Offline and Online Scheduling of Concurrent Bags-of-Tasks on Heterogeneous Platforms

Transcription

1 Offline and Online Schedling of Concrrent Bags-of-Tasks on Heterogeneos Platforms Anne Benoit, Loris Marchal, Jean-François Pinea, Yves Robert, Frédéric Vivien To cite this version: Anne Benoit, Loris Marchal, Jean-François Pinea, Yves Robert, Frédéric Vivien. Offline and Online Schedling of Concrrent Bags-of-Tasks on Heterogeneos Platforms. [Research Report] 2007, pp.49. <inria v1> HAL Id: inria Sbmitted on 20 Dec 2007 (v1), last revised 20 Dec 2007 (v2) HAL is a mlti-disciplinary open access archive for the deposit and dissemination of scientific research docments, whether they are pblished or not. The docments may come from teaching and research instittions in France or abroad, or from pblic or private research centers. L archive overte plridisciplinaire HAL, est destinée a dépôt et à la diffsion de docments scientifiqes de nivea recherche, pbliés o non, émanant des établissements d enseignement et de recherche français o étrangers, des laboratoires pblics o privés.

2 INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE Offline and Online Schedling of Concrrent Bags-of-Tasks on Heterogeneos Platforms Anne Benoit Loris Marchal Jean-François Pinea Yves Robert Frédéric Vivien N???? December 2007 Thème NUM apport de recherche ISSN ISRN INRIA/RR--????--FR+ENG

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4 Offline and Online Schedling of Concrrent Bags-of-Tasks on Heterogeneos Platforms Anne Benoit, Loris Marchal, Jean-François Pinea, Yves Robert, Frédéric Vivien Thème NUM Systèmes nmériqes Projet GRAAL Rapport de recherche n???? December pages Abstract: Schedling problems are already difficlt on traditional parallel machines. They become extremely challenging on heterogeneos clsters, even when embarrassingly parallel applications are considered. In this paper we deal with the problem of schedling mltiple applications, made of collections of independent and identical tasks, on a heterogeneos master-worker platform. The applications are sbmitted online, which means that there is no a priori (static) knowledge of the workload distribtion at the beginning of the exection. The objective is to minimize the maximm stretch, i.e. the maximm ratio between the actal time an application has spent in the system and the time this application wold have spent if exected alone. On the theoretical side, we design an optimal algorithm for the offline version of the problem (when all release dates and application characteristics are known beforehand). We also introdce several heristics for the general case of online applications. On the practical side, we have condcted extensive simlations and MPI experiments, showing that we are able to deal with very large problem instances in a few seconds. Also, the soltion that we compte totally otperforms classical heristics from the literatre, thereby flly assessing the seflness of or approach. Key-words: Heterogeneos master-worker platform, online schedling, mltiple applications. This text is also available as a research report of the Laboratoire de l Informatiqe d Parallélisme Unité de recherche INRIA Rhône-Alpes 655, avene de l Erope, Montbonnot Saint Ismier (France) Téléphone : Télécopie

5 Ordonnancement hors-ligne et en-ligne d applications concrrentes de types sacs de tâches sr plates-formes hétérogènes Résmé : Les problèmes liés à l ordonnancement de tâches sont déjà difficiles sr des machines traditionnelles. Ils deviennent encore pls inextricables sr des machines hétérogènes, même lorsqe les applications considérées sont facilement parallélisables (de type tâches indépendantes). Nos nos intéressons ici à l ordonnancement d applications mltiples, sos forme de collections de tâches indépendantes et identiqes, sr ne plate-forme maître-esclave hétérogène. Les reqêtes de calcl srviennent a cors d temps, ce qi signifie qe nos ne disposons pas de connaissance sr la charge de travail a tot débt de l exéction. Notre objectif est de minimiser l étirement (stretch) maximm des applications, c est-à-dire le rapport entre le temps qe l application passe dans le système avant d être terminée et le temps q elle y arait passé si elle disposait de la plate-forme por elle sele. D n point de ve théoriqe, nos concevons n algorithme optimal por le cas hors-ligne (offline), lorsqe totes les dates d arrivée et les caractéristiqes des applications sont connes à l avance. Nos proposons également plsiers méthodes heristiqes por le cas en-ligne (online), sans connaissance sr l arrivée ftre des applications. D n point ve expérimental, nos avons mené des expérimentations approfondies sos la forme de simlations avec SimGrid mais assi dans n environment parallèle réel, en tilisant MPI. Ces expérimentations montrent qe nos sommes capables d ordonnancer des problèmes de grande taille en qelqes secondes. Enfin, la soltion qe nos proposons srpasse les méthodes heristiqes classiqes, ce qi démontre l intérêt de notre démarche. Plate-forme maître-esclave hétérogène, ordonnancement en-ligne, applications con- Mots-clés : crrentes.

