There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

Stable non-Gaussian random processes , by G. Samorodnitsky and M. S. Taqqu. Pp. 632. £49.50. 1994. ISBN 0-412-05171-0 (Chapman and Hall).

A number of empirical studies of traffic measurements from a variety of working packet networks have demonstrated that actual network traffic is self-similar or long-range dependent in nature-in sharp contrast to commonly made traffic modeling assumptions. We provide a plausible physical explanation for the occurrence of self-similarity in local-area network (LAN) traffic. Our explanation is based on convergence results for processes that exhibit high variability and is supported by detailed statistical analyzes of real-time traffic measurements from Ethernet LANs at the level of individual sources. This paper is an extended version of Willinger et al. (1995). We develop here the mathematical results concerning the superposition of strictly alternating ON/OFF sources. Our key mathematical result states that the superposition of many ON/OFF sources (also known as packet-trains) with strictly alternating ON- and OFF-periods and whose ON-periods or OFF-periods exhibit the Noah effect produces aggregate network traffic that exhibits the Joseph effect. There is, moreover, a simple relation between the parameters describing the intensities of the Noah effect (high variability) and the Joseph effect (self-similarity). An extensive statistical analysis of high time-resolution Ethernet LAN traffic traces confirms that the data at the level of individual sources or source-destination pairs are consistent with the Noah effect. We also discuss implications of this simple physical explanation for the presence of self-similar traffic patterns in modern high-speed network traffic.

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Self-similarity through high-variability: statistical analysis of Ethernet LAN traffic at the source level

Future wireless networks are expected to constitute a distributed intelligent wireless communications, sensing, and computing platform, which will have the challenging requirement of interconnecting the physical and digital worlds in a seamless and sustainable manner. Currently, two main factors prevent wireless network operators from building such networks: (1) the lack of control of the wireless environment, whose impact on the radio waves cannot be customized, and (2) the current operation of wireless radios, which consume a lot of power because new signals are generated whenever data has to be transmitted. In this paper, we challenge the usual “more data needs more power and emission of radio waves” status quo, and motivate that future wireless networks necessitate a smart radio environment: a transformative wireless concept, where the environmental objects are coated with artificial thin films of electromagnetic and reconfigurable material (that are referred to as reconfigurable intelligent meta-surfaces), which are capable of sensing the environment and of applying customized transformations to the radio waves. Smart radio environments have the potential to provide future wireless networks with uninterrupted wireless connectivity, and with the capability of transmitting data without generating new signals but recycling existing radio waves. We will discuss, in particular, two major types of reconfigurable intelligent meta-surfaces applied to wireless networks. The first type of meta-surfaces will be embedded into, e.g., walls, and will be directly controlled by the wireless network operators via a software controller in order to shape the radio waves for, e.g., improving the network coverage. The second type of meta-surfaces will be embedded into objects, e.g., smart t-shirts with sensors for health monitoring, and will backscatter the radio waves generated by cellular base stations in order to report their sensed data to mobile phones. These functionalities will enable wireless network operators to offer new services without the emission of additional radio waves, but by recycling those already existing for other purposes. This paper overviews the current research efforts on smart radio environments, the enabling technologies to realize them in practice, the need of new communication-theoretic models for their analysis and design, and the long-term and open research issues to be solved towards their massive deployment. In a nutshell, this paper is focused on discussing how the availability of reconfigurable intelligent meta-surfaces will allow wireless network operators to redesign common and well-known network communication paradigms.

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Smart radio environments empowered by reconfigurable AI meta-surfaces: an idea whose time has come

A storage model with self-similar input process is studied. A relation coupling together the storage requirement, the achievable utilization and the output rate is derived. A lower bound for the complementary distribution function of the storage level is given.

A storage model with self-similar input

Random (pseudo)graphs G N with the following structure are studied: first, independent and identically distributed capacities Λ i are drawn for vertices i = 1 N; then, each pair of vertices (i, j) is connected, independently of the other pairs, with E(i, j) edges, where E(i, j) has distribution Poisson(Λ i Λ j /Σ N k=1 Λ k ). The main result of the paper is that when P(Λ 1 > x) ≥ x -τ+1 , where τ ∈ (2, 3), then, asymptotically almost surely, G N has a giant component, and the distance between two randomly selected vertices of the giant component is less than (2 + o(N))(log log Ν)/(-log (r - 2)). It is also shown that the cases τ > 3, τ ∈ (2, 3), and r ∈ (1, 2) present three qualitatively different connectivity architectures.

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On a conditionally Poissonian graph process

Accident risk of road and weather conditions on different road types

This study was designed to investigate the relative accident risk of different road weather conditions and combinations of conditions. The study applied a recently developed method which is based on the notion of Palm probability, originating in the theory of random point processes, which in this case corresponds to picking a random vehicle from the traffic. The method consists of calculating the Palm distribution of different conditions and comparing it with the distribution of the same conditions as seen by the accidents. The condition affects the accident risk statistically, when these two distributions differ. The study included all police reported single- and multi-vehicle accidents (N = 10,646) occurring on 43 main roads in Finland during the years 2014–2016. A major contribution of this paper is the demonstration of the method on national scale by using estimated hourly traffic volumes on road segments instead of measured ones, which would have been available for few roads only. Accident risks are commonly examined in relation to traffic volume. This paper includes the speed of the traffic and thus, the paper examines accident risk in relation to the time spent on the road segment in certain conditions. The hour-level weather and road condition data per segment were obtained from nearby road weather stations. The relative accident risks were increased for poor road weather conditions; however, they were highest for icy rain and slippery and very slippery road conditions. When comparing the relative accident risk based on road type, the results showed that the risk in poor weather and road conditions was higher on motorways compared to two-lane and multiple-lane roads even though the overall risk was lower on motorways. Furthermore, the corresponding relative accident risks were generally higher for single-vehicle accidents compared to multi-vehicle accidents.

This document is protected by copyright and other intellectual property rights, and duplication or sale of all or part of any of this document is not permitted, except duplication for research use or educational purposes in electronic or print form. You must obtain permission for any other use. Electronic or print copies may not be offered for sale. Title Simulation of fractional Brownian motion with conditionalized random midpoint displacement Author(s) Norros, Ilkka; Mannersalo, Petteri; Wang, Jonathan Citation Advances in Performance Analysis vol. 2(1999):1, pp. 77-101 Date 1999

Ilkka Norros

Papers

A storage model with self-similar input

On a conditionally Poissonian graph process

Accident risk of road and weather conditions on different road types

Accident risk of road and weather conditions on different road types

Simulation of fractional Brownian motion with conditionalized random midpoint displacement