EDF-schedulability of synchronous periodic task systems is coNP-hard

In the synchronous periodic task model, a set \tau_1,...,\tau_n of tasks is given, each releasing jobs of running time c_i with relative deadline d_i, at each integer multiple of the period p_i. It is a classical result that Earliest Deadline First (EDF) is an optimal preemptive uni-processor scheduling policy. For constrained deadlines, i.e. d_i = 0: \sum_{i=1}^n (floor(Q-d_i)/p_i) + 1) * c_i <= Q. Though an enormous amount of literature deals with this topic, the complexity status of this test has remained unknown. We prove that testing EDF-schedulability of such a task system is (weakly) coNP-hard. This solves Problem 2 from the survey "Open Problems in Real-time Scheduling" by Baruah & Pruhs. The hardness result is achieved by applying recent results on inapproximability of Diophantine approximation

EDF-schedulability of synchronous periodic task systems is coNP-hard

In the synchronous periodic task model, a set τ1,...,τn of tasks is given, each releasing jobs of running time ci with relative deadline di, at each integer multiple of the period pi. It is a classical result that Earliest Deadline First (EDF) is an optimal preemptive uniprocessor scheduling policy. For constrained deadlines, i.e. di ≤ pi, the EDF-schedule is feasible if and only if n∑ Q − d

A generalized parallel task model for recurrent real-time processes

A model is considered for representing recurrent precedence-constrained tasks that are to execute on multiprocessor platforms. A recurrent task is specified as a directed acyclic graph (DAG), a period, and a relative deadline. Each vertex of the DAG represents a sequential job, while the edges of the DAG represent precedence constraints between these jobs. All the jobs of the DAG are released simultaneously and need to complete execution within the specified relative deadline of their release. Each task may release j

Approximation Algorithms for Modern Multi-Processor Scheduling Problems

This thesis is devoted to the design and analysis of algorithms for scheduling problems. These problems are ubiquitous in the modern world. Examples include the optimization of local transportation, managing access to concurrent resources like runways at airports and efficient execution of computing tasks on server systems. Problem instances that appear in the real world often are so large and complex that it is not possible to solve them “by hand”. This rises the need for strong algorithmic approaches, which motivates our focus of study. In this work we consider two types of scheduling problems which gained in importance due to recent technological advances. The first problem comes from the avionics industry and deals with scheduling periodically recurring tasks in a parallel computer network on a plane: Each task comes with a period p and execution time c, and needs to use a processor exclusively for c time units every p time units. The scheduling problem is to assign starting offsets for the first execution of the tasks so that no collision occurs. The second problem is a scheduling problem that arises in highly parallelized processing environments with a shared common resource, e.g., modern multi-core computer architectures. In addition to classical makespan minimization problems such as scheduling on identical machines, each job has an additional resource constraint. The scheduler must ensure that at no time, the accumulated requirement of all active jobs at that time exceeds a given limit. For both types of problems we study their algorithmic complexity in a mathematical, rigorous way by designing approximation algorithms and establishing inapproximability results. We thereby give a characterization of the approximation landscape of these problems. We also consider a more practical perspective: For an engineer from the industry, a rigorous proof that an algorithm finds a solution of certain guaranteed quality for all possible kinds of problem instances is usually not that relevant. It is rather of interest to find “good enough” or even optimal solutions for particular instances that actually appear in the real world in “reasonable” time. We show that structural insights gained in the more theoretical process of designing approximation algorithms can be highly beneficial also for obtaining practical results. In particular, we develop integer programming formulations for the avionics problem based on structural properties revealed in the design of approximation algorithms. These formulations lead to strong tools that, for the first time, enable to algorithmically solve real-world instances from our industrial partner

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

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

Efficient Allocation And Enforcement Of Interfaces In Compositional Real-Time Systems

