Limit theorems for one class of Polling models

The asymptotic independence of any finite set of random processes describing states of nodes is proved for an open transportation network with $N$ nodes and $M$ servers moving between them provided $N\to\infty$ and $M/N\to r$ ($0<r<\infty$). We determine that the limit flow of servers to a fixed node is a nonstationary Poisson process and describe behavior of the system in the thermodynamic limit. A system with Poisson input flow and a special kind of travel time distribution is examined as an example of such networks. The convergence to a limit dynamic system is proved for it. Also the steady point is specified and it is discovered that this point depends on the shape of travel time distribution only through its mean.