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Spatial Spectrum and Energy Efficiency of Random Cellular Networks
Xiaohu Ge, Bin Yang, Junliang Ye, Guoqiang Mao, Cheng-Xiang Wang, Tao Han
TL;DR
Evaluating cellular-network performance is difficult because spectrum and energy efficiency depend on random geometry, channel access, traffic, and propagation. The paper integrates a Markov-chain access model with PVT random networks, derives outage and blocking probabilities, and analyzes resulting efficiencies. Numerical results show systematic dependence on path loss exponent, base-station density, call arrival rate, and SINR threshold.
Problem
Evaluating cellular mobile communication network performance, including spectrum and energy efficiency, remains challenging.
Method
The paper integrates a Markov-chain wireless channel-access model with PVT random cellular networks and derives outage, blocking, spatial spectrum, and energy-efficiency models.
Results
Numerical results show that spatial spectrum efficiency increases with path loss exponent and base-station density but is lower for PVT than grid networks, while call-arrival effects have efficiency maxima.
Takeaways & Limitations
Optimal spectrum and energy efficiency requires considering call arrival rate and SINR threshold together.
Abstract
from arXiv · showhide
It is a great challenge to evaluate the network performance of cellular mobile communication systems. In this paper, we propose new spatial spectrum and energy efficiency models for Poisson-Voronoi tessellation (PVT) random cellular networks. To evaluate the user access the network, a Markov chain based wireless channel access model is first proposed for PVT random cellular networks. On that basis, the outage probability and blocking probability of PVT random cellular networks are derived, which can be computed numerically. Furthermore, taking into account the call arrival rate, the path loss exponent and the base station (BS) density in random cellular networks, spatial spectrum and energy efficiency models are proposed and analyzed for PVT random cellular networks. Numerical simulations are conducted to evaluate the network spectrum and energy efficiency in PVT random cellular networks.
I. INTRODUCTION
The paper addresses the challenge of combining Markov-chain access modeling with stochastic-geometry cellular networks to evaluate spectrum and energy efficiency. It develops PVT-based channel, outage, blocking, spectrum, and energy-efficiency models.
- Motivation: Energy efficiency is important because cellular base stations consume substantial power, with reported annual costs of 3,000 dollars for on-grid and 30,000 dollars for off-grid operation.The paper therefore evaluates both spectrum efficiency and energy efficiency in cellular systems.
- Motivation: Regular cellular deployment models can bias performance evaluation because real base-station locations exhibit structural fluctuations.Poisson point processes are used to model base-station spatial structure more realistically.
- Research gap: Combining Markov-chain models with random cellular networks remains an open modeling challenge, particularly for representing user access.The paper targets this obstacle in a Poisson-Voronoi tessellation cellular scenario.
- Contributions: The paper proposes a Markov-chain-based channel-access model for a PVT random cellular network and extends the one-cell model network-wide using Palm theory.This extension supports analysis of spatial spectrum and energy efficiencies across the random network.
- Contributions: The proposed framework derives outage and blocking probabilities while accounting for fading, shadowing, and a spatial Poisson distribution of interfering transmitters.The propagation model covers scenarios including Rayleigh fading, Gamma-distributed effects, and log-normal shadowing.
B. User Association Scheme
The paper models user association in an irregular PVT cellular network, where each mobile user is assigned to the nearest base station. PVT cell geometry supports extending typical-cell results to the whole network.
- Association schemes: User association can use nearest-base-station, highest-SINR, or maximum-received-signal-power schemes.The paper selects nearest-base-station association for its PVT analysis.
- Nearest-BS association: Under the selected scheme, the association weight is configured as T_yj = 1.The associated base station is the nearest base station in the plane.
- PVT geometry: PVT cells form a stochastic, irregular-topology network with base stations represented as blue points and mobile users as red points.Each cell is denoted C_yj in the illustration.
- Typical-cell analysis: Palm theory states that any PVT cell has the same geometric characteristics as a typical cell with its base station fixed at the origin.Therefore, analytical results for the typical cell can be extended to the whole PVT random cellular network.
C. Wireless Channel Allocation Strategy
The channel allocation strategy centrally assigns traffic channels after user association and uses perfect channel-state information to evaluate channel SINR against a threshold.
- Centralized allocation: Traffic channels are centrally allocated to a mobile user after the user associates with a specific base station.The strategy evaluates channel availability for the associated base station.
- Channel state: A channel is marked available when its SINR is greater than or equal to γ0 and unavailable when its SINR is below γ0.The associated mobile user is assumed to know the SINR over all channels available to its base station.
