Optimal Time-Critical Scheduling via Resource Augmentation

Abstract. We consider two fundamental problems in dynamic scheduling: scheduling to meet deadlines in a preemptive multiprocessor setting, and scheduling to provide good response time in a number of scheduling environments. When viewed from the perspective of traditional worst-case analysis, no good on-line algorithms exist for these problems, and for some variants no good off-line algorithms exist unless P = NP . We study these problems using a relaxed notion of competitive analysis, introduced by Kalyanasundaram and Pruhs, in which the on-line algorithm is allowed more resources than the optimal off-line algorithm to which it is compared. Using this approach, we establish that several well-known on-line algorithms, that have poor performance from an absolute worst-case perspective, are optimal for the problems in question when allowed moderately more resources. For optimization of average flow time, these are the first results of any sort, for any NP -hard version of the problem, that indicate that it might be possible to design good approximation algorithms.     

Balanced Scheduling toward Loss-Free Packet Queuing and Delay Fairness

In current networks, packet losses can occur if routers do not provide sufficiently large buffers. This paper studies how many buffers should be provided in a router to eliminate packet losses. We assume a network router has m incoming queues, each corresponding to a single traffic stream, and must schedule at any time on-line from which queue to take the next packet to send out. To exclude packet losses with a small amount of buffers, the maximum queue length must be kept low over the entire scheduling period. We call this new on-line problem the balanced scheduling problem (BSP). By competitive analysis, we measure the power of on-line scheduling algorithms to prevent packet losses. We show that a simple greedy algorithm is Θ(log m)-competitive which is asymptotically optimal, while Round-Robin scheduling is not better than m-competitive, as actually is any deterministic on-line algorithm for BSP. We also give a polynomial time algorithm for solving off-line BSP optimally. We also study another on-line balancing problem that tries to balance the delay among the m traffic streams.     

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.     

Classifying the multi robot path finding problem into a quadratic competitive complexity class

We explore two motion planning problems where a group of mobile robots has to reach a target located in an a priori unknown environment while on-line planning the next step. In the first problem the target position is unknown and should be found by the robots, while in the second problem the target position is known and only a path to it should be found. We focus on optimizing the cost of the task in terms of motion time, which, under the assumption of uniform velocity of all the robots, correlates to the path length passed by the robot which reaches the target. The performance of an on-line algorithm is usually expressed in terms of Competitiveness, the constant ratio between the on-line and the optimal off-line solutions. Specifically, the ratio between the lengths of the actual path made by the robot which reached the target to the shortest path to the target. We use generalized competitiveness, i.e., the ratio is not necessarily constant, but could be any function. Classification of a motion planning task in the sense of performance is done by finding an upper and a lower bounds on the competitiveness of all algorithms solving that task. If the two bounds belong to the same functional class this is the Competitive Complexity Class of the task. We find the two bounds for the aforementioned common on-line motion planning problems, and classify them into competitive classes. It is shown that in general any on-line motion planning algorithm that tries to solve these problems must have at least a quadratic competitive performance. This is a lower bound of the problems. This paper describes two new on-line navigation algorithm which solve the problems under discussion. The first is called MRSAM, short for Multi-Robot Search Area Multiplication, and the second is called MRBUG, short for Multi-Robot BUG which extends Lumelsky famous BUG algorithm. Both algorithms have quadratic upper bounds, which prove that the problems they solve have quadratic upper bounds. Thus it is shown that navigation in an unknown environment by a group of robots belongs to a quadratic competitive class. MRSAM and MRBUG have a quadratic competitive performance and thus have optimal competitiveness. The algorithms’ performance is simulated in office-like environments.     

Allocating fixed-priority periodic tasks on multiprocessor systems

In this paper, we study the problem of allocating a set of periodic tasks on a multiprocessor system such that tasks are scheduled to meet their deadlines on individual processors by the Rate-Monotonic scheduling algorithm. A new schedulability condition is developed for the Rate-Monotonic scheduling that allows us to develop more efficient on-line allocation algorithms. Two on-line allocation algorithms—RM-FF and RM-BF are presented, and shown that their worst-case performance, over the optimal allocation, is upper bounded by 2.33 and lower bounded by 2.28. Then RM-FF and RM-BF are further improved to form two new algorithms: Refined-RM-FF (RRM-FF) and Refined-RM-BF (RRM-BF), both of which have a worst-case performance bound of 2. We also show that when the maximum allowable utilization of a task is small, the worst-case performance of all the new algorithms can be significantly improved. The worst-case performance bounds of RRM-FF and RRM-BF are currently the best bounds in the class of on-line scheduling algorithms proposed to solve the same scheduling problem. Simulation studies show that the average-case performance of the newly proposed algorithms is significantly superior to those in the existing literature.     

