This paper defines a new model which incorporates three key ingredients of a large class of wireless communication systems: (1) spatial interactions through interference, (2) dynamics of the queueing type, with users joining and leaving, and (3) carrier sensing and collision avoidance as used in, e.g., WiFi. In systems using (3), rather than directly accessing the shared resources upon arrival, a customer is considerate and waits to access them until nearby users in service have left. This new model can be seen as a missing piece of a larger puzzle that contains such dynamics as spatial birth-and-death processes, the Poisson-Hail model, and wireless dynamics as key other pieces. It is shown that, under natural assumptions, this model can be represented as a Markov process on the space of counting measures. The main results are then two-fold. The first result is on the shape of the stability region and, more precisely, on the characterization of the critical value of the arrival rate that separates stability from instability. The second result is of a more qualitative or perhaps even ethical nature. There is evidence that for natural values of the system parameters, the implementation of sensing and the delayed access for collision avoidance can stabilize a system that would be unstable if immediate access to the shared resources would be granted. In other words, for these parameters, renouncing greedy access makes sharing sustainable, whereas indulging in greedy access kills the system.

The reconfigurable intelligent surface (RIS) technology allows one to engineer spatial diversity in complex cellular networks. This paper provides a stochastic geometry framework for the system-level performance assessment of RIS-assisted networks. To account for the inherent randomness in the spatial deployments of base stations (BSs) and RISs, we model the RIS placements as point processes (PPs) conditioned on the associated BSs, which are modeled by a Poisson point process (PPP). We assume that the system uses the orthogonal frequency division multiplexing (OFDM) technique to exploit the multipath diversity provided by RISs. The downlink coverage probability and ergodic rate can be evaluated when RISs operate as batched powerless beamformers. The resulting analytical expressions provide a general methodology for assessing the impact of a parameterized RIS model on system performance. These RIS PPs can be adapted based on the deployment strategy. We focus on modeling the RISs as a Matérn cluster process (MCP), where each RIS cluster is a finite PPP within a ring centered on its associated BS. This model connects link-level knowledge to system-level impacts, such as overall interference and the effects of imperfect channel state information (CSI). It also evaluates key RIS deployment parameters, including batch size and RIS density. Furthermore, we analyze a variant of RIS placement in which RISs are deployed around coverage holes to demonstrate the framework’s flexibility and applicability. Numerical evaluations of the analytical expressions and Monte-Carlo simulations jointly validate the proposed analytical approach and provide valuable insights into the design of future RIS-assisted cellular networks.

Providing performance guarantees is one of the critical objectives of recent and future communication networks, toward which regulations, i.e., constraints on key system parameters, have played an indispensable role. This is the case for large wireless communication networks, where spatial regulations (e.g., constraints on intercell distance) have recently been shown, through a spatial network calculus, to be essential for establishing provable wireless link-level guarantees. In this work, we focus on performance guarantees for the downlink of cellular networks where we impose a hardcore (spatial) regulation on base station (BS) locations and evaluate how BS scheduling (which controls which BSs can transmit at a given time) impacts performance. Hardcore regulation is the simplest form of spatial regulation that enforces a minimal distance between any pair of transmitters in the network. Within this framework of spatial network calculus, we first provide an upper bound on the power of total interference for a spatially regulated cellular network, and then, identify the regimes where scheduling BSs yields better link-level rate guarantees compared to scenarios where base stations are always active. The hexagonal cellular network is analyzed as a special case. The results offer insights into what spatial regulations are needed, when to choose scheduling, and how to potentially reduce the network power consumption to provide a certain target performance guarantee.

Reconfigurable intelligent surfaces (RISs) enhance wireless communication by creating engineered signal reflection paths in addition to direct links. This work presents a stochastic geometry framework using point processes (PPs) to model multiple randomly deployed RISs conditioned on their associated base station (BS) locations. By characterizing aggregated reflections from multiple RISs using the Laplace transform, we analytically assess the performance impact of RIS-reflected signals by integrating this characterization into well-established stochastic geometry frameworks. Specifically, we derive closedform expressions for the Laplace transform of the reflected signal power in several deployment scenarios. These analytical results facilitate performance evaluation of RIS-enabled enhancements. Numerical simulations validate that optimal RIS placement favors proximity to BSs or user equipment (UEs), and further quantify the impact of reflected interference, various fading assumptions, and diverse spatial deployment strategies. Importantly, our analytical approach shows superior computational efficiency compared to Monte Carlo simulations.

Reconfigurable intelligent surfaces (RISs) are a promising technology for enhancing cellular network performance and yielding additional value to network operators. This paper proposes a techno-economic analysis of RIS-assisted cellular networks to guide operators in deciding between deploying additional RISs or base stations (BS). We assume a relative cost model that considers the total cost of ownership (TCO) of deploying additional nodes, either BSs or RISs. We assume a return on investment (RoI) that is proportional to the system's spectral efficiency. The latter is evaluated based on a stochastic geometry model that gives an integral formula for the ergodic rate in cellular networks equipped with RISs. The marginal RoI for any investment strategy is determined by the partial derivative of this integral expression with respect to node densities. We investigate two case studies: throughput enhancement and coverage hole mitigation. These examples demonstrate how operators could determine the optimal investment strategy in scenarios defined by the current densities of BSs and RISs, and their relative costs. Numerical results illustrate the evolution of ergodic rates based on the proposed investment strategy, demonstrating the investment decision-making process while considering technological and economic factors. This work quantitatively demonstrates that strategically investing in RISs can offer better system-level benefits than solely investing in BS densification.

Even when driven by the same stimulus, neuronal responses are well-known to exhibit a striking level of spiking variability. In-vivo electrophysiological recordings also reveal a surprisingly large degree of variability at the subthreshold level. In prior work, we considered biophysically relevant neuronal models to account for the observed magnitude of membrane voltage fluctuations. We found that accounting for these fluctuations requires weak but nonzero synchrony in the spiking activity, in amount that are consistent with experimentally measured spiking correlations. Here we investigate whether such synchrony can explain additional statistical features of the measured neural activity, including neuronal voltage covariability and voltage skewness. Addressing this question involves conducting a generalized moment analysis of conductance-based neurons in response to input drives modeled as correlated jump processes. Technically, we perform such an analysis using fixed-point techniques from queuing theory that are applicable in the stationary regime of activity. We found that weak but nonzero synchrony can consistently explain the experimentally reported voltage covariance and skewness. This confirms the role of synchrony as a primary driver of cortical variability and supports that physiological neural activity emerges as a population-level phenomenon, especially in the spontaneous regime.

This work develops an analytical framework for downlink low Earth orbit (LEO) or medium Earth orbit (MEO) satellite communications, leveraging tools from stochastic geometry. We propose a tractable approach to the analysis of such satellite communication systems, accounting for the fact that satellites are located on circular orbits. We accurately incorporate this geometric property of LEO or MEO satellite constellations by developing a Cox point process model that jointly produces orbits and satellites on these orbits. Our work contrasts with previous modeling studies that presumed satellite locations to be entirely random, thereby overlooking the fundamental fact that satellites are jointly positioned on orbits. Employing this Cox model, we analyze the network performance experienced by users located on Earth. Specifically, we evaluate the no-satellite probability of the proposed network and the Laplace transform of the interference created by such a network. Using it, we compute its SIR (signal-to-interference) distribution, namely its coverage probability. By presenting fundamental network performance as functions of key parameters, this model allows one to assess the statistical properties of downlink LEO or MEO satellite communications and can thus be used as a system-level design tool to operate and optimize forthcoming complex LEO or MEO satellite networks.

In this paper, we study a multiclass spatial birth-and-death (SBD) processes on a compact region of the Euclidean plane modeling wireless interactions. In such a setup, users arrive at a constant rate and leave at a rate inversely proportional to a shot noise created by interfering users in the network. The novelty of this work lies in the addition of service differentiation, inspired by bandwidth partitioning present in 5G networks: users are allocated a fixed number of frequency bands and only interfere with transmissions on these bands. The first result of our work lies in the determination of the critical user arrival rate below which the system is always stochastically stable, and above which it is unstable. The analysis requires symmetry assumptions which are defined in the paper. The proof for this result uses stochastic monotonicity and fluid limit models to bound the dynamics from above and below by two adequate discrete-state Markov jump processes, for which we obtain stability and instability results. This leads to a closed form formula for the critical arrival rate. In a second part, we propose two heuristics to estimate the steady-state densities of users in the network: the first one relies on a Poisson approximation of the steady-state processes. The second one uses a cavity approximation leveraging second-order moment measures, which leads to more accurate estimates of the steady-state user densities of all classes of users. The Poisson heuristic also gives a good estimate for the critical arrival rate.

This paper uses the theory of point processes and stochastic geometry to quantify the sky visibility experienced by users located in an outdoor environment. The general idea is to represent the buildings of this environment as a stationary marked point process, where the points represent the building locations and the marks their heights. The point process framework is first used to characterize the distribution of the blockage angle, which limits the visibility of a typical user into the sky due to the obstruction by buildings. In the context of communications, this distribution is useful when users try to connect to the nodes of an aerial or non-terrestrial network in a Line-of-Sight way. Within this context, the point process framework can also be used to investigate the gain of connectivity obtained thanks to Reconfigurable Intelligent Surfaces. Assuming that such surfaces are installed on the top of buildings to extend the user’s sky visibility, this point process approach allows one to quantify the gain in visibility and hence the gain in connectivity obtained by the typical user. The distributional properties of visibility-related metrics are cross-validated by comparison to simulation results and 3GPP measurements.

This work develops a novel approach toward performance guarantees for all links in arbitrarily large wireless networks. It introduces a spatial network calculus, consisting of spatial regulation properties for stationary point processes and the first steps of a calculus for this regulation, which can be seen as an extension to space of the classical network calculus. Specifically, two classes of regulations are defined: one includes ball regulation and shot-noise regulation, which are shown to be equivalent and upper constraint interference; the other one includes void regulation, which lower constraints the signal power. These regulations are defined both in the strong and weak sense: the former requires the regulations to hold everywhere in space, whereas the latter only requires the regulations to hold as observed by a jointly stationary point process. Using this approach, we derive performance guarantees in device-to-device, ad hoc, and cellular networks under proper regulations. We give universal bounds on the SINR for all links, which give link service guarantees based on information-theoretic achievability. They are combined with classical network calculus to provide end-to-end latency guarantees for all packets in wireless queuing networks. Such guarantees do not exist in networks that are not spatially regulated, e.g., Poisson networks.

This paper investigates the benefits of employing vehicle relays by analyzing the increment of the line-of-sight (LOS) coverage based on vehicle relays to enable high-speed communications between roadside user devices, vehicles, and roadside infrastructure. We characterize a unique spatial relationship between roadside units (RSUs) and vehicles by employing the Cox point processes. Then, the LOS coverage from these RSUs is modeled by a Boolean model on the Cox point process. Assuming vehicle relays provide an additional LOS coverage when they are within the RSU LOS coverage, we quantify the growth of the LOS coverage by separately deriving the mean area fractions of the RSU LOS coverage and RSU-plus-relay LOS coverage, respectively. We explicitly provide the gain in the LOS coverage as an integral formula and show that relays increase the LOS coverage area by nearly 50 percent.

This work develops a novel approach towards performance guarantees for all links in arbitrarily large wireless networks. It introduces spatial regulation properties for stationary spatial point processes and develops the first steps of a calculus for this regulation, which can be seen as an extension to space of the classical network calculus. Specifically, two classes of regulations are defined: one includes ball regulation and shot-noise regulation, which are shown to be equivalent and leads to upper bounds on the interference power; the other one includes void regulation, which lower constraints the signal power. These regulations are defined both in the strong and weak sense: the former requires the regulations to hold everywhere in space, whereas the latter only requires the regulations to hold as observed by a jointly stationary point process. Focusing on device-to-device networks, we then derive universal bounds on the SINR based on spatial regulations and, in turn, link service guarantees assuming information theoretic achievability. They are combined with classical network calculus to provide end-to-end latency guarantees for all packets in such wireless queuing networks. Such guarantees do not exist in networks that are not spatially regulated, e.g., Poisson networks.

This paper presents a novel spatially consistent approach for addressing blockage and line-of-sight (LOS) paths in highway vehicular networks. We use stochastic geometry to model transmitters, obstacles, and receivers located in three lines, respectively. Then, their geometric interactions characterize the existence of LOS paths. Specifically, the proposed approach focuses on the role of obstacles in blocking one or more LOS paths, which has been overlooked in most statistical models for blockage. Under the proposed framework, we derive the probability that a typical vehicle is in LOS with respect to transmitters. The proposed framework and LOS analysis are important to the LOS-critical applications such as positioning or mmWave communications in vehicular networks.

