upon arrval, aperodc tasks are servced f a controller can ncrease the CPU clock frequency to acheve the proper slack, or are rejected otherwse. The frequency s always kept as low as possble to meet the temporal constrants and mnmze energy consumpton. As far as we know, ths s the frst soluton based on TCPNs to accommodate aperodc real-tme tasks whle preservng temporal and thermal constrants, mnmzng energy. Our prelmnary results, leveragng a deadlne parttonng scheme, yeld a good compromse n terms of optmal utlzaton of CPU, context swtches and mgratons. Secton 2 provdes basc defntons and concepts on Tmed Contnuous Petr Nets (T CP N), the formal model used n ths work to represent tasks, CPUs, energy consumpton and temporal and thermal behavor. Secton 3 formulates the schedulng problem. Secton 4 explans the system model. The off-lne schedule calculaton s explaned n Secton 5, whereas Secton 6 presents the the on-lne scheduler. Secton 7 shows the procedure to deal wth aperodc aperodc tasks. Secton 8 shows some examples, and Secton 9 summarzes conclusons and further work. 2. PERI NETS BACKGROUND Ths secton provdes basc defntons and concepts on Tmed Contnuous Petr Nets (T CP N), the formal model used n ths work to represent tasks, CPUs, energy consumpton, temporal and thermal behavor. For a deeper

2 nsght on Petr Nets see Slva and Recalde (27), Davd and Alla (28), Slva et al. (211). Defnton 2.1. A Petr net (P N) s a 4-tuple N = (P, T, P re, P ost) where P = {p 1,, p P } and T = {t 1,, t T } are fnte dsjont sets of places and transtons. P re and P ost are P T P re and P ost ncdence matrces, where P re[, j] > (resp. P ost[, j] > ) f there s an arc gong from p to t j (or gong from t j to p ), P re[, j] = (or P ost[, j] = ) otherwse. Defnton 2.2. A contnuous Petr net (ContPN ) s a par ContP N = (N, m ) where N = (P, T, P re, P ost) s a P N (PN ) and m {R + } P s the ntal markng. A transton t s enabled at m f p j t, m[p j ] > ; and ts enablng degree s defned as enab(t, m) = m[p mn j] p j t P re[p. Frng t j,t ] n a certan amount α enab(t, m) yelds a markng m = m + αc[p, t ], where C = P ost P re. If m s reachable from m by frng the fnte sequence σ of enabled transtons, then m = m + C σ s named the fundamental equaton where σ {R + } T s the frng count vector,.e σ [j] s the cumulatve amount of frngs of t j n the sequence σ. Defnton 2.3. A tmed contnuous P N (TCPN ) s a tme-drven contnuous-state system descrbed by the tuple (N, λ, m ) where (N, m ) s a contnuous PN and the vector λ {R + } T represents the transtons rates determnng the temporal evoluton of the system. Transtons fre accordng to a certan speed, whch s generally a functon of the transton rates and the current markng. Such functon depends on the semantcs assocated to the transtons. Under nfnte server semantcs, the flow through a transton t (or t frng speed, denoted as f(m)) s the product of the rate, λ, and enab(t, m), the nstantaneous enablng of the transton,.e., f (m) = λ enab(t, m). The frng rate matrx s defned by Λ = dag(λ 1,, λ T ). For the flow to be well defned, every contnuous transton must have at least one nput place, so we assume t T, t 1. The mn n the above defnton leads to the concept of confguraton. A confguraton of a TCPN at m s a set of (p, t) arcs descrbng the effectve flow of each transton, and say that p constrans t j for each arc (p, t j ) n the confguraton. A confguraton matrx s defned for each confguraton as follows: Π j, (m) = 1 P re[, j] f p s constranng t j otherwse f(m) = ΛΠ(m)m s the vectoral form of the flow of a transton. The followng fundamental equaton