Automated competitive analysis of real-time scheduling with graph games

This paper is devoted to automatic competitive analysis of real-time scheduling algorithms for firm-deadline tasksets, where only completed tasks contribute some utility to the system. Given such a taskset \({\mathcal {T}}\), the competitive ratio of an on-line scheduling algorithm \({\mathcal {A}}\) for \({\mathcal {T}}\) is the worst-case utility ratio of \({\mathcal {A}}\) over the utility achieved by a clairvoyant algorithm. We leverage the theory of quantitative graph games to address the competitive analysis and competitive synthesis problems. For the competitive analysis case, given any taskset \({\mathcal {T}}\) and any finite-memory on-line scheduling algorithm \({\mathcal {A}}\), we show that the competitive ratio of \({\mathcal {A}}\) in \({\mathcal {T}}\) can be computed in polynomial time in the size of the state space of \({\mathcal {A}}\). Our approach is flexible as it also provides ways to model meaningful constraints on the released task sequences that determine the competitive ratio. We provide an experimental study of many well-known on-line scheduling algorithms, which demonstrates the feasibility of our competitive analysis approach that effectively replaces human ingenuity (required for finding worst-case scenarios) by computing power. For the competitive synthesis case, we are just given a taskset \({\mathcal {T}}\), and the goal is to automatically synthesize an optimal on-line scheduling algorithm \({\mathcal {A}}\), i.e., one that guarantees the largest competitive ratio possible for \({\mathcal {T}}\). We show how the competitive synthesis problem can be reduced to a two-player graph game with partial information, and establish that the computational complexity of solving this game is Np-complete. The competitive synthesis problem is hence in Np in the size of the state space of the non-deterministic labeled transition system encoding the taskset. Overall, the proposed framework assists in the selection of suitable scheduling algorithms for a given taskset, which is in fact the most common situation in real-time systems design.     

Balanced Scheduling toward Loss-Free Packet Queuing and Delay Fairness

In current networks, packet losses can occur if routers do not provide sufficiently large buffers. This paper studies how many buffers should be provided in a router to eliminate packet losses. We assume a network router has m incoming queues, each corresponding to a single traffic stream, and must schedule at any time on-line from which queue to take the next packet to send out. To exclude packet losses with a small amount of buffers, the maximum queue length must be kept low over the entire scheduling period. We call this new on-line problem the balanced scheduling problem (BSP). By competitive analysis, we measure the power of on-line scheduling algorithms to prevent packet losses. We show that a simple greedy algorithm is (log m)-competitive which is asymptotically optimal, while Round-Robin scheduling is not better than m-competitive, as actually is any deterministic on-line algorithm for BSP. We also give a polynomial time algorithm for solving off-line BSP optimally. We also study another on-line balancing problem that tries to balance the delay among the m traffic streams.     

A Competitive Analysis for Balanced Transactional Memory Workloads

We consider transactional memory contention management in the context of balanced workloads, where if a transaction is writing, the number of write operations it performs is a constant fraction of its total reads and writes. We explore the theoretical performance boundaries of contention management in balanced workloads from the worst-case perspective by presenting and analyzing two new polynomial time contention management algorithms. We analyze the performance of a contention management algorithm by comparison with an optimal offline contention management algorithm to provide a competitive ratio. The first algorithm Clairvoyant is $O(\sqrt{s})$ -competitive, where s is the number of shared resources. This algorithm depends on explicitly knowing the conflict graph at each time step of execution. The second algorithm Non-Clairvoyant is $O(\sqrt{s} \cdot \log n)$ -competitive, with high probability, which is only a O(log?n) factor worse, but does not require knowledge of the conflict graph, where n is the number of transactions. Both of these algorithms are greedy. We also prove that the performance of Clairvoyant is close to optimal, since there is no polynomial time contention management algorithm for the balanced transaction scheduling problem that is better than $O((\sqrt{s})^{1-\varepsilon})$ -competitive for any constant ε>0, unless NP?ZPP. To our knowledge, these results are significant improvements over the best previously known O(s) competitive ratio bound.     