6 Offline and online schedling of concrrent bags-of-tasks 3 1 Introdction Schedling problems are already difficlt on traditional parallel machines. They become extremely challenging on heterogeneos clsters, even when embarrassingly parallel applications are considered. For instance, consider a bag-of-tasks [1], i.e., an application made of a collection of independent and identical tasks, to be schedled on a master-worker platform. Althogh simple, this kind of framework is typical of a large class of problems, inclding parameter sweep applications [21] and BOINC-like comptations [18]. If the master-worker platform is homogeneos, i.e., if all workers have identical CPUs and same commnication bandwidths to/from the master, then elementary greedy strategies, sch as prely demanddriven approaches, will achieve an optimal throghpt. On the contrary, if the platform gathers heterogeneos processors, connected to the master via different-speed links, then the previos strategies are likely to fail dramatically. This is becase it is crcial to select which resorces to enroll before initiating the comptation [5, 41]. In this paper, we still target flly parallel applications, bt we introdce a mch more complex (and more realistic) framework than schedling a single application. We envision a sitation where sers, or clients, sbmit several bags-of-tasks to a heterogeneos masterworker platform, sing a classical client-server model. Applications are sbmitted online, which means that there is no a priori (static) knowledge of the workload distribtion at the beginning of the exection. When several applications are exected simltaneosly, they compete for hardware (network and CPU) resorces. What is the schedling objective in sch a framework? A greedy approach wold execte the applications seqentially in the order of their arrival, thereby optimizing the exection of each application onto the target platform. Sch a simple approach is not likely to be satisfactory for the clients. For example, the greedy approach may delay the exection of the second application for a very long time, while it might have taken only a small fraction of the resorces and few time-steps to execte it concrrently with the first one. More strikingly, both applications might have sed completely different platform resorces (being assigned to different workers) and wold have rn concrrently at the same speed as in exclsive mode on the platform. Sharing resorces to execte several applications concrrently has two key advantages: (i) from the clients point of view, the average response time (the delay between the arrival of an application and the completion of its last task) is expected to be mch smaller; (ii) from the resorce sage perspective, different applications will have different characteristics, and are likely to be assigned different resorces by the schedler. Overall, the global tilization of the platform will increase. The traditional measre to qantify the benefits of concrrent schedling on shared resorces is the maximm stretch. The stretch of an application is defined as the ratio of its response time nder the concrrent schedling policy over its response time in dedicated mode, i.e., when it is the only application exected on the platform. The objective is then to minimize the maximm stretch of any application, thereby enforcing a fair trade-off between all applications. The aim of this paper is to provide a schedling strategy which minimizes the maximm stretch of several concrrent bags-of-tasks which are sbmitted online. Or schedling algorithm relies on complicated mathematical tools bt can be compted in time polynomial to RR n

7 4 A. Benoit, L. Marchal, J.-F. Pinea, Y. Robert, F. Vivien the problem size. On the theoretical side, we prove that or strategy is optimal for the offline version of the problem (when all release dates and application characteristics are known beforehand). We also introdce several heristics for the general case of online applications. On the practical side, we have condcted extensive simlations and MPI experiments, showing that we are able to deal with very large problem instances in a few seconds. Also, the soltion that we compte totally otperforms classical heristics from the literatre, thereby flly assessing the seflness of or approach. The rest of the paper is organized as follows. Section 2 describes the platform and application models. Section 3 is devoted to the derivation of the optimal soltion in the offline case, and to the presentation of heristics for online applications. In Section 4 we report an extensive set of simlations and MPI experiments, and we compare the optimal soltion against several classical heristics from the literatre. Section 5 is devoted to an overview of related work. Finally, we state some conclding remarks in Section 6. 2 Framework In this section, we otline the model for the target platforms, as well as the characteristics of the applicative framework. Next we srvey steady-state schedling techniqes and we introdce the objective fnction, namely the maximm stretch of the applications. 2.1 Platform Model We target a heterogeneos master-worker platform (see Figre 1), also called star network or single-level tree in the literatre. The master P master is located at the root of the tree, and there are p workers P (1 p). The link between P master and P has a bandwidth b. We assme a linear cost model, hence it takes X/b time-nits to send (resp. receive) a message of size X to (resp. from) P. The comptational speed of worker P is s, meaning that it takes X/s time-nits to execte X floating point operations. Withot any loss of generality, we assme that the master has no processing capability. Otherwise, we can simlate the comptations of the master by adding an extra worker paying no commnication cost. P master b p s p P 1 P 2 P p Figre 1: A star network. INRIA

8 Offline and online schedling of concrrent bags-of-tasks Commnication models Traditional schedling models enforce the rle that comptations cannot progress faster than processor speeds wold allow: limitations of comptation resorces are well taken into accont. Criosly, these models do not make similar assmptions for commnications: in the literatre, an arbitrary nmber of commnications may take place at any time-step [50, 20]. In particlar, a given processor can send an nlimited nmber of messages in parallel, and each of these messages is roted as if was alone in the system (no sharing of resorces). Obviosly, these models are not realistic, and we need to better take commnication resorces into accont. To this prpose, we present two different models, which cover a wide range of practical sitations. Under the bonded mltiport commnication model [33], the master can send/receive data to/from all workers at a given time-step. However, there is a limit on the amont of data that the master can send per time-nit, denoted as BW. In other words, the total amont of data sent by the master to all workers each time-nit cannot exceed BW. Intitively, the bond BW corresponds to the bandwidth capacity of the master s network card; the flow of data ot of the card can be either directed to a single link or split among several links indifferently, hence the mltiport hypothesis. The bonded mltiport model flly acconts for the heterogeneity of the platform, as each link has a different bandwidth. Simltaneos sends and receives are allowed (all links are assmed bi-directional, or fll-dplex). Another, more restricted model, is the one-port model [16, 17]. In this model the master can send data to a single worker at a given time, so that the sending operations have to be serialized. Sppose for example that the master has a message of size X to send to worker P. We recall that the bandwidth of the commnication link between both processors is b. If the transfer starts at time t, then the master cannot start another sending operation before time t + X/b. Usally, a processor is spposed to be able to perform one send and one receive operation at the same time. However, this hypothesis will not be sefl in or stdy, as the master processor is the only one to send data. The one-port model seems to fit the performance of some crrent MPI implementations, which serialize asynchronos MPI sends as soon as message sizes exceed a few hndreds of kilobytes [44]. However, recent mlti-threaded commnication libraries sch as MPICH [32, 34] allow for initiating mltiple concrrent send and receive operations, thereby providing practical realizations of the mltiport model. Finally, for both the bonded mltiport and the one-port models, we assme that comptation can be overlapped by independent commnication, withot any interference Comptation models We propose two models for the comptation. Under the flid comptation model, we assme that several tasks can be exected at the same time on a given worker, with a time-sharing mechanism. Frthermore, we assme that we totally control the comptation rate for each task. For example, sppose that two tasks A and B are exected on the same worker at respective rates α and β. Dring a time period t, α t nits of work of task A and β t RR n

