BZPEERPapers, people, evidence and research paths
Menu

Discover

SearchFind papers, people, topics, institutions and opportunities from one place.GraphSee how papers, authors, topics and evidence connect.

Research tools

HomeContinue from the papers, people and topics that matter now.CollectionsTurn saved papers into reading lists, evidence maps and shareable context.ResearchReconstruct the path that led you from search to paper to question.CollaborationMove from a paper, topic or expert request into a focused thread.PublishCreate full-text research articles with metadata, formulas and versions.

Network

ExpertsFind specialists whose work and reviews explain why they can help.CommunitiesJoin research groups organized around papers, topics and shared evidence.

Opportunities

ListingsBrowse roles, calls and collaborations connected to real research context.

Account

Log inReturn to your saved papers, graph views and active threads.Create accountStart saving papers, building a research trail and joining teams.
Log inCreate account

Source author record

Radha Krishna Ganti

Radha Krishna Ganti appears in the imported research catalog. Authorship, coauthor and topic links are available while profile ownership is still unclaimed.

ResearcherUnclaimed source record
Information Theorymath.ITNetworking and Internet ArchitectureApplicationsmath.PRArtificial IntelligenceDatabaseseess.SPHardware ArchitectureMachine Learningmath.STStatistics Theory

Catalog footprint

What is connected

24works
12topics
4close collaborators

Actions

Connect this record

Log in to claim
Open graphBrowse works

Research graph

See the researcher in context

Open full explorer

Inspect adjacent papers, topics, institutions and collaborators without losing the researcher page.

Building this map preview

BZPEER is loading the nearby papers, people, topics and institutions for this page.

Published work

24 published item(s)

preprint2026arXiv

Enabling AI-Native Mobility in 6G: A Real-World Dataset for Handover, Beam Management, and Timing Advance

To address the issues of high interruption time and measurement report overhead under user equipment (UE) mobility especially in high speed 5G use cases the use of AI/ML techniques (AI/ML beam management and mobility procedures) have been proposed. These techniques rely heavily on data that are most often simulated for various scenarios and do not accurately reflect real deployment behavior or user traffic patterns. Therefore, there is an utmost need for realistic datasets under various conditions. This work presents a dataset collected from a commercially deployed network across various modes of mobility (pedestrian, bike, car, bus, and train) and at multiple speeds to depict real time UE mobility. When collecting the dataset, we focused primarily on handover (HO) scenarios, with the aim of reducing the HO interruption time and maintaining continuous throughput during and immediately after HO execution. To support this research, the dataset includes timing advance (TA) measurements at various signaling events such as RACH trigger, MAC CE, and PDCCH grant which are typically missing in existing works. We cover a detailed description of the creation of the dataset; experimental setup, data acquisition, and extraction. We also cover an exploratory analysis of the data, with a primary focus on mobility, beam management, and TA. We discuss multiple use cases in which the proposed dataset can facilitate understanding of the inference of the AI/ML model. One such use case is to train and evaluate various AI/ML models for TA prediction.

0 citations0 reviewsNo code linkedOpen access
preprint2016arXiv

A Linearization Technique for Self-Interference Cancellation in Full-Duplex Radios

The fundamental problem in the design of a full-duplex radio is the cancellation of the self-interference (SI) signal generated by the transmitter.Current techniques for suppressing SI rely on generating a copy of the SI signal and subtracting it partly in the RF (radio frequency) and digital domains. A critical step in replicating the self-interference is the estimation of the multi-path channel through which the transmitted signal propagates to the antenna. Since there is no prior model on the number of multipath reflections, current techniques assume a tap delay line filter (in the RF and digital domain) with a large number of taps, and estimate the taps in the analog and the digital domain. Assuming such a model leads to a large form-factor for the analog and RF circuits and increased complexity in the digital domain. In this paper, using a linearization technique, we show that the self-interference channel in an indoor environment can be effectively modelled as $H(f)=C_0 + C_1f$ in the frequency domain. Thus, the effective self-interference channel can be represented by two parameters $C_0$ and $C_1$, irrespective of the multipath environment. We also provide experimental evidence to verify the above channel model and propose novel low-complexity designs for self-interference cancellation. Linearization not only aids in the practicality of analog cancellation by reducing the form factor, but also results in a simpler SI filter model in the digital domain due to dimensionality reduction of the channel parameters. Therefore this method can enable the widespread adoption of full-duplex techniques to portable devices in addition to infrastructure base-stations.

