Elsevier

Parallel Computing

Volume 32, Issue 9, October 2006, Pages 701-709
Parallel Computing

Auction algorithms for decentralized parallel machine scheduling

https://doi.org/10.1016/j.parco.2006.03.002Get rights and content

Abstract

Computational grids are highly complex distributed systems (involving multiple organizations with different goals and policies) which aim at providing computing services without the users need to know the location and features of the required resources. A key issue in managing and scheduling grid resources is the coordination among multiple administrative domains. In this paper, we present a preliminary study which aims at developing auction mechanisms for decentralized scheduling which exhibit minimal communication overhead and an efficient usage of resources.

Introduction

Computational grids are emerging as promising next generation computational platforms for executing large-scale resource intensive applications arising in science, engineering and commerce [1]. They provide transparent access to large-scale flexible resources (supercomputers, storage systems, data sources, specialized devices) that are then shared among dynamic virtual organizations. In grid environments, one of the major challenges is resource management and scheduling. This is mainly due to the fact that resources are heterogeneous and geographically distributed and are often owned by different individuals or organizations with different policies, priorities and cost models.

In a grid, users submit their jobs from any one of a number of entry points (EPs). Some privileged users have direct access to the resources of some administrative domain while others are managed by an external scheduler (ES) or superscheduler. An ES acts as a broker, i.e. it is in charge of dividing the job into a number of tasks and allocating each task to a site (in such a way quality of service (QoS) constraints are satisfied). For each site (an administrative domain or a part of it), a local scheduler (LS) is responsible for determining job sequencing, local resource allocation and data transfer scheduling. In general, on receipt of a job request, the ES interrogates a number of controlled LSs to ascertain whether the task can be executed on the available resources and meet the user-specified deadline (feasibility check). If this is not the case the ES attempts to locate a LS, managed by another ES, that can meet the task requirements. If a LS cannot be located within a preset number of search steps, the task request is either rejected or passed to a scheduler that can minimize the deadline failure. When a suitable site is located, the task request is passed from the ES to the selected LS. The choice of specific algorithms for each component defines a particular grid resource management systems (GRMS).

Economy-based GRMSs (see, e.g., [2]) provide a distributed mechanism for allocating resources to jobs at a superscheduler level while enforcing QoS constraints. Indeed such approaches allow resource providers to pursuit the best possible return on their investment, and resource users to pursuit the solution of their problems within a required timeframe and budget.

In principle, various economic models, ranging from commodity market to auction based, can be adopted for resource trading in grid computing environments. However, because of the peculiar nature of grid resources and services [3], auctions are the most suitable paradigm. In an auction-based economic model, pricing is driven by how much users value the service and the highest bidder wins the access to grid services.

Auctions are important market mechanisms, used since the earliest of times for the allocation of goods and services [4]. Auctions are used for products that have no standard value (e.g., when price depends on supply and demand at a specific moment in time). An auction consists of four parts: players, objects, payoff functions and strategies. Players are (potential) buyers and sellers each with his/her/its utility function. An auction may involve a single or large quantity of indivisible or divisible objects whose true value may or may not be known to the bidder. In addition, an object’s value may vary among bidders. The payoff function may include an award mechanism, a reservation price, as well as a participation, preparation and information cost. A rational player chooses strategies that maximize his/her/its expected gain. The basic type auction is the bid auction, where buyers submit bids and sellers may accept bids but cannot make offers. In an offer auction the roles of buyers and sellers are reversed. As a general rule, one-side auctions tend to favour the silent side, e.g. sellers in a bid auction and buyers in an offer auction. From an economic perspective, a key feature of auctions is the presence of asymmetric information. In other words, each player has some private information that are not shared with other players. Four basic types of bid auctions are widely used and analyzed: the ascending-bid auction (also called the open or oral or English auction), the descending-bid auction (also called the Dutch auction), the first-price sealed-bid auction, and the second-price sealed-bid auction (also called the Vickrey auction).

Any distributed mechanism for resource coordination and scheduling must exhibit two main properties: (i) the protocol must require minimal communication overhead; (ii) the mechanism must not waste resources. As far as issue (ii) is concerned, the solutions provided by the GRMS should be required to maximize some social performance measure or, at least, to be Pareto optimal (i.e., no players can better off without harming other players). Wellman et al. [5] observe that straightforward distributed scheduling policies, such as first-come first served, shortest job first, priority first, and combinations of thereof, do not generally posses these properties. The same authors present two auction mechanisms that compute optimal or near-optimal solutions to the single unit distributed scheduling problem in a computationally efficient manner.

In this paper, we show that there exist strong links between auction and Lagrangean-based decomposition. In addition, we describe and test different versions of the auction mechanism in case of independent jobs on parallel machines. For a quite different approach, see Seredynski et al. [6] that develop competitive coevolutionary genetic algorithms (loosely coupled genetic algorithms) for scheduling a parallel and distributed system.

The remainder of the paper is organized as follows. In Section 2, we model the distributed scheduling of independent jobs as an Integer Linear Program and devise a tailored Lagrangean relaxation. Section 3 presents and analyzes some Lagrangean heuristics which can be used as auction mechanisms. Section 4 illustrates an extensive set of computational results on randomly generated instances. Finally, in Section 5, we summarize our main results.

Section snippets

Problem formulation and relaxation

We assume there are n jobs to be processed on m parallel machines with the aim of minimizing the makespan V, i.e. the maximum machine completion time [7]. A machine whose completion time equals the makespan is named critical. Let pij be the expected processing time (ETC) of job i (i = 1, …, n) on machine j (j = 1, …, m) and let xij be a binary variable equal to 1 if and only if job i (i = 1, …, n) is allocated on machine j (j = 1, …, m). Our problem can be formulated as follows:minV,s.t.,i=1npijxijV(j=1,,m),

Auction mechanisms

In this section, we show that some Lagrangean heuristics [8] based on relaxation RL(P) can be interpreted as auction mechanisms and used as distributed scheduling algorithms.

Computational results

The above described auction mechanisms have been coded in C++ and compared with a number of commonly used centralized heuristics (WorkQueue, Min–Min, Max–Min and Sufferage) [9], [10]. Tests were performed on eight classes of instances generated as follows. For a given number of jobs and machines, three instances were generated for each combination of the following parameters:

  • 1.

    Consistent versus Inconsistent ETC matrix;

  • 2.

    High job heterogeneity (Hi) versus Low job heterogeneity (Lo);

  • 3.

    High machine

Conclusion

In this paper, we have presented a preliminary study which aims at developing auction mechanisms for decentralized scheduling which exhibit minimal communication overhead and an efficient usage of resources. Computational results have shown that an auction mechanism based on a progressive Lagrangean heuristic is able to provide results comparable to those provided by centralized heuristics. In our opinion, future research in this field should be focussed on auction mechanisms that are able to

Acknowledgement

This research was partially supported by the Center of Excellence on High-Performance Computing, University of Calabria, Italy. This support is gratefully acknowledged.

References (10)

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