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Ergodic Spectral Efficiency in MIMO Cellular Networks
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Abstract
This paper shows how the application of stochastic geometry to the analysis of wireless networks is greatly facilitated by (i) a clear separation of time scales, (ii) the abstraction of small-scale effects via ergodicity, and (iii) an interference model that reflects the receiver's lack of knowledge of how each individual interference term is faded. These procedures render the analysis both more manageable and more precise, as well as more amenable to the incorporation of subsequent features. In particular, the paper presents analytical characterizations of the ergodic spectral efficiency of cellular networks with single-user multiple-input multiple-output (MIMO) and sectorization. These characterizations, in the form of easy-to-evaluate expressions, encompass the coverage, the distribution of spectral efficiency over the network locations, and the average thereof.
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This paper presents a tutorial on stochastic geometry (SG) based analysis for cellular networks. This tutorial is distinguished by its depth with respect to wireless communication details and its focus on cellular networks. The paper starts by modeling and analyzing the baseband interference in a basic cellular network model. Then, it characterizes signal-to-interference-plus-noise-ratio (SINR) and its related performance metrics. In particular, a unified approach to conduct error probability, outage probability, and rate analysis is presented. Although the main focus of the paper is on cellular networks, the presented unified approach applies for other types of wireless networks that impose interference protection around receivers. The paper then extends the baseline unified approach to capture cellular network characteristics (e.g., frequency reuse, multiple antenna, power control, etc.). It also presents numerical examples associated with demonstrations and discussions. Finally, we point out future research directions.
This paper derives Gaussian approximation bounds for the standardized aggregate wireless interference (AWI) in the downlink of K-tier heterogeneous cellular networks when base stations in each tier are distributed over the plane according to a (possibly non-homogeneous) Poisson process. The proposed methodology is general enough to account for general bounded path-loss models and fading statistics. The deviations of the distribution of the standardized AWI from the standard normal distribution are measured in terms of the Kolmogorov-Smirnov distance. An explicit expression bounding the Kolmogorov-Smirnov distance between these two distributions is obtained as a function of a broad range of network parameters such as per-tier transmission power levels, base station locations, fading statistics and the path-loss model. A simulation study is performed to corroborate the analytical results. In particular, a good statistical match between the standardized AWI distribution and its normal approximation occurs even for moderately dense heterogeneous cellular networks. These results are expected to have important ramifications on the characterization of performance upper and lower bounds for emerging 5G network architectures.
We derive a general and closed-form result for the success probability in
multiple-antenna (MIMO) heterogeneous cellular networks (HetNets), utilizing a
novel Toeplitz matrix representation. This main result, which is equivalently
the signal-to-interference ratio (SIR) distribution, includes multiuser MIMO,
single-user MIMO and per-tier biasing for K different tiers of randomly
placed base stations (BSs). The large SIR limit of this result admits a simple
closed form that is accurate at moderate SIRs, e.g., above 5-10 dB. These
results reveal that the SIR-invariance property of SISO HetNets does not hold
for MIMO HetNets; instead the success probability may decrease as the network
density increases. We prove that the maximum success probability is achieved by
activating only one tier of BSs, while the maximum area spectral efficiency
(ASE) is achieved by activating all the BSs. This reveals a unique tradeoff
between the ASE and link reliability in multiuser MIMO HetNets. To achieve the
maximum ASE while guaranteeing a certain link reliability, we develop efficient
algorithms to find the optimal BS densities. It is shown that as the link
reliability requirement increases, more BSs and more tiers should be
deactivated.
This paper aims to validate the -Ginibre point process as a model for
the distribution of base station locations in a cellular network. The
-Ginibre is a repulsive point process in which repulsion is controlled
by the parameter. When tends to zero, the point process
converges in law towards a Poisson point process. If equals to one it
becomes a Ginibre point process. Simulations on real data collected in Paris
(France) show that base station locations can be fitted with a -Ginibre
point process. Moreover we prove that their superposition tends to a Poisson
point process as it can be seen from real data. Qualitative interpretations on
deployment strategies are derived from the model fitting of the raw data.
It has recently been observed that the SIR distributions of a variety of
cellular network models and transmission techniques look very similar in shape.
As a result, they are well approximated by a simple horizontal shift (or gain)
of the distribution of the most tractable model, the Poisson point process
(PPP). To study and explain this behavior, this paper focuses on general
single-tier network models with nearest-base station association and studies
the asymptotic gain both at 0 and at infinity.