Compositional real-time research has become one of the emerging trends in embedded and real-time systems due to the increasing scale and complexity of such systems. In this design paradigm, a large system is decomposed into smaller and simpler components, each of which abstracts their temporal requirements via interfaces. Such systems are mostly implemented by resource partitions to ensure that the components receive resources according to their interfaces. Potential implementations of a resource partition are via server-based interfaces or demand-based interfaces. In this context, our thesis in this dissertation is as follows: Currently, server-based interfaces ensure strong temporal isolation among components at the cost of resource over-provisioning whereas demand-based interfaces precisely model the resource demand of a component without the guarantee of temporal isolation. For both these models, efficient and effective resource allocation as well as strict temporal isolation among components can be achieved. Specifically, we can obtain efficient and near-optimal bandwidth allocation schemes and admission controllers for periodic resource model and arbitrary demand-based interface respectively. Furthermore, efficient slack reclamation technique can be obtained to allocate unused processing resources at runtime while still enforcing the given interface. To support our thesis, we address efficient resource allocation among components with server-based interfaces by providing fully-polynomial-time approximation schemes (FPTAS) for allocating processing resource to components scheduled by earliest-deadline-first (EDF) or fixed-priority (FP) scheduling algorithm. For enforcing temporal isolation of demand-based interfaces, we provide a parametric approximate admission control algorithm, which has polynomial-time complexity in terms of number of active jobs in the system and the approximation parameter. Finally, to address efficient reclamation of unused processing resources, we give a novel technique to optimally and efficiently determine maximum allowable runtime slack for a component with arbitrary interface, considering active jobs in the system and guaranteeing system schedulability even for worst-case future job arrival scenarios. We expect that these techniques can ultimately be used to minimize the size, weight, and power requirements of real-time and embedded systems by reducing the processing resource requirements of such systems

Blocking analysis of spin locks under partitioned fixed-priority scheduling

Partitioned fixed-priority scheduling is widely used in embedded multicore real-time systems. In multicore systems, spin locks are one well-known technique used to synchronize conflicting accesses from different processor cores to shared resources (e.g., data structures). The use of spin locks can cause blocking. Accounting for blocking is a crucial part of static analysis techniques to establish correct temporal behavior. In this thesis, we consider two aspects inherent to the partitioned fixed-priority scheduling of tasks sharing resources protected by spin locks: (1) the assignment of tasks to processor cores to ensure correct timing, and (2) the blocking analysis required to derive bounds on the blocking. Heuristics commonly used for task assignment fail to produce assignments that ensure correct timing when shared resources protected by spin locks are used. We present an optimal approach that is guaranteed to find such an assignment if it exists (under the original MSRP analysis). Further, we present a well-performing and inexpensive heuristic. For most spin lock types, no blocking analysis is available in prior work, which renders them unusable in real-time systems. We present a blocking analysis approach that supports eight different types and is less pessimistic than prior analyses, where available. Further, we show that allowing nested requests for FIFO- and priority-ordered locks renders the blocking analysis problem NP-hard.Partitioned Fixed-Priority Scheduling ist in eingebetteten Multicore-Echtzeitsystemen weit verbreitet. In Multicore-Systemen sind Spinlocks ein bekannter Mechanismus um konkurrierende Zugriffe von unterschiedlichen Prozessorkernen auf geteilte Resourcen (z.B. Datenstrukturen) zu koordinieren. Bei der Nutzung von Spinlocks können Blockierungen auftreten, die in statischen Analysetechniken zum Nachweis des korrekten zeitlichen Verhaltens eines Systems zu berücksichtigen sind. Wir betrachten zwei Aspekte von Partitioned Fixed-Priority Scheduling in Verbindung mit Spinlocks zum Schutz geteilter Resourcen: (1) die Zuweisung von Tasks zu Prozessorkernen unter Einhaltung zeitlicher Vorgaben und (2) die Analyse zur Entwicklung oberer Schranken für die Blockierungsdauer. Übliche Heuristiken finden bei der Nutzung von Spinlocks oft keine Taskzuweisung, bei der die Einhaltung zeitlicher Vorgaben garantiert ist. Wir stellen einen optimalen Ansatz vor, der dies (mit der ursprünglichen MSRP Analyse) garantiert, falls eine solche Zuweisung existiert. Zudem präsentieren wir eine leistungsfähige Heuristik. Die meisten Arten von Spinlocks können mangels Analyse der Blockierungsdauer nicht für Echtzeitsysteme verwendet werden. Wir stellen einen Analyseansatz vor, der acht Spinlockarten unterstützt und weniger pessimistische Schranken liefert als vorherige Analysen, soweit vorhanden. Weiterhin zeigen wir, dass die Analyse bei verschachtelten Zugriffen mit FIFO- und prioritäts-geordneten Locks ein NP-hartes Problem ist