Even though it is usually assumed that data centers can always operate at maximum capacity, there have been recent scenarios where the amount of electricity that can be used by data centers evolve over time. Hence, the number of available processors is not a constant anymore. In this work, we assume that jobs can be checkpointed before a resource change. Indeed, in the scenarios that we consider, the resource provider warns the user before a change in the number of processors. It is thus possible to anticipate and take checkpoints before the change happens, such that no work is ever lost. The goal is then to maximize the goodput and/or the minimum yield of jobs within the next section (time between two changes in the number of processors). We model the problem and design greedy solutions and sophisticated dynamic programming algorithms with some optimality results for jobs of infinite duration, and adapt the algorithms to finite jobs. A comprehensive set of simulations, building on real-life job sets, demonstrates the performance of the proposed algorithms. Most algorithms achieve a useful platform utilization (goodput) of over 95%. With infinite jobs, the algorithms also keep fairness by having a relative minimum yield above 0.8, meaning that each job gets a good access to the platform (80% of the time that it would have had if each job had its perfect share of the platform). For finite jobs, the minimum yield can be low since very short new jobs may have to wait until the beginning of the next section to start (and finish), significantly impacting their yield. However, for 75% of the jobs within each workload, the yield ratio between these jobs is at most at a factor two, hence demonstrating the fairness of the proposed algorithms.

The analysis of massive scientific data often happens in the form of workflows with interdependent tasks. When such a scientific workflow needs to be scheduled on a parallel or distributed system, one usually represents the workflow as a directed acyclic graph (DAG). The vertices of the DAG represent the tasks, while its edges model the dependencies between the tasks (usually data to be communicated to successor tasks). When executed, each task requires a certain amount of memory and if that exceeds the available memory, the execution fails. The typical goal is to execute the workflow without failures (i.e., satisfying the memory constraints) and with the shortest possible execution time (i.e., to minimize its makespan). To address this problem, we investigate the memory-aware scheduling of DAG-shaped workflows on heterogeneous platforms, where each processor can have a different speed and a different memory size. We propose a variant of HEFT (Heterogeneous Earliest Finish Time) that, in contrast to the original, accounts for memory and includes eviction strategies for cases when it might be beneficial to remove some data from memory in order to have enough memory to execute other tasks. Furthermore, while HEFT assumes perfect knowledge of the execution time and memory usage of each task, the actual values might differ upon execution. Thus, we propose an adaptive scheduling strategy, where a schedule is recomputed when there has been a significant variation in terms of execution time or memory. The scheduler has been closely integrated with a runtime system, allowing us to perform a thorough experimental evaluation on real-world workflows. The runtime system warns the scheduler when the task parameters have changed, and a schedule can be recomputed on the fly. The memory-aware strategy allows us to schedule task graphs that would run out of memory with a state-of-the-art scheduler, and the adaptive setting allows us to significantly reduce the makespan.

Large data and computing centers consume a significant share of the world’s energy consumption. A prominent subset of the workloads in such centers are workflows with interdependent tasks, usually represented as directed acyclic graphs (DAGs). To reduce the carbon emissions resulting from executing such workflows in centers with a mixed (renewable and non-renewable) energy supply, it is advisable to move task executions to time intervals with sufficient green energy when possible. To this end, we formalize the above problem as a scheduling problem with a given mapping and ordering of the tasks. We show that this problem can be solved in polynomial time in the uniprocessor case. For at least two processors, however, the problem becomes NP-hard. Hence, we propose a heuristic framework called CaWoSched that combines several greedy approaches with local search. To assess the 16 heuristics resulting from different combinations, we also devise a simple baseline algorithm and an exact ILP-based solution. Our experimental results show that our heuristics provide significant savings in carbon emissions compared to the baseline.

Scientific workflows are often represented as directed acyclic graphs (DAGs), where vertices correspond to tasks and edges represent the dependencies between them. Since these graphs are often large in both the number of tasks and their resource requirements, it is important to schedule them efficiently on parallel or distributed compute systems. Typically, each task requires a certain amount of memory to be executed and needs to communicate data to its successor tasks. The goal is thus to execute the workflow as fast as possible (i.e., to minimize its makespan) while satisfying the memory constraints.

Abstract Directed acyclic graphs are commonly used to model scientific workflows, by expressing dependencies between tasks, as well as the resource requirements of the workflow. As a special case, rooted directed trees occur in several applications, for instance in sparse matrix computations. Since typical workflows are modeled by large trees, it is crucial to schedule them efficiently, so that their execution time (or makespan) is minimized. Furthermore, it is usually beneficial to distribute the execution on several compute nodes, hence increasing the available memory, and allowing us to parallelize parts of the execution. To exploit the heterogeneity of modern clusters in this context, we investigate the partitioning and mapping of tree‐shaped workflows on two types of target architecture models: in AM1, each processor can have a different memory size, and in AM2, each processor can also have a different speed (in addition to a different memory size). We design a three‐step heuristic for AM1, which adapts and extends previous work for homogeneous clusters [Gou C, Benoit A, Marchal L. Partitioning tree‐shaped task graphs for distributed platforms with limited memory. IEEE Trans Parallel Dist Syst 2020; 31(7): 1533–1544]. The changes we propose concern the assignment to processors (accounting for the different memory sizes) and the availability of suitable processors when splitting or merging subtrees. For AM2, we extend the heuristic for AM1 with a two‐phase local search approach. Phase A is a swap‐based hill climber, while (the optional) Phase B is inspired by iterated local search. We evaluate our heuristics for AM1 and AM2 with extensive simulations, and we demonstrate that exploiting the heterogeneity in the cluster significantly reduces the makespan, compared to the state of the art for homogeneous processors.

The problem of scheduling moldable tasks on multiprocessor systems with the objective of minimizing the overall completion time (or makespan) has been widely studied, in particular when tasks have dependencies (i.e., task graphs), or when tasks are released on-the-fly (i.e., online). However, few studies have focused on both (i.e., online scheduling of moldable task graphs). In this paper, we design a new online algorithm and derive constant competitive ratios for this problem under several common yet realistic speedup models (i.e., roofline, communication, Amdahl, and a general combination). We also prove, for each model, a lower bound on the competitiveness of our algorithm, which is very close to the constant competitive ratio. Finally, we provide the first lower bound on the competitive ratio of any deterministic online algorithm for the arbitrary speedup model, which is not constant but depends on the number of tasks in the longest path of the graph.

