On Fault-Tolerant Bin Packing for Online Resource Allocation

We study an online fault-tolerant bin packing problem that models reliable resource allocation. In this problem, each item is replicated and has $f+1$f+1 replicas including one primary and $f$f standbys. The packing of items is required to tolerate up to $f$f faulty bins, i.e., to guarantee that at least one correct replica of each item is available regardless of which $f$f bins turn to be faulty. Any feasible packing algorithm must satisfy an exclusion constraint and a space constraint. The exclusion constraint is generalized from the fault-tolerance requirement and the space constraint comes from the capacity planning. The target of bin packing is to minimize the number of bins used. We first derive a lower bound on the number of bins needed by any feasible packing algorithm. We then study three heuristic algorithms named mirroring, shifting and mixing under a particular setting where all items have the same size. The mirroring algorithm has a low utilization of the bin capacity. Compared with the mirroring algorithm, the shifting algorithm requires fewer bins. However, in online packing, the process of opening bins by the shifting algorithm is not smooth. It turns out that even for packing a few items, the shifting algorithm needs to quickly open a large number of bins. The mixing algorithm adopts a dual average strategy to gradually open new bins for incoming items. We prove that the mixing algorithm is feasible and show that it balances the number of bins used and the process of opening bins. Finally, to pack items with different sizes, we extend the mirroring algorithm by adopting the First-Fit strategy and extend both the shifting and mixing algorithms by involving the harmonic strategy. The asymptotic competitive ratios of the three extended algorithms are analyzed respectively.