Group-EDF: A New Approach and an Efficient Non-Preemptive Algorithm for Soft Real-Time Systems

Hard real-time systems in robotics, space and military missions, and control devices are specified with stringent and critical time constraints. On the other hand, soft real-time applications arising from multimedia, telecommunications, Internet web services, and games are specified with more lenient constraints. Real-time systems can also be distinguished in terms of their implementation into preemptive and non-preemptive systems. In preemptive systems, tasks are often preempted by higher priority tasks. Non-preemptive systems are gaining interest for implementing soft-real applications on multithreaded platforms. In this dissertation, I propose a new algorithm that uses a two-level scheduling strategy for scheduling non-preemptive soft real-time tasks. Our goal is to improve the success ratios of the well-known earliest deadline first (EDF) approach when the load on the system is very high and to improve the overall performance in both underloaded and overloaded conditions. Our approach, known as group-EDF (gEDF), is based on dynamic grouping of tasks with deadlines that are very close to each other, and using a shortest job first (SJF) technique to schedule tasks within the group. I believe that grouping tasks dynamically with similar deadlines and utilizing secondary criteria, such as minimizing the total execution time can lead to new and more efficient real-time scheduling algorithms. I present results comparing gEDF with other real-time algorithms including, EDF, best-effort, and guarantee scheme, by using randomly generated tasks with varying execution times, release times, deadlines and tolerances to missing deadlines, under varying workloads. Furthermore, I implemented the gEDF algorithm in the Linux kernel and evaluated gEDF for scheduling real applications

New Results on Online Resource Minimization

We consider the online resource minimization problem in which jobs with hard deadlines arrive online over time at their release dates. The task is to determine a feasible schedule on a minimum number of machines. We rigorously study this problem and derive various algorithms with small constant competitive ratios for interesting restricted problem variants. As the most important special case, we consider scheduling jobs with agreeable deadlines. We provide the first constant ratio competitive algorithm for the non-preemptive setting, which is of particular interest with regard to the known strong lower bound of n for the general problem. For the preemptive setting, we show that the natural algorithm LLF achieves a constant ratio for agreeable jobs, while for general jobs it has a lower bound of Omega(n^(1/3)). We also give an O(log n)-competitive algorithm for the general preemptive problem, which improves upon the known O(p_max/p_min)-competitive algorithm. Our algorithm maintains a dynamic partition of the job set into loose and tight jobs and schedules each (temporal) subset individually on separate sets of machines. The key is a characterization of how the decrease in the relative laxity of jobs influences the optimum number of machines. To achieve this we derive a compact expression of the optimum value, which might be of independent interest. We complement the general algorithmic result by showing lower bounds that rule out that other known algorithms may yield a similar performance guarantee

Energy-efficient algorithms for non-preemptive speed-scaling

We improve complexity bounds for energy-efficient speed scheduling problems for both the single processor and multi-processor cases. Energy conservation has become a major concern, so revisiting traditional scheduling problems to take into account the energy consumption has been part of the agenda of the scheduling community for the past few years. We consider the energy minimizing speed scaling problem introduced by Yao et al. where we wish to schedule a set of jobs, each with a release date, deadline and work volume, on a set of identical processors. The processors may change speed as a function of time and the energy they consume is the $\alpha$th power of its speed. The objective is then to find a feasible schedule which minimizes the total energy used. We show that in the setting with an arbitrary number of processors where all work volumes are equal, there is a $2(1+\varepsilon)(5(1+\varepsilon))^{\alpha -1}\tilde{B}_{\alpha}=O_{\alpha}(1)$ approximation algorithm, where $\tilde{B}_{\alpha}$ is the generalized Bell number. This is the first constant factor algorithm for this problem. This algorithm extends to general unequal processor-dependent work volumes, up to losing a factor of $(\frac{(1+r)r}{2})^{\alpha}$ in the approximation, where $r$ is the maximum ratio between two work volumes. We then show this latter problem is APX-hard, even in the special case when all release dates and deadlines are equal and $r$ is 4. In the single processor case, we introduce a new linear programming formulation of speed scaling and prove that its integrality gap is at most $12^{\alpha -1}$. As a corollary, we obtain a $(12(1+\varepsilon))^{\alpha -1}$ approximation algorithm where there is a single processor, improving on the previous best bound of $2^{\alpha-1}(1+\varepsilon)^{\alpha}\tilde{B}_{\alpha}$ when $\alpha \ge 25$

Throughput Maximization in Multiprocessor Speed-Scaling

We are given a set of $n$ jobs that have to be executed on a set of $m$ speed-scalable machines that can vary their speeds dynamically using the energy model introduced in [Yao et al., FOCS'95]. Every job $j$ is characterized by its release date $r_j$, its deadline $d_j$, its processing volume $p_{i,j}$ if $j$ is executed on machine $i$ and its weight $w_j$. We are also given a budget of energy $E$ and our objective is to maximize the weighted throughput, i.e. the total weight of jobs that are completed between their respective release dates and deadlines. We propose a polynomial-time approximation algorithm where the preemption of the jobs is allowed but not their migration. Our algorithm uses a primal-dual approach on a linearized version of a convex program with linear constraints. Furthermore, we present two optimal algorithms for the non-preemptive case where the number of machines is bounded by a fixed constant. More specifically, we consider: {\em (a)} the case of identical processing volumes, i.e. $p_{i,j}=p$ for every $i$ and $j$, for which we present a polynomial-time algorithm for the unweighted version, which becomes a pseudopolynomial-time algorithm for the weighted throughput version, and {\em (b)} the case of agreeable instances, i.e. for which $r_i \le r_j$ if and only if $d_i \le d_j$, for which we present a pseudopolynomial-time algorithm. Both algorithms are based on a discretization of the problem and the use of dynamic programming

Throughput Maximization in the Speed-Scaling Setting

We are given a set of $n$ jobs and a single processor that can vary its speed dynamically. Each job $J_j$ is characterized by its processing requirement (work) $p_j$, its release date $r_j$ and its deadline $d_j$. We are also given a budget of energy $E$ and we study the scheduling problem of maximizing the throughput (i.e. the number of jobs which are completed on time). We propose a dynamic programming algorithm that solves the preemptive case of the problem, i.e. when the execution of the jobs may be interrupted and resumed later, in pseudo-polynomial time. Our algorithm can be adapted for solving the weighted version of the problem where every job is associated with a weight $w_j$ and the objective is the maximization of the sum of the weights of the jobs that are completed on time. Moreover, we provide a strongly polynomial time algorithm to solve the non-preemptive unweighed case when the jobs have the same processing requirements. For the weighted case, our algorithm can be adapted for solving the non-preemptive version of the problem in pseudo-polynomial time.Comment: submitted to SODA 201

On the periodic behavior of real-time schedulers on identical multiprocessor platforms

This paper is proposing a general periodicity result concerning any deterministic and memoryless scheduling algorithm (including non-work-conserving algorithms), for any context, on identical multiprocessor platforms. By context we mean the hardware architecture (uniprocessor, multicore), as well as task constraints like critical sections, precedence constraints, self-suspension, etc. Since the result is based only on the releases and deadlines, it is independent from any other parameter. Note that we do not claim that the given interval is minimal, but it is an upper bound for any cycle of any feasible schedule provided by any deterministic and memoryless scheduler