Online Scheduling Problem Research Articles

Online scheduling with immediate and reliable lead-time quotation

This paper studies an online scheduling problem with immediate and reliable lead-time quotation. A manufacturer either accepts an order by quoting a reliable lead-time on its arrival or rejects it immediately. The objective is to maximize the total revenue of completed orders. Keskinocak et al. (Management Science 47(2):264---279, 2001) studied a linear revenue function in a discrete model with integer release time of order, and proposed a competitive strategy Q-FRAC. This paper investigates a relaxed revenue function in both discrete and continuous models where orders are released at integer and real time points, respectively. For the discrete model, we present a revised Q-FRAC strategy that is optimal in competitiveness for concave and linear revenue functions with unit length and uniform weight of order, improving the previous results in Keskinocak et al. (Management Science 47(2):264---279, 2001). For the scenario with uniform length $$p$$ p and nonuniform weight of order, we prove an optimal strategy for the case with $$p=1$$ p = 1 and the nonexistence of competitive strategies for the case with $$p>1$$ p > 1 . For the continuous model, we present an optimal strategy in competitiveness for the case with uniform weight of order and linear revenue functions, and prove the nonexistence of competitive strategies for the other case with nonuniform weight of order.

Clairvoyance and online scheduling in real-time energy harvesting systems

Real-time energy harvesting systems are designed using a microprocessor, a rechargeable energy storage unit and an energy harvester. The theoretical analysis shows that an optimal solution to the underlying online scheduling problem requires time lookahead which can be incompatible with the common stochastic nature of ambient energy.

추가 머신들을 이용한 동일 길이 작업들의 온라인 마감시간 스케줄링

본 논문은 마감시간을 가진 작업들의 온라인 스케줄링 문제를 다룬다. 작업들이 시간이 지남에 따라 도착하고 스케줄링 알고리즘은 앞으로 도착할 작업들의 정보를 미리 알지 못한다. 작업들은 동일한 수행시간만큼 실행되고 알고리즘의 목표는 마감 시간 안에 수행을 완료한 작업들의 수를 최대화하는 것이다. 온라인 알고리즘의 성능은 모든 작업 정보를 미리 알고 최적의 답을 줄 수 있는 최적 알고리즘의 성능과 비교하는데 두 알고리즘 성능의 비를 경쟁비라고 한다. 일반적으로 정보의 부재로 인해 온라인 알고리즘은 큰 경쟁비를 가진다. 따라서 온라인 알고리즘에 보다 많은 머신 또는 보다 빠른 머신을 제공했을 때의 경쟁비를 계산하는 자원추가 분석을 수행할 수 있다. 본 논문에서는 온라인 알고리즘이 보다 많은 머신들을 이용할 수 있을 때 최적 알고리즘과 같은 성능을 낼 수 있음을 보일 것이다. In this paper, we consider the online scheduling problem of jobs with deadlines. The jobs arrive over time and the scheduling algorithm has no information about the arriving jobs in advance. The jobs have the processing time of the equal length and the goal of the scheduling algorithm is to maximize the number of jobs completed in their deadlines. The performance of the online algorithm is compared with that of the optimal algorithm which has the full information about all the jobs. The raio of the two performances is called the competitive ratio. In general, the ratio is unbouned. So the case that the online algorithm can have more resources than the optimal algorithm is considered, which is called the resource augmentation analysis. In this paper, the online algorithm have more machines. We show that the online algorithm can have the same performance as the optimal algorithm.

Competitive ratios for preemptive and non-preemptive online scheduling with nondecreasing concave machine cost

We consider an online scheduling problem where jobs arrive one by one and each job must be irrevocably scheduled on the machines. No machine is available initially. When a job arrives, we either purchase a new machine to process it or schedule it for processing on an existing machine. The objective is to minimize the sum of the makespan and the total cost of all the purchased machines. We assume that the total machine cost function is concave in the number of purchased machines. Considering both non-preemptive and preemptive variants of the problem, we prove that the competitive ratio of any non-preemptive or preemptive algorithm is at least 1.5. For the non-preemptive variant, we present an online algorithm and show that its competitive ratio is 1.6403. For the preemptive variant, we propose an online algorithm and show that its competitive ratio is 1.5654. We further prove that both competitive ratios are tight.