This article presents a stochastic geometric framework to model and analyze spatially correlated blockage in a vehicular network. First, we model the vehicular obstacles as a Boolean model on a line using stochastic geometry. Then, we represent signal paths from transmitters to receivers as a graph from the transmitter point process to the receiver point process. The blockage of a signal path occurs if and only if any obstacle obstructs the corresponding edge. Since the signal blockage occurs by obstacles, the proposed determination of blockage preserves the spatial correlation of signal paths. Under the proposed framework, we derive the blockage probability of a typical vehicle. In addition, we derive the probability that a typical vehicle is in the line of sight (LOS) with respect to at least one transmitter. The proposed framework and blockage analysis will be instrumental to the analysis of LOS-critical applications such as positioning or mmWave communications in vehicular networks.

This paper proposes a cascade blockage model for analyzing the vision that a user has of a wireless network. This model, inspired by the classical multiplicative cascade models, has a radial structure meant to analyze blockages seen by the receiver at the origin in different angular sectors. The main novelty is that it is based on the geometry of obstacles and takes the joint blockage phenomenon into account. We show on a couple of simple instances that the Laplace transforms of total interference satisfies a functional equation that can be solved efficiently by an iterative scheme. This is used to analyze the coverage probability of the receiver and the effect of blockage correlation and penetration loss in both dense and sparse blockage environments. Furthermore, this model is used to investigate the effect of blockage correlation on user beamforming techniques. Another functional equation and its associated iterative algorithm are proposed to derive the coverage performance of the best beam selection in this context. In addition, the conditional coverage probability is also derived to evaluate the effect of beam switching. The results not only show that beam selection is quite efficient for multi-beam terminals, but also show how the correlation brought by blockages can be leveraged to accelerate beam sweeping and pairing.

Beam management is central in the operation of beamformed wireless cellular systems such as 5G New Radio (NR) networks. Focusing the energy radiated to mobile terminals (MTs) by increasing the number of beams per cell increases signal power and decreases interference, and has hence the potential to bring major improvements on area spectral efficiency (ASE). This paper proposes a first system-level stochastic geometry model encompassing major aspects of the beam management problem: frequencies, antenna configurations, and propagation; physical layer, wireless links, and coding; network geometry, interference, and resource sharing; sensing, signaling, and mobility management. This model leads to a simple analytical expression for the effective rate that the typical user gets in this context. This in turn allows one to find the number of beams per cell and per MT that maximizes the effective ASE by offering the best tradeoff between beamforming gains and beam management operational overheads and costs, for a wide variety of 5G network scenarios including millimeter wave (mmWave) and sub-6 GHz. As part of the system-level analysis, we define and analyze several underlying new and fundamental performance metrics that are of independent interest. The numerical results discuss the effects of different systemic tradeoffs and performance optimizations of mmWave and sub-6 GHz 5G deployments.

Mobility is a key challenge for beam management in 5G cellular networks due to the overhead incurred at beam switching and base station (BS) handover events. This paper focuses on a network that has a multi-tier structure with two types of BSs operating in the same frequency bands, namely macro BSs that are sparser but with higher transmit power, and micro BSs that are denser and with lower transmit power. We propose a downlink user association policy which is a function of the user mobility. Typically, high mobility users should associate with macro BSs so as to incur less beam switching overhead, whereas low mobility ones should be associated with micro BSs. The main contribution of the paper is a formalization of the optimal threshold association policy, when the optimality is understood with respect to the mean effective Shannon rate. The analysis is based on stochastic geometry and on an exact representation of the mean effective Shannon rate of the typical user in this beamforming multi-tier context. Two models are discussed. The simplest one focuses on a single-user optimization problem. We also discuss a more realistic model with bandwidth sharing between all users in the cell. Finally, we identify the mobility and user-density patterns where the velocity-based threshold association policy outperforms the classical best mean power association policy.

This paper concerns the performance of vehicle-to-everything (V2X) communications. More precisely, we analyze the broadcast of safety-related V2X communications in cellular networks where base stations and vehicles are assumed to share the same spectrum and vehicles broadcast their safety messages to neighboring users. We model the locations of vehicles as a Poisson line Cox point process and the locations of users as a planar Poisson point process. We assume that users are associated with their closest base stations when there is no vehicle within a certain distance ρ. On the other hand, users located within a distance ρ from vehicles are associated with the vehicles to receive their safety messages. We quantify the properties of this vehicle-prioritized association using the stochastic geometry framework. We derive the fractions of users that receive safety messages from vehicles. Then, we obtain the expression for the signal-to-interference ratio of the typical user evaluated on each association type. To address the impact of vehicular broadcast on the cellular network, the paper also derives the effective rate offered to the typical user in this setting.

This paper considers the time evolution of a queue that is embedded in a Poisson point process of moving wireless interferers. The queue is driven by an external arrival process and is subject to a time-varying service process that is a function of the SINR that it sees. Static configurations of interferers result in an infinite queue workload with positive probability. In contrast, a generic stability condition is established for the queue in the case where interferers possess any non-zero mobility that results in displacements that are both independent across interferers and oblivious to interferer positions. The proof leverages the mixing property of the Poisson point process. The effect of an increase in mobility on queuing metrics is also studied. Convex ordering tools are used to establish that faster moving interferers result in a queue workload that is smaller for the increasing-convex stochastic order. As a corollary, mean workload and mean delay decrease as network mobility increases. This stochastic ordering as a function of mobility is explained by establishing positive correlations between SINR level-crossing events at different time points, and by determining the autocorrelation function for interference and observing that it decreases with increasing mobility. System behaviour is empirically analyzed using discrete-event simulation and the performance of various mobility models is evaluated using heavy-traffic approximations.

Beam management is central in the operation of dense 5G cellular networks. Focusing the energy radiated to mobile terminals (MTs) by increasing the number of beams per cell increases signal power and decreases interference, and has hence the potential to bring major improvements on area spectral efficiency (ASE). This benefit, however, comes with unavoidable overheads that increase with the number of beams and the MT speed. This paper proposes a first system-level stochastic geometry model encompassing major aspects of the beam management problem: frequencies, antennas, and propagation; physical layer, wireless links, and coding; network geometry, interference, and resource sharing; sensing, signaling, and mobility management. This model leads to a simple analytical expression for the effective ASE that the typical user gets in this context. This in turn allows one to find, for a wide variety of 5G network scenarios including millimeter wave (mmWave) and sub-6 GHz, the number of beams per cell that offers the best global trade-off between these benefits and costs. We finally provide numerical results that discuss the effects of different systemic trade-offs and performances of mmWave and sub-6 GHz 5G deployments.

Inspired by a new feature in 5G NR called bandwidth part (BWP), this paper presents a bandwidth allocation (BA) model that allows one to adapt the bandwidth allocated to users depending on their data rate needs. Specifically, in adaptive BA, a wide bandwidth is divided into chunks of smaller bandwidths and the number of bandwidth chunks allocated to a user depends on its needs or type. Although BWP in 5G NR mandates allocation of a set of contiguous bandwidth chunks, our BA model also allows other assumptions on chunk allocation such as the allocation of any set of bandwidth chunks, as in, e.g., LTE resource allocation, where chunks are selected uniformly at random. The BA model studied here is probabilistic in that the user locations are assumed to form a realization of a Poisson point process and each user decides independently to be of a certain type with some probability. This model allows one to quantify spectrum sharing and service differentiation in this context, namely to predict what performance a user gets depending on its type as well as the overall performance. This is based on exact representations of key performance metrics for each user type, namely its success probability, the meta distribution of its signal-to-interference ratio, and its Shannon throughput. We show that, surprisingly, the higher traffic variability stemming from adaptive BA is beneficial: when comparing two networks using adaptive BA and having the same mean signal and the same mean interference powers, the network with higher traffic variability performs better for all these performance metrics. With respect to Shannon throughput, we observe that our BA model is roughly egalitarian per Hertz and leads to a linear service differentiation in aggregated throughput value.

This paper explores an emerging wireless Internet-of-things (IoT) architecture based on unmanned aerial vehicles (UAVs). We consider a network where a fleet of UAVs at a fixed altitude flies on planned trajectories and IoT devices on the ground are scheduled to transmit their data to the UAVs when the latter are nearby. In such a system, the UAVs' motion triggers the uplink transmissions of the IoT devices. As a result, network performance is determined by the geometric and dynamic characteristics of the system. We propose a joint stationary model for UAVs and IoT devices and then evaluate the interference, the coverage probability, and the data rate of the typical UAV. To assess the harvesting capability of the proposed architecture, we derive a formula for the amount of data uploaded from each IoT device to a UAV. We also establish a linear relationship between the UAV coverage and the harvesting capability of the network, which provides insights into the design of the proposed harvesting scheme. In addition, we use our analytical results to numerically show that there exists a trade-off between the uploaded data and the size of the IoT scheduling window. Specifically, for a given UAV and IoT geometry, there exists an optimal scheduling window that maximizes the harvesting capability of the proposed network.

This paper uses Poisson geometry to define a power control mechanism for large mobile ad hoc networks. It considers a homogeneous Poisson bipole network on the Euclidean plane. Bipoles, which represent transmitter-receiver pairs, adapt their power in a local and distributed way. The common aim of the bipoles is to maximize network wide proportional fairness. The mechanism is based on a heuristic which consists of simplifying the interference field seen by each receiver to its dominant term. This max-interference approximation decomposes the global (and infinite dimensional) optimization problem into a collection of finite problems involving the so called descending chains of the Poisson nearest neighbor graph. In addition, in the Rayleigh fading and interference limited case, the problem further factorizes into a collection of pairwise optimization problems with closed form solution. The properties of this power control mechanism are studied both analytically and by discrete event simulation. Significant performance gains are seen despite these approximations, for the motivating metric and other metrics.

This paper explores an emerging wireless architecture based on unmanned aerial vehicles (UAVs), i.e., drones. We consider a network where UAVs at fixed altitude harvest data from Internet-of-Things (IoT) devices on the ground. Each UAV serves IoT devices within its coverage area. In such a system, the UAVs' motion activates IoT uplink transmissions and so the motion triggers the interference field and determines the network performance. To analyze the performance, we propose a stochastic geometry model. The coverage area of each UAV, referred to as the activation window, is modeled for simplicity as a rectangle where at most one IoT device is scheduled to transmit at any given time. In this setting, we analyze the signal-to-interference and data rate from two typical perspectives, namely from a typical UAV's and from a typical IoT device's points of view.

This paper studies the asymptotic properties of average area spectral efficiency (ASE) of a downlink cellular network in the limit of very dense base station (BS) and user densities. This asymptotic analysis relies on three assumptions: 1) interference is treated as noise; 2) the BS locations are drawn from a Poisson point process; and 3) the path loss function is bounded above satisfying mild regularity conditions. We consider three possible definitions of the average ASE, all of which give units of bits per second per unit bandwidth per unit area. When there is no constraint on the minimum operational signal-to-interference-plus-noise ratio (SINR) and instantaneous full channel state information (CSI) is available at the transmitter, the average ASE is proven to saturate to a constant, which we derive in a closed form. For the other two ASE definitions, wherein either a minimum SINR is enforced or CSI is not available, the average ASE is instead shown to collapse to zero at high BS density. We provide several familiar case studies for the class of considered path loss models, and demonstrate that our results cover most previous models and results on ultradense networks as special cases.

We analyze a class of vector multiple access channels with additive noise, where the sum of the dimensions of the transmitted signals matches that of the received signal. We first focus on the case without power constraints, in the Poltyrev sense, using point process techniques. We find the Poltyrev capacity region for noise processes that are independent and identically distributed over channel uses. For each rate vector strictly in the Poltyrev capacity region, we study, for each subset of the transmitters, the exponent of the decay in block length of the smallest possible probability that decoding results in error for each transmitter in that subset. In the case of independent and identically distributed Gaussian noise, with arbitrary positive definite covariance matrix, we derive random coding exponents for each type of error event-these are lower bounds to the true error exponents. This also leads to random coding error exponents in the traditional power-constrained case, where the power constraint at each transmitter is defined by an arbitrary positive definite matrix.