0 citations0 reviewsNo code linkedOpen access
preprint2016arXiv

Adaptive Modulation with Impulsive Interference

In this letter, we analyze power and rate adaptation in a point-to-point link with Rayleigh fading and impulsive interference. We model the impulsive interference as a Bernoulli-Gaussian random process. Adaptation is used to maximize the average spectral efficiency by changing power and rate of the transmission subject to an average power and instantaneous probability of error constraints. Without impulsive interference, it is well known that water-filling is optimal for block fading. We provide two simple schemes that show that the conventional water-filling algorithm is not optimal in an impulsive interference channel.

0 citations0 reviewsNo code linkedOpen access
preprint2016arXiv

Coverage Analysis In Downlink Poisson Cellular Network With $κ$-$μ$ Shadowed Fading

The downlink coverage probability of a cellular network, when the base station locations are modelled by a Poisson point process (PPP), is known when the desired channel is Nakagami distributed with an integer shape parameter. However, for many interesting fading distributions such as Rician, Rician shadowing, $κ$-$μ$, $η$-$μ$, etc., the coverage probability is unknown. $κ$-$μ$ shadowed fading is a generic fading distribution whose special cases are many of these popular distributions known so far. In this letter, we derive the coverage probability when the desired channel experiences $κ$-$μ$ shadowed fading. Using numerical simulations, we verify our analytical expressions.

0 citations0 reviewsNo code linkedOpen access
preprint2016arXiv

Maximal Packing with Interference Constraints

In this work, we study the problem of scheduling a maximal set of transmitters subjected to an interference constraint across all the nodes. Given a set of nodes, the problem reduces to finding the maximum cardinality of a subset of nodes that can concurrently transmit without violating interference constraints. The resulting packing problem is a binary optimization problem and is NP hard. We propose a semi-definite relaxation (SDR) for this problem and provide bounds on the relaxation.

0 citations0 reviewsNo code linkedOpen access
preprint2016arXiv

SIR Asymptotics in General Network Models

In the performance analyses of wireless networks, asymptotic quantities and properties often pro- vide useful results and insights. The asymptotic analyses become especially important when complete analytical expressions of the performance metrics of interest are not available, which is often the case if one departs from very specific modeling assumptions. In this paper, we consider the asymptotics of the SIR distribution in general wireless network models, including ad hoc and cellular networks, simple and non-simple point processes, and singular and bounded path loss models, for which, in most cases, finding analytical expressions of the complete SIR distribution seems hopeless. We show that the lower tails of the SIR distributions decay polynomially with the order solely determined by the path loss exponent or the fading parameter, while the upper tails decay exponentially, with the exception of cellular networks with singular path loss. In addition, we analyze the impact of the nearest interferer on the asymptotic properties of the SIR distributions, and we formulate three crisp conjectures that -if true- determine the asymptotic behavior in many cases based on the large-scale path loss properties of the desired signal and/or nearest interferer only.

0 citations0 reviewsNo code linkedOpen access
preprint2015arXiv

Approximation of Capacity for ISI Channels with One-bit Output Quantization

Motivated by recent high bandwidth communication systems, Inter-Symbol Interference (ISI) channels with 1-bit quantized output are considered under an average-power-constrained continuous input. While the exact capacity is difficult to characterize, an approximation that matches with the exact channel output up to a probability of error is provided. The approximation does not have additive noise, but constrains the channel output (without noise) to be above a threshold in absolute value. The capacity under the approximation is computed using methods involving standard Gibbs distributions. Markovian achievable schemes approaching the approximate capacity are provided. The methods used over the approximate ISI channel result in ideas for practical coding schemes for ISI channels with 1-bit output quantization.

0 citations0 reviewsNo code linkedOpen access
preprint2015arXiv

Performance of Cloud Radio Networks

Cloud radio networks coordinate transmission among base stations (BSs) to reduce the interference effects, particularly for the cell-edge users. In this paper, we analyze the performance of a cloud network with static clustering where geographically close BSs form a cloud network of cooperating BSs. Because, of finite cooperation, the interference in a practical cloud radio cannot be removed and in this paper, the distance based interference is taken into account in the analysis. In particular, we consider centralized zero forcing equalizer and dirty paper precoding for cancelling the interference. Bounds are developed on the signal-to-interference ratio distribution and achievable rate with full and limited channel feedback from the cluster users. The adverse effect of finite clusters on the achievable rate is quantified. We show that, the number of cooperating BSs is more crucial than the cluster area when full channel state information form the cluster is available for precoding. Also, we study the impact of limiting the channel state information on the achievable rate. We show that even with a practically feasible feedback of about five to six channel states from each user, significant gain in mean rate and cell edge rate compared to conventional cellular systems can be obtained.