The expected asymptotical ratio for preemptive stochastic online problem

This paper considers the preemptive stochastic online scheduling problem on m uniform parallel machines with the objective to minimize the total weighted completion times, where the processing times of jobs are independent random variables. The actual processing time of each job is only learned upon its completion. In online environment, the existence and all parameters of each job, including its weight, release date and the distribution of processing time are unknown until its release date. During the processing of each job, it could be preempted and resumed later without extra cost. For this problem, a scheduling decision has to be made each time as the information of the processing time of the job is released step by step. Building on the Gittins index, we devise the Semi-Preemptive Policy for Uniform Machine Problem (SP) policy. With the assumption that all weights and processing times are bounded, we prove the expected asymptotical ratio of SP is one, which means E[∑wjCjOPT] approaches to one as the number of jobs increases to infinity, where Cj denotes the completion time of job Jj and OPT the offline optimum value.

Stochastic Online Scheduling With Preemption Penalties

This paper considers a stochastic online scheduling problem in which a set of independent jobs are to be processed on a single machine. Each job has a processing time, which is a random variable with normal distribution. All the jobs arrive overtime, which means that the existence and the parameters of each job including its processing time specifications and weight are unknown until its release date. Moreover, the actual processing time of each job is unknown until its completion. During the processing, jobs are allowed to be preempted and restarted later. So, the processing time devoted to the job before the preemption is lost and considered as preemption penalty. The objective is to minimize the expected value of the total weighted completion time. Since the problem is strongly NP-hard, a heuristic algorithm is proposed in this paper and is validated using numerical examples. The proposed method utilizes the properties of the normal distribution but it can be used as a heuristic for other distributions, as long as their means and variances are available.

An on-line scheduling problem of parallel machines with common maintenance time

In this paper, the authors consider an on-line scheduling problem of m (m ≥ 3) identical machines with common maintenance time interval and nonresumable availability. For the case that the length of maintenance time interval is larger than the largest processing time of jobs, the authors prove that any on-line algorithm has not a constant competitive ratio. For the case that the length of maintenance time interval is less than or equal to the largest processing time of jobs, the authors prove a lower bound of 3 on the competitive ratio. The authors give an on-line algorithm with competitive ratio \(4 - \tfrac{1} {m} \). In particular, for the case of m = 3, the authors prove the competitive ratio of the on-line algorithm is \(\tfrac{{10}} {3} \).

Speed Scaling with an Arbitrary Power Function

This article initiates a theoretical investigation into online scheduling problems with speed scaling where the allowable speeds may be discrete, and the power function may be arbitrary, and develops algorithmic analysis techniques for this setting. We show that a natural algorithm, which uses Shortest Remaining Processing Time for scheduling and sets the power to be one more than the number of unfinished jobs, is 3-competitive for the objective of total flow time plus energy. We also show that another natural algorithm, which uses Highest Density First for scheduling and sets the power to be the fractional weight of the unfinished jobs, is a 2-competitive algorithm for the objective of fractional weighted flow time plus energy.

Scheduling problems in transportation networks of line topology

In this paper we consider online scheduling problems for linear topology under various objective functions: minimizing the maximum completion time, minimizing the largest delay, and minimizing the sum of completion times. We give optimal solutions for uni-directional version of the problem for each of the objectives and show that for the two-directional versions of each problem, no online algorithm can deterministically achieve the optimal solution for any of the considered objective functions. We also propose 2-approximation on-line algorithms for the MinMakespan and the MinSum minimization objectives. We also prove that no online algorithm can deterministically achieve the optimal solution for any of the considered objective functions for the weighted case of uni-directional scenarios.

Online scheduling of parallel jobs with preemption on two identical machines

In this paper we consider the problem of online scheduling of parallel jobs with preemption on identical machines, where jobs arrive over time. The objective is to minimize the makespan. For the problem with two machines, we present a 1-competitive online algorithm.