This paper proposes a framework to analyze a wireless architecture where vehicles collect data from devices. Roads and vehicles are modeled by a Poisson line process and a Cox point process, respectively. At any given time, each vehicle is assumed to communicate with a roadside device in a disk of radius ν centered at the vehicle, which is referred to as the coverage disk. We study these direct communications from roadside devices to vehicles by investigating the network performance in both space and time domains. For the space domain analysis, we explicitly derive the signal-to-interference ratio distribution of the typical vehicle and the area spectral efficiency of the proposed network. For the time domain analysis, we characterize the evolution of the area fraction of the coverage disks over time and then evaluate the minimum association delay of the proposed network by deriving the distribution of the minimum time required for an arbitrarily located roadside device to be covered by a disk. Leveraging the derived network performance, we investigate the optimization of network utility functions given by linear combinations of the performance metrics.

We propose and analyze a new shadowing field model meant to capture spatial correlations. The interference field associated with this new model is compared to that of the widely used independent shadowing model. Independent shadowing over links is adopted because of the resulting closed forms for performance metrics, and in spite of the well-known fact that the shadowing fields of networks are spatially correlated. The main purpose of this paper is to challenge this independent shadowing approximation. For this, we analyze the interference measured at the origin in networks where 1) nodes which are in the same cell of some random shadowing tessellation share the same shadow, or 2) nodes which share a common mother point in some cluster process share the same shadow. By leveraging stochastic comparison techniques, we give the order relation of the three main user performance metrics, namely coverage probability, Shannon throughput and local delay, under both the correlated and the independent shadowing assumptions. We show that the evaluation of the considered metrics under the independent approximation is systematically pessimistic compared to the correlated shadowing model. The improvement in each metric when adopting the correlated shadow model is quantified and shown to be quite significant.

Ultra densification along with the use of wider bands at higher frequencies are likely to be key elements towards meeting the throughput/coverage objectives of 5G wireless networks. In addition to increased parallelism, densification leads to improved, but eventually bounded, benefits from proximity of users to base stations, while resulting in increased aggregate interference. Such networks are expected to be interference limited, and in higher frequency regimes, the interference is expected to become spatially variable due to the increased sensitivity of propagation to obstructions and the proximity of active interferers. This paper studies the characteristics of the spatial random fields associated with interference and Shannon capacity in ultra-dense limiting regimes. They rely on the theory of Gaussian random fields which arise as natural limits under densification. Our models show how densification and operation at higher frequencies, could lead to increasingly rough temporal variations in the interference process. This is characterized by the Holder exponent of the interference field. We show that these fluctuations make it more difficult for mobile users to adapt modulation and coding. We further study how the spatial correlations in users' rates impact backhaul dimensioning. Therefore, this paper identifies and quantifies challenges associated with densification in terms of the resulting unpredictability and the correlation of interference on the achievable rates.

This paper studies the asymptotic properties of area spectral efficiency (ASE) of a downlink cellular network in the limit of very dense base station (BS) and user densities. This asymptotic analysis relies on three assumptions: (1) interference is treated as noise; (2) the BS locations are drawn from a Poisson point process; (3) the path loss function is bounded above satisfying mild regularity conditions. When there is no constraint on the minimum operational SINR and instantaneous full channel state information is available at the transmitter, the ASE is proven to saturate to a constant, which we derive in closed form. We provide several familiar case studies for the class of considered path loss models, and demonstrate that our results cover most previous models and results on ultradense networks as special cases.

This paper analyzes the coverage probability of a typical user in a cellular network comprised of planar base stations, planar users, vehicular base stations, and vehicular users. Planar base stations and users in the Euclidean space are modeled by independent stationary planar Poisson point processes. Then, conditionally on a stationary Poisson line process, vehicular base stations and vehicular users are modeled by linear Poisson point processes on the lines. Utilizing the Palm distribution of each user point process, we derive the association probability and the coverage probability of each typical user, namely that of the typical planar user and the typical vehicular user. Using the association and coverage expressions, we fully characterize the Shannon rate distributions of all downlink combinations present in the proposed network based on their association types: vehicle-to-vehicle, vehicle-to-planar, planar-to-vehicle, and planar-to-planar.

Disruptive changes are underway in the automotive industry as large-scale platforms based on vehicular fleets are deployed to deliver ride sharing and delivery services. Such platforms can also be leveraged to deliver wireless connectivity services, e.g., large-scale connectivity for the Internet of Things (IoT). This paper examines a network architecture based on a mesh of IoT devices, roadside repositories and vehicular mobile gateways -- referred to as mesh+vehicular. We propose a system-level model to study its relative merits versus conventional infrastructure-based IoT architectures-- referred to as mesh+cellular. The model reflects the salient properties of the architectures including the key interplay among the variability in the network geometries, routing trees, wireless capacity and eventually IoT queue stability.

This paper analyzes an emerging architecture of cellular network utilizing both planar base stations uniformly distributed in the Euclidean plane and base stations located on roads. An example of this architecture is that where, in addition to conventional planar cellular base stations and users, vehicles also play the role of both base stations and users. A Poisson line process is used to model the road network and, conditionally on the lines, linear Poisson point processes are used to model the vehicles on the roads. The conventional planar base stations and users are modeled by the independent planar Poisson point processes. We use Palm calculus to investigate the statistical properties of a typical user in such a network. Specifically, this paper discusses two different Palm distributions, with respect to the user point processes depending on its type: planar or vehicular. We derive the distance to the nearest base station, the association of the typical users, and the coverage probability of the typical user. Furthermore, we provide a comprehensive characterization of coverage of all possible cellular transmissions in this setting, namely, vehicle-to-vehicle, vehicle-to-infrastructure, infrastructure-to-vehicle, and infrastructure-to-infrastructure.

Cell search is the process for a user to detect its neighboring base stations (BSs) and make a cell selection decision. Due to the importance of beamforming in 5G cellular networks including both the millimeter wave and sub-6 GHz networks, there is a need for a better understanding of the directional cell search delay performance. A cellular network with fixed BS and user locations is considered, so as to take into account the strong temporal correlations that exist for the SINR experienced by each BS and user in this context. For Poisson cellular networks with Rayleigh fading channels, a closed-form expression for the spatially averaged mean cell search delay of all users is derived. This mean cell search delay for a noise-limited network is proved to be infinite whenever the non-line-of-sight path loss exponent is larger than two. For interference-limited networks, a phase transition for the mean cell search delay is shown to exist in terms of the number of BS beams M: the mean cell search delay is infinite when M is smaller than a threshold and finite otherwise. Beam-sweeping is also demonstrated to be effective in decreasing the cell search delay, especially for cell edge users.

In this paper, we examine the benefits of multiple antenna communication in random wireless networks, the topology of which is modeled by stochastic geometry. The setting is the Poisson bipolar model introduced in [1], which is a natural model for ad-hoc and device-to-device networks. The primary finding is that, with the knowledge of channel state information between a receiver and its associated transmitter, by zero-forcing successive interference cancellation, and for appropriate antenna configurations, the ergodic spectral efficiency can be made to scale linearly with both: 1) the minimum of the number of transmit and receive antennas and 2) the density of nodes. This scaling law is achieved by using the multiple transmit antennas to send multiple data streams (e.g., through an openloop transmission method) and by exploiting the receive antennas to cancel interference. Furthermore, when a receiver is able to learn channel state information from a certain number of near interferers, higher scaling gains can be achieved when a successive interference cancellation method is used. Both results require rich scattering environments. A major implication of the derived scaling laws is that, under this scattering assumption, spatial multiplexing transmission methods are essential for obtaining better and eventually optimal scaling laws in random wireless networks with multiple antennas.

This paper focuses on the modeling and analysis of the temporal performance variation experienced by a mobile user in a wireless network and its impact on system-level design. We consider a simple stochastic geometry model: the infrastructure nodes are Poisson distributed while the user’s motion is the simplest possible, i.e., constant velocity on a straight line. We first characterize variations in the signal-to-noise ratio (SNR) process and associated downlink Shannon rate, resulting from variations in the infrastructure geometry seen by the mobile. Specifically, by making a connection between stochastic geometry and queuing theory, the level crossings of the SNR process are shown to form an alternating renewal process whose distribution is completely characterized. For large/small SNR levels, and associated rare events, we further derive simple distributional (exponential) models. We then characterize the second major contributor to such variations, namely, changes in the number of other users sharing the infrastructure. Combining these two phenomena, we study what are the dominant factors (infrastructure geometry or sharing number) when a mobile experiences a very high/low shared rate. These results are then used to evaluate and optimize the system-level quality of experience of the mobile users sharing such a wireless infrastructure, including mobile devices streaming video which proactively buffer content to prevent rebuffering and mobiles which are downloading large files. Finally, we use simulation to assess the fidelity of this model and its robustness to factors which are presently not taken into account.

Distance-based attenuation is a critical aspect of wireless communications. As opposed to the ubiquitous powerlaw path loss model, this paper proposes a stretched exponential path loss model that is suitable for short-range communication. In this model, the signal power attenuates over a distance r as e-αrβ, where α and β are tunable parameters. Using experimental propagation measurements, we show that the proposed model is accurate for short to moderate distances in the range r ∈ (5, 300) meters and so is a suitable model for dense and ultradense networks. We integrate this path loss model into a downlink cellular network with base stations modeled by a Poisson point process, and derive expressions for the coverage probability, potential throughput, and area spectral efficiency. Although the most general result for coverage probability has a double integral, several special cases are given, where the coverage probability has a compact or even closed form. We then show that the potential throughput is maximized for a particular BS density and then collapses to zero for high densities, assuming a fixed signal-to-interference-plus-noise ratio (SINR) threshold. We next prove that the area spectral efficiency, which assumes an adaptive SINR threshold, is nondecreasing with the BS density and converges to a constant for high densities.

This paper considers a multi-antenna wireless net- work where the locations of transmitters are distributed as a Poisson point process, which is a natural model for ad-hoc and device-to-device networks. We show that, in such a network, the ergodic spectral efficiency scales linearly with respect to the network density under appropriate multiple antenna configurations and diversity assumptions. This scaling law is achieved by a simple zero-forcing decoder, which eliminates inter-stream interference using spatial multiplexing transmissions. We also show that when each receiver knows channel state information from some interferers, a higher scaling law holds for ergodic spectral efficiency than that without channel state information when using a partial zero-forcing method which eliminates dominant interference signals while boosting the desired signal power. Further, we show that spatial multiplexing transmission methods are essential for obtaining better scaling laws in certain regions of network parameters.

We propose and study a novel continuous space-time model for wireless networks, which considers the stochastic interactions in both space through interference and in time due to randomness in traffic. Our model consists of an interacting particle birth-death dynamics incorporating information-theoretic spectrum-sharing. Roughly speaking, particles (or more generally wireless links) arrive according to a Poisson point process on space-time, and stay for a duration governed by the local configuration of points present, and then exit the network after completion of a file transfer. We analyze this particle dynamics to derive an explicit condition for time ergodicity (i.e., stability), which is tight. We also prove that when the dynamics is ergodic, the steady-state point process of links (or particles) exhibits a form statistical clustering. Based on the clustering, we propose a conjecture, which we leverage to derive approximations, bounds, and asymptotics on performance characteristics, such as delay and mean number of links per unit-space in the stationary regime. The mathematical analysis is combined with discrete event simulation to study the performance of this type of networks.

Initial access is the process which allows a mobile user to first connect to a cellular network. It consists of two main steps: cell search (CS) on the downlink and random access (RA) on the uplink. Millimeter wave (mm-wave) cellular systems typically must rely on directional beamforming (BF) in order to create a viable connection. The BF direction must, therefore, be learned-as well as used-in the initial access process for mm-wave cellular networks. This paper considers four simple but representative initial access protocols that use various combinations of directional BF and omnidirectional transmission and reception at the mobile and the BS, during the CS and RA phases. We provide a system-level analysis of the success probability for CS and RA for each one, as well as of the initial access delay and user-perceived downlink throughput (UPT). For a baseline exhaustive search protocol, we find the optimal BS beamwidth and observe that in terms of initial access delay it is decreasing as blockage becomes more severe, but is relatively constant (about π/12) for UPT. Of the considered protocols, the best tradeoff between initial access delay and UPT is achieved under a fast CS protocol.

The Poisson building features a Poisson collection of random planes orthogonal to the axes of the 3-D Euclidean space. It divides the space into rectangular rooms of random sizes. The addition of wireless small cell base stations, deployed as Poisson point processes along the ceiling and corner lines of these rooms, provides a first stochastic geometric model representing in-building 3-D wireless networks. The main challenge for analyzing interference in such an environment is the fact that electromagnetic signals originating from different locations are blocked by common obstacles like walls and floors, which makes the path loss highly correlated in space. We propose a natural propagation model taking this phenomenon into account. We give analytical expressions for the interference field and its correlation within this framework. We illustrate the tractability of this model by providing analytical expressions for the spectral efficiency of the downlink in such indoor cellular networks and for D2D communications in such an environment. We combine the model and spatial simulations to show that classical 2-D and distance based attenuation models cannot be used in this context and to argue for the need of such 3-D models to assess the performance of this type of indoor communications.