Optimal On-Line Algorithms to Minimize Makespan on Two Machines with Resource Augmentation

We study the problem of on-line scheduling on two uniformly related machines where the on-line algorithm has resources different from those of the off-line algorithm. We consider three versions of this problem, preemptive semi-online, non-preemptive on-line and preemptive on-line scheduling. For all these cases we design algorithms with best possible competitive ratios as functions of the machine speeds. This work was submitted as a part of the M.Sc. thesis of the second author. A preliminary version of this paper appeared in the proceedings of The First Workshop on Approximation and Online Algorithms (WAOA’03), pages 109–122.     

Two-machine flow-shop scheduling with rejection

We study a scheduling problem with rejection on a set of two machines in a flow-shop scheduling system. We evaluate the quality of a solution by two criteria: the first is the makespan and the second is the total rejection cost. We show that the problem of minimizing the makespan plus total rejection cost is NP-hard and for its solution we provide two different approximation algorithms, a pseudo-polynomial time optimization algorithm and a fully polynomial time approximation scheme (FPTAS). We also study the problem of finding the entire set of Pareto-optimal points (this problem is NP-hard due to the NP-hardness of the same problem variation on a single machine [20]). We show that this problem can be solved in pseudo-polynomial time. Moreover, we show how we can provide an FPTAS that, given that there exists a Pareto optimal schedule with a total rejection cost of at most R and a makespan of at most K, finds a solution with a total rejection cost of at most (1+?)R and a makespan value of at most (1+?)K. This is done by defining a set of auxiliary problems and providing an FPTAS algorithm to each one of them.     

Measuring the impact of adversarial errors on packet scheduling strategies

In this paper, we explore the problem of achieving efficient packet transmission over unreliable links with worst-case occurrence of errors. In such a setup, even an omniscient offline scheduling strategy cannot achieve stability of the packet queue, nor is it able to use up all the available bandwidth. Hence, an important first step is to identify an appropriate metric to measure the efficiency of scheduling strategies in such a setting. To this end, we propose an asymptotic throughput metric which corresponds to the long-term competitive ratio of the algorithm with respect to the optimal. We then explore the impact of the error detection mechanism and feedback delay on our measure. We compare instantaneous with deferred error feedback, which requires a faulty packet to be fully received in order to detect the error. We propose algorithms for worst-case adversarial and stochastic packet arrival models, and formally analyze their performance. The asymptotic throughput achieved by these algorithms is shown to be close to optimal by deriving lower bounds on the metric and almost matching upper bounds for any algorithm in the considered settings. Our collection of results demonstrate the potential of using instantaneous feedback to improve the performance of communication systems in adverse environments.     

周期性任务调度的装箱算法

针对基于时间触发的CAN控制系统，给出了确定周期性任务表中的基本周期的两种策略，提出了构造周期性任务调度表的下次适应、降序下次适应、最佳适应和降序最佳适应四种算法，分析了这四种不同算法的时间复杂度和最坏渐近性能比，最后对不同规模下的四种算法进行了仿真比较，结果表明文中给出的四种算法效果均优于经典的一维装箱算法。     

On controlling prioritized discrete event systems with real-time constraints

We study a class of prioritized Discrete Event Systems (DESs) that involve the control of resources allocated to tasks under real-time constraints. Our work is motivated by applications in communication systems, computing systems, and manufacturing systems where the objective is to minimize energy consumption while guaranteeing that task deadlines are always met. In the off-line setting, we discover several structural properties of the optimal sample path of such DESs. Using the structural properties, we also propose a greedy algorithm which is shown numerically near optimal. For on-line control, we design a Receding Horizon (RH) controller. Using worst-case estimation, the RH control is able to guarantee feasibility (when the off-line problem is feasible) and achieve good performance.     

