Trade-offs between speed and processor in hard-deadline scheduling

This paper revisits the problem of on-line scheduling of sequential jobs with hard deadlines in a preemptive, multiprocessor setting. An on-line scheduling algorithm is said to be optimal if it can schedule any set of jobs to meet their deadlines whenever it is feasible in the off-line sense. It is known that the earliest-deadline-first strategy (EDF) is optimal in a one-processor setting, and there is no optimal on-line algorithm in an m-processor setting where m≥2. Recent work however reveals that if the on-line algorithm is given faster processors, EDF is actually optimal for all m (e.g., when m = 2, it suffices to use processors 1.5 times as fast). This paper initiates the study of the trade-off between increasing the speed and using more processors in deriving optimal on-line scheduling algorithms. Several upper bound and lower bound results are presented. For example, the speed requirement of EDF can be reduced to 2-1+p/m+p when it is given p≥0 extra processors. The main result is a new on-line algorithm which demands less speedy processors so as to attain optimality (e.g., when m = 2, the speed requirement is 1 1/3) and admits a better speed-processor trade-off than EDF (e.g., when m = 2 and p = 1, the speed requirement is 1.2). In general, no optimal algorithm exists when the speed factor is less than 1/(2√2+p/m-2).published_or_final_versio

Performance guarantee for online deadline scheduling in the presence of overload

Earliest deadline first (EDF) is a widely-used online algorithm for scheduling jobs with deadlines in real-time systems. Yet, existing results on the performance guarantee of EDF are limited to underloaded systems [6,12,14]. This paper initiates the study of EDF for overloaded systems, attaining similar performance guarantees as in the underloaded setting. Specifically, we show that EDF with a simple form of admission control is optimal for scheduling on both uniprocessor and multiprocessors when moderately faster processors are available (our analysis actually admits a tradeoff between speed and extra processors). This is the first result attaining optimality under overload. Another contribution of this paper is an improved analysis of the competitiveness for weighted deadline scheduling.published_or_final_versio

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

On Packet Scheduling with Adversarial Jamming and Speedup

In Packet Scheduling with Adversarial Jamming packets of arbitrary sizes arrive over time to be transmitted over a channel in which instantaneous jamming errors occur at times chosen by the adversary and not known to the algorithm. The transmission taking place at the time of jamming is corrupt, and the algorithm learns this fact immediately. An online algorithm maximizes the total size of packets it successfully transmits and the goal is to develop an algorithm with the lowest possible asymptotic competitive ratio, where the additive constant may depend on packet sizes. Our main contribution is a universal algorithm that works for any speedup and packet sizes and, unlike previous algorithms for the problem, it does not need to know these properties in advance. We show that this algorithm guarantees 1-competitiveness with speedup 4, making it the first known algorithm to maintain 1-competitiveness with a moderate speedup in the general setting of arbitrary packet sizes. We also prove a lower bound of $\phi+1\approx 2.618$ on the speedup of any 1-competitive deterministic algorithm, showing that our algorithm is close to the optimum. Additionally, we formulate a general framework for analyzing our algorithm locally and use it to show upper bounds on its competitive ratio for speedups in $[1,4)$ and for several special cases, recovering some previously known results, each of which had a dedicated proof. In particular, our algorithm is 3-competitive without speedup, matching both the (worst-case) performance of the algorithm by Jurdzinski et al. and the lower bound by Anta et al.Comment: Appeared in Proc. of the 15th Workshop on Approximation and Online Algorithms (WAOA 2017

A Constant Factor Approximation Algorithm for Unsplittable Flow on Paths

In the unsplittable flow problem on a path, we are given a capacitated path $P$ and $n$ tasks, each task having a demand, a profit, and start and end vertices. The goal is to compute a maximum profit set of tasks, such that for each edge $e$ of $P$, the total demand of selected tasks that use $e$ does not exceed the capacity of $e$. This is a well-studied problem that has been studied under alternative names, such as resource allocation, bandwidth allocation, resource constrained scheduling, temporal knapsack and interval packing. We present a polynomial time constant-factor approximation algorithm for this problem. This improves on the previous best known approximation ratio of $O(\log n)$. The approximation ratio of our algorithm is $7+\epsilon$ for any $\epsilon>0$. We introduce several novel algorithmic techniques, which might be of independent interest: a framework which reduces the problem to instances with a bounded range of capacities, and a new geometrically inspired dynamic program which solves a special case of the maximum weight independent set of rectangles problem to optimality. In the setting of resource augmentation, wherein the capacities can be slightly violated, we give a $(2+\epsilon)$-approximation algorithm. In addition, we show that the problem is strongly NP-hard even if all edge capacities are equal and all demands are either~1,~2, or~3.Comment: 37 pages, 5 figures Version 2 contains the same results as version 1, but the presentation has been greatly revised and improved. References have been adde

Tians scheduling: Using partial processing in best-effort applications

Abstract-To service requests with high quality, interactive services such as web search, on-demand video and online gaming keep average server utilization low. As servers become busy, queuing delays increase, and requests miss their deadlines, resulting in degraded quality of service with poor user experience and potential revenue loss. In this paper, we propose Tians scheduling, a group of scheduling algorithms for interactive services that can produce partial answers during overload. A Tians scheduler allocates processing time to each request based on system load with the objective of maximizing overall quality of responses. We propose three Tians scheduling algorithms -offline, online clairvoyant and online nonclairvoyant. For interactive applications with concave quality profile, we prove that the offline algorithm is optimal. We show the effectiveness of the online algorithms by conducting a simulation study modeling important applications -a web search engine and video-ondemand (VOD) system. Simulation results show a significant improvement of Tians over traditional server models: average response quality improves and the variance of responses decreases. Keywords-interactive services, best-effort applications, offline, online clairvoyant, online nonclairvoyant, partial results, quality profile, scheduling, VOD bandwidth allocation, web search engine

On packet scheduling with adversarial jamming and speedup

In Packet Scheduling with Adversarial Jamming, packets of arbitrary sizes arrive over time to be transmitted over a channel in which instantaneous jamming errors occur at times chosen by the adversary and not known to the algorithm. The transmission taking place at the time of jamming is corrupt, and the algorithm learns this fact immediately. An online algorithm maximizes the total size of packets it successfully transmits and the goal is to develop an algorithm with the lowest possible asymptotic competitive ratio, where the additive constant may depend on packet sizes. Our main contribution is a universal algorithm that works for any speedup and packet sizes and, unlike previous algorithms for the problem, it does not need to know these parameters in advance. We show that this algorithm guarantees 1-competitiveness with speedup 4, making it the first known algorithm to maintain 1-competitiveness with a moderate speedup in the general setting of arbitrary packet sizes. We also prove a lower bound of ϕ+1≈2.618 on the speedup of any 1-competitive deterministic algorithm, showing that our algorithm is close to the optimum. Additionally, we formulate a general framework for analyzing our algorithm locally and use it to show upper bounds on its competitive ratio for speedups in [1, 4) and for several special cases, recovering some previously known results, each of which had a dedicated proof. In particular, our algorithm is 3-competitive without speedup, matching both the (worst-case) performance of the algorithm by Jurdzinski et al. (Proceedings of the 12th workshop on approximation and online algorithms (WAOA), LNCS 8952, pp 193–206, 2015. http://doi.org/10.1007/978-3-319-18263-6_17) and the lower bound by Anta et al. (J Sched 19(2):135–152, 2016. http://doi.org/10.1007/s10951-015-0451-z)