We propose a novel framework for wireless network scheduling with fairness using determinantal (point) processes. Our approach incorporates the repulsive nature of determinantal processes, generalizing traditional Aloha protocols that schedule transmissions independently. We formulate the scheduling problem with an utility function representing fairness. We then recast this formulation as a convex optimization problem over a certain class of determinantal point processes called$L$-ensembles, which are particularly suited for statistical and numerical treatments. These determinantal processes, which have already proven valuable in subset learning, offer an attractive approach to network resource scheduling and allocating. We demonstrate the suitability of determinantal processes for network models based on the signal-to-interference-plus-noise ratio (SINR). Our results highlight the potential of determinantal scheduling coupled with fairness. This work bridges recent advances in machine learning with wireless communications, providing a mathematically elegant and computationally tractable approach to network scheduling.

To study the overall connectivity in device-to-device networks in cities, we incorporate a signal-to-interference-plus-noise connectivity model into a Poisson-Voronoi tessellation model representing the streets of a city. Relays are located at crossroads (or street intersections), whereas (user) devices are scattered along streets. Between any two adjacent relays, we assume data can be transmitted either directly between the relays or through users, given they share a common street. Our simulation results reveal that the network connectivity is ensured when the density of users (on the streets) exceeds a certain critical value. But then the network connectivity disappears when the user density exceeds a second critical value. The intuition is that for longer streets, where direct relay-to-relay communication is not possible, users are needed to transmit data between relays, but with too many users the interference becomes too strong, eventually reducing the overall network connectivity. This observation on the user density evokes previous results based on another wireless network model, where transmitter-receivers were scattered across the plane. This effect disappears when interference is removed from the model, giving a variation of the classic Gilbert model and recalling the lesson that neglecting interference in such network models can give overly optimistic results. For physically reasonable model parameters, we show that crowded streets (with more than six users on a typical street) lead to a sudden drop in connectivity. We also give numerical results outlining a relationship between the user density and the strength of any interference reduction techniques.

It has been shown that deploying device-to-device (D2D) networks in urban environments requires equipping a considerable proportion of crossroads with relays. This represents a necessary economic investment for an operator. In this work, we tackle the problem of the economic feasibility of such relay assisted D2D networks. First, we propose a stochastic model taking into account a positive surface for streets and crossroads, thus allowing for a more realistic estimation of the minimal number of needed relays. Secondly, we introduce a cost model for the deployment of relays, allowing one to study operators' D2D deployment strategies. We investigate the example of an uberizing neo-operator willing to set up a network entirely relying on D2D and show that a return on the initial investment in relays is possible in a realistic period of time, even if the network is funded by a very low revenue per D2D user. Our results bring quantitative arguments to the discussion on possible uberization scenarios of telecommunications networks.

Motivated by the prediction of cell loads in cellular networks, we formulate the following new, fundamental problem of statistical learning of geometric marks of point processes: An unknown marking function, depending on the geometry of point patterns, produces characteristics (marks) of the points. One aims at learning this function from the examples of marked point patterns in order to predict the marks of new point patterns. To approximate (interpolate) the marking function, in our baseline approach, we build a statistical regression model of the marks with respect to some local point distance representation. In a more advanced approach, we use a global data representation via the scattering moments of random measures, which build informative and stable to deformations data representation, already proven useful in image analysis and related application domains. In this case, the regression of the scattering moments of the marked point patterns with respect to the non-marked ones is combined with the numerical solution of the inverse problem, where the marks are recovered from the estimated scattering moments. Considering some simple, generic marks, often appearing in the modeling of wireless networks, such as the shot-noise values, nearest neighbour distance, and some characteristics of the Voronoi cells, we show that the scattering moments can capture similar geometry information as the baseline approach, and can reach even better performance, especially for non-local marking functions. Our results motivate further development of statistical learning tools for stochastic geometry and analysis of wireless networks, in particular to predict cell loads in cellular networks from the locations of base stations and traffic demand.