Improving on-line construction of two-dimensional suffix trees for square matrices

The two-dimensional (2-D) suffix tree of an n×n square matrix A is a compacted trie that represents all square submatrices of A. We consider constructing 2-D suffix trees on-line, which means, instead of giving the whole matrix A in advance, A is separated and each part of A is given at different time as algorithms proceed. In general, developing an on-line algorithm is more difficult than developing an off-line algorithm. Moreover, the smaller the input grain size is, the harder it is to develop an on-line algorithm. In the case of 2-D suffix tree construction, dealing with a character at a time is harder than dealing with a row or a column at a time.In this paper we propose a randomized linear-time algorithm for constructing 2-D suffix trees on-line. This algorithm is superior to previous algorithms in two ways: (1) This is the first linear-time algorithm for constructing 2-D suffix trees on-line. Although there have been some linear-time algorithms for off-line construction, there were no linear-time algorithms for on-line construction. (2) We deal with the most fine-grain on-line case, i.e., our algorithm can construct a 2-D suffix tree even though only one character of A is given at a time, while previous on-line algorithms require at least a row and/or a column at a time.     

Polynomial-Time Algorithms for Energy Games with Special Weight Structures

Energy games belong to a class of turn-based two-player infinite-duration games played on a weighted directed graph. It is one of the rare and intriguing combinatorial problems that lie in NP∩co-NP, but are not known to be in P. The existence of polynomial-time algorithms has been a major open problem for decades and apart from pseudopolynomial algorithms there is no algorithm that solves any non-trivial subclass in polynomial time. In this paper, we give several results based on the weight structures of the graph. First, we identify a notion of penalty and present a polynomial-time algorithm when the penalty is large. Our algorithm is the first polynomial-time algorithm on a large class of weighted graphs. It includes several worst-case instances on which previous algorithms, such as value iteration and random facet algorithms, require at least sub-exponential time. Our main technique is developing the first non-trivial approximation algorithm and showing how to convert it to an exact algorithm. Moreover, we show that in a practical case in verification where weights are clustered around a constant number of values, the energy game problem can be solved in polynomial time. We also show that the problem is still as hard as in general when the clique-width is bounded or the graph is strongly ergodic, suggesting that restricting the graph structure does not necessarily help.     

On-line multiversion database concurrency control

This paper presents a new model for studying the concurrency vs. computation time tradeoffs involved in on-line multiversion database concurrency control. The basic problem that is studied in our model is the following: Given:a current database system state which includes information such as which transaction previously read a version from which other transaction; which transaction has written which versions into the database; and the ordering of versions previously written; anda set of read and write requests of requesting transactions. Question: Does there exist a new database system state in which the requesting transactions can be immediately put into execution (their read and write requests satisfied, or in the case of predeclared writeset transactions, write requests are guaranteed to be satisfied) while preserving consistency under a given set of additional constraints? (The amount of concurrency achieved is defined by the set of additional constraints). In this paper we derive “limits” of performance achievable by polynomial time concurrency control algorithms. Each limit is characterized by a minimal set of constraints that allow the on-line scheduling problem to be solved in polynomial time. If any one constraint in that minimal set is omitted, although it could increase the amount of concurrency, it would also have the dramatic negative effect of making the scheduling problem NP-complete; whereas if we do not omit any constraint in the minimal set, then the scheduling problem can be solved in polynomial time. With each of these limits, one can construct an efficient scheduling algorithm that achieves an optimal level of concurrency in polynomial computation time according to the constraints defined in the minimal set.     

