Achieving Stability for Aloha Networks with Multiple Receivers

TL;DR

This work extends stability analysis of slotted Aloha networks to scenarios with multiple capture receivers under an SINR-based capture model. By deriving fixed-point equations for the HOL-packet success probabilities, it obtains exact transmitter service rates and characterizes stability via the all-unsaturated region $S_{U^K}$ and the stability region $S_Q$, with explicit results for two T-R pairs and for $K$ symmetric T-R pairs. The framework applies to multi-cell and ad-hoc topologies and is validated through simulations, showing that proper per-transmitter transmission control—adjusted to traffic and topology—stabilizes the network whenever input rates lie within $S_Q$. Importantly, the results challenge the conventional wisdom that a fixed, identical transmission probability across all nodes cannot stabilize positive traffic, highlighting the need for topology-aware control in dense wireless networks.

Abstract

Slotted Aloha has been widely adopted in various communication networks. Yet if the transmission probabilities and traffic input rates of transmitters are not properly regulated, their data queues may easily become unstable. For stability analysis of Aloha networks with multiple receivers, the focus of previous studies has been placed on the maximum input rate of each transmitter, below which the network is guaranteed to be stabilized under any given topology. By assuming a fixed and identical transmission probability across the network, however, network stability is found to be unachievable when the input rate exceeds zero. As we will demonstrate in this paper, the key to stabilizing the network lies in proper selection of transmission probabilities according to the traffic input rates and locations of all transmitters and receivers. Specifically, for an Aloha network with multiple capture receivers, by establishing and solving the fixed-point equations of the steady-state probabilities of successful transmissions of Head-of-Line (HOL) packets, the exact service rates of all transmitters' queues are obtained, based on which the operating region of transmission probabilities for achieving stability and the stability region of input rates are further characterized. The results are illustrated in various scenarios of multi-cell and ad-hoc networks. Simulation results validate the analysis and corroborate that the network can be stabilized as long as the traffic input rates are within the stability region, and the transmission probabilities are properly adjusted according to the traffic input rates and network topology.

Figures (15)

• Figure 1: Graphic illustration of (a) the uplink of a single-cell network, (b) the uplink of a multi-cell network, (c) the downlink of a multi-cell network, and (d) an ad-hoc network.
• Figure 2: The percentage of stable transmitters versus the boundary length $L$ of the network. Transmitters' locations are generated in $[0, L]^2$ according to PPP with density $\xi=10^{-4}$ m$^{-2}$. Each transmitter has a receiver with $d_{i,i}= 25$ m with random orientation, $i\in \mathcal{K}$. $\lambda_i = 0.2$, $q_i = 1$, $\theta_i = 0$ dB, $P_i = 17$ dBm, $i\in\mathcal{K}$, $\sigma^2 =-90$ dBm, and $\alpha = 3.8$.
• Figure 3: Embedded Markov chain $\mathbf{X}_k$ of the state transition process of HOL packets of Transmitter $k$, $k\in\mathcal{K}$.
• Figure 4: (a)(c) Topologies of two T-R pairs. $\rho_{1,1} = -3$ dB. $\rho_{2,2} = -1.3$ dB. $\rho_{2,1} = 5.1$ dB. (a) $\rho_{1,2} = 8.8$ dB. (c) $\rho_{1,2} = -3.4$ dB. (b)(d) All-unsaturated region $S_{UU} (\bm{q}, \bm{\lambda})$ of transmission probabilities $\bm{q}$ obtained via Algorithm \ref{['alg:findUU']} with topologies given in (a) and (c), respectively. $\theta_1 = -5$ dB. $\theta_2 = -7$ dB. $\lambda_1 = 0.2$. $\lambda_2 = 0.27$.
• Figure 5: (a) A single-cell network and (c) a two-cell network, where locations of transmitters are generated according to PPP with density $\xi = 10^{-3} \text{ m}^{-2}$ in $[0, 150 \text{ m}]^2$, as represented by dots. The location of each BS is randomly generated in $[0, 150 \text{ m}]^2$, as represented by the triangle, and each transmitter is associated with its closest BS. (b) All-unsaturated region $S_{U^{K}}(q, \lambda)$ of transmission probability $q$ obtained via Algorithm \ref{['alg:findUU']} for the single-cell network given in (a) with symmetric setting. $\rho = 10$ dB. $\theta = 0$ dB. (d) All-unsaturated region $S_{U^K}(\bm{q}, \bm{\lambda})$ of transmission probabilities $\bm{q}$ obtained via Algorithm \ref{['alg:findUU']} for the two-cell network given in (c). $\rho_{i,i^{\ast}} = 0$ dB, $\lambda_i = 0.1$, $i\in \mathcal{K}$. $\theta_1 = \theta_2 = -8$ dB. $\alpha = 4$.

Theorems & Definitions (15)

• remark 1
• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3
• proof
• Theorem 4
• proof
• lemma 1