A new type of dependent thinning for point processes in continuous space is proposed, which leverages the advantages of determinantal point processes defined on finite spaces and, as such, is particularly amenable to statistical, numerical, and simulation techniques. It gives a new point process that can serve as a network model exhibiting repulsion. The properties and functions of the new point process, such as moment measures, the Laplace functional, the void probabilities, as well as conditional (Palm) characteristics can be estimated accurately by simulating the underlying (non-thinned) point process, which can be taken, for example, to be Poisson. This is in contrast (and preference to) finite Gibbs point processes, which, instead of thinning, require weighting the Poisson realizations, involving usually intractable normalizing constants. Models based on determinantal point processes are also well suited for statistical (supervised) learning techniques, allowing the models to be fitted to observed network patterns with some particular geometric properties. We illustrate this approach by imitating with determinantal thinning the well-known Matérn II hard-core thinning, as well as a soft-core thinning depending on nearest-neighbour triangles. These two examples demonstrate how the proposed approach can lead to new, statistically optimized, probabilistic transmission scheduling schemes.

In this work, we use percolation theory to study the feasibility of large-scale connectivity of relay-augmented device-to-device (D2D) networks in an urban scenario featuring a haphazard system of streets and canyon shadowing allowing only for line-of-sight (LOS) communications in a finite range. We use a homogeneous Poisson-Voronoi tessellation (PVT) model of streets with homogeneous Poisson users (devices) on its edges and independent Bernoulli relays on the vertices. Using this model, we demonstrate the existence of a minimal threshold for relays below which large-scale connectivity of the network is not possible, regardless of all other network parameters. Through simulations, we estimate this threshold to 71.3%. Moreover, if the mean street length is not larger than some threshold (predicted to 74.3% of the communication range; which might be the case in a typical urban scenario) then any (whatever small) density of users can be compensated by equipping more crossroads with relays. Above this latter threshold, good connectivity requires some minimal density of users, compensated by the relays in a way we make explicit. The existence of the above regimes brings interesting qualitative arguments to the discussion on the possible D2D deployment scenarios.

In this paper, we consider the problem of location-dependent opportunistic bandwidth sharing between static and mobile (i.e., moving) downlink users in a cellular network. Each cell of the network has some fixed number of static users. Mobile users enter the cell, move inside the cell for some time, and then leave the cell. In order to provide higher data rate to the highly mobile users whose fast fading channel variation is difficult to track, we propose location dependent bandwidth sharing between the two classes of static and mobile users; the idea is to provide higher bandwidth to the mobile users at favourable locations, and provide higher bandwidth to the static users in other times. Our approach is agnostic to the way the bandwidth is further shared within the same class of users; it can be combined with any particular bandwidth allocation policy employed for one of these two classes of users. We formulate the problem as a long run average reward Markov decision process (MDP) where the per-step reward is a linear combination of instantaneous data volumes received by static and mobile users, and find the optimal policy. The optimal policy is binary in nature; it allocates the entire bandwidth either to the static users or to the mobile users at any given time. The reward structure of this MDP is not known in general, and it may change with time. To alleviate these issues, we propose a learning algorithm based on single timescale stochastic approximation. Also, noting that the MDP problem can be used to maximize the long run average data rate for mobile users subject to a constraint on the long run average data rate of static users, we provide a learning algorithm based on multi-timescale stochastic approximation. We prove asymptotic convergence of the bandwidth sharing policies under these learning algorithms to the optimal policy. The results are extended to address the issue of fair bandwidth sharing between the two classes of static and mobile users, where the notion of fairness is motivated by the popular notion of α-fairness in the literature. Numerical results exhibit significant performance improvement by our scheme, as well as fast convergence, and also demonstrate the trade-off between performance gain and fairness requirement.