A. Markov Chain Model of PVT Random Cellular Networks
The Markov-chain model represents channel access in a typical PVT cell using states that track occupied and available channels. Call arrivals, service completions, and time-varying channel conditions drive state transitions.
- Channel-access model: The continuous Gilbert-Elliott model describes transitions between available and unavailable channels, while call arrivals and holding times are modeled probabilistically.Call arrivals follow a Poisson distribution, and session and dwelling times follow exponential distributions.
- Transition rates: The unavailable-to-available and available-to-unavailable transition rates are denoted α and β, respectively.Their values are determined using the channel-unavailability probability and outage probability.
- State representation: A state is represented by (m, n), where m is occupied channels and n is total allocatable available channels, with m ≤ n ≤ C.C denotes the maximum number of available channels in the typical cell.
- Call dynamics: New calls increase occupied channels when m < n, while successfully serviced calls release occupied channels.These transitions change m while leaving n unchanged.
- Channel dynamics: Time-varying interference changes channel availability, increasing n when an unavailable channel becomes available and decreasing n when an available channel becomes unavailable.These transitions change n while preserving the number of occupied channels.
- Stationary analysis: The stationary probabilities and channel-use probabilities are obtained by solving linear equations once the outage probability is known.The unavailable-channel probability averages outage probability over different numbers of interferers.
B. Outage Probability Model
The paper derives outage probability for a typical PVT cell by combining the SINR model with Laplace-transform analysis of noise and interference conditioned on the number of interferers.
- SINR and association: The SINR at a mobile user associated with its nearest base station includes desired signal, Gaussian noise, and co-channel interference from adjacent base stations.The user association is nearest-BS based, and interference is analyzed through the number of aggregated interferers.
- Conditional outage probability: The outage probability conditioned on Δ interferers is obtained from the complementary success probability.The success probability uses the independence of noise and interference and is further expressed through Laplace transforms.
- Probability computation: Parseval’s theorem and the Laplace transforms of signal, noise, and interference provide a numerically computable success-probability expression.Under Rayleigh fading, the signal-power distribution is handled through its Laplace transform.
- Interference analysis: For a finite disk containing Δ interferers, their independent locations yield a product-form Laplace transform for aggregate interference.The derivation conditions on the number of nodes, uses uniform radial density, and then lets the disk radius approach infinity while preserving density.
- Interference analysis: Assuming the path-loss law L(r) = r^b, the individual interferer transform is derived using the path-loss exponent and Gamma function.The individual transforms are combined to obtain the aggregate-interference transform used in the outage expression.
C. Blocking Probability with Rayleigh Fading
Under Rayleigh fading, the paper derives blocking probability from channel availability and active-user capacity, with mean sojourn time obtained through Little’s theorem.
- Rayleigh fading: Under Rayleigh fading, the signal-power probability density is exponential, enabling its Laplace transform to be used in the outage derivation.The paper sets the normalized received signal power to an exponential distribution and derives its transform.
- Outage derivation: The aggregate-interference Laplace transform is substituted into the conditional outage expression to obtain the outage probability.This connects the fading-specific signal model with the interference analysis.
- Blocking probability: A call is blocked when it cannot be served immediately because active mobile users exceed the available channels in the typical cell.The blocking event is attributed to insufficient channel resources.
- Mean sojourn time: The mean sojourn time of a mobile user is derived using Little’s theorem.This provides the delay-related performance measure after blocking and channel-access modeling.
D. Performance Analysis
The numerical analysis evaluates blocking probability and mean sojourn time in PVT random cellular networks as SINR threshold, channel capacity, path-loss exponent, and call arrival rate vary.
- Simulation configuration: The numerical evaluation uses Rayleigh fading and a default PVT configuration with BS intensity λB = 0.2 per square kilometers and C = 20.Other defaults include λ = 1minute−1, b = 4, and BS transmitting power Pyi = 30dBm.
- Blocking probability: Blocking probability increases with SINR threshold when the maximum channel number C is fixed.A higher threshold reduces the number of successfully decoded signals available to mobile users.
- Blocking probability: Blocking probability increases as the maximum available channel number decreases at a fixed SINR threshold.The paper attributes this trend to insufficient channel resources and a higher likelihood of dropped calls.
- Blocking probability: Blocking probability decreases with increasing path-loss exponent b at a fixed SINR threshold.The reported curves imply stronger attenuation of aggregate interference than of desired signals in the PVT network.
