ILP-based approaches to partitioning recurrent workloads upon heterogeneous multiprocessors

The problem of partitioning systems of independent constrained-deadline sporadic tasks upon heterogeneous multiprocessor platforms is considered. Several different integer linear program (ILP) formulations of this problem, offering different tradeoffs between effectiveness (as quantified by speedup bound) and running time efficiency, are presented

Algorithms for Hierarchical and Semi-Partitioned Parallel Scheduling

We propose a model for scheduling jobs in a parallel machine setting that takes into account the cost of migrations by assuming that the processing time of a job may depend on the specific set of machines among which the job is migrated. For the makespan minimization objective, the model generalizes classical scheduling problems such as unrelated parallel machine scheduling, as well as novel ones such as semi-partitioned and clustered scheduling. In the case of a hierarchical family of machines, we derive a compact integer linear programming formulation of the problem and leverage its fractional relaxation to obtain a polynomial-time 2-approximation algorithm. Extensions that incorporate memory capacity constraints are also discussed

Real-time scheduling with resource sharing on heterogeneous multiprocessors

Consider the problem of scheduling a task set τ of implicit-deadline sporadic tasks to meet all deadlines on a t-type heterogeneous multiprocessor platform where tasks may access multiple shared resources. The multiprocessor platform has m k processors of type-k, where k∈{1,2,…,t}. The execution time of a task depends on the type of processor on which it executes. The set of shared resources is denoted by R. For each task τ i , there is a resource set R i ⊆R such that for each job of τ i , during one phase of its execution, the job requests to hold the resource set R i exclusively with the interpretation that (i) the job makes a single request to hold all the resources in the resource set R i and (ii) at all times, when a job of τ i holds R i , no other job holds any resource in R i . Each job of task τ i may request the resource set R i at most once during its execution. A job is allowed to migrate when it requests a resource set and when it releases the resource set but a job is not allowed to migrate at other times. Our goal is to design a scheduling algorithm for this problem and prove its performance. We propose an algorithm, LP-EE-vpr, which offers the guarantee that if an implicit-deadline sporadic task set is schedulable on a t-type heterogeneous multiprocessor platform by an optimal scheduling algorithm that allows a job to migrate only when it requests or releases a resource set, then our algorithm also meets the deadlines with the same restriction on job migration, if given processors 4×(1+MAXP×⌈|P|×MAXPmin{m1,m2,…,mt}⌉) times as fast. (Here MAXP and |P| are computed based on the resource sets that tasks request.) For the special case that each task requests at most one resource, the bound of LP-EE-vpr collapses to 4×(1+⌈|R|min{m1,m2,…,mt}⌉). To the best of our knowledge, LP-EE-vpr is the first algorithm with proven performance guarantee for real-time scheduling of sporadic tasks with resource sharing on t-type heterogeneous multiprocessors

Scheduling techniques to improve the worst-case execution time of real-time parallel applications on heterogeneous platforms

The key to providing high performance and energy-efficient execution for hard real-time applications is the time predictable and efficient usage of heterogeneous multiprocessors. However, schedulability analysis of parallel applications executed on unrelated heterogeneous multiprocessors is challenging and has not been investigated adequately by earlier works. The unrelated model is suitable to represent many of the multiprocessor platforms available today because a task (i.e., sequential code) may exhibit a different work-case-execution-time (WCET) on each type of processor on an unrelated heterogeneous multiprocessors platform. A parallel application can be realistically modeled as a directed acyclic graph (DAG), where the nodes are sequential tasks and the edges are dependencies among the tasks. This thesis considers a sporadic DAG model which is used broadly to analyze and verify the real-time requirements of parallel applications. A global work-conserving scheduler can efficiently utilize an unrelated platform by executing the tasks of a DAG on different processor types. However, it is challenging to compute an upper bound on the worst-case schedule length of the DAG, called makespan, which is used to verify whether the deadline of a DAG is met or not. There are two main challenges. First, because of the heterogeneity of the processors, the WCET for each task of the DAG depends on which processor the task is executing on during actual runtime. Second, timing anomalies are the main obstacle to compute the makespan even for the simpler case when all the processors are of the same type, i.e., homogeneous multiprocessors. To that end, this thesis addresses the following problem: How we can schedule multiple sporadic DAGs on unrelated multiprocessors such that all the DAGs meet their deadlines. Initially, the thesis focuses on homogeneous multiprocessors that is a special case of unrelated multiprocessors to understand and tackle the main challenge of timing anomalies. A novel timing-anomaly-free scheduler is proposed which can be used to compute the makespan of a DAG just by simulating the execution of the tasks based on this proposed scheduler. A set of representative task-based parallel OpenMP applications from the BOTS benchmark suite are modeled as DAGs to investigate the timing behavior of real-world applications. A simulation framework is developed to evaluate the proposed method. Furthermore, the thesis targets unrelated multiprocessors and proposes a global scheduler to execute the tasks of a single DAG to an unrelated multiprocessors platform. Based on the proposed scheduler, methods to compute the makespan of a single DAG are introduced. A set of representative parallel applications from the BOTS benchmark suite are modeled as DAGs that execute on unrelated multiprocessors. Furthermore, synthetic DAGs are generated to examine additional structures of parallel applications and various platform capabilities. A simulation framework that simulates the execution of the tasks of a DAG on an unrelated multiprocessor platform is introduced to assess the effectiveness of the proposed makespan computations. Finally, based on the makespan computation of a single DAG this thesis presents the design and schedulability analysis of global and federated scheduling of sporadic DAGs that execute on unrelated multiprocessors

