The hierarchical model for load balancing on two machines

Chassid, Orion; Epstein, Leah

doi:10.1007/s10878-007-9078-0

“…For the cases of related machines and restricted assignment, we extend the known negative results of [4,9], and show that a buffer of size at most m − 2 does not admit an algorithm of finite approximation ratio. For an arbitrary sized buffer, we show lower bounds of m on the approximation ratio of any algorithm.…”

Section: Introductionmentioning

confidence: 81%

“…To the best of our knowledge, no positive results are known for the case of restricted assignment. It is known that no finite approximation ratio can be achieved for a purely online algorithm, even for two machines [9], and even in the model of hierarchical machines, where for every j, the set M j is a prefix of the machine set.…”

Section: Introductionmentioning

confidence: 99%

Max-min Online Allocations with a Reordering Buffer

Epstein¹,

Levin²,

Stee³

2011

SIAM J. Discrete Math.

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Abstract. We consider online scheduling so as to maximize the minimum load, using a reordering buffer which can store some of the jobs before they are assigned irrevocably to machines. For m identical machines, we show an upper bound of Hm−1 + 1 for a buffer of size m − 1. A competitive ratio below Hm is not possible with any finite buffer size, and it requires a buffer of sizeΩ(m) to get a ratio of O(log m). For uniformly related machines, we show that a buffer of size m + 1 is sufficient to get an approximation ratio of m, which is best possible for any finite sized buffer. Finally, for the restricted assignment model, we show lower bounds identical to those of uniformly related machines, but using different constructions. In addition, we design an algorithm of approximation ratio O(m) which uses a finite sized buffer. We give tight bounds for two machines in all the three models. These results sharply contrast to the (previously known) results which can be achieved without the usage of a reordering buffer, where it is not possible to get a ratio below an approximation ratio of m already for identical machines, and it is impossible to obtain an algorithm of finite approximation ratio in the other two models, even for m = 2. Our results strengthen the previous conclusion that a reordering buffer is a powerful tool and it allows a significant decrease in the competitive ratio of online algorithms for scheduling problems. Another interesting aspect of our results is that our algorithm for identical machines imitates the behavior of the greedy algorithm on (a specific set of) related machines, whereas our algorithm for related machines completely ignores the speeds until the end, and then only uses the relative order of the speeds.

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“…, 1 m }. For the cases of related machines and restricted assignment, we extend the known negative results of [4,9], and show that a buffer of size at most m − 2 does not admit an algorithm of finite approximation ratio. For an arbitrary sized buffer, we show lower bounds of m on the approximation ratio of any algorithm.…”

Section: Introductionmentioning

confidence: 81%

Max-min Online Allocations with a Reordering Buffer

Epstein

¹

,

Levin

²

,

Stee³

2010

Automata, Languages and Programming

Self Cite

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Abstract. We consider online scheduling so as to maximize the minimum load, using a reordering buffer which can store some of the jobs before they are assigned irrevocably to machines. For m identical machines, we show an upper bound of Hm−1 + 1 for a buffer of size m − 1. A competitive ratio below Hm is not possible with any finite buffer size, and it requires a buffer of sizeΩ(m) to get a ratio of O(log m). For uniformly related machines, we show that a buffer of size m + 1 is sufficient to get an approximation ratio of m, which is best possible for any finite sized buffer. Finally, for the restricted assignment model, we show lower bounds identical to those of uniformly related machines, but using different constructions. In addition, we design an algorithm of approximation ratio O(m) which uses a finite sized buffer. We give tight bounds for two machines in all the three models. These results sharply contrast to the (previously known) results which can be achieved without the usage of a reordering buffer, where it is not possible to get a ratio below an approximation ratio of m already for identical machines, and it is impossible to obtain an algorithm of finite approximation ratio in the other two models, even for m = 2. Our results strengthen the previous conclusion that a reordering buffer is a powerful tool and it allows a significant decrease in the competitive ratio of online algorithms for scheduling problems. Another interesting aspect of our results is that our algorithm for identical machines imitates the behavior of the greedy algorithm on (a specific set of) related machines, whereas our algorithm for related machines completely ignores the speeds until the end, and then only uses the relative order of the speeds.

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“…Note that it is exactly the case of our problem without resource augmentation constraint. For this problem, Chassid and Epstein [4] considered fractional model where each job can be arbitrarily split between the machines and parts of the same job can run on different machines in parallel, and a semi-online model with known the total job size in advance. For both two models, optimal algorithms were proposed.…”

Section: Introductionmentioning

confidence: 99%

Optimal Online Algorithms on Two Hierarchical Machines With Resource Augmentation

Jiang

¹

,

Zhang

²

,

Hu

³

2012

Discrete Math. Algorithm. Appl.

0

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This paper investigates an online hierarchical scheduling problem with resource augmentation, i.e., the resources of the online algorithms are different from those of the offline algorithms. The machines are provided with different capacity according to their hierarchies. One with the hierarchy 1 has a speed of s(q) in the online (offline) algorithms and can process all the jobs. The other with hierarchy 2 has a speed of 1 in the online/offline algorithms and can only process partial jobs. The objective is to minimize makespan. For any 0 < q, s < ∞, we present optimal online algorithms with parametric competitive ratios.

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The hierarchical model for load balancing on two machines

Cited by 38 publications

References 16 publications

Max-min Online Allocations with a Reordering Buffer

Max-min Online Allocations with a Reordering Buffer

Max-min Online Allocations with a Reordering Buffer

Optimal Online Algorithms on Two Hierarchical Machines With Resource Augmentation

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