Combining stochastic geometric approach with some classical results from queueing theory regarding generalized processor sharing queues, in this paper we extend the synthetic framework for the performance study of large cellular networks, previously proposed in Błaszczyszyn et al., by allowing it to take into account opportunistic scheduling. Rapid and verifiable with respect to real data, our approach is particularly useful for network dimensioning and long term economic planning. It is based on a detailed network model combining an information-theoretic representation of the link layer, a queueing-theoretic representation of the users' scheduler, and a stochastic-geometric representation of the signal propagation and the network cells. It allows one to evaluate principal characteristics of the individual cells, such as loads the mean number of users and the user throughput. A simplified, Gaussian approximate model is also proposed to facilitate study of the spatial distribution of these metrics across the network. Using of both models requires only simulations of the point process of base stations and the shadowing field to estimate the expectations of some stochastic-geometric functionals not admitting explicit expressions. A key observation of our approach, bridging spatial and temporal analysis, relates the SINR distribution of the typical user to the load of the typical cell of the network. The former is a static characteristic of the network related to its spectral efficiency while the latter characterizes the performance of the (generalized) processor sharing queue serving the dynamic population of users of this cell.

In small cell networks, the high mobility of users results in frequent handoff and thus severely restricts the data rate for mobile users (MUs). To alleviate this problem, we propose use of the heterogeneous two-tier network structure where static users (SUs) are served by both macro and micro base stations (BSs), whereas the mobile (i.e., moving) users are served only by the macro BSs having larger cells; the idea is to prevent frequent data outage for MUs due to handoff. We use the classical two-tier Poisson network model with different transmit powers, and assume the independent Poisson process of SUs and doubly stochastic Poisson process of MUs moving at a constant speed along infinite straight lines generated by a Poisson line process. Using tools from stochastic geometry, we calculate the average downlink data rate of the typical static and mobile (i.e., moving) users, and the latter accounted for handoff outage periods. We consider also the average throughput of these two types of users defined as their average data rates divided by the mean total number of users co-served by the same base station. We find that if the density of a homogeneous network and/or the speed of MUs is high, it is advantageous to let the MUs connect only to some optimal fraction of BSs (i.e., an optimal random subset of BSs) to reduce the frequency of handoffs during which the connection is not assured. If a heterogeneous structure of the network is allowed, one can further jointly optimize the mean throughput of MUs and SUs. This joint optimization is done by appropriately tuning the powers of micro and macro BSs subject to some aggregate power constraint ensuring unchanged mean data rates of SUs via the network equivalence property.

In this paper, we develop Gibbs sampling-based techniques for learning the optimal placement of contents in a cellular network. We consider the situation where a finite collection of base stations are scattered on the plane, each covering a cell (possibly overlapping with other cells). Mobile users request downloads from a finite set of contents according to some popularity distribution which may be known or unknown to the base stations. Each base station has a fixed memory space that can store only a strict subset of the contents at a time; hence, if a user requests content that is not stored at any of its serving base stations, the content has to be downloaded from the backhaul. Hence, we consider the problem of optimal content placement which minimizes the rate of download from the backhaul, or equivalently maximize the cache hit rate. It is known that, when multiple cells can overlap with one another (e.g., under dense deployment of base stations in small cell networks), it is not optimal to place the most popular contents in each base station. However, the optimal content placement problem is NP-complete. Using the ideas of Gibbs sampling, we propose simple sequential content update rules that decide whether to store content at a base station (if required from the base station) and which content has to be removed from the corresponding cache, based on the knowledge of contents stored in its neighboring base stations. The update rule is shown to be asymptotically converging to the optimal content placement for all nodes under the knowledge of content popularity. Next, we extend the algorithm to address the situation where content popularities and cell topology are initially unknown, but are estimated as new requests arrive to the base stations; we show that our algorithm working with the running estimates of content popularities and cell topology also converges asymptotically to the optimal content placement. Finally, we demonstrate the improvement in cache hit rate compared with the most popular content placement and independent content placement strategies via numerical exploration.

Results are presented for optimizing device-to-device communications in cellular networks, while maintaining spectral efficiency of the base-station-to-device downlink channel. We build upon established and tested stochastic geometry models of signal-to-interference ratio in wireless networks based on the Poisson point process, which incorporate random propagation effects such as fading and shadowing. A key result is a simple formula, allowing one to optimize the device-to-device spatial throughput by suitably adjusting the proportion of active devices. These results can lead to further investigation as they can be immediately applied to more sophisticated models such as studying multi-tier network models to address coverage in closed access networks.