TCP is Competitive with Resource Augmentation

The well-known Transport Control Protocol (TCP) is a crucial component of the TCP/IP architecture on which the Internet is built, and is a de facto standard for reliable communication on the Internet. At the heart of the TCP protocol is its congestion control algorithm. While most practitioners believe that the TCP congestion control algorithm performs very well, a complete analysis of the congestion control algorithm is yet to be done. A lot of effort has, therefore, gone into the evaluation of different performance metrics like throughput and average latency under TCP. In this paper, we approach the problem from a different perspective and use the competitive analysis framework to provide some answers to the question “how good is the TCP/IP congestion control algorithm?” We describe how the TCP congestion control algorithm can be viewed as an online, distributed scheduling algorithm. We observe that existing lower bounds for non-clairvoyant scheduling algorithms imply that no online, distributed, non-clairvoyant algorithm can be competitive with an optimal offline algorithm if both algorithms were given the same resources. Therefore, in order to evaluate TCP using competitive analysis, we must limit the power of the adversary, or equivalently, allow TCP to have extra resources compared to an optimal, offline algorithm for the same problem. In this paper, we show that TCP is competitive to an optimal, offline algorithm provided the former is given more resources. Specifically, we prove first that for networks with a single bottleneck (or point of congestion), TCP is ${\mathcal{O}}(1)The well-known Transport Control Protocol (TCP) is a crucial component of the TCP/IP architecture on which the Internet is built, and is a de facto standard for reliable communication on the Internet. At the heart of the TCP protocol is its congestion control algorithm. While most practitioners believe that the TCP congestion control algorithm performs very well, a complete analysis of the congestion control algorithm is yet to be done. A lot of effort has, therefore, gone into the evaluation of different performance metrics like throughput and average latency under TCP. In this paper, we approach the problem from a different perspective and use the competitive analysis framework to provide some answers to the question “how good is the TCP/IP congestion control algorithm?” We describe how the TCP congestion control algorithm can be viewed as an online, distributed scheduling algorithm. We observe that existing lower bounds for non-clairvoyant scheduling algorithms imply that no online, distributed, non-clairvoyant algorithm can be competitive with an optimal offline algorithm if both algorithms were given the same resources. Therefore, in order to evaluate TCP using competitive analysis, we must limit the power of the adversary, or equivalently, allow TCP to have extra resources compared to an optimal, offline algorithm for the same problem. In this paper, we show that TCP is competitive to an optimal, offline algorithm provided the former is given more resources. Specifically, we prove first that for networks with a single bottleneck (or point of congestion), TCP is O(1){\mathcal{O}}(1)-competitive to an optimal centralized (global) algorithm in minimizing the user-perceived latency or flow time of the sessions, provided we allow TCP O(1){\mathcal{O}}(1) times as much bandwidth and O(1){\mathcal{O}}(1) extra time per session. Second, we show that TCP is fair by proving that the bandwidths allocated to sessions quickly converge to fair sharing of network bandwidth.     

The Seat Reservation Problem

We investigate the problem of giving seat reservations on-line. We assume that a train travels from a start station to an end station, stopping at k stations, including the first and last. Reservations can be made for any trip going from any station to any later station. The train has a fixed number of seats. The seat reservation system attempts to maximize income. We consider the case in which all tickets have the same price and the case in which the price of a ticket is proportional to the length of the trip. For both cases we prove upper and lower bounds of Θ(1/k) on the competitive ratio of any ``fair' deterministic algorithm. We also define the accommodating ratio which is similar to the competitive ratio except that the only sequences of requests allowed are sequences for which the optimal off-line algorithm could accommodate all requests. We prove upper and lower bounds of Θ(1) on the accommodating ratio of any ``fair' deterministic algorithm, in the case in which all tickets have the same price, but Θ(1/k) in the case in which the ticket price is proportional to the length of the trip. The most surprising of these results is that all ``fair' algorithms are at least 1/2 -accommodating when all tickets have the same price. We prove similar results bounding the performance of any ``fair' randomized algorithm against an adaptive on-line adversary. We also consider concrete algorithms; more specifically, First-Fit and Best-Fit. Received February 6, 1997; revised November 6, 1997.     

An Experimental Study of Algorithms for Weighted Completion Time Scheduling

We consider the total weighted completion time scheduling problem for parallel identical machines and precedence constraints, P| prec|\sum w _i C _i . This important and broad class of problems is known to be NP-hard, even for restricted special cases, and the best known approximation algorithms have worst-case performance that is far from optimal. However, little is known about the experimental behavior of algorithms for the general problem. This paper represents the first attempt to describe and evaluate comprehensively a range of weighted completion time scheduling algorithms. We first describe a family of combinatorial scheduling algorithms that optimally solve the single-machine problem, and show that they can be used to achieve good performance for the multiple-machine problem. These algorithms are efficient and find schedules that are on average within 1.5\percent of optimal over a large synthetic benchmark consisting of trees, chains, and instances with no precedence constraints. We then present several ways to create feasible schedules from nonintegral solutions to a new linear programming relaxation for the multiple-machine problem. The best of these linear programming-based approaches finds schedules that are within 0.2\percent of optimal over our benchmark. Finally, we describe how the scheduling phase in profile-based program compilation can be expressed as a weighted completion time scheduling problem and apply our algorithms to a set of instances extracted from the SPECint95 compiler benchmark. For these instances with arbitrary precedence constraints, the best linear programming-based approach finds optimal solutions in 78\percent of cases. Our results demonstrate that careful experimentation can help lead the way to high quality algorithms, even for difficult optimization problems. Received October 30, 1998; revised March 28, 2001.