This paper is focused on information theoretic properties of point processes. Firstly, we discuss the entropy of a point process and the entropy rate of a stationary point process. Then we give explicit formulas for these quantities in the Poisson case, as well as maximal entropy properties for homogeneous Poisson point processes. Secondly, we define the mutual information rate of two stationary point processes. We then give explicit formulas for the mutual information rate between a homogeneous Poisson point process and its displacement.

This paper proposes a stochastic geometry framework to study the temporal performance variations experienced by a mobile user in a cellular network. The focus is on the variations of the Signal to Noise Ratio (SNR) and the downlink Shannon rate experienced when the user moves across Poisson cellular network of the Euclidean plane. The motion is the simplest possible i.e., a user moving at a constant velocity on a straight line. The level crossings of the associated SNR process are shown to form an alternating-renewal process. The two distributions characterizing this process are derived in closed form. The theory of rare events provides simplified expressions for the law of this process for extremes (very large and very small) SNR thresholds. The framework is then leveraged to predict the quality of service experienced by mobile users in two concrete scenarios: that of streaming a video on the downlink and partially buffered on the hand-set to prevent video freezing; and that of downloading a large file, where the main question is the download delay. Finally, discrete event simulation is used to assess practical use of this model on its robustness to perturbations that cannot presently be taken into account in the analysis.

This paper considers large random wireless networks, where transmit-and-receive node pairs communicate within a certain range while sharing a common spectrum. By modeling the spatial locations of nodes as a Poisson point process, analytical expressions for the ergodic spectral efficiency of a typical node pair are derived as a function of the channel state information available at a receiver (CSIR) in terms of relevant system parameters: the density of communication links, the number of receive antennas, the path loss exponent, and the operating signal-to-noise ratio. One key finding is that when the receiver only exploits CSIR for the direct link, the sum spectral efficiency increases linearly with the density, provided the number of receive antennas increases as a certain superlinear function of the density. When each receiver exploits CSIR for a set of dominant interfering links in addition to that of the direct link, the sum spectral efficiency increases linearly with both the density and the path loss exponent if the number of antennas is a linear function of the density. This observation demonstrates that having CSIR for dominant interfering links provides an order gain in the scaling law. It is also shown that this linear scaling holds for direct CSIR when incorporating the effect of the receive antenna correlation, provided that the rank of the spatial correlation matrix scales superlinearly with the density. These scaling laws are derived from integral representations of the distribution of the signal to interference and noise ratio, which are of independent interest and which in turn derived from stochastic geometry and more precisely from the theory of shot noise fields. Simulation results back the scaling laws and the integral representations.

Consider orthogonal planes in the 3-D space representing floors and walls in a large building. These planes divide the space into rooms, where a wireless infrastructure is deployed. This paper is focused on the analysis of the correlated shadowing field created by this wireless infrastructure through the set of walls and floors. When the locations of the planes and wireless nodes are governed by Poisson processes, we obtain a simple stochastic model which captures the non-uniform nature of node deployment and room sizes. This model, which we propose to call the Poisson building, captures the complex in-building shadowing correlations, is scalable in the number of dimensions, and can be used for network performance analysis. It allows an exact mathematical characterization of the interference distribution in both infinite and finite buildings, which further leads to closed-form expressions for the coverage probabilities in in-building cellular networks and the success probability of in-building underlay D2D transmissions.

We leverage stochastic geometry to characterize key performance metrics for neighboring Wi-Fi and LTE networks in unlicensed spectrum. Our analysis focuses on a single unlicensed frequency band, where the locations for the Wi-Fi access points and LTE eNodeBs are modeled as two independent homogeneous Poisson point processes. Three LTE coexistence mechanisms are investigated: 1) LTE with continuous transmission and no protocol modifications; 2) LTE with discontinuous transmission; and 3) LTE with listen-before-talk and random back-off. For each scenario, we derive the medium access probability, the signal-to-interference-plus-noise ratio coverage probability, the density of successful transmissions (DST), and the rate coverage probability for both Wi-Fi and LTE. Compared with the baseline scenario where one Wi-Fi network coexists with an additional Wi-Fi network, our results show that Wi-Fi performance is severely degraded when LTE transmits continuously. However, LTE is able to improve the DST and rate coverage probability of Wi-Fi while maintaining acceptable data rate performance when it adopts one or more of the following coexistence features: a shorter transmission duty cycle, lower channel access priority, or more sensitive clear channel assessment thresholds.

A Manhattan Poisson line process divides the plane into an infinite number of rectangular rooms with walls extending infinitely along the axes. When the path loss is dominated by the penetration through each of the walls, a Poisson field of transmitters creates a heavy tailed interference at a randomly picked room, whose distribution is tractable in the Laplace domain. Interference correlation at different rooms is explicitly available. This model gives the first tractable mathematical abstraction to indoor physical environments where wireless signals are shadowed by (common) walls. Applying the analytical results leads to a formula for success probabilities of a transmission attempt between two given rooms.

Traditional models in opinion dynamics involve agents updating their opinions based on the opinions of their neighbors in a static social-graph, regardless of their differences in opinions. In contrast, the bounded confidence opinion dynamics does not presume a static interaction graph, and instead models interactions between those agents that share similar opinions (i.e., are close to one another, capturing online discussion groups and conventional meetings). We generalize the bounded confidence opinion dynamics model by incorporating pairwise stochastic interactions based on opinion differences as well as the self or endogenous evolution of the agent opinions, which is represented by a random process. We analytically characterize the conditions under which this stochastic dynamics is stable in an appropriate sense. This characterization relates well to what is observed in social systems. Moreover, this generalization sheds light on dynamics that combine aspects of graph-based updates and bounded confidence models.

This paper presents an analytically tractable stochastic geometry model for urban wireless networks, where the locations of the nodes and the shadowing are highly correlated and different path loss functions can be applied to line-of-sight (LOS) and non-line-of-sight (NLOS) links. Using a distance-based LOS path loss model and a blockage (shadowing)-based NLOS path loss model, we are able to derive the distribution of the interference observed at a typical location and the joint distribution at different locations. When applied to cellular networks, this model leads to tractable expressions for the coverage probability (SINR distribution). We show that this model captures important features of urban wireless networks, which cannot be analyzed using existing models. The numerical results also suggest that even in the presence of significant penetration loss, ignoring the NLOS interference can lead to erroneous estimations on coverage. They also suggest that allowing users to be associated with NLOS BSs may provide a non-trivial gain on coverage.

The CSMA/CA protocol is based on the “Interference as Noise” (IAN) paradigm i.e. it always gets rid of strong interference near a receiver to ensure quality of reception. However, it is well known from Multi-user Information Theory that treating Interference as Noise is not optimal. This paper proposes a class of protocols that employ the Successive Interference Cancellation (SIC) technique in a systematic fashion to move beyond always treating interference as noise. Such protocols allow one to pack more links than the classical CSMA. We describe the protocols along with their signaling mechanism to implement them in a distributed fashion. We then perform Monte Carlo simulations to evaluate the performance and show significant gains over the IAN based CSMA/CA protocol in large random networks.

We analyze a class of vector multiple access channels with additive noise, where the sum of the dimensions of the transmitted signals matches that of the received signal. We first focus on the case without power constraints, using point process techniques. We derive the capacity region in the Poltyrev sense, a representation of the error probabilities for each subset of transmitters based on Palm theory, and random coding exponents for each type of error event in the case without power constraints, focusing on the case of independent and identically distributed Gaussian noise, with arbitrary positive definite covariance matrix at each time. This also leads to random coding error exponents in the traditional power-constrained case, where the power constraint at each transmitter is defined by an arbitrary positive definite matrix at each time.

Although the Poisson point process (PPP) has been widely used to model base station (BS) locations in cellular networks, it is an idealized model that neglects the spatial correlation among BSs. This paper proposes the use of the determinantal point process (DPP) to take into account these correlations, in particular the repulsiveness among macro BS locations. DPPs are demonstrated to be analytically tractable by leveraging several unique computational properties. Specifically, we show that the empty space function, the nearest neighbor function, the mean interference, and the signal-to-interference ratio (SIR) distribution have explicit analytical representations and can be numerically evaluated for cellular networks with DPP-configured BSs. In addition, the modeling accuracy of DPPs is investigated by fitting three DPP models to real BS location data sets from two major U.S. cities. Using hypothesis testing for various performance metrics of interest, we show that these fitted DPPs are significantly more accurate than popular choices such as the PPP and the perturbed hexagonal grid model.

Cooperation in cellular networks is a promising scheme to improve system performance, especially for cell-edge users. In this work, stochastic geometry is used to analyze cooperation models where the positions of base stations follow a Poisson point process distribution and where Voronoi cells define the planar areas associated with them. For the service of each user, either one or two base stations are involved. If two, these cooperate by exchange of user data and channel related information with conferencing over some backhaul link. Our framework generally allows for variable levels of channel information at the transmitters. This paper is focused on a case of limited information based on Willems' encoding. The total per-user transmission power is split between the two transmitters and a common message is encoded. The decision for a user to choose service with or without cooperation is directed by a family of geometric policies, depending on its relative position to its two closest base stations. An exact expression of the network coverage probability is derived. Numerical evaluation shows average coverage benefits of up to 17% compared to the non-cooperative case. Various other network problems of cellular cooperation, like the fully adaptive case, can be analyzed within our framework.

The macro base station (BS) deployments in modern cellular networks are neither regular nor completely random. We use determinantal point process (DPP) models to study the repulsiveness among macro base stations observed in cellular networks. Three DPP models are fitted to base station location data sets from two major US cities. Hypothesis testing is used to validate the goodness-of-fit for these DPP models. Based on performance metrics including the K-function, the L-function and coverage probability, DPP models are shown to be accurate in modeling real BS deployments. On the contrary, the Poisson point process and perturbed hexagonal grid model are shown to be less realistic. Different DPP models are compared, and several computational properties of these models are also discussed.

The proportionally fair sharing of the capacity of a Poisson network using Spatial-Aloha leads to closed-form performance expressions in two extreme cases: (1) the case without topology information, where the analysis boils down to a parametric optimization problem leveraging stochastic geometry; (2) the case with full network topology information, which was recently solved using shot-noise techniques. We show that there exists a continuum of adaptive controls between these two extremes, based on local stopping sets, which can also be analyzed in closed form. We also show that these control schemes are implementable, in contrast to the full information case which is not. As local information increases, the performance levels of these schemes are shown to get arbitrarily close to those of the full information scheme. The analytical results are combined with discrete event simulation to provide a detailed evaluation of the performance of this class of medium access controls.

We investigate the benefits of channel-aware (opportunistic) scheduling of transmissions in ad hoc networks. The key challenge in optimizing the performance of such systems is finding a good compromise among three interdependent quantities: 1) the density of scheduled transmitters; 2) the quality of transmissions; and 3) the long term fairness among nodes. We propose two new channel-aware slotted CSMA protocols opportunistic CSMA and quantile-based CSMA (QT-CSMA) and develop new stochastic geometric models to quantify their performance in terms of spatial reuse and spatial fairness. When properly optimized, these protocols offer substantial improvements in performance relative to CSMA—particularly, when the density of nodes is moderate to high. In addition, we show that a simple version of QT-CSMA can achieve robust performance gains without requiring careful parameter optimization. The quantitative results in this paper suggest that channel-aware scheduling in ad hoc networks can provide substantial benefits which might far outweigh the associated implementation overheads.

We propose a new model for peer-to-peer networking which takes the network bottlenecks into account beyond the access. This model can cope with key features of P2P networking like degree or locality constraints together with the fact that distant peers often have a smaller rate than nearby peers. Using a network model based on rate functions, we give a closed form expression of peers download performance in the system's fluid limit, as well as approximations for the other cases. Our results show the existence of realistic settings for which the average download time is a decreasing function of the load, a phenomenon that we call super-scalability.

The broadcast of periodic messages is a key functionality in vehicular ad hoc networks. In the emerging vehicular networks, IEEE 802.11p is the standard of choice to support the PHY and MAC layer functionalities. The broadcast process in IEEE 802.11p is based on the CSMA mechanism where a device transmitting a packet senses the channel for ongoing transmissions and performs a random back-off before accessing the channel. Without RTS/CTS mechanisms, carrier sensing is expected to provide a protection region around the transmitter where no other transmitters are allowed. The point process characterizing the concurrent transmitters is expected to enforce a minimum separation between concurrent transmitters. However, at increasing densities, the CSMA behavior breaks down to an ALOHA-like transmission pattern where concurrent transmitters are distributed as a Poisson point process, indicating the lack of protection around transmitters. In this paper, we model the CSMA mechanism as a slotted system and analytically characterize the critical node/packet arrival density where the CSMA mechanism approaches an ALOHAlike behavior. Further, we use tools from stochastic geometry to establish closed-form expressions for the performance metrics of the broadcast mechanism in the ALOHA regime. Finally, using ns2 (an unslotted asynchronous simulator), we compare the theoretical results with simulations.