We state and solve a problem of the optimal geographic caching of content in cellular networks, where linear combinations of contents are stored in the caches of base stations. We consider a general content popularity distribution and a general distribution of the number of stations covering the typical location in the network. We are looking for a policy of content caching maximizing the probability of serving the typical content request from the caches of covering stations. The problem has a special form of monotone sub-modular set function maximization. Using dynamic programming, we find a deterministic policy solving the problem. We also consider two natural greedy caching policies. We evaluate our policies considering two popular stochastic geometric coverage models: the Boolean one and the Signal-to-Interference-and-Noise-Ratio one, assuming Zipf popularity distribution. Our numerical results show that the proposed deterministic policies are in general not worse than some randomized policy considered in the literature and can further improve the total hit probability in the moderately high coverage regime.

The main focus of this paper is to explicitly characterize the disparity of quality of service (QoS) metrics between base stations in large heterogeneous wireless cellular networks. The considered QoS metrics are cell load, users' number, and user throughput. The spatial disparity of these metrics is due to the irregularity of the cells' geometry. In order to consider these irregularities, we assume a Poisson point process of base station locations, random transmission powers, and log-normal shadowing. The interdependency between the performances of the base stations is characterized by a system of load equations. The typical cell simulation model consists in resolving this system in order to find the loads and then deduce the remaining characteristics for each cell of the network. Using stochastic geometric and queueing theoretic techniques, we define the QoS averages, variances, and distributions. Inspired by the analysis of the typical cell model, several investigations lead us to propose a fully analytic approach, called mean cell model, that approximates the averages, variances, and distributions of these QoS metrics. Numerical experiments show a good agreement between the proposed approximations, simulation results, and real-life network measurements.

A model of cellular networks where the base station locations constitute a Poisson point process and each base station is equipped with three sectorial antennas is proposed. This model permits studying the spatial distribution of the signal-to-interference-and-noise ratio (SINR) in the downlink. In particular, this distribution is shown to be insensitive to the distribution of antenna azimuths. Moreover, the effect of horizontal sectorization is shown to be equivalent to that of shadowing. Assuming ideal vertical antenna pattern, an explicit expression of the Laplace transform of the inverse of SINR is given. The model is validated by comparing its results to measurements in an operational network. It is observed numerically that, in the case of dense urban regions where interference is preponderant, one may neglect the effect of the vertical sectorization when calculating the distribution of the SINR, which provides considerable tractability. Combined with queuing theory results, the SINR's distribution permits to express the user's quality of service as function of the traffic demand. This permits in particular to operators to predict the required investments to face the continual increase of traffic demand.

In this work we consider the problem of an optimal geographic placement of content in wireless cellular networks modelled by Poisson point processes. Specifically, for the typical user requesting some particular content and whose popularity follows a given law (e.g. Zipf), we calculate the probability of finding the content cached in one of the base stations. Wireless coverage follows the usual signal-to-interference-and noise ratio (SINR) model, or some variants of it. We formulate and solve the problem of an optimal randomized content placement policy, to maximize the user's hit probability. The result dictates that it is not always optimal to follow the standard policy “cache the most popular content, everywhere”. In fact, our numerical results regarding three different coverage scenarios, show that the optimal policy significantly increases the chances of hit under high-coverage regime, i.e., when the probabilities of coverage by more than just one station are high enough.

We propose a model for heterogeneous cellular networks assuming a space-time Poisson process of call arrivals, independently marked by data volumes, and served by different types of base stations (having different transmission powers) represented by the superposition of independent Poisson processes on the plane. Each station applies a processor sharing policy to serve users arriving in its vicinity, modeled by the Voronoi cell perturbed by some random signal propagation effects (shadowing). Users' peak service rates depend on their signal-to-interference-and-noise ratios (SINR) with respect to the serving station. The mutual-dependence of the cells (due to the extra-cell interference) is captured via some system of cell-load equations impacting the spatial distribution of the SINR. We use this model to study in a semi-analytic way (involving only static simulations, with the temporal evolution handled by the queuing theoretic results) network performance metrics (cell loads, mean number of users) and the quality of service perceived by the users (mean throughput) served by different types of base stations. Our goal is to identify macroscopic laws regarding these performance metrics, involving averaging both over time and the network geometry. The reveled laws are validated against real field measurement in an operational network.

