Poisson Hail on a Wireless Ground

TL;DR

The paper studies a spatial queuing model for wireless systems where arrivals carry exclusion footprints and service rates depend on interference, captured by the Poisson hail on a wireless ground (PHWG). It develops a Markov representation on the space of counting measures and identifies a stability threshold: there exists a critical arrival rate $\lambda_c$ such that the system is positive recurrent for $\lambda<\lambda_c$ and transient for $\lambda>\lambda_c$, with a constructive method to estimate $\lambda_c$. A key insight is that carrier sensing and exclusion can stabilize systems that would be unstable under greedy, immediate access, and simulations reveal an optimal exclusion radius $R^*$ in some regimes. The analysis hinges on block decomposition via zigzag events, reduction to a Lindley-type recurrence for block waiting times, and regenerative arguments, with extensions to spatial interaction birth-death processes and discussions of practical CSMA/CA implications. The results provide quantitative guidelines for designing shared wireless protocols that balance protection zones and throughput to achieve finite delays in steady state.

Abstract

This paper defines a new model which incorporates three key ingredients of a large class of wireless communication systems: (1) spatial interactions through interference, (2) dynamics of the queueing type, with users joining and leaving, and (3) carrier sensing and collision avoidance as used in, e.g., WiFi. In systems using (3), rather than directly accessing the shared resources upon arrival, a customer is considerate and waits to access them until nearby users in service have left. This new model can be seen as a missing piece of a larger puzzle that contains such dynamics as spatial birth-and-death processes, the Poisson-Hail model, and wireless dynamics as key other pieces. It is shown that, under natural assumptions, this model can be represented as a Markov process on the space of counting measures. The main results are then two-fold. The first is on the shape of the stability region and, more precisely, on the characterization of the critical value of the arrival rate that separates stability from instability. The second is of a more qualitative or perhaps even ethical nature. There is evidence that for natural values of the system parameters, the implementation of sensing and collision avoidance stabilizes a system that would be unstable if immediate access to the shared resources would be granted. In other words, for these parameters, renouncing greedy access makes sharing sustainable, whereas indulging in greedy access kills the system.

Figures (4)

• Figure 1: A discrete-time illustration of the dynamics on $\cal X \subset \mathbb{R}$. The hot ground (or equivalently, $\cal X$) is illustrated by the red line segment. Each rectangle represents an arrival, where the height represents the file size, and the projection of the rectangle on the ground represents the exclusion set.
• Figure 2: An illustration of an underlying deterministic sequence $\{B_i\}_{i=1}^k$ with the corresponding zigzag and block structure for $\cal X \subset\mathbb{R}$. For simplicity, the arrivals are not illustrated, and here the height of $\{B_i\}_{i=1}^k$ has no physical meaning.
• Figure 3: $\cal X$ = $[-2,2]^2$. $\mathbb{E} h = 1$, i.i.d. exponential radius, and function $l(r) = \min(1, r^{-4}), w=0.05$.
• Figure 4: $\cal X$ = $[-10,10]^2$. $\mathbb{E} h = 1$, fixed constant radius, and function $l(r) = \min(1, r^{-4}), w=0.05$.

Theorems & Definitions (25)

• Remark 1
• Lemma 2
• proof
• Remark 3
• Remark 4
• Definition 5
• Proposition 6
• Remark 7
• Proposition 8
• proof