9 6 A. Benoit, L. Marchal, J.-F. Pinea, Y. Robert, F. Vivien nits of work of task B are completed. These comptation rates may be changed at any time dring the comptation of a task. Or second comptation model, the atomic comptation model, assmes that only a single task can be compted on a worker at any given time, and this exection cannot be stopped before its completion (no preemption). Under both comptation models, a worker can only start compting a task once it has completely received the message containing the task. However, for the ease of proofs, we add a variant to the flid comptation model, called synchronos start comptation: in this model, the comptation on a worker can start at the same time as the reception of the task starts, provided that the comptation rate is smaller than, or eqal to, the commnication rate (the commnication mst complete before the comptation). This models the fact that, in several applications, only the first bytes of data are needed to start execting a task. In addition, the theoretical reslts of this paper are more easily expressed nder this model, which provides an pper bond on the achievable performance Proposed platform model taxonomy We smmarize here the varios platform and application models nder stdy: Bonded Mltiport with Flid Comptation and Synchronos Start (BMP-FC-SS). This is the ttermost simple model: commnication and comptation start at the same time, commnication and comptation rates can vary over time within the limits of link and processor capabilities. We inclde this model in or stdy becase it provides a good and intitive framework to nderstand the reslts presented here. This model also provides an pper bond on the achievable performance, which we se as a reference for other models. Bonded Mltiport with Flid Comptation (BMP-FC). This model is a step closer to reality, as it allows comptation and commnication rates to vary over time, bt it imposes that a task inpt data is completely received before its exection can start. Bonded Mltiport with Atomic Comptation (BMP-AC). In this model, two tasks cannot be compted concrrently on a worker. This model takes into accont the fact that controlling precisely the compting rate of two concrrent applications is practically challenging, and that it is sometimes impossible to rn simltaneosly two applications becase of memory constraints. One-Port Model with Atomic Comptation (OP-AC). This is the same model as the BMP-AC, bt with one-port commnication constraint on the master. It represents systems where concrrent sends are not allowed. In the following, we mainly focs on the variants of the bonded mltiport model. We explain the reslts obtained with the one-port model in Section INRIA

10 Offline and online schedling of concrrent bags-of-tasks 7 There is a hierarchy among all the mltiport models: intitively, in terms of hardness, BMP-FC-SS < BMP-FC < BMP-AC Formally, a valid schedle for BMP-AC is valid for BMP-FC and a valid schedle for BMP- FC is valid for BMP-FC-SS. This is why stdying BMP-FC-SS is sefl for deriving pper bonds for all other models. 2.2 Application model We consider n bags-of-tasks A k, 1 k n. The master P master holds the inpt data of each application A k pon its release time. Application A k is composed of a set of Π (k) independent, same-size tasks. In order to completely execte an application, all its constittive tasks mst be compted (in any order). We let w (k) be the amont of comptations (expressed in flops) reqired to process a task of A k. The speed of a worker P may well be different for each application, depending pon the characteristics of the processor and pon the type of comptations needed by each application. To take this into accont, we refine the platform model and add an extra parameter, sing s (k) instead of s in the following. In other words, we move from the niform machine model to the nrelated machine model of schedling theory [20]. The time reqired to process one task of A k on processor P is ths w (k) /s (k). Each task of A k has a size δ (k) (expressed in bytes), which means that it takes a time δ (k) /b to send a task of A k to processor P (when there are no other ongoing transfers). For simplicity we do not consider any retrn message: either we assme that the reslts of the tasks are stored on the workers, or we merge the retrn message of the crrent task with the inpt message of the next one (and pdate the commnication volme accordingly). 2.3 Steady-state schedling Assme for a while that a niqe bag-of-tasks A k is exected on the platform. If Π (k), the nmber of independent tasks composing the application, is large (otherwise, why wold we deploy A k on a parallel platform?), we can relax the problem of minimizing the total exection time. Instead, we aim at maximizing the throghpt, i.e., the average (fractional) nmber of tasks exected per time-nit. We design a cyclic schedle, that reprodces the same schedle every period, except possibly for the very first (initialization) and last (clean-p) periods. It is shown in [9, 5] how to derive an optimal schedle for throghpt maximization. The idea is to characterize the optimal throghpt as the soltion of a linear program over rational nmbers, which is a problem with polynomial time complexity. Throghot the paper, we denote by ρ (k) the throghpt of worker P for application A k, i.e., the average nmber of tasks of A k that P exectes each time-nit. In the special case where application A k is exected alone in the platform, we denote by ρ (k) the vale of this throghpt in the soltion which maximizes the total throghpt: ρ (k) = p =1 ρ (k). RR n

11 8 A. Benoit, L. Marchal, J.-F. Pinea, Y. Robert, F. Vivien We write the following linear program (see Eqation (1)), which enables s to compte an asymptotically optimal schedle. The maximization of the throghpt is bonded by three types of constraints: (1) The first set of constraints state that the processing capacity of P is not exceeded. The second set of constraints states that the bandwidth of the link from P master to P is not exceeded. The last constraint states that the total otgoing capacity of the master is not exceeded. Maximize ρ (k) = p =1 ρ (k) sbject to 1 p, 1 p, p =1 ρ (k) w (k) s (k) ρ (k) δ (k) BW 1 1 ρ (k) δ (k) b 1 The formlation in terms of a linear program is simple when considering a single application. In this case, a closed-form expression can be derived. First, the first two sets of constraints can be transformed into: 1 p ρ (k) Then, the last constraint can be rewritten: p =1 ρ (k) min =1 { BW δ (k). s (k) w (k), b δ (k) So that the optimal throghpt is { { }} ρ (k) BW p = min δ, s (k) min (k) w, b. (k) δ (k) It can be shown [9, 5] that any feasible schedle nder one of the mltiport model has to enforce the previos constraints. Hence the optimal vale ρ (k) is an pper bond of the achievable throghpt. Moreover, we can constrct an actal schedle, based on an optimal soltion of the linear program and which approaches the optimal throghpt. The reconstrction is particlarly easy. For example the following procedre bilds an asymptotic optimal schedle for the BMP-AC model (bonded mltiport commnication with atomic comptation). As this is the most constrained mltiport model, this schedle is feasible in any mltiport model: }. INRIA