This paper studies the performance of Mobile Ad hoc Networks (MANETs) when the nodes, that form a Poisson point process, selfishly choose their Medium Access Probability (MAP). We consider goodput and delay as the performance metric that each node is interested in optimizing taking into account the transmission energy costs. We introduce a pricing scheme based on the transmission energy requirements and compute the symmetric Nash equilibria of the game in closed form. It is shown that by appropriately pricing the nodes, the selfish behavior of the nodes can be used to achieve the social optimum at equilibrium. The price of anarchy is then analyzed for these games. For the game with delay based utility, we bound the price of anarchy and study the effect of the price factor. For the game with goodput based utility, it is shown that price of anarchy is infinite at the price factor that achieves the global optima.

In order to represent the set of transmitters simultaneously accessing a wireless network using carrier sensing based medium access protocols, one needs tractable point processes satisfying certain exclusion rules. Such exclusion rules forbid the use of Poisson point processes within this context. It has been observed that Matérn point processes, which have been advocated in the past because of their exclusion based definition, are rather conservative within this context. The present paper confirms that it would be more appropriate to use the point processes induced by the Random Sequential Algorithm in order to describe such point patterns. It also shows that this point process is in fact as tractable as the Matérn model. The generating functional of this point process is shown to be the solution of a differential equation, which is the main new mathematical result of the paper. In comparison, no equivalent result is known for the Matérn hard-core model. Using this differential equation, a new heuristic method is proposed, which leads to simple bounds and estimates for several important network performance metrics. These bounds and estimates are evaluated by Monte Carlo simulation.

We propose a probabilistic model based on stochastic geometry to analyze cognitive radio in a large wireless network with randomly located users sharing the medium with carrier sensing multiple access. Analytical results are derived on the impact of the interaction between primary and secondary users, on their medium access probability, coverage probability and throughput. These results can be seen as the continuation of the theory of priorities in queueing theory to spatial processes. They give insight into the guarantees that can be offered to primary users and more generally on the possibilities offered by cognitive radio to improve the effectiveness of spectrum utilization in such networks.

Cellular networks are in a major transition from a carefully planned set of large tower-mounted base-stations (BSs) to an irregular deployment of heterogeneous infrastructure elements that often additionally includes micro, pico, and femtocells, as well as distributed antennas. In this paper, we develop a tractable, flexible, and accurate model for a downlink heterogeneous cellular network (HCN) consisting of K tiers of randomly located BSs, where each tier may differ in terms of average transmit power, supported data rate and BS density. Assuming a mobile user connects to the strongest candidate BS, the resulting Signal-to-Interference-plus-Noise-Ratio (SINR) is greater than 1 when in coverage, Rayleigh fading, we derive an expression for the probability of coverage (equivalently outage) over the entire network under both open and closed access, which assumes a strikingly simple closed-form in the high SINR regime and is accurate down to -4 dB even under weaker assumptions. For external validation, we compare against an actual LTE network (for tier 1) with the other K-1 tiers being modeled as independent Poisson Point Processes. In this case as well, our model is accurate to within 1-2 dB. We also derive the average rate achieved by a randomly located mobile and the average load on each tier of BSs. One interesting observation for interference-limited open access networks is that at a given \sinr, adding more tiers and/or BSs neither increases nor decreases the probability of coverage or outage when all the tiers have the same target-SINR.

This paper studies the performance of a wireless network when the nodes, that form a Poisson point process, selfishly choose their Medium Access Probability (MAP). We define the utility of each node as a weighted difference between a performance metric and some transmission costs. We consider expected goodput and expected delay as the performance metrics. The relative preference of nodes for their performance metrics and the transmission costs is represented by a tradeoff factor. We first consider a scenario in which nodes can be priced for the channel access. We relate the tradeoff factor to some pricing mechanism and compute the symmetric Nash equilibria of the game in closed form as a function of the price factor. We show that simple pricing mechanisms can be used to maximize system efficiency. In particular, we show that for a specific value of price factor, the selfish behavior of the nodes can be used to achieve the same performance as social optima at equilibrium. In the case without pricing where the dis-utility coincides with the transmission energy costs, we analyze the Price of Anarchy for these games. For the game with goodput based utility, we show that the Price of Anarchy is infinite at the tradeoff factor that achieves the global optimal goodput. For the game with delay based utility, we bound the Price of Anarchy and study the effect of the tradeoff factor.

The spatial correlations in transmitter node locations introduced by common multiple access protocols make the analysis of interference, outage, and other related metrics in a wireless network extremely difficult. Most works therefore assume that nodes are distributed either as a Poisson point process (PPP) or a grid, and utilize the independence properties of the PPP (or the regular structure of the grid) to analyze interference, outage and other metrics. But, the independence of node locations makes the PPP a dubious model for nontrivial MACs which intentionally introduce correlations, e.g., spatial separation, while the grid is too idealized to model real networks. In this paper, we introduce a new technique based on the factorial moment expansion of functionals of point processes to analyze functions of interference, in particular outage probability. We provide a Taylor-series type expansion of functions of interference, wherein increasing the number of terms in the series provides a better approximation at the cost of increased complexity of computation. Various examples illustrate how this new approach can be used to find outage probability in both Poisson and non-Poisson wireless networks.

It is common practice to model the base station (BS) locations in a cellular system by a grid, such as a hexagonal or square lattice. This model is usually analytically intractable as well as quite idealized. Therefore, system designers resort to complex simulations to evaluate network performance. In this paper, we introduce a new model for the base station locations based on a homogeneous Poisson point process (PPP), whereby the mobiles communicate with their nearest base stations. We obtain the distribution of the signal-to-interference-noise ratio (SINR), compute the average ergodic rate, and analytically verify the trade-off between coverage and rate with frequency reuse. We compare our results with actual BS locations as well as the grid model. In addition to being tractable, we also observe that the performance predicted by the PPP model lower bounds the actual performance, and is about as predictive as the grid model which provides upper bounds.

The defining characteristic of wireless and mobile networking is user mobility, and related to it is the ability for the network to capture information on where users are located and how users change location over time. Information about location is becoming critical, and therefore valuable, for an increasingly larger number of location-based or location-aware services. One key open question, however, is how valuable exactly this information is. Our goal in this paper is to develop an analytic framework, namely models and the techniques to solve them, to help quantify the economics of location information. Our aim is to derive models which can be used as decision making tools for entities interested in or involved in the location data economics chain, such as mobile operators or providers of location aware services (mobile advertising, etc). We consider in particular the fundamental problem of quantifying the value of different granularities of location information, for example how much more valuable is it to know the GPS location of a mobile user compared to only knowing the access point, or the cell tower, that the user is associated with. We illustrate our approach by considering what is arguably the quintessential location-based service, namely proximity-based advertising. We make three main contributions. First, we develop several novel models, based on stochastic geometry, which capture the location-based economic activity of mobile users with diverse sets of preferences or interests. Second, we derive closed-form analytic solutions for the economic value generated by those users. Third, we augment the models to consider uncertainty about the users' location, and derive expressions for the economic value generated with different granularities of location information. To our knowledge, this paper is the first one to present and analyze this class of economic models.

The paper establishes the capacity region of the Gaussian interference channel with many transmitter-receiver pairs constrained to use point-to-point codes. The capacity region is shown to be strictly larger in general than the achievable rate regions when treating interference as noise, using successive interference cancellation decoding, and using joint decoding. In a spatial network where the nodes are distributed according to a Poisson point process and the channel path loss exponent is β >; 2, it is shown that the density of users that can be supported by treating interference as noise can scale no faster than B2/βas the bandwidth B grows, while the density of users can scale linearly with B under optimal decoding.

We propose and analyze a class of distributed algorithms performing the joint optimization of radio resources in heterogeneous cellular networks made of a juxtaposition of macro and small cells. We show that within this context, it is essential to use algorithms able to simultaneously solve the problems of channel selection, user association and power control. In such networks, the unpredictability of the cell and user patterns also requires self-optimized schemes. The proposed solution is inspired from statistical physics and is based on Gibbs sampler. It can be implemented in a fully distributed way and nevertheless achieves minimal system-wide potential delay. Simulation results show that it outperforms today's default operational methods in both throughput and energy efficiency.

Recent deployments of data-rich smart phones has provided a fresh impetus for designing, deploying and understanding the performance of wide area ad-hoc networks. The most popular medium access mechanism for such ad hoc networks is CSMA/CA with RTS/CTS. In this paper, using tools from stochastic geometry, we study and optimize the throughput performance of such networks. We show that in ad-hoc networks enabled with SIR based scheduling, a simple modification to the transmit power level - setting it to be inversely proportional to the square root of the link gain - leads to large improvements in network throughput. This simple power-level selection is optimal over the class of all ”local” transmit power selection strategies when channels are stationary, and further is at most a factor of two away from optimality in the fading case. Using stochastic geometric techniques, we also provide analytical expressions for the medium access probability in different scenarios.

We investigate the benefits of channel-aware (opportunistic) scheduling of transmissions in ad-hoc networks. The key challenge in optimizing the performance of such systems is finding a good compromise among three interdependent quantities, the density and channel quality of the scheduled transmitters, and the resulting interference at receivers. We propose two new channel-aware slotted CSMA protocols: opportunistic CSMA (O-CSMA) and quantile-based CSMA (QT-CSMA) and develop stochastic geometric models allowing us to quantify their performance in terms of spatial reuse and spatial fairness. When properly optimized these protocols offer substantial improvements in terms of both of these metrics relative to CSMA — particularly when the density of nodes is moderate to high. Moreover, we show that a simple version of QT-CSMA can achieve robust performance gains without requiring careful parameter optimization. The paper supports the case that the benefits associated with channel-aware scheduling in ad hoc networks, as in centralized base station scenarios, might far outweigh the associated overhead, and this can be done robustly using a QT-CSMA like protocol.

Cellular networks are usually modeled by placing the base stations on a grid, with mobile users either randomly scattered or placed deterministically. These models have been used extensively but suffer from being both highly idealized and not very tractable, so complex system-level simulations are used to evaluate coverage/outage probability and rate. More tractable models have long been desirable. We develop new general models for the multi-cell signal-to-interference-plus-noise ratio (SINR) using stochastic geometry. Under very general assumptions, the resulting expressions for the downlink SINR CCDF (equivalent to the coverage probability) involve quickly computable integrals, and in some practical special cases can be simplified to common integrals (e.g., the Q-function) or even to simple closed-form expressions. We also derive the mean rate, and then the coverage gain (and mean rate loss) from static frequency reuse. We compare our coverage predictions to the grid model and an actual base station deployment, and observe that the proposed model is pessimistic (a lower bound on coverage) whereas the grid model is optimistic, and that both are about equally accurate. In addition to being more tractable, the proposed model may better capture the increasingly opportunistic and dense placement of base stations in future networks.

The paper establishes the capacity region of the Gaussian interference channel with many transmitter-receiver pairs constrained to use point-to-point codes. The capacity region is shown to be strictly larger in general than the achievable rate regions when treating interference as noise, using successive interference cancellation decoding, and using joint decoding. The gains in coverage and achievable rate using the optimal decoder are analyzed in terms of ensemble averages using stochastic geometry. In a spatial network where the nodes are distributed according to a Poisson point process and the channel path loss exponent is β >; 2, it is shown that the density of users that can be supported by treating interference as noise can scale no faster than B2/βas the bandwidthBgrows, while the density of users can scale linearly with B under optimal decoding.

In this paper, we formulate a delay-power control (DPC) scheme for wireless networking, which efficiently balances delay against transmitter power on each wireless link. The DPC scheme is scalable, as each link autonomously updates its power based on the interference observed at its receiver; no cross-link communication is required. It is shown that DPC converges to a unique equilibrium power and several key properties are established, concerning the nature of channel bandwidth sharing achieved by the links. The DPC scheme is contrasted to the well-known Foschini-Miljanic (FM) formulation for transmitter power control in wireless networks, and some key advantages are established. Based on the DPC and FM schemes, two protocols are developed, which leverage adaptive tuning of DPC parameters. One of them is inspired by TCP and exhibits analogous behavior. This paper primarily focuses on the theoretical underpinnings of DPC and their practical implications for efficient protocol design. The DPC dynamics are also investigated numerically.

