Ant colony optimisation and local search for bin-packing and cutting stock problems

The Bin Packing Problem and the Cutting Stock Problem are two related classes of NP-hard combinatorial optimization problems. Exact solution methods can only be used for very small instances, so for real-world problems, we have to rely on heuristic methods. In recent years, researchers have started to apply evolutionary approaches to these problems, including Genetic Algorithms and Evolutionary Programming. In the work presented here, we used an ant colony optimization (ACO) approach to solve both Bin Packing and Cutting Stock Problems. We present a pure ACO approach, as well as an ACO approach augmented with a simple but very effective local search algorithm. It is shown that the pure ACO approach can compete with existing evolutionary methods, whereas the hybrid approach can outperform the best-known hybrid evolutionary solution methods for certain problem classes. The hybrid ACO approach is also shown to require different parameter values from the pure ACO approach and to give a more robust performance across different problems with a single set of parameter values. The local search algorithm is also run with random restarts and shown to perform significantly worse than when combined with ACO

Complexity and Inapproximability Results for Parallel Task Scheduling and Strip Packing

We study the Parallel Task Scheduling problem $Pm|size_j|C_{\max}$ with a constant number of machines. This problem is known to be strongly NP-complete for each $m \geq 5$, while it is solvable in pseudo-polynomial time for each $m \leq 3$. We give a positive answer to the long-standing open question whether this problem is strongly $NP$-complete for $m=4$. As a second result, we improve the lower bound of $\frac{12}{11}$ for approximating pseudo-polynomial Strip Packing to $\frac{5}{4}$. Since the best known approximation algorithm for this problem has a ratio of $\frac{4}{3} + \varepsilon$, this result narrows the gap between approximation ratio and inapproximability result by a significant step. Both results are proven by a reduction from the strongly $NP$-complete problem 3-Partition

Performance Analysis of Data Traffic in Small Cells Networks with User Mobility

We analyze the impact of inter-cell mobility on data traffic performance in dense networks with small cells. To this end, a multi-class queuing system with impatience is proposed as a generic model that captures mobility through the sojourn time of users in the cell. We provide mathematical proofs for the stability and the regularity of this multi-class queuing system. We then show how the performance of a homogeneous network is amenable to the application of the generic model to a single representative cell. This model is applied to derive the throughput of both mobile and static users, along with the handover probability. Numerical evaluation and simulation results are provided to assess the accuracy of the approach; we show, in particular, that both classes of users benefit from a throughput gain induced by the opportunistic displacement of mobile users among cells

Sparse, continuous policy representations for uniform online bin packing via regression of interpolants

Online bin packing is a classic optimisation problem, widely tackled by heuristic methods. In addition to human-designed heuristic packing policies (e.g. first- or best- fit), there has been interest over the last decade in the automatic generation of policies. One of the main limitations of some previously-used policy representations is the trade-off between locality and granularity in the associated search space. In this article, we adopt an interpolation-based representation which has the jointly-desirable properties of being sparse and continuous (i.e. exhibits good genotype-to-phenotype locality). In contrast to previous approaches, the policy space is searchable via real-valued optimization methods. Packing policies using five different interpolation methods are comprehensively compared against a range of existing methods from the literature, and it is determined that the proposed method scales to larger instances than those in the literature

Moment-Generating Algorithm for Response Time in Processor Sharing Queueing Systems

Response times are arguably the most representative and important metric for measuring the performance of modern computer systems. Further, service level agreements (SLAs), ranging from data centres to smartphone users, demand quick and, equally important, predictable response times. Hence, it is necessary to calculate moments, at least, and ideally response time distributions, which is not straightforward. A new moment-generating algorithm for calculating response times analytically is obtained, based on M/M/1 processor sharing (PS) queueing models. This algorithm is compared against existing work on response times in M/M/1-PS queues and extended to M/M/1 discriminatory PS queues. Two real-world case studies are evaluated

Asymptotic Expansions for the Sojourn Time Distribution in the M/G/1M/G/1-PS Queue

We consider the $M/G/1$ queue with a processor sharing server. We study the conditional sojourn time distribution, conditioned on the customer's service requirement, as well as the unconditional distribution, in various asymptotic limits. These include large time and/or large service request, and heavy traffic, where the arrival rate is only slightly less than the service rate. Our results demonstrate the possible tail behaviors of the unconditional distribution, which was previously known in the cases $G=M$ and $G=D$ (where it is purely exponential). We assume that the service density decays at least exponentially fast. We use various methods for the asymptotic expansion of integrals, such as the Laplace and saddle point methods.Comment: 45 page

Iterative approximation of k-limited polling systems

The present paper deals with the problem of calculating queue length distributions in a polling model with (exhaustive) k-limited service under the assumption of general arrival, service and setup distributions. The interest for this model is fueled by an application in the field of logistics. Knowledge of the queue length distributions is needed to operate the system properly. The multi-queue polling system is decomposed into single-queue vacation systems with k-limited service and state-dependent vacations, for which the vacation distributions are computed in an iterative approximate manner. These vacation models are analyzed via matrix-analytic techniques. The accuracy of the approximation scheme is verified by means of an extensive simulation study. The developed approximation turns out be accurate, robust and computationally efficient

Immigration Rates in Fragmented Landscapes – Empirical Evidence for the Importance of Habitat Amount for Species Persistence

BACKGROUND: The total amount of native vegetation is an important property of fragmented landscapes and is known to exert a strong influence on population and metapopulation dynamics. As the relationship between habitat loss and local patch and gap characteristics is strongly non-linear, theoretical models predict that immigration rates should decrease dramatically at low levels of remaining native vegetation cover, leading to patch-area effects and the existence of species extinction thresholds across fragmented landscapes with different proportions of remaining native vegetation. Although empirical patterns of species distribution and richness give support to these models, direct measurements of immigration rates across fragmented landscapes are still lacking. METHODOLOGY/PRINCIPAL FINDINGS: Using the Brazilian Atlantic forest marsupial Gray Slender Mouse Opossum (Marmosops incanus) as a model species and estimating demographic parameters of populations in patches situated in three landscapes differing in the total amount of remaining forest, we tested the hypotheses that patch-area effects on population density are apparent only at intermediate levels of forest cover, and that immigration rates into forest patches are defined primarily by landscape context surrounding patches. As expected, we observed a positive patch-area effect on M. incanus density only within the landscape with intermediate forest cover. Density was independent of patch size in the most forested landscape and the species was absent from the most deforested landscape. Specifically, the mean estimated numbers of immigrants into small patches were lower in the landscape with intermediate forest cover compared to the most forested landscape. CONCLUSIONS/SIGNIFICANCE: Our results reveal the crucial importance of the total amount of remaining native vegetation for species persistence in fragmented landscapes, and specifically as to the role of variable immigration rates in providing the underlying mechanism that drives both patch-area effects and species extinction thresholds