We propose an analytic approach to the frequency bandwidth dimensioning problem, faced by cellular network operators who deploy/upgrade their networks in various geographical regions (countries) with an inhomogeneous urbanization. We present a model allowing one to capture fundamental relations between users' quality of service parameters (mean downlink throughput), traffic demand, the density of base station deployment, and the available frequency bandwidth. These relations depend on the applied cellular technology (3G or 4G impacting user peak bit-rate) and on the path-loss characteristics observed in different (urban, sub-urban and rural) areas. We observe that if the distance between base stations is kept inversely proportional to the distance coefficient of the path-loss function, then the performance of the typical cells of these different areas is similar when serving the same (per-cell) traffic demand. In this case, the frequency bandwidth dimensioning problem can be solved uniformly across the country applying the mean cell approach proposed in [1]. We validate our approach by comparing the analytical results to measurements in operational networks in various geographical zones of different countries.

Based on a stationary Poisson point process, a wireless network model with random propagation effects (shadowing and/or fading) is considered in order to examine the process formed by the signal-to-interference-plus-noise ratio (SINR) values experienced by a typical user with respect to all the base stations in the down-link channel. This SINR process is completely characterized by deriving its factorial moment measures, which involve numerically tractable, explicit integral expressions. This novel framework naturally leads to expressions for the k-coverage probability, including the case of random SINR threshold values considered in multi-tier network models. While the k-coverage probabilities correspond to the marginal distributions of the order statistics of the SINR process, a more general relation is presented, connecting the factorial moment measures of the SINR process to the joint densities of these order statistics. This gives a way for calculating the exact values of the coverage probabilities arising in a general scenario of signal combination and interference cancellation between base stations. The presented framework consisting of the mathematical representations of SINR characteristics with respect to the factorial moment measures holds for the whole domain of SINR, and is amenable to considerable model extension.

Geographic locations of cellular base stations sometimes can be well fitted with spatial homogeneous Poisson point processes. In this paper, we make a complementary observation. In the presence of the log-normal shadowing of sufficiently high variance, the statistics of the propagation loss of a single user with respect to different network stations are invariant with respect to their geographic positioning, whether regular or not, for a wide class of empirically homogeneous networks. Even in a perfectly hexagonal case they appear as though they were realized in a Poisson network model, i.e., form an inhomogeneous Poisson point process on the positive half-line with a power-law density characterized by the path-loss exponent. At the same time, the conditional distances to the corresponding base stations, given their observed propagation losses, become independent and log-normally distributed, which can be seen as a decoupling between the real and model geometry. The result applies also to the Suzuki (Rayleigh-log-normal) propagation model. We use the Kolmogorov-Smirnov test to empirically study the quality of the Poisson approximation and use it to build a linear-regression method for the statistical estimation of the value of the path-loss exponent.

In our basic model, we study a stationary Poisson pattern of nodes on a line embedded in an independent planar Poisson field of interfering nodes. Assuming slotted Aloha and the signal-to-interference-and-noise ratio capture condition, with the usual power-law path loss model and Rayleigh fading, we explicitly evaluate several local and end-to-end performance characteristics related to the nearest-neighbor packet relaying on this line, and study their dependence on the model parameters (the density of relaying and interfering nodes, Aloha tuning and the external noise power). Our model can be applied in two cases. The first use is for vehicular ad-hoc networks, where vehicles are randomly located on a straight road. The second use is to study a “typical” route traced in a (general) planar ad-hoc network by some routing mechanism. The approach we have chosen allows us to quantify the non-efficiency of long-distance routing in “pure ad-hoc” networks and evaluate a possible remedy for it in the form of additional “fixed” relaying nodes, called road-side units in a vehicular network. It also allows us to consider a more general field of interfering nodes and study the impact of the clustering of its nodes on the routing performance. As a special case of a field with more clustering than the Poison field, we consider a Poisson-line field of interfering nodes, in which all the nodes are randomly located on random straight lines. In this case, our analysis rigorously (in the sense of Palm theory) corresponds to the typical route of this network. The comparison to our basic model reveals a paradox: clustering of interfering nodes decreases the outage probability of a single (typical) transmission on the route, but increases the mean end-to-end delay.

