The periodicity and robustness in a single-track train scheduling problem

The most important operating problem in any railway industry is to produce robust train timetables with minimum delays. The train scheduling problem is defined as an application of job shop scheduling which is considered to be one of the most interesting research topics. This paper deals with scheduling different types of trains in a single railway track. The authors have focused on the robust and periodic aspects of produced timetables. This paper is also concerned with some applicable constraints, such as the acceleration and deceleration times, station capacity and headway constraints. The periodic timetable for railways is modeled based on the periodic event scheduling problem (PESP). Furthermore, a fuzzy approach is used to reach a tradeoff among the total train delays, the robustness of schedules, and the time interval between departures of trains from the same origins. To solve large-scale problems, a meta-heuristic algorithm based on simulated annealing (SA) is utilized and validated using some numerical examples on a periodic robust train scheduling problem. Finally, a robustness measure is defined in order to assure the effectiveness of the proposed SA to find robust solutions.     

Solving a periodic single-track train timetabling problem by an efficient hybrid algorithm

Train timetabling with minimum delays is the most important operating problem in any railway industry. This problem is considered to be one of the most interesting research topics in railway optimization problems. This paper deals with scheduling different types of trains in a single railway track. The primary focus of this paper is on the periodic aspects of produced timetables and the proposed modeling is based on the periodic event scheduling problem (PESP). To solve large-scale problems, a hybrid meta-heuristic algorithm based on simulated annealing (SA) and particle swarm optimization (PSO) is proposed and validated using some numerical examples and an Iranian case study that covers the railway line between two cities of Isfahan and Tehran.     

Single-machine scheduling with periodic maintenance to minimize makespan

We consider a single-machine scheduling problem with periodic maintenance activities. Although the scheduling problem with maintenance has attracted researchers’ attention, most of past studies considered only one maintenance period. In this research several maintenance periods are considered where each maintenance activity is scheduled after a periodic time interval. The objective is to find a schedule that minimizes the makespan, subject to periodic maintenance and nonresumable jobs. We first prove that the worst-case ratio of the classical LPT   algorithm is 2. Then we show that there is no polynomial time approximation algorithm with a worst-case ratio less than 2 unless P=NP$P=\text{<italic>NP</italic>}$, which implies that the LPT algorithm is the best possible.     

Deadline-based scheduling of periodic task systems on multiprocessors

We consider the problem of scheduling periodic task systems on multiprocessors and present a deadline-based scheduling algorithm for solving this problem. We show that our algorithm successfully schedules on m processors any periodic task system with utilization at most m²/(2m−1).     

Applying a finite-horizon numerical optimization method to a periodic optimal control problem

Computing a numerical solution to a periodic optimal control problem can be difficult, especially when the period is unknown. A method of approximating a solution to a stochastic optimal control problem using Markov chains was developed in [Krawczyk, J. B. (2001). A Markovian approximated solution to a portfolio management problem. Information Technology for Economics and Management, 1, http://www.item.woiz.polsl.pl/issue/journal1.htm]. This paper describes the application of that method to a periodic optimal control problem formulated in [Gaitsgory, V. & Rossomakhine, S. (2006). Linear programming approach to deterministic long run average problems of optimal control. SIAM Journal on Control and Optimization, 44(6), 2006-2037]. As a result, approximately optimal feedback rules are computed that can control the system both on and off the optimal orbit.     

An optimal boundary fair scheduling algorithm for multiprocessor real-time systems

With the emergence of multicore processors, the research on multiprocessor real-time scheduling has caught more researchers’ attention recently. Although the topic has been studied for decades, it is still an evolving research field with many open problems. In this work, focusing on periodic real-time tasks with quantum-based computation requirements and implicit deadlines, we propose a novel optimal scheduling algorithm, namely boundary fair (Bfair), which can achieve full system utilization as the well-known Pfair scheduling algorithms. However, different from Pfair algorithms that make scheduling decisions and enforce proportional progress (i.e., fairness) for all tasks at each and every time unit, Bfair makes scheduling decisions and enforces fairness to tasks only at tasks’ period boundaries (i.e., deadlines of periodic tasks). The correctness of the Bfair algorithm to meet the deadlines of all tasks’ instances is formally proved and its performance is evaluated through extensive simulations. The results show that, compared to that of Pfair algorithms, Bfair can significantly reduce the number of scheduling points (by up to 94%) and the overhead of Bfair at each scheduling point is comparable to that of the most efficient Pfair algorithm (i.e., PD²). Moreover, by aggregating the time allocation of tasks for the time interval between consecutive period boundaries, the resulting Bfair schedule can dramatically reduce the number of required context switches and task migrations (as much as 82% and 85%, respectively) when compared to those of Pfair schedules, which in turn reduces the run-time overhead of the system.     