- Mean sojourn time: Mean sojourn time decreases as call arrival rate λ or SINR threshold increases.The paper links the threshold effect to fewer available channels and reduced network service ability.
IV. SPATIAL SPECTRUM AND ENERGY EFFICIENCY OF PVT RANDOM CELLULAR NETWORKS
This section evaluates spatial spectrum and energy efficiency for PVT random cellular networks.
- Section scope: The paper further evaluates the spatial spectrum and energy efficiency of PVT random cellular networks.The section introduces the scope of the subsequent performance analysis.
A. Spatial Spectrum and Energy Efficiency
The paper develops spatial spectrum efficiency and energy-efficiency formulations for PVT random cellular networks. It defines link capacity and cell throughput, then combines lifetime throughput with operation and embodied energy to extend typical-cell efficiency to the network.
- Energy Efficiency: The typical-cell throughput is defined from its bandwidth, B, and the energy efficiency is evaluated over the cell’s whole lifetime.The model relates lifetime throughput to lifetime base-station energy consumption.
- Spatial Spectrum Efficiency: Link capacity between a mobile user and its associated base station is defined under an assumed number of interferers, Δ.The capacity formulation uses SINR for the user–base-station link.
- Spatial Spectrum Efficiency: The spatial spectrum efficiency of the PVT random cellular network is derived from the link-capacity formulation and depends on base-station density, λ_B.The derivation uses the cumulative distribution function and the nonnegative SINR property.
- Energy Efficiency: Lifetime base-station energy includes operation energy and embodied energy, with embodied energy divided into initial and maintenance components.Operation energy is modeled as a linear function of total transmission power over occupied channels.
- Energy Efficiency: The typical-cell energy-efficiency result is extended to the whole PVT random cellular network using Palm theory.This connects the cell-level lifetime energy and throughput model to network-level analysis.
B. Numerical Results and Discussion
Numerical results compare PVT random and regular grid cellular networks across path loss, BS density, call arrival rate, and SINR threshold. Spatial spectrum and energy efficiency generally improve with path loss exponent, while traffic and deployment density create trade-offs and PVT networks underperform grid networks in spatial spectrum efficiency.
- Simulation setup: The simulations compare PVT random cellular networks with regular hexagonal grid networks using configured bandwidth, embodied energy, and transmission parameters.The comparison setup uses B = 0.1MHz and embodied energy EEMinit + EEMma int = 85GJ.
- Spatial spectrum efficiency: Spatial spectrum efficiency increases with both path loss exponent and BS density in PVT and grid cellular networks.The path loss and BS-density trends are reported for Fig. 6.
- Spatial spectrum efficiency: PVT random cellular networks have lower spatial spectrum efficiency than grid cellular networks when transmission bandwidth is fixed.The paper relates this difference to lower average transmission rates in PPP random networks than in grid networks.
- Traffic effects: Spatial spectrum efficiency decreases as call arrival rate increases when the path loss exponent is fixed, because blocking probability increases with traffic.The increased blocking probability reduces spatial spectrum efficiency in both network models.
- Traffic and SINR effects: For fixed call arrival rate, spatial spectrum efficiency decreases with SINR threshold; across arrival rates, it rises below a threshold and falls above it, producing a maximum.The simulations validate maximum spatial spectrum efficiency values for different call arrival rates.
- Energy efficiency: Energy efficiency rises below a call-arrival threshold and falls above it, so operators should consider call arrival rate and SINR threshold together.The results identify maximum energy efficiency values for different call arrival rates.
- Energy efficiency: Energy efficiency increases with path loss exponent but decreases with BS density when the path loss exponent is fixed.The larger attenuation of interference than desired signals as path loss increases reduces outage probability and supports the efficiency increase.
V. CONCLUSIONS
The paper integrates a Markov-chain channel-access model into PVT random cellular networks to analyze spatial spectrum and energy efficiency. Numerical results identify how call arrival rate, BS density, path loss exponent, and SINR threshold affect these metrics.
- A Markov chain models wireless channel access in a typical PVT cell, incorporating path loss and Rayleigh fading effects.
- The model derives numerically computable outage and blocking probabilities for a typical PVT cell.
- Spatial spectrum and energy efficiency models are obtained for PVT random cellular networks.
- Call arrival rate and BS density have adverse effects on spatial spectrum efficiency in PVT random cellular networks.
- Path loss exponent and SINR threshold strongly affect energy efficiency in PVT cellular networks.