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 ρ-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.
With the introduction of ρ-reversibility, the basic notion of reversible Markov chain has been relaxed by allowing a wider range of scenarios. Specifically, the reversibility properties are not just sought on the chain itself, but also on all the possible topology-preserving renamings of its state space. Such renamings, called Renaming Functions, exhibit many interesting properties which can be exploited in different contexts. Unfortunately, finding a renaming function for a Markov chain is a very computationally intensive task. Using a naive approach it could require to check for all the possible state space permutations, which is unfeasible for all but the most trivial chains. As a matter of fact, we prove that the corresponding decision problem is polynomially equivalent to Graph Isomorphism. Nevertheless, we introduce an algorithm that, exploiting some necessary conditions for ρ-reversibility, is able to efficiently prune the search space and then verify the remaining renaming candidates. The correctness of the method is theoretically demonstrated and its practical effectiveness is shown over a significant set of discrete and continuous ρ-reversible Markov chains.
<?tight?>Fork-join systems play a pivotal role in the analysis of distributed systems, telecommunication infrastructures, and storage systems. In this article, we consider a fork-join system consisting of K parallel servers, each of which works on one of the K tasks that form each job. The system allocates a fixed amount of computational resources among the K servers, hence determining their service speed. The goal of this article is that of studying the resource allocation policies among the servers. We assume that the queueing disciplines of the fork- and join-queues are First Come First Served. At each epoch, at most K tasks are in service while the others wait in the fork-queues. We propose an algorithm with a very simple implementation that allocates the computational resources in a way that aims at minimizing the join-queue lengths, and hence at reducing<?brk?> the expected job service time. We study its performance in saturation and under exponential service time. The model has an elegant closed-form stationary distribution. Moreover, we provide an algorithm to numerically or symbolically derive the marginal probabilities for the join-queue lengths. Therefore, the expressions for the expected join-queue length and the expected response time under immediate join can be derived. Finally, we compare the performance of the proposed resource allocation algorithm with that of other strategies.
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