The empirical mean position of a branching Lévy process

We consider a supercritical branching L\'evy process on the real line. Under mild moment assumptions on the number of offspring and their displacements, we prove a second-order limit theorem on the empirical mean position

A Random Multiple Access Protocol with Spatial Interactions

We analyse an ALOHA-type random multiple-access protocol where users have local interactions. We show that the fluid model of the system workload satisfies a certain differential equation. We obtain a sufficient condition for the stability of this differential equation and deduce from that a sufficient condition for the stability of the protocol. We discuss the necessary condition. Further, for the underlying Markov chain, we estimate the rate of convergence to the stationary distribution. Then we establish an interesting and unexpected result showing that the main diagonal is locally unstable if the input rate is sufficiently small. Finally, we consider two generalisations of the model.Comment: 29 page

Stochastic stability of monotone economies in regenerative environments

We introduce and analyze a new class of monotone stochastic recursions in a regenerative environment which is essentially broader than that of Markov chains. We prove stability theorems and apply our results to three canonical models in recursive economics, generalizing some known stability results to the cases when driving sequences are not independent and identically distributed

Un Protocole d'Accès Multiple Aléatoire avec Interactions Spatiales

Nous étudions un protocole d'accès de type aloha où les utilisateurs ont des interactions locales. Nous établissons que le modèle fluide de la charge du système satisfait une équation différentielle. Nous établissons une condition sur la stabilité de cette équation différentielle et en déduisons une condition suffisante pour la stabilité du protocole. La condition nécéssaire est discutée

CSMA/CA in Time and Frequency Domains

It has recently been shown that flexible channelization, whereby wireless stations adapt their spectrum bands on a per-frame basis, is feasible in practice. In this paper, we propose TF-CSMA/CA, an algorithm for flexible channelization that schedules packets in time and frequency domains. TFCSMA/CA is a simple extension of the CSMA/CA protocol used by IEEE 802.11. Contrary to existing channelization schemes, it is entirely distributed and it reacts only to packet collisions, successful transmissions and carrier sensing. With TF-CSMA/CA, when a station is involved in a collision, it performs backoff in both time and frequency domains. Backing off also in the frequency domain allows the transmitters to be much more efficient and aggressive in the time domain, which significantly reduces the severe overheads present with recent 802.11 PHY layers. The main challenge, however, is that the stations need some level of self-organization in order to find spectrum bands of variable widths that minimize interference, while still efficiently using the available spectrum. Using analysis and simulations, we show that such an extension of CSMA/CA to the frequency domain drastically improves both throughput and fairness. Notably, it enables the stations to find interference-free spectrum bands of appropriate size using no communication – relying only on collisions and successes as implicit signals

Structural bias in population-based algorithms

Challenging optimisation problems are abundant in all areas of science and industry. Since the 1950s, scientists have responded to this by developing ever-diversifying families of 'black box' optimisation algorithms. The latter are designed to be able to address any optimisation problem, requiring only that the quality of any candidate solution can be calculated via a 'fitness function' specific to the problem. For such algorithms to be successful, at least three properties are required: (i) an effective informed sampling strategy, that guides the generation of new candidates on the basis of the fitnesses and locations of previously visited candidates; (ii) mechanisms to ensure efficiency, so that (for example) the same candidates are not repeatedly visited; and (iii) the absence of structural bias, which, if present, would predispose the algorithm towards limiting its search to specific regions of the solution space. The first two of these properties have been extensively investigated, however the third is little understood and rarely explored. In this article we provide theoretical and empirical analyses that contribute to the understanding of structural bias. In particular, we state and prove a theorem concerning the dynamics of population variance in the case of real-valued search spaces and a 'flat' fitness landscape. This reveals how structural bias can arise and manifest as non-uniform clustering of the population over time. Critically, theory predicts that structural bias is exacerbated with (independently) increasing population size, and increasing problem difficulty. These predictions, supported by our empirical analyses, reveal two previously unrecognised aspects of structural bias that would seem vital for algorithm designers and practitioners. Respectively, (i) increasing the population size, though ostensibly promoting diversity, will magnify any inherent structural bias, and (ii) the effects of structural bias are more apparent when faced with (many classes of) 'difficult' problems. Our theoretical result also contributes to the 'exploitation/exploration' conundrum in optimisation algorithm design, by suggesting that two commonly used approaches to enhancing exploration - increasing the population size, and increasing the disruptiveness of search operators - have quite distinct implications in terms of structural bias

Rare events and stability for random walks and stochastic communication systems

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