descrbes the dynamc behavour of a PN system: (1) m = Cf(m) = CΛΠ(m)m (2) A control acton can be appled to (2) by addng a term u to every transton t such that u f, ndcatng that ts flow can be reduced. Thus, the controlled flow of transton t becomes w = f u and the forced state equaton s: m = C[f u] = Cw. 3. PROBLEM DEFINITION Defnton 3.1. Let T = {τ 1,, τ n } be a set of n ndependent perodc tasks. Each task s dentfed by the 3 tuple τ = (cc, d, ω ), where cc s the worst-case executon n CPU cycles, ω the perod and d s the relatve mplct deadlne (d = ω ) (Baruah et al. (215)). Let P = {CP U 1,..., CP U m } be a set of m dentcal processors wth an homogeneous clock frequency F F = [F 1,, F max ]. We assume that all task parameters, ncludng task perod and CPU cycles are ntegers and that any task can be preempted at any tme. The hyper-perod s defned as the perod equal to the least common multple of perods H = lcm(ω 1, ω 2,..., ω n ) of the n perodc tasks. A task τ executed on a processor at ts maxmum frequency F max, requres c = cc F max processor tme at every ω nterval. The system utlzaton s defned as the fracton of tme durng whch the processor s busy runnng the task set.e., U = n c ω. The CPU utlzaton must be computed and should be less or equal to the number of processors.e., U m (Baruah et al. (1996)). Problem 3.1. Mnmum Energy Thermal Aware Real Tme Scheduler (M ET ART S). Gven the sets of tasks T and CPUs P, the M ET ART S problem conssts n desgnng an algorthm to allocate wthn the hyperperod H the tasks n T to the m dentcal CPUs n such a way that the deadlnes for T are always satsfed and the CPU temperatures are kept always below a gven temperature bound T max and the consumed energy s mnmum. In addton to prevous problem, aperodc tasks, that must be allocated to CPUs arrve to the system. Defnton 3.2. Aperodc Task. Let T a = {τ1 a,..., τz a } be an ndependent aperodc task set. Each τ a s defned as a 3-tuple (cc a, da, ra ) where cca (requred CPU cycles), d a (ts deadlne) are known a prory, and ra s the arrval tme, whch s unknown. 4. SYSTEM MODEL The CP U thermal actvty and the tasks flud allocaton to CP Us happens n a contnuous space and honor a set of dfferental equatons. The use of a TCPN allows modellng ths contnuous actvty whle provdng data about the system s state at any moment. Ths secton summarzes the TCPN model of the system (CPUs, tasks, thermal behavor, energy). For detals on the thermal model we refer to Desrena-Lopez et al. (214). The task model s based on Desrena-Lopez et al. (216). Fg. 1 detals a part of the model correspondng to a one task (τ 1 ) and one CPU (CP U 1 ). Task and CPU Model.- The places p ω, pcc and t ω, together wth ther arcs, model the th task. The places p busy 1,j,, p busy n,j, pexec 1,j,, pexec n,j, pdle j, and transtons t alloc 1,j,, t alloc n,j and t exec 1,j,, texec n,j together wth ther arcs, model the j th CPU. The weghted arc cc corresponds to the duraton of τ (n CPU cycles). The CPU cycles requred to run a task s job are stored n place p cc. The transtons t exec,j represent the executon on the CP U j. The current markng of places p exec,j (m exec,j ) represents the CPU worst-case executon tme of task τ currently