0 citations0 reviewsNo code linkedOpen access
preprint2015arXiv

Spatial CSMA: A Distributed Scheduling Algorithm for the SIR Model with Time-varying Channels

Recent work has shown that adaptive CSMA algorithms can achieve throughput optimality. However, these adaptive CSMA algorithms assume a rather simplistic model for the wireless medium. Specifically, the interference is typically modelled by a conflict graph, and the channels are assumed to be static. In this work, we propose a distributed and adaptive CSMA algorithm under a more realistic signal-to-interference ratio (SIR) based interference model, with time-varying channels. We prove that our algorithm is throughput optimal under this generalized model. Further, we augment our proposed algorithm by using a parallel update technique. Numerical results show that our algorithm outperforms the conflict graph based algorithms, in terms of supportable throughput and the rate of convergence to steady-state.

0 citations0 reviewsNo code linkedOpen access
preprint2014arXiv

Performance of PZF and MMSE Receivers in Cellular Networks with Multi-User Spatial Multiplexing

This paper characterizes the performance of cellular networks employing multiple antenna open-loop spatial multiplexing techniques. We use a stochastic geometric framework to model distance depended inter cell interference. Using this framework, we analyze the coverage and rate using two linear receivers, namely, partial zero-forcing (PZF) and minimum-mean-square-estimation (MMSE) receivers. Analytical expressions are obtained for coverage and rate distribution that are suitable for fast numerical computation. In the case of the PZF receiver, we show that it is not optimal to utilize all the receive antenna for canceling interference. With $α$ as the path loss exponent, $N_t$ transmit antenna, $N_r$ receive antenna, we show that it is optimal to use $N_t\left\lceil\left(1-\frac{2}α\right) \left( \frac{N_r}{N_t} -\frac{1}{2}\right)\right\rceil$ receive antennas for interference cancellation and the remaining antennas for signal enhancement (array gain). For both PZF and MMSE receivers, we observe that increasing the number of data streams provides an improvement in the mean data rate with diminishing returns. Also transmitting $N_r$ streams is not optimal in terms of the mean sum rate. We observe that increasing the SM rate with a PZF receiver always degrades the cell edge data rate while the performance with MMSE receiver is nearly independent of the SM rate.

0 citations0 reviewsNo code linkedOpen access
preprint2013arXiv

Load-Aware Modeling and Analysis of Heterogeneous Cellular Networks

Random spatial models are attractive for modeling heterogeneous cellular networks (HCNs) due to their realism, tractability, and scalability. A major limitation of such models to date in the context of HCNs is the neglect of network traffic and load: all base stations (BSs) have typically been assumed to always be transmitting. Small cells in particular will have a lighter load than macrocells, and so their contribution to the network interference may be significantly overstated in a fully loaded model. This paper incorporates a flexible notion of BS load by introducing a new idea of conditionally thinning the interference field. For a K-tier HCN where BSs across tiers differ in terms of transmit power, supported data rate, deployment density, and now load, we derive the coverage probability for a typical mobile, which connects to the strongest BS signal. Conditioned on this connection, the interfering BSs of the $i^{th}$ tier are assumed to transmit independently with probability $p_i$, which models the load. Assuming - reasonably - that smaller cells are more lightly loaded than macrocells, the analysis shows that adding such access points to the network always increases the coverage probability. We also observe that fully loaded models are quite pessimistic in terms of coverage.

0 citations0 reviewsNo code linkedOpen access
preprint2013arXiv

Towards Understanding the Fundamentals of Mobility in Cellular Networks

Despite the central role of mobility in wireless networks, analytical study on its impact on network performance is notoriously difficult. This paper aims to address this gap by proposing a random waypoint (RWP) mobility model defined on the entire plane and applying it to analyze two key cellular network parameters: handover rate and sojourn time. We first analyze the stochastic properties of the proposed model and compare it to two other models: the classical RWP mobility model and a synthetic truncated Levy walk model which is constructed from real mobility trajectories. The comparison shows that the proposed RWP mobility model is more appropriate for the mobility simulation in emerging cellular networks, which have ever-smaller cells. Then we apply the proposed model to cellular networks under both deterministic (hexagonal) and random (Poisson) base station (BS) models. We present analytic expressions for both handover rate and sojourn time, which have the expected property that the handover rate is proportional to the square root of BS density. Compared to an actual BS distribution, we find that the Poisson-Voronoi model is about as accurate in terms of mobility evaluation as hexagonal model, though being more pessimistic in that it predicts a higher handover rate and lower sojourn time.

