Distribution of Cell Area in Bounded Poisson Voronoi Tessellations with Application to Secure Local Connectivity

Poisson Voronoi tessellations have been used in modeling many types of systems across different sciences, from geography and astronomy to telecommunications. The existing literature on the statistical properties of Poisson Voronoi cells is vast, however, little is known about the properties of Voronoi cells located close to the boundaries of a compact domain. In a domain with boundaries, some Voronoi cells would be naturally clipped by the boundary, and the cell area falling inside the deployment domain would have different statistical properties as compared to those of non-clipped Voronoi cells located in the bulk of the domain. In this paper, we consider the planar Voronoi tessellation induced by a homogeneous Poisson point process of intensity $\lambda\!>\!0$ in a quadrant, where the two half-axes represent boundaries. We show that the mean cell area is less than $\lambda^{-1}$ when the seed is located exactly at the boundary, and it can be larger than $\lambda^{-1}$ when the seed lies close to the boundary. In addition, we calculate the second moment of cell area at two locations for the seed: (i) at the corner of a quadrant, and (ii) at the boundary of the half-plane. We illustrate that the two-parameter Gamma distribution, with location-dependent parameters calculated using the method of moments, can be of use in approximating the distribution of cell area. As a potential application, we use the Gamma approximations to study the degree distribution for secure connectivity in wireless sensor networks deployed over a domain with boundaries.Comment: to be publishe

Co-primary inter-operator spectrum sharing over a limited spectrum pool using repeated games

We consider two small cell operators deployed in the same geographical area, sharing spectrum resources from a common pool. A method is investigated to coordinate the utilization of the spectrum pool without monetary transactions and without revealing operator-specific information to other parties. For this, we construct a protocol based on asking and receiving spectrum usage favors by the operators, and keeping a book of the favors. A spectrum usage favor is exchanged between the operators if one is asking for a permission to use some of the resources from the pool on an exclusive basis, and the other is willing to accept that. As a result, the proposed method does not force an operator to take action. An operator with a high load may take spectrum usage favors from an operator that has few users to serve, and it is likely to return these favors in the future to show a cooperative spirit and maintain reciprocity. We formulate the interactions between the operators as a repeated game and determine rules to decide whether to ask or grant a favor at each stage game. We illustrate that under frequent network load variations, which are expected to be prominent in small cell deployments, both operators can attain higher user rates as compared to the case of no coordination of the resource utilization.Comment: To be published in proceedings of IEEE International Conference on Communications (ICC) at London, Jun. 201

Counterfactual Explainer Framework for Deep Reinforcement Learning Models Using Policy Distillation

Deep Reinforcement Learning (DRL) has demonstrated promising capability in solving complex control problems. However, DRL applications in safety-critical systems are hindered by the inherent lack of robust verification techniques to assure their performance in such applications. One of the key requirements of the verification process is the development of effective techniques to explain the system functionality, i.e., why the system produces specific results in given circumstances. Recently, interpretation methods based on the Counterfactual (CF) explanation approach have been proposed to address the problem of explanation in DRLs. This paper proposes a novel CF explanation framework to explain the decisions made by a black-box DRL. To evaluate the efficacy of the proposed explanation framework, we carried out several experiments in the domains of automated driving systems and Atari Pong game. Our analysis demonstrates that the proposed framework generates plausible and meaningful explanations for various decisions made by deep underlying DRLs. Source codes are available at: \url{https://github.com/Amir-Samadi/Counterfactual-Explanation

Multimodal Manoeuvre and Trajectory Prediction for Autonomous Vehicles Using Transformer Networks

Predicting the behaviour (i.e. manoeuvre/trajectory) of other road users, including vehicles, is critical for the safe and efficient operation of autonomous vehicles (AVs), a.k.a. automated driving systems (ADSs). Due to the uncertain future behaviour of vehicles, multiple future behaviour modes are often plausible for a vehicle in a given driving scene. Therefore, multimodal prediction can provide richer information than single-mode prediction enabling AVs to perform a better risk assessment. To this end, we propose a novel multimodal prediction framework that can predict multiple plausible behaviour modes and their likelihoods. The proposed framework includes a bespoke problem formulation for manoeuvre prediction, a novel transformer-based prediction model, and a tailored training method for multimodal manoeuvre and trajectory prediction. The performance of the framework is evaluated using two public benchmark highway driving datasets, namely NGSIM and highD. The results show that the proposed framework outperforms the state-of-the-art multimodal methods in the literature in terms of prediction error and is capable of predicting plausible manoeuvre and trajectory modes.Comment: 8 pages, 3 figures, submitted to IEEE RA

Two-Hop Connectivity to the Roadside in a VANET Under the Random Connection Model

We compute the expected number of cars that have at least one two-hop path to a fixed roadside unit in a one-dimensional vehicular ad hoc network in which other cars can be used as relays to reach a roadside unit when they do not have a reliable direct link. The pairwise channels between cars experience Rayleigh fading in the random connection model, and so exist, with probability function of the mutual distance between the cars, or between the cars and the roadside unit. We derive exact equivalents for this expected number of cars when the car density $\rho$ tends to zero and to infinity, and determine its behaviour using an infinite oscillating power series in $\rho$, which is accurate for all regimes. We also corroborate those findings to a realistic situation, using snapshots of actual traffic data. Finally, a normal approximation is discussed for the probability mass function of the number of cars with a two-hop connection to the origin. The probability mass function appears to be well fitted by a Gaussian approximation with mean equal to the expected number of cars with two hops to the origin.Comment: 21 pages, 7 figure