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.

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.

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.

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.

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.

The objective of the present paper is to build analytical methods for the evaluation of the quality of service (QoS) perceived by the users in the downlink of OFDMA wireless cellular networks serving streaming and elastic traffic. To do so, we first describe the resource (power and bandwidth) allocation problem and characterize its feasibility by some reference feasibility condition (FC). The QoS for FC may only by evaluated by simulations. To cope with this difficulty, we propose some sufficient feasibility condition (SFC) having the multi-Erlangform which permits analytical evaluation of the QoS. In particular, the blocking probability for streaming users can be calculated using Kaufman-Roberts algorithm. For elastic users, explicit expressions of the throughputs are obtained by using a multi-class processor sharing model. Moreover, we study the QoS in a network serving simultaneously streaming and elastic traffic. We validate this approach by comparing SFC's blocking probabilities to these simulated for FC. Moreover, we illustrate the proposed approach by solving the dimensioning problem; i.e., evaluating the minimal density of base stations assuring acceptable QoS of a given traffic demand per surface unit.

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.

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.