An EPTAS for Scheduling on Unrelated Machines of Few Different Types

In the classical problem of scheduling on unrelated parallel machines, a set of jobs has to be assigned to a set of machines. The jobs have a processing time depending on the machine and the goal is to minimize the makespan, that is the maximum machine load. It is well known that this problem is NP-hard and does not allow polynomial time approximation algorithms with approximation guarantees smaller than $1.5$ unless P$=$NP. We consider the case that there are only a constant number $K$ of machine types. Two machines have the same type if all jobs have the same processing time for them. This variant of the problem is strongly NP-hard already for $K=1$. We present an efficient polynomial time approximation scheme (EPTAS) for the problem, that is, for any $\varepsilon > 0$ an assignment with makespan of length at most $(1+\varepsilon)$ times the optimum can be found in polynomial time in the input length and the exponent is independent of $1/\varepsilon$. In particular we achieve a running time of $2^{\mathcal{O}(K\log(K) \frac{1}{\varepsilon}\log^4 \frac{1}{\varepsilon})}+\mathrm{poly}(|I|)$, where $|I|$ denotes the input length. Furthermore, we study three other problem variants and present an EPTAS for each of them: The Santa Claus problem, where the minimum machine load has to be maximized; the case of scheduling on unrelated parallel machines with a constant number of uniform types, where machines of the same type behave like uniformly related machines; and the multidimensional vector scheduling variant of the problem where both the dimension and the number of machine types are constant. For the Santa Claus problem we achieve the same running time. The results are achieved, using mixed integer linear programming and rounding techniques

Provably good task assignment on heterogeneous multiprocessor platforms for a restricted case but with a stronger adversary

Consider the problem of scheduling a set of implicit-deadline sporadic tasks to meet all deadlines on a heterogeneous multiprocessor platform. We consider a restricted case where the maximum utilization of any task on any processor in the system is no greater than one. We use an algorithm proposed in [1] (we refer to it as LP-EE) from state-of-the-art for assigning tasks to heterogeneous multiprocessor platform and (re-)prove its performance guarantee for this restricted case but for a stronger adversary. We show that if a task set can be scheduled to meet deadlines on a heterogeneous multiprocessor platform by an optimal task assignment scheme that allows task migrations then LP-EE meets deadlines as well with no migrations if given processors twice as fast

Approximate feasibility in real-time scheduling: Speeding up in order to meet deadlines

Stougie, L. [Promotor]Marchetti-Spaccamela, A. [Promotor

A PTAS for assigning sporadic tasks on two-type heterogeneous multiprocessors

Consider the problem of determining a task-toprocessor assignment for a given collection of implicit-deadline sporadic tasks upon a multiprocessor platform in which there are two distinct kinds of processors. We propose a polynomialtime approximation scheme (PTAS) for this problem. It offers the following guarantee: for a given task set and a given platform, if there exists a feasible task-to-processor assignment, then given an input parameter, ϵ, our PTAS succeeds, in polynomial time, in finding such a feasible task-to-processor assignment on a platform in which each processor is 1+3ϵ times faster. In the simulations, our PTAS outperforms the state-of-the-art PTAS [1] and also for the vast majority of task sets, it requires significantly smaller processor speedup than (its upper bound of) 1+3ϵ for successfully determining a feasible task-to-processor assignment

A conjecture about provably good task assignment on heterogeneous multiprocessor platforms but with a stronger adversary

Consider the problem of scheduling a set of implicit-deadline sporadic tasks to meet all deadlines on a heterogeneous multiprocessor platform. We use an algorithm proposed in [1] (we refer to it as LP-EE) from state-of-the-art for assigning tasks to heterogeneous multiprocessor platform and (re-)prove its performance guarantee but for a stronger adversary.We conjecture that if a task set can be scheduled to meet deadlines on a heterogeneous multiprocessor platform by an optimal task assignment scheme that allows task migrations then LP-EE meets deadlines as well with no migrations if given processors twice as fast. We illustrate this with an example

Assigning real-time tasks on heterogeneous multiprocessors with two types of processors

Consider the problem of scheduling a set of implicitdeadline sporadic tasks on a heterogeneous multiprocessor so as to meet all deadlines. Tasks cannot migrate and the platform is restricted in that each processor is either of type-1 or type-2 (with each task characterized by a different speed of execution upon each type of processor). We present an algorithm for this problem with a timecomplexity of O(n·m), where n is the number of tasks and m is the number of processors. It offers the guarantee that if a task set can be scheduled by any non-migrative algorithm to meet deadlines then our algorithm meets deadlines as well if given processors twice as fast. Although this result is proven for only a restricted heterogeneous multiprocessor, we consider it significant for being the first realtime scheduling algorithm to use a low-complexity binpacking approach to schedule tasks on a heterogeneous multiprocessor with provably good performance