We show that the gain at 0 is determined by the so-called mean
interference-to-signal ratio (MISR) between the PPP and the network model under
consideration, while the gain at infinity is determined by the expected
fading-to-interference ratio (EFIR).
The analysis of the MISR is based on a novel type of point process, the
so-called relative distance process, which is a one-dimensional point process
on the unit interval [0,1] that fully determines the SIR. A comparison of the
gains at 0 and infinity shows that the gain at 0 indeed provides an excellent
approximation for the entire SIR distribution. Moreover, the gain is mostly a
function of the network geometry and barely depends on the path loss exponent
and the fading. The results are illustrated using several examples of repulsive
point processes.
Capitalizing on the analytical potency of stochastic geometry and on some new ideas to model intercell interference, this paper presents analytical expressions that enable quantifying the spectral efficiency of interference alignment (IA) in cellular networks without the need for simulation. From these expressions, the benefits of IA are characterized. Even under favorable assumptions, IA is found to be beneficial only in very specific and relatively infrequent network situations, and a blanket utilization of IA is found to be altogether detrimental. Applied only in the appropriate situations, IA does bring about benefits that are significant for the users involved but relatively small in terms of average spectral efficiency for the entire system.
We model and analyze heterogeneous cellular networks with multiple antenna BSs (multi-antenna HetNets) with K classes or tiers of base stations (BSs), which may differ in terms of transmit power, deployment density, number of transmit antennas, number of users served, transmission scheme, and path loss exponent. We show that the cell selection rules in multi-antenna HetNets may differ significantly from the single-antenna HetNets due to the possible differences in multi-antenna transmission schemes across tiers. While it is challenging to derive exact cell selection rules even for maximizing signal-to-interference-plus-noise-ratio (SINR) at the receiver, we show that adding an appropriately chosen tier-dependent cell selection bias in the received power yields a close approximation. Assuming arbitrary selection bias for each tier, simple expressions for downlink coverage and rate are derived. For coverage maximization, the required selection bias for each tier is given in closed form. Due to this connection with biasing, multi-antenna HetNets may balance load more naturally across tiers in certain regimes compared to single-antenna HetNets, where a large cell selection bias is often needed to offload traffic to small cells.
We study multiple-input multiple-output (MIMO) based downlink heterogeneous
cellular network (HetNets) with joint transmit-receive diversity using
orthogonal space-time block codes at the base stations (BSs) and maximal-ratio
combining (MRC) at the users. MIMO diversity with MRC is especially appealing
in cellular networks due to the relatively low hardware complexity at both the
BS and user device. Using stochastic geometry, we develop a tractable
stochastic model for analyzing such HetNets taking into account the irregular
and multi-tier BS deployment. We derive the coverage probability for both
interference-blind (IB) and interference-aware (IA) MRC as a function of the
relevant tier-specific system parameters such as BS density and power, path
loss law, and number of transmit (Tx) antennas. Important insights arising from
our analysis for typical HetNets are for instance: (i) IA-MRC becomes less
favorable than IB-MRC with Tx diversity due to the smaller interference
variance and increased interference correlation across Rx antennas; (ii)
ignoring spatial interference correlation significantly overestimates the
performance of IA-MRC; (iii) for small number of Rx antennas, selection
combining may offer a better complexity-performance trade-off than MRC.
We introduce a simple yet powerful and versatile analytical framework to
approximate the SIR distribution in the downlink of cellular systems. It is
based on the mean interference-to-signal ratio and yields the horizontal gap
(SIR gain) between the SIR distribution in question and a reference SIR
distribution. As applications, we determine the SIR gain for base station
silencing, cooperation, and lattice deployment over a baseline architecture
that is based on a Poisson deployment of base stations and strongest-base
station association. The applications demonstrate that the proposed approach
unifies several recent results and provides a convenient framework for the
analysis and comparison of future network architectures and transmission
schemes, including amorphous networks where a user is served by multiple base
stations and, consequently, (hard) cell association becomes obsolete.
In cellular network models, the base stations are usually assumed to form a
lattice or a Poisson point process (PPP). In reality, however, they are
deployed neither fully regularly nor completely randomly. Accordingly, in this
paper, we consider the very general class of motion-invariant models and
analyze the behavior of the outage probability (the probability that the
signal-to-interference-plus-noise-ratio (SINR) is smaller than a threshold) as
the threshold goes to zero. We show that, remarkably, the slope of the outage
probability (in dB) as a function of the threshold (also in dB) is the same for
essentially all motion-invariant point processes. The slope merely depends on
the fading statistics. Using this result, we introduce the notion of the
asymptotic deployment gain (ADG), which characterizes the horizontal gap
between the success probabilities of the PPP and another point process in the
high-reliability regime (where the success probability is near 1). To
demonstrate the usefulness of the ADG for the characterization of the SINR
distribution, we investigate the outage probabilities and the ADGs for
different point processes and fading statistics by simulations.
