On Stochastic Geometry Modeling of Cellular Uplink Transmission with Truncated Channel Inversion Power Control (1401.6145v1)

Abstract: Using stochastic geometry, we develop a tractable uplink modeling paradigm for outage probability and spectral efficiency in both single and multi-tier cellular wireless networks. The analysis accounts for per user equipment (UE) power control as well as the maximum power limitations for UEs. More specifically, for interference mitigation and robust uplink communication, each UE is required to control its transmit power such that the average received signal power at its serving base station (BS) is equal to a certain threshold $\rho_o$. Due to the limited transmit power, the UEs employ a truncated channel inversion power control policy with a cutoff threshold of $\rho_o$. We show that there exists a transfer point in the uplink system performance that depends on the tuple: BS intensity ($\lambda$), maximum transmit power of UEs ($P_u$), and $\rho_o$. That is, when $P_u$ is a tight operational constraint with respect to [w.r.t.] $\lambda$ and $\rho_o$, the uplink outage probability and spectral efficiency highly depend on the values of $\lambda$ and $\rho_o$. In this case, there exists an optimal cutoff threshold $\rho^*_o$, which depends on the system parameters, that minimizes the outage probability. On the other hand, when $P_u$ is not a binding operational constraint w.r.t. $\lambda$ and $\rho_o$, the uplink outage probability and spectral efficiency become independent of $\lambda$ and $\rho_o$. We obtain approximate yet accurate simple expressions for outage probability and spectral efficiency which reduce to closed-forms in some special cases.

• The paper introduces a robust model that extends stochastic geometry to uplink transmissions by incorporating truncated channel inversion power control.
• It derives tractable expressions for outage probability and spectral efficiency, identifying an optimal power control threshold.
• The framework applies to both single and multi-tier networks, offering practical insights for base station density and power control optimization.

An Analytical Framework for Cellular Uplink Transmission Using Stochastic Geometry

The paper by Hesham ElSawy and Ekram Hossain presents a comprehensive mathematical framework for analyzing uplink performance in cellular networks, specifically considering the aspects of truncated channel inversion power control through stochastic geometry. It provides a tractable approach to model the uplink behavior in both single and multi-tier cellular network topologies under constraints of power control, offering insights into outage probability and spectral efficiency.

Key Contributions

The paper introduces several key contributions to the existing body of knowledge in wireless communications:

1. Modeling Framework: The authors extend stochastic geometry, historically utilized for modeling downlink communications, to establish a robust framework for analyzing uplink transmissions. This is crucial for modern heterogeneous networks where base stations (BSs) are not organized in regular grids.
2. Truncated Channel Inversion Power Control: A novel approach is proposed that requires user equipment (UE) to adjust its transmission power to meet a received power threshold at its serving BS, considering maximum power constraints.
3. Outage Probability and Spectral Efficiency: The paper derives simple yet accurate expressions to assess these metrics. Significant is the identification of a 'transfer point', beyond which the system transitions from being affected by the maximum transmit power constraint to being independent of it.
4. Extension to Multi-tier Networks: The analysis considers multi-tier networks with possibly different infrastructure designs, like varying BS densities across the tiers, which is vital in an era where macro, micro, and femto cells coexist.

Numerical Results and Analysis

The paper provides a rigorous mathematical apparatus supported by extensive numerical analyses. Key findings include:

• Outage Minimization: There exists an optimal power control threshold, denoted as $\rho^*_o$, which minimizes the probability of outages. This aspect introduces a balance between maintaining a high enough received power to decode signals successfully and ensuring UE power constraints are not overly restrictive.
• Spectral Efficiency: Results show that spectral efficiency is improved by optimizing the cutoff threshold $\rho_o$. The work suggests that changes in $\rho_o$ affect the balance between conservation of power in transmission and overall network efficiency.
• Transfer Point and BS Intensity: Findings highlight that once a threshold of base station density is crossed, self-interference between UEs becomes a non-issue, aligning the uplink performance more closely with paradigms observed in downlink scenarios.

Implications and Future Developments

The implications of this research span both theoretical enhancements and practical deployments:

• Network Planning: The proposed model can guide operators in determining base station densities and power control settings for optimal uplink performance, ensuring a robust service without excessive power consumption.
• Comparison with Downlink Models: The demonstrated similarities with stochastic models used for downlink communications underscore potential avenues for harmonizing theoretical approaches across up and downlink analyses.
• Universality across Network Tiers: Since cellular network environments increasingly comprise multiple tier architectures, embracing this model could standardize performance evaluations across assorted deployment landscapes.

Conclusion

By extending stochastic geometry to the cellular uplink field, this paper bridges a crucial analytical gap, addressing the more intricate interference characteristics and power control challenges unique to such scenarios. This framework opens opportunities for enhancing uplink designs and equipping cellular networks for the increased demand in traffic and heterogeneity consistent with next-generation wireless technology deployments.