12 Offline and online schedling of concrrent bags-of-tasks 9 While there are tasks to process on the master, send tasks to processor P with rate ρ (k). As soon as processor P starts receiving a task it processes at the rate ρ (k). De to the constraints of the linear program, this schedle is always feasible and it is asymptotically optimal, not only among periodic schedles, bt more generally among any possible schedles. More precisely, its exection time differs from the minimm exection time by a constant factor, independent of the total nmber of tasks Π (k) to process [5]. This allows s to accrately approximate the total exection time, also called makespan, as: MS (k) = Π(k) ρ (k). We often se MS (k) as a comparison basis to approximate the makespan of an application when it is alone on the compting platform. If MS (k) opt is the optimal makespan for this single application, then we have MS (k) opt M k MS (k) MS (k) opt where M k is a fixed constant, independent of Π (k). 2.4 Stretch We come back to the original scenario, where several applications are exected concrrently. Becase they compete for resorces, their throghpt will be lower. Eqivalently, their exection rate will be slowed down. Informally, the stretch [12] of an application is the slowdown factor. Let r (k) be the release date of application A k on the platform. Its exection will terminate at time C (k) r (k) +MS (k), where MS (k) is the time to execte all Π (k) tasks of A k. Becase there might be other applications rnning concrrently to A k dring part or whole of its exection, we expect that MS (k) MS (k). We define the average throghpt ρ (k) achieved by A k dring its (concrrent) exection sing the same eqation as before: MS (k) = Π(k) ρ (k). In order to process all applications fairly, we wold like to ensre that their actal (concrrent) exection is as close as possible to their exection in dedicated mode. The stretch of application A k is its slowdown factor S k = MS(k) ρ (k) = MS (k) ρ (k) RR n

13 10 A. Benoit, L. Marchal, J.-F. Pinea, Y. Robert, F. Vivien Or objective fnction is defined as the max-stretch S, which is the maximm of the stretches of all applications: S = max 1 k n S k Minimizing the max-stretch S ensres that the slowdown factor is kept as low as possible for each application, and that none of them is ndly favored by the schedler. 3 Theoretical stdy The main contribtion of this paper is a polynomial algorithm to schedle several bag-of-task applications arriving online, while minimizing the maximm stretch. We start this section with the presentation of an asymptotically optimal algorithm for the offline setting, when application release dates and characteristics are known in advance. Then we present or soltion for the online framework. 3.1 Offline setting for the flid model Defining the set of possible soltions In this section, we assme that all characteristics of the n applications A k, 1 k n are known in advance. The schedling algorithm is the following. Given a candidate vale for the max-stretch, we have a procedre to determine whether there exists a soltion that can achieve this vale. The optimal vale will then be fond sing a binary search on possible vales. Consider a candidate vale S l for the max-stretch. If this objective is feasible, all applications will have a max-stretch smaller than S l, hence: 1 k n, MS (k) MS (k) Sl 1 k n, C (k) = r (k) + MS (k) r (k) + S l MS (k) Ths, given a candidate vale S l, we have a deadline: (2) d (k) = r (k) + S l MS (k) for each application A k, 1 k n. This means that the application mst complete before this deadline in order to ensre the expected max-stretch. If this is not possible, no soltion is fond, and a larger max-stretch shold be tried by the binary search. Once a candidate stretch vale S has been chosen, we divide the total exection time into time-intervals whose bonds are epochal times, that is, applications release dates or deadlines. Epochal times are denoted t j {r (1),..., r (n) } {d (1),..., d (n) }, sch that t j t j+1, 1 j 2n 1. Or algorithm consists in rnning each application A k dring its whole exection window [r (k), d (k) ], bt with a different throghpt on each time-interval [t j, t j+1 ] sch that r (k) t j and t j+1 d (k). Some release dates and deadlines may be eqal, leading INRIA

14 Offline and online schedling of concrrent bags-of-tasks 11 to empty time-intervals, for example if there exists j sch that t j = t j+1. We do not try to remove these empty time-intervals so as to keep simple indices. Note that contrarily to the steady-state operation with only one application, in the different time-intervals, the commnication throghpt may differ from the comptation throghpt: when the commnication rate is larger than the comptation rate, extra tasks are stored in a bffer. On the contrary, when the comptation rate is larger, tasks are extracted from the bffer and processed. We introdce new notations to take both rates, as well as bffer sizes, into accont: ρ (k) M (t j, t j+1 ) denotes the commnication throghpt from the master to the worker P dring time-interval [t j, t j+1 ] for application A k, i.e., the average nmber of tasks of A k sent to P per time-nits. ρ (k) (t j, t j+1 ) denotes the comptation throghpt of worker P dring time-interval [t j, t j+1 ] for application A k, i.e., the average nmber of tasks of A k compted by P per time-nits. B (k) (t j ) denotes the (fractional) nmber of tasks of application A k stored in a bffer on P at time t j. We write the linear constraints that mst be satisfied by the previos variables. Or aim is to find a schedle with minimm stretch satisfying those constraints. Later, based on rates satisfying these constraints, we show how to constrct a schedle achieving the corresponding stretch. All tasks sent by the master. The first set of constraints ensres that all the tasks of a given application A k are actally sent by the master: (3) 1 k n, p 1 j 2n 1 =1 t j r (k) t j+1 d (k) ρ (k) M (t j, t j+1 ) (t j+1 t j ) = Π (k). Non-negative bffers. Each bffer shold always have a non-negative size: (4) 1 k n, 1 p, 1 j 2n, B (k) (t j ) 0. Bffer initialization. At the beginning of the comptation of application A k, all corresponding bffers are empty: (5) 1 k n, 1 p, B (k) (r(k) ) = 0. Emptying Bffer. After the deadline of application A k, no tasks of this application shold remain on any node: (6) 1 k n, 1 p, B (k) (d (k) ) = 0. RR n