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.
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.
Inter-cell interference coordination (ICIC) and intra-cell diversity (ICD)
play important roles in improving cellular downlink coverage. Modeling cellular
base stations (BSs) as a homogeneous Poisson point process (PPP), this paper
provides explicit finite-integral expressions for the coverage probability with
ICIC and ICD, taking into account the temporal/spectral correlation of the
signal and interference. In addition, we show that in the high-reliability
regime, where the user outage probability goes to zero, ICIC and ICD affect the
network coverage in drastically different ways: ICD can provide order gain
while ICIC only offers linear gain. In the high-spectral efficiency regime
where the SIR threshold goes to infinity, the order difference in the coverage
probability does not exist, however the linear difference makes ICIC a better
scheme than ICD for realistic path loss exponents. Consequently, depending on
the SIR requirements, different combinations of ICIC and ICD optimize the
coverage probability.
The spatial structure of transmitters in wireless networks plays a key role
in evaluating the mutual interference and hence the performance. Although the
Poisson point process (PPP) has been widely used to model the spatial
configuration of wireless networks, it is not suitable for networks with
repulsion. The Ginibre point process (GPP) is one of the main examples of
determinantal point processes that can be used to model random phenomena where
repulsion is observed. Considering the accuracy, tractability and
practicability tradeoffs, we introduce and promote the -GPP, an
intermediate class between the PPP and the GPP, as a model for wireless
networks when the nodes exhibit repulsion. To show that the model leads to
analytically tractable results in several cases of interest, we derive the mean
and variance of the interference using two different approaches: the Palm
measure approach and the reduced second moment approach, and then provide
approximations of the interference distribution by three known probability
density functions. Besides, to show that the model is relevant for cellular
systems, we derive the coverage probability of the typical user and also find
that the fitted -GPP can closely model the deployment of actual base
stations in terms of the coverage probability and other statistics.
New research directions will lead to fundamental changes in the design of future fifth generation (5G) cellular networks. This article describes five technologies that could lead to both architectural and component disruptive design changes: device-centric architectures, millimeter wave, massive MIMO, smarter devices, and native support for machine-to-machine communications. The key ideas for each technology are described, along with their potential impact on 5G and the research challenges that remain.
This paper presents a tractable model for analyzing non-coherent
joint-transmission base station (BS) cooperation, taking into account the
irregular BS deployment typically encountered in practice. Besides
cellular-network specific aspects such as BS density, channel fading, average
path loss and interference, the model also captures relevant cooperation
mechanisms including user-centric BS clustering and channel-dependent
scheduling. The locations of all BSs are modeled by a Poisson point process.
Using tools from stochastic geometry, the signal-to-interference-plus-noise
ratio (SINR) distribution with cooperation is precisely characterized in a
generality-preserving form. The result is then applied to practical design
problems of recent interest. We find that increasing the network-wide BS
density improves the SINR, while the gains increase with the path loss
exponent. For pilot-based channel estimation, the averaged spectral efficiency
saturates at cluster sizes of around 7 BSs for typical values, irrespective of
backhaul quality. Finally, it is shown that intra-cluster frequency reuse is
favorable in moderately-loaded cells with generous triggering of a
joint-transmission, while intra-cluster coordinated scheduling may be better in
lightly-loaded cells with conservative triggering of a joint transmission.
Pushing data traffic from cellular to WiFi is an example of inter radio
access technology (RAT) offloading. While this clearly alleviates congestion on
the over-loaded cellular network, the ultimate potential of such offloading and
its effect on overall system performance is not well understood. To address
this, we develop a general and tractable model that consists of M different
RATs, each deploying up to K different tiers of access points (APs), where
each tier differs in transmit power, path loss exponent, deployment density and
bandwidth. Each class of APs is modeled as an independent Poisson point process
(PPP), with mobile user locations modeled as another independent PPP, all
channels further consisting of i.i.d. Rayleigh fading. The distribution of rate
over the entire network is then derived for a weighted association strategy,
where such weights can be tuned to optimize a particular objective. We show
that the optimum fraction of traffic offloaded to maximize \SINR coverage is
not in general the same as the one that maximizes rate coverage, defined as the
fraction of users achieving a given rate.
