On dynamic bin packing for resource allocation in the cloud

Dynamic Bin Packing (DBP) is a variant of classical bin packing, which assumes that items may arrive and depart at arbitrary times. Existing works on DBP generally aim to minimize the maximum number of bins ever used in the packing. In this paper, we consider a new version of the DBP problem, namely, the MinTotal DBP problem which targets at minimizing the total cost of the bins used over time. It is motivated by the request dispatching problem arising in cloud gaming systems. We analyze the competitive ratios of the commonly used First Fit, Best Fit, and Any Fit packing (the family of packing algorithms that open a new bin only when no currently opened bin can accommodate the item to be packed) algorithms for the MinTotal DBP problem. We show that the competitive ratio of Any Fit packing cannot be better than the max/min item interval length ratio μ. The competitive ratio of Best Fit packing is not bounded for any given μ. For First Fit packing, if all the item sizes are smaller than W⁄k (W is the bin capacity and k≥1 is a constant), it has a competitive ratio of k⁄k-1μ + 6k⁄k-1 + 1. For the general case, First Fit packing has a competitive ratio of 2μ + 13. We also propose a Modified First Fit packing algorithm that can achieve a competitive ratio of 8⁄7μ + 55⁄7 when μ is not known and can achieve a competitive ratio of μ + 8 when μ is known.