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.
When the nodes always have packets to transmit, the network is said to be saturated and the macroscopic activity behavior is amenable to analysis under the assumption of an interference graph. In particular, the activity process has an elegant product-form stationary distribution @cite_17 @cite_38 @cite_11 . The computation of the stationary distribution of the activity process reduces to the identification of all the subsets of nodes which may transmit simultaneously, namely the independent sets of the interference graph.
@cite_38 @cite_11 @cite_17
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This work started out with our discovery of a pattern of throughput distributions among links in IEEE 802.11 networks from experimental results. This pattern gives rise to an easy computation method, which we term back-of-the-envelop (BoE) computation. For many network configurations, very accurate results can be obtained by BoE within minutes, if not seconds, by simple hand computation. This allows us to make shortcuts in performance evaluation, bypassing complicated stochastic analysis. To explain BoE, we construct a theory based on the model of an “ideal CSMA network” (ICN). The BoE computation method emerges from ICN when we take the limit c → 0, where c is the ratio of the mean backoff countdown time to the mean transmission time in the CSMA protocol. Importantly, we derive a new mathematical result: the link throughputs of ICN are insensitive to the distributions of the backoff countdown time and transmission time (packet duration) given the ratio of their means c. This insensitivity result explains why BoE works so well for practical 802.11 networks, in which the backoff countdown process is one that has memory, and in which the packet size can be arbitrarily distributed. Our results indicate that BoE is a good approximation technique for modest-size networks such as those typically seen in 802.11 deployments. Beyond explaining BoE, the theoretical framework of ICN is also a foundation for fundamental understanding of very-large-scale CSMA networks. In particular, ICN is similar to the Ising model in statistical physics used to explain phenomena arising out of the interactions of a large number of entities. Many new research directions arise out of the ICN model. Random-access algorithms such as the Carrier-Sense Multiple-Access (CSMA) protocol provide a popular mechanism for distributed medium access control in large-scale wireless networks. In recent years, fairly tractable models have been shown to yield remarkably accurate throughput estimates for CSMA networks. These models typically assume that both the transmission durations and the back-off periods are exponentially distributed. We show that the stationary distribution of the system is in fact insensitive with respect to the transmission durations and the back-off times. These models primarily pertain to a saturated scenario where nodes always have packets to transmit. In reality however, the buffers may occasionally be empty as packets are randomly generated and transmitted over time. The resulting interplay between the activity states and the buffer contents gives rise to quite complicated queueing dynamics, and even establishing the stability criteria is usually a serious challenge. We explicitly identify the stability conditions in a few relevant scenarios, and illustrate the difficulties arising in other cases. In this paper, we use a Markov model to develop a product form solution to efficiently analyze the throughput of arbitrary topology multihop packet radio networks that employ a carrier sensing multiple access (CSMA) protocol with perfect capture. We consider both exponential and nonexponential packet length distributions. Our method preserves the dependence between nodes, characteristic of CSMA, and determines the joint probability that nodes are transmitting. The product form analysis provides the basis for an automated algorithm that determines the maximum throughput in networks of size up to 100 radio nodes. Numerical examples for several networks are presented. This model has led to many theoretical and practical extensions. These include determination of conditions for product form analysis to hold, extension to other access protocols, and consideration of acknowledgments.