We study the resilient scheduling of moldable parallel jobs on high-performance computing (HPC) platforms. Moldable jobs allow for choosing a processor allocation before execution, and their execution time obeys various speedup models. The objective is to minimize the overall completion time or the makespan, when jobs can fail due to silent errors and hence may need to be re-executed after each failure until successful completion. Our work generalizes the classical scheduling framework for failure-free jobs. To cope with silent errors, we introduce two resilient scheduling algorithms,Lpa-ListandBatch-List, both of which use theListstrategy to schedule the jobs. Without knowing a priori how many times each job will fail,Lpa-Listrelies on a local strategy to allocate processors to the jobs, whileBatch-Listschedules the jobs in batches and allows only a restricted number of failures per job in each batch. We prove approximation ratios for the two algorithms under several prominent speedup models (e.g., roofline, communication, Amdahl, power, monotonic, and a mix model). An extensive set of simulations is conducted to evaluate different variants of the two algorithms, and the results show that they consistently outperform some baseline heuristics. Overall, our best algorithm is within a factor of 1.6 of a lower bound on average over the entire set of experiments, and within a factor of 4.2 in the worst case.

We consider the problem of scheduling independent jobs that are generated by processing units at the edge of the network. These jobs can either be executed locally, or sent to a centralized cloud platform that can execute them at greater speed. Such edge-generated jobs may come from various applications, such as e-health, disaster recovery, autonomous vehicles or flying drones. The problem is to decide where and when to schedule each job, with the objective to minimize the maximum stretch incurred by any job. The stretch of a job is the ratio of the time spent by that job in the system, divided by the minimum time it could have taken if the job was alone in the system. We formalize the problem and explain the differences with other models that can be found in the literature. We prove that minimizing the max-stretch is NP-complete, even in the simpler instance with no release dates (all jobs are known in advance). This result comes from the proof that minimizing the max-stretch with homogeneous processors and without release dates is NP-complete, a complexity problem that was left open before this work. We design several algorithms to propose efficient solutions to the general problem, and we conduct simulations based on real platform parameters to evaluate the performance of these algorithms.

Scheduling independent tasks on a parallel platform is a widely-studied problem, in particular when the goal is to minimize the total execution time, or makespan ($P\Vert C_{max}$problem in Graham's notations). Also, many applications do not consist of sequential tasks, but rather parallel moldable tasks that can decide their degree of parallelism at execution (i.e., on how many processors they are executed). Furthermore, since the energy consumption of data centers is a growing concern, both from an environmental and economical point of view, minimizing the energy consumption of a schedule is a main challenge to be addressed. One can then decide, for each task, on how many processors it is executed, and at which speed the processors are operated, with the goal to minimize the total energy consumption. We further focus on co-schedules, where tasks are partitioned into shelves, and we prove that the problem of minimizing the energy consumption remains NP-complete when static energy is consumed during the whole duration of the application. We are however able to provide an optimal algorithm for the schedule within one shelf, i.e., for a set of tasks that start at the same time. Several approximation results are derived, and simulations are performed to show the performance of the proposed algorithms.

This paper focuses on the resilient scheduling of moldable parallel jobs on high-performance computing (HPC) platforms. Moldable jobs allow for choosing a processor allocation before execution, and their execution time obeys various speedup models. The objective is to minimize the overall completion time of the jobs, or makespan, assuming that jobs are subject to arbitrary failure scenarios, and hence need to be re-executed each time they fail until successful completion. This work generalizes the classical framework where jobs are known offline and do not fail. We introduce a list-based algorithm, and prove new approximation ratios for three prominent speedup models (roofline, communication, Amdahl). We also introduce a batch-based algorithm, where each job is allowed a restricted number of failures per batch, and prove a new approximation ratio for the arbitrary speedup model. We conduct an extensive set of simulations to evaluate and compare different variants of the two algorithms. The results show that they consistently outperform some baseline heuristics. In particular, the list algorithm performs better for the roofline and communication models, while the batch algorithm has better performance for the Amdahl's model. Overall, our best algorithm is within a factor of 1.47 of a lower bound on average over the whole set of experiments, and within a factor of 1.8 in the worst case.

Streaming applications come from various application fields such as physics, and many can be represented as a series-parallel dependence graph. We aim at minimizing the energy consumption of such applications when executed on a hierarchical platform, by proposing novel mapping strategies. Dynamic voltage and frequency scaling (DVFS) is used to reduce the energy consumption, and we ensure a reliable execution by either executing a task at maximum speed, or by triplicating it. In this paper, we propose a structure rule to partition the series-parallel applications, and we prove that the optimization problem is NP-complete. We are able to derive a dynamic programming algorithm for the special case of linear chains, which provides an interesting heuristic and a building block for designing heuristics for the general case. The heuristics performance is compared to a baseline solution, where each task is executed at maximum speed. Simulations demonstrate that significant energy savings can be obtained.

Scientific applications are commonly modeled as the processing of directed acyclic graphs of tasks, and for some of them, the graph takes the special form of a rooted tree. This tree expresses both the computational dependencies between tasks and their storage requirements. The problem of scheduling/traversing such a tree on a single processor to minimize its memory footprint has already been widely studied. The present article considers the parallel processing of such a tree and studies how to partition it for a homogeneous multiprocessor platform, where each processor is equipped with its own memory. We formally state the problem of partitioning the tree into subtrees, such that each subtree can be processed on a single processor (i.e., it must fit in memory), and the goal is to minimize the total resulting processing time. We prove that this problem is NP-complete, and we design polynomial-time heuristics to address it. An extensive set of simulations demonstrates the usefulness of these heuristics.

When scheduling a directed acyclic graph (DAG) of tasks with communication costs on computational platforms, a good trade-off between load balance and data locality is necessary. List-based scheduling techniques are commonly-used greedy approaches for this problem. The downside of list-scheduling heuristics is that they are incapable of making short term sacrifices for the global efficiency of the schedule. In this work, we describe new list-based scheduling heuristics based on clustering for homogeneous platforms, under the realistic duplex single-port communication model. Our approach uses an acyclic partitioner for DAGs for clustering. The clustering enhances the data locality of the scheduler with a global view of the graph. Furthermore, since the partition is acyclic, we can schedule each part completely once its input tasks are ready to be executed. We present an extensive experimental evaluation showing the tradeoffs between the granularity of clustering and the parallelism, and how this affects the scheduling. Furthermore, we compare our heuristics to the best state-of-the-art list-scheduling and clustering heuristics, and obtain more than three times better makespan in cases with many communications.

