2010/08/20 by Graham, Carl, Robert, Philippe
#FOS: Computer and information sciences #FOS: Mathematics #Networking and Internet Architecture (cs.NI) #Probability (math.PR)
paper · doi:10.48550/arxiv.1008.3505
A Markovian model of the evolution of intermittent connections of various classes in a communication network is established and investigated. Any connection evolves in a way which depends only on its class and the state of the network, in particular as to the route it uses among a subset of the network nodes. It can be either active (ON) when it is transmitting data along its route, or idle (OFF). The congestion of a given node is defined as a functional of the transmission rates of all ON connections going through it, and causes losses and delays to these connections. In order to control this, the ON connections self-adaptively vary their transmission rate in TCP-like fashion. The connections interact through this feedback loop. A Markovian model is provided by the states (OFF, or ON with some transmission rate) of the connections. The number of connections in each class being potentially huge, a mean-field limit result is proved with an appropriate scaling so as to reduce the dimensionality. In the limit, the evolution of the states of the connections can be represented by a non-linear system of stochastic differential equations, of dimension the number of classes. Additionally, it is shown that the corresponding stationary distribution can be expressed by the solution of a fixed-point equation of finite dimension.