Online dynamic bin packing

In this thesis we study online algorithms for dynamic bin packing. An online algorithm is presented with input throughout time and must make irrevocable decisions without knowledge of future input. The classical bin packing problem is a combinatorial optimization problem in which a set of items must be packed into a minimum number of uniform-sized bins without exceeding their capacities. The problem has been studied since the early 1970s and many variants continue to attract researchers’ attention today. The dynamic version of the bin packing problem was introduced by Coffman, Garey and Johnson in 1983. The problem is a generalization of the bin packing problem in which items may arrive and depart dynamically. In this setting, an online algorithm for bin packing is presented with one item at a time, without knowledge of its departure time, nor arrival and departure times of future items, and must decide in which bin the item should be packed. Migration of items between bins is not allowed, however rearrangement of items within a bin is permitted. The objective of problem is to minimize the maximum number of bins used over all time. In multi-dimensional generalizations of the problem, multi-dimensional items must be packed without overlap in multi-dimensional bins of uniform size in each dimension. In this work, we study the setting where items are oriented and cannot be rotated. We first consider online one-dimensional dynamic bin packing and present a lower bound of 8/3 ∼ 2.666 on the achievable competitive ratio of any deterministic online algorithm, improving the best known 2.5-lower bound. Since the introduction of the problem by Coffman, Garey and Johnson, the progress on the problem has focused on improving the original lower bound of 2.388 to 2.428, and to the best known 2.5-lower bound. Our improvement from 2.5 to 8/3 ∼ 2.666 makes a big step forward in closing the gap between the lower bound and the upper bound, which currently stands at 2.788. Secondly we study the online two- and three-dimensional dynamic bin packing problem by designing and analyzing algorithms for special types of input. Bar-Noy et al. initiated the study of the one-dimensional unit fraction bin packing problem, a restricted version where all sizes of items are of the form 1/k, for some integer k > 0. Another related problem is for power fraction items, where sizes are of the form 1/2k, for some integer k ≥ 0. We initiate the study of online multi-dimensional dynamic bin packing of unit fraction items and power fraction items, where items have lengths unit fraction and power fraction in each dimension, respectively. While algorithms for general input are suitable for unit fraction and power fraction items, their worst-case performance guarantees are the same for special types of input. For unit fraction and power fraction items, we design and analyze online algorithms that achieve better worst-case performance guarantees compared to their classical counterparts. Our algorithms give careful consideration to unit and power fraction items, which allows us to reduce the competitive ratios for these types of inputs. Lastly we focus on obtaining lower bounds on the performance of the family of Any- Fit algorithms (Any-Fit, Best-Fit, First-Fit, Worst-Fit) for online multi-dimensional dynamic bin packing. Any-Fit algorithms are classical online algorithms initially studied for the one-dimensional version of the bin packing problem. The common rule that the algorithms use is to never pack a new item to a new bin if the item can be packed in any of the existing bins. While the family of Any-Fit algorithms is always O(1)-competitive for one-dimensional dynamic bin packing, we show that this is no longer the case for multi-dimensional dynamic bin packing when using Best-Fit and Worst-Fit, even if the input consists of power fraction items or unit fraction items. For these restricted inputs, we prove that Best-Fit and Worst-Fit have unbounded competitive ratios, while for First-Fit we provide lower bounds that are higher than the lower bounds for any online algorithm. Furthermore, for general input we show that all classical Any-Fit algorithms are not competitive for online multi-dimensional dynamic bin packing

Overcommitment in Cloud Services -- Bin packing with Chance Constraints

This paper considers a traditional problem of resource allocation, scheduling jobs on machines. One such recent application is cloud computing, where jobs arrive in an online fashion with capacity requirements and need to be immediately scheduled on physical machines in data centers. It is often observed that the requested capacities are not fully utilized, hence offering an opportunity to employ an overcommitment policy, i.e., selling resources beyond capacity. Setting the right overcommitment level can induce a significant cost reduction for the cloud provider, while only inducing a very low risk of violating capacity constraints. We introduce and study a model that quantifies the value of overcommitment by modeling the problem as a bin packing with chance constraints. We then propose an alternative formulation that transforms each chance constraint into a submodular function. We show that our model captures the risk pooling effect and can guide scheduling and overcommitment decisions. We also develop a family of online algorithms that are intuitive, easy to implement and provide a constant factor guarantee from optimal. Finally, we calibrate our model using realistic workload data, and test our approach in a practical setting. Our analysis and experiments illustrate the benefit of overcommitment in cloud services, and suggest a cost reduction of 1.5% to 17% depending on the provider's risk tolerance

Online Bin Covering: Expectations vs. Guarantees

Bin covering is a dual version of classic bin packing. Thus, the goal is to cover as many bins as possible, where covering a bin means packing items of total size at least one in the bin. For online bin covering, competitive analysis fails to distinguish between most algorithms of interest; all "reasonable" algorithms have a competitive ratio of 1/2. Thus, in order to get a better understanding of the combinatorial difficulties in solving this problem, we turn to other performance measures, namely relative worst order, random order, and max/max analysis, as well as analyzing input with restricted or uniformly distributed item sizes. In this way, our study also supplements the ongoing systematic studies of the relative strengths of various performance measures. Two classic algorithms for online bin packing that have natural dual versions are Harmonic and Next-Fit. Even though the algorithms are quite different in nature, the dual versions are not separated by competitive analysis. We make the case that when guarantees are needed, even under restricted input sequences, dual Harmonic is preferable. In addition, we establish quite robust theoretical results showing that if items come from a uniform distribution or even if just the ordering of items is uniformly random, then dual Next-Fit is the right choice.Comment: IMADA-preprint-c

Performance-oriented Cloud Provisioning: Taxonomy and Survey

Cloud computing is being viewed as the technology of today and the future. Through this paradigm, the customers gain access to shared computing resources located in remote data centers that are hosted by cloud providers (CP). This technology allows for provisioning of various resources such as virtual machines (VM), physical machines, processors, memory, network, storage and software as per the needs of customers. Application providers (AP), who are customers of the CP, deploy applications on the cloud infrastructure and then these applications are used by the end-users. To meet the fluctuating application workload demands, dynamic provisioning is essential and this article provides a detailed literature survey of dynamic provisioning within cloud systems with focus on application performance. The well-known types of provisioning and the associated problems are clearly and pictorially explained and the provisioning terminology is clarified. A very detailed and general cloud provisioning classification is presented, which views provisioning from different perspectives, aiding in understanding the process inside-out. Cloud dynamic provisioning is explained by considering resources, stakeholders, techniques, technologies, algorithms, problems, goals and more.Comment: 14 pages, 3 figures, 3 table