Energy-Aware Algorithms for Tasks and Bandwidth Co-Allocation under Real-Time and Redundancy Constraints

Transcription

1 Proceedngs of the 17th IEEE Internatonal Conference on Emergng Technology and Factory Automaton (ETFA 2012) Energy-Aware Algorthms for Tasks and Bandwdth Co-Allocaton under Real-Tme and Redundancy Constrants Francesco Prosper, Maro Bambagn, Gorgo Buttazzo, Mauro Marnon, Ganluca Franchno Scuola Superore Sant Anna, Psa, Italy Abstract The energy consumpton n dstrbuted systems depends on several nter-related factors, ncludng task parttonng, process redundancy, fault tolerance, task and message schedulng, and communcaton bandwdth allocaton. Although some of these ssues have been consdered n the lterature n solaton, a systematc approach consderng all the constrants s stll mssng. Ths paper addresses the problem of allocatng a task set and the requred communcaton bandwdth on a dstrbuted embedded system, amng at reducng energy consumpton whle guaranteeng tmng and redundancy constrants. Two heurstc approaches are proposed and compared aganst a complete method and smulated annealng. Smulaton results show the effectveness of the proposed approaches. I. INTRODUCTION Today, a large number of embedded systems conssts of a set of computng nodes nterconnected va wred or wreless networks. In such systems, both node responsveness and energy consumpton are affected by the polces used to allocate and schedule tasks on the varous nodes, schedule messages over the network, and manage the other avalable resources, such as the I/O devces. In several stuatons (e.g., wreless sensor networks) redundancy s requred to guarantee hgh qualty and relablty n events detecton. In fact, sensor nodes are typcally affected by hgh falure rates, so that a certan level of replcaton s requred n the system to ncrease relablty n the measurements. Another context where redundancy ssues are relevant s an assembly lne consstng of many robots n charge of performng and supervsng the varous producton stages, where each robot may nclude dfferent sensors, actuators and control unts that need to exchange data and sgnals. Snce a fault may jeopardze the qualty of products, sensor redundancy would ncrease relablty and allow earler falure detecton and avodance. Another beneft that comes from redundancy s the possblty of ncreasng measurement accuracy, where multple observatons from dfferent nodes are ntegrated to produce more relable aggregated values, used to reduce statstcal uncertanty n the measurements. hle redundancy mproves relablty and precson, t ncreases energy consumpton due to CPU utlzaton, network communcaton, etc. The problem s complcated by the fact that there are stuatons n whch reducng the energy consumpton on a node may ncrease the energy dsspated n other nodes, so shortenng the lfe of the entre system. A way to address ths ssue s to consder the problem of co-schedulng tasks and messages wth the objectve of mnmzng energy consumpton whle meetng tmng constrants. Contrbutons: Ths paper addresses the problem of parttonng tasks and communcaton messages on a dstrbuted embedded system wth the objectve of guaranteeng tmng constrants whle mnmzng the overall energy consumpton and maxmzng task duplcaton. A trade-off between such two conflctng objectves s explored, ensurng task real- requrements and guaranteeng a mnmum number of task copes and system lfe. Bandwdth and communcaton buffer constrants are also taken nto account. Two heurstc approaches are presented and evaluated wth respect to smulated annealng and a complete method based on branch and bound search. Organzaton of the paper: Secton II presents the system model, n terms of processng nodes, communcaton bandwdth, power model, computatonal and communcaton workload. The problem statement s ntroduced n Secton III, ncludng the specfcaton of all the system constrants and the performance metrc used to evaluate the goodness of an allocaton. Secton IV llustrates the proposed approaches, whereas Secton V reports the expermental results. Fnally, Secton VI ends the paper wth the concludng remarks. A. Related work Although a lot of research has been focused on schedulng on networked/multcore embedded systems [5], lttle work has been done to optmze allocaton takng multple objectves nto account, such as real- constrants for tasks and messages, energy consumpton, and redundancy ssues. Km et al. [11] consdered the problem of mappng mulda applcatons onto a mult-processor system-onchp (MPSoC) and proposed a framework for desgn space exploraton based on heterogeneous schedulng polces and DPM technques. Schranzhofer et al. [19] addressed a smlar problem proposng a dynamc mappng strategy able to mnmze energy consumpton to prolong the system lfe for heterogeneous processng unts. 1