3 Task & CPU Model (x = {T, a, T, P} ) of the thermal, task and processors subnet. CT alloc, CT alloc and CP alloc stand for the connectons of transtons t alloc,j from (to) places n the thermal model, task and CPU throughput model, respectvely. Matrx CP exec holds the columns of the transtons t exec,j of the ncdence matrx C P.w alloc s the controlled flow of the allocaton transtons (.e. the task allocaton rate to CPUs). Eq. (3a) represents the system s temperature evoluton. Eq. (3b) ndcates that the envronmental temperature keeps constant at all tmes (ts dervatve s neglected). Eq. (3c) descrbes the arrval of perodc tasks to the system. Eq. (3d) models the CPUs cycles that are assgned to tasks. Fnally, Eq. (3e) models task executon. T2 T1 Tk 5. OFF-LINE STAGE T T3 Thermal Model In our approach, pror to consderng the aperodc tasks, we need to fnd the mnmum and maxmum frequences to execute the perodc task set subject to temporal and thermal constrants. Ths requres frst to study the system thermal behavor. Later, we wll compute a schedule applyng a deadlne parttonng approach. Fg. 1. Equvalent TCPN model for a multprocessor system. executed at a frequency F. Places p dle j represent the dle state and the busy state of the processor. The and p busy,j markng of place p dle j models the avalable CPU cycles (throughput capacty). The ntal markng at p dle s set to 1, ndcatng that CP U j s dle. The arcs gong from transtons t exec 1,j,, texec n,j to place p dle j and from place p dle j to transtons t alloc 1,j,, t alloc n,j are weghted by a constant value η, to ensure that the flow n transtons t alloc 1,j,, t alloc n,j s lmted by the throughput CPU capacty modeled by place p dle j. Thermal model.- The executon of cycles of τ n CP U j s modeled by frng transtons t exec,j (Fg. 1), whch adds tokens to places p comj, actvatng the thermal model for CP U j (whose temperature wll ncrease because of the actvty). The rest of transtons and places n ths thermal model represent heat transport by conducton and convecton, for a deeper explanaton see Desrena-Lopez et al. (214). Task and thermal model evoluton.- The dynamc behavor of the global model (Fg. 1) s provded by the followng equatons: ṁ T =C T Λ T Π T (m)m T + C aλ aπ a(m)m a +CP exec (3a) ṁ a = (3b) ṁ T =C T Λ T Π T (m)m T CT alloc w alloc (3c) ṁ P =C P Λ P Π P (m)m P + CP alloc w alloc (3d) ṁ exec =CP exec (3e) C x, Λ x, and Π x (m) are the ncdence matrx, the frng rate transtons and the confguraton matrx respectvely j Steady state Thermal Analyss.- Task executon generates a thermal actvty gven by Eq. (3a), where CP exec = F 3 C exec P. Ths can be rewrtten as a space state equaton: ṁ T = Am T + B m a + F 3 Bf exec and Y T = Sm T (the CPUs temperature), where A = C T Λ T Π T (m), B = C exec P and B = C a Λ a Π a (m). Snce the schedule s perodc, the temperature s a non-decreasng functon reachng a steady state temperature (m Tss ),.e. ṁ T = when tme tends to nfnte. Hence m Tss = A 1 (F 3 Bw alloc + B m a ). Snce Sm Tss T max then: SA 1 F 3 Bw alloc T max + SA 1 B m a (4) Ths equaton provdes the thermal constrants that the allocaton of tasks to the processors (w alloc ) must fulfll. Energy consumed by a schedule.- If the CP U j clock frequency s F durng the tme nterval (ζ 1, ζ 2 ], then the average energy consumed durng ths nterval by the tasks runnng on CP U j s defned as: E j = ζ2 ζ 1 P CP Uj (F )d ζ (5) P CP Uj (F ) s the power consumed by a CP U j. It depends on P dyn (F ), the dynamc power due to computatonal actvtes of tasks, and P leak, the statc power due to leakage. It s computed as: P CP Uj (F ) = P dynj (F ) + P leakj = λ exec,j m busy,j (ζ)f 3 + P leakj. Where P leakj can be modeled as a lnear functon of temperature (Ahmed et al. (216)): P leak = δt +ρ, where T s the CPUs temperature and δ and ρ are modelng constants. The consumed energy s mnmzed ff the clock frequency F s mnmzed, but F must be hgh enough to ensure that temporal constrants are met. Next, we compute ths frequency.