This paper revisits replication coupled with checkpointing for fail-stop errors. Replication enables the application to survive many fail-stop errors, thereby allowing for longer checkpointing periods. Previously published works use replication with the no-restart strategy, which works as follows: (i) compute the application Mean Time To Interruption (MTTI) M as a function of the number of processor pairs and the individual processor Mean Time Between Failures (MTBF); (ii) use checkpointing period [EQUATION] à la Young/Daly, where C is the checkpoint duration; and (iii) never restart failed processors until the application crashes. We introduce the restart strategy where failed processors are restarted after each checkpoint. We compute the optimal checkpointing period [EQUATION] for this strategy, which is much larger than [EQUATION], thereby decreasing I/O pressure. We show through simulations that using [EQUATION] and the restart strategy, instead of [EQUATION] and the usual no-restart strategy, significantly decreases the overhead induced by replication.

Co-scheduling techniques are used to improve the throughput of applications on chip multiprocessors (CMP), but sharing resources often generates critical interferences. We focus on the interferences in the last level of cache (LLC) and use the Cache Allocation Technology (CAT) recently provided by Intel to partition the LLC and give each co-scheduled application their own cache area. We consider m iterative HPC applications running concurrently and answer the following questions: (i) how to precisely model the behavior of these applications on the cache partitioned platform? and (ii) how many cores and cache fractions should be assigned to each application to maximize the platform efficiency? Here, platform efficiency is defined as maximizing the performance either globally, or as guaranteeing a fixed ratio of iterations per second for each application. Through extensive experiments using CAT, we demonstrate the impact of cache partitioning when multiple HPC application are co-scheduled onto CMP platforms.

Multi-Processor System-on-Chip (MPSoC) has emerged as a promising platform to meet the increasing performance demand of embedded applications. However, due to limited energy budget, it is hard to guarantee that applications on MPSoC can be accomplished on time with a required throughput. The situation becomes even worse for applications with high reliability requirements, since extra energy will be inevitably consumed by task re-executions or duplicated tasks. Based on Dynamic Voltage and Frequency Scaling (DVFS) and task duplication techniques, this paper presents a novel energy-efficient scheduling model, which aims at minimizing the overall energy consumption of MPSoC applications under both throughput and reliability constraints. The problem is shown to be NP-complete, and several polynomial-time heuristics are proposed to tackle this problem. Comprehensive simulations on both synthetic and real application graphs show that our proposed heuristics can meet all the given constraints, while reducing the energy consumption.

This work presents a realistic performance model to execute scientific workflows on high-bandwidth-memory architectures such as the Intel Knights Landing. We provide a detailed analysis of the execution time on such platforms, taking into account transfers from both fast and slow memory and their overlap with computations. We discuss several scheduling and mapping strategies: not only tasks must be assigned to computing resources, but also one has to decide which fraction of input and output data will reside in fast memory and which will have to stay in slow memory. We use extensive simulations to assess the impact of the mapping strategies on performance. We also conduct experiments for a simple 1D Gauss-Seidel kernel, which assess the accuracy of the model and further demonstrate the importance of a tuned memory management. Our model and results lay the foundations for further studies and experiments on dual-memory systems.

Scientific applications are commonly modeled as the processing of directed acyclic graphs of tasks, and for some of them, the graph takes the special form of a rooted tree. This tree expresses both the computational dependencies between tasks and their storage requirements. The problem of scheduling/traversing such a tree on a single processor to minimize its memory footprint has already been widely studied. Hence, we move to parallel processing and study how to partition the tree for a homogeneous multiprocessor platform, where each processor is equipped with its own memory. We formally state the problem of partitioning the tree into subtrees such that each subtree can be processed on a single processor and the total resulting processing time is minimized. We prove that the problem is NP-complete, and we design polynomial-time heuristics to address it. An extensive set of simulations demonstrates the usefulness of these heuristics.

We provide a framework to analyze multi-level checkpointing protocols, by formally defining a$k$-level checkpointing pattern. We provide a first-order approximation to the optimal checkpointing period, and show that the corresponding overhead is in the order of$\sum _{\ell =1}^{k}\sqrt{2\lambda _\ell C_\ell}$, where$\lambda _\ell$is the error rate at level$\ell$, and$C_\ell$the checkpointing cost at level$\ell$. This nicely extends the classical Young/Daly formula on single-level checkpointing. Furthermore, we are able to fully characterize the shape of the optimal pattern (number and positions of checkpoints), and we provide a dynamic programming algorithm to determine the optimal subset of levels to be used. Finally, we perform simulations to check the accuracy of the theoretical study and to confirm the optimality of the subset of levels returned by the dynamic programming algorithm. The results nicely corroborate the theoretical study, and demonstrate the usefulness of multi-level checkpointing with the optimal subset of levels.

Recently, the benefits of co-scheduling several applications have been demonstrated in a fault-free context, both in terms of performance and energy savings. However, large-scale computer systems are confronted to frequent failures, and resilience techniques must be employed to ensure the completion of large applications. Indeed, failures may create severe imbalance between applications, and significantly degrade performance. In this paper, we propose to redistribute the resources assigned to each application upon the striking of failures, in order to minimize the expected completion time of a set of co-scheduled applications. First, we introduce a formal model and establish complexity results. When no redistribution is allowed, we can minimize the expected completion time in polynomial time, while the problem becomes NP-complete with redistributions, even in a fault-free context. Therefore, we design polynomial-time heuristics that perform redistributions and account for processor failures. A fault simulator is used to perform extensive simulations that demonstrate the usefulness of redistribution and the performance of the proposed heuristics.

This work focuses on resilience techniques at extreme scale. Many papers deal with fail-stop errors. Many others deal with silent errors (or silent data corruptions). But very few papers deal with fail-stop and silent errors simultaneously. However, HPC applications will obviously have to cope with both error sources. This paper presents a unified framework and optimal algorithmic solutions to this double challenge. Silent errors are handled via verification mechanisms(either partially or fully accurate) and in-memory checkpoints. Fail-stop errors are processed via disk checkpoints. All verification and checkpoint types are combined into computational patterns. We provide a unified model, and a full characterization of the optimal pattern. Our results nicely extend several published solutions and demonstrate how to make use of different techniques to solve the double threat of fail-stop and silent errors. Extensive simulations based on real data confirm the accuracy of the model, and show that patterns that combine all resilience mechanisms are required to provide acceptable overheads.

International audience

A significant percentage of the computing capacity of large-scale platforms is wasted because of interferences incurred by multiple applications that access a shared parallel file system concurrently. One solution to handling I/O bursts enlarge-scale HPC systems is to absorb them at an intermediate storage layer consisting of burst buffers. However, our analysis of the Argonne's Mira system shows that burst buffers cannot prevent congestion at all times. Consequently, I/O performances dramatically degraded, showing in some cases a decrease in I/O throughput of 67%. In this paper, we analyze the effects of interference on application I/O bandwidth and propose several scheduling techniques to mitigate congestion. We show through extensive experiments that our global I/O scheduler is able to reduce the effects of congestion, even on systems where burst buffers are used, and can increase the overall system throughput up to 56%. We also show that it outperforms current Mira I/O schedulers.