2 Aydn et al. [2] proposed several methods for reducng the overall energy consumpton explotng frequency scalng n parttonng-based systems where EDF s the scheduler used for each node. More precsely, they modfed and compared several well-known heurstcs to consder the energy consumpton nto the performance metrc. Kandhalu et al. [8] proposed an approach for reducng the energy dsspaton n mult-core platforms wth a sngle voltage/frequency doman usng a parttoned fxed-prorty scheduler. They showed that balancng the workload s crucal as the core wth the hghest load s lkely to mpose the common frequency. Other parttoned approaches take nto account precedence dependences among tasks [20], [15]. The major dffculty n ths problem s to take precedence relatons nto account n the schedulablty analyss. Buttazzo et al. [4] approached ths ssue by assgnng ntermedate deadlnes to tasks accordng to the dependency graph, and then schedulng them accordng to EDF. In ths context, other authors [9], [18] appled genetc algorthms to allocate tasks and generate a statc schedule usng DVFS technques to reduce energy consumpton. Huang et al. [7] addressed the problem of allocatng and schedulng dependent tasks on a set of heterogeneous cores also consderng communcaton costs. Three solutons were compared to solve the problem, one based on Integer Lnear Programmng (ILP) and two based on smulated annealng. Concernng task redundancy, tmng and precedence constrants, Ramamrtham [17] proposed an approach that guarantees the allocaton of a gven number of subtask copes, where each task s descrbed by a precedence graph. Auluck and Agrawal [1] proposed a method for allocatng and schedulng non-preemptve tasks on a heterogeneous system to guarantee the executon of a real- task set n case of a processor fault. Once the task graph has been allocated, the algorthm keeps allocatng selected task duplcates untl all the processors are utlzed. Qn et al. [16] proposed a fault-tolerant algorthm to handle a sngle processor falure n a fully connected network of heterogeneous nodes runnng real- dependent tasks. In ther model, falure condtons are supposed to be detected after a fxed perod. Other researches focused on networked embedded systems. Xe et al. [21] consdered a sngle-hop network composed by heterogeneous devces wthout DVS capablty and provded a task allocaton method amed at reducng the latency and the energy consumpton of the system, takng nto account both computaton and communcaton actvtes. Km et al. [10] proposed a method to dynamcally assgn tasks n a sngle-hop wreless network of moble nodes wth DVS mechansm, guaranteeng constrants and a mnmum lfe of eght hours. To ths end, the authors proposed and compared a set of dynamc mappng heurstcs. Xue et al. [22] consdered a network-wde dynamc energy management mechansm for a system composed by wreless nodes wth DVS capablty, executng real- tasks wth precedence constrants. The proposed method assgns tasks and DVS levels to each node, takng nto account both rado sleep ntervals and the applcaton parallelsm. Grd computng systems wth moble nodes have been consdered by Chunln and Layuan [13], who proposed a method to guarantee both lfe and constrants of grd applcatons. The authors modeled the problem as a dstrbuted energy constraned resources allocaton, and proposed a prce-based algorthm to provde an optmal soluton. None of the papers exstng n the lterature, however, addressed all the constrants consdered n ths work, hence they cannot be compared wth the proposed approaches. II. SYSTEM MODEL Ths work consders a dstrbuted system of m homogeneous nodes Ψ 1,...,Ψ m logcally connected as llustrated n Fgure 1. Each node performs sensory acquston and sends messages to a coordnator node (Ψ 0 ), accordng to a star topology, usng a TDMA-based communcaton protocol that guarantees a bounded transmsson delay. In each TDMA wheel, each node Ψ s assgned a bandwdth slot of lengthw. Note that, thecoordnatorψ 0 s not subject to the analyss carred out n ths work, snce t s assumed to be always on and avalable to the nodes. Snce the system s ntended as a network of nodes, throughout the rest of the paper, the terms system and network are used nterchangeably. Fgure 1. Ψ 0 w 1,Γ 1 w 2,Γ 2 w 