In this work, we develop mathematical and algorithmic tools for the self-optimization of mobile cellular networks. Scalable algorithms which are based on local measurements and do not require heavy coordination among the wireless devices are proposed. We focus on the optimization of transmit power and of user association. The method is applicable to both joint and separate optimizations. The global utility minimized is linked to potential delay fairness. The distributed algorithm adaptively updates the system parameters and achieves global optimality by measuring SINR and interference. It is built on Gibbs' sampler and offers a unified framework that can be easily reused for different purposes. Simulation results demonstrate the effectiveness of the algorithm.

We study a slotted version of the Aloha Medium Access (MAC) protocol in a Mobile Ad-hoc Network (MANET). Our model features transmitters randomly located in the Euclidean plane, according to a Poisson point process and a set of receivers representing the next-hop from every transmitter. We concentrate on the so-called outage scenario, where a successful transmission requires a Signal-to-Interference-and-Noise (SINR) larger than some threshold. We analyze the local delays in such a network, namely the number of times slots required for nodes to transmit a packet to their prescribed next-hop receivers. The analysis depends very much on the receiver scenario and on the variability of the fading. In most cases, each node has finite-mean geometric random delay and thus a positive next hop throughput. However, the spatial (or large population) averaging of these individual finite mean-delays leads to infinite values in several practical cases, including the Rayleigh fading and positive thermal noise case. In some cases it exhibits an interesting phase transition phenomenon where the spatial average is finite when certain model parameters (receiver distance, thermal noise, Aloha medium access probability) are below a threshold and infinite above. To the best of our knowledge, this phenomenon, which we propose to call the wireless contention phase transition, has not been discussed in the literature. We comment on the relationships between the above facts and the heavy tails found in the so-called "RESTART" algorithm. We argue that the spatial average of the mean local delays is infinite primarily because of the outage logic, where one transmits full packets at time slots when the receiver is covered at the required SINR and where one wastes all the other time slots. This results in the "RESTART" mechanism, which in turn explains why we have infinite spatial average. Adaptive coding offers another nice way of breaking the outage/RESTART logic. We show examples where the average delays are finite in the adaptive coding case, whereas they are infinite in the outage case.

In this paper, we analyze phenomena related to user clumps and hot spots occuring in mobile networks at the occasion of large urban mass gatherings. Our analysis is based on observations made on mobility traces of GSM users in several large cities. Classical mobility models, such as the random waypoint, do not allow one to represent the observed dynamics of clumps in a proper manner. This motivates the introduction and the mathematical analysis of a new interaction-based mobility model, which is the main contribution of the present paper. This model is shown to allow one to describe the dynamics of clumps and in particular to predict key phenomena such as the building of hot spots and the scattering between hot spots, which play a key role in the engineering of wireless networks. We show how to obtain the main parameters of this model from simple communication activity measurements and we illustrate this calibration process on real cases.

In this paper, we present a general technique based on Bayesian inference to locate mobiles in cellular networks. We study the problem of localizing users in a cellular network for calls with information regarding only one base station and hence triangulation or trilateration cannot be performed. In our call data records, this happens more than 50% of time. We show how to localize mobiles based on our knowledge of the network layout and how to incorporate additional information such as round-trip-time and signal to noise and interference ratio (SINR) measurements. We study important parameters used in this Bayesian method through mining call data records and matching GPS records and obtain their distribution or typical values. We validate our localization technique in a commercial network with a few thousand emergency calls. The results show that the Bayesian method can reduce the localization error by 20% compared to a blind approach and the accuracy of localization can be further improved by refining the a priori user distribution in the Bayesian technique.

This paper bears on the design and the quantitative evaluation of MAC mechanisms for wireless ad-hoc networks with performance guarantees. By this, we mean mechanisms where each accepted connection obtains a minimum rate or equivalently a minimum SINR level - which is not guaranteed by CSMA/CA - and which are adapted to the wireless ad-hoc network framework, namely are decentralized, power efficient and provide a good spatial reuse. Two such access control algorithms are defined and compared. Both take the interference level into account to decide on the set of connections which can access the shared channel at any given time. The main difference between the two is the possibility or not of adjusting the transmission power of the nodes. A thorough comparison of the performance of these two mechanisms and CSMA/CA is presented, based on a mix of analytical models and simulation and on a comprehensive set of performance metrics which include spatial reuse and power efficiency. Different network topologies, propagation environments and traffic scenarios are considered. The main aim of our study is to identify which of the proposed mechanisms outperforms CSMA/CA best depending on the scenario.

This paper is meant to be an illustration of the use of stochastic geometry for analyzing the performance of routing in large wireless ad hoc (mobile or mesh) networks. In classical routing strategies used in such networks, packets are transmitted on a pre-defined route that is usually obtained by a shortest-path routing protocol. In this paper we review some recent ideas concerning a new routing technique which is opportunistic in the sense that each packet at each hop on its (specific) route from an origin to a destination takes advantage of the actual pattern of nodes that captured its recent (re)transmission in order to choose the next relay. The paper focuses both on the distributed algorithms allowing such a routing technique to work and on the evaluation of the gain in performance it brings compared to classical mechanisms. On the algorithmic side, we show that it is possible to implement this opportunistic technique in such a way that the current transmitter of a given packet does not need to know its next relay a priori, but the nodes that capture this transmission (if any) perform a self-selection procedure to choose the packet relay node and acknowledge the transmitter. We also show that this routing technique works well with various medium access protocols (such as Aloha, CSMA, TDMA). Finally, we show that the above relay self-selection procedure can be optimized in the sense that it is the node that optimizes some given utility criterion (e.g. minimize the remaining distance to the final destination), which is chosen as the relay. The performance evaluation part is based on stochastic geometry and combines simulation as analytical models. The main result is that such opportunistic schemes very significantly outperform classical routing schemes when properly optimized and provided at least a small number of nodes in the network know their geographical positions exactly.

The unsatisfactory performance of TCP in high speed wide area networks has led to several versions of TCP- like H-TCP, Fast TCP, Scalable TCP, Compound or CUBIC, all aimed at speeding up the window update algorithm. In this paper we focus on Scalable TCP (STCP), a TCP version which belongs to the class of Multiplicative Increase Multiplicative Decrease (MIMD) congestion protocols. We present a new stochastic model for the evolution of the instantaneous throughput of a single STCP flow in the Congestion Avoidance phase, under the assumption of a constant per-packet loss probability. This model allows one to derive several closed-form expressions for the key stationary distributions associated with this protocol: we characterize the throughput obtained by the flow, the time separating Multiplicative Decrease events, the number of bits transmitted over certain time intervals and the size of rate decrease. Several applications leveraging these closed form expressions are considered with a particular emphasis on QoS guarantees in the context of dimensioning. A set of ns2 simulations highlights the model accuracy.

Summary form only given, as follows. In this talk, we will discuss some problems related to cooperative spectrum sensing, and show how random matrix theory can help to address them. We will propose a simple test for frequency band sensing in wireless networks. The test is based on the analysis of the ratio of the extreme eigenvalues related to the gain matrix of the channel. The novelty relies in the fact that the test does not require the knowledge of the noise statistics. Large random matrix results allow us to build the threshold for the test, and also to study its type II error. This in particular enables us to compare this test with a different although popular test already proposed in the literature. We will show that our test is uniformly more powerful.

Wireless networks are fundamentally limited by the intensity of the received signals and by their interference. Since both of these quantities depend on the spatial location of the nodes, mathematical techniques have been developed in the last decade to provide communication-theoretic results accounting for the networks geometrical configuration. Often, the location of the nodes in the network can be modeled as random, following for example a Poisson point process. In this case, different techniques based on stochastic geometry and the theory of random geometric graphs -including point process theory, percolation theory, and probabilistic combinatorics-have led to results on the connectivity, the capacity, the outage probability, and other fundamental limits of wireless networks. This tutorial article surveys some of these techniques, discusses their application to model wireless networks, and presents some of the main results that have appeared in the literature. It also serves as an introduction to the field for the other papers in this special issue.

Spatial Aloha is probably the simplest medium access protocol to be used in a large mobile ad hoc network: each station tosses a coin independently of everything else and accesses the channel if it gets heads. In a network where stations are randomly and homogeneously located in the Euclidean plane, there is a way to tune the bias of the coin so as to obtain the best possible compromise between spatial reuse and per transmitter throughput. This paper shows how to address this questions using stochastic geometry and more precisely Poisson shot noise field theory. The theory that is developed is fully computational and leads to new closed form expressions for various kinds of spatial averages (like e.g. outage, throughput or transport). It also allows one to derive general scaling laws that hold for general fading assumptions. We exemplify its flexibility by analyzing a natural variant of Spatial Aloha that we call Opportunistic Aloha and that consists in replacing the coin tossing by an evaluation of the quality of the channel of each station to its receiver and a selection of the stations with good channels (e.g. fading) conditions. We show how to adapt the general machinery to this variant and how to optimize and implement it. We show that when properly tuned, Opportunistic Aloha very significantly outperforms Spatial Aloha, with e.g. a mean throughput per unit area twice higher for Rayleigh fading scenarios with typical parameters.

The one tutorial and 22 papers in this special issue focus on geometry and random graph for the analysis and design of wireless networks. The papers are organized into five groups: Topology; Outage, throughput, capacity, and scaling laws; Connectivity and coverage; Co-existence of disparate wireless networks and cognitive radio; and Distributed algorithms.

The split of a multihop, point to point TCP connection consists in replacing a plain, end-to-end TCP connection by a cascade of TCP connections. In such a cascade, connection n feeds connection n+1 through some proxy node n. This technique is used in a variety of contexts. In overlay networks, proxies are often peers of the underlying peer to peer network. Split TCP is also already proposed and largely adopted in wireless networks at the wired/wireless interface to separate links with vastly different characteristics. In order to avoid losses in the proxies, a backpressure mechanism is often used in this context. In this paper we develop a model for such a split TCP connection aimed at the analysis of the throughput dynamics on both links and that of the buffer occupancy in the proxy for long file transfers. The two main variants of Split TCP are considered: that with backpressure and that without. The study consists of two parts: the first part is purely experimental and is based on ns2 simulations. It allows us to identify complex interaction phenomena between TCP flow rates and proxy buffer occupancy, which seem to have been ignored by previous work on Split TCP. The second part of the paper is of mathematical nature. We establish the basic equations that govern the evolution of such a cascade and prove some of the experimental observations made in the first part. In particular, we give the conditions for system stability and we show the possibility of heavy tail asymptotics for proxy buffer occupancy and delays in the stationary regime.

We define a class of problems in the theory of Euclidean point processes, motivated by the study of the error exponent (reliability function) for additive noise channels. For the case of Gaussian noise this gives an interesting perspective on the Poltyrev exponent. It also suggests an approach to attack the long standing gap between the best known upper and lower bounds on the reliability function of the traditional AWGN channel, using techniques from point process theory.

Past approaches to routing in mesh networks either (i) do not account for the MAC-layer interactions between the links in a tractable manner, or (ii) are agnostic to load-balancing across gateways. Our answer to these problems is MaLB (MAC-aware and Load Balanced routing algorithm), a greedy, tractable, and distributed mesh routing algorithm. Since the underlying objective function has high combinatorial complexity, MaLB uses a greedy approach. MaLB finds an optimum routing forest (union of trees rooted at the gateways) by taking into account MAC-layer interaction between links, as well as optimum multi-hop association of mesh nodes to gateways. MaLB builds on top of ETP (Expected Through-Put), a recently proposed MAC-aware routing metric. We also propose a low complexity variant of MaLB called LB (Load Balanced routing algorithm) which performs load balancing in a MAC-agnostic manner. MaLB performs especially well in networks with skewed topologies that result from unplanned mesh network deployment, as well as in the presence of gateway failures. Simulations with an enhanced version of ns-2 show that MaLB results in up to 60% higher throughput than a shortest path algorithm with ETX (Expected Transmission Count). Furthermore, MaLB results in up to 30% improvement over the LB algorithm, as well as a shortest path algorithm with ETT (Expected Transmission Time).

Packet delay and loss are two fundamental measures of performance. Using active probing to measure delay and loss typically involves sending Poisson probes, on the basis of the PASTA property (Poisson Arrivals See Time Averages), which ensures that Poisson probing yields unbiased estimates. Recent work, however, has questioned the utility of PASTA for probing and shown that, for delay measurements, i) a wide variety of processes other than Poisson can be used to probe with zero bias and ii) Poisson probing does not necessarily minimize the variance of delay estimates.