We consider techniques to improve the performance of parallel sparse triangular solution on non-uniform memory architecture multicores by extending earlier coloring and level set schemes for single-core multiprocessors. We develop STS-k, where k represents a small number of transformations for latency reduction from increased spatial and temporal locality of data accesses. We propose a graph model of data reuse to inform the development of STS-k and to prove that computing an optimal cost schedule is NP-complete. We observe significant speed-ups with STS-3 on 32-core Intel Westmere-Ex and 24-core AMD `MagnyCours' processors. Incremental gains solely from the 3-level transformations in STS-3 for a fixed ordering, correspond to reductions in execution times by factors of 1.4(Intel) and 1.5(AMD) for level sets and 2(Intel) and 2.2(AMD) for coloring. On average, execution times are reduced by a factor of 6(Intel) and 4(AMD) for STS-3 with coloring compared to a reference implementation using level sets.

Summary We revisit the classical problem of the reduction collective operation in a heterogeneous environment. We discuss and evaluate four algorithms that are non‐clairvoyant, that is, they do not know in advance the computation and communication costs. On the one hand, Binomial‐stat and Fibonacci‐stat are static algorithms that decide in advance which operations will be reduced, without adapting to the environment; they were originally defined for homogeneous settings. On the other hand, Tree‐dyn and Non‐Commut‐Tree‐dyn are fully dynamic algorithms, for commutative or non‐commutative reductions. We show that these algorithms are approximation algorithms with constant or asymptotic ratios. We assess the relative performance of all four non‐clairvoyant algorithms with heterogeneous costs through a set of simulations. Our conclusions hold for a variety of distributions. Copyright © 2014 John Wiley & Sons, Ltd.

In this paper, we revisit traditional check pointing and rollback recovery strategies, with a focus on silent data corruption errors. Contrarily to fail-stop failures, such latent errors cannot be detected immediately, and a mechanism to detect them must be provided. We consider two models: (i) errors are detected after some delays following a probability distribution (typically, an Exponential distribution), (ii) errors are detected through some verification mechanism. In both cases, we compute the optimal period in order to minimize the waste, i.e., the fraction of time where nodes do not perform useful computations. In practice, only a fixed number of checkpoints can be kept in memory, and the first model may lead to an irrecoverable failure. In this case, we compute the minimum period required for an acceptable risk. For the second model, there is no risk of irrecoverable failure, owing to the verification mechanism, but the corresponding overhead is included in the waste. Finally, both models are instantiated using realistic scenarios and application/architecture parameters.

SUMMARY We consider a task graph to be executed on a set of processors. We assume that the mapping is given, say by an ordered list of tasks to execute on each processor, and we aim at optimizing the energy consumption while enforcing a prescribed bound on the execution time. Although it is not possible to change the allocation of a task, it is possible to change its speed. Rather than using a local approach such as backfilling, we consider the problem as a whole and study the impact of several speed variation models on its complexity. For continuous speeds, we give a closed‐form formula for trees and series–parallel graphs, and we cast the problem into a geometric programming problem for general directed acyclic graphs. We show that the classical dynamic voltage and frequency scaling (DVFS) model with discrete modes leads to an NP‐complete problem, even if the modes are regularly distributed (an important particular case in practice, which we analyze as the incremental model). On the contrary, the Vdd‐hopping model that allows to switch between different supply voltages ( V DD ) while executing a task leads to a polynomial solution. Finally, we provide an approximation algorithm for the incremental model, which we extend for the general DVFS model. Copyright © 2012 John Wiley & Sons, Ltd.

We consider a task graph mapped on a set of homogeneous processors. We aim at minimizing the energy consumption while enforcing two constraints: a prescribed bound on the execution time (or makespan), and a reliability threshold. Dynamic voltage and frequency scaling (DVFS) is an approach frequently used to reduce the energy consumption of a schedule, but slowing down the execution of a task to save energy is decreasing the reliability of the execution. In this work, to improve the reliability of a schedule while reducing the energy consumption, we allow for the re-execution of some tasks. We assess the complexity of the tri-criteria scheduling problem (makespan, reliability, energy) of deciding which task to re-execute, and at which speed each execution of a task should be done, with two different speed models: either processors can have arbitrary speeds (CONTINUOUS model), or a processor can run at a finite number of different speeds and change its speed during a computation (VDD-HoPPING model). We propose several novel tri-criteria scheduling heuristics under the continuous speed model, and we evaluate them through a set of simulations. The two best heuristics turn out to be very efficient and complementary.

In this paper, we study the problem of replica placement in tree networks subject to server capacity and distance constraints. The client requests are known beforehand, while the number and location of the servers are to be determined. The Single policy enforces that all requests of a client are served by a single server in the tree, while in the Multiple policy, the requests of a given client can be processed by multiple servers, thus distributing the processing of requests over the platform. For the Single policy, we prove that all instances of the problem are NP-hard, and we propose approximation algorithms. The problem with the Multiple policy was known to be NP-hard with distance constraints, but we provide a polynomial time optimal algorithm to solve the problem in the particular case of binary trees when no request exceeds the server capacity.

We investigate the routing of communications in chip multiprocessors (CMPs). The goal is to find a valid routing in the sense that the amount of data routed between two neighboring cores does not exceed the maximum link bandwidth while the power dissipated by communications is minimized. Our position is at the system level: we assume that several applications, described as task graphs, are executed on a CMP, and each task is already mapped to a core. Therefore, we consider a set of communications that have to be routed between the cores of the CMP. We consider a classical model, where the power consumed by a communication link is the sum of a static part and a dynamic part, with the dynamic part depending on the frequency of the link. This frequency is scalable and it is proportional to the throughput of the link. The most natural and widely used algorithm to handle all these communications is XY routing: for each communication, data is first forwarded horizontally, and then vertically, from source to destination. However, if it is allowed to use all Manhattan paths between the source and the destination, the consumed power can be reduced dramatically. Moreover, some solutions may be found while none existed with the XY routing. In this paper, we compare XY routing and Manhattan routing, both from a theoretical and from a practical point of view. We consider two variants of Manhattan routing: in single-path routing, only one path can be used for each communication, while multi-paths routing allows to split a communication between different routes. We establish the NP-completeness of the problem of finding a Manhattan routing that minimizes the dissipated power, we exhibit the minimum upper bound of the ratio power consumed by an XY routing over power consumed by a Manhattan routing, and finally we perform simulations to assess the performance of Manhattan routing heuristics that we designed.