3,Γ 3 wm,γm... Ψ 1 Ψ 2 Ψ 3 Ψ m A sample network wth a node coordnator. The system s assumed to run an applcaton Γ consstng of a set of n sporadc real- tasks Γ = {τ 1,...,τ n }. Each task τ j s characterzed by a worst-case executon C j, a mnmum nter-arrval T j, and a relatve deadlne D j (assumed to be equal to T j ). Each task generates an nfnte sequence of jobs, τ j,k, each havng release r j,k and absolute deadlne d j,k = r j,k + D j. Each job τ j,k sends a message of payloadm j to node Ψ 0 at ts termnaton, whch can occur any n the nterval (r j,k,r j,k + D j ]. The overhead of transferrng a message generated by a task nto the transmsson buffer s accounted n C j. The utlzaton of task τ j s denoted by u j and s computed as C j /T j. Each node Ψ hosts a subset Γ of Γ. The utlzaton of node Ψ and the system utlzaton are denoted as U and

3 U tot, respectvely. Note that a task τ j can run n more than one node, but each node can execute at most one copy of τ j. The number of runnng nstances of task τ j on the entre network s denoted as µ j. For each task τ j, the applcaton specfes a reward functon denoted as γ j, whch grows wth the number of task nstances and, hence, measures the satsfacton of the task redundancy across the system. Tasks n each node are scheduled by the Earlest Deadlne Frst (EDF) [14] algorthm, but the analyss can easly be extended to dfferent schedulng polces. The tmng parameters ntroduced above are summarzed n Fgure 2. r j C j A. Bandwdth model Fgure 2. T j w M j Tmng parameters. d j The communcaton between the nodes and the coordnator s managed by a TDMA-based communcaton protocol, accordng to a star topology. Each node Ψ s assgned a slot w n each TDMA wheel where t can transmt ts messages. The messages produced by the tasks runnng n node Ψ are enqueued n a transmsson buffer of length Θ adoptng a FIFO polcy and then sent as soon as the bandwdth becomes avalable. A necessary condton to guarantee that each node can transmt ts messages wthout overflowng the transmsson buffer requres the wheel length to be no shorter than the mnmum perod n the task set, hence s set as = mn j [1..n] T j. (1) Each task τ j of node Ψ s assgned a budget Q j suffcent to ensure the correct delvery of the related message wthn 2T j from the assocated job release r j,k. Then, w s set as the sum of the budgets Q j s of the tasks allocated to node Ψ : w = τ j Γ Q j. (2) The task budget Q j s computed as follows: Tj Q j = M j. (3) Tj Observe that represents the number of complete TDMA wheels avalable durng a task perod T j,.e., the number of slots that the node runnng τ j can explot to transmt the message generated by such a task. Snce M j s the length of any message generated by τ j, the ratonale behnd the computaton of Q j s to assgn a budget suffcent to send a message of length M j every T j unts. If w com denotes the sum of the slots w assgned to the nodes, the overall communcaton load on the network s w com /. Snce for each node Ψ, the slot w s computed n such a way the node can send all messages generated by ts tasks, t follows that a necessary and suffcent condton to guarantee all message deadlnes s: w com. (4) To save energy, the adopted bandwdth scheme reserves a slot S = w com over each n whch the communcaton s not allowed and all the nodes turn ther transcever off. Consder a sample scenaro featurng three nodes (Ψ 1,Ψ 2,Ψ 3 ) and a task set composed by two tasks (τ 1, τ 2 ), where node Ψ 1 hosts both the tasks, node Ψ 2 hosts τ 1, and node Ψ 3 hosts τ 2. The resultng TDMA wheel partton s llustrated n Fgure 3. Note that, throughout the paper, the slot lengths w are ordered by the node ndexes nsde the wheel, whle the budgets Q j nsde the slot are ordered by the task ndexes. w com w 1 w 2 w 3 Q 1 Q 2 Q 1 Q 2 Fgure 3. B. Power Model Bandwdth allocaton example. S Snce ths paper focuses on a network-wde approach for reducng energy consumpton, a smplfed power model s adopted for the nodes to estmate ther power consumpton. A fner model would be unnecessarly detaled n ths context, consderng that each node could apply further run strateges to save energy. In partcular, each devce (.e., CPU or transcever) s assumed to be n one of the followng states: actve. In ths state, the devce performs ts job executng tasks or handlng messages. The power consumed n ths state s denoted by Pa CPU and Pa com, respectvely. sleep. In ths state, the devce s completely turned off and consumes the least amount of power, denoted as Ps CPU and Ps com, respectvely.