15 12 A. Benoit, L. Marchal, J.-F. Pinea, Y. Robert, F. Vivien Task conservation. Dring time-interval [t j, t j+1 ], some tasks of application A k are received and some are consmed (compted), which impacts the size of the bffer: (7) 1 k n, 1 j 2n 1, 1 p, B (k) (t j+1) = B (k) (t j) + ( ρ (k) M (t j, t j+1 ) ρ (k) (t j, t j+1 ) ) ( t j+1 t j ) Bonded compting capacity. The compting capacity of a node shold not be exceeded on any time-interval: (8) 1 j 2n 1, 1 p, n k=1 ρ (k) (t j, t j+1 ) w(k) s (k) 1. Bonded link capacity. The bandwidth of each link shold not be exceeded: (9) 1 j 2n 1, 1 p, n k=1 ρ (k) M (t j, t j+1 ) δ(k) b 1. Limited sending capacity of master. The total otgoing bandwidth of the master shold not be exceeded: (10) 1 j 2n 1, p n =1 k=1 ρ (k) M (t j, t j+1 ) δ(k) BW 1. Non-negative throghpts. (11) 1 p, 1 k n, 1 j 2n 1, ρ (k) M (t j, t j+1 ) 0 and ρ (k) (t j, t j+1 ) 0. We obtain a convex polyhedron (K) defined by the previos constraints. The problem trns now into checking whether the polyhedron is empty and, if not, into finding a point in the polyhedron. (K) { ρ (k) M (t j, t j+1 ), ρ (k) (t j, t j+1 ), k,, j sch that 1 k n, 1 p, 1 j 2n 1 nder the constraints (3), (7), (5), (6), (4), (8), (9), (10) and (11) Nmber of tasks processed At first sight, it may seem srprising that in this set of linear constraints, we do not have an eqation establishing that all tasks of a given application are eventally processed. Indeed, sch a constraint can be derived from the constraints related to the nmber of tasks sent from the master and the size of bffers. Consider the constraints on task conservation INRIA

16 Offline and online schedling of concrrent bags-of-tasks 13 (Eqation (7)) on a given processor P, and for a given application A k ; these eqations can be written: 1 j 2n 1, B (k) (t j+1) B (k) (t j) = ( ρ (k) M (t j, t j+1 ) ρ (k) (t j, t j+1 ) ) ( ) t j+1 t j. If we sm all these constraints for all time-interval bonds between t start = r (k) and t stop = d (k), we obtain: B (k) (t stop ) B (k) (t start ) = ρ (k) M (t j, t j+1 ) ( ) t j+1 t j ρ (k) (t j, t j+1 ) ( ) t j+1 t j [t j, t j+1] t j r (k) t j+1 d (k) [t j, t j+1] t j r (k) t j+1 d (k) Thanks to constraints (5) and (6), we know that B (k) (t start ) = 0 and B (k) (t stop ) = 0. So the overall nmber of tasks sent to a processor P is eqal to the total nmber of tasks compted: [t j, t j+1] t j r (k) t j+1 d (k) ρ (k) M (t j, t j+1 ) ( t j+1 t j ) = [t j, t j+1] t j r (k) t j+1 d (k) ρ (k) (t j, t j+1 ) ( t j+1 t j ) This is tre for all processors, and constraints (3) tells s that the total nmber of tasks sent for application A k is Π (k), so: p =1 [t j, t j+1] t j r (k) t j+1 d (k) ρ (k) (t j, t j+1 ) ( ) t j+1 t j = Π (k) Therefore in any soltion in Polyhedron (K), all tasks of each application are processed Bonding the bffer size The size of the bffers cold also be bonded by adding constraints: 1 p, 1 j 2n, n k=1 B (k) (t j )δ (k) M where M is the size of the memory available on node P. We bond the needed memory only at time-interval bonds, bt the above argment can be sed to prove that the bffer size on P never exceeds M. We choose not to inclde this constraint in or basic set of constraints, as this bffer size limitation only applies to the flid model. Indeed, we have earlier proven that limiting the bffer size for independent tasks schedling leads to NP-complete problems [10]. RR n