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.
In wireless networks with random node distribution, the underlying point process model and the channel fading process are usually considered separately. A unified framework is introduced that permits the geometric characterization of fading by incorporating the fading process into the point process model. Concretely, assuming nodes are distributed in a stationary Poisson point process in Rd , the properties of the point processes that describe the path loss with fading are analyzed. The main applications are single-hop connectivity and broadcasting.
The performance of wireless networks depends critically on their spatial configuration, because received signal power and interference depend critically on the distances between numerous transmitters and receivers. This is particularly true in emerging network paradigms that may include femtocells, hotspots, relays, white space harvesters, and meshing approaches, which are often overlaid with traditional cellular networks. These heterogeneous approaches to providing high-capacity network access are characterized by randomly located nodes, irregularly deployed infrastructure, and uncertain spatial configurations due to factors like mobility and unplanned user-installed access points. This major shift is just beginning, and it requires new design approaches that are robust to spatial randomness, just as wireless links have long been designed to be robust to fading. The objective of this article is to illustrate the power of spatial models and analytical techniques in the design of wireless networks, and to provide an entry-level tutorial.
This paper analyzes the carrier-to-interference ratio of the so-called shotgun cellular system (SCS). In the SCS, base-stations are placed randomly according to a two-dimensional Poisson point process. Such a system can model a dense cellular or wireless data network deployment, where the base station locations end up being close to random due to constraints other than optimal coverage. The SCS is a simple cellular system where we can introduce several variations and design scenarios such as shadow fading, power control features, and multiple channel reuse groups, and assess their impact on the performance. We first derive an analytical expression for the characteristic function of the inverse of the carrier-to-interference ratio. Using this result, we show that the carrier-to-interference ratio is independent of the base station density and further, we derive a semi-analytical expression for the tail-probability. These results enable a complete characterization of the cellular performance of the SCS. Next, we incorporate shadow fading into the SCS and demonstrate that it merely scales the base station density by a constant. Hence, the cellular performance of the SCS is independent of shadow fading. These results are further used to analyze dense cellular scenarios.
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 that of the Poisson bipolar model introduced in [1], which is a natural model for ad-hoc and device-to-device (D2D) networks. The primary finding is that, with 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, 2) the density of nodes and 3) the path-loss exponent. This linear gain is achieved by using the transmit antennas to send multiple data streams (e.g. through an open-loop 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 using a successive interference cancellation method. A major implication of the derived scaling laws is that spatial multiplexing transmission methods are essential for obtaining better and eventually optimal scaling laws in multiple antenna random wireless networks. Simulation results support this analysis.
One of the principal underlying assumptions of current approaches to the analysis of heterogeneous cellular networks (HetNets) with random spatial models is the uniform distribution of users independent of the base station (BS) locations. This assumption is not quite accurate, especially for user-centric capacity-driven small cell deployments where low-power BSs are deployed in the areas of high user density, thus inducing a natural correlation in the BS and user locations. In order to capture this correlation, we enrich the existing K-tier Poisson Point Process (PPP) HetNet model by considering user locations as Poisson Cluster Process (PCP) with the BSs at the cluster centers. In particular, we provide the formal analysis of the downlink coverage probability in terms of a general density functions describing the locations of users around the BSs. The derived results are specialized for two cases of interest: (i) Thomas cluster process, where the locations of the users around BSs are Gaussian distributed, and (ii) Mat\'ern cluster process, where the users are uniformly distributed inside a disc of a given radius. Tight closed-form bounds for the coverage probability in these two cases are also derived. Our results demonstrate that the coverage probability decreases as the size of user clusters around BSs increases, ultimately collapsing to the result obtained under the assumption of PPP distribution of users independent of the BS locations when the cluster size goes to infinity. Using these results, we also handle mixed user distributions consisting of two types of users: (i) uniformly distributed, and (ii) clustered around certain tiers.
We consider the point process of signal strengths emitted from transmitters in a wireless network and observed at a fixed position. In our model, transmitters are placed deterministically or randomly according to a hard core or Poisson point process and signals are subjected to power law path loss and random propagation effects that may be correlated between transmitters. We provide bounds on the distance between the point process of signal strengths and a Poisson process with the same mean measure, assuming correlated log-normal shadowing. For "strong shadowing" and moderate correlations, we find that the signal strengths are close to a Poisson process, generalizing a recently shown analogous result for independent shadowing.