4 The overhead to swtch between power states for both the devces s assumed to be neglgble, both n terms of and energy. The node power consumpton s estmated as follows: P = U Pa CPU +(1 U )Ps CPU + w ( Pcom a + 1 w ) Ps com. (5) The node and system mean power consumpton s estmated assumng that each node explots the dle ntervals n task and message schedules turnng the CPU and the transcever off durng dle perods and outsde bandwdth slots, respectvely, as follows: P = U tot Pa CPU w com Pcom a + +(m U tot )P CPU s + (m wcom ) P com s. (6) The current energy level of node Ψ s denoted by E and every node s provded wth the same ntal energy E (0). C. Lfe model The system lfe s a key element for evaluatng sensor networks. Its defnton s not unvocal because t depends from a lot of applcaton parameters (e.g., number of sensors, network coverage, qualty of servce, etc) and ts value s an aggregaton of all nodes statuses. The network can only fulfll ts purpose as long as t s consdered alve, but not after that. Therefore, the lfe s an ndcator for the maxmum utlty a sensor network can provde. There are a lot of lfe defntons [6], but the most common and most frequently used n the lterature s the m- of-m lfe. In ths defnton, the dstrbuted system lfe L ends as soon as the frst node fals, thus L = mnl, wth L representng the lfe of node Ψ computed as E /P. A common constrant on the lfe mposes a mnmum amount of (L) n whch the system has to stay alve. The mnmum lfe could be formalzed as: mn E P > L. D. Allocaton specfc parameters An allocaton descrbes the dstrbuton of the tasks nstances among the nodes of the system. Snce a task τ j can run on dfferent nodes smultaneously, an allocaton matrx A {0,1} m n s defned to keep track of such a dstrbuton on the network. The generc element a j of the allocaton matrx A s a boolean varable ndcatng whether task τ j s present on node Ψ or not. Note that each node can execute at most one copy of τ j. In the followng, a set of prevously defned parameters are formalzed based on A. The subset Γ of tasks runnng on node Ψ s expressed as Γ = {τ j a j = 1}, (7) whle the length w of the communcaton bandwdth slot assgned to the node Ψ s computed as n w = a j Q j. (8) j=1 The total utlzaton U of node Ψ and the system utlzaton U tot are formally defned as n U = a j u j (9) and U tot = j=1 m U. (10) =1 The actual number µ j of nstances of task τ j across the system s computed as m µ j = a j. (11) =1 The worst case scenaro for the transcever buffer occurs when two nstances of the same task generate a message between the end of the prevous Q j and the start of the next one, as depcted n Fgure 4. In the example llustrated n the fgure, the Q j s supposed to occur at the begnnng of each wheel. More than two generatons are not possble because the wheel s equal to the mnmum task perod. Ths leads to consder that the mnmum requred sze Θ mn of the transcever nternal buffer on node Ψ s computed as: buffer CPU comm. slots 2M j M j T j Fgure 4. Θ mn = 2 T j m a j M j. j=1 τ j,1 τ j,2 τ j,3 T j Q j,1 Q j,2 Example of worst-case buffer usage. III. PROBLEM STATEMENT Ths paper addresses the problem of allocatng a real- task set over a dstrbuted system composed by homogeneous embedded nodes. Such an allocaton has to meet a set of constrants:

5 Task schedulablty: All the tasks runnng n each node have to termnate wthn ther deadlnes. Under the assumpton that all the tasks have relatve deadlnes equal to perods and are scheduled by EDF, such a constrant s expressed as follows: [1,m] U 1. (12) Bandwdth: To guarantee the schedulablty of the messages produced by the tasks, the sum w com of all the slots assgned to the nodes must not exceed the wheel length : w com. (13) Buffer: For each node, the transmsson buffer (whose length Θ s equal for each node) must be long enough to contan all the messages generated n the worst case, that s: [1,m] Θ Θ mn. (14) Redundancy: To mprove the accuracy level of measurements, a mnmum number µ mn j of runnng nstances for each task τ j must be guaranteed to be executed n the system, that s: j [1,n] µ j µ mn j. (15) Lfe: To guarantee a desred system lfe L, the ntal energy E (0) avalable n each node must be suffcent to keep the entre network alve for the requred duraton, that s: E (0) [1,m] L. (16) P A. Performance evaluaton To evaluate the performance of a generc allocaton, three dstnct normalzed ndexes ( [0, 1]) are ntroduced to measure redundancy, energy savng, and unformty. Such factors are combned to obtan a normalzed scalar functon able to compare the goodness of dfferent allocatons. The performance functon s modeled to operate also n the case of unfeasble allocatons. Such a functon s bult so the co-doman range of the performance functon s [ 3, 1], equally dstrbuted between the feasble stuatons ([0, 1]) and the unfeasble ones ([ 3, 0)). As already explaned n Secton II, more nstances of each task can execute on the network. Note that an mplct lmt on the runnng nstances of a task τ j exsts as at most m nodes can run a sngle nstance of τ j. For each task τ j, a monotonc functon γ j s provded by the applcaton desgner to specfy the reward accordng to the actual number of task nstances runnng n the system, from µ mn j to m. γ j s a functon wth bounded output value n [0,1). The redundancy ndex ρ s defned as n j=1 ρ γ j. n The overall energy consumpton of the system s a key parameter taken nto account by the analyss carred out n ths work. The ndex ξ measures the energy savng of an allocaton wth respect to the hghest power consumpton P max that the system can experence, that s: ξ max(0,p max P). P max Notce that P max refers to an deal allocaton n whch all the nodes are fully loaded (.e. all nodes have utlzaton equal to 1) and the bandwdth slots are assgned n such a way that w com s equal to. Based on the defnton of system lfe assumed n ths work, how task nstances are spread across the system nodes s of key mportance, as a more unform load dstrbuton results n a longer global lfe (due to the proportonalty between utlzaton and power consumpton at node level). The utlzaton unformty of the allocaton s evaluated through the varance σ 2 of the total utlzaton U of each node Ψ computed as follows: σ 2 1 m ( U U ) 2 tot. m m =1 As for the prevous ndexes, the unformty of the allocaton s evaluated through an ndex α normalzed n the nterval [0, 1]: α max(0,σ2 max σ2 ) σmax 2, where σmax 2 denotes the varance n the worst-case scenaro n whch half nodes are fully loaded and half are empty, leadng to a value of σ 2 equal to Consderng the trade-off between redundancy and power consumpton, a parameter η [0, 1] s ntroduced to balance the two contrbutons. Notce that such a parameter remans constant throughout all the optmzaton process. For a feasble allocaton, the performance functon Φ s defned as follows: Φ + η (A) = α[ηρ+(1 η)ξ], (17) whereas, for an unfeasble allocaton, t s defned as follows: Φ (A) = 1 n max(0,µ mn j µ j ) + n j µ mn j m max(0,u 1) + (U tot 1)m max(0,wcom ) m n j Q. (18) j Such a functon s composed by three terms, each ntroduced to wegh the volaton of one of the evaluated constrants. Note that whle the second term s defned at a node level, snce t concerns CPU overrun, the last term operates at a system level, as the communcaton wheel s shared among all the nodes.