17 14 A. Benoit, L. Marchal, J.-F. Pinea, Y. Robert, F. Vivien Eqivalence between non-emptiness of Polyhedron (K) and achievable stretch Finding a point in Polyhedron (K) allows to determine whether the candidate vale for the stretch is feasible. Depending on whether Polyhedron (K) is empty, the binary search will be contined with a larger or smaller stretch vale: If the polyhedron is not empty, then there exists a schedle achieving stretch S. S becomes the pper bond of the binary search interval and the search proceeds. On the contrary, if the polyhedron is empty, then it is not possible to achieve S. S becomes the lower bond of the binary search. This binary search and its proof are described below. For now, we concentrate on proving that the polyhedron is not empty if and only if the stretch S is achievable. Note that the previos stdy assmes a flid framework, with flexible compting and commnicating rates. This is particlarly convenient for the totally flid model (BMP-FC- SS) and we prove below that the algorithm comptes the optimal stretch nder this model. The strength of or method is that this stdy is also valid for the other models. The reslts are slightly different, leading to asymptotic optimality reslts and the proofs detailed below are slightly more involved. However, this techniqe allows to approach optimality. Theorem 1. Under the totally flid model, Polyhedron (K) is not empty if and only if there exists a schedle with stretch S. In practice, to know if the polyhedron is empty or to obtain a point in (K), we can se classical tools for linear programs, jst by adding a fictitios linear objective fnction to or set of constraints. Some solvers allow the ser to limit the nmber of refinement steps once a point is fond in the polyhedron; this cold be helpfl to redce the rnning time of the schedler. Proof. Assme that the polyhedron is not empty, and consider a point in (K), given by the vales of the ρ (k) M (t j, t j+1 ) and ρ (k) (t j, t j+1 ). We constrct a schedle which obeys exactly these vales. Dring time-interval [t j, t j+1 ], the master sends tasks of application A k to processor P with rate ρ (k) M (t j, t j+1 ), and this processor comptes these tasks at a rate ρ (k) (t j, t j+1 ). To prove that this schedle is valid nder the flid model, and that it has the expected stretch, we define ρ (k) M (t) as the instantaneos commnication rate, and ρ(k) (t) as the instantaneos comptation rate. Then the (fractional) nmber of tasks of A k sent to P in interval [0, T ] is T 0 ρ (k) M (t)dt With the same argment as in the previos remark, applied on interval [0, T ], we have B (k) (T ) = T 0 ρ (k) M (t)dt T 0 ρ (k) (t)dt INRIA

18 Offline and online schedling of concrrent bags-of-tasks 15 Since the bffer size is positive for all t j and evolves linearly in each interval [t j, t j+1 ], it is not possible that a bffer has a negative size, so T 0 T ρ (k) (t)dt ρ (k) M (t)dt 0 Hence data is always received before being processed. With the constraints of Polyhedron (K), it is easy to check that no processor or no link is over-tilized and the otgoing capacity of the master is never exceeded. All the deadlines compted for stretch S are satisfied by constrction, so this schedle achieves stretch S. Now we prove that if there exists a schedle S 1 with stretch S, Polyhedron (K) is not empty. We consider sch a schedle, and we call ρ (k) M (t) (and ρ(k) (t)) the commnication (and comptation) rate in this schedle for tasks of application A k on processor P at time t. We compte as follows the average vales for commnication and comptation rates dring time interval [t j, t j+1 ]: ρ (k) M (t j, t j+1 ) = tj+1 t j ρ (k) M (t)dt t j+1 t j and ρ (k) (t j, t j+1 ) = tj+1 In this schedle, all tasks of application A k are sent by the master, so t j ρ (k) (t)dt. t j+1 t j d (k) r (k) ρ (k) M (t)dt = Π(k). With the previos definitions, Eqation (3) is satisfied. Along the same line, we can prove that the task conservation constraints (Eqation (7)) are satisfied. Constraints on bffers (Eqations 5, 6 and 4) are necessarily satisfied by the size of the bffer in schedle S 1 since it is feasible. Similarly, we can check that the constraints on capacities are verified Binary search To find the optimal stretch, we perform a binary search sing the emptiness of Polyhedron (K) to determine whether it is possible to achieve the crrent stretch. The initial pper bond for this binary search is compted sing a naive schedle where all applications are compted seqentially. For the sake of simplicity, we consider that all applications are released at time 0 and terminate simltaneosly. This is clearly a worst case scenario. We recall that the throghpt for a single application on the whole platform can be compted as: { { }} ρ (k) BW p = min δ, s (k) min (k) w, b (k) δ (k) =1 RR n

19 16 A. Benoit, L. Marchal, J.-F. Pinea, Y. Robert, F. Vivien Then the exection time for application A k is simply Π (k) /ρ (k). We consider that all applications terminate at time k Π(k) /ρ (k), so that the worst stretch is S max = max k Π (k) /ρ (k) k Π(k) /ρ (k). The lower bond on the achievable stretch is 1. Determining the termination criterion of the binary search, that is the minimm gap ɛ between two possible stretches, is qite involved, and not very sefl in practice. We focs here on the case where this precision ɛ is given by the ser. Please refer to Section 3.4 for a low-complexity techniqe (a binary search among stretch-intervals) to compte the optimal maximm stretch. Algorithm 1: Binary search begin S inf 1 S sp S max while S sp S inf > ɛ do S (S sp + S inf )/2 if Polyhedron (K) is empty then S inf S else S sp S retrn S sp end Sppose that we are given ɛ > 0. The binary search is condcted sing Algorithm 1. This algorithm allows s to approach the optimal stretch, as stated by the following theorem. Theorem 2. For any ɛ > 0, Algorithm 1 comptes a stretch S sch that there exists a schedle achieving S and S S opt + ɛ, where S opt is the optimal stretch. The complexity of Algorithm 1 is O(log Smax ɛ ). Proof. We prove that at each step, the optimal stretch is contained in the interval [S inf, S sp ] and S sp is achievable. This is obvios at the beginning. At each step, we consider the set of constraints for a stretch S in the interval. If the corresponding polyhedron is empty, Theorem 1 tells s that stretch S is not achievable, so the optimal stretch is greater than S. If the polyhedron is not empty, there exists a schedle achieving this stretch, ths the optimal stretch is smaller than S. The size of the work interval is divided by 2 at each step, and we stop when this size is smaller than ɛ. Ths the nmber of steps is O(log Smax ɛ ). At the end, S opt [S inf, S sp ] with S sp S inf ɛ, so that S sp S opt + ɛ, and S sp is achievable. INRIA