Periodic sensor switching and the random walk

Assume the random walk system is subjected to periodic sensor switching between a low-quality sensor with no delay, and a high-quality sensor with a delay. The low-quality sensor is utilized N−1$N-1$ consecutive times followed by one utilization of the high-quality sensor. A periodic Kalman filter is employed. Necessary and sufficient conditions are derived for the existence of scheduling sizes N$N$ such that the resulting periodic system has a steady state error covariance strictly smaller than both the steady state error covariances of the systems obtained with each sensor applied individually. The relationship between this result and recent developments in event driven scheduling is also provided.     

Converging to periodic schedules for cyclic scheduling problems with resources and deadlines

Cyclic scheduling has been widely studied because of the importance of applications in manufacturing systems and in computer science. For this class of problems, a finite set of tasks with precedence relations and resource constraints must be executed repetitively while maximizing the throughput. Many applications also require that execution schedules be periodic i.e. the execution of each task is repeated with a fixed global period w.The present paper develops a new method to build periodic schedules with cumulative resource constraints, periodic release dates and deadlines. The main idea is to fix the period w, to unwind the cyclic scheduling problem for some number of iterations, and to add precedence relations so that the minimum time lag between two successive executions of any task equals w. Then, using any usual (not cyclic) scheduling algorithm to compute task starting times for the unwound problem, we prove that either the method converges to a periodic schedule of period w or it fails to compute a schedule. A non-polynomial upper bound on the number of iterations to unwind in order to guarantee that cyclic precedence relations and resource constraints are fulfilled is also provided. This method is successfully applied to a real-life problem, namely the software pipelining of inner loops on an embedded VLIW processor core by using a Graham list scheduling algorithm.     

Fuzzy job shop scheduling problem with availability constraints

This paper presents the fuzzy job shop scheduling problem with availability constraints. The objective is to find a schedule that maximizes the minimum agreement index subject to periodic maintenance, non-resumable jobs and fuzzy due-date. A random key genetic algorithm (RKGA) is proposed for the problem, in which a novel random key representation, a new decoding strategy incorporating maintenance operation and discrete crossover (DX) are used. RKGA is applied to some fuzzy scheduling problem with availability constraints and compared with other algorithms. Computational results show that RKGA performs better than other algorithms.     

DP-Fair: a unifying theory for optimal hard real-time multiprocessor scheduling

We consider the problem of optimal real-time scheduling of periodic and sporadic tasks on identical multiprocessors. A number of recent papers have used the notions of fluid scheduling and deadline partitioning to guarantee optimality and improve performance. This article develops a unifying theory with the DP-Fair scheduling policy and examines how it overcomes problems faced by greedy scheduling algorithms. In addition, we present DP-Wrap, a simple DP-Fair scheduling algorithm which serves as a least common ancestor to other recent algorithms. The DP-Fair scheduling policy is extended to address the problem of scheduling sporadic task sets with arbitrary deadlines.     

Static scheduling of directed acyclic data flow graphs onto multiprocessors using particle swarm optimization

An efficient method based on particle swarm optimization (PSO) is developed to solve the Multiprocessor Task Scheduling Problem (MPTSP). To efficiently execute parallelized programs on a multiprocessor environment, a scheduling problem must be solved to determine the assignment of tasks to the processors, the execution order of the tasks, and the starting time of each task, such that some optimality criteria are met. The scheduling problem is known as an NP-complete problem even when the target processors are fully connected and no communication delay is considered among the tasks in the task graph. The complexity of the scheduling problem depends on the number of tasks (N), the number of processors (M), the task processing time and the precedence constraints. The Directed Acyclic Graph (DAG) was exploited to represent the tasks and their precedence constraints. The proposed algorithm was compared with the Genetic Algorithm (GA) and the Duplication Scheduling Heuristic (DSH). We also provide a systematic investigation on the effect of varying problem settings. The results show that the proposed algorithm could not outperform the DSH while it could outperform the GA in some cases.     

A new algorithm for scheduling periodic, real-time tasks

We consider the problem of preemptively scheduling a set of periodic, real-time tasks on a multiprocessor computer system. We give a new scheduling algorithm, the so-called Slack-Time Algorithm, and show that it is more effective than the known Deadline Algorithm. We also give an (exponential-time) algorithm to decide if a task system is schedulable by the Slack-Time or the Deadline Algorithm. The same algorithm can also be used to decide if a task system is schedulable by any given fixed-priority scheduling algorithm. This resolves an open question posed by Leung and Whitehead. Finally, it is shown that the problem of deciding if a task system is schedulable by the Slack-Time, the Deadline, or any given fixed-priority scheduling algorithm is co-NP-hard for each fixedm.     

