Analysis of d-ary Tree Algorithms with Successive Interference Cancellation

TL;DR

The paper analyzes d-ary tree-based random access with successive interference cancellation (SICTA), addressing throughput and several related observables. It introduces a Markovian branching model with a split index, derives a functional equation for the moment generating function of the CRI length, and obtains closed-form means and asymptotic expressions for throughput, collisions, and idle slots. It shows that the maximum throughput of $1/\log 2$ is achievable with a specific splitting distribution and reveals a tractable trade-off between throughput and collisions, complemented by a delay-analysis framework. The results extend the understanding of SICTA performance beyond the binary case and provide a blueprint for analyzing additional observables in similar random-tree algorithms.

Abstract

In this article, we calculate the mean throughput, number of collisions, successes, and idle slots for random tree algorithms with successive interference cancellation. Except for the case of the throughput for the binary tree, all the results are new. We furthermore disprove the claim that only the binary tree maximises throughput. Our method works with many observables and can be used as a blueprint for further analysis.

Figures (2)

• Figure 1: The tree illustration of the ternary ($\tf=3$) tree algorithm. The number inside the nodes in the tree represents the number of users transmitting in that slot. The number outside the node represents the slot number. Slots 5,8,9 and 10 will be skipped in the SICTA.
• Figure 2: The minimal obtainable collision rate, constrained by achieving a certain throughput rate. The figure was obtained numerically using a standard solver for constraint non-linear optimisation problems. $\ps$ was used as a the initial value.

Theorems & Definitions (13)

• Theorem 1
• Proposition 1
• Corollary 1
• proof
• Proposition 2
• proof
• Lemma 1
• proof
• Lemma 2
• proof