The Meta Distribution of the SIR in Poisson Bipolar and Cellular Networks (1506.01644v1)

Abstract: The calculation of the SIR distribution at the typical receiver (or, equivalently, the success probability of transmissions over the typical link) in Poisson bipolar and cellular networks with Rayleigh fading is relatively straightforward, but it only provides limited information on the success probabilities of the individual links. This paper introduces the notion of the meta distribution of the SIR, which is the distribution of the conditional success probability $P$ given the point process, and provides bounds, an exact analytical expression, and a simple approximation for it. The meta distribution provides fine-grained information on the SIR and answers questions such as "What fraction of users in a Poisson cellular network achieve 90% link reliability if the required SIR is 5 dB?". Interestingly, in the bipolar model, if the transmit probability $p$ is reduced while increasing the network density $\lambda$ such that the density of concurrent transmitters $\lambda p$ stays constant as $p\to 0$, $P$ degenerates to a constant, i.e., all links have exactly the same success probability in the limit, which is the one of the typical link. In contrast, in the cellular case, if the interfering base stations are active independently with probability $p$, the variance of $P$ approaches a non-zero constant when $p$ is reduced to $0$ while keeping the mean success probability constant.

• The paper introduces the meta distribution of SIR, offering a refined analysis over traditional average success probability methods.
• It derives closed-form expressions and employs a beta approximation to accurately quantify link success probabilities in Poisson bipolar and cellular settings.
• The study provides practical insights for network design by highlighting uniform performance in dense bipolar networks versus persistent variability in cellular networks.

The Meta Distribution of the SIR in Poisson Bipolar and Cellular Networks

In this paper, the author examines the signal-to-interference ratio (SIR) in Poisson bipolar and cellular networks, focusing on the meta distribution of the SIR. The meta distribution provides a more refined perspective compared to traditional methods that involve calculating the distribution of the SIR at the typical receiver. The typical calculation only offers limited insight into individual link success probabilities within the network. By contrast, the meta distribution enables an analysis of the conditional success probability distribution given a point process, offering a granular view of network performance.

Key Contributions

1. Introduction of the Meta Distribution: The paper introduces the concept of the meta distribution of the SIR, which captures the distribution of the conditional success probability. This provides detailed insight beyond average success probabilities and addresses questions regarding specific user link reliability within stochastic wireless networks.
2. Closed-form Expressions: The work derives closed-form expressions for the moments of the meta distribution in Poisson bipolar networks with ALOHA and Poisson cellular networks with Rayleigh fading. These analytical results facilitate a deeper understanding of network performance under varying conditions.
3. Analytical and Approximate Expressions: The author presents an exact analytical expression for the meta distribution and proposes the beta distribution as an accurate approximation. This is particularly significant as it allows for practical application by providing bounds for real networks.
4. Observations on Network Models: A notable finding is that in dense bipolar networks with low transmission probability, all links tend to achieve the same success probability, highlighting uniform performance under specific configurations. However, this finding does not translate to cellular networks with independently active interfering base stations, where variance persists.
5. Implications for Network Design: The conditions for finite mean local delay given specific SIR thresholds and transmission probabilities are also delineated, offering practical insights for designing networks that meet delay performance requirements.

Theoretical and Practical Implications

The paper has both theoretical and practical implications. Theoretically, it enhances the understanding of interference and link reliability variability in random networks, contributing to stochastic geometry's application in wireless network analysis. Practically, understanding the meta distribution allows network designers to set realistic expectations on network reliability and optimize configurations to improve user experience. For instance, in ultra-dense networks, reducing transmission probability can lead to more uniform performance across users.

Future Directions

This paper lays the groundwork for future research in several directions. The methods and results could be extended to more complex network structures, such as multi-tier systems (e.g., heterogeneous networks), or networks with different fading models (e.g., Nakagami fading). Additionally, exploring efficient approximations and simulations for larger, more intricate network topologies could be a vital area of future work. As AI and machine learning increasingly interface with network design, investigating how the meta distribution can aid in more intelligent network management and resource allocation.

This research enhances the analytical toolkit for network designers, providing a foundation for leveraging stochastic geometry to address the challenges presented by the spatial and temporal variability in wireless networks. The meta distribution of the SIR offers a comprehensive view that is essential for next-generation networks where fine-grained performance guarantees are crucial.