Source-linked AI summary
Downlink Capacity and Base Station Density in Cellular Networks
Seung Min Yu, Seong-Lyun Kim
TL;DR
Regular cellular models can overestimate performance and offer limited tractable analysis for arbitrary users. This paper uses stochastic geometry with user-density modeling to derive service probabilities and capacity, finding diminishing gains as base-station density increases.
Problem
Regular base-station models can overestimate capacity and do not readily provide tractable analysis for arbitrary user locations or multiple interfering users.
Method
The paper models base stations and mobile users with homogeneous Poisson point processes and derives inactive-base-station, user-selection, service-success, and service-capacity quantities.
Results
Success-transmission density increases with base-station density, but its increasing rate diminishes because some small cells lack users and additional base stations increase co-channel interference.
Takeaways & Limitations
The framework provides closed-form cellular-network performance analysis for dense access-point deployments with less reliance on system-level simulations.
Takeaways & Limitations
The analysis omits shadow fading and assumes equally likely user selection at each base station.
Abstract
from arXiv · showhide
There have been a bulk of analytic results about the performance of cellular networks where base stations are regularly located on a hexagonal or square lattice. This regular model cannot reflect the reality, and tends to overestimate the network performance. Moreover, tractable analysis can be performed only for a fixed location user (e.g., cell center or edge user). In this paper, we use the stochastic geometry approach, where base stations can be modeled as a homogeneous Poisson point process. We also consider the user density, and derive the user outage probability that an arbitrary user is under outage owing to low signal-to-interference-plus-noise ratio or high congestion by multiple users. Using the result, we calculate the density of success transmissions in the downlink cellular network. An interesting observation is that the success transmission density increases with the base station density, but the increasing rate diminishes. This means that the number of base stations installed should be more than $n$-times to increase the network capacity by a factor of $n$. Our results will provide a framework for performance analysis of the wireless infrastructure with a high density of access points, which will significantly reduce the burden of network-level simulations.
I. INTRODUCTION
The paper develops closed-form cellular-capacity analysis for randomly located base stations and users, addressing limitations of regular-grid models and simulation-heavy evaluation. It incorporates user density and interference to quantify service capacity as base-station density changes.
- The framework aims to answer capacity questions analytically rather than relying only on simulations.
- Regular base-station models can overestimate capacity because they neglect weak outer-tier interference and restrict analysis to fixed user locations.
- The paper applies stochastic geometry with a homogeneous Poisson point process to model random base-station locations and arbitrary user positions.
- User density is included to derive closed-form probabilities for outage caused by low SINR or congestion from multiple users.
- The analysis assumes frequency reuse factor 1 and accounts for fluctuating radio channels through fading and pathloss.
II. SYSTEM MODEL
The system model represents base stations and mobile users as independent spatial Poisson processes, with users served by their nearest base station. Each base station serves at most one randomly selected user per resource block.
- Base stations and mobile users follow independent homogeneous PPPs with densities λ_b and λ_u, respectively.
- Each mobile user is served by its nearest base station, producing a Voronoi tessellation of cellular areas.
- The channel model uses pathloss and Rayleigh fading, while log-normal shadowing is included in simulations.
- Only one mobile user is scheduled in each base station’s resource block, with equal selection probability among users in that cell.
III. INACTIVE BASE STATION PROBABILITY AND USER SELECTION PROBABILITY
This section derives the inactive-base-station and user-selection probabilities needed to characterize interference and whether an arbitrary user receives service.
- The inactive base station probability is the chance that a randomly chosen base station has no mobile user in its Voronoi cell.It is used to calculate aggregate inter-cell interference.
- The user selection probability is the chance that a randomly chosen mobile user is assigned the resource block and served by its nearest base station.
A. Inactive Base Station Probability
The paper derives inactive-base-station probability by modeling the number of mobile users in a typical Voronoi cell from the cell-area distribution.
- The analysis begins with the probability density of a typical Voronoi-cell size normalized by 1/λ_b.
- The number of mobile users in a typical Voronoi cell is then characterized through its probability mass function.
- The derivation uses the law of total probability and the Laplace transform of the cell-area density.
- The inactive base station probability is obtained as the probability that the typical cell contains zero mobile users.
B. User Selection Probability
The paper derives the Voronoi-cell-size distribution seen by a randomly chosen mobile user and uses it to obtain user selection probabilities. Simulations with λu = 30 match the analytical results.