4 Mnmum frequency Modern CPUs vary ther clock frequency accordng to a number of preset values,.e. F = [F 1,, F max ]. We normalze ths set as φ = [φ mn = F 1 F max,., 1]. The next proposton obtans the mnmum clock frequency that fulflls temporal constrants. Proposton 5.1. Assumng that the task utlzaton s less than the number of processors n the METARTS problem, the normalzed clock frequency that mnmzes the total energy consumpton whle meetng temporal constrants s constant: Φ = max{φ mn, 1 n cc } (6) m ω F max Proof 5.1. Accordng to Eq. (5), the energy has a mnmum ff the consumer power s mnmum. Ths occurs when φ 3 s mnmum and fulflls that n cc φw F max = m, and φ φ mn. Usng Lagrange multplers, the Lagrangan functon s L = φ 3 + µ 1 ( 1 n cc φ w F max m) + µ 2 (φ mn φ). The soluton yelds four cases: a) Both multplers are nactve (µ 1,2 = ); b) Both multplers are actve (µ 1,2 ); c) µ 1 = and µ 2 ; and d) µ 1 and µ 2 =. The frst case s unfeasble, because φ cannot be zero. In the second case, the only soluton s φ = φ mn = 1 n cc m w F max. Fnally, f one multpler s actve whle the other one s nactve there are two possble solutons: φ = φ mn or φ = 1 n cc m w F max. Consequently, n order to fulfll both constrants, the normalzed clock frequency that mnmze the total energy consumpton n becomes Φ = max{φ mn, 1 cc m ω F max }. The normalzed frequency Φ meets the temporal constrants. To guarantee that the thermal constrants are also fulflled, we must compute w alloc and solve Eq. (4). At frequency Φ the total system utlzaton s U = n cc ω φ F max = m and the processor frequency s F = mn{f F F Φ F max }, gven the nature of the dscrete set of frequences. Snce we have a fully utlzed system, the dstrbuton of the CPU cycles requred to execute all tasks must be homogeneous,.e., w alloc = [ 1 n cc m ω F,..., 1 n cc m ω F ] T. Moreover, f w alloc satsfes Eq. (4), then the thermal constrants are also satsfed. Otherwse, the M ET ART S problem does not have a soluton. If t has a soluton (Φ s feasble), then we can compute the maxmum CPU cycles avalable for aperodc tasks, and the maxmum clock frequency that can be used subject to thermal constrants. Maxmum CPU cycles and clock frequency The maxmum thermal frequency F + s the greatest frequency at whch all CPUs operate at 1% of utlzaton and the temperature never exceeds the maxmum thermal constrants. F + can be computed by usng the next programmng problem. The frst constrant s thermal. CC j represents the cycles that CP U j must execute per hyperperod. Snce all CPUs must work at ther maxmum capacty, the second constrant mples that the CPU utlzaton s 1%. The last constrant bounds F + to the actual clock frequency range n the MPSoC. max F + s.t. SA 1 F +3 B [ CC1 F + H,..., CCm F + H CC j F + H = 1 j = 1,..., m ] T T max + SA 1 B m a F F + F max (7) The soluton for F + has to be n the set F of dscrete frequences. Thus the processor frequency s updated as F + = max{f F F F + }. 5.1 Deadlne parttonng We consder the ordered set of all tasks jobs deadlnes to defne schedulng ntervals, as n deadlne parttonng (Funk et al. (211)). Each task τ must be executed n = H ω tmes wthn the hyperperod H. Thus every q ω, where q = 1,, n s a deadlne that must be consdered n the analyss. These deadlnes can be ordered and joned n the set SD = {sd 1,, sdn }. A general set of deadlnes s defned as SD = SD SD T where SD = {}. The elements of SD can be arranged n ascendant order and renamed as SD = {sd,, sd α }, where α s the last deadlne. The schedulng nterval ISD k = [sd k 1, sd k ] s defned and ISD k = sd k sd k 1 represents the schedulng nterval duraton. The proposed deadlne parttonng problems assumes a 1% utlzaton on every CPU, but recallng that