We revisit the well-known greedy algorithm for scheduling independent jobs on parallel processors, with the objective of minimizing the power consumption. We assess the performance of the online version, as well as the performance of the offline version, which sorts the jobs by non-increasing size before execution. We derive new approximation factors, as well as examples that show that these factors cannot be improved, thereby completely characterizing the performance of the algorithms.

This paper studies the problem of mapping streaming applications that can be modeled by a series-parallel graph, onto a 2-dimensional tiled CMP architecture. The objective of the mapping is to minimize the energy consumption, using dynamic voltage scaling techniques, while maintaining a given level of performance, reflected by the rate of processing the data streams. This mapping problem turns out to be NP-hard, but we identify simpler instances, whose optimal solution can be computed by a dynamic programming algorithm in polynomial time. Several heuristics are proposed to tackle the general problem, building upon the theoretical results. Finally, we assess the performance of the heuristics through a set of comprehensive simulations.

This paper deals with optimal strategies to place replicas in tree networks, with the double objective to minimize the total cost of the servers, and/or to optimize power consumption. The client requests are known beforehand, and some servers are assumed to pre-exist in the tree. Without power consumption constraints, the total cost is an arbitrary function of the number of existing servers that are reused, and of the number of new servers. Whenever creating and operating a new server has higher cost than reusing an existing one (which is a very natural assumption), cost optimal strategies have to trade-off between reusing resources and load-balancing requests on new servers. We provide an optimal dynamic programming algorithm that returns the optimal cost, thereby extending known results without pre-existing servers. With power consumption constraints, we assume that servers operate under a set of M different modes depending upon the number of requests that they have to process. In practice M is a small number, typically 2 or 3, depending upon the number of allowed voltages. Power consumption includes a static part, proportional to the total number of servers, and a dynamic part, proportional to a constant exponent of the server mode, which depends upon the model for power. The cost function becomes a more complicated function that takes into account reuse and creation as before, but also upgrading or downgrading an existing server from one mode to another. We show that with an arbitrary number of modes, the power minimization problem is NP-complete, even without cost constraint, and without static power. Still, we provide an optimal dynamic programming algorithm that returns the minimal power, given a threshold value on the total cost, it has exponential complexity in the number of modes M, and its practical usefulness is limited to small values of M. Still, experiments conducted with this algorithm show that it can process large trees in reasonable time, despite its worst-case complexity.

We consider a task graph to be executed on a set of processors. We assume that the mapping is given, say by an ordered list of tasks to execute on each processor, and we aim at optimizing the energy consumption while enforcing a prescribed bound on the execution time. While it is not possible to change the allocation of a task, it is possible to change its speed. We study the complexity of the problem for different models: continuous speeds, discrete modes, distributed either arbitrarily or regularly, and VDD-hopping.

This paper deals with the problem of mapping pipelined applications on heterogeneous platforms whose processors are subject to failures. We address a difficult bi-criteria problem, namely deciding which stages to replicate, and on which resources, in order to optimize the reliability of the schedule, while guaranteeing a minimal throughput. Previous work had addressed the complexity of interval mappings, where the application is partitioned into intervals of consecutive stages (which are then replicated and assigned to processors). In this paper we investigate general mappings, where stages may be partitioned without any constraint, thereby allowing a better usage of processors and communication network capabilities. The price to pay for general mappings is a dramatic increase in the problem complexity. We show that computing the period of a given general mapping is an NP-complete problem, and we provide polynomial bounds to determine a (conservative) approximated value. The bi-criteria mapping problem itself becomes NP-complete on homogeneous platforms, while it is polynomial with interval mappings. We design a set of efficient heuristics, which we compare with interval mapping strategies through extensive simulations.

We consider pipelined real-time systems, commonly found in assembly lines, consisting of a chain of tasks executing on a distributed platform. Their processing is pipelined: each processor executes only one interval of consecutive tasks. We are therefore interested in minimizing both the input-output latency and the period. For dependability reasons, we are also interested in maximizing the reliability of the system. We therefore assign several processors to each interval of tasks, so as to increase the reliability of the system. We assume that both processors and communication links are unreliable and subject to transient failures, the arrival of which follows a constant parameter Poisson law. We also assume that the failures are statistically independent events. We study several variants of this multiprocessor mapping problem with several hypotheses on the target platform (homogeneous/heterogeneous speeds and/or failure rates). We provide NP-hardness complexity results, and optimal mapping algorithms for polynomial problem instances.

Pipelined workflows are a popular programming paradigm for parallel applications. In these workflows, the computation is divided into several stages, and these stages are connected to each other through first-in first-out channels. In order to execute these workflows on a parallel machine, we must first determine the mapping of the stages onto the various processors on the machine. After finding the mapping, we must compute the schedule, i.e., the order in which the various stages execute on their assigned processors. In this paper, we assume that the mapping is given and explore the latter problem of scheduling, particularly for linear workflows. Linear workflows are those in which dependencies between stages can be represented by a linear graph. The objective of the scheduling algorithm is either to minimize the period (the inverse of the throughput), or to minimize the latency (response time), or both. We consider two realistic execution models: the one-port model (all operations are serialized) and the multi-port model (bounded communication capacities and communication/computation overlap). In both models, finding a schedule to minimize the latency is easy. However, computing the schedule to minimize the period is NP-hard in the one-port model, but can be done in polynomial time in the multi-port model. We also present an approximation algorithm to minimize the period in the one-port model. Finally, the bi-criteria problem, which consists in finding a schedule respecting a given period and a given latency, is NP-hard in both models.

In this paper, we study the problem of finding optimal mappings for several independent but concurrent workflow applications, in order to optimize performance-related criteria together with energy consumption. Each application consists in a linear chain graph with several stages, and processes successive data sets in pipeline mode, from the first to the last stage. We study the problem complexity on different target execution platforms, ranking from fully homogeneous platforms to fully heterogeneous ones. The goal is to select an execution speed for each processor, and then to assign stages to processors, with the aim of optimizing several concurrent optimization criteria. There is a clear trade-off to reach, since running faster and/or more processors leads to better performance, but the energy consumption is then very high. Energy savings can be achieved at the price of a lower performance, by reducing processor speeds or enrolling fewer resources. We consider two mapping strategies: in one-to-one mappings, a processor is assigned a single stage, while in interval mappings, a processor may process an interval of consecutive stages of the same application. For both mapping strategies and all platform types, we establish the complexity of several multi-criteria optimization problems, whose objective functions combine period, latency and energy criteria. In particular, we exhibit cases where the problem is NP-hard with concurrent applications, while it can be solved in polynomial time for a single application. Also, we demonstrate the difficulty of performance/energy trade-offs by proving that the tri-criteria problem is NP-hard, even with a single application on a fully homogeneous platform.