Online scheduling with rejection and reordering: exact algorithms for unit size jobs

We study an online scheduling problem with rejection on $$m\ge 2$$ identical machines, in which we deal with unit size jobs. Each arriving job has a rejection value (a rejection cost or penalty for minimization problems, and a rejection profit for maximization problems) associated with it. A buffer of size $$K$$ is available to store $$K$$ jobs. A job which is not stored in the buffer must be either assigned to a machine or rejected. Upon the arrival of a new job, the job can be stored in the buffer if there is a free slot (possibly created by evicting other jobs and assigning or rejecting every evicted job). At termination, the buffer must be emptied. We study four variants of the problem, as follows. We study the makespan minimization problem, where the goal is to minimize the sum of the makespan and the penalty of rejected jobs, and the $$\ell _p$$ norm minimization problem, where the goal is to minimize the sum of the $$\ell _p$$ norm of the vector of machine completion times and the penalty of rejected jobs. We also study two maximization problems, where the goal in the first version is to maximize the sum of the minimum machine load (the cover value of the machines) and the total rejection profit, and in the second version the goal is to maximize a function of the machine completion times (which measures the balance of machine loads) and the total rejection profit. We show that an optimal solution (an exact solution for the offline problem) can always be obtained in this environment, and determine the required buffer size. Specifically, for all four variants we present optimal algorithms with $$K=m-1$$ and prove that in each case, using a buffer of size at most $$m-2$$ does not allow the design of an optimal algorithm, which makes our algorithms optimal in this respect as well. The lower bounds hold even for the special case where the rejection value is equal for all input jobs.

WSPT's Competitive Performance for Minimizing the Total Weighted Flow Time: From Single to Parallel Machines

We consider the classical online scheduling problem over single and parallel machines with the objective of minimizing total weighted flow time. We employ an intuitive and systematic analysis method and show that the weighted shortest processing time (WSPT) is an optimal online algorithm with the competitive ratio ofP+1for the case of single machine, and it is (P+(3/2)−(1/2m))-competitive for the case of parallel machines(m>1), wherePis the ratio of the longest to the shortest processing time.

A data‐driven rolling‐horizon online scheduling model for diesel production of a real‐world refinery

A rolling‐horizon optimal control strategy is developed to solve the online scheduling problem for a real‐world refinery diesel production based on a data‐driven model. A mixed‐integer nonlinear programming (MINLP) scheduling model considering the implementation of nonlinear blending quality relations and quantity conservation principles is developed. The data variations which drive the MINLP model come from different sources of certain and uncertain events. The scheduling time horizon is divided into equivalent discrete time intervals, which describe regular production and continuous time intervals which represent the beginning and ending time of expected and unexpected events that are not restricted to the boundaries of discrete time intervals. This rolling‐horizon optimal control strategy ensures the dimension of the diesel online scheduling model can be accepted in industry use. LINGO is selected to be the solution software. Finally, the daily diesel scheduling scheme of one entire month for a real‐world refinery is effectively solved. © 2012 American Institute of Chemical Engineers AIChE J, 59: 1160–1174, 2013

A best possible online algorithm for scheduling equal-length jobs on two machines with chain precedence constraints

We consider the online scheduling on two identical parallel machines with chain precedence constraints to minimize makespan, where jobs arrive over time and have identical processing times. For this online scheduling problem, Huo and Leung [Y. Huo and J.Y.-T. Leung, Online scheduling of precedence constrained tasks, SIAM Journal on Computing, 34 (2005), 743–762] proved that it is impossible to have an online algorithm of a competitive ratio 1. We provide a best possible online algorithm of competitive ratio 13−12.

Markov Decision Process Based Energy-Efficient On-Line Scheduling for Slice-Parallel Video Decoders on Multicore Systems

