Single-machine ready times scheduling with group technology and proportional linear deterioration

Scheduling research has increasingly taken the concept of deterioration into consideration. In this paper, we study a single machine group scheduling problem with deterioration effect, where the jobs are already put into groups, before any optimization. We assume that the actual processing times of jobs are increasing functions of their starting times, i.e., the job processing times are described by a function which is proportional to a linear function of time. The setup times of groups are assumed to be fixed and known. For some special cases of minimizing the makespan with ready times of the jobs, we show that the problem can be solved in polynomial time for the proposed model. For the general case, a heuristic algorithm is proposed, and the computational experiments show that the performance of the heuristic is fairly accurately in obtaining near-optimal solutions. The results imply that the average percentage error of the proposed heuristic algorithm from optimal solutions is less than 3%.     

Single-machine scheduling with linear decreasing deterioration to minimize earliness penalties

We consider a single-machine scheduling problem with linear decreasing deterioration in which the due dates are determined by the equal slack (SLK) method. By the linear decreasing deterioration, we mean that the job’s processing time is a decreasing function of its starting time. The objective is to minimize the total weighted earliness penalty subject to no tardy jobs. We prove that two special cases of the problem remain polynomially solvable. The first case is the problem with equally weighted monotonous penalty objective function and the other case is the problem with weighted linear penalty objective function.     

On the law of the iterated logarithm in multiserver open queueing networks

The paper is devoted to the analysis of queueing systems in the context of the network and communication theory. We investigate a theorem on the law of the iterated logarithm for a queue of jobs in a multiserver open queueing network under heavy traffic conditions.     

Heavy-tailed limits for medium size jobs and comparison scheduling

We study the conditional sojourn time distributions of processor sharing (PS), foreground background processor sharing (FBPS) and shortest remaining processing time first (SRPT) scheduling disciplines on an event where the job size of a customer arriving in stationarity is smaller than exactly k≥0 out of the preceding m≥k arrivals. Then, conditioning on the preceding event, the sojourn time distribution of this newly arriving customer behaves asymptotically the same as if the customer were served in isolation with a server of rate (1−ρ)/(k+1) for PS/FBPS, and (1−ρ) for SRPT, respectively, where ρ is the traffic intensity. Hence, the introduced notion of conditional limits allows us to distinguish the asymptotic performance of the studied schedulers by showing that SRPT exhibits considerably better asymptotic behavior for relatively smaller jobs than PS/FBPS. Inspired by the preceding results, we propose an approximation to the SRPT discipline based on a novel adaptive job grouping mechanism that uses relative size comparison of a newly arriving job to the preceding m arrivals. Specifically, if the newly arriving job is smaller than k and larger than m−k of the previous m jobs, it is routed into class k. Then, the classes of smaller jobs are served with higher priorities using the static priority scheduling. The good performance of this mechanism, even for a small number of classes m+1, is demonstrated using the asymptotic queueing analysis under the heavy-tailed job requirements. We also discuss refinements of the comparison grouping mechanism that improve the accuracy of job classification at the expense of a small additional complexity. This work is supported by NSF Grant 0615126.     

分组截尾数据下离散型寿命概率分布的估计方法

目前针对离散型未知寿命分布的分组数据研究较少．为避免求解复杂的非线性极大似然方程组,由加入拉格朗日乘子的似然方程组中推导具有递推关系的概率分布计算公式,并通过退化的单区间模型得到P1的极大似然估计,从而完成概率分布律Pi的递推计算．实验说明该方法有效．     

Minimum weighted number of tardy jobs on an m-machine flow-shop with a critical machine

We study a flow-shop problem, where each of the jobs is limited to no more than two operations. One of these operations is common for all the jobs, and is performed on the same (”critical”) machine. Reflecting many applications, jobs are assumed to be processed in blocks on the critical machine. All the jobs share a common due-date, and the objective is minimum weighted number of tardy jobs. We prove that the problem is NP-hard. Then we formulate the problem as an integer program, and introduce a pseudo-polynomial dynamic programming algorithm, proving that the problem is NP-hard in the ordinary sense. We also propose an efficient heuristic, which is shown numerically to produce very close-to-optimal schedules. Finally, we show that the special case of identical weights is polynomially solvable.     

一类具有资源限制的调度问题

研究了具有线性退化及学习效应作用下的单台机器调度问题,对于工件的到达时间是其资源消耗量的正的严格单调递减函数时,考虑了总资源消耗量限定情形下求最大完工时间最小化问题给出了最优算法.     

加工时间恶化的两个成组加工排序问题

This paper considers single-machine scheduling problems in group technology with the jobs‘ processing times being simple linear functions of their start times. The objective functions are the minimizing of makespan and total weighted completion time. Some optimal conditions and algorithms are given and the fact that the problem of total weighted completion times is NP-hard is proved.     

Exact analysis of queueing networks with multiple job classes and blocking-after-service

Two models of closed queueing networks with blocking-after-service and multiple job classes are analyzed. The first model is a network withN stations and each station has either type II or type III. The second model is a star-like queueing network, also called a central server model, in which the stations may have either type I or type IV, with the condition that the neighbors of these stations must be of type II or type III such that blocking will be caused only by this set of station types. Exact product form solutions are obtained for the equilibrium state probabilities in both models. Formulae for performance measures such as throughput and the mean number of jobs are also derived.This work was supported by the National Science Foundation (NSF) under Grant No. CCR-90-11981.