Spatial Fluid Limits for Stochastic Mobile Networks

We consider Markov models of large-scale networks where nodes are characterized by their local behavior and by a mobility model over a two-dimensional lattice. By assuming random walk, we prove convergence to a system of partial differential equations (PDEs) whose size depends neither on the lattice size nor on the population of nodes. This provides a macroscopic view of the model which approximates discrete stochastic movements with continuous deterministic diffusions. We illustrate the practical applicability of this result by modeling a network of mobile nodes with on/off behavior performing file transfers with connectivity to 802.11 access points. By means of an empirical validation against discrete-event simulation we show high quality of the PDE approximation even for low populations and coarse lattices. In addition, we confirm the computational advantage in using the PDE limit over a traditional ordinary differential equation limit where the lattice is modeled discretely, yielding speed-ups of up to two orders of magnitude

Formal lumping of polynomial differential equations through approximate equivalences

It is well known that exact notions of model abstraction and reduction for dynamical systems may not be robust enough in practice because they are highly sensitive to the specific choice of parameters. In this paper we consider this problem for nonlinear ordinary differential equations (ODEs) with polynomial derivatives. We introduce a model reduction technique based on approximate differential equivalence, i.e., a partition of the set of ODE variables that performs an aggregation when the variables are governed by nearby derivatives. We develop algorithms to (i) compute the largest approximate differential equivalence; (ii) construct an approximately reduced model from the original one via an appropriate perturbation of the coefficients of the polynomials; and (iii) provide a formal certificate on the quality of the approximation as an error bound, computed as an over-approximation of the reachable set of the reduced model. Finally, we apply approximate differential equivalences to case studies on electric circuits, biological models, and polymerization reaction networks

A Product-Form Model for the Performance Evaluation of a Bandwidth Allocation Strategy in WSNs

Wireless Sensor Networks (WSNs) are important examples of Collective Adaptive System, which consist of a set of motes that are spatially distributed in an indoor or outdoor space. Each mote monitors its surrounding conditions, such as humidity, intensity of light, temperature, and vibrations, but also collects complex information, such as images or small videos, and cooperates with the whole set of motes forming the WSN to allow the routing process. The traffic in the WSN consists of packets that contain the data harvested by the motes and can be classified according to the type of information that they carry. One pivotal problem in WSNs is the bandwidth allocation among the motes. The problem is known to be challenging due to the reduced computational capacity of the motes, their energy consumption constraints, and the fully decentralised network architecture. In this article, we study a novel algorithm to allocate the WSN bandwidth among the motes by taking into account the type of traffic they aim to send. Under the assumption of a mesh network and Poisson distributed harvested packets, we propose an analytical model for its performance evaluation that allows a designer to study the optimal configuration parameters. Although the Markov chain underlying the model is not reversible, we show it to be.-reversible under a certain renaming of states. By an extensive set of simulations, we show that the analytical model accurately approximates the performance of networks that do not satisfy the assumptions. The algorithm is studied with respect to the achieved throughput and fairness. We show that it provides a good approximation of the max-min fairness requirements

QoS provision in a dynamic channel allocation based on admission control decisions

El PDF se corresponde a la versión de los autores, no obstante los datos de la cita son los de la versión definitiva, tal cual figuran en la página web de la revista.PostprintCognitive Radio Networks have emerged in the last decades as a solution of two problems: spectrum underutilization and spectrum scarcity. In this work, we propose a dynamic spectrum sharing mechanism, where primary users have strict priority over secondary ones in order to improve the mean spectrum utilization with the objective of providing to secondary users a satisfactory grade of service with a small interruption probability. We study a stochastic model for Cognitive Radio Networks with fluid limits techniques. Our main findings consist in a Gaussian limit theorem in the sub-critical case, and a non-Gaussian limit theorem, under a different scaling scheme, in the critical case. These results provide us practical QoS criteria for sharing policies. We support our analysis with representative simulated examples in both scenarios

Fluid aggregations for Markovian process algebra

Quantitative analysis by means of discrete-state stochastic processes is hindered by the well-known phenomenon of state-space explosion, whereby the size of the state space may have an exponential growth with the number of objects in the model. When the stochastic process underlies a Markovian process algebra model, this problem may be alleviated by suitable notions of behavioural equivalence that induce lumping at the underlying continuous-time Markov chain, establishing an exact relation between a potentially much smaller aggregated chain and the original one. However, in the modelling of massively distributed computer systems, even aggregated chains may be still too large for efficient numerical analysis. Recently this problem has been addressed by fluid techniques, where the Markov chain is approximated by a system of ordinary differential equations (ODEs) whose size does not depend on the number of the objects in the model. The technique has been primarily applied in the case of massively replicated sequential processes with small local state space sizes. This thesis devises two different approaches that broaden the scope of applicability of efficient fluid approximations. Fluid lumpability applies in the case where objects are composites of simple objects, and aggregates the potentially massive, naively constructed ODE system into one whose size is independent from the number of composites in the model. Similarly to quasi and near lumpability, we introduce approximate fluid lumpability that covers ODE systems which can be aggregated after a small perturbation in the parameters. The technique of spatial aggregation, instead, applies to models whose objects perform a random walk on a two-dimensional lattice. Specifically, it is shown that the underlying ODE system, whose size is proportional to the number of the regions, converges to a system of partial differential equations of constant size as the number of regions goes to infinity. This allows for an efficient analysis of large-scale mobile models in continuous space like ad hoc networks and multi-agent systems