0 citations0 reviewsNo code linkedOpen access
preprint2012arXiv

MIMO Interference Alignment in Random Access Networks

In this paper, we analyze a multiple-input multiple-output (MIMO) interference channel where nodes are randomly distributed on a plane as a spatial Poisson cluster point process. Each cluster uses interference alignment (IA) to suppress intra-cluster interference but unlike most work on IA, we do not neglect inter-cluster interference. We also connect the accuracy of channel state information to the distance between the nodes, i.e. the quality of CSI degrades with increasing distance. Accounting for the training and feedback overhead, we derive the transmission capacity of this MIMO IA ad hoc network and then compare it to open-loop (interference-blind) spatial multiplexing. Finally, we present exemplary system setups where spatial multiplexing outperforms IA due to the imperfect channel state information or the non-aligned inter-cluster interference.

0 citations0 reviewsNo code linkedOpen access
preprint2012arXiv

Modeling and Analysis of K-Tier Downlink Heterogeneous Cellular Networks

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

0 citations0 reviewsNo code linkedOpen access
preprint2012arXiv

Modeling Non-Uniform UE Distributions in Downlink Cellular Networks

A recent way to model and analyze downlink cellular networks is by using random spatial models. Assuming user equipment (UE) distribution to be uniform, the analysis is performed at a typical UE located at the origin. While this method of sampling UEs provides statistics averaged over the UE locations, it is not possible to sample cell interior and cell edge UEs separately. This complicates the problem of analyzing deployment scenarios involving non-uniform distribution of UEs, especially when the locations of the UEs and the base stations (BSs) are dependent. To facilitate this separation, we propose a new tractable method of sampling UEs by conditionally thinning the BS point process and show that the resulting framework can be used as a tractable generative model to study cellular networks with non-uniform UE distribution.

0 citations0 reviewsNo code linkedOpen access
preprint2011arXiv

A Tractable Approach to Coverage and Rate in Cellular Networks

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.

0 citations0 reviewsNo code linkedOpen access
preprint2011arXiv

Analytical Evaluation of Fractional Frequency Reuse for Heterogeneous Cellular Networks

Interference management techniques are critical to the performance of heterogeneous cellular networks, which will have dense and overlapping coverage areas, and experience high levels of interference. Fractional frequency reuse (FFR) is an attractive interference management technique due to its low complexity and overhead, and significant coverage improvement for low-percentile (cell-edge) users. Instead of relying on system simulations based on deterministic access point locations, this paper instead proposes an analytical model for evaluating Strict FFR and Soft Frequency Reuse (SFR) deployments based on the spatial Poisson point process. Our results both capture the non-uniformity of heterogeneous deployments and produce tractable expressions which can be used for system design with Strict FFR and SFR. We observe that the use of Strict FFR bands reserved for the users of each tier with the lowest average SINR provides the highest gains in terms of coverage and rate, while the use of SFR allows for more efficient use of shared spectrum between the tiers, while still mitigating much of the interference. Additionally, in the context of multi-tier networks with closed access in some tiers, the proposed framework shows the impact of cross-tier interference on closed access FFR, and informs the selection of key FFR parameters in open access.

0 citations0 reviewsNo code linkedOpen access
preprint2011arXiv

Analytical Evaluation of Fractional Frequency Reuse for OFDMA Cellular Networks

Fractional frequency reuse (FFR) is an interference management technique well-suited to OFDMA-based cellular networks wherein the cells are partitioned into spatial regions with different frequency reuse factors. To date, FFR techniques have been typically been evaluated through system-level simulations using a hexagonal grid for the base station locations. This paper instead focuses on analytically evaluating the two main types of FFR deployments - Strict FFR and Soft Frequency Reuse (SFR) - using a Poisson point process to model the base station locations. The results are compared with the standard grid model and an actual urban deployment. Under reasonable special cases for modern cellular networks, our results reduce to simple closed-form expressions, which provide insight into system design guidelines and the relative merits of Strict FFR, SFR, universal reuse, and fixed frequency reuse. We observe that FFR provides an increase in the sum-rate as well as the well-known benefit of improved coverage for cell-edge users. Finally, a SINR-proportional resource allocation strategy is proposed based on the analytical expressions, showing that Strict FFR provides greater overall network throughput at low traffic loads, while SFR better balances the requirements of interference reduction and resource efficiency when the traffic load is high.