- The user-occupied cell-size density fY(y) is defined using the cell-size variable Y normalized by 1/λb.The density is derived by conditioning on a randomly chosen user being located in the cell and normalizing the result.
- Large Voronoi cells are more likely to contain a randomly chosen mobile user than typical cells.This size bias produces different distributions for typical cells and user-occupied cells.
- User selection probability is derived from the number of other mobile users in the selected user’s Voronoi cell.Given n other users, equal-likelihood scheduling selects the tagged user with probability 1/(n + 1).
- The analysis derives the probability that a randomly chosen mobile user is assigned a resource block and served by the nearest base station.The derivation integrates over the user-occupied Voronoi-cell distribution and the number of other users.
- With λu = 30, simulations exactly coincide with the analytical Voronoi-cell, inactive-base-station, and user-selection results shown in Figure 2.The simulations use 10^5 independent samples of base-station and mobile-user locations.
IV. PERFORMANCE ANALYSIS OF CELLULAR NETWORKS
The paper analyzes cellular-network capacity as a function of mobile-user density, base-station density, and target service quality using service success probability and service capacity.
- Service success probability is the probability that the cellular network successfully serves an arbitrary mobile user.It combines user selection probability with transmission success probability.
- Service capacity is the density of mobile users with successful transmissions.
A. Service Success Probability
The paper derives service success probability by combining user selection and transmission success under a stochastic interference model. The formulation includes an average independent-thinning approximation and has a closed form when α = 4 in an interference-limited system.
- Service success probability combines user selection probability with transmission success probability above target SINR γ̂.Transmission success is the event that the received SINR γ exceeds γ̂.
- The definition of pservice assumes independence between user selection and transmission success, although the paper reports that their dependency is negligible.The negligible dependency is assessed through the match between theoretical and simulation results in Figure 3.
- The interfering-base-station density is λi = λb · (1 − pinactive).
- Interfering base stations are approximated by an independently thinned PPP with thinning probability pinactive.The actual interfering process is dependent, but independent thinning is assumed for tractability using the average inactive-base-station probability.
- When noise is negligible and α = 4, the service success probability reduces to a closed-form formula.For other pathloss exponents, the paper states that a closed form is not available to the authors’ knowledge.
B. Service Capacity
The paper defines service capacity as the density of mobile users with successful transmissions and derives it from service success probability. Capacity increases with base-station density, but its marginal increase diminishes because of inactive small cells and co-channel interference.
- Service capacity Cservice is defined as the density of mobile users with successful transmissions.
- The authors derive Cservice from the service success probability using a closed-form proposition.
- 10^5 independent simulations evaluate service probability and service capacity under α = 4, λu = 30, target SINR ˆγ = 0dB, and an interference-limited system.
- The analysis and simulations have closely matching curve shapes, with a small gap as base-station density increases because simulations add shadow fading.
- Service capacity is a concave function of the number of base stations, so marginal capacity decreases as installations grow.
- Higher pathloss exponents increase capacity by filtering co-channel interference, while diminishing marginal capacity remains.
C. Asymptotic Cases
The paper analyzes two density-asymptotic regimes to simplify service probability and capacity formulas. In highly congested networks, transmitting-base-station density approaches the existing base-station density, while Figure 4 compares capacity across pathloss exponents.
- High base-station density: When base-station density greatly exceeds mobile-user density, user selection probability approaches one and transmitting-base-station density approaches mobile-user density.
- High base-station density: This high-base-station-density regime models femtocell-like deployments and supports simpler closed-form expressions.
- Pathloss comparison: Figure 4 reports service capacity for pathloss exponents α under λu = 30, target SINR ˆγ = 0dB, and an interference-limited system.
- High user density: When mobile-user density greatly exceeds base-station density, inactive probability approaches zero and transmitting-base-station density approaches base-station density.
- High user density: The high-user-density regime represents highly congested areas such as downtowns and yields simplified service probability and capacity formulas.
V. CONCLUSIONS
The paper uses stochastic geometry to derive cellular-network distributions and probabilities, then calculates downlink service capacity. Capacity rises with base-station density but with diminishing returns, except that saturated interference-limited traffic yields linear growth.
- The stochastic-geometry analysis derives useful cellular-network distributions and probabilities.
- The resulting service capacity is the density of successful downlink transmissions.
- Service capacity increases with base-station density, but its increasing rate diminishes.
- Under saturated traffic and negligible noise, success-transmission density increases linearly with base-station density.
- The current analysis omits shadow fading and assumes equally likely user selection within each base station.