Mapping a pipelined application onto a distributed and parallel platform is a challenging problem. The problem becomes even more difficult when multiple optimization criteria are involved, and when the target resources are heterogeneous (processors and communication links) and subject to failures. This paper investigates the problem of mapping pipelined applications, consisting of a linear chain of stages executed in a pipeline way, onto such platforms. The objective is to optimize the reliability under a performance constraint, i.e., while guaranteeing a threshold throughput. In order to increase reliability, we replicate the execution of stages on multiple processors. We present complexity results, proving that this bi-criteria optimization problem is NP-hard. We then propose some heuristics, and discuss extensive experiments evaluating their performance.

Mapping and scheduling an application onto the processors of a parallel system is a difficult problem. This is true when performance is the only objective, but becomes worse when a second optimization criterion like reliability is involved. In this paper we investigate the problem of mapping an application consisting of several consecutive stages, i.e., a pipeline, onto heterogeneous processors, while considering both the performance, measured as throughput, and the reliability. The mechanism of replication, which refers to the mapping of an application stage onto more than one processor, can be used to increase throughput but also to increase reliability. Finding the right replication trade-off plays a pivotal role for this bi-criteria optimization problem. Our formal model includes heterogeneous processors, both in terms of execution speed as well as in terms of reliability. We study the complexity of the various sub problems and show how a solution can be obtained for the polynomial cases. For the general NP-hard problem, heuristics are presented and experimentally evaluated. We further propose the design of an exact algorithm based on A* state space search which allows us to evaluate the performance of our heuristics for small problem instances.

We aim at finding optimal mappings for concurrent streaming applications. Each application consists of a linear chain with several stages, and processes successive data sets in pipeline mode. The objective is to minimize the energy consumption of the whole platform, while satisfying given performance-related bounds on the period and latency of each application. The problem is to decide which processors to enroll, at which speed (or mode) to use them, and which stages they should execute. We distinguish two mapping categories, interval mappings without reuse, and fully arbitrary general mappings. On the theoretical side, we establish complexity results for this tri-criteria mapping problem (energy, period, latency). Furthermore, we derive an integer linear program that provides the optimal solution in the most general case. On the experimental side, we design polynomial-time heuristics, and assess their absolute performance thanks to the linear program. One main goal is to evaluate the impact of processor sharing on the quality of the solution.

In this paper, we investigate how to compute the throughput of probabilistic and replicated streaming applications. We are given (i) a streaming application whose dependence graph is a linear chain; (ii) a one-to-many mapping of the application onto a fully heterogeneous target, where a processor is assigned at most one application stage, but where a stage can be replicated onto a set of processors; and (iii) a set of IID (Independent and Identically-Distributed) variables to model each computation and communication time in the mapping. How can we compute the throughput of the application, i.e., the rate at which data sets can be processed? We consider two execution models, the STRICT model where the actions of each processor are sequentialized, and the OVERLAP model where a processor can compute and communicate in parallel. The problem is easy when application stages are not replicated, i.e., assigned to a single processor: in that case the throughput is dictated by the critical hardware resource. However, when stages are replicated, i.e., assigned to several processors, the problem becomes surprisingly complicated: even in the deterministic case, the optimal throughput may be lower than the smallest internal resource throughput. To the best of our knowledge, the problem has never been considered in the probabilistic case. The first main contribution of the paper is to provide a general method (although of exponential cost) to compute the throughput when mapping parameters follow IID exponential laws. This general method is based upon the analysis of timed Petri nets deduced from the application mapping; it turns out that these Petri nets exhibit a regular structure in the OVERLAP model, thereby enabling to reduce the cost and provide a polynomial algorithm. The second main contribution of the paper is to provide bounds for the throughput when stage parameters are arbitrary IID and NBUE (New Better than Used in Expectation) variables: the throughput is bounded from below by the exponential case and bounded from above by the deterministic case.

Latency, fault tolerance and reliability are important requirements for several applications that are time critical in nature: such applications require guarantees in terms of latency, even when processors are subject to failures. In this paper, we propose a fault-tolerant scheduling heuristic for mapping precedence task graphs on heterogeneous systems. Our approach is based on an active replication scheme, capable of supporting ε arbitrary fail-silent/fail-stop processor failures, and hence valid results will be provided even if ε processors fail. First we focus on a bi-criteria approach, where we aim at minimizing the latency given a fixed number of failures supported in the system, or the other way round. Next we derive a more complex algorithm in which we not only minimize latency and support a fixed number of failures, but also improve the overall reliability. Major achievements include low complexity of the new algorithms, and a drastic reduction of the number of additional communications induced by the replication mechanism. Experimental results demonstrate that our heuristics, despite their lower complexity, outperform their direct competitor, the fault-tolerance based active replication scheduling algorithm FTBAR.

Scheduling problems are already difficult on traditional parallel machines, and they become extremely challenging on heterogeneous clusters. In this paper, we deal with the problem of scheduling multiple applications, made of collections of independent and identical tasks, on a heterogeneous master-worker platform. The applications are submitted online, which means that there is no a priori (static) knowledge of the workload distribution at the beginning of the execution. The objective is to minimize the maximum stretch, i.e., the maximum ratio between the actual time an application has spent in the system and the time this application would have spent if executed 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 introduce a heuristic for the general case of online applications. On the practical side, we have conducted extensive simulations and MPI experiments, showing that we are able to deal with very large problem instances in a few seconds. Also, the solution that we compute totally outperforms classical heuristics from the literature, thereby fully assessing the usefulness of our approach.

Supporting high-performance computing pipelines in wide-area networks is crucial to enabling large-scale distributed scientific applications that require minimizing end-to-end delay for fast user interaction or maximizing frame rate for smooth data flow. We formulate and categorize the linear pipeline configuration problems into six classes with two mapping objectives, i.e. minimum end-to-end delay and maximum frame rate, and three network constraints, i.e. no, contiguous, and arbitrary node reuse. We design a dynamic programming-based optimal solution to the problem of minimum end-to-end delay with arbitrary node reuse and prove the NP-completeness of the rest five problems, for each of which, a heuristic algorithm based on a similar optimization procedure is proposed. These heuristics are implemented and tested on a large set of simulated networks of various scales and their performance superiorities are illustrated by extensive experimental results in comparison with existing methods.

In this paper, we focus on computing the throughput of replicated workflows. Given a streaming application whose dependence graph is a linear chain, and a mapping of this application onto a fully heterogeneous platform, how can we compute the optimal throughput, or equivalently the minimal period? The problem is easy when workflow stages are not replicated, i.e., assigned to a single processor: in that case the period is dictated by the critical hardware resource. But when stages are replicated, i.e., assigned to several processors, the problem gets surprisingly complicated, and we provide examples where the optimal period is larger than the largest cycle-time of any resource. We then show how to model the problem as a timed Petri net to compute the optimal period in the general case, and we provide a polynomial algorithm for the one-port communication model with overlap. Finally, we report comprehensive simulation results on the gap between the optimal period and the largest resource cycle-time.

