Divisible Job Research Articles

Scheduling with divisible jobs and subcontracting option

In electronics industry, aircraft manufacturing, distributed computer systems and supply chains, it is common that many jobs are divisible and can be considered as a batch of potentially infinitely small and independent items. Subcontracting divisible jobs means that a job can be partially processed by an in-house machine and the remaining of it can be processed by a subcontractor’s machine. Considering three subcontracting pricing strategies: non-increasing, non-decreasing and constant over time, this paper studies a scheduling problem with divisible jobs and subcontracting option in which both the manufacturer and the subcontractor are in single-machine environment. The objective is to minimize the sum of total weighted tardiness and total subcontracting costs. A mixed integer programming (MIP) model is formulated. Then a Lagrangian-based Benders Dual Decomposition (denoted by LB-BDD) method is developed based on the MIP formulation. Extensive computational experiments are conducted on five groups of randomly generated problem instances, and the results show that the proposed LB-BDD method outperforms the basic Benders Decomposition (denoted by BD) method and solving the MIP model directly in Gurobi solver.

Decentralized subcontractor scheduling with divisible jobs

Subcontracting allows manufacturer agents to reduce completion times of their jobs and thus obtain savings. This paper addresses the coordination of decentralized scheduling systems with a single subcontractor and several agents having divisible jobs. Assuming complete information, we design parametric pricing schemes that strongly coordinate this decentralized system, i.e., the agents' choices of subcontracting intervals always result in efficient schedules. The subcontractor's revenue under the pricing schemes depends on a single parameter which can be chosen to make the revenue as close to the total savings as required. Also, we give a lower bound on the subcontractor's revenue for any coordinating pricing scheme. Allowing private information about processing times, we prove that the pivotal mechanism is coordinating, i.e., agents are better off by reporting their true processing times, and by participating in the subcontracting. We show that the subcontractor's maximum revenue with any coordinating mechanism under private information equals the lower bound of that with coordinating pricing schemes under complete information. Finally, we address the asymmetric case where agents obtain savings at different rates per unit reduction in completion times. We show that coordinating pricing schemes do not always exist in this case.

Performance Evaluation of Matrix Multiplication on a Network of Work Stations with Communication Delay

Network-based multicomputer systems merge as a potential economical candidate to replace supercomputers. Despite enormous effort to evaluate the performance of those systems, we still lack a complete performance model to analyze distributed computing systems. The model is complete if all system parameters, network parameters, communication overhead parameters, and application parameters are considered explicitly in the solution. In this paper we develop a closed form solution for a finish time of executing a matrix-multiplication on a distributed network of processors (cloud computing). Matrix multiplication represents a paradigm of divisible jobs, which has a wide area of applications. The closed form solution is derived considering one-dimensional decomposition and two-dimensional decomposition of the matrices involved in the operation. The analytical results combine all system parameters, network parameters, and matrix dimension in the solution, which helps the designer to study the effect of each individual parameter on the overall system performance. This then becomes a tool for a designer of a multicomputer system to mange limited resources in an optimal manner paying attention only to those parameters that are most critical. The results show the superiority of the row-wise one-dimensional decomposition. We also show that, because of the communication overhead, adding more processors will not incessantly improve the system performance. We derived a closed form solution for the optimum number of processors (nodes) to be used to obtain the maximum speed up (equally the minimum finish time). Process allocation tradeoff is very important for optimizing the multicomputer system performance.

DIVISIBLE LOAD DISTRIBUTION IN A NETWORK OF PROCESSORS

The paper presents a closed form solution for an optimum scheduling of a divisible job on an optimum number of processor arranged in an optimum sequence in a multilevel tree networks. The solution has been derived for a single divisible job where there is no dependency among subtasks and the root processor can either perform communication and computation at the same time. The solution is carried out through three basic theorems. One of the theorems selects the optimum number of available processors that must participate in executing a divisible job. The other solves the sequencing problem in load distribution by which we are able to find the optimum sequence for load distribution in a generalized form. Having the optimum number of processors and their sequencing for load distribution, we have developed a closed form solution that determines the optimum share of each processor in the sequence such that the finish time is minimized. Any alteration of the number of processors, their sequences, or their shares that are determined by the three theorems will increase the finish time.

Performance limits for processor networks with divisible jobs

Ultimate performance limits to the aggregate processing speed of networks of processors that are processing a divisible job are described. These take the form of either closed-form expressions or numerical procedures to calculate the equivalent processing speed of an infinite number of processors. These processors are interconnected in either a linear daisy chain with load origination from the network interior or a tree topology. The tree topology is particularly general as a natural way to perform load distribution in a professor network topology with cycles (e.g., hypercube, toroidal network) is to use an embedded spanning tree. Such limits on performance are important as they provide an ideal baseline against which to compare the performance of finite configurations of processors.

Distributed processing of divisible jobs with communication startup costs

In this work we analyze the problem of an optimal distribution of a computational task among a set of processors. We assume that the task can be arbitrarily divided and its parts can be processed in parallel on different processors. A wide range of interconnection architectures of distributed computer systems is taken into consideration: a chain, a loop, a tree, a star of processors, a set of processors using shared buses, and a hypercube of processors. It is assumed that the communication time is equal to some startup value plus some amount proportional to the volume of transferred data. Using a uniform methodology we present a method to find the distribution of the load so that the minimum completion time is achieved for the considered data distribution scheme. The results can also be used to find such parameters of the processor network as equivalent speed, speedup and utilization. Moreover, the methodology presented here can be a model of the application roll-in time, and can be applied in load balancing in a heterogeneous multiprocessor system.

Optimal divisible job load sharing for bus networks

Optimal load allocation for load sharing a divisible job over N processors interconnected in bus-oriented network is considered. The processors are equipped with front-end processors. It is analytically proved, for the first time, that a minimal solution time is achieved when the computation by each processor finishes at the same time. Closed form solutions for the minimum finish time and the optimal data allocation for each processor are also obtained.

Scheduling divisible jobs on hypercubes†

In this work a problem of finding an optimal distribution of a divisible computational job among a set of processors is considered. In the model of parallel computer systems two important factors must be taken into account: speeds of processors and speeds of communications links. With regard to this, we propose a deterministic approach finding an optimal distribution of the job's load on a hypercube of processors. The method used allows also the determination of performance bounds on the hypercube architecture.

Closed form solutions for bus and tree networks of processors load sharing a divisible job

Optimal load allocation for load sharing a divisible job over processors interconnected in either a bus or a tree network is considered. The processors are either equipped with front-end processors or not so equipped. Closed form solutions for the minimum finish time and the optimal data allocation for each processor are obtained. The performance of large symmetric tree networks is examined by aggregating the component links and processors into a single equivalent processor. This allows an easy examination of large tree networks. In addition, it becomes possible to find a closed form solution for the optimal amount of data that is to be assigned to each processor in the tree network in order to achieve the minimum finish time.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>