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 assume a space-time Poisson process of call arrivals on the infinite plane, independently marked by data volumes and served by a cellular network modeled by an infinite ergodic point process of base stations. Each point of this point process represents the location of a base station that applies a processor sharing policy to serve users arriving in its vicinity, modeled by the Voronoi cell, possibly perturbed by some random signal propagation effects. User service rates depend on their signal-to-interference-and-noise ratios with respect to the serving station.

We present a new stochastic service model with capacity sharing and interruptions, appropriate for the evaluation of the quality of real-time streaming (e.g. mobile TV) in wireless cellular networks. It takes into account multi-class Markovian process of call arrivals (to capture different radio channel conditions, requested streaming bit-rates and call-durations) and allows for a general resource allocation policy saying which users are temporarily denied the requested fixed streaming bit-rates (put in outage) due to resource constraints. We develop general expressions for the performance characteristics of this model, including the mean outage duration and the mean number of outage incidents for a typical user of a given class, involving only the steady-state of the traffic demand. We propose also a natural class of least-effort-served-first resource allocation policies, which cope with optimality and fairness issues known in wireless networks, and whose performance metrics can be easily calculated using Fourier analysis of Poisson variables. We specify and use our model to analyze the quality of real time streaming in 3GPP Long Term Evolution (LTE) cellular networks. Our results can be used for the dimensioning of these networks.

An almost ubiquitous assumption made in the stochastic-analytic approach to study of the quality of user-service in cellular networks is Poisson distribution of base stations, often completed by some specific assumption regarding the distribution of the fading (e.g. Rayleigh). The former (Poisson) assumption is usually (vaguely) justified in the context of cellular networks, by various irregularities in the real placement of base stations, which ideally should form a lattice (e.g. hexagonal) pattern. In the first part of this paper we provide a different and rigorous argument justifying the Poisson assumption under sufficiently strong lognormal shadowing observed in the network, in the evaluation of a natural class of the typical-user service-characteristics (including path-loss, interference, signal-to-interference ratio, spectral efficiency). Namely, we present a Poisson-convergence result for a broad range of stationary (including lattice) networks subject to log-normal shadowing of increasing variance. We show also for the Poisson model that the distribution of all these typical-user service characteristics does not depend on the particular form of the additional fading distribution. Our approach involves a mapping of 2D network model to 1D image of it “perceived” by the typical user. For this image we prove our Poisson convergence result and the invariance of the Poisson limit with respect to the distribution of the additional shadowing or fading. Moreover, in the second part of the paper we present some new results for Poisson model allowing one to calculate the distribution function of the SINR in its whole domain. We use them to study and optimize the mean energy efficiency in cellular networks.

We give numerically tractable, explicit integral expressions for the distribution of the signal-to-interference-and-noise-ratio (SINR) experienced by a typical user in the downlink channel from the k-th strongest base stations of a cellular network modelled by Poisson point process on the plane. Our signal propagation-loss model comprises of a power-law path-loss function with arbitrarily distributed shadowing, independent across all base stations, with and without Rayleigh fading. Our results are valid in the whole domain of SINR, in particular for SINR <; 1, where one observes multiple coverage. In this latter aspect our paper complements previous studies reported in [1].