20 Offline and online schedling of concrrent bags-of-tasks Property of the one-dimensional load-balancing schedle Before showing how to extend the previos reslt to more complex platform models, we introdce a tool that will prove helpfl for the proofs: the one-dimensional load-balancing schedle and its properties. A significant part of this paper is devoted to comparing reslts nder different models. One of the major differences between these models is whether they allow or not preemption and time-sharing. On the one hand, we stdy flid models, where a resorce (processor or commnication link) can be simltaneosly sed by several tasks, provided that the total tilization rate is below one. On the other hand, we also stdy atomic models, where a resorce can be devoted to only one task, which cannot be preempted: once a task is started on a given resorce, this resorce cannot perform other tasks before the first one is completed. In this section, we show how to constrct a schedle withot preemption from flid schedles, in a way that keeps the interesting properties of the original schedle. Namely, we aim at constrcting atomic-model schedles in which tasks terminate not later, or start not earlier, than in the original flid schedle. We consider a general case of n applications A 1,..., A n to be schedled on the same resorce, typically a given processor, and we denote by t k the time needed to process one task of application A k at fll speed. We start from a flid schedle S flid where each application A k is processed at a rate of α k tasks per time-nits, sch that n k=1 α k 1. Figre 2(a) illstrates sch a schedle. T i!!!!! " " " " " " " " " " " " " " " " " " " " " " # # # # # # # # # # # # # # # # # # # # # # # # % % % $ $ $ % % % $ $ $ % % % $ $ $ % % % $ $ $ % % %! $ $ $ % % % & & & & & & & & & & & & & & & & & & & & & & ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ) ( ) ( ) ( ) ( ) ( ) * * * * * * * * * * * ,,, - - -,,, - - -,,, - - -,,, - - -,,, / / / / / / / / / / / / T ; ; ; ; ; ; : : : ; ; ; : : : ; ; ; : : : ; ; ; : : : ; ; ; : : : < < < < < < < < < < < = = = = = = = = = = = = time time (a) flid schedle S flid (b) atomic schedle S 1D Figre 2: Gantt charts for the proof illstrating the one-dimensional load-balancing algorithm. From S flid, we bild an atomic-model schedle S 1D sing a one-dimensional loadbalancing algorithm [19, 6]: at any time step, if n k is the nmber of tasks of application A k that have already been schedled, the next task to be schedled is the one which minimizes the qantity (n k+1) t k α k. Figre 2(b) illstrates the schedle obtained. We now prove that this schedle has the nice property that a task is not processed later in S 1D than in S flid. Lemma 1. In the schedle S 1D, a task T does not terminate later than in S flid. Proof. First, we point ot that t k /α k is the time needed to process one task of application A k in S flid (with rate α k ). So n k t k α k is the time needed to process the first n k tasks of application A k. The schedling decision which chooses the application minimizing (n k+1) t k α k RR n>

21 18 A. Benoit, L. Marchal, J.-F. Pinea, Y. Robert, F. Vivien consists in choosing the task which is not yet schedled and which terminates first in S flid. Ths, in S 1D, the tasks are exected in the order of their termination date in S flid. Note that if several tasks terminate at the very same time in S flid, then these tasks can be exected in any order in S 1D, and the partial order of their termination date is still observed in S 1D. T other T before t ki T before t ki d flid Then, consider a task T i of a given application A ki, its termination date d flid in S flid, and its termination date d 1D in S 1D. We call S before the set of tasks which are exected before T i in S 1D. Becase S 1D exectes the tasks in the order of their termination date in S flid, S before is made of tasks which are completed before T i in S flid, and possibly some tasks completed at the same time as T i (at time d flid ). We denote by T before the time needed to process the tasks in S before. In S 1D, we have d 1D = T before +t ki whereas in S flid, we have d flid = T before +t ki +T other where T other is the time spent processing tasks from other application than A k and which are not completed at time d flid, or tasks completing at time d flid and schedled later than T i in S 1D. Since T other 0, we have d 1D d flid. d 1D The previos property is sefl when we want to constrct an atomic-model schedle, that is a schedle withot preemption, in which task reslts are available no later than in a flid schedle. On the contrary, it can be sefl to ensre that no task will start earlier in an atomic-model schedle than in the original flid schedle. Here is a procedre to constrct a schedle with the latter property. 1. We start again from a flid schedle S flid, of makespan M. We transform this schedle into a schedle S 1 flid by reversing the time: a task starting at time d and finishing at time f in S flid is schedled to start at time M f and to terminate at M d in S 1 flid, and is processed at the same rate as in S flid. Note that this is possible since we have no precedence constraints between tasks. 2. Then, we apply the previos one-dimensional load-balancing algorithm on S 1 flid, leading to the schedle S 1 1D. Thanks to the previos reslt, we know that a task T does not terminate later in S 1 1D than in S 1 flid. 3. Finally, we transform S 1 1D by reverting the time one last time: we obtain the schedle S 2 1D. A task starting at time d and finishing at time f in S 1 1D starts at time M f and finishes at time M d in S 2 1D. Note that S 1 1D may have a makespan smaller that M (if the resorce was not totally sed in the original schedle S flid ). In this case, or INRIA

