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On the Stability of a N-class Aloha Network

P. S. Dester, P. Cardieri, J. M. C. Brito

TL;DR

Necessary and sufficient conditions are established for the stability of a high-mobility N-class Aloha network, where the position of the sources follows a Poisson point process, each source has an infinity capacity buffer and packets arrive according to a Bernoulli distribution.

Abstract

Necessary and sufficient conditions are established for the stability of a high-mobility N-class Aloha network, where the position of the sources follows a Poisson point process, each source has an infinity capacity buffer, packets arrive according to a Bernoulli distribution and the link distance between source and destination follows a Rayleigh distribution. It is also derived simple formulas for the stationary packet success probability and mean delay.

On the Stability of a N-class Aloha Network

TL;DR

Necessary and sufficient conditions are established for the stability of a high-mobility N-class Aloha network, where the position of the sources follows a Poisson point process, each source has an infinity capacity buffer and packets arrive according to a Bernoulli distribution.

Abstract

Necessary and sufficient conditions are established for the stability of a high-mobility N-class Aloha network, where the position of the sources follows a Poisson point process, each source has an infinity capacity buffer, packets arrive according to a Bernoulli distribution and the link distance between source and destination follows a Rayleigh distribution. It is also derived simple formulas for the stationary packet success probability and mean delay.

Paper Structure

This paper contains 11 sections, 10 theorems, 48 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. System Model
  3. Single User Class Network Analysis
  4. Stability Conditions and Stationary Analysis
  5. Multiple-Class Network
  6. Interpretation and Application of the Results
  7. Optimization problem
  8. Cellular and D2D
  9. Conclusions
  10. Proof of Lemma \ref{['lem:sufficient']}
  11. Proof of Theorem \ref{['th:stability']}

Key Result

Proposition 1

The queueing system $\{Q_{i}(t)\}$ is stable in the sense defined by szpankowski1994stability if and only if where $\phi \triangleq 4\,\Gamma(1+2/\alpha)\,\Gamma(1-2/\alpha)\, \overline{R}^2\,\theta^{2/\alpha}$ and $\zeta \triangleq \sum_{n=2}^N (P_n/P_1)^{2/\alpha}\,\lambda_n\,p_n$. Then, the closure of arrival rates is given by

Figures (4)

  • Figure 1: Delay $D$ as a function of the arrival rate of packets $a$ per time slot, at the optimum medium access probability ($p=1$) and $\zeta = 0$. Simulation results are shown in crosses. Dashed curves correspond to the model presented in stamatiou2010random, where $R$ is constant.
  • Figure 2: Delays of two-class D2D network, for $a_1=a_2=0.7$, $\phi_1\,\lambda_1 = \phi_2\,\lambda_2 = 0.15$, and $c_1 = 1$.
  • Figure 3: These figures represent the optimization of a 2-class D2D network with the following parameters: $a_1=0.7$, $\phi_1\,\lambda_1 = \phi_2\,\lambda_2 = 0.15$ and $c_1 = 1$. In the left figure, the dashed curve is $D_1$ and the full curve is $D_2$.
  • Figure 4: Left figure shows the maximum arrival rate achievable for the first traffic class (D2D), such that the constraints of Proposition \ref{['prop:max_a1']} are satisfied; $\Psi_2$ represents the use of the channel by the second traffic class (cellular); we used $\phi_1\,\lambda_1 = 1$. Right figure shows the transmit power ratio to achieve the maximum $a_1$; we used $\phi_1\,\lambda_1 = \phi_2\,\lambda_2 = 1$ and $D_2^* = 3$ [slots].

Theorems & Definitions (22)

  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • Lemma 1
  • proof
  • ...and 12 more