F = mn{f F F Φ F max }, n most cases F Φ F max consequently dle cycles appear. To solve ths problem we ntroduce a dummy task wth perod H and utlzaton m n c F ω. Then the cycles that each task must execute n the ISD k,.e xk, can be computed as follows. Let cc = ω F cc be the cycles that task τ can be dle. Thus, the total amount of cycles (sd k F ) n sd k can be rewrtten as sd k F = q ω F + r, where r < ω F and q Z, where q represents the occurrences of a task (the task s jobs) n the system. If r =, t means that τ has ts deadlne n the schedulng nterval. Then the followng LPP can be posed to compute x k. n mn k = 1,..., α s.t k x k n x k = m Ik SD F f r = f r k x γ = q cc γ=1 k x γ q cc + max{, k I γ SD F cc } γ=1 γ=1 x k Iγ SD F (8) The frst constrant mples that the CPU utlzaton s 1%. It s requred snce Φ ndcates that CPU utlzaton s 1%. The second constrant guarantees that

5 those tasks that must complete executon n ths nterval actually end. The last constrant guarantees deadlne fulfllment. The followng proposton guarantees that f the former LLPs are orderly solved accordng to the k th nterval, then the computed amount of tme that each task must run per nterval yelds a feasble schedule. Proposton 5.2. Gven a task set T presented n Defnton 3.1, where the task utlzaton at F s equal to the number of CPUs, the soluton of the lnear programmng problems n Eq. (8) s always nteger and f each task τ s executed exactly x k cycles durng the k th nterval, then a feasble schedule s obtaned. Proof 5.2. Let T k = T1 k T2 k be a task set partton, where T1 k = {τ 1,, τ v } s the set of tasks that have ther deadlnes at sd k and T2 k = {τ v+1,, τ n } = T T1 k. Some slack varables h are added to form a set of constrants that can be represented as My = b where y = [x h] T (.e., equalty constrants are obtaned). Notce that vector b s always nteger. By constructon, [ the restrcton ] matrx L M has the form: M = (v+1) n, and L has Q (2n v) n I (2n v) [ the form: L (v+1) n = 1 1 I v v ]. It s easly seen that rank(l) = v + 1 and rank(m) = rank(l) + rank(i) = 2n + 1,.e M s a full row rank. The soluton s always nteger f M s unmodular,.e., the determnant of every square submatrx (M s ) of M s ether,+1 or -1. All M s are obtaned deletng columns, and there are three possble scenaros: frst, f any of the frst v columns s removed, M s loses rank, hence det(m s ) =. Second, f any deleted column contans a nonzero entre where ts correspondng row has a nonzero element among the frst v columns, M s loses rank snce the resultng row s duplcated among the frst v rows. Thus, det(m s ) =. Fnally, when any other column not lsted before s deleted, [ the resultng ] matrx A always can be arranged as M s =, where matrx B I A s always TUM, accordng to Theorem 3.4 reported n Serksma (21), thus det(a) =, ±1. Therefore, det(m s ) = det(a) det(i) =, ±1. 6. ON-LINE STAGE: SCHEDULER The prevous secton computed the CPU clock frequency F, maxmum clock frequency F + and task executon tme per schedulng nterval. The on-lne scheduler uses these data to mplement a Fxed Prorty Zero-Laxty (FPZL) algorthm (Davs and Burns (211a)). It allocates tasks jobs durng ther respectve schedulng nterval upon the occurrence of three possble events: a zero-laxty event (a job must mmedately execute lest t msses ts deadlne), job completon or the arrval of an aperodc task. Durng the ISD k nterval task τ k must execute x k cycles at a gven F n clock frequency. Prorty Levels Whenever an event occurs, the task prorty s updated as follows. Jobs reachng ther zero-laxty tme are gven the maxmum prorty (= 1). Jobs beng executed and wth laxty dfferent from zero receve prorty equal to 2. The remanng jobs receve prorty level equal to 3 (the lowest one). Thus, zero laxty tasks have the hghest prorty and must be executed mmedately. Algorthm 1 On-lne schedule 1: Input ISD k Schedulnng ntervals; Xk tasks CP U cycles per nterval; ex k executon CP U cycles per nterval 2: Output A feasble schedule; k =, ζ = 3: Compute the ordered set of laxtes as: SL = {l l = sd k+1 (F n x k exk ) ζ} 4: whle ζ H do 5: whle ζ sd k+1 do 6: Compute task prortes usng Prorty Levels 7: Execute the m tasks wth hgher prorty untl an event occurs (An event occurs f a task reaches ts zero laxty, task ends or aperodc task arrves.) 8: Compute the ordered set of laxtes as: SL = {l l = sd k+1 (F n x k execk ) ζ} 9: ζ = ζ+ current tme 1: end whle 11: k = k : end whle Executon of m tasks wth the hghest prorty In Alg. 1 step 8, m tasks must be executed (.e. allocated to a CPU). In order to reduce the number of mgratons, tasks that are executed durng two consecutve events are allocated to the same CPU. 7. APERIODIC TASKS Aperodc tasks arrve asynchronously to the system. The system determnes f these tasks can be executed wthout compromsng the hard real-tme constrants of the perodc task set. If so, a new CP U clock frequency s computed to allow the executon of the aperodc task. Ths frequency must be n the range F s = {F... F + } (every frequency n ths range meets the thermal constrants), and t s kept as low as possble to guarantee a mnmum energy consumpton whle meetng the temporal constrants. Fg. 2 shows the scheme to schedule aperodc tasks wthout compromsng the hard real-tme constrants of the perodc task set. The off-lne stage computes the optmal and maxmum allowed frequences F, F +, the schedulng ntervals ISD k and the perodc task CP U cycles that must be executed per schedulng nterval X = {X 1,, X α }, where X k = {x k 1,..., x k n} represents the set of CP U cycles x k that must be executed durng the k th schedulng nterval of task τ. Job executon s tracked durng the k th nterval by the system and passed as the nput to the adaptve scheduler (AS). AS s actvated at the arrval or endng tme of an aperodc task τ a. At ths tme, the task s CP U cycles cc a and ts relatve deadlne d a are sent to AS. Ths nformaton together wth the outputs of the offlne stage are used to compute the new frequency and the CP U cycles (x k τ ) requred to execute τ a a per schedulng nterval, such that X k = X k {x k τ }. When τ a a fnshes ts executon, the frequency s recalculated and the CP U cycles demanded by τ a are dscarded. Complexty The complexty of the On-lne stage depends on two algorthms. The prorty level and the computaton of laxty n Alg. 1 s lnear n the number of tasks. At most n = T tasks wll end ts executon x k n the k th nterval (there at most n tasks). Also at most n tasks wll reach ther zero laxty. If q aperodc tasks arrve n the k th nterval, then the nested whle loop ends n (n + n + q)

6 8. SIMULATIONS RESULTS Fg. 2. Mnmum Energy Thermal Aware scheme for perodc and aperodc tasks. (n + n) (number of events number of operatons). Consderng that the outer loop runs α = I SD tmes, then the number of steps of ths algorthm s polynomal n the order of tasks. Alg. 2 runs on the arrval of an aperodc task and s polynomal n the order of tasks and ndependent of the number of CPUs. Thus the proposed algorthm s polynomal n the order of tasks. Algorthm 2 Adaptve Scheduler (AS) 1: Input cc a, da Aperodc tasks parameters; exk Cycles executed n the system for all actve tasks. 