Fluid limits for QB-CSMA with polynomial rates, homogenization and reflection

We study in this paper a variation of the acclaimed carrier sense multiple access (CSMA) protocol. A random access algorithm is where back-off rates depend on the state of the network through queue lengths. We provide the first case where full proof of convergence to fluid limits can be obtained. Although this has been often justified informally, this is the first proof dedicated to obtaining this result formally. We then outline the difficulties arising when queues reach zero and solve them in the case of a complete interference graph. The main contribution of this paper is to provide a formal justification of the fluid limits for a complete interference graph in the case where some initial queue lengths are zero. This is done with a coupling argument. This paper also proves convergence to the fluid limits in the general case up to the time a queue reaches 0 on the fluid scale.