In this paper, we deal with the problem of scheduling streaming applications on unreliable heterogeneous platforms. We use the realistic one-port model with full computation/communication overlap. We deal with three optimization objectives. The first two, latency and throughput, are performance-related while the third, tolerating a given number of processor failures, is reliability-oriented. The major contribution of this paper is the design of a new scheduling algorithm to minimize latency under both throughput and reliability constraints. We provide a comprehensive set of experimental results, that fully demonstrate the usefulness of the proposed algorithm.

We consider the operator mapping problem for in-network stream processing, i.e., the application of a tree of operators in steady-state to multiple data objects that are continuously updated at various locations in a network. Examples of in-network stream processing include the processing of data in a sensor network, or of continuous queries on distributed relational databases. Our aim is to provide the user a set of processors that should be bought or rented in order to ensure that the application achieves a minimum steady-state throughput, and with the objective of minimizing platform cost. We prove that even the simplest variant of the problem is NP-hard, and we design several polynomial time heuristics, which are evaluated via extensive simulations and compared to theoretical bounds.

In this paper, we explore the problem of mapping filtering services on large-scale heterogeneous platforms. Two important optimization criteria should be considered in such a framework. The period, which is the inverse of the throughput, measures the rate at which data sets can enter the system. The latency measures the response time of the system in order to process one single data set entirely. Both criteria are antagonistic. For homogeneous platforms, the complexity of period minimization is already known [12]; we derive an algorithm to solve the latency minimization problem in the general case with service precedence constraints; we also show that the bi-criteria problem (latency minimization without exceeding a prescribed value for the period) is of polynomial complexity. However, when adding heterogeneity to the platform, we prove that minimizing the period or the latency becomes NP-complete, and that these problems cannot be approximated by any constant factor (unless P=NP). The latter results hold true even for services without precedence constraints.

In this paper, we explore the problem of mapping filtering query services on chains of heterogeneous processors. Two important optimization criteria should be considered in such a framework. The period, which is the inverse of the throughput, measures the rate at which data sets can enter the system. The latency measures the response time of the system in order to process one single data set entirely. We provide a comprehensive set of complexity results for period and latency optimization problems, with proportional or arbitrary computation costs, and without or with communication costs. We present polynomial algorithms for problems whose dependence graph is a linear chain (hence a fixed ordering of the filtering services). For independent services, the problems are all NP-complete except latency minimization with proportional computation costs, which was shown polynomial in [6].

In this paper, we deal with the large-scale divisible load problem studied in. We show how to reduce this problem to a classical preemptive scheduling problem on a single machine, thereby establishing new complexity results, and providing new approximation algorithms and heuristics that subsume those presented in. We also give some hints on how to extend the results to a more realistic framework where communication costs are taken into account.

One has a large workload that is ldquodivisiblerdquo-its constituent work's granularity can be adjusted arbitrarily;-and one has access to p remote computers that can assist in computing the workload. The problem is that the remote computers are subject to interruptions of known likelihood that kill all work in progress. One wishes to orchestrate sharing the workload with the remote computers in a way that maximizes the expected amount of work completed. Strategies for achieving this goal, by balancing the desire to checkpoint often, in order to decrease the amount of vulnerable work at any point, vs. the desire to avoid the context-switching required to checkpoint, are studied. Strategies are devised that provably maximize the expected amount of work when there is only one remote computer (the case p = 1). Results suggest the intractability of such maximization for higher values of p, which motivates the development of heuristic approaches. Heuristics are developed that replicate works on several remote computers, in the hope of thereby decreasing the impact of work-killing interruptions. The quality of these heuristics is assessed through exhaustive simulations.

In this paper, we study the problem of optimizing the throughput for micro-factories subject to failures. The challenge consists in mapping several tasks onto a set of machines. The originality of our approach is the failure model for such applications in which tasks are subject to failures rather than machines. If there is exactly one task per machine in the mapping, then we prove that the optimal solution can be computed in polynomial time. However, the problem becomes NP-hard if several tasks can be assigned to the same machine. Several polynomial time heuristics are presented for the most realistic specialized setting, in which tasks of a same type can be mapped onto the same machine. Experimental results show that the best heuristics obtain a good throughput, much better than the throughput obtained with a random mapping. Moreover, we obtain a throughput close to the optimal solution in the particular cases on which the optimal throughput can be computed.

Supporting high-performance computing pipelines in wide-area networks is crucial to enabling large-scale distributed scientific applications that require minimizing end-to-end delay for fast user interaction or maximizing frame rate for smooth data flow. We formulate and categorize the linear pipeline configuration problems into six classes with two mapping objectives, i.e. minimum end-to-end delay and maximum frame rate, and three network constraints, i.e. no, contiguous, and arbitrary node reuse. We design a dynamic programming-based optimal solution to the configuration problem for minimum end-to-end delay with arbitrary node reuse and prove the NP-completeness of the rest five problems, for each of which, a heuristic algorithm based on a similar optimization procedure is proposed. Performance superiorities of these heuristics are illustrated by extensive experimental results in comparison with existing methods.

In this paper, we explore the problem of mapping filtering streaming applications on large-scale homogeneous platforms, with a particular emphasis on communication models and their impact. Filtering application are streaming applications where each node also has a selectivity which either increases or decreases the size of its input data set. This selectivity makes the problem of scheduling these applications more challenging than the more studied problem of scheduling "non-filtering" streaming workflows. We identify three significant realistic communication models. For each of them, we address the complexity of the following important problems:

This paper presents theoretical results for mapping and scheduling linear workflows onto heterogeneous platforms. We use a realistic architectural model, representative of current multi-threaded systems. Our model has bounded communication capabilities and full computation/communication overlap. In these workflow applications, the goal is often to maximize throughput or to minimize latency. We present several complexity results, and approximation algorithms, for these two criteria. We also consider the implications of adding feedback loops to linear chain applications.

Most list scheduling heuristics rely on a simple platform model where communication contention is not taken into account. In addition, it is generally assumed that processors in the systems are completely safe. To schedule precedence graphs in a more realistic framework, we introduce an efficient fault tolerant scheduling algorithm that is both contention-aware and capable of supporting epsiv arbitrary fail-silent/fail-stop processor failures. We focus on a bi- criteria approach, where we aim at minimizing the total execution time, or latency, given a fixed number of failures supported in the system. Our algorithm has a low time complexity, and drastically reduces the number of additional communications induced by the replication mechanism. Experimental results fully demonstrate the usefulness of the proposed algorithm, which leads to efficient execution schemes while guaranteeing a prescribed level of fault tolerance.

