Parametric analysis of the quality of single preemption schedules on three uniform parallel machines

For a scheduling problem to minimize the makespan on three uniform parallel machines we present a parametric analysis of the quality of a schedule with at most one preemption compared to the global optimal schedule with any number of preemptions. A tight bound is derived as a function of the relative speeds of the machines, provided that two of the machines have the same speed

Split Scheduling with Uniform Setup Times

We study a scheduling problem in which jobs may be split into parts, where the parts of a split job may be processed simultaneously on more than one machine. Each part of a job requires a setup time, however, on the machine where the job part is processed. During setup a machine cannot process or set up any other job. We concentrate on the basic case in which setup times are job-, machine-, and sequence-independent. Problems of this kind were encountered when modelling practical problems in planning disaster relief operations. Our main algorithmic result is a polynomial-time algorithm for minimising total completion time on two parallel identical machines. We argue why the same problem with three machines is not an easy extension of the two-machine case, leaving the complexity of this case as a tantalising open problem. We give a constant-factor approximation algorithm for the general case with any number of machines and a polynomial-time approximation scheme for a fixed number of machines. For the version with objective minimising weighted total completion time we prove NP-hardness. Finally, we conclude with an overview of the state of the art for other split scheduling problems with job-, machine-, and sequence-independent setup times

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

Power of preemption on uniform parallel machines

For a scheduling problem on parallel machines, the power of preemption is defined as the ratio of the makespan of an optimal non-preemptive schedule over the makespan of an optimal preemptive schedule. For m uniform parallel machines, we give the necessary and sufficient conditions under which the global bound of 2-1/m is tight. If the makespan of the optimal preemptive schedule is defined by the ratio of the total processing times of r < m longest jobs over the total speed of r fastest machines, we show that the tight bound on the power of preemption is 2-1/min{r,m-r}

Dynamic Windows Scheduling with Reallocation

We consider the Windows Scheduling problem. The problem is a restricted version of Unit-Fractions Bin Packing, and it is also called Inventory Replenishment in the context of Supply Chain. In brief, the problem is to schedule the use of communication channels to clients. Each client ci is characterized by an active cycle and a window wi. During the period of time that any given client ci is active, there must be at least one transmission from ci scheduled in any wi consecutive time slots, but at most one transmission can be carried out in each channel per time slot. The goal is to minimize the number of channels used. We extend previous online models, where decisions are permanent, assuming that clients may be reallocated at some cost. We assume that such cost is a constant amount paid per reallocation. That is, we aim to minimize also the number of reallocations. We present three online reallocation algorithms for Windows Scheduling. We evaluate experimentally these protocols showing that, in practice, all three achieve constant amortized reallocations with close to optimal channel usage. Our simulations also expose interesting trade-offs between reallocations and channel usage. We introduce a new objective function for WS with reallocations, that can be also applied to models where reallocations are not possible. We analyze this metric for one of the algorithms which, to the best of our knowledge, is the first online WS protocol with theoretical guarantees that applies to scenarios where clients may leave and the analysis is against current load rather than peak load. Using previous results, we also observe bounds on channel usage for one of the algorithms.Comment: 6 figure

Power of preemption for minimizing total completion time for uniform parallel machines

For scheduling problems on parallel machines, the power of preemption is defined as the supremum ratio of the cost of an optimal nonpreemptive schedule over the cost of an optimal preemptive schedule (for the same input), where the cost is defined by a fixed common cost function. We present a tight analysis of the power of preemption for the problem of minimizing the total completion time on m ≥ 2 uniformly related machines, showing that its value for m = 2 is equal to 1.2, and its overall value is approximately 1.39795

Non-preemptive Scheduling in a Smart Grid Model and its Implications on Machine Minimization

We study a scheduling problem arising in demand response management in smart grid. Consumers send in power requests with a flexible feasible time interval during which their requests can be served. The grid controller, upon receiving power requests, schedules each request within the specified interval. The electricity cost is measured by a convex function of the load in each timeslot. The objective is to schedule all requests with the minimum total electricity cost. Previous work has studied cases where jobs have unit power requirement and unit duration. We extend the study to arbitrary power requirement and duration, which has been shown to be NP-hard. We give the first online algorithm for the general problem, and prove that the problem is fixed parameter tractable. We also show that the online algorithm is asymptotically optimal when the objective is to minimize the peak load. In addition, we observe that the classical non-preemptive machine minimization problem is a special case of the smart grid problem with min-peak objective, and show that we can solve the non-preemptive machine minimization problem asymptotically optimally