The popularity of IEEE 802.11 WLANs has led to dense deployments in urban areas. High density leads to sub-optimal performance unless the interfering networks learn how to optimally use and share the spectrum. This paper proposes two fully distributed algorithms that allow (i) multiple interfering 802.11 access points to select their operating frequency in order to minimize interference, and (ii) users to choose the access point they attach to, in order to get their fair share of the whole network bandwidth. The proposed algorithms rely on Gibbs sampler, and do not require explicit coordination among the wireless devices. They only require the participating wireless nodes to measure local quantities such as interference and transmission delay. The algorithms are shown to lead to optimal bandwidth sharing, where optimality is defined according to the minimal potential delay. We analytically prove the convergence of the proposed algorithms, and study their performance by simulation.

The low cost and the ease of deployment of WiFi devices, as well as the need to support high bandwidth applications over 802.11 WLANs has led to the emergence of high density 802.11 networks in urban areas and enterprises. High density wireless networks, by design, face significant challenges due to increased interference resulting from the close proximity of co-channel cells. We demonstrate that power control can be used to mitigate interference in such an environment. It is well-known that variable transmit powers result in asymmetric links in the network, and can potentially lead to throughput starvation of some nodes. We first show that in order to perform starvation-free power control in 802.11 networks, across-layerapproach is required, whereby the transmit powers and the carrier sensing parameter of the MAC layer of the nodes should bejointlytuned. We then propose a framework that determines optimum settings for these parameters with the objective of maximizing the network-wide throughput for elastic traffic. Within this framework, we devise a distributed power control algorithm that uses a Gibbs sampler. OPNET simulations and experiments over a proof of concept testbed demonstrate that in a dense network the proposed power control algorithm yields significant improvement in client throughput.

This paper presents a stochastic geometry model for the performance analysis and the planning of dense IEEE 802.11 networks. This model allows one to propose heuristic formulas for various properties of such networks like the probability for users to be covered, the probability for access points to be granted access to the channel or the average long term throughput provided to end-users. The main merit of this model is to take the effect of interferences and that of CSMA into account within this dense network context. This analytic model, which is based on Matern point processes, is partly validated against simulation. It is then used to assess various properties of such networks. We show for instance how the long term throughput obtained by end-users behaves when the access point density increases. We also briefly show how to use this model for the planning of managed networks and for the economic modeling of unplanned networks.

Poisson Arrivals See Time Averages (PASTA) is a well known property applicable to many stochastic systems. In active probing, PASTA is invoked to justify the sending of probe packets (or trains) at Poisson times in a variety of contexts. However, due to the diversity of aims and analysis techniques used in active probing, the benefits of Poisson based measurement, and the utility and role of PASTA, are unclear. Using a combination of rigorous results and carefully constructed examples and counter-examples, we map out the issues involved, and argue that PASTA is of very limited use in active probing. In particular, Poisson probes are not unique in their ability to sample without bias. Furthermore, PASTA ignores the issue of estimation variance, and the central need for an inversion phase to estimate the quantity of interest ased on what is directly observable. We give concrete examples of when Poisson probes should not be used, and explain why, and offer initial guidelines on suitable alternative sending processes.

An Aloha-type access control mechanism for large mobile, multihop, wireless networks is defined and analyzed. This access scheme is designed for the multihop context, where it is important to find a compromise between the spatial density of communications and the range of each transmission. More precisely, the analysis aims at optimizing the product of the number of simultaneously successful transmissions per unit of space (spatial reuse) by the average range of each transmission. The optimization is obtained via an averaging over all Poisson configurations for the location of interfering mobiles, where an exact evaluation of signal over noise ratio is possible. The main mathematical tools stem from stochastic geometry and are spatial versions of the so-called additive and max shot noise processes. The resulting medium access control (MAC) protocol exhibits some interesting properties. First, it can be implemented in a decentralized way provided some local geographic information is available to the mobiles. In addition, its transport capacity is proportional to the square root of the density of mobiles which is the upper bound of Gupta and Kumar. Finally, this protocol is self-adapting to the node density and it does not require prior knowledge of this density.

This paper builds upon the scalable admission control schemes for CDMA networks developed in F. Baccalli et al. (2003, December 2004). These schemes are based on an exact representation of the geometry of both the downlink and the uplink channels and ensure that the associated power allocation problems have solutions under constraints on the maximal power of each station/user. These schemes are decentralized in that they can be implemented in such a way that each base station only has to consider the load brought by its own users to decide on admission. By load we mean here some function of the configuration of the users and of their bit rates that is described in the paper. When implemented in each base station, such schemes ensure the global feasibility of the power allocation even in a very large (infinite number of cells) network. The estimation of the capacity of large CDMA networks controlled by such schemes was made in these references. In certain cases, for example for a Poisson pattern of mobiles in an hexagonal network of base stations, this approach gives explicit formulas for the infeasibility probability, defined as the fraction of cells where the population of users cannot be entirely admitted by the base station. In the present paper we show that the notion of infeasibility probability is closely related to the notion of blocking probability, defined as the fraction of users that are rejected by the admission control policy in the long run, a notion of central practical importance within this setting. The relation between these two notions is not bound to our particular admission control schemes, but is of more general nature, and in a simplified scenario it can be identified with the well-known Erlang loss formula. We prove this relation using a general spatial birth-and-death process, where customer locations are represented by a spatial point process that evolves over time as users arrive or depart. This allows our model to include the exact representation of the geometry of inter-cell and intra-cell interferences, which play an essential role in the load indicators used in these cellular network admission control schemes.

We consider reliable multicast in overlay networks where nodes have finite-size buffers and are subject to failures. We address issues of end-to-end reliability and throughput scalability in this framework. We propose a simple architecture which consists of using distinct point-to-point TCP connections between adjacent pairs of end-systems, together with a back-pressure control mechanism regulating the transfers of adjacent TCP connections, as well as a back-up buffering system handling node failures. This architecture, that we call the one-to-many TCP overlay, is a natural extension of TCP to the one-to-many case, in that it adapts the rate of the group communication to local congestion in a decentralized way via the window back-pressure mechanism. Using theoretical investigations, experimentations in the Internet, and large network simulations, we show that this architecture provides end-to-end reliability and can tolerate multiple simultaneous node failures, provided the backup buffers are sized appropriately. We also show that under random perturbations caused by cross traffic described in the paper, the throughput of this reliable group communication is always larger than a positive constant, that does not depend on the group size. This scalability result contrasts with known results about the non-scalability of IP-supported multicast for reliable group communication.

This paper analyzes the performance of a large population composed of several classes of long lived TCP flows experiencing packet losses due to random transmission errors and to congestion created by the sharing of a common tail-drop or RED bottleneck router. Each class has a different transmission error rate. This setting is used to analyze the competition between wired and wireless users in an access network, where one class (the wired class) has no or small (like BER in DSL) transmission error losses whereas the other class has higher transmission error losses, or the competition between DSL flows using different coding schemes. We propose a natural and simple model for the joint throughput evolution of several classes of TCP flows under such a mix of losses. Two types of random transmission error losses are considered: one where losses are Poisson and independent of the rate of the flow, and one where the losses are still Poisson but with an intensity that is proportional to the rate of the source. We show that the large population model where the population tends to infinity has a threshold (given in closed form) below which there are no congestion losses at all in steady state, and above which there is a stationary limiting regime in which we can compute both the mean value and the distribution of the rate obtained by each class of flow. We also show that the maximum mean value for the aggregated rate is achieved at the threshold.

Active probing heuristics are usually based on queuing systems. However, a rigorous probabilistic treatment of probing methods has been lacking. For instance, it is not known even in principle, what can and cannot be measured in general, nor the true limitations of existing methods. We provide a probabilistic treatment for the measurement of cross traffic in the 1-hop case. We derive inversion formulae for the cross traffic process, and explain their fundamental limits, using an intuitive geometric framework.

The aim of this paper is to analyze the performance of a large number of long-lived TCP controlled flows sharing many routers (or links), from the knowledge of the network parameters (capacity, buffer size, topology) and of the characteristics of each TCP flow (RTT, route etc.) when taking synchronization into account. It is shown that in the small buffer case, the dynamics of such a network can be described in terms of iterate of random piecewise affine maps, or geometrically as a billiards in the Euclidean space with as many dimensions as the number of flow classes and as many reflection facets as there are routers. This class of billiards exhibits both periodic and nonperiodic asymptotic oscillations, the characteristics of which are extremely sensitive to the parameters of the network. It is also shown that for large populations and in the presence of synchronization, aggregated throughputs exhibit fluctuations that are due to the network as a whole, that follow some complex fractal patterns, and that come on top of other and more classical flow or packet level fluctuations. The consequences on TCP's fairness are exemplified on a few typical cases of small dimension.

We study the impact of interferences on the connectivity of large-scale ad hoc networks, using percolation theory. We assume that a bi-directional connection can be set up between two nodes if the signal to noise ratio at the receiver is larger than some threshold. The noise is the sum of the contribution of interferences from all other nodes, weighted by a coefficient /spl gamma/, and of a background noise. We find that there is a critical value of /spl gamma/ above which the network is made of disconnected clusters of nodes. We also prove that if /spl gamma/ is nonzero but small enough, there exist node spatial densities for which the network contains a large (theoretically infinite) cluster of nodes, enabling distant nodes to communicate in multiple hops. Since small values of /spl gamma/ cannot be achieved without efficient CDMA codes, we investigate the use of a very simple TDMA scheme, where nodes can emit only every nth time slot. We show that it achieves connectivity similar to the previous system with a parameter /spl gamma//n.

This study provides some new insights into the scalability of reliable group communication mechanisms using overlays. These mechanisms use individual TCP connections for packet transfers between end-systems. End-systems store incoming packets and forward them to downstream nodes using different unicast TCP connections. In this paper we assume that buffers in end-systems are large enough for the transfers. It is shown that the throughput of the reliable overlay group communication scales in the sense that for all multicast tree sizes and topologies, the group throughput is strictly positive under natural conditions. This is in contrast with the IP supported multicast paradigm where reliable protocols have vanishing throughput when the group size tends to infinity. The scalability of packet delay and buffer occupancy is then investigated. In the absence of additional control, the occupancy of the buffer and the latency in the end-systems explodes with time. It is then shown that proactive rate throttle mechanism implemented at the source leads to finite packet latency and buffer occupancy in any end-system of the network provided certain moment conditions are satisfied by cross traffic in the routers.

This paper analyzes the performance of a large population of long lived TCP flows experiencing random packet losses due to both random transmission errors and congestion created by the sharing of a common tail drop bottleneck router. We propose a natural and simple model for the joint throughput evolution of the set of TCP sessions under such a mix of losses. For the case of Poisson transmission errors, we show that the asymptotic model where the population tends to infinity leads to a well defined and tractable dynamical system. In particular, we get the mean value of the throughput of each session as a function of the transmission error rate and the synchronization rate in the bottleneck router. The large population asymptotic model has two interesting and non-intuitive properties: 1) there exists a positive threshold (given in closed form) on the transmission error rate above which there are no congestion losses at all in steady state; 2) below this threshold, the mean throughput of each flow is an increasing function of the transmission error rate, so that the maximum mean value is in fact achieved when the transmission error rate is equal to this threshold. The finite population model and models based on other classes of point processes are also studied. In particular, a sufficient condition is obtained for the existence of congestion times in the case of arbitrary transmission error point processes.

In this paper, we consider a set of HTTP flows using TCP over a common drop-tail link to download files. After each download, a flow waits for a random think time before requesting the download of another file, whose size is also random. When a flow is active its throughput is increasing with time according to the additive increase rule, but if it suffers losses created when the total transmission rate of the flows exceeds the link rate, its transmission rate is decreased. The throughput obtained by a flow, and the consecutive time to download one file are then given as the consequence of the interaction of all the flows through their total transmission rate and the link's behavior.We study the mean-field model obtained by letting the number of flows go to infinity. This mean-field limit may have two stable regimes : one without congestion in the link, in which the density of transmission rate can be explicitly described, the other one with periodic congestion epochs, where the inter-congestion time can be characterized as the solution of a fixed point equation, that we compute numerically, leading to a density of transmission rate given by as the solution of a Fredholm equation. It is shown that for certain values of the parameters (more precisely when the link capacity per user is not significantly larger than the load per user), each of these two stable regimes can be reached depending on the initial condition. This phenomenon can be seen as an analogue of turbulence in fluid dynamics: for some initial conditions, the transfers progress in a fluid and interaction-less way; for others, the connections interact and slow down because of the resulting fluctuations, which in turn perpetuates interaction forever, in spite of the fact that the load per user is less than the capacity per user. We prove that this phenomenon is present in the Tahoe case and both the numerical method that we develop and simulations suggest that it is present in the Reno case too. It translates into a bi-stability phenomenon for the finite population model within this range of parameters.