This paper presents a unified mathematical paradigm, based on stochastic geometry, for downlink cellular networks with multiple-input-multiple-output (MIMO) base stations (BSs). The developed paradigm accounts for signal retransmission upon decoding errors, in which the temporal correlation among the signal-to-interference plus-noise-ratio (SINR) of the original and retransmitted signals is captured. In addition to modeling the effect of retransmission on the network performance, the developed mathematical model presents twofold analysis unification for MIMO cellular networks literature. First, it integrates the tangible decoding error probability and the abstracted (i.e., modulation scheme and receiver type agnostic) outage probability analysis, which are largely disjoint in the literature. Second, it unifies the analysis for different MIMO configurations. The unified MIMO analysis is achieved by abstracting unnecessary information conveyed within the interfering signals by Gaussian signaling approximation along with an equivalent SISO representation for the per-data stream SINR in MIMO cellular networks. We show that the proposed unification simplifies the analysis without sacrificing the model accuracy. To this end, we discuss the diversity-multiplexing tradeoff imposed by different MIMO schemes and shed light on the diversity loss due to the temporal correlation among the SINRs of the original and retransmitted signals. Finally, several design insights are highlighted.
This paper combines Poisson Cluster Process (PCP) with a Poisson Hole Process (PHP) to develop a new spatial model for an inband device-to-device (D2D) communications network, where D2D and cellular transmissions share the same spectrum. The locations of the devices engaging in D2D communications are modeled by a modified Thomas cluster process in which the cluster centers are modeled by a PHP instead of more popular homogeneous Poisson Point Process (PPP). While the clusters capture the inherent proximity in the devices engaging in D2D communications, the holes model exclusion zones where D2D communication is prohibited in order to protect cellular transmissions. For this setup, we characterize network performance in terms of coverage probability and area spectral efficiency.
In this paper, mathematical frameworks for system-level analysis and design of uplink heterogeneous cellular networks with multiple antennas at the base station (BS) are introduced. Maximum ratio combining (MRC) and optimum combining (OC) at the BSs are studied and compared. A generalized cell association criterion and fractional power control scheme are considered. The locations of all tiers of BSs are modeled as points of homogeneous and independent Poisson point processes. With the aid of stochastic geometry, coverage probability and average rate are formulated in integral but mathematically and computationally tractable expressions. Based on them, performance trends for small-and large-scale multiple-antenna BSs are discussed. Coverage and rate are shown to highly depend on several parameters, including the path-loss exponent, the fractional power control compensation factor, and the maximum transmit power of the mobile terminals. The gain of OC compared with MRC is proved to increase, if a more aggressive power control is used and if the number of BS antennas increases but is finite. For the same number of BS antennas, OC is shown to reach the noise-limited asymptote faster than MRC. All findings are validated via Monte Carlo simulations.
Explicit derivation of interferences in hexagonal wireless networks has been widely considered intractable and requires extensive computations with system level simulations. In this paper, we fundamentally tackle this problem and explicitly evaluate the downlink interference-to-signal ratio (ISR) for any mobile location m in a hexagonal wireless network, whether composed of omni-directional or tri-sectorized sites. The explicit formula of ISR is a very convergent series on m and involves the use of Gauss hypergeometric and Hurwitz Riemann zeta functions. Besides, we establish simple identities that well approximate this convergent series and turn out quite useful compared to other approximations in literature. The derived expression of ISR is easily extended to any frequency reuse pattern. Moreover, it is also exploited in the derivation of an explicit form of SINR distribution for any arbitrary distribution of mobile user locations, reflecting the spatial traffic density in the network. Knowing explicitly about interferences and SINR distribution is very useful information in capacity and coverage planning of wireless cellular networks and particularly for macro-cells' layer that forms almost a regular point pattern.
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.
Due to its tractability, a multitier model of mutually independent Poisson point processes (PPPs) for heterogeneous cellular networks (HCNs) has recently been attracting much attention. However, in reality, the locations of the BSs, within each tier and across tiers, are not fully independent. Accordingly, in this paper, we propose two HCN models with inter-tier dependence (Case 1) and intra-tier dependence (Case 2), respectively. In Case 1, the macro-base station (MBS) and the pico-base station (PBS) deployments follow a Poisson point process (PPP) and a Poisson hole process (PHP), respectively. Under this setup and conditioning on a fixed serving distance (distance between a user and its nearest serving BS), we derive bounds on the outage probabilities of both macro and pico users. We also use a fitted Poisson cluster process to approximate the PHP, which is shown to provide a good approximation of the interference and outage statistics. In Case 2, the MBSs and the PBSs follow a PPP and an independent Matern cluster process, respectively. Explicit expressions of the interference and the outage probability are derived first for fixed serving distance and second with random distance, and we derive the outage performance, the per-user capacity, and the area spectral efficiency (ASE) for both cases. It turns out that the proposed Case 2 model is a more appropriate and accurate model for a HCN with hotspot regions than the multitier independent PPP model since the latter underestimates some key performance metrics, such as the per-user capacity and the ASE, by a factor of 1.5 to 2. Overall, the two models proposed provide good tradeoffs between the accuracy, tractability, and practicability.