6 IV. PROPOSED ALGORITHMS Ths secton presents three types of algorthms that try to maxmze a user-defned performance functon under the constrants formalzed n Secton III. The cost of each approach s measured through the number of performance evaluatons, rather than through the total processng, snce the executon of an algorthm s affected by several factors (e.g., processor speed, memory, and mplementaton effcency). The complexty to evaluate a specfc allocaton s O(nm). A. Complete Search A branch and bound search has been mplemented to evaluate the dstance of the optmal soluton wth respect to the other methods, at least for a small system sze. The exhaustve search uses an EDF feasblty test on sngle nodes to prune unfeasble branches. Note that, at low total utlzatons, the number of feasble (sngle node) allocatons s exponental on the number of tasks, snce very low tasks utlzatons make any task combnaton feasble, and drops down at hgher utlzatons. Snce the embedded nodes composng the system are homogeneous, several solutons are symmetrc, thus leadng to the same performance. To avod the generaton of symmetrc allocatons, ths approach combnes the solutons prevously found to reduce the number of possble confguratons. B. Heurstcs Ths secton presents two heurstc approaches, Heurstc A and Heurstc B. Heurstc A starts from an empty allocaton and, at each step, generates all the possble confguratons that dffer from the current one by a sngle task (by swtchng a sngle element of matrx A from 0 to 1). Then, the confguraton wth the hghest performance s selected. In the case of multple confguratons featurng the hghest performance, the frst occurrence s selected. The algorthm stops as soon as t fals to mprove the current best performance. Snce at each step the approach performs at most n m evaluatons, ts worst-case complexty s O(n 3 m 3 ), as t s possble to nsert at mostn m nstances to fll theamatrx. Heurstc B s a varaton of Heurstc A, snce at each step t selects the node wth the lowest utlzaton, testng the resultng performance at each addton. Snce at each step the approach has to evaluate n dfferent confguratons, the worst-case complexty s O(n 3 m 2 ). Lke Heurstc A, the approach stops as soon as t fals to mprove the current best performance. C. Smulated Annealng Smulated Annealng [12] s an effectve technque for fndng an acceptably good soluton n a fxed amount of, even n large search spaces. In ths approach, a generc pont s of the search space s called a state, and the neghborhood of a state s s defned as the set of states produced from s by sutable alteratons. In ths mplementaton, a state s represented by a global allocaton, and ts neghborhood s the set of allocatons that dffer from the startng one only for one boolean element of the matrx A. Ths means that the neghborhood of a state s another allocaton n whch a sngle task s added (f t was not present) or removed (f t was present). The other parameters, that s the coolng factor, the maxmum number of tres at fxed temperature, and the temperature threshold, are selected to mprove performance. In ths work, the smulated annealng has been mplemented accordng to two approaches, whch dffer for ther ntal state. The frst approach starts from the empty allocaton matrx and t s referred to as Basc Smulated Annealng (BSA), whle the second one les upon the result of Heurstc A and, hence, t s referred to as Heurstc Smulated Annealng (HSA). V. EXPERIMENTAL RESULTS Ths secton presents a set of expermental results of the approaches proposed n Secton IV. Such results are obtaned by smulaton on synthetc workloads adoptng the power profle taken by the Mcrochp dspic 1 datasheet, nterpolatng the typcal consumptons. To test the behavor of the varous approaches, a C smulator has been developed. An executon scenaro s characterzed by the tuple (n, m, U, B), where n denotes the number of tasks, m the number of nodes n the network, U the total task set utlzaton, and B the total communcaton bandwdth requred by the task set (B = n j=1 M j/t j ). Gven a total utlzaton factor (U and B), the task set features (C j, T j and M j ) are computed accordng to a unform dstrbuton [3]. From such values, the wheel length and task bandwdth slots (Q j ) are computed accordng to Equaton (1) and Equaton (3). As already exposed n Secton III-A, the applcaton desgner has to specfy the reward functon γ j for each task τ j. If µ sat j denotes the number of runnng nstances of τ j beyond whch the measures have no gan on accuracy, n ths set of experments γ j s set as where γ j = ( 1 e j), j ) µ sat j µ mn j j = 5(µ j µ mn For all the experments, µ sat j s set equal to the number of nodes m. The balancng parameter η between redundancy and energy savng s chosen to be 0.5. Due to the complexty of consderng all the parameters ntroduced n the analyss, the set of experments descrbed n ths secton do not check the volaton of the constrants 1 dspic33fj256mc710 mcrocontroller.