This paper describes and analyzes SCORE, a scalable multicast-based communication protocol for large-scale virtual environments (LSVE) on the Internet. Today, many of these applications have to handle an increasing number of participants and deal with the difficult problem of scalability. We propose an approach at the transport layer, using multiple multicast groups and multiple agents. This approach involves the dynamic partitioning of the virtual environment into spatial areas and the association of these areas with multicast groups. It uses a method based on the theory of planar point processes to determine an appropriate cell size, so that the incoming traffic at the receiver side remains with a given probability below a sufficiently low threshold. We evaluate the performance of our scheme and show that it allows to significantly improve the participants' satisfaction while adding very low overhead.

This paper is focused on the influence of geometry on the combination of intercell and intracell interferences in the downlink of large CDMA networks. We use an exact representation of the geometry of the downlink channels to define scalable admission and congestion control schemes, namely schemes that allow each base station to decide independently of the others what set of voice users to serve and/or what bit rates to offer to elastic traffic users competing for bandwidth. We then study the load of these schemes when the size of the network tends to infinity using stochastic geometry tools. By load, we mean here the distribution of the number of voice users that each base station can serve and that of the bit rate offered to each elastic traffic user.

The aim of this paper is to analyze the performance of a large number of long lived TCP controlled flows sharing many routers (or links), from the knowledge of the network parameters (capacity, buffer size, topology) and of the characteristics of each TCP flow (RTT, route etc.) when taking synchronization into account. It is shown that the dynamics of such a network can be described in terms of iterate of random piecewise affine maps, or geometrically as a billiards in the Euclidean space with as many dimensions as the number of flow classes and as many reflection facets as there are routers. This class of billiards exhibits both periodic and non-periodic asymptotic oscillations, the characteristics of which are extremely sensitive to the parameters of the network. It is also shown that for large populations and in the presence of synchronization, aggregated throughputs exhibit fluctuations that are due to the network as a whole, that follow some complex fractal patterns, and that come on top of other and more classical flow or packet level fluctuations. The consequences on TCP's fairness are exemplified on a few typical cases of small dimension.

The aim of this paper is to simulate the interaction of a large number of TCP controlled flows and UDP flows sharing many routers/links, from the knowledge of the network parameters (capacity, buffer size, topology, scheduling) and of the characteristics of each TCP (RTT, route etc.) and UDP flow. This work is based on the description via some fluid evolution equations, of the joint evolution of the window sizes of all flows over a single bottleneck router/link, as function of the synchronization rate. It is shown that the generalization of this fluid dynamics to a network composed of several routers can be described via equations allowing one to simulate the interaction of e.g. millions of TCP flows on networks composed of tens of thousands of links and routers on a standard workstation. The main output of the simulator are the mean value and the fluctuations of the throughput obtained by each flow, the localization of the bottleneck routers/links, the losses on each of them and the time evolution of aggregated input traffic at each router or link. The method is validated against NS simulations. We show that several important statistical properties of TCP traffic which were identified on traces are also present on traffic generated by our simulator: for instance, aggregated traffic generated by this representation exhibits the same short time scale statistical properties as those observed on real traces. Similarly, the experimental laws describing the fairness of the bandwidth sharing operated by TCP over a large network are also observed on the simulations.

The impact of interferences on the connectivity of large-scale ad-hoc networks is studied using percolation theory. We assume that a bi-directional connection can be set up between two nodes if the signal to noise ratio at the receiver is larger than some threshold. The noise is the sum of the contribution of interferences from all other nodes, weighted by a coefficient /spl gamma/, and of a background noise. We find that there is a critical value of /spl gamma/ above which the network is made of disconnected clusters of nodes. We also prove that if /spl gamma/ is nonzero but small enough, there exist node spatial densities for which the network contains a large (theoretically infinite) cluster of nodes, enabling distant nodes to communicate in multiple hops. Since small values of /spl gamma/ cannot be achieved without efficient CDMA codes, we investigate the use of a very simple TDMA scheme, where nodes can emit only every n-th time slot. We show qualitatively that it even achieves a better connectivity than the previous system with a parameter /spl gamma//n.

The aim of the present paper is to show that stochastic geometry provides an efficient computational framework allowing one to predict geometrical characteristics of large CDMA networks such as coverage or soft-handoff level. The general idea consists in representing the location of antennas and/or mobile stations as realizations of stochastic point processes in the plane within a simple parametric class, which takes into account the irregularities of antenna/mobile patterns in a statistical way. This approach leads to new formulas and simulation schemes allowing one to compute/estimate the spatial averages of these local characteristics in function of the model parameters (density of antennas or mobiles, law of emission power, fading law etc.) and to perform various parametric optimizations.

We propose a natural and simple model for the joint throughput evolution of a set of TCP sessions sharing a common tail drop bottleneck router, via products of random matrices. This model allows one to predict the fluctuations of the throughput of each session, as a function of the synchronization rate in the bottleneck router; several other and more refined properties of the protocol are analyzed such as the instantaneous imbalance between sessions, the autocorrelation function or the performance degradation due to synchronization of losses. When aggregating traffic obtained from this model, one obtains, for certain ranges of the parameters, short time scale statistical properties that are consistent with a fractal scaling similar to what was identified on real traces using wavelets.

We study the impact of random queueing delays stemming from traffic variability on the performance of a multicast session. With a simple analytical model, we analyze the throughput degradation within a multicast (one-to-many) tree under TCP-like congestion and flow control. We use the (max,plus) formalism together with methods based on stochastic comparison (association and convex ordering) and on the theory of extremes to prove various properties of the throughput. We first prove that the throughput predicted by a deterministic model is systematically optimistic. In the presence of light-tailed random delays, we show that the throughput decreases according to the inverse of the logarithm of the number of receivers. We find analytically an upper and a lower bound for the throughput degradation. Within these bounds, we characterize the degradation which is obtained for various tree topologies. In particular, we observe that a class of trees commonly found in IP multicast sessions is significantly more sensitive to traffic variability than other topologies.

We study the impact of random noise (queueing delay) on the performance of a multicast session. With a simple analytical model, we analyze the throughput degradation within a multicast (one-to-many) tree under TCP-like congestion and flow control. We use the (max, plus) formalism together with methods based on stochastic comparison (association and convex ordering) and on the theory of extremes (Lai and Robbins' (1978) notion of maximal characteristics) to prove various properties of the throughput. We first prove that the throughput obtained from Golestani and Sabnani's (1999) deterministic model is systematically optimistic. In presence of light tailed random noise, we show that the throughput decreases like the inverse of the logarithm of the number of receivers. We find analytically an upper and a lower bound for the throughput degradation. Within these bounds, we characterize the degradation which is obtained for various tree topologies. In particular, we observe that a class of trees commonly found in IP multicast sessions (which we call umbrella trees) is significantly more sensitive to network noise than other topologies.

We consider a tandem of queues fed by a superposition of on-off sources. The service process at each of the queues in the tandem are also independent on-off fluid processes. The off duration in the service process may be due to a server breakdown or due to the unavailability of the server because of the presence of higher priority cross-traffic. We obtain sufficient conditions for the tail of the stationary end-to-end backlog distribution to be dominated by one of the service processes or by one of the source processes.

No abstract available.

Multicast routing protocols suitable for wide-area networks are being developed for the Internet. Protocols based on hierarchical trees appear to be well suited for their superior scalability and flexibility. We show how to construct a class of hierarchical multicast trees and we analyze their performances. This study gives insight into how the chosen hierarchical structure impacts the tree performance with respect to network resource consumption. Optimal structures which minimize resource consumption are deduced, which allows for simple dimensioning rules, such as how many hierarchical levels should be used. A stochastic geometric approach turned out to be well adapted for this study. This approach leads to explicit expressions for the average tree cost, as a function of the hierarchical clustering, from which the optimal tree structure can then be easily deduced.

Parallel execution of an arrival stream of jobs with and without real-time constraints on a (possibly heterogeneous) multiprocessor system IS considered.A job consists of a set of tasks and a partial order specifying the precedence constraints between the tasks.The real-time constraints are specified by due times, also called so~t real time deadlines.It is assumed that there is a predefine mapping from the set of tasks onto the set of machines that is identical for all jobs.Associated with each task is a service time that may depend on the machine that it is allocated to.The problem of scheduling tasks into execution at each machine is the subject of this paper.Dynamic nonpreemptive scheduling policies that do not use service-time information is examined and a class of Local Order Preserving (LOP) policies that contains the class of nonidling First Come Fust Serve (FCFS) policies is defined.It is shown that policies from this last class, along with the classes of LOP Shortest Due Time First (S DTF), LOP Largest Due Time First (LDTF), and LOP Last Come First Serve (LCFS) policies stochastically minimize the number of jobs in the system and maximize the job throughput.The class of FCFS policies is further shown to minimize the vector of transient response times in the increasing Schur convex sense.Last, we consider the job lateness, the difference between the due time and the completion time of the job, and prove that within the class of LOP policies, the SDTF and LDTF policies bound, respectively, from below and from above the transient vector of the job latenesses, in the Schur convex sense.The paper concludes with extensions to the steady state performance metrics, to the class of preemptive-resume policies, and to jobs having random task graphs.All of the results, except those concerned with preemptive policies, assume that task service times form mutually independent sequences of independent and identically distributed random variables.In the latter case, service times are further assumed to be exponential random variables.

Petri nets provide a powerful modeling formalism, which allows one to describe and study various classes of systems, such as synchronous and asynchronous processes, and/or parallel or sequential ones.

The new class of queuing models, calledSynchronized Queuing Networks, is proposed for evaluating the performance of multiprogrammed and multitasked multiprocessor systems, where workloads consists of parallel programs of similar structure and where the scheduling discipline is first-come-first-serve. Pathwise evolution equations are established for these networks that capture the effects of competition for processors and the precedence constraints governing tasks executions. A general expression is deduced for the stability condition of such queuing networks under general statistical assumptions (basically the stationarity and the ergodicity of input sequences), which yields the maximum program throughput of the multiprocessor system, or equivalently, the maximum rate at which programs can be executed or submitted. The proof is based on the ergodic theory of queues. Basic integral equations are also derived for the stationary distribution of important performance criteria such as the workload of the queues and program response times. An iterative numerical schema that converges to this solution is proposed and various upper and lower bounds on moments are derived using stochastic ordering techniques.

In this paper the class of acyclic fork-join queuing networks that arise in various applications, including parallel processing and flexible manufacturing are studied. In such queuing networks, a fork describes the simultaneous creation of several new customers, which are sent to different queues. The corresponding join occurs when the services of all these new customers are completed. The evolution equations that govern the behavior of such networks are derived. From this, the stability conditions are obtained and upper and lower bounds on the network response times are developed. These bounds are based on various stochastic ordering principles and on the notion of association of random variables.

The authors consider queueing that occur naturally in the study of a class of resource-sharing problems under synchronization constraints such as resequencing and fork-join primitives. These queueing models are amenable to a representation in terms of a state recursion. The proposed methods of analysis are complementary and draw on classical ideas of queuing theory as well as on mathematical tools from the theory of stochastic ordering and ergodic theory. The state recursion is at the center of all aspects of the analysis, be it for developing the exact solutions, obtaining bounds on system performance or establishing the stability conditions. The ideas are illustrated on simple models of resequencing and fork-join synchronization, which emphasis put on deriving computable bounds on the performance measures.>

We define some decomposition schema of the derivation trees of an arithmetic infix grammar into a set of subtrees. From the parenthesis nesting of an arithmetic expression this decomposition determines a cover of its derivation tree by a set of subtrees with known syntactic properties. We define from this a recursive interpretation actor. Roughly speaking, for a given arithmetic expression, this interpreter creates one actor for each of the subtrees in its derivation tree decomposition, each created actor being in charge of local (i.e., concerning its own subtree only) parsing and semantic tasks which involve synchronization with the other actors.

The resequencing or serialization problem is of basic interest in distributed systems and computer communication systems.This is because a flow of packets, messages, or updates entering a communication system in chronological order from the same port or from different ports may be disordered.The receiving port must then ensure that these objects are resequenced in the appropriate order before they are fed to the output of the system.In this paper we analyze the end-to-end delay recurred by objects traversing such a system, including the d~sordering delay, the delay introduced by the resequencing algorithm, and the delay due to the output server at the receiving port.The analysis is carded out via factorization methods.