Process Completing Sequences for Resource Allocation Systems with Synchronization

This paper considers the problem of establishing live resource allocation in workflows with synchronization stages. Establishing live resource allocation in this class of systems is challenging since deciding whether a given level of resource capacities is sufficient to complete a single process is NP-complete. In this paper, we develop two necessary conditions and one sufficient condition that provide quickly computable tests for the existence of process completing sequences. The necessary conditions are based on the sequence of completions of � subprocesses that merge together at a synchronization. Although the worst case complexity is O(2�), we expect the number of subprocesses combined at any synchronization will be sufficiently small so that total computation time remains manageable. The sufficient condition uses a reduction scheme that computes a sufficient capacity level of each resource type to complete and merge all � subprocesses. The worst case complexity is O(�⋅�), where � is the number of synchronizations. Finally, the paper develops capacity bounds and polynomial methods for generating feasible resource allocation sequences for merging systems with single unit allocation. This method is based on single step look-ahead for deadly marked siphons and is O(2�). Throughout the paper, we use a class of Petri nets called Generalized Augmented Marked Graphs to represent our resource allocation systems

Inhomogeneous percolation models for spreading phenomena in random graphs

Percolation theory has been largely used in the study of structural properties of complex networks such as the robustness, with remarkable results. Nevertheless, a purely topological description is not sufficient for a correct characterization of networks behaviour in relation with physical flows and spreading phenomena taking place on them. The functionality of real networks also depends on the ability of the nodes and the edges in bearing and handling loads of flows, energy, information and other physical quantities. We propose to study these properties introducing a process of inhomogeneous percolation, in which both the nodes and the edges spread out the flows with a given probability. Generating functions approach is exploited in order to get a generalization of the Molloy-Reed Criterion for inhomogeneous joint site bond percolation in correlated random graphs. A series of simple assumptions allows the analysis of more realistic situations, for which a number of new results are presented. In particular, for the site percolation with inhomogeneous edge transmission, we obtain the explicit expressions of the percolation threshold for many interesting cases, that are analyzed by means of simple examples and numerical simulations. Some possible applications are debated.Comment: 28 pages, 11 figure

Redundancy Scheduling with Locally Stable Compatibility Graphs

Redundancy scheduling is a popular concept to improve performance in parallel-server systems. In the baseline scenario any job can be handled equally well by any server, and is replicated to a fixed number of servers selected uniformly at random. Quite often however, there may be heterogeneity in job characteristics or server capabilities, and jobs can only be replicated to specific servers because of affinity relations or compatibility constraints. In order to capture such situations, we consider a scenario where jobs of various types are replicated to different subsets of servers as prescribed by a general compatibility graph. We exploit a product-form stationary distribution and weak local stability conditions to establish a state space collapse in heavy traffic. In this limiting regime, the parallel-server system with graph-based redundancy scheduling operates as a multi-class single-server system, achieving full resource pooling and exhibiting strong insensitivity to the underlying compatibility constraints.Comment: 28 pages, 4 figure

Graphical Markov models, unifying results and their interpretation

Graphical Markov models combine conditional independence constraints with graphical representations of stepwise data generating processes.The models started to be formulated about 40 years ago and vigorous development is ongoing. Longitudinal observational studies as well as intervention studies are best modeled via a subclass called regression graph models and, especially traceable regressions. Regression graphs include two types of undirected graph and directed acyclic graphs in ordered sequences of joint responses. Response components may correspond to discrete or continuous random variables and may depend exclusively on variables which have been generated earlier. These aspects are essential when causal hypothesis are the motivation for the planning of empirical studies. To turn the graphs into useful tools for tracing developmental pathways and for predicting structure in alternative models, the generated distributions have to mimic some properties of joint Gaussian distributions. Here, relevant results concerning these aspects are spelled out and illustrated by examples. With regression graph models, it becomes feasible, for the first time, to derive structural effects of (1) ignoring some of the variables, of (2) selecting subpopulations via fixed levels of some other variables or of (3) changing the order in which the variables might get generated. Thus, the most important future applications of these models will aim at the best possible integration of knowledge from related studies.Comment: 34 Pages, 11 figures, 1 tabl

Data granulation by the principles of uncertainty

Researches in granular modeling produced a variety of mathematical models, such as intervals, (higher-order) fuzzy sets, rough sets, and shadowed sets, which are all suitable to characterize the so-called information granules. Modeling of the input data uncertainty is recognized as a crucial aspect in information granulation. Moreover, the uncertainty is a well-studied concept in many mathematical settings, such as those of probability theory, fuzzy set theory, and possibility theory. This fact suggests that an appropriate quantification of the uncertainty expressed by the information granule model could be used to define an invariant property, to be exploited in practical situations of information granulation. In this perspective, a procedure of information granulation is effective if the uncertainty conveyed by the synthesized information granule is in a monotonically increasing relation with the uncertainty of the input data. In this paper, we present a data granulation framework that elaborates over the principles of uncertainty introduced by Klir. Being the uncertainty a mesoscopic descriptor of systems and data, it is possible to apply such principles regardless of the input data type and the specific mathematical setting adopted for the information granules. The proposed framework is conceived (i) to offer a guideline for the synthesis of information granules and (ii) to build a groundwork to compare and quantitatively judge over different data granulation procedures. To provide a suitable case study, we introduce a new data granulation technique based on the minimum sum of distances, which is designed to generate type-2 fuzzy sets. We analyze the procedure by performing different experiments on two distinct data types: feature vectors and labeled graphs. Results show that the uncertainty of the input data is suitably conveyed by the generated type-2 fuzzy set models.Comment: 16 pages, 9 figures, 52 reference