Resource allocation for cloud services is a complex task due to the diversity of the services and the dynamic workloads. One way to address this is by overprovisioning which results in high cost due to the unutilized resources. A much more economical approach, relying on the stochastic nature of the demand, is to allocate just the right amount of resources and use additional more expensive mechanisms in case of overflow situations where demand exceeds the capacity. In this paper we study this approach and show both by comprehensive analysis for independent normal distributed demands and simulation on synthetic data that it is significantly better than currently deployed methods.
A naive approach to such a problem is to reduce it to classical bin packing as follows: for the @math 'th service define the as the number @math such that the probability that @math is larger than @math is small; then solve the classical bin packing problem (or a variant of it) with item sizes @math . However, @cite_3 showed this approach can be quite wasteful, mostly because it adds extra space per service and not per bin. To demonstrate the issue, think about unbiased, independent coin tosses. The probability one coin toss significantly deviates from its mean is @math , while the probability @math independent coin tosses significantly deviate from the mean is exponentially small. When running independent trials there is a effect that considerably reduces the chance of high deviations. This can also be seen from the fact that the standard deviation of @math independent, identical processes is only @math times the standard deviation of one process, and so the standard deviation grows much slower than the number of processes.
@cite_3
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The bin packing problem is to find the minimum number of bins of size one to pack a list of items with sizes a 1 ,…, a n in (0,1]. Using uniform sampling, which selects a random element from the input list each time, we develop a randomized @math time (1+e )-approximation scheme for the bin packing problem. We show that every randomized algorithm with uniform random sampling needs @math time to give an (1+e )-approximation. For each function s (n ): N →N , define ∑(s (n )) to be the set of all bin packing problems with the sum of item sizes equal to s (n ). We show that ∑(n b ) is NP-hard for every b ∈(0,1]. This implies a dense sublinear time hierarchy of approximation schemes for a class of NP-hard problems, which are derived from the bin packing problem. We also show a randomized streaming approximation scheme for the bin packing problem such that it needs only constant updating time and constant space, and outputs an (1+e )-approximation in @math time. Let S (δ )-bin packing be the class of bin packing problems with each input item of size at least δ . This research also gives a natural example of NP-hard problem (S (δ )-bin packing) that has a constant time approximation scheme, and a constant time and space sliding window streaming approximation scheme, where δ is a positive constant. We revisit the classic online bin packing problem studied in the half-century. In this problem, items of positive sizes no larger than 1 are presented one by one to be packed into subsets called bins of total sizes no larger than 1, such that every item is assigned to a bin before the next item is presented. We use online partitioning of items into classes based on sizes, as in previous work, but we also apply a new method where items of one class can be packed into more than two types of bins, where a bin type is defined according to the number of such items grouped together. Additionally, we allow the smallest class of items to be packed in multiple kinds of bins, and not only into their own bins. We combine this with the approach of packing of sufficiently big items according to their exact sizes. Finally, we simplify the analysis of such algorithms, allowing the analysis to be based on the most standard weight functions. This simplified analysis allows us to study the algorithm which we defined based on all these ideas. This leads us to the design and analysis of the first algorithm of asymptotic competitive ratio strictly below 1.58, specifically, we break this barrier by providing an algorithm AH (Advanced Harmonic) whose asymptotic competitive ratio does not exceed 1.57829. Our main contribution is the introduction of the simple analysis based on weight function to analyze the state of the art online algorithms for the classic online bin packing problem. The previously used analytic tool named weight system was too complicated for the community in this area to adjust it for other problems and other algorithmic tools that are needed in order to improve the current best algorithms. We show that the weight system based analysis is not needed for the analysis of the current algorithms for the classic online bin packing problem. The importance of a simple analysis is demonstrated by analyzing several new features together with all existing techniques, and by proving a better competitive ratio than the previously best one. Bin covering takes as input a list of items with sizes in (0, 1) and places them into bins of unit demand so as to maximize the number of bins whose demand is satisfied. This is in a sense a dual problem to the classical one-dimensional bin packing problem, but has for many years lagged behind the latter in terms of the quality of the best approximation algorithms. We design algorithms for this problem that close the gap, both in terms of worst- and average-case results. We present (1) the first asymptotic approximation scheme for the offline version, (2) algorithms that have bounded worst-case behavior for instances with discrete item sizes and expected behavior that is asymptotically optimal for all discrete “perfect-packing distributions” (ones for which optimal packings have sublinear expected waste), and (3) a learning algorithm that has asymptotically optimal expected behavior for all discrete distributions. The algorithms of (2) and (3) are based on the recently-developed online Sum-of-Squares algorithm for bin packing. We also present experimental analysis comparing the algorithms of (2) and suggesting that one of them, the Sum-of-Squares-with-Threshold algorithm, performs quite well even for discrete distributions that do not have the perfect-packing property. In the dual bin packing problem, the objective is to assign items of given size to the largest possible number of bins, subject to the constraint that the total size of the items assigned to any bin is at least equal to 1. We carry out a probabilistic analysis of this problem under the assumption that the items are drawn independently from the uniform distribution on [0, 1] and reveal the connection between this problem and the classical bin packing problem as well as to renewal theory.