2: Output New Frequency F n F s = {F,..., F + }, task CP U cycles per ISD k, xk τ a Per-nterval CPU cycles for executon of the aperodc task. 3: Intalze n = T, m = P, F n = F, q the aperodc tasks currently beng attended. BEGIN 4: f aperodc task arrves then 5: Let r a = current tme when τ a arrves; 6: Compute requred CP U cycles for actve tasks from r a to r a + da ; X k Γ X γ C u = (x k exk ) + x γ ; γ=k+1 where k s the current schedulng nterval at r a and and Γ s the schedulng nterval at r a + da ; 7: C free = m d a F + C u; the free CP U cycles n the nterval [r a, ra + da ]; 8: f C free cc a then 9: Accept task τ a; 1: F n = mnmum F F s such that F Cu+cca m d a ; cc r = cc a ; { X k x k τ a = mn m( ISD k ra )Fn (x k exk }; ), ccr n the k th nterval; for γ = k + 1 to Γ do - n other ntervals cc r = cc r x γ 1 τ a ; x γ τ a = mn {m( I γsd X γ )Fn x γ };, ccr end for 11: else 12: Reject task; 13: end f 14: end f 15: f an aperodc task fnshes then Dscard the CP U cycles assocated to the aperodc task Recalculate the new frequency; end f END In order to show how to use the proposed scheme, a proof of concept s presented. It conssts of a set of sporadc perodc tasks T = {τ 1, τ 2, τ 3 }, where τ 1 = (2, 4), τ 2 = (5, 8), τ 3 = (6, 12), the hyperperod s H = 24. These tasks run on two homogeneous mcroprocessors where the sotropc thermal propertes and dmensons of the materals are taken from Desrena-Lopez et al. (214). The processor supports four operatng frequency levels F = {.5,.85,.95, 1}KHz. The temperature of the surroundng ar s set to o C and t s constant. The maxmum operatng temperature level s set to T max1,2 = 5 o C. The smulatons heren presented consder CPUs wth caches and speculatve mechansms non-exstent or turned off. Frst, the mnmum frequency for the perodc task set s off-lne computed accordng to Eq. (6), obtanng Φ =.8125, hence the selected frequency s F =.85kHz. Eq. (7) provdes the maxmum clock frequency (F + = 1kHz), so that the MET ART S problem has a soluton. We assume that shcedulng and context swtch overheads are ncluded n the tasks W CET. Then, solvng the LPP n Eq. (8) for F yelds the CPU cycles of each task to be executed at each nterval (x k ). Fg. 3 provdes the schedule and temperature evoluton produced by the algorthm wthout consderng aperodc tasks. Fg. 4 depcts the outcome of the algorthm and the evoluton of the temperature when an aperodc task τ1 a = (2, 1) arrves at ζ = 2, durng the ISD 1 nterval, thus τ1 a has an absolute deadlne at ζ = 12. Snce C free cc a 1, the AS accepts the aperodc task, and computes F n = khz F s as the frequency at whch the processors must execute durng nterval [2, 12]. Fg. 4 shows that temperature ncreases durng ths nterval because of the executon of the tasks at frequency F n, and then t decreases after ζ = 12 because a new (lower) frequency has been calculated for the next nterval. In both experments, wth and wthout the aperodc tasks, CP U 1 acheves full utlzaton, whereas CP U 2 shows a slack (dle tme) at about ζ = 18, whch translates nto a temperature valley. Ths slack appears because the exact optmal frequency calculated n Eq. (6) s celed to a frequency belongng the dscrete set of frequences avalable n the mcroprocessor (F n F s ). 9. CONCLUSIONS Ths work shows that the TCPN formalsm s a sutable way to model real-tme task schedulng problems consderng thermal, temporal and energy restrctons. Upon a TCPN model, we buld a two-stage thermal-aware realtme system n whch a perodc task set executes at mnmum clock frequency to save energy and acheve maxmum CPU utlzaton whle honorng thermal constrants. Aperodc tasks are also dynamcally accepted as long as ncreasng the clock frequency allows to obtan a sutable slack. Tme and thermal constrants are preserved n all cases. We have assumed that the aperodc task (τ a ) fts n the slack nsde the hyperperod (r a + da H). Immedate further work ncludes to relax that condton, and to mprove our slack reclamng approach to decrease context swtchng when acceptng aperodc tasks. Also, the fact that our underlyng TCPN model allows modelng