We consider the point process of signal strengths from transmitters in a
wireless network observed from a fixed position under models with general
signal path loss and random propagation effects. We show via coupling arguments
that under general conditions this point process of signal strengths can be
well-approximated by an inhomogeneous Poisson or a Cox point processes on the
positive real line. We also provide some bounds on the total variation distance
between the laws of these point processes and both Poisson and Cox point
processes. Under appropriate conditions, these results support the use of a
spatial Poisson point process for the underlying positioning of transmitters in
models of wireless networks, even if in reality the positioning does not appear
Poisson. We apply the results to a number of models with popular choices for
positioning of transmitters, path loss functions, and distributions of
propagation effects.
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 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 become independent and
log-normally distributed, which can be seen as a decoupling between the real
and model geometry. The result applies also to Suzuki (Rayleigh-log-normal)
propagation model. We use 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.
This paper presents an analytical framework that enables characterizing
analytically the spectral efficiency achievable by D2D (device-to-device)
communication links integrated within a cellular network. This framework is
based on a stochastic geometry formulation with a novel approach to the
modeling and spatial averaging of interference, which facilitates obtaining
compact expressions, and with the added possibility of incorporating exclusion
regions to protect cellular users from excessive interference from active D2D
transmitters. To illustrate the potential of the framework, a number of
examples are provided. These examples confirm the hefty potential of D2D
communication in situations of strong traffic locality as well as the
effectiveness of properly sized exclusion regions.
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.
For more than three decades, stochastic geometry has been used to model large-scale ad hoc wireless networks, and it has succeeded to develop tractable models to characterize and better understand the performance of these networks. Recently, stochastic geometry models have been shown to provide tractable yet accurate performance bounds for multi-tier and cognitive cellular wireless networks. Given the need for interference characterization in multi-tier cellular networks, stochastic geometry models provide high potential to simplify their modeling and provide insights into their design. Hence, a new research area dealing with the modeling and analysis of multi-tier and cognitive cellular wireless networks is increasingly attracting the attention of the research community. In this article, we present a comprehensive survey on the literature related to stochastic geometry models for single-tier as well as multi-tier and cognitive cellular wireless networks. A taxonomy based on the target network model, the point process used, and the performance evaluation technique is also presented. To conclude, we discuss the open research challenges and future research directions.
The spatial structure of base stations (BSs) in cellular networks plays a key role in evaluating the downlink performance. In this paper, different spatial stochastic models (the Poisson point process (PPP), the Poisson hard-core process (PHCP), the Strauss process (SP), and the perturbed triangular lattice) are used to model the structure by fitting them to the locations of BSs in real cellular networks obtained from a public database. We provide two general approaches for fitting. One is fitting by the method of maximum pseudolikelihood. As for the fitted models, it is not sufficient to distinguish them conclusively by some classical statistics. We propose the coverage probability as the criterion for the goodness-of-fit. In terms of coverage, the SP provides a better fit than the PPP and the PHCP. The other approach is fitting by the method of minimum contrast that minimizes the average squared error of the coverage probability. This way, fitted models are obtained whose coverage performance matches that of the given data set very accurately. Furthermore, we introduce a novel metric, the deployment gain, and we demonstrate how it can be used to estimate the coverage performance and average rate achieved by a data set.
We develop a general downlink model for multi-antenna heterogeneous cellular networks (HetNets), where base stations (BSs) across tiers may differ in terms of transmit power, target signal-to-interference-ratio (SIR), deployment density, number of transmit antennas and the type of multi-antenna transmission. In particular, we consider and compare space division multiple access (SDMA), single user beamforming (SU-BF), and baseline single-input single-output (SISO) transmission. For this general model, the main contributions are: (i) ordering results for both coverage probability and per user rate in closed form for any BS distribution for the three considered techniques, using novel tools from stochastic orders, (ii) upper bounds on the coverage probability assuming a Poisson BS distribution, and (iii) a comparison of the area spectral efficiency (ASE). The analysis concretely demonstrates, for example, that for a given total number of transmit antennas in the network, it is preferable to spread them across many single-antenna BSs vs. fewer multi-antenna BSs. Another observation is that SU-BF provides higher coverage and per user data rate than SDMA, but SDMA is in some cases better in terms of ASE.