We consider the problem of energy-efficient on-line scheduling for slice-parallel video decoders on multicore systems. We assume that each of the processors are Dynamic Voltage Frequency Scaling (DVFS) enabled such that they can independently trade off performance for power, while taking the video decoding workload into account. In the past, scheduling and DVFS policies in multi-core systems have been formulated heuristically due to the inherent complexity of the on-line multicore scheduling problem. The key contribution of this report is that we rigorously formulate the problem as a Markov decision process (MDP), which simultaneously takes into account the on-line scheduling and per-core DVFS capabilities; the power consumption of the processor cores and caches; and the loss tolerant and dynamic nature of the video decoder's traffic. In particular, we model the video traffic using a Direct Acyclic Graph (DAG) to capture the precedence constraints among frames in a Group of Pictures (GOP) structure, while also accounting for the fact that frames have different display/decoding deadlines and non-deterministic decoding complexities. The objective of the MDP is to minimize long-term power consumption subject to a minimum Quality of Service (QoS) constraint related to the decoder's throughput. Although MDPs notoriously suffer from the curse of dimensionality, we show that, with appropriate simplifications and approximations, the complexity of the MDP can be mitigated. We implement a slice-parallel version of H.264 on a multiprocessor ARM (MPARM) virtual platform simulator, which provides cycle-accurate and bus signal-accurate simulation for different processors. We use this platform to generate realistic video decoding traces with which we evaluate the proposed on-line scheduling algorithm in Matlab.

Competitive analysis for make-to-order scheduling with reliable lead time quotation

This paper makes extended studies on the discrete problem of online scheduling and reliable lead time quotation (discrete Q-SLTQ) introduced by Keskinocak et al. (Manag. Sci. 47(2):264–279, 2001). We first relax the assumption on revenue function from a linear decreasing function to any decreasing function. We present an online deterministic strategy which is optimal in competitiveness for concave revenue functions. The above results are further extended to the continuous Q-SLTQ model where orders are released at arbitrary time points. For the discrete Q-SLTQ problem, if orders are with nonuniform lengths, we prove the nonexistence of online strategies with bounded competitive ratios; otherwise if orders are with unit length but various weights, we present an optimal online strategy.

On the optimality of the TLS algorithm for solving the online-list scheduling problem with two job types on a set of multipurpose machines

In this paper we study the optimality of the TLS algorithm for solving the online scheduling problem of minimizing the makespan on a set of m multipurpose machines, where there are two different job types and each job type can only be processed on a unique subset of machines. The literature shows that the TLS algorithm is optimal for the special cases where either m=2 or where all processing times are restricted to unity. We show that the TLS algorithm is optimal also for the special cases where the job processing times are either job type or machine set dependent. For both cases, the optimality of the TLS algorithm is proven by showing that its competitive ratio matches the lower bound for any processing set and processing time parameters.

Temperature aware online algorithms for scheduling equal length jobs

We study the online scheduling problem of maximising job completion subject to temperature constraints. In our setting, jobs are of equal length and have deadlines and heat contributions. The algorithm tries to complete as many jobs as possible before their deadlines while keeping the temperature of the system within an acceptable limit. We give an optimal algorithm for the case where preemption is not allowed. Then we consider the case of preemption and prove a number of lower bounds, showing that for many combination of system parameters, preemption (with restart) is not helpful in improving the competitiveness.

A note on “An optimal online algorithm for single machine scheduling to minimize total general completion time”

In this paper we study the problem of online scheduling of jobs with release times on a single machine to minimize the total general completion time ∑ C j α , where C j is the completion time of job J j and α ⩾ 1 is a constant. It has been shown in the paper by Liu et al., namely, “An optimal online algorithm for single machine scheduling to minimize total general completion time” (Liu et al., 2010) [2] that the lower bound on the competitive ratio of any online algorithm is 2 α . The authors also attempted to prove that the online algorithm DSPT (Delayed Shortest Processing Time) is 2 α -competitive. Unfortunately, Lemma 2 in the paper, which is crucial in the proof, is indeed incorrect. This leaves the performance of DSPT as an open question. The contribution of this paper is two-fold. First, we give a counterexample to this lemma and an explanation of the flaw in the argument. Second, we give a proof that DSPT is 2 α -competitive. Together with the lower bound by Liu et al., this implies that DSPT is an optimal online algorithm for minimizing the total general completion time.

Online scheduling on uniform machines with two hierarchies

In this paper we study online scheduling problem on m parallel uniform machines with two hierarchies. The objective is to minimize the maximum completion time (makespan). Machines are provided with different capability. The machines with speed s can schedule all jobs, while the other machines with speed 1 can only process partial jobs. Online algorithms for any 0<s<? are provided in the paper. For the case of k=1 and m=2, and the case of some values of s, k=1 and m=3, the algorithms are the best possible, where k is the number of machines with hierarchy 1, and m is the number of machines. Lower bounds for some special cases are also presented.