7 T 1 CPU 1 T 2 CPU 2 τ 1 τ 2 τ 3 Temperature tme (sec) Fg. 3. Temperature evoluton (upper plot) for the perodc schedule (lower plot) at CP U 1 (above) and CP U 2 (below). The maxmum temperature produced by ths schedule s T CP U1,2 = 46.5 o C. T 1 CPU 1 T 2 CPU 2 48 τ 1 τ 2 τ 3 τ a 1 Temperature 48 tme (sec) Fg. 4. Temperature evoluton (upper plot) for the perodc schedule (lower plot) at CP U 1 (above) and CP U 2 (below) upon acceptance of the aperodc task τ1 a. The maxmum temperature produced by ths schedule s T CP U1,2 =.6 o C resource sharng, a tough problem n multcore real-tme schedulng, opens up a promsng venue. Last, we stll have to measure how much energy we can save wth respect to other schedulng technques for aperodc tasks. ACKNOWLEDGEMENTS Ths work was partally supported by grants TIN C2-1-R (AEI/FEDER, UE), gaz: T48 research group (Aragón Gov. and European ESF), and HPEAC4 (European H22/687698) REFERENCES Ahmed, R., Ramanathan, P., and Saluja, K.K. (216). Necessary and suffcent condtons for thermal schedulablty of perodc real-tme tasks under flud schedulng model. ACM Transactons on Embedded Computng Systems (TECS), 15(3), 49. Baruah, S., Bertogna, M., and Butazzo, G. (215). Multprocessor Schedulng for Real-Tme Systems. Sprnger- Verlag New York, Inc., Secaucus, NJ, USA. Baruah, S.K., Cohen, N.K., Plaxton, C.G., and Varvel, D.A. (1996). Proportonate progress: A noton of farness n resource allocaton. Algorthmca, 15(6), Davd, R. and Alla, H. (28). Dscrete, contnuous and hybrd Petr nets (davd, r. and alla, h.; 24). Control Systems, IEEE, 28(3), Davs, R.I. and Burns, A. (211a). Fpzl schedulablty analyss. In Proceedngs of the th IEEE Real- Tme and Embedded Technology and Applcatons Symposum, RTAS 11, IEEE Computer Socety, Washngton, DC, USA. Davs, R.I. and Burns, A. (211b). A survey of hard real-tme schedulng for multprocessor systems. ACM computng surveys (CSUR), 43(4), 35. Desrena-Lopez, G., Brz, J.L., Vázquez, C.R., Ramírez- Trevño, A., and Gómez-Gutérrez, D. (216). On-lne schedulng n multprocessor systems based on contnuous control usng tmed contnuous petr nets. In 13th Internatonal Workshop on Dscrete Event Systems, Desrena-Lopez, G., Vázquez, C.R., Ramírez-Trevño, A., and Gómez-Gutérrez, D. (214). Thermal modellng for temperature control n MPSoC s usng flud Petr nets. In IEEE Conference on Control Applcatons part of Mult-conference on Systems and Control. Funk, S., Levn, G., Sadowsk, C., Pye, I., and Brandt, S. (211). Dp-far: a unfyng theory for optmal hard realtme multprocessor schedulng. Real-Tme Systems, (5), Hettarachch, P.M., Fsher, N., Ahmed, M., Wang, L.Y., Wang, S., and Sh, W. (214). A desgn and analyss framework for thermal-reslent hard real-tme systems. Embedded Computng Systems, ACM Transactons on, 13(5s), 146:1 146:25. Kong, J., Chung, S.W., and Skadron, K. (214). Recent thermal management technques for mcroprocessors. ACM Computng Surveys, 44(3), 13:1 13:42. Schor, L., Bacvarov, I., Yang, H., and Thele, L. (212). Worst-case temperature guarantees for real-tme applcatons on mult-core systems. In 212 IEEE 18th Real Tme and Embedded Technology and Applcatons Symposum, IEEE. Serksma, G. (21). Lnear and nteger programmng: theory and practce. CRC Press. Slva, M., Júlvez, J., Mahulea, C., and Vázquez, C.R. (211). On fludzaton of dscrete event models: observaton and control of contnuous Petr nets. Dscrete Event Dynamc Systems, 21(4)(3), Slva, M. and Recalde, L. (27). Redes de Petr contnuas: Expresvdad, análss y control de una clase de sstemas lneales conmutados. Revsta Iberoamercana de Automátca e nformátca Industral, 4(3), 5 33.