Cooperation is viewed as a key ingredient for interference management in wireless networks. This paper shows that cooperation has fundamental limitations. First, it is established that in systems that rely on pilot-assisted channel estimation, the spectral efficiency is upper-bounded by a quantity that does not depend on the transmit powers; in this framework, cooperation is possible only within clusters of limited size, which are subject to out-of-cluster interference whose power scales with that of the in-cluster signals. Second, an upper bound is also shown to exist if the cooperation extends to an entire (large) system operating as a single cluster; here, pilot-assisted transmission is necessarily transcended. Altogether, it is concluded that cooperation cannot in general change an interference-limited network to a noise-limited one. Consequently, the existing literature that routinely assumes that the high-power spectral efficiency scales with the log-scale transmit power provides only a partial characterization. The complete characterization proposed in this paper subdivides the high-power regime into a degree-of-freedom regime, where the scaling with the log-scale transmit power holds approximately, and a saturation regime, where the spectral efficiency hits a ceiling that is independent of the power. Using a cellular system as an example, it is demonstrated that the spectral efficiency saturates at power levels of operational relevance.
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 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 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 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.
In this book modern algorithmic techniques for summation, most of which have been introduced within the last decade, are developed and carefully implemented in the computer algebra system Maple. The algorithms of Gosper, Zeilberger and Petkovsek on hypergeometric summation and recurrence equations and their q-analogues are covered, and similar algorithms on differential equations are considered. An equivalent theory of hyperexponential integration due to Almkvist and Zeilberger completes the book.
The combination of all results considered gives work with orthogonal polynomials and (hypergeometric type) special functions a solid algorithmic foundation. Hence, many examples from this very active field are given.
Consider a cognitive radio network with two types of users: primary users (PUs) and cognitive users (CUs), whose locations follow two independent Poisson point processes. The cognitive users follow the policy that a cognitive transmitter is active only when it is outside the primary user exclusion regions. We found that under this setup the active cognitive users form a point process called the Poisson hole process. Due to the interaction between the primary users and the cognitive users through exclusion regions, an exact calculation of the interference and the outage probability seems unfeasible. Instead, two different approaches are taken to tackle this problem. First, bounds for the interference (in the form of Laplace transforms) and the outage probability are derived, and second, it is shown how to use a Poisson cluster process to model the interference in this kind of network. Furthermore, the bipolar network model with different exclusion region settings is analyzed.
Recent results [1], [2] on the distribution of the downlink SINR in heterogeneous wireless networks assume that the serving base station (BS) for a given user (UE) location is either (a) the BS that is geographically nearest to the UE location [1], or (b) the one that has the highest received power at the UE location [2]. For (a), the distribution of the downlink SINR at an arbitrary UE location can be derived exactly. For (b), the best result for the cumulative distribution function (CDF) of the downlink SINR [2] is exact only for arguments that exceed unity. In this paper, we extend the results in [2] to derive an exact expression for the CDF of the downlink SINR at an arbitrary UE location in a multi-tier heterogeneous network when the serving BS is chosen according to (b). We then explore some interesting implications of the result for coverage probabilities.
Covering point process theory, random geometric graphs and coverage processes, this rigorous introduction to stochastic geometry will enable you to obtain powerful, general estimates and bounds of wireless network performance and make good design choices for future wireless architectures and protocols that efficiently manage interference effects. Practical engineering applications are integrated with mathematical theory, with an understanding of probability the only prerequisite. At the same time, stochastic geometry is connected to percolation theory and the theory of random geometric graphs and accompanied by a brief introduction to the R statistical computing language. Combining theory and hands-on analytical techniques with practical examples and exercises, this is a comprehensive guide to the spatial stochastic models essential for modelling and analysis of wireless network performance.