22 Offline and online schedling of concrrent bags-of-tasks 19 method atomatically introdces idle time in the one-dimensional schedle, to avoid that a task is started too early. Lemma 2. A task does not start sooner in S 2 1D than in S flid. Proof. Consider a task T, call f 1 its termination date in S 1 flid, and f 2 its termination date in S 1 1D. Thanks to Lemma 1, we know that f 2 f 1. By constrction of the reverted schedles, the starting date of task T in S flid is M f 1. Similarly, its starting date in S 2 1D is M f 2 and we have M f 2 M f Qasi-optimality for more realistic models In this section, we explain how the previos optimality reslt can be adapted to the other models presented in Section As expected, the more realistic the model, the less tight the optimality garanty. Fortnately, we are always able to reach asymptotic optimality: or schedles get closer to the optimal as the nmber of tasks per application increases. We describe the delay indced by each model in comparison to the flid model: starting from a schedle optimal nder the flid model (BMP-FC-SS), we try to bild a schedle with comparable performance nder a more constrained scenario. In the following, we consider a schedle S 1, with stretch S, valid nder the totally flid model (BMP-FC-SS). For the sake of simplicity, we consider that this schedle has been bilt from a point in Polyhedron (K) as explained in the previos section: the comptation and commnication rates (ρ (k) (t j, t j+1 ) and ρ (k) M (t j, t j+1 )) are constant dring each interval, and are defined by the coordinates of the point in Polyhedron (K). We assess the delay indced by each model. Given the stretch S, we can compte a deadline d (k) for each application A k. By moving to more constrained models, we will not be able to ensre that the finishing time MS (k) is smaller than d (k). We call lateness for application A k the qantity max{0, MS (k) d (k) }, that is the time between the de date of an application and its real termination. Once we have compted the maximm lateness for each model, we show how to obtain asymptotic optimality in Section Withot simltaneos start: the BMP-FC model We consider here the BMP-FC model, which differs from the previos model only by the fact that a task cannot start before it has been totally received by a processor. Theorem 3. From schedle S 1, we can bild a schedle S 2 obeying the BMP-FC model n w (k) where the maximm lateness for each application is max. 1 p k=1 Proof. From the schedle S 1, valid nder the flid model (BMP-FC-SS), we aim at bilding S 2 with a similar stretch where the exection of a task cannot start before the end of the corresponding commnication. We first bild a schedle as follows, for each processor P (1 p): s (k) RR n

23 20 A. Benoit, L. Marchal, J.-F. Pinea, Y. Robert, F. Vivien 1. Commnications to P are the same as in S 1 ; 2. By comparison to S 1, the comptations on P are shifted for each application A k : the comptation of the first task of A k is not really performed (P is kept idle instead of compting this task), and we replace the comptation of task i by the comptation of task i 1. Becase of the shift of the comptations, the last task of application A k is not exected in this schedle at time d (k). We complete the constrction of S 2 by adding some delay after deadline d (k) to process this last task of application A k at fll speed, which takes a time w (k) s (k). All the following comptations on processor P (in the next time-intervals) are shifted by this delay. The lateness for any application A k on processor P is at most the sm of the delays for all applications on this processor, n k=1 w(k), and the total lateness of A s (k) k is bonded by the maximm lateness between all processors: lateness (k) max n 1 p k=1 w (k) An example of sch a schedle S 2 is shown on Figre 3 (on a single processor). s (k) ρ 1 ρ 1 0 r (0) d (0) t 0 r (0) d (0) 11 t (a) Schedle S 1 (BMP-FC-SS model) (b) Schedle S 2 (BMP-FC model) Figre 3: Example of the constrction of a schedle S 2 for BMP-FC model from a schedle S 1 for BMP-FC-SS model. We plot only the compting rate. Each box corresponds to the exection of one task Atomic exection of tasks: the BMP-AC model We now move to the BMP-AC model, where a given processor cannot compte several tasks in parallel, and the exection of a task cannot be preempted: a started task mst be completed before any other task can be processed. INRIA

24 Offline and online schedling of concrrent bags-of-tasks 21 Theorem 4. From schedle S 1, we can bild a schedle S 3 obeying the BMP-AC model where the maximm lateness for each application is max 2n n 1 p k=1 Proof. Starting from a schedle S 1 valid nder the flid model (BMP-FC-SS), we want to bild S 3, valid in BMP-AC. We take here advantage of the properties described in Section 3.2 of one-dimensional load-balancing schedles, and especially of S 2 1D. Schedle S 3 is bilt as follows: 1. Commnications are kept nchanged; 2. We consider the comptations taking place in S 1 on processor P dring time-interval [t j, t j+1 ]. A rational nmber of tasks of each application may be involved in the flid schedle. We first compte the integer nmber of tasks of application A k to be compted in S 3 : w (k) s (k) n,j,k = ρ (k) (t j, t j+1 ) (t j+1 t j ). The first n,j,k tasks of A k schedled in time-interval [t j, t j+1 ] on P are organized sing the transformation to bild S 2 1D in Section Then, the comptations are shifted as for S 2 : for each application A k, the comptation of the first task of A k is not really performed (the processor is kept idle instead of compting this task), and we replace the comptation of task i by the comptation of task i 1. Lemma 2 proves that, dring time-interval [t j, t j+1 ], on processor P, a comptation does not start earlier in S 3 than in S 1. As S 1 obeys the totally flid model (BMP-FC-SS), a comptation of S 1 does not start earlier than the corresponding commnication, so a comptation of task i of application A k in S 1 does not start earlier than the finish time of the commnication for task i 1 of A k. Together with the shifting of the comptations, this proves that in S 3, the comptation of a task does not start earlier than the end of the corresponding commnication, on each processor. Becase of the ronding down to the closest integer, on each processor P, at each timeinterval, S 3 comptes at most one task less than S 1 of application A k. Moreover, one more task comptation of application A k is not performed in S 3 de to the comptation shift. On the whole, as there are at most 2n 1 time-intervals, at most 2n tasks of A k remain to be compted on P at time d (k). The delay for application A k is: lateness (k) max 2n n w(k). 1 p s (k). k=1 This is obviosly not the most efficient way to constrct a schedle for the BMP-AC model: in particlar, each processor is idle dring each interval (becase of the ronding down). It wold certainly be more efficient to sometimes start a task even if it cannot be RR n