The periodic optimality of LQ controllers satisfying strong stabilization

An LQ strong stabilization problem is proposed. To determine when a controller with periodic gains is locally superior to a linear time invariant compensator for this problem, a Π test is presented. For systems with strictly proper transfer functions, it is proven that the frequency range where stable periodic controllers based on weak variations about the LTI case can give better performance than stable LTI compensators is finite. In the development, a means to evaluate the second partials of functions with respect to matrix-valued parameters is introduced. For those systems where periodic control is warranted, techniques for designing optimal periodic strongly stabilizing controllers are presented. Two examples detailing the application of the Π test are provided, as well as an optimal periodic controller design example.     

Two-machine flowshop scheduling with bicriteria problem

This paper attempts to solve a two-machine flowshop bicriteria scheduling problem with release dates for the jobs, in which the objective function is to minimize a weighed sum of total flow time and makespan. To tackle this scheduling problem, an integer programming model with N²+3N variables and 5N constraints where N is the number of jobs, is formulated. Because of the lengthy computing time and high computing complexity of the integer programming model, a heuristic scheduling algorithm is presented. Experimental results show that the proposed heuristic algorithm can solve this problem rapidly and accurately. The average solution quality of the heuristic algorithm is above 99% and is much better than that of the SPT rule as a benchmark. A 15-job case requires only 0.018 s, on average, to obtain an ultimate or even optimal solution. The heuristic scheduling algorithm is a more practical approach to real world applications than the integer programming model.     

Preference-oriented real-time scheduling and its application in fault-tolerant systems

In this paper, we consider a set of real-time periodic tasks where some tasks are preferably executed as soon as possible (ASAP) and others as late as possible (ALAP) while still meeting their deadlines. After introducing the idea of preference-oriented (PO) execution, we formally define the concept of PO-optimality. For fully-loaded systems (with 100% utilization), we first propose a PO-optimal scheduler, namely ASAP-Ensured Earliest Deadline (SEED), by focusing on ASAP tasks where the optimality of ALAP tasks’ preference is achieved implicitly due to the harmonicity of the PO-optimal schedules for such systems. Then, for under-utilized systems (with less than 100% utilization), we show the discrepancies between different PO-optimal schedules. By extending SEED, we propose a generalized Preference-Oriented Earliest Deadline (POED) scheduler that can obtain a PO-optimal schedule for any schedulable task set. The application of the POED scheduler in a dual-processor fault-tolerant system is further illustrated. We evaluate the proposed PO-optimal schedulers through extensive simulations. The results show that, comparing to that of the well-known EDF scheduler, the scheduling overheads of SEED and POED are higher (but still manageable) due to the additional consideration of tasks’ preferences. However, SEED and POED can achieve the preference-oriented execution objectives in a more successful way than EDF.     

Algorithms and reductions for rewriting problems. II

In this paper we give polynomial-time reductions between a version of joinability for rewrite systems and the word problem for rewrite systems. We prove log-space hardness or completeness for P for several problems of ground rewrite systems. We show that matching (and unification) modulo ground equations is NP-hard even when variables are restricted to at most two occurrences in the pattern and the subject is just a constant. Finally, we give the first polynomial-time algorithms for matching modulo ground equations with linear pattern and for joinability problem with ground rewrite systems. The joinability result leads to polynomial time algorithms for: local confluence of ground rewrite systems, confluence of terminating ground rewrite systems, and completeness of ground rewrite systems. The results for matching modulo ground equations are optimal with respect to occurrences of variables.     

Minimizing the makespan on a single machine with flexible maintenances and jobs’ release dates

This paper addresses the production scheduling problem on a single machine with flexible periodic preventive maintenances (PM), where jobs’ release dates are also considered. Both resumable and non-resumable cases are studied. For the resumable case, it is proved that the problem can be solved in polynomial time with Earliest Release Date (ERD) rule. For the non-resumable case, it is proved to be NP-Hard in strong sense. And, a mixed integer programming (MIP) mathematical model is provided. Then, an effective heuristic ERD-LPT based on the properties of optimal solution is proposed. Meanwhile, a branch-and-bound algorithm (B and B) that utilizes several dominance rules is developed to search the optimal schedule for small-to-medium sized problems. Computational results indicate that the proposed heuristic is highly accurate and the two algorithms are complementary in dealing with different sized problems. Furthermore, the improvement of the integration between production scheduling and PM is significant compared with the First-in-First-out (FIFO) rule which is adopted commonly in industry.     

避免模调度中cache代价的优化方法

软件流水能够加快循环的执行速度.模调度是一种被广泛采用的软件流水的启发式.为了改善存储系统,cache使用了分级机制,但这也带来了额外的存储延迟-cache代价.证明了模调度可能导致cache代价,并提出了一种可以避免模调度的cache代价的PCPMS(prevent cache penalty in modulo scheduling)算法.实验结果表明,PCPMS能够避免模调度中的cache代价,提高程序性能.