The Signal to Interference Plus Noise Ratio (SINR) on a wireless link is an important basis for consideration of outage, capacity, and throughput in a cellular network. It is therefore important to understand the SINR distribution within such networks, and in particular heterogeneous cellular networks, since these are expected to dominate future network deployments [1]. Until recently the distribution of SINR in hetero-geneous networks was studied almost exclusively via simulation, for selected scenarios representing pre-defined arrangements [2] of users and the elements of the heterogeneous network such as macro-cells, femto-cells, etc. However, the dynamic nature of heterogeneous networks makes it difficult to design a few rep-resentative simulation scenarios from which general inferences can be drawn that apply to all deployments. In this paper, we examine the downlink of a heterogeneous cellular network made up of multiple tiers of transmitters (e.g., macro-, micro-, pico-, and femto-cells) and provide a general theoretical analysis of the distribution of the SINR at an arbitrarily-located user. Using physically realistic stochastic models for the locations of the base stations (BSs) in the tiers, we can compute the general SINR distribution in closed form. We illustrate a use of this approach for a three-tier network by calculating the probability of the user being able to camp on a macro-cell or an open-access (OA) femto-cell in the presence of Closed Subscriber Group (CSG) femto-cells. We show that this probability depends only on the relative densities and transmit powers of the macro-and femto-cells, the fraction of femto-cells operating in OA vs. Closed Subscriber Group (CSG) mode, and on the parameters of the wireless channel model. For an operator considering a femto overlay on a macro network, the parameters of the femto deployment can be selected from a set of universal curves.
The interference factor, defined for a given location in the network as the ratio of the sum of the path-gains form interfering base-stations (BS) to the path-gain from the serving BS is an important ingredient in the analysis of wireless cellular networks. It depends on the geometric placement of the BS in the network and the propagation gains between these stations and the given location. In this paper we study the mean interference factor taking into account the impact of these two elements. Regarding the geometry, we consider both the perfect hexagonal grid of BS and completely random Poisson pattern of BS. Regarding the signal propagation model, we consider not only a deterministic, signal-power-loss function that depends only on the distance between a transmitter and a receiver, and is mainly characterized by the so called path-loss exponent, but also random shadowing that characterizes in a statistical manner the way various obstacles on a given path modify this deterministic function. We present a detailed analysis of the impact of the path loss exponent, variance of the shadowing and the size of the network on the mean interference factor in the case of hexagonal and Poisson network architectures. We observe, as commonly expected, that small and moderate shadowing has a negative impact on regular networks as it increases the mean interference factor. However, as pointed out in the seminal paper, this impact can be largely reduced if the serving BS is chosen as the one which offers the smallest path-loss. Revisiting the model studied in this latter paper, we obtain a perhaps more surprising result saying that in large irregular (Poisson) networks the shadowing does not impact at all the interference factor, whose mean can be evaluated explicitly in a simple expression depending only on the path-loss exponent. Moreover, in small and moderate size networks, a very strong variability of the shadowing can be even beneficial in both hexagonal and Poisson networks-
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We present a mathematical model for communication subject to both network interference and noise. We introduce a framework where the interferers are scattered according to a spatial Poisson process, and are operating asynchronously in a wireless environment subject to path loss, shadowing, and multipath fading. We consider both cases of slow and fast-varying interferer positions. The paper is comprised of two separate parts. In Part I, we determine the distribution of the aggregate network interference at the output of a linear receiver. We characterize the error performance of the link, in terms of average and outage probabilities. The proposed model is valid for any linear modulation scheme (e.g., M-ary phase shift keying or M-ary quadrature amplitude modulation), and captures all the essential physical parameters that affect network interference. Our work generalizes the conventional analysis of communication in the presence of additive white Gaussian noise and fast fading, allowing such results to account for the effect of network interference. In Part II of the paper, we derive the capacity of the link when subject to network interference and noise, and characterize the spectrum of the aggregate interference.
A contemporary perspective on transmit antenna diversity and spatial multiplexing is provided. It is argued that, in the context of most modern wireless systems and for the operating points of interest, transmission techniques that utilize all available spatial degrees of freedom for multiplexing outperform techniques that explicitly sacrifice spatial multiplexing for diversity. Reaching this conclusion, however, requires that the channel and some key system features be adequately modeled and that suitable performance metrics be adopted; failure to do so may bring about starkly different conclusions. As a specific example, this contrast is illustrated using the 3GPP long-term evolution system design.
The analysis of flat-fading channels is often performed under the assumption that the additive noise is white and Gaussian, and that the receiver has precise knowledge of the realization of the fading process. These assumptions imply the optimality of Gaussian codebooks and of scaled nearest-neighbor decoding. Here we study the robustness of this communication scheme with respect to errors in the estimation of the fading process. We quantify the degradation in performance that results from such estimation errors, and demonstrate the lack of robustness of this scheme. For some situations we suggest the rule of thumb that, in order to avoid degradation, the estimation error should be negligible compared to the reciprocal of the signal-to-noise ratio (SNR)