We establish mean-field limits for large-scale random-access networks with buffer dynamics and arbitrary interference graphs. While saturated-buffer scenarios have been widely investigated and yield useful throughput estimates for persistent sessions, they fail to capture the fluctuations in buffer contents over time, and provide no insight in the delay performance of flows with intermittent packet arrivals. Motivated by that issue, we explore in the present paper random-access networks with buffer dynamics, where flows with empty buffers refrain from competition for the medium. The occurrence of empty buffers thus results in a complex dynamic interaction between activity states and buffer contents, which severely complicates the performance analysis. Hence we focus on a many-sources regime where the total number of nodes grows large, which not only offers mathematical tractability but is also highly relevant with the densification of wireless networks as the Internet of Things emerges. We exploit time scale separation properties to prove that the properly scaled buffer occupancy process converges to the solution of a deterministic initial-value problem, and establish the existence and uniqueness of the associated fixed point. This approach simplifies the performance analysis of networks with huge numbers of nodes to a low-dimensional fixed-point calculation. For the case of a complete interference graph, we demonstrate asymptotic stability, provide a simple closed-form expression for the fixed point, and prove interchange of the mean-field and steady-state limits. This yields asymptotically exact approximations for key performance metrics, in particular the stationary buffer content and packet delay distributions. The methodological framework that we develop easily extends to various model refinements as will be illustrated by several examples.
A thorough survey of mean-field analysis of random-access protocols is presented in @cite_24 . The work of Bianchi @cite_7 is a landmark paper which assumed a propagation of chaos property to hold so as to derive tractable formulae for the key performance measures of the system. The papers surveyed in @cite_24 mostly use mean-field theory to provide either evidence or objection for the propagation of chaos assumption of Bianchi. Among these papers, it is worth mentioning @cite_26 , where the authors investigated the existence of a global attractor for the mean-field system and provided sufficient conditions for its existence, deducing the validity of Bianchi's assumption. Further papers which deserve to be mentioned are @cite_25 @cite_9 , where the authors exploited mean-field theory so as to obtain approximations for key performance measures of large systems. In particular @cite_25 focuses on the characterization of the stability region, while @cite_9 examines the throughput performance of the system. None of the above-mentioned papers considered scenarios with unsaturated buffers, with the exception of @cite_25 , which however dealt with systems evolving in discrete time and did not consider performance metrics like packet delays.
@cite_26 @cite_7 @cite_9 @cite_24 @cite_25
1411539474 2166657206 1968517161 2033020629
In 1998, Giuseppe Bianchi introduced a mean field Markov model of the fundamental medium access control protocol used in Wireless Local Area Networks (WLANs). Due to the model’s intuitive appeal and the accuracy of its predictions, since then there has been a vast body of material published that extends and analyzes models of a similar character. As the majority of this development has taken place within the culture and nomenclature of the telecommunications community, the aim of the present article is to review this work in a way that makes it accessible to probabilists. In doing so, we hope to illustrate why this modeling approach has proved so popular, to explain what is known rigorously, and to draw attention to outstanding questions of a mathematical nature whose solution would be of interest to the telecommunications community. For non-saturated WLANs, these questions include rigorous support for its fundamental decoupling approximation, determination of the properties of the self-consistent equations and the identification of the queueing stability region. We study a fixed-point formalization of the well-known analysis of Bianchi. We provide a significant simplification and generalization of the analysis. In this more general framework, the fixed-point solution and performance measures resulting from it are studied. Uniqueness of the fixed point is established. Simple and general throughput formulas are provided. It is shown that the throughput of any flow will be bounded by the one with the smallest transmission rate. The aggregate throughput is bounded by the reciprocal of the harmonic mean of the transmission rates. In an asymptotic regime with a large number of nodes, explicit formulas for the collision probability, the aggregate attempt rate, and the aggregate throughput are provided. The results from the analysis are compared with ns2 simulations and also with an exact Markov model of the backoff process. It is shown how the saturated network analysis can be used to obtain TCP transfer throughputs in some cases. Performance evaluation of the 802.11 MAC protocol is classically based on the decoupling assumption, which hypothesizes that the backoff processes at different nodes are independent. This decoupling assumption results from mean field convergence and is generally true in transient regime in the asymptotic sense (when the number of wireless nodes tends to infinity), but, contrary to widespread belief, may not necessarily hold in stationary regime. The issue is often related with the existence and uniqueness of a solution to a fixed point equation; however, it was also recently shown that this condition is not sufficient; in contrast, a sufficient condition is a global stability property of the associated ordinary differential equation. In this paper, we give a simple condition that establishes the asymptotic validity of the decoupling assumption for the homogeneous case (all nodes have the same parameters). We also discuss the heterogeneous and the differentiated service cases and formulate a new ordinary differential equation. We show that the uniqueness of a solution to the associated fixed point equation is not sufficient; we exhibit one case where the fixed point equation has a unique solution but the decoupling assumption is not valid in the asymptotic sense in stationary regime. Analytical models of IEEE 802.11-based WLANs are invariably based on approximations, such as the well-known mean-field approximations proposed by Bianchi for saturated nodes. In this paper, we provide a new approach for modeling the situation when the nodes are not saturated. We study a State Dependent Attempt Rate (SDAR) approximation to model M queues (one queue per node) served by the CSMA CA protocol as standardized in the IEEE 802.11 DCF. The approximation is that, when n of the M queues are non-empty, the attempt probability of the n non-empty nodes is given by the long-term attempt probability of n saturated nodes as provided by Bianchi's model. This yields a coupled queue system. When packets arrive to the M queues according to independent Poisson processes, we provide an exact model for the coupled queue system with SDAR service. The main contribution of this paper is to provide an analysis of the coupled queue process by studying a lower dimensional process and by introducing a certain conditional independence approximation. We show that the numerical results obtained from our finite buffer analysis are in excellent agreement with the corresponding results obtained from ns-2 simulations. We replace the CSMA CA protocol as implemented in the ns-2 simulator with the SDAR service model to show that the SDAR approximation provides an accurate model for the CSMA CA protocol. We also report the simulation speed-ups thus obtained by our model-based simulation.