0 citations0 reviewsNo code linkedOpen access
preprint2011arXiv

On the Throughput Cost of Physical Layer Security in Decentralized Wireless Networks

This paper studies the throughput of large-scale decentralized wireless networks with physical layer security constraints. In particular, we are interested in the question of how much throughput needs to be sacrificed for achieving a certain level of security. We consider random networks where the legitimate nodes and the eavesdroppers are distributed according to independent two-dimensional Poisson point processes. The transmission capacity framework is used to characterize the area spectral efficiency of secure transmissions with constraints on both the quality of service (QoS) and the level of security. This framework illustrates the dependence of the network throughput on key system parameters, such as the densities of legitimate nodes and eavesdroppers, as well as the QoS and security constraints. One important finding is that the throughput cost of achieving a moderate level of security is quite low, while throughput must be significantly sacrificed to realize a highly secure network. We also study the use of a secrecy guard zone, which is shown to give a significant improvement on the throughput of networks with high security requirements.

0 citations0 reviewsNo code linkedOpen access
preprint2011arXiv

Series Expansion for Interference in Wireless Networks

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

0 citations0 reviewsNo code linkedOpen access
preprint2010arXiv

High-SIR Transmission Capacity of Wireless Networks with General Fading and Node Distribution

In many wireless systems, interference is the main performance-limiting factor, and is primarily dictated by the locations of concurrent transmitters. In many earlier works, the locations of the transmitters is often modeled as a Poisson point process for analytical tractability. While analytically convenient, the PPP only accurately models networks whose nodes are placed independently and use ALOHA as the channel access protocol, which preserves the independence. Correlations between transmitter locations in non-Poisson networks, which model intelligent access protocols, makes the outage analysis extremely difficult. In this paper, we take an alternative approach and focus on an asymptotic regime where the density of interferers $η$ goes to 0. We prove for general node distributions and fading statistics that the success probability $\p \sim 1-γη^κ$ for $η\rightarrow 0$, and provide values of $γ$ and $κ$ for a number of important special cases. We show that $κ$ is lower bounded by 1 and upper bounded by a value that depends on the path loss exponent and the fading. This new analytical framework is then used to characterize the transmission capacity of a very general class of networks, defined as the maximum spatial density of active links given an outage constraint.

0 citations0 reviewsNo code linkedOpen access
preprint2010arXiv

Outage Probability of General Ad Hoc Networks in the High-Reliability Regime

Outage probabilities in wireless networks depend on various factors: the node distribution, the MAC scheme, and the models for path loss, fading and transmission success. In prior work on outage characterization for networks with randomly placed nodes, most of the emphasis was put on networks whose nodes are Poisson distributed and where ALOHA is used as the MAC protocol. In this paper we provide a general framework for the analysis of outage probabilities in the high-reliability regime. The outage probability characterization is based on two parameters: the intrinsic spatial contention $γ$ of the network, introduced in [1], and the coordination level achieved by the MAC as measured by the interference scaling exponent $κ$ introduced in this paper. We study outage probabilities under the signal-to-interference ratio (SIR) model, Rayleigh fading, and power-law path loss, and explain how the two parameters depend on the network model. The main result is that the outage probability approaches $γη^κ$ as the density of interferers $η$ goes to zero, and that $κ$ assumes values in the range $1\leq κ\leq α/2$ for all practical MAC protocols, where $α$ is the path loss exponent. This asymptotic expression is valid for all motion-invariant point processes. We suggest a novel and complete taxonomy of MAC protocols based mainly on the value of $κ$. Finally, our findings suggest a conjecture that tightly bounds the outage probability for all interferer densities.

0 citations0 reviewsNo code linkedOpen access
preprint2009arXiv

Spatial and Temporal Correlation of the Interference in ALOHA Ad Hoc Networks

Interference is a main limiting factor of the performance of a wireless ad hoc network. The temporal and the spatial correlation of the interference makes the outages correlated temporally (important for retransmissions) and spatially correlated (important for routing). In this letter we quantify the temporal and spatial correlation of the interference in a wireless ad hoc network whose nodes are distributed as a Poisson point process on the plane when ALOHA is used as the multiple-access scheme.

0 citations0 reviewsNo code linkedOpen access
preprint2008arXiv

The Transport Capacity of a Wireless Network is a Subadditive Euclidean Functional

The transport capacity of a dense ad hoc network with n nodes scales like \sqrt(n). We show that the transport capacity divided by \sqrt(n) approaches a non-random limit with probability one when the nodes are i.i.d. distributed on the unit square. We prove that the transport capacity under the protocol model is a subadditive Euclidean functional and use the machinery of subadditive functions in the spirit of Steele to show the existence of the limit.

0 citations0 reviewsNo code linkedOpen access