We study the fault-tolerant variant of the online bin packing problem. Similar to the classic bin packing problem, an online sequence of items of various sizes should be packed into a minimum number of bins of uniform capacity. For applications such as server consolidation, where bins represent servers and items represent jobs of different loads, it is required to maintain fault-tolerant solutions. In a fault-tolerant packing, any job is replicated into f+1 servers, for some integer f > 1, so that the failure of up to f servers does not interrupt service. We build over a practical model introduced by Li and Tang [SPAA 2017] in which each job of load $x$ has a primary replica of load $x$ and $f$ standby replicas, each of load $x/\eta$, where $\eta >1$ is a parameter of the problem. Upon failure of up to $f$ servers, any primary replica in a failed bin should be replaced by one of its standby replicas so that the extra load of the new primary replica does not cause an overflow in its bin. We study a general setting in which bins might fail while the input is still being revealed. Our main contribution is an algorithm, named Harmonic-Stretch, which maintains fault-tolerant packings under this general setting. We prove that Harmonic-Stretch has an asymptotic competitive ratio of at most 1.75. This is an improvement over the best existing asymptotic competitive ratio 2 of an algorithm by Li and Tang [TPDS 2020], which works under a model in which bins fail only after all items are packed.
翻译:我们研究在线垃圾包装问题的错误容忍变量。 类似经典的垃圾包装问题, 各种大小项目的在线序列应该被包装到最小数量的统一容量的垃圾桶中。 对于服务器整合等应用程序, 垃圾代表服务器和项目代表不同负荷的工作, 需要保持错误容忍解决方案。 在错误容忍包装中, 任何工作都复制到f+1服务器, 某些整数 f > 1 的服务器中, 使得顶级服务器的失败不会中断服务 。 我们建在由李和唐[ 推出的实用模型上, 各种大小项目的在线序列应该被包装到最小数量的统一能力。 对于服务器整合等应用程序, 包括服务器和项目代表不同负荷的工作。 对于服务器的错误容忍选项, 任何工作都可以复制到 f+1 服务器, 这样, 任何失败服务器的原始复制应该被一个备用复制器替换, 这样, 新的主要复制器的存储量不会溢出。 我们研究一个总模型设置的硬度, 将所有的硬度比值比重值都比重 $x/\ 。 。 在总的解算法中, 我们的计算中, 将一个总的精度中, 将一个总的精确比值显示为 。 。 。 在总的精度中, 我们的精度中, 我们的精度的精度的精度的精度的精度是的精度的精度的精度的精度的精度的精度的精度是, 。