Shadowing is believed to degrade the quality of service (QoS) in wireless cellular networks. Assuming log-normal shadowing, and studying mobile's path-loss with respect to the serving base station (BS) and the corresponding interference factor (the ratio of the sum of the path-gains form interfering BS's to the path-gain from the serving BS), which are two key ingredients of the analysis and design of the cellular networks, we discovered a more subtle reality. We observe, as commonly expected, that a strong variance of the shadowing increases the mean path-loss with respect to the serving BS, which in consequence, may compromise QoS. However, in some cases, an increase of the variance of the shadowing can significantly reduce the mean interference factor and, in consequence, improve some QoS metrics in interference limited systems, provided the handover policy selects the BS with the smallest path loss as the serving one. We exemplify this phenomenon, similar to stochastic resonance and related to the "single big jump principle" of the heavy-tailed log-nornal distribution, studying the blocking probability in regular, hexagonal networks in a semi-analytic manner, using a spatial version of the Erlang's loss formula combined with Kaufman-Roberts algorithm. More detailed probabilistic analysis explains that increasing variance of the log-normal shadowing amplifies the ratio between the strongest signal and all other signals thus reducing the interference. The above observations might shed new light, in particular on the design of indoor communication scenarios.

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 propose two analytically tractable stochastic models of non-slotted Aloha for Mobile Ad-hoc NETworks (MANETs): one model assumes a static pattern of nodes while the other assumes that the pattern of nodes varies over time. Both models feature transmitters randomly located in the Euclidean plane, according to a Poisson point process with the receivers randomly located at a fixed distance from the emitters. We concentrate on the so-called outage scenario, where a successful transmission requires a Signal-to-Interference-and-Noise Ratio (SINR) larger than a given threshold. With Rayleigh fading and the SINR averaged over the duration of the packet transmission, both models lead to closed form expressions for the probability of successful transmission. We show an excellent matching of these results with simulations. Using our models we compare the performances of non-slotted Aloha to previously studied slotted Aloha. We observe that when the path loss is not very strong both models, when appropriately optimized, exhibit similar performance. For stronger path loss non-slotted Aloha performs worse than slotted Aloha, however when the path loss exponent is equal to 4 its density of successfully received packets is still 75% of that in the slotted scheme. This is still much more than the 50% predicted by the well-known analysis where simultaneous transmissions are never successful. Moreover, in any path loss scenario, both schemes exhibit the same energy efficiency.

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.

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.

We consider a hybrid wireless sensor network with regular and transmit-only sensors. The transmit-only sensors do not have the receiver circuit (or have a very low data-rate one), hence are cheaper and less energy consuming, but their transmissions cannot be coordinated. Regular sensors, also called cluster-heads, are responsible for receiving information from the transmit-only sensors and forwarding it to sinks. The main goal of such a hybrid network is to reduce the cost of deployment while achieving some performance goals (minimum coverage, sensing rate, etc). In this paper we are interested in the communication between the transmit-only sensors and the cluster-heads. Since the sensors have no feedback, their transmission schedule is random. The cluster-heads, on the contrary, adapt their reception policy to achieve the performance goals. Using a mathematical model of random access networks developed in [1] we define and evaluate packet admission policies for different performance criteria. We show that the proposed hybrid network architecture, using the optimal policies, can achieve substantial dollar cost and power consumption savings as compared to conventional architectures while providing the same performance guarantees.

This paper deals with the performance evaluation of some congestion control schemes for elastic traffic in wireless cellular networks with power allocation/control. These schemes allow us to identify the feasible configurations of instantaneous up-and downlink bit-rates of users; i.e., such that can be obtained by allocating respective powers, taking into account in an exact way the interference created in the whole, multicellular network. We consider the bit-rate configurations identified by these schemes as feasible sets for some classical, maximal fair resource allocation policies, and study their performance in the long-term evolution of the system. Specifically, we assume Markovian arrivals, departures and mobility of customers, which transmit some given data-volumes, as well as some temporal channel variability (fading), and study the mean number of users, the mean throughput i.e., the mean bit-rates, and the mean delay that these policies offer in different parts of a given cell. Explicit formulas are obtained in the case of proportional fair policies, which may or may-not take advantage of the fading, for null or infinitely rapid customer mobility. This approach applies also to a channel shared by the elastic traffic and a streaming, with predefined customer bit-rates, regulated by the respective admission policy.

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.

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 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.