Fault tolerance and latency are important requirements in several applications which are time critical in nature: such applications require guaranties in terms of latency, even when processors are subject to failures. In this paper, we propose a fault tolerant scheduling heuristic for mapping precedence task graphs on heterogeneous systems. Our approach is based on an active replication scheme, capable of supporting epsiv arbitrary fail-silent (fail-stop) processor failures, hence valid results will be provided even if epsiv processors fail. We focus on a bi-criteria approach, where we aim at minimizing the latency given a fixed number of failures supported in the system, or the other way round. Major achievements include a low complexity, and a drastic reduction of the number of additional communications induced by the replication mechanism. Experimental results demonstrate that our heuristics, despite their lower complexity, outperform their direct competitor, the FTBAR scheduling algorithm [3].

Scheduling problems are already difficult on traditional parallel machines. They become extremely challenging on heterogeneous clusters, even when embarrassingly parallel applications are considered. In this paper we deal with the problem of scheduling multiple applications, made of collections of independent and identical tasks, on a heterogeneous master-worker platform. The applications are submitted online, which means that there is no a priori (static) knowledge of the workload distribution at the beginning of the execution. The objective is to minimize the maximum stretch, i.e. the maximum ratio between the actual time an application has spent in the system and the time this application would have spent if executed 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 introduce several heuristics for the general case of online applications. On the practical side, we have conducted extensive simulations and MPI experiments, showing that we are able to deal with very large problem instances in a few seconds. Also, the solution that we compute totally outperforms classical heuristics from the literature, thereby fully assessing the usefulness of our approach.

Mapping applications onto heterogeneous platforms is a difficult challenge, even for simple application patterns such as pipeline graphs. The problem is even more complex when processors are subject to failure during the execution of the application. In this paper, we study the complexity of a bi-criteria mapping which aims at optimizing the latency (i.e., the response time) and the reliability (i.e., the probability that the computation will be successful) of the application. Latency is minimized by using faster processors, while reliability is increased by replicating computations on a set of processors. However, replication increases latency (additional communications, slower processors). The application fails to be executed only if all the processors fail during execution. While simple polynomial algorithms can be found for fully homogeneous platforms, the problem becomes NP- hard when tackling heterogeneous platforms. This is yet another illustration of the additional complexity added by heterogeneity.

In this paper, we discuss and compare several policies to place replicas in tree networks, subject to server capacity and Quality of Service (QoS) constraints. The client requests are known beforehand, while the number and location of the servers are to be determined. The standard approach in the literature is to enforce that all requests of a client be served by the closest server in the tree. We introduce and study two new policies. In the first policy, all requests from a given client are still processed by the same server, but this server can be located anywhere in the path from the client to the root. In the second policy, the requests of a given client can be processed by multiple servers. One major contribution of this paper is to assess the impact of these new policies on the total replication cost. Another important goal is to assess the impact of server heterogeneity. In this paper, we establish several new complexity results, and provide several efficient polynomial heuristics for NP-complete instances of the problem. The absolute performance of these heuristics is assessed by comparison with the optimal solution provided by the formulation of the problem in terms of the solution of an integer linear program.

Mapping applications onto parallel platforms is a challenging problem, even for simple application patterns such as pipeline or fork graphs. Several antagonist criteria should be optimized for workflow applications, such as throughput and latency (or a combination). In this paper, we consider a simplified model with no communication cost, and we provide an exhaustive list of complexity results for different problem instances. Some instances are shown to be NP-hard, thereby exposing the inherent complexity of the mapping problem. We provide polynomial algorithms for other problem instances. Altogether, we provide solid theoretical foundations for the study of mono-criterion or bi-criteria mapping optimization problems.

Mapping workflow applications onto parallel platforms is a challenging problem, even for simple application patterns such as pipeline graphs. Several antagonist criteria should be optimized, such as throughput and latency (or a combination). In this paper, we study the complexity of the bi-criteria mapping problem for pipeline graphs on communication homogeneous platforms. In particular, we assess the complexity of the well-known chains-to-chains problem for different-speed processors, which turns out to be NP-hard. We provide several efficient polynomial bi-criteria heuristics, and their relative performance is evaluated through extensive simulations.

In this paper, we discuss and compare several policies to place replicas in tree networks, subject to server capacity constraints. The client requests are known beforehand, while the number and location of the servers are to be determined. The standard approach in the literature is to enforce that all requests of a client he served by the closest server in the tree. We introduce and study two new policies. In the first policy, all requests from a given client are still processed by the same server, but this server can be located anywhere in the path from the client to the root. In the second policy, the requests of a given client can be processed by multiple servers. One major contribution of this paper is to assess the impact of these new policies on the total replication cost. Another important goal is to assess the impact of server heterogeneity, both from a theoretical and a practical perspective. In this paper, we establish several new complexity results, and provide several efficient polynomial heuristics for NP-complete instances of the problem. These heuristics are compared to an absolute lower bound provided by the formulation of the problem in terms of the solution of an integer linear program.

In grid applications the heterogeneity and potential failures of the computing infrastructure poses significant challenges to efficient scheduling. Performance models have been shown to be useful in providing predictions on which schedules can be based (N. Furmento et al., 2002) and most such techniques can also take account of failures and degraded service. However, when several alternative schedules are to be compared it is vital that the analysis of the models does not become so costly as to outweigh the potential gain of choosing the best schedule. Moreover, it is vital that the modelling approach can scale to match the size and complexity of realistic applications. In this paper, we present a novel method of modelling job execution on grid compute clusters. As previously we use performance evaluation process algebra (PEPA) (J. Hillston, 1996) as the system description formalism, capturing both workload and computing fabric. The novel feature is that we make a continuous approximation of the state space underlying the PEPA model and represent it as a set of ordinary differential equations (ODEs) for solution, rather than a continuous time, but discrete state space, Markov chain.

Any scheduling scheme for grid applications must make implicit or explicit assumptions about both the future behaviour of the application and the future availability and performance of grid resources. This paper describes an approach in which the future application behaviour is constrained by the use of algorithmic skeletons, facilitating modelling with a performance oriented process algebra, and future grid resource performance is predicted by the Network Weather Service (NWS) tool. The concept is illustrated through a case study involving Pipeline and Deal skeletons. A tool is presented which automatically generates and solves a set of models which are parameterised with information obtained from NWS. Some numerical results and timing information on the use of the tool are provided, illustrating the efficacy of this approach.