7 on the mnmum buffer and the mnmum lfe formalzed n Equaton (14) and Equaton (16), respectvely. The frst experment evaluates the complexty of the algorthms as a functon of the product n m, for all the approaches, settng U = m/2 and B = 0.3. Table I reports the number of performance evaluatons averaged on 50 runs of each approach. As descrbed n Secton IV-A, the Complete Search (CS) approach represents an effectve method to fnd the optmal soluton only for small problem sze, as the number of evaluatons becomes too hgh already at n m = 10x6. The number of evaluatons performed by each approach reflects the complexty values reported n Secton IV. Smulated annealng technques have a hgher number of evaluatons snce they explore the search space untl the system s cooled but wth a condtonal stop crteron on the stablty of the result. The performance dfference between CS and the other approaches s under 5%. n m CS Heu A Heu B BSA HSA 5x x x x x Table I PROBLEM COMPLEXITY. The second experment evaluates the performance ndex Φ as a functon of the task set utlzaton U, and results are shown n Fgure 5. All the approaches have been tested wth n = 25, m = 10 and B = 0.3. The utlzaton s normalzed on the number of nodes m. analyzng the three components of the performance ndex as a functon of the utlzaton U, normalzed on the number of nodes m α ρ ξ U/m Fgure 6. Performance components. The results are reported n Fgure 6, whch clearly shows that the α factor s about constant, revealng that the solutons found by Heurstc B are characterzed by unform allocatons. As expected, both the redundancy and energy savng ndexes decrease wth the growng of the normalzed CPU load. Ths occurs because tasks wth hgher utlzaton are dffcult to spread across the network wthout volatng the schedulablty constrant and causng a sensble ncrease of power consumpton. The fourth and last experment analyzes Heurstc B as a functon of both n and m, for a bandwdth utlzaton B = 0.3. The values of n range n {5,10,15,20,25}, whle the values of m range n {4, 6, 8, 10, 12}. In ths experment, the utlzaton U s always set to m/ Φ Heurstc B HSA Heurstc A BSA U/m Fgure 5. Performance analyss of the approaches. Φ # of nodes Fgure # of tasks Performance analyss of Heurstc B when B= As Fgure 5 shows, Heurstc B outperforms all the approaches. BSA performs poorly at utlzatons hgher than 0.5. Heurstc A s often able to fnd a reasonable soluton, although t can be mproved by HSA, whose performance s very smlar to that of Heurstc B. The thrd experment deeply nvestgates Heurstc B, As reported n Fgure 7, the performance obtaned by Heurstc B mproves as the number of tasks ncreases. Ths s due to the hgher fragmentaton of the task set, that results n a wder feasble soluton space. Instead, the performance ndex decreases as the number of nodes grows. Snce the utlzaton U s always set to the half of m, the tasks result

8 n a hgher utlzaton. As for the thrd experment, the approach s able to spread only a few tasks nstances across the network payng a hgher power consumpton. VI. CONCLUSIONS Ths paper addressed the problem of allocatng a set of real- tasks on a dstrbuted system consstng of sensors nodes sendng data to a coordnator, wth the objectve of satsfyng feasblty constrants, whle mnmzng the overall energy consumpton and maxmzng task duplcaton. Trade-offs between such two conflctng goals have been explored, by comparng two heurstc approaches aganst a complete method and smulated annealng. Smulaton results show that one of the heurstc algorthms provdes a reasonable soluton, outperformng smulated annealng both n terms of effcency and performance, especally for larger task set utlzatons. REFERENCES [1] N. Auluck and D. P. Agrawal. A scalable task duplcaton based algorthm for mprovng the schedulablty of real- heterogeneous multprocessor systems. In ICPP orkshops, pages 89 96, [2] H. Aydn and Q. Yang. 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